Ekkehard Ramm

Abstract

Consistent treatment of large rotations in common Reissner–Mindlin formula-tions is a complicated task. Reissner–Mindlin formulations that use a hierarchicparametrization provide an elegant way to facilitate large rotation shell anal-yses. This can be achieved by the assumption of linearized transverse shearstrains, resulting in an additive split of strain components, which technicallysimplifies implementation of corresponding shell finite elements. The presentstudy aims at validating this assumption by systematically comparing numeri-cal solutions with those of a newly implemented hierarchic and fully nonlinearReissner–Mindlin shell element.

Abstract

Bei der Berechnung der Seilnetztragwerke für die Überdachungen der Sportstätten für die Olympischen Spiele 1972 in München spielte ein bislang unveröffentlichtes und auch in Fachkreisen bislang weithin unbekanntes Manuskript des französischen Bauingenieurs Marc Biguenet, damals Mitarbeiter von Jörg Schlaich im Ingenieurbüro Leonhardt und Andrä, eine wesentliche Rolle. Es wird in zwei Veröffentlichungen von Klaus Linkwitz und Hans-Jörg Schek aus 1971 zur Berechnung und Formfindung von Seilnetztragwerken erwähnt und liefert wichtige Vorarbeiten. Das Manuskript war in Archiven und Bibliotheken allerdings nicht aufzufinden und wurde dem ersten Autor nach intensiven Recherchen schließlich von Marc Biguenet persönlich zur Verfügung gestellt. Im Rahmen dieses Berichts wird der Inhalt des Manuskripts mit dem Ziel der Quellen- und Wissenssicherung erstmals veröffentlicht, vergleichend kommentiert und in den technikhistorischen Kontext eingebettet. Die logisch-historischen Wurzeln der Berechnung von Seilnetztragwerken stehen im Zusammenhang mit der Entwicklungsgeschichte nichtlinearer baustatischer Theorien, dem Bau weitgespannter Hängebrücken sowie der Herausbildung computergestützter Berechnungsmethoden.

Abstract

The degree of statical indeterminacy as a fundamental property in structural mechanics is today mainly known as a property of a complete system without any information about its spatial distribution. The redundancy matrix provides information about the distribution of statical indeterminacy in the system and by this gives an additional valuable insight into the load-bearing behaviour. The derivation and definition of the redundancy matrix are presented based on truss systems and its mathematical properties and their mechanical interpretations are provided. The definition of the redundancy matrix is extended to other discrete systems like beam structures and a definition of the redundancy density is given for the continuous 1D case. Potential applications of the concept include robust design of structures, quantification of imperfection sensitivity as well as assessment of optimal actuator placement in adaptive structures.

Abstract

n the present work intentionally polygonal particles with a regular hexagonal geometry are investigated. This allows removing the complexity of randomly shaped particles thus concentrating on the interaction between adjacent particles. For this purpose, conceptual compression experiments on assemblies of hexagonal steel nuts are performed and subsequently simulated by a discrete element method. The interaction models for contact of two particles are as follows: in normal direction an elastic model augmented by a viscous supplement and in tangential direction an elasto-plastic model are applied; furthermore, an elasto-plastic model describes the contact of a particle with a plane underground. For an adhering bond between particles an elasto-damage beam including an axial force is introduced between the centers of adjacent particles. It allows modeling gradual failure of the bond. In order to test the capability of these models in a direct way, the conceptual experiments on simple regular particle arrangements are compared with their corresponding simulations. For samples of unglued particles relevant characteristics like shear bands are reproduced. For assemblies of particles glued together by an adhesive the study describes important failure properties like localization in cracks as well as ductile failure.

Abstract

We present a variational method for problems in solid and structural mechanics that is designed to be intrinsically free from locking when using equal order interpolation for all involved fields. The specific feature of the formulation is that it avoids all geometrical locking effects (as opposed to material locking effects, e.g. Poisson locking) for any type of structural or solid model, independent of the underlying discretization scheme. The possibility to employ equal order interpolation for all involved fields circumvents the task of finding particular function spaces to remove locking and avoid artificial stress oscillations. This is particularly attractive for instance for isogeometric analysis using unstructured meshes or T-splines. Comprehensive numerical tests underline the promising behaviour of the proposed method for geometrically linear and non-linear problems in terms of displacements and stress resultants using standard finite elements, isogeometric finite elements and a meshless method.

Abstract

Two novel hierarchic finite element formulations for geometrically nonlinear shell analysis including the effects of transverse shear are presented. Both methods combine a fully nonlinear Kirchhoff-Love shell model with hierarchically added linearized transverse shear components. Thus, large rotations can be taken into account while circumventing the peculiar task of finding a corresponding parametrization of the rotation tensor. The two formulations differ in the way the transverse shear effects are included, either using hierarchic rotations or hierarchic displacements. The underlying assertion is that in most practical applications the transverse shear angles are small even for large deformations. This is confirmed by various numerical experiments. The hierarchic construction results in an additive strain decomposition into parts resulting from membrane and bending deformation and additional contributions from transverse shear. It requires at least C1-continuous shape functions, which can be easily established within the isogeometric context using spline based finite elements. As reported earlier, this concept is intrinsically free from transverse shear locking. In the nonlinear case it dramatically facilitates representation of large rotations in shell analysis.

Abstract

A finite element formulation for a geometrically linear, shear deformable (Reissner–Mindlin type) shell theory is presented, which exclusively uses displacement degrees of freedom. The total displacement is subdivided into a part representing the membrane and bending deformation, enriched by two extra “shear displacements”, representing transverse shear deformation. This rotation-free approach is accomplished within the isogeometric concept, using C1-continuous, quadratic NURBS as shape functions. The particular displacement parametrization decouples transverse shear from bending and thus the formulation is free from transverse shear locking by construction, i.e. locking is avoided on the theory level, not by choice of a particular discretization. Compared to the hierarchic formulation proposed earlier within the group of the authors (Echter et al., 2013), the method presented herein avoids artificial oscillations of the transverse shear forces. Up to now, a similar, displacement based method to avoid membrane locking has not been found. Thus, in the present formulation the mixed method from Echter et al. (2013) is used to avoid membrane locking.

Abstract

The present contribution deals with the question how structures with softening material behavior can be controlled in a numerical analysis beyond limit points, when conventional path following schemes fail. For nonlinear problems with localized cracks, adaptive path following schemes that increase numerical robustness, minimize user interference and avoid nonphysical (artificial) unloading are presented. In the methods proposed, a control region is identified where control parameters are evaluated. This control region adapts with the continuation of the crack tip. Robustness and applicability of the schemes are illustrated by numerical examples.

Abstract

The contribution addresses the specific physical sensitivity inherent in the interaction of slender structures like membranes or shells and instationary incompressible flows and its consequences for numerical solution schemes. Different partitioned strong coupling approaches are presented and compared with respect to their efficiency. In addition to the classical block Gauss-Seidel iteration and a Newton-Krylov approach, a two level coupling scheme is introduced. It is based on a coarse grid predictor followed by the usual iterative fine scale solution. The performance of the three coupling schemes is investigated applying a two-dimensional model problem of a membrane roof. Three-dimensional numerical examples demonstrate the mechanical and numerical sensitivity for these especially delicate fluid-structure interaction problems.

Abstract

The present contribution deals with an optimization strategy of fiber reinforced composites. Although the methodical concept is very general we concentrate on Fiber Reinforced Concrete with a complex failure mechanism resulting from material brittleness of both constituents matrix and fibers. Because of these unfavorable characteristics the interface between fiber and matrix plays a particularly important role in the structural response. A prominent objective for this kind of composite is the improvement of ductility.The influential factors on the entire structural response of this composite are (i) material parameters involved in the interface, (ii) the material layout at the small scale level, and (iii) the fiber geometry on the macroscopic structural level. The purpose of the present paper is to improve the structural ductility of the fiber reinforced composites applying an optimization method with respect to the geometrical layout of continuous long textile fibers.The method proposed is achieved by applying a so-called embedded reinforcement formulation. This methodology is extended to a damage formulation in order to represent a realistic structural behavior. For the optimization problem a gradient-based optimization scheme is assumed. An optimality criteria method is applied because of its numerically high efficiency and robustness. The performance of the method is demonstrated by a series of numerical examples; it is verified that the ductility can be substantially improved.

Abstract

The present paper addresses an optimization strategy of textile fiber reinforced concrete (FRC) with emphasis on its special failure behavior. Since both concrete and fiber are brittle materials a prominent objective for FRC structures is concerned with the improvement of ductility. Despite above unfavorable characteristics the interface between fiber and matrix plays a substantial role in the structural response. This favorable ‘composite effect’ is related to material parameters involved in the interface and the material layout on the small scale level. Therefore the purpose of the present paper is to improve the structural ductility of FRC at the macroscopic level applying an optimization method with respect to significant material parameters at the small scale level. The method discussed is based on multiphase material optimization. This methodology is extended to a damage formulation. The performance of the proposed method is demonstrated in a series of numerical examples; it is verified that the ductility can considerably be improved.

Abstract

Many existing algorithms for the analysis of large deformation contact problems use a so-called node-to-segment approach to discretize the contact interface between dissimilar meshes. It is well known, that this discretization strategy may lead to problems like loss of convergence or jumps in the contact forces. Additionally it is popular to use penalty methods to satisfy the contact constraints. This necessitates a user defined penalty parameter the choice of which is somehow arbitrary, problem dependent and might influence the accuracy of the analysis. In this work, a frictionless segment-to-segment contact formulation is presented that does not require any user defined parameter to handle the non-linearity of the contact conditions. The approach is based on the mortar method enforcing the compatibility condition along the contact interface in a weak integral sense. The application of dual spaces for the interpolation of the Lagrange multiplier allows for a nodal decoupling of the contact constraints. A local basis transformation in combination with a primal¿dual active set strategy enables the exact enforcement of the contact constraints via prescribed incremental boundary conditions. Due to the biorthogonality condition of the basis functions the Lagrange multipliers can be locally eliminated. A static condensation leads to a reduced system of equations to be solved solely for the unknown nodal displacements. Thus the size of the system of equations remains constant during the whole calculation. The discrete Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the analysis of dynamic contact problems the proposed contact description is combined with the implicit Generalized Energy-Momentum Method. Several numerical examples illustrate the performance of the suggested contact formulation.

Abstract

Composites or multi-phase materials are characterized by a distinct heterogeneous microstructure. The failure modes of these materials are governed by several micromechanical effcts like debonding phenomena and matrix cracks. The overall mechanical behavior of composites in the linear as well as the nonlinear regime is not only governed by the material properties of the components and their bonds but also by the material layout. In the present contribution the material structure is resolved and modeled on a small scale allowing to deal with these effcts. For the numerical simulation we apply a combination of the eXtended Finite-element method (X-FEM) and the Level Set Method (LSM). In the X-FEM the Finite-element approximation is enriched by appropriate functions through the concept of partition of unity. The geometry of material interfaces and cracks is described by the LSM. The combination of both, X-FEM and LSM, turns out to be very natural since the enrichment can be described and even constructed in terms of level set functions. In order to project the material behavior modeled on a small scale onto the large or structural scale, we employ the Variational Multiscale Method (VMM). This concept is based on an additive split of the displacement field into large and small scale parts. For an efficient solution of the discrete problem we postulate that the small scale displacements are locally supported; in order to achieve this objective one has to assume appropriate constraint conditions. It can be shown that the applied numerical model allows a considerable flexibility concerning the variation of the material design and consequently of the mechanical behavior of a composite.

Abstract

Within this paper the so-called artificial added mass effect is investigated which is responsible for devastating instabilities within sequentially staggered Fluid-structure Interaction (FSI) simulations where incompressible fluids are considered. A discrete representation of the added mass operator M_A is given and ‘instability conditions’ are evaluated for different temporal discretisation schemes. It is proven that for every sequentially staggered scheme and given spatial discretisation of a problem, a mass ratio between fluid and structural mass density can be found at which the coupled system becomes unstable. The analysis is quite general and does not depend upon the particular spatial discretisation schemes used. However here special attention is given to stabilised finite elements employed on the fluid partition. Numerical investigations further highlight the results.

Abstract

The efficient modelling of three-dimensional contact problems is still a challenge in non-linear implicit structural analysis. We use a primal-dual active set strategy, based on dual Lagrange multipliers to handle the non-linearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to achieve a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the application to thin-walled structures we adapt a three-dimensional non-linear shell formulation including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface-oriented shell element, which allows the application of contact conditions directly to nodes lying on the contact surface. Shell typical locking phenomena are treated with the enhanced-assumed-strain-method and the assumed-natural-strain-method. The discretization in time is done with the implicit Generalized-alpha method and the Generalized Energy-Momentum Method to compare the development of energies within a frictionless contact description. In order to conserve the total energy within the discretized frictionless contact framework, we follow an approach from Laursen and Love, who introduced a discrete contact velocity to update the velocity field in a post-processing step. Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.

Abstract

This paper presents a two-scale approach for the mechanical and numerical modeling of materials with microstructure like concrete or fiber reinforced concrete in the nonlinear regime. It addresses applications, where the assumption of scale separation as the basis for classical homogenization methods does not hold. This occurs when the resolution of micro and macro scale does not differ ab initio or when evolving fluctuations in the macro-fields are in the order of the micro scale during the loading progress. Typical examples are localization phenomena. The objective of the present study is to develop an efficient solution method exploiting the physically existing multiscale character of the problem. The proposed method belongs to the superposition based methods with local enrichment of the large scale solution by a small scale part. The main focus of the present formulation is to allow for locality of the small scale solution within the large scale elements to achieve an efficient solution strategy. At the same time the small scale information exchange over the large scale element boundaries is facilitated while maintaining the accuracy of a refined complete solution. Thus the emphasis lies on finding appropriate locality constraints for the small scale solution. To illustrate the method the formulation is applied to a damage mechanics based material model for concrete-like materials.

Abstract

Fluid–structure interaction (FSI) problems are of great relevance to many fields in engineering and applied sciences. One wide spread and complex FSI-subclass is the category that studies the instationary behavior of incompressible viscous flows and thin-walled structures exhibiting large deformations. Free surfaces often present an essential additional challenge for this class of problems. Prominent application areas are fluid sloshing in tanks and numerable problems in offshore engineering and naval architecture. Especially when partitioned strong coupling schemes are used in order to solve the coupled FSI problem the design of an appropriate overall computational approach including free surface effects is not trivial. In this paper a new so-called partitioned implicit free surface approach is introduced and embedded into a strong coupling FSI solver. For complex problem classes this approach is combined with the general elevation equation that is closed through a dimensionally reduced pseudo-structural approach. The presented approach shows the same stability properties as a full implicit approach but is by far more efficient—especially in the partitioned coupled case.

Abstract

The present contribution introduces enhanced discrete element simulation methodologies (DEM) with a special focus on a microstructure-based model environment. Therewith, it is possible to represent the failure of cohesive granular materials like concrete, ceramics or marl in a qualitative as well as quantitative manner. Starting from a basic polygonal two-dimensional particle model for non-cohesive granular materials, more complex models for cohesive materials are obtained by inclusion of beam or interface elements between corresponding particles. In particular, we will introduce an interface enhanced DEM methodology where a standard ingredient of computational mechanics, namely interface elements, are combined with the particle methodology contained in the DEM. The last step in the series of increasing complexity is the realization of a microstructure-based simulation environment which utilizes the interface enhanced DEM methodology. With growing model complexity a wide variety of failure features of cohesive as well as non-cohesive geomaterials can be represented. Finally, the plan of the paper is enriched by the validation of the newly introduced and re-developed discrete models with regard to qualitative and quantitative aspects.

Abstract

In this paper, an adaptive approach, with remeshing as essential ingredient, towards robust and efficient simulation techniques for fast transient, highly non-linear processes including contact is discussed. The necessity for remeshing stems from two sources: the capability to deal with large deformations that might even require topological changes of the mesh and the desire for an error driven distribution of computational resources. The overall computational approach is sketched, the adaptive remeshing is presented and the crucial aspect, the choise of suitable error indicator(s). is discussed in more detail. Several numerical examples demonstrate the performance of the approach.

Abstract

This note revisits the derivation of the ALE form of the incompressible Navier-Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the flow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretisation. The geometric conservation law is introduced into the equations and the resulting formulation is discretised in time and space without loss of stability and accuracy compared to the fixed grid version. There is no need for temporal averaging remaining. The formulation applies equally to different time integration schemes within a finite element context.

Abstract

Multi-phase materials are characterized by the fact that they possess a specific heterogeneous microstructure. This feature often necessitates complex formulations in order to obtain realistic mechanical behavior with macroscopic material models. Furthermore, such complex models cause substantial difficulties when the corresponding material parameters need to be identified and their physical meaning interpreted. In multiscale analyses, usually the microstructure of a material point is resolved directly. Thus, the material behavior at the material point can be modeled in an elegant, natural and simple way. The present contribution aims at a detailed geometric modeling of multi-phase materials, as well as at a local mechanical modeling of material interfaces and interfacial failure. The geometry of the material distribution within the microstructure is described through level set functions. The mechanical modeling of material interfaces and interfacial cracks is accomplished by the extended finite-element method (X-FEM). The combination of both, the level set method and the X-FEM, allows one to model such internal features of the microstructure without the adaptation of the mesh. Thus, expensive meshing procedures can be avoided and considerable exibility is obtained with respect to the variation of the material design. In the X-FEM, the finite-element approximation is enriched by appropriate functions through the concept of partition of unity. Remarkably, the level set method can not only be applied to describe the geometry of the material interfaces and cracks, respectively but also to support and even to develop the corresponding enrichment function.

Abstract

The present paper addresses the so-called microplane formulation which became recently more and more popular for the description of quasi-brittle materials. The essential feature of this material formulation is a split of the local microplane strains and stresses allowing one to resort to simplified or in certain cases even unidirectional constitutive laws. The main attraction of the microplane concept is that an initial or evolving anisotropic material behavior can be described in a natural and simple way. Motivated from a macroscopic viewpoint, it is advocated to restrict the microplane concept to the pure volumetric-deviatoric split, as a constraint subset of the most often applied volumetric-deviatoric-tangential split. This variant has the particular advantage that typical macroscopic responses are directly reflected on the mesoscale. It will be shown that in certain cases the present version of a microplane formulation is closely related to well-known macroscopic models although being much more general than those macroscopic formulations. This close relation is exploited to derive physically sound microplane constitutive laws. Therefore the characteristic damage mechanisms of materials at two levels of observation, (1) at the macroscale in the sense of classical continuum damage mechanics, and (2) at the mesoscale utilizing the so-called microplane concept, are examined. The comparison of the microplane formulation to a well-known macroscopic one-parameter damage model enables the identification and interpretation of the microplane constitutive laws. The constitutive formulations are embedded in a thermodynamically consistent framework. Finally, the performance of the attained microplane formulation is analyzed in a mixed-mode fracture simulation of concrete.

Abstract

Powders are challenging materials for many engineers and scientists, since they often show unexpected behavior which is quite different from the behavior of gases, liquids or solids. In addition, powderlike materials appear quite often in real applications. Our study is driven by the need to appropriately grasp the behavior of dry powder and our main focus is on the damping and energy absorbing behavior of dry powder under impact loading. The overall approach, including both the model and the algorithmic setup, that we developed for this purpose is presented in this paper. Applicability to quasi static as well as highly transient real world problems and robustness are crucial constraints for the whole undertaking. These requirements are met through a model with relatively few and, more importantly, easy-to-obtain material parameters and through some special algorithmic developments. After a general introduction into powder, our continuum model based on finite strain elasto-plasticity for the simulation of quasi static and transient dynamic processes is presented. Then the algorithmic setup, i.e. the required return mapping algorithm formulated in principal stresses, is presented. Finally, the parameter determination from standard laboratory tests is described and appropriate numerical results are shown for both quasi-static and highly transient impact cases.

Abstract

The analysis of large-scale nonlinear shell problems asks for parallel simulation approaches. One crucial part of efficient and well scalable parallel FE-simulations is the solver for the system of equations. Due to the inherent suitability for parallelization one is very much directed towards preconditioned iterative solvers. However thin-walled-structures discretized by finite elements lead to ill-conditioned system matrices and therefore performance of iterative solvers is generally poor. This situation further deteriorates when the thickness change of the shell is taken into account. A preconditioner for this challenging class of problems is presented combining two approaches in a parallel framework. The first approach is a mechanically motivated improvement called 'Scaled Director Conditioning' (SDC) and is able to remove the extra - ill conditioning that appears with three dimensional shell formulations as compared to formulations that neglect thickness change of the shell. It is introduced at the element level and harmonizes well with the second approach utilizing a multilevel algorithm. Here a hierarchy of coarse grids is generated in a semi-algebraic sense using an aggregation concept. Thereby the complicated and expensive explicit generation of course triangulations can be avoided. The formulation of this combined preconditioning approach is given and the effects on the performance of iterative solvers is demonstrated via numerical examples.

Abstract

The variational multiscale method provides a methodical framework for large eddy simulation of turbulent flows. In this work, a particular implementation in form of a three-level finite element method separating large resolved, small resolved, and unresolved scales is proposed. Residual-free bubbles are used for the numerical approximation of the small-scale momentum equation. A stabilizing term is added, in order to take into account the effect of the small-scale continuity equation. This implementation guarantees the stability of the method without further provisions and offers substantial computational savings on the small-scale level. Furthermore, it is accounted for the unresolved scales by a specific dynamic modeling procedure. The method is tested for two different turbulent flow situations.

Abstract

A layered shell formulation allowing to apply three-dimensional material laws within the different layers is described. The formulation is combined with a rate independent plasticity constitutive model for plain and reinforced concrete. A fracture energy regularization in the softening branch is used in the constitutive law of concrete. The reinforcement is described as a smeared layer within the shell-discretization representing also the effects from tension stiffening if necessary. Several examples demonstrate the applicability of the proposed model.

Abstract

The present contribution focuses on the influence of geometrical nonlinearities on the structural behavior in the design process. The notion of the stiffest structure loses its clear definition in the case of nonlinear kinematics; here we will discuss this concept on the basis of different objectives. Apparently topology optimization is often a generator of slender struts, which tend to buckle before the structure is completely loaded. To include the instability phenomena into the design process, the critical load level will be determined directly; this condition will be included as an inequality constraint. Further on, to reduce the imperfection sensitivity, a geometrically modified structure including the imperfection shape is also introduced. The present optimization procedures are demonstrated by examples showing rather the principal effects of the enhancements than real practical design problems.

Abstract

The mechanical behavior of granular materials can be well described on the particle scale with discrete element models (DEM) if the motion of individual grains is taken into account. Microscopic quantities like shape and stiffness of the grains are input parameters and relative deformations as well as contact forces follow as an output from the DEM simulation. In order to relate the microscopic kinematic and dynamic quantities of the DEM to macroscopic variables like strains and stresses, homogenization methods are needed. In the present contribution, a homogenization strategy presented recently is extended in order to evaluate stresses, strains and further kinematic quantities of representative elementary volumes (REV) in granular assemblies. Based on the definition of volume averages, expressions for macroscopic stress, couple stress, strain and curvature tensors are derived for an arbitrary REV. DEM simulations of two different test setups including cohesionless and cohesive granular assemblies are used as a validation of the proposed homogenization technique. A non-symmetric macroscopic stress tensor, as well as couple stresses are obtained following the proposed procedure, even if a single particle is described as a standard continuum on the microscopic scale. The numerical implementation of the homogenization procedure with respect to a two-dimensional discrete element code with polygonal particles is discussed in detail. Special attention is given to the definition of the REV volume. The proposed homogenization procedure is verified by the evaluation of different boundary value problems: The first example is a biaxial test of a cohesionless particle sample, followed by a brazilian test simulation of a cohesive particle assembly. In both examples, it is found that the onset of localization zones of finite width dominates the macroscopic failure of the samples.

Abstract

We present a two- and a three-level finite element method for the numerical simulation of incompressible flow governed by the Navier-Stokes equations. Within the theoretical framework of the variational multiscale method and exploiting the concept of residual-free bubbles we propose a separation of three different scale groups, namely large (resolved) scales, small (resolved) scales and unresolved scales. The resolution of the large and small scales takes place on level 1 and level 2 with the aid of diverse approaches. The dynamic calculation of a subgrid viscosity representing the effect of the unresolved scales constitutes level 3 of our algorithm. The proposed algorithm is tested for various laminar flow situations and compared to the results obtained via an unusual stabilized finite element method. It is supposed to be the basic scheme also for turbulent flow in a subsequent study.

Abstract

It is apparent that physics and numerics are strongly linked in every serious undertaking in computational mechanics. This is in particular pronounced when the matter of study excels through a very sophisticated, even sometimes tricky physical behavior. Such a delicate characteristic is the trademark of shell structures, which are the most often used structural components in nature and technology. This outstanding position in the hierarchy of all structures is due to their curvature allowing to carry transverse loading in an optimal way by in-plane membrane actions, despite an often extreme slenderness. As typical for optimized systems their performance might be on the one hand excellent, but can also be extremely sensitive to certain parameter changes on the other hand. This prima donna like mechanical behavior with all its sensitivities is of course carried over to any numerical scheme. In other words it is a fundamental precondition to understand the principle features of the load carrying mechanisms of shells before designing and applying any numerical formulation. The present study addresses this peculiar interrelation between physics and numerics. At first typical characteristics of shell structures are described; this include their benefits but also their extreme sensitivities. In the second part these aspects are reflected on related computational models and numerical procedures. This discussion is carried through a number of selected problems and examples. It need to be said that the paper is the outcome of a general plenary lecture addressing fundamental aspects rather than concentrating on a specific formulation or numerical scheme.

Abstract

In the present contribution, a transition from the dynamics of single particles to a Cosserat continuum is discussed. Based on the definiton of volume averages, expressions for the macroscopic stress tensors and for the couple stress tensors are derived. It is found that an ensemble of particles allows for a non-symmetric macroscopic stress tensor and, thus, for the existence of couple stresses, even if the single particles are considered as standard continua. DEM simulations of a biaxial box are used for the validation of the proposed homogenization technique.

Abstract

This contribution aims at characteristic damage mechanisms of materials at two levels of observation, (a) at the macro-scale in the sense of classical continuum damage mechanics, and (b) at the meso-scale utilizing the so-called microplane concept. The constitutive formulations are embedded in a thermodynamically consistent framework. The microplane formulation is based on a volumetric-deviatoric split (V-D), which is motivated from a macroscopic viewpoint. The main advantage of this formulation is that the material behavior at a material point is characterized by constitutive laws formulated on the individual microplanes, allowing to describe anisotropic material behavior in a very natural and simple way. One particular advantage of the present version of a microplane formulation is also its close relation to macroscopic models which enables the interpretation and identification of the microplane constitutive laws in terms of well-known macroscopic constitutive laws. The appropriate choice of the microplane loading functions is illustrated by a comparison of the microplane damage formulation with a macroscopic two-parameter damage model.

Abstract

In this paper we present a combined beam-particle model as an improved discrete element simulation scheme. In our two-dimensional approach the material is considered as a cohesive granular frame represented by polygonal mesostructural elements bonded together by elastic beams. The convex unbreakable polygons are used to model the grains in the material which exert an elastic restoring force when they are pressed against each other. We combine a particle and a lattice model by connecting the centre of mass of the convex polygons by elastic beams. Under loading within the model solid the overstressed beams can break in a brittle manner according to a physical breaking rule. This paper is intended as a first step in view of the qualitative and quantitative modelling of geomaterials including the full range of material states (from a continuous to a discontinuous representation). The model is applied to study the fracture of a model material that in the present form oughts to represent a basic form of a type of cohesive granular material. Various loading conditions, i.~e. uniaxial compression, elongation, and shear of a rectangular sample will be considered. One of the main advantages of our coupled beam-particle approach with respect to other models is that it naturally accounts for the fact that during the loading process crack surfaces may exert repulsive force on each other. An effect which is impossible to take into account in the framework of lattice models.

Abstract

The present work deals with the post strengthening of concrete slabs and the numerical and analytical calculation of their load carrying behavior. The used strengthening materials are Fiber Reinforced Polymers which are of increasing interest in the civil engineering applications for example textile reinforced concrete tubes, cables of cable-stayed bridges or even entire bridges. A test series comprising 13 model scale slabs and one full scale slab was undertaken at the University of California, San Diego. These slabs were enhanced using different strengthening systems and composite material concepts. The aim of these tests was to investigate the influence of distinct parameters on the load and deformation capacities. Moreover, a simplified method calculating the load-displacement response is given which provides good agreement with the test results. In addition, a finite element investigation is performed at the University of Stuttgart. The slabs are modelled firstly in a two-dimensional design space, assuming a plane stress condition for the concrete and the Fiber Reinforced Polymer, and secondly in a three-dimensional design space with multilayered shell elements allowing to include the varying response across the depth of the slabs.

Abstract

It is a common practice to base both, material topology optimization as well as a subsequent shape optimization on linear elastic response. However, in order to obtain a realistic design, it might be essential to base the optimization on a more realistic physical behavior, i.e. to consider geometrically or/and materially nonlinear effects. In the present paper, an elastoplastic von Mises material model with linear isotropic hardening/softening for small strains is used. The objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed. With respect to the specific features of either topology or shape optimization, for example the number of optimization variables or their local–global influence on the structural response, different methods are applied. For topology optimization problems, the gradient of the ductility is determined by the variational adjoint approach. In shape optimization, the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach. The topology optimization problem is solved by an optimality criteria (OC) method and the shape optimization problem by a mathematical programming (MP) method. In topology optimization, a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities. The procedures are verified by 2D design problems under plane stress conditions.

Abstract

The present contribution deals with the sensitivity analysis and optimization of structures for path-dependent structural response. Geometrically as well as materially non-linear behavior with hardening and softening is taken into account. Prandtl-Reuss-plasticity is adopted so that not only the state variables but also their sensitivities are path-dependent. Because of this the variational direct approach is preferred for the sensitivity analysis. For accuracy reasons the sensitivity analysis has to be consistent with the analysis method evaluating the structural response. The proposed sensitivity analysis as well as its application in structural optimization is demonstrated by several examples.

Abstract

In recent years, considerable attention has been given to the development of higher order plate and shell models. These models are able to approximately represent three-dimensional effects, while pertaining the efficiency of a two-dimensional formulation due to pre-integration of the structural stiffness matrix across the thickness. Especially, the possibility to use unmodified, complete three-dimensional material laws within shell analysis has been a major motivation for the development of such models. While the theoretical and numerical formulation of so-called 7-parameter shell models, including a thickness stretch of the shell, has been discussed in numerous papers, no thorough investigation of the physical significance of the additional kinematic and static variables, coming along with the extension into three dimensions, is known to the authors. However, realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures, e.g. concerning loading and kinematic boundary conditions, and a correct interpretation of the results. In the present paper, the significance of kinematic and static variables, appearing in a 7-parameter model proposed by Büchter and Ramm (1992a) are discussed. It is shown, how these quantities ‘refine’ the model behavior and how they can be related to the ‘classical’ variables, such as ‘curvatures’ and ‘stress resultants’. Furthermore, the special role of the material law within such a formulation is addressed. It is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness, to ensure correct results. In this context it is found, that the considered 7-parameter model can be regarded as ‘optimal’ with respect to the number of degrees of freedom involved.

Abstract

A new concept for the construction of locking-free finite elements for bending of shear deformable plates and shells, called DSG (Discrete Shear Gap) method, is presented. The method is based on a pure displacement formulation and utilizes only the usual displacement and rotational degrees of freedom (dof) at the nodes, without additional internal parameters, bubble modes, edge rotations or whatever. One unique rule is derived which can be applied to both triangular and rectangular elements of arbitrary polynomial order. Due to the nature of the method, the order of numerical integration can be reduced, thus the elements are actually cheaper than displacement elements with respect to computation time. The resulting triangular elements prove to perform particularly well in comparison with existing elements. The rectangular elements have a certain relation to the Assumed Natural Strain (ANS) or MITC-elements, in the case of a bilinear interpolation, they are even identical.

Abstract

We present a posteriori error estimators and adaptive methods for the finite element approximation of non-linear problems and especially elastoplasticity. The main characteristic of the proposed method is the introduction of duality techniques or in other notions the reciprocal theorem. For error estimation at an equilibrium point the non-linear boundary value problem and an additional linearized dual problem are considered. The loading of the dual problem is specifically designed for capturing the influence of the errors of the entire domain to the considered variable. Our approach leads to easy computable refinement indicators for locally or integrally defined variables. For instationary problems as elastoplasticity, in a first step, we neglect the errors due to time discretization, and evaluate the error indicators within each time step for a stationary problem. The versatility of the presented framework is demonstrated with numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.We present a posteriori error estimators and adaptive methods for the finite element approximation of non-linear problems and especially elastoplasticity. The main characteristic of the proposed method is the introduction of duality techniques or in other notions the reciprocal theorem. For error estimation at an equilibrium point the non-linear boundary value problem and an additional linearized dual problem are considered. The loading of the dual problem is specifically designed for capturing the influence of the errors of the entire domain to the considered variable. Our approach leads to easy computable refinement indicators for locally or integrally defined variables. For instationary problems as elastoplasticity, in a first step, we neglect the errors due to time discretization, and evaluate the error indicators within each time step for a stationary problem. The versatility of the presented framework is demonstrated with numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.

Abstract

The postbuckling behavior of an axially compressed circular cylindrical shell is exceedingly complicated due to an infinite number of closely spaced postbuckling branches and bifurcation points. The minimum strength existing in the deep bottom of the postbuckling region may serve as a design limit. The primary concern in this present paper is to compute this stable postbuckling equilibrium solution by two different approaches: One is to repeat the procedures of tracing unstable branches, pinpointing bifurcation points and branch-switching in order to carefully approach to the target. The other is to trigger a static jump to the target by two-parametric loading. As a numerical example, a perfect circular cylindrical panel is analyzed to show that a direct jump from the undeformed state to a stable postbuckling solution is possible with a proper choice of the load perturbation.

Abstract

The main objective of this paper is the discussion of two different strategies of simulating the constitutive behavior of granular assemblies. For this, we will focus on discrete particle methods which are widely used in physical science and on continuum-based microplane models which are applied by the engineering community. After deriving the overall constitutive equations based on Voigt's hypothesis, special focus will be dedicated to the comparison of the relations between the microscopic and macroscopic quantities of each model. It will be demonstrated, that the two basically different modelling techniques lead to remarkably similar results for elastic as well as elasto-plastic material behavior.

Abstract

An anisotropic continuum damage model based on the microplane concept is elaborated. Scalar damage laws are formulated on several individual microplanes representing the planes of potential failure. These uniaxial constitutive laws can be cast into a fourth-order damage formulation such that anisotropy of the overall constitutive law is introduced in a natural fashion. Strain gradients are incorporated in the constitutive equations in order to account for microstructural interaction. Consequently, the underlying boundary value problem remains well-posed even in the softening regime. The gradient continuum enhancement results in a set of additional partial differential equations which are satisfied in a weak form. Additional nodal degrees of freedom are introduced which leads to a modified element formulation. The governing equations can be linearized consistently and solved within an incremental-iterative Newton–Raphson solution procedure. The capability of the present model to properly simulate the localized failure of quasi-brittle materials will be demonstrated by means of several examples.

Abstract

This paper aims at characteristic failure mechanisms of cohesive frictional materials at two levels of observation: (a) at the macroscale of continuum elastoplasticity, and (b) at the microscale of active microplanes with arbitrary orientation. Thereby, the criteria for the loss of uniqueness and the loss of ellipticity will be discussed for a macroscopic as well as a microplane-based anisotropic plasticity formulation. In addition, conditions for shear dilatancy will be developed at the two scales which illustrate the necessity to couple normal and shear components at each microplane.

Abstract

The paper first focuses on the historical context in which Reissner's famous shear deformation plate theory was derived. Here essentially Eric REISSNER's own view on this matter, in particular the relation to Mindlin's contribution, is followed 15. The significance of shear deformable plate and shell theories for the derivation of finite elements is briefly described. As a major aspect it is shown how these formulations can be easily extended to a completely three-dimensional model allowing to apply unmodified 3D constitutive equations and to include large strain effects but keeping the essential features of a thin-walled structure. Finally, the importance of Reissner's variational principle for the development of hybrid finite element models is pointed out.

Abstract

In structural optimization the quality of the optimization result strongly depends on the reliability of the underlying structural analysis. This comprises the quality and range of the mechanical model, e.g. linear elastic or geometrically and materially nonlinear, as well as the accuracy of the numerical model, e.g. the discretization error of the FE-model. The latter aspect is addressed in the present contribution. In order to guarantee the quality of the numerical results the discretization error of the finite element solution is controlled and the finite element discretization is adaptively refined during the optimization process. Conventionally, so-called global error estimates are applied in structural optimization which estimate the error of the total strain energy. In the present paper local error estimates are introduced in shape optimization which allow to control directly the discretization error of local optimization criteria. In general, the adaptive refinement of the finite element discretization by remeshing affects the convergence of the optimization process if a gradient-based optimization algorithm is applied. In order to reduce this effect the sensitivity of the discretization error must also be restricted. Suitable refinement indicators are developed for globally and locally adaptive procedures. Finally, the potential of two techniques, which may improve the numerical efficiency of adaptive FE-procedures within the optimization process, is studied. The proposed methods and procedures are verified by 2-D shape optimization examples.

Abstract

The study discusses the concept of error estimation in linear elastodynamics. Two different types of error estimators are presented. First ‘classical’ methods based on post-processing techniques are discussed starting from a semidiscrete formulation. The temporal error due to the finite difference discretization is measured independently of the spatial error of the finite element discretization. The temporal error estimators are applied within one time step and the spatial error estimators at a time point. The error is measured in the global energy norm. The temporal evolution of the error cannot be reflected. Furthermore the estimators can only evaluate the mean error of the whole spatial domain. As the second scheme local error estimators are presented. These estimators are designed to evaluate the error of local variables in a certain region by applying duality techniques. Local estimators are known from linear elastostatics and have later on been extended to nonlinear problems. The corresponding dual problem represents the influence of the local variable on the initial problem and may be related to the reciprocal theorem of Betti–Maxwell. In the present study this concept is transferred to linear structural dynamics. Because the dual problem is established over the total space–time domain, the spatial and temporal error of all time steps can be accumulated within one procedure. In this study the space–time finite element method is introduced as a single field formulation.

Abstract

Stabilized finite element methods have been developed mainly in the context of Computational Fluid Dynamics (CFD) and have shown to be able to add stability to previously unstable formulations in a consistent way. In this contribution a deformation dependent stabilization technique, conceptually based on the above mentioned developments in the CFD area, is developed for Solid Mechanics to cure the well-known enhanced assumed strain (EAS) method from artificial instabilities (hourglass modes) that have been observed in the range of large compressive strains. In investigating the defect of the original formulation the dominating role of the kinematic equation as cause for the instabilities is revealed. This observation serves as key ingredient for the design of the stabilizing term, introduced on the level of the variational equation. A proper design for the stabilization parameter is given based on a mechanical interpretation of the underlying defect as well as of the stabilizing action. This stabilizing action can be thought of an additional constraint, introduced into the reparametrized Hu–Washizu functional in a least-square form, together with a deformation dependent stabilization parameter. Numerical examples show the capability of this approach to effectively eliminate spurious hourglass modes, which otherwise may appear in the presence of large compressive strains, while preserving the advantageous features of the EAS method, namely the reduction of the stiffness for an ‘in-plane bending' mode, i.e. when plane stress elements are used in a bending situation.

Abstract

Fracture and fragmentation of solids are difficult problems to handle numerically due to the creation and continuous motion of new surfaces. Commonly used numerical methods solve partial differential equations of continuum mechanics. With classical numerical methods such as the Finite Elements, Finite Differences or Boundary Elements a small number of discontinuities may be considered but these methods cannot encompass the entire fracturing process. An alternative approach is the Discrete Element Method in which the elastic medium is considered to be fully discontinuous, i.e. the elastic solid is assembled of discrete elements. In this paper we present a two-dimensional discrete model of solids based on the Discrete Element Method that allows us to follow the behaviour of the solid body and of the fragments well beyond the formation of simple cracks. The model, consisting of polygonal cells connected via beams, is an extension of discrete models used to study granular flows. This modelling is particularly suited for the simulation of fracture and fragmentation processes. After calculating the macroscopic elastic moduli from the cell and beam parameters, we present a detailed study of an uniaxial compression test of a rectangular block, and of the dynamic fragmentation processes of solids in various experimental situations, like the catastrophic failure of solids due to impact and explosion. The model proved to be succesful in reproducing the experimentally observed subtleties of fragmenting solids.

Abstract

In this contribution a class of formulations for beams, plates and shells is presented, which intrinsically avoids locking, independent of the utilized discretization scheme. The key idea is the reparametrization of the kinematic equations to avoid locking on theory level – prior to discretization. Thus, the resulting formulations are locking‐free for any equal‐order interpolation. As demonstrator, we present both mixed and primal concepts for Timoshenko beams in both weak and strong form, as well as their theoretical relationships. Besides a weak form Galerkin‐type solution using B‐Splines, we show the generality of the presented concepts by employing isogeometric collocation based on the corresponding Euler‐Lagrange equations of the boundary value problem. The quality of both stress resultants and displacements is investigated. Although the underlying concept addresses beams, plates and shells, the present contribution illustrates the methodology for the Timoshenko beam.

Abstract

The higher inter-element continuity of the Isogeometric Analysis (IGA) applying NURBS functions for geometry as well as mechanics opens up new possibilities in the analysis of thin-walled structures, i.e. beams, plates and shells. An important advantage is the straightforward implementation of classical theories, which require C1-continuity, e. g. Kirchhoff-Love shell formulations. Based on these "simplest" models shear deformable theories, introducing Timoshenko and Reissner-Mindlin kinematics, are formulated in a hierarchic manner. In contrast to using total rotations the present formulations introduce incremental transverse shear rotations as primary variables. This reparametrization of the kinematic equations can be established in several different fashions, which is addressed in this contribution. All parametrizations have in common, that they lead to shear deformable beam, plate and shell formulations being intrinsically free from transverse shear locking. This is a remarkable feature since the spectrum of different smooth discretization schemes in IGA and other fields of computational shell analysis is continuously growing and the proposed hierarchic shell formulations are locking-free, independent of the underlying discretization.

Abstract

The higher inter-element continuity of the Isogeometric Analysis (IGA) applying NURBS functions for geometry as well as mechanics opens up new possibilities in the analysis of thin-walled structures, i.e. beams, plates and shells. The contribution addresses the straightforward implementation of classical theories requiring C1-continuity, such as the Euler-Bernoulli beam and Kirchhoff-Love shell theory. Based on these “simplest” models shear deformable theories, introducing Timoshenko and Reissner-Mindlin kinematics, are formulated in a hierarchic manner. In contrast to the usual Finite Element concept using total rotations the present model picks up traditional formulations introducing incremental rotations as primary variables. Furthermore an alternative version is discussed with a split of the displacements into bending and transverse shear parts. Both hierarchic concepts can be easily extended to 3D–shell models. The key aspect of this alternative parameterization is the complete a-priori removal of the transverse shear locking and curvature thickness locking (in the case of 3D-shells).

Abstract

The present study discusses optimization strategies for maximizing the structural ductility of Fiber Reinforced Concrete (FRC) with long textile fibers. Due to material brittleness of both concrete and fibers in addition to complex interfacial behavior between above constituents the structural response of FRC is highly nonlinear. This material nonlinearity including the interface response has to be taken into account for an optimal fiber layout in the structural domain. In the present contribution three kinds of optimization strategies based on a damage formulation are described. The performance of the proposed method is demonstrated by a series of numerical examples; it is verified that the ductility can be substantially improved.

Abstract

In this paper a two-dimensional discrete element method with rigid, polygonal particles is used to model material failure of granular as well as quasi-brittle materials. Different models for soft contact as well as cohesion between the particles are presented. The capabilities of the method are demonstrated simulating simplistic granular model materials as well as complex concrete specimens with an artificial microstructure.

Abstract

The present study addresses an optimization strategy for maximizing the structural ductility of Fiber Reinforced Concrete (FRC) with long textile fibers. Due to material brittleness of both concrete and fiber in addition to complex interfacial behavior between above constituents the structural response of FRC is highly nonlinear. Consideration of this material nonlinearity including interface is mandatory to deal with this kind of composite. In the present contribution three kinds of optimization strategies based on a damage formulation are described. The performance of the proposed method is demonstrated by a series of numerical examples; it is verified that the ductility can be substantially improved.

Abstract

The study addresses the material layout optimization for heterogeneous materials considering non-linear response. In particular two different materials are investigated, namely cellular materials like polymer or metal foams and Fiber Reinforced Composites. The contribution describes all necessary methodical details, such as non-linear material formulations, optimisation building blocks like optimality criteria and sensitivity analyses in the context of non-linear formulations and related algorithmic aspects. Several numerical examples underline the basic concept 1, 2, 3.

Abstract

The coupling of thin, light-weight structures to incompressible flows is a particular challenge within the broad field of computational fluid-structure interaction. The problem is governed by the sensitive and highly non-linear dynamics of slender structures such as shells and membranes in conjunction with the omnipotent incompressibility condition. The contribution presents a partitioned fluid-structure interaction algorithm based on a second order accurate structural solver employing solid shell elements and an Arbitrary Lagrangean Eulerian flow formulation. Stabilized finite elements are used on the fluid domain. Selected aspects of the individual constituents and their interaction are discussed; among them is the problem of conditioning for the shell model, the flow solver satisfying the geometric conservation law and the reliability of the fluid formulation in case of small time steps and distorted mesh. In view of the incompressibility condition the coupling of the two partitions is in particular addressed. It is well-known that sequential coupling approaches, often used for their computational efficiency, may fail in certain situations when the incompressible flow interacts with an extremely slender low mass structure. It could be shown by a rigorous analysis that this is due to the so-called artificial added mass effect. This is an inherent instability of such schemes that unfortunately cannot be removed by reducing the time steps size. As a remedy a strongly coupled partitioning algorithm is applied iteratively adjusting the interface conditions; here different dynamically adapted relaxation methods can be utilized to accelerate the convergence of the iteration. The use of a fully converged coarse grid solution of the coupled problem as a predictor to the iteration scheme can significantly accelerate its convergence and increase the efficiency of the whole coupling scheme. A selection of two- and three-dimensional numerical examples demonstrates the capabilities of the formulation.

Abstract

In many areas of mechanical engineering contact problems of thin-walled structures play a crucial role. Car crash tests and incremental sheet metal forming can be named as examples. But also in civil engineering, for instance when determining the moment-rotation characteristics of a bolted beam-column joint, contact occurs. Effective simulation of these and other contact problems, especially in three-dimensional non-linear implicit structural mechanic is still a challenging task. Modelling of those problems needs a robust method, which takes the thin-walled character and dynamic effects into account. We use a segment-to-segment approach for discretization of the contact and introduce Lagrange Multipliers, which physically represent the contact pressure. The geometric impenetrability condition is formulated in a weak, integral sense. Choosing dual shape functions for the interpolation of the Lagrange Multipliers, we obtain decoupled nodal constraint conditions. Combining this with an active set strategy, an elimination of the Lagrange multipliers is easily possible, so that the size of the resulting system of equations remains constant. Discretization in time is done with the implicit Generalized--$\alpha$ Method and the Generalized Energy-Momentum Method. Using the ”Velocity-Update” Method, the total energy is conserved for frictionless contact. Various examples show the performance of the presented strategies.

Abstract

Numerical simulation of large scale computational fluid dynamics (CFD) and fluid-structure interaction (FSI) problems is still today a very challenging task. The correct modeling of fluid flow, governed by the instationary incompressible Navier-Stokes equations, and the nonlinear structural behavior are challenges of their own. In addition coupling of both physical fields introduces further requirements on stability and efficiency of the involved algorithms. For this class of problems computing time is still a limiting factor for size and complexity of the problem. Especially for FSI simulations the necessary iterative coupling schemes dramatically increase the required computing time. Here, the use of advanced coupling strategies, reducing either the time needed for one iteration or the number of iterations, can considerably speed up the simulation. We will introduce a class of coupling schemes enhanced by a coarse grid solution to reduce calculation time and accelerate convergence. In addition, the single fields – especially the fluid – demand high efficiency of the solution algorithm. Here, assembly of the element matrices and in particular the iterative solver for the global system of linear equations are the most time consuming parts of the computational process. Very often these algorithms only use a small fraction of the available computer power in scientific codes. Therefore it is highly advisable to take a closer look at the efficiency of algorithms and improve them to make the best out of the available computer power. We will present approaches to significantly increase efficiency in the assembly and solution part of a finite element code. Here, the special features of vector super computers will be exploited.

Abstract

A numerical approach to simulate the coupled problem of slender structures and incompressible Newtonian flows is described. The formulation is based on a partitioned scheme while finite elements are employed on the single fields. The structural field is discretized in space by finite elements based on a seven-parameter shell formulation capable of dealing with the complex dynamics of very slender structures. An efficient solver for the structural system of equations exploiting an algebraic multigrid approach has been developed. A stabilized fluid element following an ALE formulation of the incompressible Navier-Stokes equations is used to model the flow on a time dependent domain. The geometric conservation law is built in into the formulation to obtain a stable scheme which is second order accurate in time. The accuracy and reliability of the approach with respect to critical parameters such as very small time steps, distorted meshes or steep gradients in particular within the flow field has been analyzed. The approach is able to also deal with free fluid surfaces. The inherent instability of weakly coupled approaches is analyzed which reveals the devastating influence of the so-called `artificial added mass' effect. The convergence of the strongly coupled approach depends on an appropriately chosen relaxation parameter. Ways to automatically adjust the proper amount of relaxation are given. Special emphasis is also put on the efficient implementation particularly of the fluid problem. Vectorization is employed to significantly speed up the time spent for calculating element matrices on vector machines.

Abstract

The artificial added mass effct inherent in sequentially staggered coupling schemes is investigated by means of a fluid-structure interaction problem. A discrete representation of a simpliflied added mass operator in terms of the participating coefficient matrices is given and `instability conditions' are evaluated for different temporal discretisation schemes. With respect to the time discretisation two different cases are distinguished. Discretisation schemes with stationary characteristics might allow for stable computations when good natured problems are considered. Such schemes yield a constant instability limit. Temporal discretisation schemes which exhibit recursive characteristics however yield an instability condition which is increasingly restrictive with every further step. Such schemes will therefore definitively fail in long time simulations irrespective of the problem parameters. It is also shown that for any sequentially staggered scheme and given spatial discretisation of a problem, a mass ratio between fluid and structural mass density exists at which the coupled system becomes unstable. Numerical observations confirm the theoretical results.

Abstract

The efficient modeling of 3D contact problems is still a challenge in non-linear implicit structural analysis. Most of the existing contact algorithms use penalty methods to satisfy the contact constraints, which necessitates a user defined penalty parameter. As it is well known, the choice of this additional parameter is somehow arbitrary, problem dependent and influences the accuracy of the analysis. We use a primal-dual active set strategy, based on dual Lagrange multipliers to handle the nonlinearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to get a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For our application to thin-walled structures we adapt a three-dimensional non-linear shell formulation, including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface oriented shell element, which allows to apply the contact conditions directly to nodes lying on the contact surface. The discretization in time is done with the implicit Generalized Energy-Momentum Method. To conserve the total energy within our contact framework, we follow an approach from Laursen and Love who introduce a discrete contact velocity to update the velocity field in a post processing step. Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.

Abstract

The efficient modeling and analysis of 3D contact problems is still a difficult aspect in non-linear implicit structural dynamics. Most of the existing contact algorithms use some kind of penalty method to enforce the contact constraints. This procedure necessitates a user defined penalty parameter, the choice or which is somehow arbitrary, problem dependent and might influence the accuracy of the analysis. We use a primal-dual active set strategy, based on dual Lagrange multipliers to handle the non-linearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without introducing any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers, which represent the contact pressure, can be locally eliminated. A static condensation will lead to a reduced system of equations for the unknown displacements. The Lagrange multipliers can be easily recovered from the displacements in a variationally consistent way. It is well known, that the combination of a contact algorithm with a numerical time integration scheme will not yield an energy conserving procedure, unless some additional discrete constraints are satisfied. The most popular way to design an energy conserving contact algorithm is to enforce the so called algorithmic persistency condition. Various strategies to fulfill this additional constraint lead to a compromise of violating the geometric admissibility. In this work we will pick up an idea by Laursen and Love, who introduce a discrete contact velocity to update the velocity field in a post processing step. This procedure leads to an energy conserving scheme while still satisfying the geometric contact constraints. Using the Generalized Energy-Momentum Method for the time discretization, we generalize the velocity update method accordingly. For our application to thin-walled structures we adapt a 7-parameter non-linear shell formulation for the application of contact problems. A reparametrization of the geometric description of the shell body will lead to a surface oriented shell element. Various examples will be presented to show the performance of the proposed algorithms.

Abstract

The efficient modeling and analysis of 3D contact problems is still a chalenge in non-linear implicit structural dynamics. Many of the existing contact algorithms use some kind of penalty method to enforce the contact constraints, which necessitates a user defined parameter. Its choice is somehow arbitrary, problem dependent and might influence the accuracy of the analysis. In this work, a primal-dual active set strategy, based on dual Lagrange multipliers is used to handle the non-linearity of the contact conditions. This allows to enforce the contact constraints in a weak, integral sense without introducing any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers, representing the contact pressure, can be locally eliminated. A static condensation leads to a reduced system of equations for the unknown displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the spatial discretization of thin-walled structures, a 7-parameter non-linear shell formulation is adapted for the application of contact problems. A reparametrization of the geometric description of the shell body results in a surface oriented finite shell element, which allows to apply the contact conditions directly to nodes lying on the contact surface. The time discretization is done whith the Generalized Energy-Momentum Method. Generalizing an idea by Laursen and Love, who introduce a discrete contact velocity to update the velocity field in a post processing step, according to the contact algorithm will lead to an energy conserving scheme while still satisfying the geometric contact constraints. Various examples will illustrate the good performance of the propsed strategy applied to the implicit dynamic analysis of thin-walled structures.

Abstract

The efficient modeling and analysis of 3D contact problems is still a chalenge in non-linear implicit structural dynamics. Many of the existing contact algorithms use some kind of penalty method to enforce the contact constraints, which necessitates a user defined parameter. Its choice is somehow arbitrary, problem dependent and might influence the accuracy of the analysis. In this work, a primal-dual active set strategy, based on dual Lagrange multipliers is used to handle the non-linearity of the contact conditions. This allows to enforce the contact constraints in a weak, integral sense without introducing any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers, representing the contact pressure, can be locally eliminated. A static condensation leads to a reduced system of equations for the unknown displacements. It is well known, that the combination of a contact algorithm with a numerical time integration scheme will not yield an energy conserving procedure, unless some additional discrete constraints are satisfied. The most popular way to design an energy conserving contact algorithm is to enforce the so-called algorithmic persistency condition. Various strategies to fulfill this additional constraint lead to a compromise of violating the geometric admissibility. Following an idea by Laursen and Love, who introduce a discrete contact velocity to update the velocity field in a post processing step, will lead to an energy conserving scheme while still satisfying the geometric contact constraints. Using the Generalized Energy-Momentum Method for the time discretization, the velocity update method is generalized accordingly. For the analysis of thin-walled structures a 7-parameter non-linear shell formulation is adapted for the application of contact problems.

Abstract

The present study addresses the preliminary steps for the optimization of fiber reinforced cementitious composites (FRC) with respect to brittle failure behavior of a single component. FRC is one of the advanced materials that consist of a concrete matrix and glass, carbon, or aramid fibers. This kind of reinforcement is corrosion free and highly durable when used in concrete, which makes it possible to produce very thin structural concrete elements. It is well known that the properties of such composites depend on the layout of the material at the microscopic level (e.g. fiber size, fiber coating, impregnation, and surface roughness among other material characteristics). The investigation of the sensitivity of the structural response of the composite material with respect to the microscopic material parameters is part of this work. To improve the structural behavior of FRC, the significant parameters of the material models for the interface and the fibers are introduced as design parameters and adjusted by a controlling optimization process. Two different objectives are investigated: strength and ductility of FRC. To perform the structural analysis, an isotropic gradient–enhanced damage model for concrete as well as fibers is used. Numerically integrated interface elements are applied for modeling the debonding between the fibers and the matrix. To obtain a reliable composite by applying structural optimization, the nonlinear failure behavior of matrix material, fiber, and interface models are considered within the optimization process 3. Gradient–based optimization algorithms are used to adjust these parameters to improve both strength and ductility of such composite structures.

Abstract

In this paper we address various efficiency aspects of finite element (FE) simulations on vector computers. Especially for the numerical simulation of large scale Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) problems efficiency and robustness of the algorithms are two key requirements. In the first part of this paper a straightforward concept is described to increase the performance of the integration of finite elements in arbitrary, unstructured meshes by allowing for vectorization. In addition the effect of different programming languages and different array management techniques on the performance will be investigated. Besides the element calculation, the solution of the linear system of equations takes a considerable part of computation time. Using the jagged diagonal format (JAD) for the sparse matrix, the average vector length can be increased. Block oriented computation schemes lead to considerably less indirect addressing and at the same time packaging more instructions. Thus, the overall performance of the iterative solver can be improved. The last part discusses the input and output facility of parallel scientific software. Next to efficiency the crucial requirements for the IO subsystem in a parallel setting are scalability, flexibility and long term reliability.

Abstract

For the numerical simulation of large scale CFD and fluid-structure interaction (FSI) problems efficiency and robustness of the algorithms are two key requirements. In this paper we would like to describe a very simple concept to increase significantly the performance of the element calculation of an arbitrary unstructured finite element mesh on vector computers. By grouping computationally similar elements together the length of the innermost loops and the vector length can be controlled. In addition the effect of different programming languages and different array management techniques will be investigated. A numerical CFD simulation will show the improvement in the overall time-to-solution on vector computers as well as on other architectures. Especially for FSI simulations also the robustness of the algorithm is very important. For the transient interaction of incompressible viscous flows and nonlinear flexible structures commonly used sequential staggered coupling schemes exhibit weak instabilities. As best remedy to this problem subiterations should be invoked to guarantee kinematic and dynamic continuity across the fluid-structure interface. To ensure the efficiency of these iterative substructuring schemes two robust and problem-independent acceleration methods are proposed.

Abstract

For the dynamic behavior of lightweight structures like thin shells and membranes exposed to fluid flow the interaction between the two fields is often essential. Computational fluid-structure interaction provides a tool to predict this interaction and complement or eventually replace expensive wind tunnel experiments. Partitioned analyses techniques enjoy great popularity for the numerical simulation of these interactions. This is due to their computational superiority over simultaneous, i.e. fully coupled monolithic approaches, as they allow the independent use of suitable discretization methods and modular analysis software.We use, for the fluid, GLS stabilized finite elements on a moving domain based on the incompressible instationary Navier-Stokes equations, where the formulation guarantees geometric conservation on the deforming domain. The structure is discretized by nonlinear, three-dimensional shell elements. Commonly used sequential staggered coupling schemes may exhibit instabilities due to the so-called artificial added mass effect. As best remedy to this problem subiterations should be invoked to guarantee kinematic and dynamic continuity across the fluid-structure interface. Since iterative coupling algorithms are computationally very costly, their convergence rate is very decisive for their usability. To ensure and accelerate the convergence of this iteration the updates of the interface position are relaxed. The time dependent, ’optimal’ relaxation parameter is determined automatically without any user-input via exploiting a gradient method or applying an Aitken iteration scheme.

Abstract

This paper presents a two-scale approach for the mechanical and numerical modeling of composites like textile Fiber Reinforced Concrete (FRC) and Fiber Reinforced Polymers (FRP) characterized by long range material failure. The objective of the present study is to develop an efficient solution method exploiting the physically existing multiscale character of the problem. The proposed method belongs to the superposition based methods with local enrichment of the large scale solution u by a small scale part u0. The question of appropriate locality constraints for u0 will be investigated. The second central point is the handling of highly localized failure zones or cracks (strong discontinuities) within the composite on the small scale level. Besides usual continuum material models like the gradient enhanced damage formulation the eXtended Finite Element Method (X-FEM) coupled to the level set method is used to model the material behavior of composites on the small scale.

Abstract

This contribution focusses on computational approaches for fluid structure interaction problems from several perspectives. Common driving force is the desire to handle even the large deformation case in a robust, efficient and straightforward way. In order to meet these requirements main subjects are on the one hand necessary improvements on coupling issues as well as on Arbitrary Lagrangian Eulerian (ALE) approaches. On the other hand we discuss pros and cons of available fixed grid approaches and start the development of new such approaches. Some numerical examples are provided along the paper.

Abstract

The present work deals with the comparison of three-dimensional (3d) finite element analysis of thin-walled structures to computations based on reduced, two-dimensional (2d) models which a priori satisfy the plane stress assumption. We apply a plasticity-based non-linear softening material model for concrete1 to simple four-node 2d plane stress elements and to a 7-parameter 3d shell formulation. Numerical results are compared to experimental data for an L-shaped panel.

Abstract

This paper addresses the mechanical and numerical modeling of materials with microstructure in the nonlinear range. One intended application is the simluation of fiber reinforced concrete (cement), a material which allows to design very thin structures. The properties of this composite depend on the one hand on the material layout, e.g. the properties of its components, the fiber length, content, coating and orientation (length scale µm) and on the other hand on the structural layout, e.g. ply thickness (length scale cm). This gives an idea of the physically intrinsic scales appearing: A ”microscale”, the scale of material heterogeneity and a ”macroscale” related to the structural dimensions. The post critical behaviour of such a composite is driven by the accumulation of the failure mechanisms on the microscale. This can be fiber or matrix cracking, debonding between the fibers in a filament, or between fiber and matrix. Those failure mechanisms, taking place on the microscale, are incorporated in the macroscopic formulation using the variational multiscale method (VMM), introduced under this name by Hughes et al. 9. A central point of the presented scheme for an efficient solution of the discrete problem is the locality assumption for the small scale part of the solution, leading to decoupled problems on the micro scale. One focus of the presentation is the description of the material behaviour of composites on the microscale whereby two potential and well-known approaches will be presented. For applications with smeared failure of the matrix material we use an isotropic gradient enhanced damage model. In addition, the numerical model comprises conventional interface elements which account for interfacial failure between the particular material components. The second approach constitutes the utilization of the extended finite-element method (X-FEM) 13 to model discrete failure of the matrix material as well as of the particular interfaces.

Abstract

For the numerical simulation of large scale CFD and fluid-structure interaction (FSI) problems efficiency and robustness of the algorithms are two key requirements. In this paper we would like to describe a very simple concept to increase significantly the performance of the element calculation of an arbitrary unstructured finite element mesh on vector computers. By grouping computationally similar elements together the length of the innermost loops and the vector length can be controlled. In addition the effect of different programming languages and different array management techniques will be investigated. A numerical CFD simulation will show the improvement in the overall time-to-solution on vector computers as well as on other architectures.

Abstract

A robust computational approach for transient dynamic impact processes will be presented. Main focus will be on an adaptive remeshing strategy and constitutive models for metals and geomaterials under impact loading. Practical relevant numerical examples will complete this contribution.

Abstract

Different discrete element models (DEM) for the simulation of cohesive geomaterials like concrete, ceramics or marl will be presented. Starting from a basic polygonal two-dimensional DEM model for non-cohesive granular materials, more complex models for cohesive materials are obtained by inclusion of beam or interface elements between corresponding particles. The last step in the series of increasing complexity is the realization of a microstructure-based simulation environment which utilizes the enhanced DEM models. A type of artificial microstructure can be included, if different properties of the cohesive components (beam or interface) are assigned with respect to their position, i.e. inside the matrix, inside the aggregate and between aggregate and matrix. With growing model complexity a wide variety of failure features of geomaterials can be represented. Furthermore, the inclusion of an artificial microstructure which regards for stiffer aggregates embedded in a less stiffer matrix enables a coherent quantification of the model. A validation of these discrete models with regard to qualitative and quantitative aspects enriches the plan of the paper. Thereby, extracts of the extensive simulation program are reported bit by bit in the context of each section in order to clarify the respective subject matter. The paper is organized as follows: The second section provides a brief outline of the theoretical background of a standard (non-cohesive) form of the DEM model supplemented by a discussion on the modeling of cohesion within particle models. Afterwards, in the third section, the extension to an interface enhanced DEM model along with some representative simulation results is sketched. The fourth section presents the formulation of a microstructure-based DEM environment which utilizes the interface enhanced model according to the third section. Again, simulation results are used to validate the proposed method. A summary in the fifth section concludes with the attained insights and an outlook on future perspectives.

Abstract

Different discrete element models (DEM) for the simulation of problems in the context of damage mechanics will be presented. The target materials to be simulated are cohesive and non-cohesive geomaterials like concrete- or sand-type materials. Starting from a basic polygonal two-dimensional DEM model for non-cohesive granular materials, more complex models for cohesive materials are obtained by inclusion of beam or interface elements between corresponding particles. The last step in the series of increasing complexity is the realization of a microstructure-based simulation environment which utilizes the enhanced DEM models. With growing model complexity a wide variety of failure features of geomaterials can be represented. Furthermore, adequate homogenization approaches are derived which supplement the definition of the discrete models. These homogenization approaches allow us to relate microscopic quantities, like contact forces or displacements, to corresponding macroscopic quantities, like stresses or strains. Representative numerical examples are used to validate the proposed DEM models along with the homogenization.

Abstract

In the present contribution three different model classes for the numerical simulation of geomaterials will be presented. These classes are related to different material scales and are discussed in the first part of the paper. At first, macroscopic models in the context of invariant-based damage and plasticity theories are introduced. Then, a microplane model applying a volumetric-deviatoric split will be discussed. This model describes the material behavior at the material point by formulating constitutive laws on different planes followed by a homogenization process. The last model class discussed concerns a discrete particle model consisting of convex polygons that are able to withstand a limited cohesion. This includes a contact description for the particles as well as a bond formulation between the particles. The related forces are inserted into the equations of motion which are solved numerically based on the discrete element methodology. Macroscopic continuum models are typically involved in the large scale computation of engineering structures; microplane models are used to efficiently model anisotropic material behavior in a natural way. Despite their extreme numerical costs discrete models like particle models become more and more mature for large scale computations. In the second part, interrelations between the models are investigated in terms of homogenization procedures that bridge effects from a lower to a higher scale. In a first approach the constitutive equations of the particle and the microplane model are inspected and reveal remarkable similarities. Afterwards, a homogenization procedure for the particle model is introduced defining representative particle assemblies that allow the determination of average quantities like stresses and strains on a higher scale. Finally, it will be shown how the present version of the microplane formulation enables the interpretation and identification of the microplane constitutive laws in terms of well-known macroscopic constitutive laws. The interrelations between the model classes are illustrated by several numerical examples.

Abstract

Dry powder, as a special case of extremely fine grade granular materials, is a collection of discrete solid grains up to 0.2 mm in size. As the space between the particles is filled with air, the density of the bulk solid is always less than the density of the discrete solid components. The shape of the particles and the formation of the packing are further properties characterizing powder-like materials. Their complex behavior is of high interest for civil, chemical, mechanical and materials engineers. Since many varying applications and processes exist in these different fields, the requirements for appropriate models are also manifold. The consolidation of a loose packing to a dense and compact mold or the comminution of brittle material under high pressure into pulverized media are only two examples. A meaningful kinematic and constitutive model is therefore crucial for every transient dynamic analysis. In this paper, a continuum model for the simulation of dry powder under quasi-static as well as dynamic loading will be presented. It is implemented in the context of multiplicative hyperelastic-based finite strain elasto-plasticity in a Lagrangean framework. In the description of the algorithm, it is focussed on a robust return mapping scheme which is formulated in principal stresses. A nonsmooth multisurface plasticity with tension cutoff, Drucker-Prager failure envelope and strain hardening cap provides the reproduction of the relevant phenomena: tensile failure, material flow under shearing and compaction under pressure. Corresponding material parameters are determined by geotechnical experiments like uniaxial and triaxial compression tests. The results of quasi-static and impact loading simulations are compared with experimental results. Computational issues related to the challenges of continuum treatment of powder-like materials will be addressed.

Abstract

In this paper, a continuum model for the simulation of dry powder under quasi-static as well as dynamic loading will be presented. It is implemented in the context of multiplicative hyperelastic-based finite strain elasto-plasticity in a Lagrangean framework. A nonsmooth multisurface plasticity with tension cutoff, Drucker-Prager failure envelope and strain hardening cap provides the reproduction of the relevant phenomena: tensile failure, material flow under shearing and compaction under pressure. Corresponding material parameters are determined by geotechnical experiments like uniaxial and triaxial compression tests. The results of quasi-static simulations are compared with experimental results.

Abstract

An appropriate treatment of instationary free surface flows is of great relevance to many problems in engineering and applied sciences. In this context the presence of structural interactions constitute an additional challenge on the one hand but on the other hand even further broadens the application area. In this study we present a partitioned implicit approach for the efficient and robust treatment of free surface flows within fluid structure interaction simulations.

Abstract

This paper describes a three-level finite element method for solving the instationary incompressible Navier-Stokes equations. Separating large resolved scales, small resolved scales and unresolved scales enables us to deal with each of these scale groups differently. Whereas the computation of the large scales is performed using a standard Galerkin method the small scales are resolved by approximate residual-free bubbles exploiting an elementwise submesh with respect to the original discretization. The unresolved scales are merely regarded in their dissipative effect onto the small scales. A dynamic modeling process for this making use of an elementwise sub-submesh slightly finer than the submesh is incorporated as level 3 of the method. Currently, applications of this method to the numerical simulation of turbulent flows are underway.

Abstract

The present work is concerned with the numerical simulation of nonlinear behaviour in advanced composite material structures using the Finite Element method. Thereby special attention is payed to nonlinear material behaviour and shell structures. Often Fibre Reinforced Polymers are utilised within shell-like and layered structures because of their superior stiffness to weight ratio and the possibility to design material and structural behaviour, respectively. Unfortunately the use of a layered laminate may introduce the important failure mode of delamination. In this study phenomenological material models known from literature are presented, which are capable of describing the overall structural response including local effects like delaminations. Based on the work of Hashagen, 1998 an anisotropic plasticity model combined with anisotropic hardening, which is capable to describe the overall nonlinear structural response, is first presented. In order to study the effect of the anisotropic hardening plasticity model structural examples are compared and assessed. Secondly the frequently observed and very dangerous type of failure delamination is considered in particular. For this the delamination process is described within a thin layer of finite thickness in the framework of three-dimensional finite shell-elements with layerwise Reissner-Mindlin kinematics Braun et al., 1994. The Hashin Hashin, 1980 delamination criterion is utilized in the framework of a softening plasticity formulation based on Sprenger, 2000. The softening behaviour is assumed to be linear with transition to an exponential function at high equivalent plastic strains. The approach is assessed through a comparison of experimental and numerical results of a DCB test. An elaboration on the overall approach can be found in Hörmann, 2002.

Abstract

Geomaterials belong to the most widely used materials in engineering. Materials like concrete, marl, cement or sand are typical examples. The failure modes of these materials range from brittle to ductile behavior and usually involve anisotropic biases through an overall stiffness degradation. This degradation results for example from a coalescence of microcracks or a typical failure mode localization. In the present contribution three different model classes for the numerical simulation of geomaterials will be presented. These classes are related to different material scales and are discussed in the first part of the paper. At first, macroscopic models in the context of invariant-based damage and plasticity theories are introduced. Then, a microplane model applying a volumetric-deviatoric split will be discussed. This model describes the material behavior at the material point by formulating constitutive laws on different planes followed by a homogenization process. The last model class discussed concerns a discrete particle model consisting of convex polygons that are able to withstand a limited cohesion. This includes a contact description for the particles as well as a bond formulation between the particles. The related forces are inserted into the equations of motion which are solved numerically based on the discrete element methodology. Macroscopic continuum models are typically involved in the large scale computation of engineering structures; microplane models are used to efficiently model anisotropic material behavior in a natural way. Despite their extreme numerical costs discrete models like particle models become more and more mature for large scale computations. In the second part, interrelations between the models are investigated in terms of homogenization procedures that bridge effects from a lower to a higher scale. In a first approach the constitutive equations of the particle and the microplane model are inspected and reveal remarkable similarities. Afterwards, a homogenization procedure for the particle model is introduced defining representative particle assemblies that allow the determination of average quantities like stresses and strains on a higher scale. Finally, it will be shown how the present version of the microplane formulation enables the interpretation and identification of the microplane constitutive laws in terms of well-known macroscopic constitutive laws. The interrelations between the model classes are illustrated by several numerical examples.

Abstract

This paper discusses partitioned analysis approaches for the transient interaction of incompressible viscous flows and nonlinear flexible structures with large deformations. It is shown that for this kind of coupled systems the commonly used sequential staggered coupling schemes exhibit weak instabilities and may eventually fail, mainly due to inconsistencies in the coupling boundary conditions. As best remedy to this problem subiterations should be invoked which ensure kinematic and dynamic continuity across the fluid-structure interface, thus ensuring stable and accurate numerical solutions even for long-time simulations. For the desirable acceleration of convergence of such iterative substructuring schemes two robust and problem-independent methods are proposed and their efficiency is demonstrated with selected numerical examples.

Abstract

Classical continuum mechanics formulations applying e.g. standard finite elements can consider only a small number of discontinuities and cannot encompass the entire fracturing process. However more enhanced continuum models, for example based on the gradient or the Cosserat theory, may provide the possibility to account for a granular microstructure in a straight forward way. On the other hand, the mechanical behaviour of granular materials can be well described on a smaller scale via discrete element models (DEM) if the motion of the individual grains is taken into account. The time evolution of such a model is followed solving the Newtonian equations of motion of the individual elements numerically. Our concern in this paper is the relation between the microscopic quantities, like the contact forces and displacements, and corresponding macroscopic quantities, like stresses and strains via corresponding homogenization techniques. In this contribution we will propose a new technique and validate it by two-dimensional DEM simulations with polygonal particles. The evaluation of different boundary value problems proves the efficiency of the homogenization technique. In this context the dynamical, kinematical and energetical quantities of a representary elementary volumes are determined. A non-symmetric macroscopic stress tensor, as well as the existence of couple stresses are obtained.

Abstract

In this work we present a Three-Level FEM for the numerical simulation of the incompressible Navier- Stokes equations. Based on the variational multiscale method and the residual-free bubble concept we pursued the technique of a Two-Level FEM in our previous work. Level 1 is therein attributed to the solution of the large-scale equation by a standard finite element space. On level 2 we seek an approximation of the elementwise residual-free bubbles with the help of a stabilized finite element method restricted to individual elements. This represents our small-scale solution. The Two-Level FEM aimed at the simulation of laminar flows. Now we want to turn our focus on the numerical simulation of turbulent flows by way of a large eddy simulation and in this work our first respective steps are described. With regard to this objective we extend our basic algorithm paying particular attention to the fulfillment of the continuity equation. Moreover, in lieu of the stabilization term we alternatively propose the inclusion of a subgrid viscosity term henceforth. This term should provide us with the required dissipative effect of the unresolved scales on the small-scale equation. The dynamic calculation of the subgrid viscosity constitutes the third level of our algorithm.

Abstract

Fiber Reinforced Polymers (FRP) are increasingly used in all kinds of engineering applications, e.g. retrofitting of structural members, structural parts of aircrafts or cars and already complete pedestrian and vehicle bridges. Due to the brittle nature of most FRP, which depends on the used materials for fibers and matrix, and in order to minimize the costs of these materials, it is of utmost interest to predict the load carrying capacities and failure modes correctly. It is widely accepted that three main failure modes of FRP exist: (i) fiber rupture, (ii) matrix micro cracking and (iii) debonding of the interface fiber/matrix. Combining several layers of equally or differently oriented FRP leads to a laminated structure eventually involving the additional and very important failure mode of delamination. In this contribution we will restrict ourselves to this failure mode, since it is predominant compared to the other ones. The delamination process between two adjacent FRP layers is geometrically described by a thin softening process zone, such that a continuum mechanics model is still applicable. Consequently the introduction of zero thickness interface elements is avoided, but the most likely delamination interface must be known in advance by means of a First Ply Failure (FPF) analysis. The present approach is prefered compared to a discrete geometrical description of the delamination since it omits remeshing of the delamination front, which has in general a complex geometry. Finite shell elements with layerwise Reissner-Mindlin kinematics are used, leading to the fact, that the number of degrees of freedom per node depends on the number of layers having its own kinematics.Three different material models are used to simulate the nonlinearity of the process zone. Firstly, the Hashin delamination criterion is utilized in the framework of a softening plasticity formulation based on the work of Sprenger and Wagner et al., respectively. The softening behaviour is assumed to be linear with transition to an exponential function at high equivalent plastic strains. Secondly, the Hashin criterion is transferred into strain space and applied in the context of an isotropic damage formulation. Linear or modified power law softening damage evolution is assumed. Thirdly, a damage formulation is derived which uses the definition of a specific damaged material tensor and leads to an energy release rate based failure criterion. All material models use the fracture energy Gc as controlling parameter for the softening regime Schellekens and de Borst.

Abstract

In the present paper, an alternative microplane formulation based on a split which is motivated from a macroscopic viewpoint will be derived. Thereby, both, elastic and inelastic material behavior is taken into account in an unique way. The constitutive formulation is embedded in a thermodynamically consistent framework for microplane models. Consequently, the material behavior at a material point is characterized by constitutive laws formulated on the individual microplanes, allowing to describe anisotropic material behavior in a very natural and simple way. Special features of the new microplane formulation will be discussed in comparison to existing microplane models presented in the literature. It will be shown that this alternative approach remedies several deficiencies of former microplane formulations. One particular advantage of the present version is that the macroscopic response is directly reflected on the mesoscale.

Abstract

The present contribution focuses on the infuence of geometrical nonlinearities on the structural behavior in the design process. The notion of the stiffest structure loses its clear definition in case of nonlinear kinematics; here we will discuss this concept on the basis of three different objectives. Apparently topology optimization is often a generator of slender struts, which tend to buckle before the structure is completely loaded. To include the instability phenomena into the design process the critical load level will be determined directly; this condition will be included as an inequality constraint. Further on in order to reduce the imperfection sensitivity a geometrically modified structure including the imperfection shape is introduced in addition. The present optimization procedures are demonstrated by examples showing rather the principle effects of the enhancements than real practical design problems.

Abstract

Ultimate load analyses and nonlinear dynamics, parallel and efficient solution approaches, adaptivity and error estimation, multi-scale analyses, structural optimization and coupled problems simulation are some of present challenges in computational mechanics. The present talk addresses peculiarities of shell structures in such an advanced computational environment. Here their dominant feature plays a central role, namely their usually large slenderness, allowing them to be optimal on the one hand but possibly leading to an extreme parameter sensitivity on the other hand. This is reflected not only in their physical behavior but also involved in their numerical models and formulations. Despite the general character the present contribution will focus on a few selected areas where these shell specific features are dominant. For example these aspects will be demonstrated in the context of structural optimization, coupled problem simulation and ultimate load analyses. The paper abandons the attempt to present and discuss details of this wide area. Rather it is intended to exemplary select a few typical characteristics and provide an introduction into the topic following the typical objective of a plenary lecture. To this aim principal shell phenomena and modeling aspects are addressed; this comes along with several remarks about nowadays computational environment and some aspects of interdependency between mechanical response and computational model. Through this introduction one should be able to internalize the underlying idea behind this topic and to be sensitized for the peculiar features of these thin-walled curved structures. For more detailed discussions of the different aspects and fields the reader is also referred to further publications of our research group that are listed under http://www.uni-stuttgart.de/ibs/

Abstract

A two-dimensional model of heterogeneous cohesive frictional solids where the material structure is idealized by a discrete granular particle assembly is presented. The discrete model is composed of convex polygons which are linked together by simple beams accounting for cohesive effects. Varying its parameters the model naturally interpolates between a continuous solid state and a discontinuous dry granular state of the material. In order to illustrate the application range of our model various numerical simulations have been carried out. The emphasis lies on the studying of quasi-static loading scenarios of rectangular solids, like compression, tension, simple shear and recently carried out biaxial simulations. In the context of biaxial setups four different lateral pressures have been investigated and their effect on the macroscopic stress-strain response was studied as well as the evolving shear zones. A comparison with experimental observations provides a verification of the qualitative application of the model.

Abstract

From a physical point of view geomaterials, like concrete, ceramics or marl can be considered as cemented granulates forming a heterogeneous macroscopic solid. The failure mechanisms of this materials are characterized by complex failure modes under various loading situations and a highly anisotropic bias due to their inhomogeneous microstructure. The growth and coalescence of microcracks lead to the formation of macroscopic crack patterns and, finally, to a fragmentation into separate particle clusters, forming a solid--granulates mix. In this paper we present a two-dimensional model of heterogeneous cohesive frictional solids where the material structure is idealized by a discrete granular particle assembly. Our discrete model is composed of convex polygons which are linked together by simple beams accounting for cohesive effects. Varying its parameters the model naturally interpolates between a continuous solid state and a discontinuous dry granular state of the material. In order to demonstrate the wide applicability of the model simulations ranging from the quasi-static uniaxial loading and shearing of a solid to the dynamic fragmentation due to explosion, impact and collision of solids, will be presented. A comparison with experimental observations is carried out in order to verify the qualitative application of the model. To provide a geometrical description of the damage state, different characteristic quantities will be introduced and their relevance will be examined. In the context of a geometric description of the state of damage different failure identicators will be introduced and their relevance will be examined. In this context especially a defect correlation length will be introduced. This is done in order to calculate the width of an evolving localization zone as well as to determine the switch-over point where the failure within a specimen changes from a crack nucleation to a crack propagation mode. By a histogrammatic representation of the position of evolving cracks with regard to the effective shear zone (line) three different regions are observed and thus lead to a determination of the shear band width. It will generally be demonstrated that this simulation method allows us to monitor quantities which are hard to measure or are not measurable at all in experiments. Hence, this model provides a deeper understanding of the fracture and fragmentation processes well beyond the description of the final state.

Abstract

The constitutive description of the mechanical behavior of granular systems is of great interest to the fields of geotechnics and various other related applications. By taking into account the discrete nature of the microstructure of a granular assembly, numerous different discrete models have been developed, Most of them are based on a finite number of discrete, semi-rigid spherical or polygon-shaped particles interacting by means of contact forces. The specification of an appropriate contact law is probably the most significant part of the discrete model. For example, the contact behavior can be formulated in a linear or nonlinear elastic fashion according to the classical Hertz model or include frictional effects along the line of Coulomb's friction law. In order to compare the results of the discrete element simulation with macroscopic measurements, different averaging techniques can be applied to derive homogenized quantities characterizing the overall behavior of the assembly. Thus, the definition of the macroscopic stress tensor has been studied intensively in the beginning of the 80s and can now be considered as well-established While discrete models take into account the individual behavior of each single particle, continuum-based approaches can only describe the material behavior in an average sense. Although, in most cases, the choice of the specific constitutive formulation is motivated by microstructural considerations, the material response is characterized exclusively in terms of stresses or strains and a set of internal variables, which represent microstructural effects in a phenomenological fashion. The microplane plasticity model is a classical representative of this class of continuum-based constitutive models. It is based on the early ideas of Mohr, who suggested to characterize the response of a material point by describing its behavior in various representative directions in space. Similar to the particle models, the choice of the constitutive assumption relating the corresponding stress and strain vector of each direction can be considered as the most important feature of the model. The overall response of the material point is obtained by integrating the resulting stress vectors over the entire solid angle. Although derived from two completely different fields, both models show significant similarities from a theoretical point of view. This contribution aims at highlighting the equivalences of the two different material formulations. Therefore, the basic ideas of the microplane model are summarized as well as a basic introduction to discrete particle modelling. Particular interest is dedicated to the fact, that the initially continuous microplane model has to be discretized for computational reasons while the initially discrete particle model must be 'continuumized' in order to relate its material parameters to macroscopic quantities. Finally, the two different approaches are compared and advantages and disadvantages of both formulations are discussed.

Abstract

Usually mechanical laws are applied for a prescribed design of a structure in order to determine its response. The external loads, boundary conditions, and also the geometry which is defined by the topology and the shape of the structure are given. However, the mechanical principles can also be used to determine the conceptual layout (topology) for a prescribed structural response. This inverse method is called ’Structural Optimization’. To obtain a reliable design, it is essential to base optimization as far as possible on realistic physical behaviour. In the present work, an elastoplastic material model with linear isotropic/kinematic hardening/ softening for small strains is used. The objective of the chosen design problem is the maximum structural ductility for a prescribed amount of mass in the design space and will be compared to results of a topology optimization analysis based on a pure elastic solution in a three–dimensional design space. Since the structural response is history–dependent, as it is in flow theory of plasticity, the sensitivities are also history–dependent. Two approaches for the calculation of sensitivities, the direct differentiation method and an efficient self–adjoint sensitivity analysis are tested with respect to their reliability. Numerical examples for the topology design of three–dimensional structures are presented.

Abstract

This paper discusses partitioned analysis approaches for the transient interaction of incompressible viscous flows and nonlinear flexible structures with large deformations. It is shown that for this kind of coupled systems the commonly used sequential staggered coupling schemes exhibit instabilities and eventually fail, mainly due to inconsistencies in the treatment of the coupling boundary conditions. As best remedy to this problem subiterations should be invoked which ensure kinematic and dynamic continuity across the fluid–structure interface, thus ensuring stable and accurate numerical solutions even for long-time simulations. For the effective convergence acceleration of such iterative substructuring schemes two robust and problem-independent methods are proposed and their efficiency is demonstrated with selected numerical examples.

Abstract

The numerical simulation of the interaction of fluids and solids not only has a wide range of applications in all different engineering disciplines and natural sciences it is still one of the present challenges in computational mechanics. This talk will first provide a short overview on different areas and approaches of Fluid–Structure Interaction (FSI). The main concern of the presentation will then be on special modeling, discretization and solution aspects of a transient and nonlinear FSI–approach for incompressible, viscous flows with thin–walled structures.

Abstract

A challenge for parallel iterative solvers in nonlinear shell analysis comes from the fact that shell analyses inherently tend to be ill conditioned problems. This challenge becomes especially pronounced when three-dimensional shell formulations - including thickness change - are used. A preconditioning idea for such models along with its advantageous influence on the performance of (parallel) iterative solvers is introduced and discussed in this paper.

Abstract

The failure mechanisms of heterogeneous geomaterials like concrete, ceramics or marl are characterized by complex failure modes under various loading situations and a highly anisotropic bias due to their inhomogeneous microstructure. The growth and coalescence of microcracks lead to the formation of macroscopic crack patterns resulting in an overall stiffness degradation and eventually in a structural failure. Behaving quasi-brittle under load, this materials are characterized by a localization of deformations in typically narrow zones. This class of materials show different physical and geometrical properties depending on the level of observation. Taking into account the point of view and the chosen scale the structural fluctuations play a more or less dominant role and thus determine the point for which a material can be regarded as homogeneous. This is accompanied by the fact, that with decreasing resolution length on a specific level the internal material structure is less identifiable up to the point when the material is considered as continuous. Basically a class of continuum models can be applied at each size scale, if the local quantities, e.g. damage, are smeared over a certain region. Emerging heterogeneities, anisotropies and discontinuities are then cast in the form of macroscopic continuum variables. From the point a localization phenomenon, like a crack, occurs due to a specific loading the material cannot be treated as continuous any longer. Due to their theoretical basis continuous simulation models cannot account for the discrete nature of material failure in a natural way. Therefore, discrete models like particle, lattice or molecular/granular dynamics models are used. Although prohibitive in large scale computations this class of models is able to predict and simulate the fracture behaviour of small scale applications of geomaterials. We will present different numerical mesolevel models within the framework of computational material mechanics allowing completely diverse insights into the material behaviour. In order to obtain an anisotropic damage evolution in a natural and conceptually simple way geomaterials are modelled as a discrete granular particle assembly composed of convex polygons that are linked together by simple beams accounting for cohesive effects. An internal length scale is incorporated intrinsically into the model by a statistically controlled particle generation process. Compression simulations with different boundary conditions are used to verify the qualitative application of the model. A transition from a continuous to a discontinuous state of the material results as a naturally output of the simulation. The continuous approach presented in the second part is based on the microplane concept allowing for directional dependent stiffness degradation at the material point level. Different concepts regarding the enhancement of the microplane formulation are outlined. Among these are versions for elasto-damage, elasto-plasticity and their combination on the one hand and a gradient enhancement on the other hand. Finally, a comparison of both, discrete and continuous models, namely the discrete element model and the continuous elastic microplane model, shows some remarkable similarties between both schemes. This allows a schematic comparison noting the corresponding inter-relations between both models.

Abstract

Thin shells are per definition optimized structures. If they are designed properly they can carry heavy loads over large spans with a minimum of construction material by activating an almost pure membrane stress state. In this context the shape of a shell plays a key role. The present paper deals with the comparison of two prominent numerical formfinding methods. The first procedure is related to the hanging model principle, traditionally used in physical experiments. Here this principle is simulated through numerical computation. As in the physical model the cutting pattern, the material used and the applied load case are important parameters. The second scheme is more general; it utilizes the structural optimization concept and allows a multitude of design criteria. In this concept the shape is described by a CAGD model and the structural analyses are supplemented by sensitivity computation and mathematical optimization. Both methods are compared for a typical free form shell over a quadrilateral plan.

Abstract

This paper discusses special aspects of a three-dimensional formulation for the modelling of shell structures. Such a formulation offers various merits compared to `classical' shell models. With respect to the realization of such a model one is on the other hand also confronted with a number of challenges. One challenge that is adressed in this paper is the ill conditioning of such three-dimensional shell models compared to `classical' models. Especially in the context of iterative solvers or in the case of explicit dynamical simulations this challenge might become in particular urgent. In this article an approach is presented and discussed that offers a remedy to this challenge. This approach is based on the use of a scaled director. It can be interpreted as a kind of `smart' preconditioning of three-dimensional shells based on an in-depth knowledge of the underlying physical model. It is able to effectively improve the condition number of the global stiffness matrix almost without additional numerical costs. The scope of it is to achieve an improved numerical behaviour of such models in nonlinear static and dynamic simulations. Details on the formulation as well as its impact on iterative solvers and explicit dynamical simulations will be given.

Abstract

This paper discusses the numerical simulation of the dynamic interaction of shells exhibiting large deformations and incompressible, viscous flows. Fluid structure interaction is herein viewed as a three–field coupled problem, consisting of the physical and/or computational fields — fluid, structure and moving mesh in the context of an Arbitrary Lagrangean Eulerian (ALE) formulation. Special focus is given to the three–dimensional shell formulation, the stabilized 3D finite element flow solver and on some coupling issues between these models. Selected numerical examples demonstrate the performance of the overall computational procedure.

Abstract

Stabilized finite element methods have been developed mainly in the context of Computational Fluid Dynamics and have shown to be able to add stability to previously unstable formulations in a consistent way. In this contribution a deformation dependent stabilization technique, conceptually based on the above mentioned developments in the CFD area, is developed for Solid Mechanics to cure the well known Enhanced Assumed Strain (EAS) method from artificial instabilities (hourglass modes) that have been observed in the range of large compressive strains. In investigating the defect of the original formulation the dominating role of the kinematic equation as cause for the instabilities is revealed. This observation serves as key ingredient for the design of the stabilizing term, introduced on the level of the variational equation. A proper design for the stabilization parameter is given based on a mechanical interpretation of the underlying defect as well as of the stabilizing action. This stabilizing action can be thought of an additional constraint, introduced into the reparametrized Hu–Washizu functional in a least square form, together with a deformation dependent stabilization parameter. Numerical examples show the capability of this approach to effectively eliminate spurious hourglass modes, which otherwise may appear in the presence of large compressive strains, while preserving the advantageous features of the EAS method, namely the reduction of the stiffness for an ‘in–plane bending’ mode, i.e. when plane stress elements are used in a bending situation.

Abstract

Two different numerical models within the framework of computational material mechanics will be presented allowing completely diverse insights into the material behaviour. In order to obtain an anisotropic damage evolution in a natural and conceptually simple way "concrete" is modelled as a discrete granular particle assembly composed by convex polygons that are linked together by simple beams accounting for cohesive effects. An internal length scale is incorporated intrinsically into the model by a statistically controlled particle generation process. The theoretical background of the model will be presented in an ample manner and its application to small samples pointing out basic features of either the elastic response of a modelled solid subjected to a fixed loading and the crack propagation up to complete failure and fragmentation will be shown. For the sake of a qualitative comparison of discrete and continuum mechanical models the theoretical background of a typical continuum-oriented formulation, namely the microplane model, is briefly outlined. This continuous approach is based on continuum damage mechanics in combination with the microplane concept, allowing for directional dependent stiffness degradation. In order to simulate nonlocal effects higher order gradients are included in the constitutive equations, thus introducing an internal length scale. This model is in its nature remarkably similar to particle models, as shown recently in a related publication of the institute. The discrete and the continuum model will be compared in a qualitative fashion by means of compressive and tensile simulations. Moreover, a comparison with experiments regarding the mesoscopic mechanisms as failure modes and the development of localization zones will be presented.

Abstract

Structural optimization based on linear material behavior and a linear kinematic relation is common practice in this�eld. In order to obtain a reliable design it is important to apply a more realistic physical behavior, i.e. to consider buckling, elastoplastic behavior and a nonlinear kinematic relation. In the present study an elastoplastic von Mises material model with linear isotropic/kinematic hardening/softening for small strains but large displacements under plane stress conditions is used. The objective of the design problem is to maximize the structural ductility for a given range of prescribed displacements. The mass is kept constant during the optimization process. The crucial point of the entire optimization procedure is to determine the derivatives with respect to the optimization variables, i.e. the sensitivity analysis. For path-dependent problems such as elastoplasticity the derivatives are also path-dependent and have to be determined after each incremental step. These sensitivities of the structural response, objective and constraints are obtained by a variational direct approach for shape as well as topology optimization problems. In order to speed up the topology optimization procedure a simpli�ed approach is applied to evaluate the sensitivity of the objective by a variational adjoint method. The optimization problems are solved by mathematical programming (MP) methods and optimality criteria (OC) methods, respectively. The numerical procedures determining the derivatives are veri�ed by test examples for 2-dim design problems under plane stress conditions.

Abstract

The development of new analysis approaches for the simulation of transient, nonlinear fluid structure interaction problems are the driving force behind this study. Fluid structure interaction is herein viewed as a three–field coupled problem, consisting of the physical and/or computational fields – fluid, structure and moving mesh. For the possibility of covering a wide range of examples and application areas, rather general models have to be used for the single fields of the coupled problem. The instationary, incompressible Navier–Stokes equations are employed herein together with a geometrically nonlinear elastodynamic model for the structure. The present computational approach is completely based on finite elements for the spatial discretization. The space time relation is adopted in a semi–discretization approach employing different direct time integration schemes. The main part of this contribution is concerned with different solution methods for coupled problems. From the computational point of view it is highly desirable to base developments of solvers for coupled problems on the philosophy of partitioned analysis. Both loose and strong partitioned coupling procedures are discussed. Algorithmic examplification is done on the one hand for fluid structure interaction problems and on the other hand for artificially partitioned problems in nonlinear structural dynamics. Numerical examples are given for both types of partitioned coupling procedures and problem types.

Abstract

The contribution takes up the concept of preventing locking a priori in the theory of thin-walled structures instead of curing it during discretization. After briefly summarizing the successful concept for avoiding transverse shear locking through reparametrization of primary variables for beams, plates and shells we concentrate on the same approach for preventing also membrane locking. Here, we describe first steps referring to the plane curved Bernoulli beam as a conceptual proof for the new method. Inspired by the so-called Mixed Displacement variational method we discuss three variants of replacing primary displacement parameters by alternative variables.

Abstract

The contribution describes a two-dimensional discrete element method with polygonal particles in a two-dimensional setting allowing the simulation of granular as well as quasi-brittle material. Different models for soft contact as well as cohesion between the particles are presented. Emphasis is put on the specific features of the polygonal particles. Simulations are compared to results of small scale experiments with regular particles of steel nuts. In addition the capabilities of the method are demonstrated simulating complex concrete specimens with a distinct heterogeneous microstructure.

Abstract

The contribution gives an overview on a discrete element model with polygonal particles in a two-dimensional setting allowing the simulation of granular as well as quasi-brittle material response. It briefly describes the basic formulation for geometry and applied contact models in normal and tangential direction, supplemented by friction on a background plate. Special emphasis is put on modeling of cohesion; three different models with an increasing complexity are introduced, namely an overlay brittle beam lattice, a beam with damage and an interface model. Homogenization of the discrete particle response is utilized deriving variables like stresses and strains for an interpretation in the context of classical and micropolar continua. Several numerical examples for different loading scenarios are added, among them the simulation of a quasi-brittle material sample with a heterogeneous microstructure. In addition conceptual small scale experiments with regular particles of steel nuts have been performed; results from tests and simulations for samples with and without cohesion are compared.

Abstract

In an engineering context, the quality of numerical analyses, like finite element calculations, often is related to the achieved accuracy of selected variables, e.g. displacements, stresses, strains, etc. This point directly motivates the development of error estimators based on goal-oriented norms, directly coupled to the values of interest. Therefore, adaptive finite element methods, can be applied to refine the mesh based on this kind of error measurements and give the optimal mesh for the target variable. Compared to the classical global error estimators the goal-oriented error estimators are often denoted as local methods since they most often are related to local variables. As an example of local error estimation a small elastic plate with a crack under concentrated load is considered. The classical global energy norm controlled adaptive finite element method is showing mesh refinement near the crack as well as near the load concentrations. The goal-oriented scheme for the stress near the crack however is refining the mesh only in the area of interest. Error estimators like these are a supplement to the traditional error estimators measuring the finite element error in the energy norm, the natural norm from a mathematical point of view, e.g. Babuska & Rheinboldt, Verfuerth, Zienkiewicz & Zhu, Ainsworth & Oden and Stewart & Hughes. For elliptic partial differential equations, PDEs, like the linear elasticity problem, error estimators in the energy norm are accepted as a general tool, in particular since they show sometimes surprisingly good results also in estimating the error of certain other variables like displacements or certain stress or strain values. But even in this rather simple case goal-oriented error estimators with specifically designed error norms which hold for a user-prescribed tolerance show distinct advantages over the general framework of the energy-norm. In an even more pronounced way the increasing efficiency of goal-oriented error estimators can be observed in the case of initial boundary value problems. In this case of hyperbolic PDEs, the solution and therefore also the discretization error propagate along problem-specific characteristics. As a consequence dealing with global error estimators often leads to very large effectivity indexes for the error of individual quantities. The in general a priori calculated global stability of the problem, measured in the stability constant, cannot capture a corresponding user-specified norm in an effective way. In general, sharp analytical estimates for the strong stability constant can not be determined a priori. Therefore the lack of standard energy norm estimators causes a loss of efficiency, indicated by very large numbers for the effectivity index. One objective of this paper is to compare global energy-norm error estimators with different goal-oriented error estimators and to show the efficiency of weighted error estimation. The outline of this paper is as follows: For the simple linear elasticity problem we give a short overview on standard error estimation techniques in the energy norm. In a straight forward manner we extend the error estimator to a goal-oriented error estimation technique. This general goal-oriented technique is then specified for selected variables with their own individual error norms. These error estimators are derived for individual quantities, e.g. for local values like stress or strain components or selected displacements or values in small sub-regions of the considered domain. To derive such goal oriented estimates, we will introduce dual problems corresponding to the strong form of the underlying primal problem, see e.g. Eriksson, Estep, Hansbo & Johnson, Becker & Rannacher, and Cirak & Ramm. After this we extend the concept to nonlinear problems, e.g. plasticity, viscoplasticity, and finally to linear elastodynamics. For all considered physical problems one-dimensional and two-dimensional numerical calculations are given to demonstrate the efficiency of the underlying error estimation and adaptive refinement technique. We conclude the paper with some remarks on the numerical implementation.