Manfred Bischoff

Abstract

Fast snapping in the carnivorous Venus flytrap (Dionaea muscipula) involves trap lobe bending and abrupt curvature inversion (snap-buckling), but how do these traps reopen? Here, the trap reopening mechanics in two different D. muscipula clones, producing normal-sized (N traps, max. ≈3 cm in length) and large traps (L traps, max. ≈4.5 cm in length) are investigated. Time-lapse experiments reveal that both N and L traps can reopen by smooth and continuous outward lobe bending, but only L traps can undergo smooth bending followed by a much faster snap-through of the lobes. Additionally, L traps can reopen asynchronously, with one of the lobes moving before the other. This study challenges the current consensus on trap reopening, which describes it as a slow, smooth process driven by hydraulics and cell growth and/or expansion. Based on the results gained via three-dimensional digital image correlation (3D-DIC), morphological and mechanical investigations, the differences in trap reopening are proposed to stem from a combination of size and slenderness of individual traps. This study elucidates trap reopening processes in the (in)famous Dionaea snap traps – unique shape-shifting structures of great interest for plant biomechanics, functional morphology, and applications in biomimetics, i.e., soft robotics.

Abstract

The opening and closing of pine cones is based on the hygroscopic behavior of the individual seed scales around the cone axis, which bend passively in response to changes in environmental humidity. Although prior studies suggest a bilayer architecture consisting of lower actuating (swellable) sclereid and upper restrictive (non- or lesser swellable) sclerenchymatous fiber tissue layers to be the structural basis of this behavior, the exact mechanism of how humidity changes are translated into global movement are still unclear. Here, the mechanical and hydraulic properties of each structural component of the scale are investigated to get a holistic picture of their functional interplay. Measurements of the wetting behavior, water uptake, and mechanical measurements are used to analyze the influence of hydration on the different tissues of the cone scales. Furthermore, their dimensional changes during actuation are measured by comparative micro-computed tomography (µ-CT) investigations of dry and wet scales, which are corroborated and extended by 3D-digital image correlation-based displacement and strain analyses, biomechanical testing of actuation force, and finite element simulations. Altogether, a model allowing a detailed mechanistic understanding of pine cone actuation is developed, which is a prime concept generator for the development of biomimetic hygromorphic systems.

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

In coreless filament winding, resin-impregnated fibre filaments are wound around anchor points without an additional mould. The final geometry of the produced part results from the interaction of fibres in space and is initially undetermined. Therefore, the success of large-scale coreless wound fibre composite structures for architectural applications relies on the reciprocal collaboration of simulation, fabrication, quality evaluation, and data integration domains. The correlation of data from those domains enables the optimization of the design towards ideal performance and material efficiency. This paper elaborates on a computational co-design framework to enable new modes of collaboration for coreless wound fibre–polymer composite structures. It introduces the use of a shared object model acting as a central data repository that facilitates interdisciplinary data exchange and the investigation of correlations between domains. The application of the developed computational co-design framework is demonstrated in a case study in which the data are successfully mapped, linked, and analysed across the different fields of expertise. The results showcase the framework’s potential to gain a deeper understanding of large-scale coreless wound filament structures and their fabrication and geometrical implications for design optimization.

Abstract

We present an objective, singularity-free, path independent, numerically robust and efficient geometrically non-linear Reissner-Mindlin shell finite element formulation. The formulation is especially suitable for higher order ansatz spaces. The formulation utilizes geometric finite elements presented by Sander 74 and Grohs 34 for the interpolation on non-linear manifolds. The proposed method is objective and free from artificial singularities and spurious path dependence. Due to the fact that the director field lives on the unit sphere, a special linearization procedure is required to obtain the stiffness matrix. Here, we use the simple constructions of as reported by Absil et al. 2, 3, which yields an easy way to obtain the correct tangent operator of the potential energy. Additionally, we compare three different interpolation schemes for the shell director that can be found in the literature, where one of them is applied for the first time for the Reissner-Mindlin shell model. Furthermore, we compare the exponential map to the radial return normalization as procedure to update the nodal directors and conclude the superiority of the latter, in terms of fewer load steps. We also investigate the construction of a consistent tangent base update scheme. Path independence, efficiency and objectivity of the formulation are verified via a set of numerical examples.

Abstract

We present a comprehensive study on the approximation power of NURBS and the significance of exact geometry in stability analyses of shells. Pre-buckling analyses are carried out to estimate the critical load levels and the initial buckling patterns. Various finite element solutions obtained with the commercial code ANSYS are compared with solutions from the isogeometric version of the finite element method, using our in-house code NumPro. In some problem setups, the isogeometric shell elements provide superior accuracy compared to standard (as opposed to isogeometric) shell finite elements, requiring only a fractional amount of degrees of freedom for the same level of accuracy. The present study systematically investigates the sources of this superior accuracy of the isogeometric approach. In particular, hypotheses are tested concerning the influence of exact geometry and smoothness of splines.

Abstract

Finite-Elemente-Modellierungsansätze nach dem aktuellen Stand der Technik stoßen bei der Simulation von bestimmten Blechumformprozessen an ihre Grenzen. Ein Lösungsansatz zur Verbesserung der Simulationsgenauigkeit dieser Blechumformprozesse wird zurzeit in einem IGF-Forschungsprojekt am Fraunhofer IWM und am Institut für Baustatik und Baudynamik der Universität Stuttgart gemeinsam entwickelt. Dieser basiert auf der Kombination von erweiterten Schalenformulierungen und 3D-Materialmodellen und soll zukünftig die Simulationsgenauigkeit dieser Blechumformprozesse verbessern.

Abstract

We show that most geometrically nonlinear three-dimensional shell elements and solid shell elements suffer from a previously unknown artificial stiffening effect that only appears in geometrically nonlinear problems, in particular in the presence of large bending deformations. It can be interpreted as a nonlinear variant of the well-known Poisson thickness locking effect. We explain why and under which circumstances this phenomenon appears and propose concepts to avoid it.

Abstract

In the present paper, theoretical foundations of redundancy in spatially continuous, elastostatic, and linear representations of structures are derived. Adopting an operator-theoretical perspective, the redundancy operator is introduced, inspired by the concept of redundancy matrices, previously described for spatially discrete representations of structures. Studying symmetry, trace, rank, and spectral properties of this operator as well as revealing the relation to the concept of statical indeterminacy, a continuum-mechanical theory of redundancy is proposed. Here, the notion “continuum-mechanical” refers to the representation being spatially continuous. Apart from the theory itself, the novel outcome is a clear concept providing information on the distribution of statical indeterminacy in space and with respect to different load carrying mechanisms. The theoretical findings are confirmed and illustrated within exemplary rod, plane beam, and plane frame structures. The additional insight into the load carrying behavior may be valuable in numerous applications, including robust design of structures, quantification of imperfection sensitivity, assessment of adaptability, as well as actuator placement and optimized control in adaptive structures.

Abstract

(1) Significance of geometry for bio-inspired hygroscopically actuated bilayer structures is well studied and can be used to fine-tune curvatures in many existent material systems. We developed a material design space to find new material combinations that takes into account unequal effective widths of the layers, as commonly used in fused filament fabrication, and deflections under self-weight. (2) For this purpose, we adapted Timoshenko’s model for the curvature of bilayer strips and used an established hygromorphic 4D-printed bilayer system to validate its ability to predict curvatures in various experiments. (3) The combination of curvature evaluation with simple, linear beam deflection calculations leads to an analytical solution space to study influences of Young’s moduli, swelling strains and densities on deflection under self-weight and curvature under hygroscopic swelling. It shows that the choice of the ratio of Young’s moduli can be crucial for achieving a solution that is stable against production errors. (4) Under the assumption of linear material behavior, the presented development of a material design space allows selection or design of a suited material combination for application-specific, bio-inspired bilayer systems with unequal layer widths.

Abstract

Adaptive structures are characterized by their ability to adjust their geometrical and other properties to changing loads or requirements during service. This contribution deals with a method for the design of quasi‐static motions of structures between two prescribed geometrical configurations that are optimal with regard to a specified quality function while taking large deformations into account. It is based on a variational formulation and the solution by two finite element discretizations, the spatial discretization (the standard finite element mesh) and an additional discretization of the deformation path or trajectory. For the investigations, an exemplary objective function, the minimization of the internal energy, integrated along the deformation path, is used. The method for motion design presented herein uses the Newton‐Raphson method as a second‐order optimization algorithm and allows for analytical sensitivity analysis. The proposed method is verified and its properties are investigated by benchmark examples including rigid body motions, instability phenomena and determination of inextensible deformations of shells.

Abstract

The design of adaptive structures is one method to improve sustainability of buildings. Adaptive structures are able to adapt to different loading and environmental conditions or to changing requirements by either small or large shape changes. In the latter case, also the mechanics and properties of the deformation process play a role for the structure’s energy efficiency. The method of variational motion design, previously developed in the group of the authors, allows to identify deformation paths between two given geometrical configurations that are optimal with respect to a defined quality function. In a preliminary, academic setting this method assumes that every single degree of freedom is accessible to arbitrary external actuation forces that realize the optimized motion. These (nodal) forces can be recovered a posteriori. The present contribution deals with an extension of the method of motion design by the constraint that the motion is to be realized by a predefined set of actuation forces. These can be either external forces or prescribed length chances of discrete, internal actuator elements. As an additional constraint, static stability of each intermediate configuration during the motion is taken into account. It can be accomplished by enforcing a positive determinant of the stiffness matrix.

Abstract

Die „Große Beschleunigung“ bei Bevölkerungszahlen, klimaschädlichen Emissionen, Wasserverbrauch und vielem anderen stellt die gesamte Menschheit vor große Herausforderungen. Dies trifft besonders auf das Bauschaffen zu. Es gilt, zukünftig für mehr Menschen mit weniger Material emissionsfrei zu bauen. Hierfür muss unsere Art des Planens, Bauens und Nutzens von Bauwerken neu gedacht und neu konzipiert werden. Auf der bautechnischen Seite bedeutet dies die konsequente flächendeckende Umsetzung von Leichtbaustrategien. Zu diesen zählt neben dem klassischen Leichtbau und den Gradientenbauweisen auch das Bauen mit adaptiven Hüllen und Strukturen. Unter Adaptivität sind dabei unterschiedliche Veränderungen der Geometrie, der physikalischen Eigenschaften von einzelnen Bauteilen oder von ganzen Bauwerken zu verstehen. Durch Adaption können Spannungsfelder homogenisiert, Bauteilverformungen reduziert und bauphysikalische Verhalten von Bauteilen verändert werden. All dies verringert nicht nur den Materialbedarf, sondern liefert auch einen wesentlichen Beitrag zur Steigerung des Nutzerkomforts. Adaptivität im weiteren Sinne bezeichnet einen ganzheitlichen Ansatz, in dem die Anpassung sozialer, kultureller und räumlicher Erfahrungen sowie architektonischer und planerischer Handlungsweisen eng mit den technologischen Entwicklungen verknüpft wird. Die Zusammenführung dieser Perspektiven ist Anspruch des SFB, um ganzheitliche Lösungen für eine zukünftige gebaute Umwelt zu finden.

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

Taking advantage of adaptivity in the field of civil engineering is a subject of ongoing research. Integration of adaptive elements in load-bearing structures is already well-established in many other engineering fields, albeit mostly for different purposes than withstanding predominantly static loads. Initial investigations have demonstrated potential for substantial material and energy savings also in the field of civil engineering, especially for high-rise buildings and wide-span structures, such as roofs or bridges. Adaptive civil structures show promise in tackling current challenges arising from emissions and shortages of materials. In this study, we compare the possible minimum-weight designs for different actuator placement approaches and for different structural topologies that satisfy various constraints for high-rise buildings. We use case studies as illustrative examples to show which advantages and disadvantages can be expected from a specific design. The overarching aim is to learn how truss and beam structures should be designed to perform well as adaptive structures.

Abstract

The enhanced assumed strain (EAS) method is one of the most frequently used methods to avoid locking in solid and structural finite elements. One issue of EAS elements in the context of geometrically non‐linear analyses is their lack of robustness in the Newton‐Raphson scheme, which is characterized by the necessity of small load increments and large numbers of iterations. In the present work we extend the recently proposed mixed integration point (MIP) method to EAS elements in order to overcome this drawback in numerous applications. Furthermore, the MIP method is generalized to generic material models, which makes this simple method easily applicable for a broad class of problems. In the numerical simulations in this work, we compare standard strain based EAS elements and their MIP improved versions to elements based on the assumed stress method in order to explain when and why the MIP method allows to improve robustness. A further novelty in the present work is an inverse stress‐strain relation for a Neo‐Hookean material model.

Abstract

In all structural models, the section or fiber response is a relation between the strain measures and the stress resultants. This relation can only be expressed in a simple analytical form when the material response is linear elastic. For other, more complex and interesting situations, kinematic and kinetic hypotheses need to be invoked, and a constrained three-dimensional constitutive relation has to be employed at every point of the section in order to implement non-linear and dissipative constitutive laws into dimensionally reduced structural models. In this article we explain in which sense reduced constitutive models can be expressed as minimization problems, helping to formulate the global equilibrium as a single optimization problem. Casting the problem this way has implications from the mathematical and numerical points of view, naturally defining error indicators. General purpose solution algorithms for constrained material response, with and without optimization character, are discussed and provided in an open-source library.

Abstract

The mechanical principles for fast snapping in the iconic Venus flytrap are not yet fully understood. In this study, we obtained time-resolved strain distributions via three-dimensional digital image correlation (DIC) for the outer and inner trap-lobe surfaces throughout the closing motion. In combination with finite element models, the various possible contributions of the trap tissue layers were investigated with respect to the trap’s movement behavior and the amount of strain required for snapping. Supported by in vivo experiments, we show that full trap turgescence is a mechanical–physiological prerequisite for successful (fast and geometrically correct) snapping, driven by differential tissue changes (swelling, shrinking, or no contribution). These are probably the result of the previous accumulation of internal hydrostatic pressure (prestress), which is released after trap triggering. Our research leads to an in-depth mechanical understanding of a complex plant movement incorporating various actuation principles.

Abstract

We develop a mixed geometrically nonlinear isogeometric Reissner–Mindlin shell element for the analysis of thin-walled structures that leverages Bézier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses. The accuracy of computed solutions over coarse meshes, that have highly non-interpolatory control meshes, is achieved through the application of a continuous rotational approach. The starting point of the formulation is the modified Hellinger–Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the corresponding Bézier dual bases. Leveraging the orthogonality property of the Bézier dual basis, the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system. The condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees of freedom. From a practical point of view, since the Bézier dual basis is completely specified through Bézier extraction, any spline space that admits Bézier extraction can utilize the proposed approach directly.

Abstract

This contribution proposes a new approach to derive geometrically parameterized, reduced order finite element models. An element formulation for geometrically parameterized finite elements is suggested. The parameterized elements are used to derive models with a parameterized geometry where the parameterized system matrices are expressed in an affine representa- tion. Parametric model order reduction can then be efficiently used to reduce the full order parameterized model to a reduced order parameterized model. The approach shows two beneficial features. First, design studies and shape optimizations can be conducted with parameterized reduced order model of much lower dimension compared to the parameterized, full order model. Second, it is possible to compute sensitivities analytically, and therefore, to avoid the computation of finite differences gradients. The approach is illustrated with two numerical examples. The first example includes a detailed error analysis. The second example is a shape optimization example of an adaptive structure.

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

The skeleton of Echinocyamus pusillus is considered as an exceptional model organism for structural strength and skeletal integrity within the echinoids as demonstrated by the absence of supportive collagenous fibres between single plates and the high preservation potential of their skeletons. The structural principles behind this remarkably stable, multi-plated, light-weight construction remain hardly explored. In this study, high-resolution X-ray micro-computed tomography, finite-element analysis and physical crushing tests are used to examine the structural mechanisms of this echinoid’s skeleton. The virtual model of E. pusillus shows that the material is heterogeneously distributed with high material accumulations in the internal buttress system and at the plate boundaries. Finite-element analysis indicates that the heterogeneous material distribution has no effect on the skeleton’s strength. This numerical approach also demonstrates that the internal buttress system is of high significance for the overall skeletal stability of this flattened echinoid. Results of the finite-element analyses with respect to the buttress importance were evaluated by physical crushing tests. These uniaxial compression experiments support the results of the simulation analysis. Additionally, the crushing tests demonstrate that organic tissues do not significantly contribute to the skeletal stability. The strength of the echinoid shell, hence, predominantly relies on the structural design.

Abstract

Smart and adaptive outer façade shading systems are of high interest in modern architecture. For long lasting and reliable systems, the abandonment of hinges which often fail due to mechanical wear during repetitive use is of particular importance. Drawing inspiration from the hinge-less motion of the underwater snap-trap of the carnivorous waterwheel plant (Aldrovanda vesiculosa), the compliant façade shading device Flectofold was developed. Based on computational simulations of the biological role-model’s elastic and reversible motion, the actuation principle of the plant can be identified. The enclosed geometric motion principle is abstracted into a simplified curved-line folding geometry with distinct flexible hinge-zones. The kinematic behaviour is translated into a quantitative kinetic model, using finite element simulation which allows the detailed analyses of the influence of geometric parameters such as curved-fold line radius and various pneumatically driven actuation principles on the motion behaviour, stress concentrations within the hinge-zones, and actuation forces. The information regarding geometric relations and material gradients gained from those computational models are then used to develop novel material combinations for glass fibre reinforced plastics which enabled the fabrication of physical prototypes of the compliant façade shading device Flectofold.

Abstract

In this contribution, a novel local, node-based time step estimate for reciprocal mass matrices is proposed. Element-based estimates turn out to be not generally conservative and are consequently inadequate. Therefore, the nodal time step estimate for diagonally lumped mass matrices based on Gershgorin’s theorem is further developed for application to reciprocal mass matrices. Additionally, simplifications of the proposed time step estimate that improve computational efficiency, especially for contact problems with the penalty method, are discussed and evaluated by numerical examples.

Abstract

Adaptive structures in civil engineering are mechanical structures with the ability to modify their response to external loads. Actuators strongly affect a structure’s adaptivity and have to be placed thoughtfully in the design process to effectively compensate external loads. For constant loads, this property is introduced as steadystate disturbance compensability. This property can be linked to concepts from structural engineering such as redundancy or statical indeterminacy, thus representing an interdisciplinary approach. Based on the disturbance compensability matrix, a scalar performance metric is derived as quantitative measure of a structure’s ability to compensate the output error for arbitrary constant disturbances with a given set of actuators. By minimizing this metric, an actuator configuration is determined. The concept is applied to an example of a truss structure.

Abstract

In 2017, the University of Stuttgart started a Collaborative Research Centre with the title Adaptive Skins and Structures for the Built Environment of Tomorrow. The goal of this research project is to find an answer to today’s most urgent social and ecological questions as the global population continuously increases and the available resources remain limited. As the central approach to the solution of this problem, adaptive elements will be included in the structure, the interior and the façade of an experimental 37 m tall building. This paper introduces the topic of adaptivity in building structures and provides an overview of the research topics applied in this globally unique adaptive high-rise building. Due to the complexity of research topics of this Collaborative Research Centre, this paper only covers the research concerning the experimental high-rise building.

Abstract

The fast motion of the snap-traps of the terrestrial Venus flytrap (Dionaea muscipula) have been intensively studied, in contrast to the tenfold faster underwater snap-traps of its phylogenetic sister, the waterwheel plant (Aldrovanda vesiculosa). Based on biomechanical and functional–morphological analyses and on a reverse biomimetic approach via mechanical modelling and computer simulations, we identify a combination of hydraulic turgor change and the release of prestress stored in the trap as essential for actuation. Our study is the first to identify and analyse in detail the motion principle of Aldrovanda, which not only leads to a deepened understanding of fast plant movements in general, but also contributes to the question of how snap-traps may have evolved and also allows for the development of novel biomimetic compliant mechanisms.

Abstract

The stability and reflection-transmission properties of the bipenalty method are studied in application to explicit finite element analysis of one-dimensional contact-impact problems. It is known that the standard penalty method, where an additional stiffness term corresponding to contact boundary conditions is applied, attacks the stability limit of finite element model. Generally, the critical time step size rapidly decreases with increasing penalty stiffness. Recent comprehensive studies have shown that the so-called bipenalty technique, using mass penalty together with standard stiffness penalty, preserves the critical time step size associated to contact-free bodies. In this paper, the influence of the penalty ratio (ratio of stiffness and mass penalty parameters) on stability and reflection-transmission properties in one-dimensional contact-impact problems using the same material and mesh size for both domains is studied. The paper closes with numerical examples, which demonstrate the stability and reflection-transmission behavior of the bipenalty method in one-dimensional contact-impact and wave propagation problems of homogeneous materials.

Abstract

In the context of isogeometric contact analysis, surfaces of objects can be described smoothly due to the high continuity of the involved shape functions. This facilitates construction of a continuous normal field which may be beneficial for modeling of contact problems. This property allows for stable collocation of the virtual contact work using considerable less evaluation points than numerical integration, while the approximation quality of the computational results is almost the same. In this context, a weighted point-based method (PTS+) is introduced as an extension of the Point-To-Segment method (PTS) proposed previously by Matzen et al. (2013). The weights enable transmission of contact stresses instead of contact forces along the contact surfaces. In normal direction the non-penetration condition is enforced by the Lagrange multiplier method, combined with the penalty method in tangential direction. Corresponding to the collocation weights, a constant or NURBS Lagrange multiplier field is introduced. A comparison between the collocation-based contact formulations PTS, PTS+ and a mortar method is drawn, showing that weighted collocation is able to achieve almost the same accuracy as the integration-based mortar formulation.

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

In this contribution, variationally consistent inertia templates for B-spline and NURBS-based finite elements are proposed as a unified concept for two different purposes: Customization of the template allows construction of masses and reciprocal masses with desired properties like higher-order accuracy or improved dispersion behavior; Inertia scaling allows substantial speed-up for explicit dynamics by increased critical time steps. The derivation of the template is based on a three-field parametrized functional as in previous works of the authors’ group, but with modified primary variables, namely displacement, velocity and mass-specific linear momentum. The latter allows for mass-preservation for non-constant density throughout the domain and is therefore an enhancement to the formulation proposed in Tkachuk and Bischoff (2015). With the focus on B-spline and NURBS-based finite elements, the proposed template provides alternatives to the row-sum- lumped mass matrix, which is only 2nd order accurate independent of the polynomial order. Earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances. Furthermore, higher-order reciprocal masses can be constructed from the template. They are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems. For non-linear problems only small overhead is expected, but this paper focuses on linear problems only and mainly undistorted meshes. Tuning the method towards inertia scaling, a reduction of the maximum eigenfrequency by 25%–40% is obtained in the examples herein, whereas the accuracy is higher than for lumped or consistent mass.

Abstract

In this paper, a new approach is proposed to improve efficiency of the integration procedure for mortar integrals within finite element mortar methods for contact. Appropriate approaches subdivide polygonal integration segments into triangular integration cells where well-established quadrature rules can be applied for numerical integration. Here, a subdivision of segments into quadrilateral integration cells is proposed and investigated in detail. By this procedure, the numerical effort is decreased because the number of integration cells is smaller and less quadrature points are needed. In all the aforementioned methods, necessary projections of integration points result in rational polynomials in the integrand. Thus, an exact numerical integration is impossible. Using quadrilateral integration cells additionally involves non-constant Jacobian determinants which further increases the polynomial degree of the integrand. Numerical experiments indicate, that the resulting increase in the error is small enough to be acceptable in consideration of the gained speed-up.

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

Purpose Current procedures of buckling load estimation for thin-walled structures may provide very conservative estimates. Their refinement offers the potential to use structure and material properties more efficiently. Due to the large variety of design variables, for example laminate layup in composite structures, a prohibitively large number of tests would be required for experimental assessment, and thus reliable numerical techniques are of particular interest. The purpose of this paper is to analyze different methods of numerical buckling load estimation, formulate simulation procedures suitable for commercial software and give recommendations regarding their application. All investigations have been carried out for cylindrical composite shells; however similar approaches are feasible for other structures as well. Design/methodology/approach We develop a concept to apply artificial load imperfections with the aim to estimate as good as possible lower bounds for the buckling loads of shells for which the actual physical imperfections are not known. Single and triple perturbation load approach, global and local dynamic perturbation approach and path following techniques are applied to the analysis of a cylindrical composite shell with known buckling characteristics. Results of simulations are compared with published experimental data. Findings A single perturbation load approach is reproduced and modified. Buckling behavior for negative values of the perturbation load is examined and a pattern similar to a positive perturbation load is observed. Simulations with three perturbation forces show a decreased (i. e. more critical) value of the buckling load compared to the single perturbation load approach. Global and local dynamic perturbation approaches exhibit a behavior suitable for lower bound estimation for structures with arbitrary geometries. Originality/value Various load imperfection approaches to buckling load estimation are validated and compared. All investigated methods do not require knowledge of the real geometrical imperfections of the structure. Simulations were performed using a commercial finite element code. Investigations of sensitivity with respect to a single perturbation load are extended to the negative range of the perturbation load amplitude. A specific pattern for a global perturbation approach was developed, and based on it a novel simulation procedure is proposed.

Abstract

Classical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non-diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element-wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed.

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

Purpose – The purpose of this paper is to develop a method to model entire structures on a large scale, at the same time taking into account localized non-linear phenomena of the discrete microstructure of cohesive-frictional materials. Design/methodology/approach – Finite element (FEM) based continuum methods are generally considered appropriate as long as solutions are smooth. However, when discontinuities like cracks and fragmentation appear and evolve, application of models that take into account (evolving) microstructures may be advantageous. One popular model to simulate behavior of cohesive-frictional materials is the discrete element method (DEM). However, even if the microscale is close to the macroscale, DEMs are computationally expensive and can only be applied to relatively small specimen sizes and time intervals. Hence, a method is desirable that combines efficiency of FEM with accuracy of DEM by adaptively switching from the continuous to the discrete model where necessary. Findings – An existing method which allows smooth transition between discrete and continuous models is the quasicontinuum method, developed in the field of atomistic simulations. It is taken as a starting point and its concepts are extended to applications in structural mechanics in this paper. The kinematics in the method presented herein is obtained from FEM whereas DEM yields the constitutive behavior. With respect to the constitutive law, three levels of resolution – continuous, intermediate and discrete – are introduced. Originality/value – The overall concept combines model adaptation with adaptive mesh refinement with the aim to obtain a most efficient and accurate solution.

Abstract

The problem of optimal selective mass scaling for linearized elasto-dynamics is discussed. Optimal selective mass scaling should provide solutions for dynamical problems that are close to the ones obtained with a lumped mass matrix, but at much smaller computational costs. It should be equally applicable to all structurally relevant load cases. The three main optimality criteria, namely eigenmode preservation, small number of non-zero entries and good conditioning of the mass matrix are explicitly formulated in the article. An example of optimal mass scaling which relies on redistribution of mass on a global system level is constructed. Alternative local mass scaling strategies are proposed and compared with existing methods using one modal and two transient numerical examples.

Abstract

A hierarchic family of isogeometric shell finite elements based on NURBS shape functions is presented. In contrast to classical shell finite element formulations, inter-element continuity of at least C1 enables a unique and continuous representation of the surface normal within one NURBS patch. This does not only facilitate formulation of Kirchhoff-Love type shell models, for which the standard Galerkin weak form has a variational index of 2, but it also offers significant advantages for shear deformable (Reissner-Mindlin type) shells and higher order shell models. For a 5-parameter shell formulation with Reissner-Mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated Kirchhoff-Love type director of the deformed configuration. This split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration, assumed natural strains or mixed finite element formulations. The third member of the hierarchy is a 7-parameter model including thickness change and allowing for application of unmodified three-dimensional constitutive laws. The phenomenon of curvature thickness locking, coming along with this kinematic extension, again is automatically avoided by the hierarchic difference vector concept without any further treatment. Membrane locking and in-plane shear locking are removed by two different approaches: firstly elimination via the Discrete Strain Gap (DSG) method and secondly removal of parasitic membrane strains using a hybrid-mixed method based on the Hellinger-Reissner variational principle. The hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach.

Abstract

Formulation of isogeometric finite elements has received a great deal of attention in the recent past. The present study deals with the treatment of problems in structural mechanics including large elastic deformations and contact. One decisive difference between isogeometric finite elements, based on NURBS functions and standard finite elements (FE), based on Lagrange polynomials, is higher inter-element continuity. This is a promising characteristics to model contact problems, as all issues associated with kinks between elements are naturally avoided. Particularly in cases with large sliding this has the potential to be an attractive feature. We present a bilateral isogeometric collocation contact formulation for geometrically non-linear two-dimensional problems, using Greville and Botella points to collocate the contact integrals. Different methods to obtain accurate and physically meaningful stress distributions are investigated and compared. The results show that the higher inter-element continuity, in context of non-smooth problems, may imply undesired effects in the numerical solution such as unphysical stress oscillations. Similar effects have been observed in standard p-FEM. Numerical experiments indicate that these oscillations may be avoided if the basis functions of contact and non-contact zones are separated by knot relocation and knot repetition.

Abstract

A new variational method for selective mass scaling is proposed. It is based on a new penalized Hamilton’s principle where relations between variables for displacement, velocity and momentum are imposed via a penalty method. Independent spatial discretization of the variables along with a local static condensation for velocity and momentum yields a parametric family of consistent mass matrices. In this framework new mass matrices with desired properties can be constructed. It is demonstrated how usage of these non-diagonal mass matrices decreases the maximum frequency of the discretized system and allows for larger steps in explicit time integration. At the same time the lowest eigenfrequencies in the range of interest and global structural response are not significantly changed. Results of numerical experiments for two-dimensional and three-dimensional problems are discussed.

Abstract

An alternative spatial semi-discretization of dynamic contact based on a modified Hamilton’s principle is proposed. The modified Hamilton’s principle uses displacement, velocity and momentum as variables, which allows their independent spatial discretization. Along with a local static condensation for velocity and momentum, it leads to an approach with a hybrid-mixed consistent mass matrix. An attractive feature of such a formulation is the possibility to construct hybrid singular mass matrices with zero components at those nodes where contact is collocated. This improves numerical stability of the semi-discrete problem: the differential index of the underlying differential-algebraic system is reduced from 3 to 1, and spurious oscillations in the contact pressure, which are commonly reported for formulations with Lagrange multipliers, are significantly reduced. Results of numerical experiments for truss and Timoshenko beam elements are discussed. In addition, the properties of the novel discretization scheme for an unconstrained dynamic problem are assessed by a dispersion analysis.

Abstract

Computational modelling of contact problems raises two basic questions: Which method should be used to enforce the contact conditions and how should this method be discretised? The most popular enforcement methods are the Lagrange multiplier method, the penalty method and combinations of these two. A frequently used discretisation method is the so called node-to-segment approach. However, this approach might lead to problems like jumps in contact forces, loss of convergence or failure to pass the patch test. Thus in the last few years, several segment-to-segment contact algorithms based on the mortar method were proposed. Combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations. On the other hand, a Lagrange multiplier mortar method requires additional unknowns. Hence, condensation of the Lagrange multipliers is desirable to preserve the initial size of the system of equations. This can be achieved by interpolating the Lagrange multipliers with so-called dual shape functions. Discretising two contacting bodies leads to opposed contact surface representations of finite element edges, called slave and master elements, respectively. In current versions of dual Lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element (instead of the entire element) belongs to the contact area. We present a modified definition of the dual shape functions in such slave elements. The basic idea is to construct dual shape functions that fulfill the so-called biorthogonality condition within the contact area. This leads to consistent mortar matrices also in the boundary region. To avoid ill-conditioning of the stiffness matrix, the modified mortar matrices are weighted with appropriate weighting factors. In doing so, the corresponding modified Lagrange multiplier nodal values are of the same order as the unmodified ones. Various examples demonstrate the performance of the modified mortar contact algorithm.

Abstract

The strategy of using finite elements with NURBS shape functions for of both geometry and displacements (“isogeometric approach”) is investigated from the point of view of finite element technology. Convergence rates are compared to those of classical finite element approaches utilizing standard Lagrange shape functions. Moreover, typical locking phenomena are examined. It is found that higher order inter-element continuity within the NURBS approach results in identical convergence rates but smaller absolute errors compared to C0-continuous approaches. However, NURBS finite elements suffer from the same locking problems as finite elements using Lagrange shape functions. The discrete shear gap (DSG) method, a general framework for formulation of locking-free elements, is applied to develop a new class of NURBS finite elements. The resulting NURBS DSG elements are absolutely free from locking and preserve the property of improved accuracy compared with standard locking-free finite elements. The method is exemplified for the Timoshenko beam model, but may be applied to more general cases.

Abstract

A generalization of the classical method of incompatible modes is presented, containing the original formulation as a special case. The method constitutes a class of finite elements which is equivalent to the class of enhanced assumed strain elements, yet based on a pure displacement formulation.

Abstract

A non-conforming finite element formulation, the Incompatible Bubbles method, is proposed for the problem of linear elasticity. As the classical Incompatible Modes method, the proposed formulation is based on an enrichment of the Galerkin solution space with non-conforming functions, the incompatible bubbles. In fact, the two formulations coincide in the particular case of non-distorted meshes. The advantage of the new formulation is that by a careful choice of the bubbles some of the "variational crimes" of the classical method become unnecessary for convergence, as the analysis reveals. Also, the relationship of the proposed method with more recent subgrid scale finite element formulations is investigated. Numerical examples illustrating the performance of the method are provided.

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

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

For most sheet metal forming simulations, shell elements that consider a reduced stress state, in particular, assuming a zero transverse normal stress sigma_33 and neglecting the shear stress components sigma_13 and sigma_23 in the yield function, are used. Moreover, certain kinematic assumptions, like cross-sectional material fibers being assumed to remain straight during deformation, are typically applied. However, for some applications, like bending with small radii and thick sheets, this approach is not a workable solution to obtain accurate and reliable results, since the prerequisites that justify the aforementioned kinematic assumptions are not met anymore. In this contribution, a 3d-shell element is presented that allows for cross-sectional warping. For the evaluation, numerical results of a metal stripe drawn through a draw bead are compared against experimental data. The results demonstrate that the 3d-shell element is able to represent warping of cross-sectional material fibers during deformation. In addition, further numerical tests conducted with this element are shown.

Abstract

Taking advantage of adaptivity in the field of civil engineering is an ongoing research topic. Integration of adaptive elements in the load-bearing structure is already well established in many other engineering fields. First investigations promise large material saving potentials also in the field of civil engineering, especially when it comes to high-rise buildings or wide spanned structures like roofs or bridges. In times of emission problems and shortage of materials, the potentials of adaptive civil structures open various new possibilities. In the design and optimization process of adaptive civil structures, we address the differences between classical approaches for passive systems and new practices considering adaptivity. By using a suitable actuator placement, it is possible to manipulate the displacements of the structure as well as the force distribution within the structure. Both material and energy savings can be accomplished with an integrated design of the adaptive structure taking into account the actuation, suitable combination of structural design and actuator placement. For demonstration of the differences in the design process and in the resulting optimized structure, we use a small case study on a truss structure, which is inspired by a high-rise building, and consider static loads.

Abstract

Die Antwort einer Struktur auf Aktuierung wird maßgeblich vom Grad der statischen Unbestimmtheit und deren Verteilung in der Struktur beeinflusst. Die Redundanzmatrix enthält diese Informationen über das Tragwerk. Aus ihr können daher Rückschlüsse für die Aktorplatzierung und für die Bewertung der Aktuierbarkeit von Strukturen gezogen werden. Die Untersuchung eines Beispieltragwerks veranschaulicht einerseits die Einsatzmöglichkeiten der Redundanzmatrix und andererseits das Masseneinsparungspotential, das mithilfe des Einsatzes aktiver Elemente in passiven Strukturen erschlossen werden kann.

Abstract

Curved folding, a method to create curved 3D structures from a flat sheet, can be used to produce material and manufacturing efficient, static or dynamic structures. However, the complex assembly and folding sequence of curved crease patterns is the bottleneck in their fabrication process. This paper presents Self-shaping Curved Folding: a material programming approach to create curved crease origami structures that self-assemble from flat into 3D folded state upon exposure to external stimuli. We propose a digital fabrication process via the 3D-printing of shape-changing materials, accompanied by a computational design workflow in which the geometry of a crease pattern is correlated with the printing toolpaths and the layup of stimuli-responsive and passive materials to achieve a target shape-change. We demonstrate our method by producing multiple prototypes and documenting their shape-change upon actuation. Lastly, we explore the functional and performance benefits of self-shaping curved folding under three application scenarios relevant to the field of industrial design and architecture.

Abstract

In this contribution, a novel local, node-based time step estimate for reciprocal mass matrices with improved sharpness is proposed. Reciprocal mass matrices are sparse matrices used in explicit dynamics that allow computation of the nodal acceleration from the total force vector. They aim for higher critical time steps or/and accuracy than the lumped mass matrix approximation. Since the eigenvalue inequality by Fried does not hold for reciprocal mass matrices, element-based estimates may be not conservative and are consequently inadequate. Therefore, nodal time step estimates are preferred for reciprocal mass matrices. A nodal time step estimate requires an eigenvalue bound using row or/and column information associated with the nodal degrees of freedom. Recently, an efficient estimate was proposed that uses Gershgorin’s circle theorem and a corresponding bound. This estimate is always conservative but it usually leaves a 10-30% gap to the true eigenvalue. This raises the question whether a sharper estimate may be constructed by using a different eigenvalue bound. A recent paper by Cottereau and Sevilla compares five eigenvalue bounds for a diagonal mass matrix with spectral finite elements and indicates that the Ostrowski’s bound is the sharpest bound for most of the considered cases. In this contribution, the latter estimate is further developed for the case of reciprocal mass matrices and an extra improvement via similarity transformation for elements with rotary degrees of freedom is discussed. Finally, the estimate is tested for a set of common finite elements in explicit dynamics.

Abstract

Beyond the shell model of Reissner and Mindlin, which is available in LS-DYNA® for example in shell ELFORM=2/16, there have been many developments in the field of 3d-shell models in recent years. 3d-shell models can be beneficial in sheet metal forming simulations because they allow for three-dimensional stress states. 3d-shell elements are available in LS-DYNA®, e.g. ELFORM=25. In the doctoral dissertation of Fleischer it has been found that under certain conditions this element formulation suffers from an artificial stiffening effect. Although this finding dates back to 2009, this phenomenon has remained unexplained so far. In this contribution, the authors explain the reason for this stiffening effect and show a possibility to remove it. Moreover, an outlook on the development of higher order shell models for sheet metal forming simulation is given.

Abstract

In this contribution, coupled, parameterized second order systems are considered where the coupled, parameterized system is derived from the assembly of several parameterized component models. Two approaches for the Parametric Model Order Reduction of such coupled systems are presented and compared in a reduced order shape optimization example. In the first approach, the coupled, parameterized system is derived by coupling the parameterized, full order component models. Then, Parametric Model Order Reduction is executed for the coupled system. In the second approach, the parameterized, component models are first reduced independently of their actual mounting situation. Afterwards, the parameterized, reduced order component models are coupled to derive the parameterized, reduced order system model. It is shown that the first approach yields smaller parameterized, reduced order system models. However, the second approach allows to reuse and to recombine the parameterized, reduced order component models arbitrarily. It therefore introduces more flexibility in the modeling process, enabling for example a toolbox based optimization with parameterized, reduced order models.

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

Within the collaborative research center Biological Design and Integrative Structures (CRC TRR 141), a research team of biologists, architects and engineers from Freiburg, Tübingen and Stuttgart is working on the development of biomimetic and bioinspired structures for implementation in architecture and building construction. One of the projects in this research center deals with the kinematics of planar, curved and corrugated plant surfaces as concept generators for deployable systems in architecture. It is an example for methods of engineering science at the interface between biology and architecture. The contribution will provide an insight into the process of analyzing the waterwheel plant Aldrovanda vesiculosa, a carnivorous plant that catches its prey by a quick closing movement of a snap trap consisting of two lobes attached to a midrib. At a first glance, the mechanism resembles the one of the famous Venus flytrap; however, it appears to be quite different from a mechanical point of view. In fact, a key aspect in the research presented here is the development of mechanical models and performing corresponding finite element analyses of the plant with the aim to obtain a better understanding of the compliant mechanism of Aldrovanda vesiculosa and its actuation. The latter is related to a change in turgor pressure in a so-called motor zone, adjacent to the midrib, possibly combined with prestressing effects. Apart from this scientific contribution to technical biology or reverse biomimetics, the abstraction of the trapping movement and its implementation in a façade element (Flectofold) as an example of biomimetic architectural design are briefly described.

Abstract

This paper presents results of the investigation of two biological role models, the shield bug (Graphosoma italicum) and the carnivorous Waterwheel plant (Aldrovanda vesiculosa). The aim was to identify biological construction and movement principles as inspiration for technical, deployable systems. The subsequent processes of abstraction and simulation of the movement and the design principles are summarized, followed by results on the mechanical investigations on various combinations of fibers and matrices with regard to taking advantage of the anisotropy of fiber-reinforced plastics (FRPs). With the results gained, it was possible to implement defined flexible bending zones in stiff composite components using one composite material, and thereby to mimic the biological role models. First small-scale demonstrators for adaptive façade shading systems – Flectofold and Flexagon – are proving the functionality.

Abstract

Part I deals with material modelling of woven fabric membranes. Due to their structure of crossed yarns embedded in coating, woven fabric membranes are characterised by a highly nonlinear stress-strain behaviour. In order to determine an accurate structural response of membrane structures, a suitable description of the material behaviour is required. A linear elastic orthotropic model approach, which is current practice, only allows a relative coarse approximation of the material behaviour. The present work focuses on two different material approaches: A first approach becomes evident by focusing on the meso-scale. The inhomogeneous, however periodic structure of woven fabrics motivates for microstructural modelling. An established microstructural model is considered and enhanced with regard to the coating stiffness. Secondly, an anisotropic hyperelastic material model for woven fabric membranes is considered. By performing inverse processes of parameter identification, fits of the two different material models w.r.t. measured data from a common biaxial test are shown. The results of the inversely parametrised material models are compared and discussed.

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

Um dynamische Finite-Elemente-Berechnungen mit expliziter Zeitintegration zu beschleunigen, ist es in der Praxis üblich, Massenskalierung anzuwenden. Damit soll die kritische Zeitschrittweite erhöht werden ohne maßgeblich an Genauigkeit in den entscheidenden niederfrequenten Moden zu verlieren. Bei der konventionellen Massenskalierung (CMS) wird künstliche Masse zu den Diagonaltermen der diagonalisierten Massenmatrix addiert und die Diagonalform der Massenmatrix bleibt erhalten. CMS wird gewöhnlich auf eine geringe Anzahl kleiner oder steifer Elemente angewandt, deren hohe Eigenfrequenzen die kritische Zeitschrittweite begrenzen. Dabei wird allerdings die Trägheit der Struktur erhöht, was zu unphysikalischen Effekten führen kann. Mit einer sogenannten selektiven Massenskalierung kann wenigstens die translatorische Trägheit (d.h. der Impuls bei gleichmäßiger Anfangsgeschwindigkeit) erhalten werden, allerdings auf Kosten von Nebendiagonaltermen in der Massenmatrix. Diese Methode lässt sich gleichmäßig auf die gesamte Struktur anwenden und hat geringere unphysikalische Nebeneffekte. Bislang werden skalierte Massenmatrizen rein algebraisch, z.B. steifigkeitsproportional konstruiert. Die Auswahl an Skalierungs-Templates ist beschränkt und ihnen liegt keine konsistente Formulierung zugrunde. In diesem Artikel werden variationelle Methoden zur selektiven Massenskalierung vorgestellt, die nicht nur mathematisch konsistent sind, sondern es auch erlauben, gezielt bestimmte Eigenschaften wie die Erhaltung der Trägheit, Reduktion der höchsten Frequenzen und Genauigkeit bei den niedrigen Frequenzen einzustellen. Die Leistungsfähigkeit der Methode wird an numerischen Beispielen mit Tetraeder- und Hexaeder-Elementen demonstriert.

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

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

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

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

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

Concrete double-curved shell constructions have been used in architectural design and building constructions since the beginning of the twentieth century. Although monolithic shells show a high stiffness as their geometry transfers loads through membrane forces, they have been mostly replaced by the more cost-efficient lattice systems. As lattice systems are covered by planar glass or metal panes, they neither reach the structural efficiency of monolithic shells, nor is their architectural elegance reflected in a continuous curvature. The shells of sand dollars’ – highly adapted sea urchins – combine a modular and multi-plated shell with a flexible, curved as well as smooth design of a monolithic construction. The single elements of the sand dollars’ skeleton are connected by calcite protrusions and can be additionally supported by organic fibres. The structural efficiency of the sea urchin’s skeleton and the principles behind them can be used for innovations in engineering sciences and architectural design while, at the same time, they can be used to illustrate the biological adaptations of these ecologically important animals within their environments. The structure of the sand dollar’s shell is investigated using modern as well as established imaging techniques such as x-ray micro-computed tomography (µCT), scanning electron microscopy and various optical imaging techniques. 3D models generated by µCT scans are the basis for Finite Element Analysis of the sand dollar’s shell to identify possible structural principles and to analyse their structural behaviour. The gained insights of the sand dollar’s mechanical properties can then be used for improving the state-of-the-art techniques of engineering sciences and architectural design.

Abstract

We present ideas and concepts towards defining a common framework unifying abstract metrics in order to quantify key features of technical load-bearing structures and biological systems. Our aim is to transfer biological concepts to technical systems at this abstract level rather than on the basis of their outward appearance or actual functionality. This means that the biological concept generators for load-bearing structures do not have to be load-bearing structures themselves but may instead achieve rather different functionalities. We intend to carry out this transference by generalizing graph-based abstractions of both technical and biological worlds to allow comparisons to be made at an abstract level. We focus in particular on the intrinsically competing aims of optimality versus multi-functionality and robustness. In this review, we present initial attempts towards defining suitable quantitative measures for robustness to serve as a common ground for studying technical systems and biological systems simultaneously. We discuss generic properties of a ubiquitous signalling network motif and potential relationships to a minimal model for a robust truss structure. These case studies suggest that topological complexity can serve as a common source for a design that is insensitive to perturbations and thus robust in the measures of both worlds.

Abstract

We present ideas and concepts towards defining a common framework unifying abstract metrics in order to quantify key features of technical load-bearing structures and biological systems. Our aim is to transfer biological concepts to technical systems at this abstract level rather than on the basis of their outward appearance or actual functionality. This means that the biological concept generators for load-bearing structures do not have to be load-bearing structures themselves but may instead achieve rather different functionalities. We intend to carry out this transference by generalizing graph-based abstractions of both technical and biological worlds to allow comparisons to be made at an abstract level. We focus in particular on the intrinsically competing aims of optimality versus multi-functionality and robustness. In this review, we present initial attempts towards defining suitable quantitative measures for robustness to serve as a common ground for studying technical systems and biological systems simultaneously. We discuss generic properties of a ubiquitous signalling network motif and potential relationships to a minimal model for a robust truss structure. These case studies suggest that topological complexity can serve as a common source for a design that is insensitive to perturbations and thus robust in the measures of both worlds.

Abstract

Plant movements can inspire deployable systems for architectural purposes which can be regarded as ideal solutions combining resilient bio-inspired functionality with elegant natural motion. Here, we first give a concise overview of various compliant mechanisms existing in technics and in plants. Then we describe two case studies from our current joint research project among biologists, architects, construction engineers and materials scientists where the aesthetic movements of such role models from the plant kingdom are analysed, abstracted and implemented in bioinspired technical structures for sustainable architecture. Both examples are based on fast snapping movements of traps of carnivorous plants. The Waterwheel plant (Aldrovanda vesiculosa) captures prey underwater and the Venus flytrap (Dionaea muscipula) snaps in the air. We present results on the motion principles gained by quantitative biomechanical and functional-morphological analyses as well as their simulation and abstraction by using e.g. Finite Element Methods. The Aldrovanda mechanism was successfully translated into a similarly aesthetic and functional technical structure, named Flectofold, which exists in a prototype state. The Flectofold can be used as a façade shading element for complex curved surfaces as existing in modern architecture.

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 7-parameter shell model as proposed by Büchter and Ramm (1992) is an extension of conventional shear deformation theories with five degrees of freedom. Application of this model is especially sensible if complete three-dimensional constitutive laws ought to be applied allowing also to solve problems involving large deformations and large strains. Based on a mixed formulation the seventh degree of freedom can be eliminated on the element level. Thus, the numerical effort is only slightly larger compared to `usual' shell elements. The model can consequently also be utilized for analyses of such shell structures, where a conventional shell formulation would be sufficient. Büchter and Ramm (1992) describe the 7-parameter shell model along with a finite element formulation. Here, the seventh degree of freedom is introduced on the element level by means of a hybrid-mixed formulation. In the present work the 7-parameter model is derived independent from a finite element formulation. In this context it can be interpreted as semi-discretization of the shell continuum in thickness direction. The decisive difference to most of the conventional shell theories is that this discretization is based on a multifield variational formulation. By this procedure a system of partial differential equations can be obtained, describing the 7-parameter model as a two-dimensional, continuous theory with seven kinematical degrees of freedom per material point of the reference surface. It is also intended to give a physical interpretation of the kinematic and static variables appearing in the model. Here, the main emphasis is put on those quantities, which do not show up in a conventional 5-parameter formulation. The investigations provide some insight into the mechanical behavior of the model. For the linear part of the transverse shear strains a new shear correction factor is proposed, which can reduce the error with respect to the three-dimensional solution in membrane dominated situations. It is shown that the 7-parameter model is `optimal' concerning the number of kinematic and static variables involved. This means that exactly those components are involved which are necessary for a complete three-dimensional material description. Finally, a concept for the formulation of efficient finite elements for the 7-parameter model is presented. Here, established methods from the literature are combined with own developments. A special feature of the proposed concept is that a unified formulation for triangular and rectangular elements of arbitrary polynomial order is obtained. In addition, an improvement in the treatment of shells with kinks is proposed. In the course of the present study the concept is realized for linear and quadratic triangular and rectangular elements. The features of the proposed elements are investigated in numerical experiments, including both linear and geometrically and materially non-linear problems.