Aufsätze in Fachzeitschriften

Zusammenfassung

The degree of static indeterminacy and its spatial distribution characterize load-bearing structures independent of a specific load case. The redundancy matrix stores the distribution of the static indeterminacy on its main diagonal, and thereby offers the possibility to use this property for the assessment of structures. It is especially suitable to be used in early planning stages for design exploration. In this paper, performance indicators with respect to robustness and assemblability are derived from the redundancy matrix. For each of the performance indicators, a detailed matrix-based derivation is given and the application is showcased with various truss examples.

Zusammenfassung

Finding the critical time step for conditionally stable time integration methods has been a decades-long problem. The apparently obvious option of directly computing it from a generalized eigenvalue analysis, identifying the largest eigenfrequency of the discrete system, is usually impractical because of its numerical expense and because a stiffness matrix is often unavailable in the context of explicit analysis. There exist two popular approaches to efficiently estimate the critical time step: A characteristic element length can be estimated based on heuristic formulas. The resulting estimate, however, cannot be guaranteed to be conservative. Another approach is to reformulate and simplify the underlying eigenvalue problem on the element level and to use certain inequalities to derive an upper bound for the largest eigenvalue. This is conservative but may show poor performance by significantly under-predicting the actual critical time step. Moreover, the necessary simplifications are usually specific to the investigated element formulation. Many works that develop time step estimators demonstrate their performance only for particular element configurations, making it difficult to compare the estimators. In this paper, data-driven approaches for time step estimation for 2d-elements that address several of the aforementioned problems are proposed. First, the set of all possible quadrilateral element geometries and its discrete representation are described. A detailed comparison of nine existing time step estimators based on more than ten million element configurations is presented. Additionally, the concept of an optimal safety factor function is introduced. This concept allows us to generate the optimal and conservative version of an existing estimator and thus solves two problems at the same time: It can be used to make non-conservative estimators conservative and to improve the performance of estimators that are conservative by construction. Finally, we formulate time step estimation as a function approximation problem. It allows us to derive customizable time step estimators solely based on data. Through two examples, we demonstrate that this data-driven approach yields time step estimators that outperform state-of-the-art estimators in terms of accuracy while also being efficient to evaluate.

Zusammenfassung

This paper presents the integration of a numerical structural model based on the redundancy matrix and experimental results of Multi-Layered Randomized Architected Materials (MLRAM). It presents a combination of the relatively new field of architected materials with a load-independent performance indicator from theoretical structural mechanics. The redundancy matrix by itself provides a measure for structural assessment that is independent of a specific load case. Various layouts of the MLRAM samples and recorded testing allow the analysis of the redundancy distribution within the structure as it undergoes failure. An in-depth analysis of the tested MLRAM samples is provided, as they show a high degree of static indeterminacy and thus, multiple different load paths. A special focus lies on the change of the redundancy distribution as global progressive failure happens. Another focus is set on the investigation of the failure initiation, meaning that the redundancy distribution can help to identify critical elements. A simple introductory example shows the interdependence between the variation of the geometric location of nodes and the redundancy distribution. The study shows, that the distribution of static indeterminacy can be used as a measure to quantify vulnerability to failure and rank the individual element's importance. Furthermore, progressive collapse is identified as a series of local effects in the highly statically indeterminate MLRAM samples, underlining the fact that the spatial distribution of static indeterminacy is of central importance for the assessment of structural safety.

Zusammenfassung

Form-finding is an essential task in the design of efficient lightweight structures. It is based on the crucial assumption of one single shape-determining load case, usually represented by self-weight. Adaptive components integrated into the structure open a way to even more efficient lightweight designs, as such structures can adapt their shapes to varying external loads and redistribute internal forces. This article presents a method for form-finding of adaptive truss structures subject to multiple, independently acting load cases, also incorporating possible design constraints. To ensure the consistency of the manufacturing lengths of passive elements in all load cases, special constraints are considered. The method enables to reduce sensitivity of the structural shape with respect to various different loads by means of actuation to meet design and serviceability requirements with a lower structural mass compared to conventional design strategies. This is demonstrated within a replaced real-world-like setting of an adaptive suspension truss bridge.

Zusammenfassung

Fiber-reinforced composites offer innovative solutions for architectural applications with high strength and low weight. Coreless filament winding extends industrial processes, reduces formwork, and allows for tailoring of fiber layups to specific requirements. A previously developed computational co-design framework for coreless filament winding is extended toward the integration of reciprocal design feedback to maximize design flexibility and inform design decisions throughout the process. A multi-scalar design representation is introduced, representing fiber structures at different levels of detail to generate feedback between computational design, engineering, and fabrication. Design methods for global, component, and material systems are outlined and feedback generation is explained. Structural and fabrication feedback are classified, and their integration is described in detail. This paper demonstrates how reciprocal feedback allows for co-evolution of domains of expertise and extends the existing co-design framework toward design problems. The developed methods are shown in two case studies at a global and component scale.

Zusammenfassung

The critical time step in explicit time integration methods depends on the highest natural angular frequency of the discretized problem. For shear deformable beam, plate and shell formulations, efficiency is therefore typically limited by the highest transverse shear frequencies, which are mostly of minor importance for the structural response. Direct parametrization using transverse shear variables within hierarchic structural element formulations allows a selective scaling of transverse shear frequencies in a simple manner, while bending frequencies remain practically unaffected. In particular, the novel concept of intrinsically selective mass scaling (ISMS) results in an efficient method that features high accuracy and preserves both linear and angular momentum for both consistent and lumped mass matrices. In addition, ISMS preserves the diagonal structure of lumped mass matrices. Similar to the underlying intrinsically locking-free, hierarchic concept for shear deformable structural element formulations, ISMS retains its beneficial properties for any smooth discretization scheme. In this contribution, we extend recent research on ISMS for beam formulations to the case of shear deformable plate formulations. We test our novel concept with respect to accuracy and efficiency by means of selected numerical experiments. We study both frequency spectra and the transient behavior in explicit time integration. To demonstrate the generality of ISMS, exemplarily both isogeometric discretizations based on B-splines and meshfree discretizations using local maximum-entropy approximants are investigated.

Zusammenfassung

This work investigates the application of an external adaptive tensioning (EAT) system for high-speed railway (HSR) bridges. The design of HSR bridges involves strict displacement and acceleration limits, which typically results in oversizing. The EAT system comprises under-deck cables deviated by compressive struts that are equipped with linear actuators. Since the cable is eccentric to the bridge neutral axis, tensioning the under-deck cables by adjusting the length of the linear actuators generates a bending moment that counteracts the effect of the external loads. The response under variable loading is reduced by computing the actuator commands with a linear quadratic regulator (LQR). Numerical results show that active control through the EAT system allows satisfying displacement and acceleration limits, which otherwise cannot be met without increasing the stiffness and mass of the bridge. A significant reduction of the response is achieved when resonance conditions occur. In addition, peak stresses are significantly reduced, showing the potential for fatigue-life extension. Parametric analyses comparing different bridge depths and spans, EAT system dimensions, controller parameters and actuator placement are carried out to investigate system efficacy. Results show that the adaptive bridge solution can achieve up to 32% mass savings compared to an equivalent passive bridge.

Zusammenfassung

In many applications in computer-aided engineering, like parametric studies, structural optimization, or virtual material design, a large number of almost similar models must be simulated. Although the individual scenarios may differ only marginally in both space and time, the same amount of effort is invested in each new simulation, without taking into account the experience and knowledge gained in previous simulations. Therefore, we have developed a method that combines data-based Model Order Reduction (MOR) and reanalysis, exploiting knowledge from previous simulation runs to accelerate computations in multi-query contexts. While MOR allows reducing model fidelity in space and time without significantly deteriorating accuracy, reanalysis uses results from previous computations as a predictor or preconditioner. In particular, this method enables acceleration of the exact computation of critical points, such as limit and bifurcation points, by the method of extended systems for systems that depend on a set of design parameters, such as material or geometric properties. Such critical points are of utmost engineering significance due to the special characteristics of the structural behavior in their vicinity. Conventional reanalysis methods, like the fold line analysis, can be used to accelerate the computation of critical points of almost similar systems but are limited in their applicability. For the fold line analysis, only small parameter variations are possible as the algorithm may not converge to the correct solution or fail to converge elsewise. Moreover, this method is only suited to finding the first critical points of limit point problems. In contrast to that, our developed data-based “reduced model reanalysis” method overcomes these problems. Thus, a larger parameter space can be covered. The efficiency of this method is demonstrated for a couple of numerical examples, including standard and isogeometric finite element models.

Zusammenfassung

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.

Zusammenfassung

Within the framework of plane strain nonlinear elasticity, we present a discussion on the stability properties of various Enhanced Assumed Strain (EAS) finite element formulations with respect to physical and artificial (hourglassing) instabilities. By means of a linearized buckling analysis we analyze the influence of element formulations on the geometric stiffness and provide new mechanical insights into the hourglassing phenomenon. Based on these findings, a simple strategy to avoid hourglassing for compression problems is proposed. It is based on a modification of the discrete Green-Lagrange strain, simple to implement and generally applicable. The stabilization concept is tested for various popular element formulations (namely EAS elements and the assumed stress element by Pian and Sumihara). A further aspect of the present contribution is a discussion on proper benchmarking of finite elements in the context of hourglassing. We propose a simple bifurcation problem for which analytical solutions are readily available in the literature. It is tailored for an in-depth stability analysis of finite elements and allows a reliable assessment of its stability properties.

Zusammenfassung

The linear design workflow for structural systems, involving a multitude of iterative loops and specialists, obstructs disruptive innovations. During design iterations, vast amounts of data in different reference systems, origins, and significance are generated. This data is often not directly comparable or is not collected at all, which implies a great unused potential for advancements in the process. In this paper, a novel workflow to process and analyze the data sets in a unified reference frame is proposed. From this, differently sophisticated iteration loops can be derived. The developed methods are presented within a case study using coreless filament winding as an exemplary fabrication process within an architectural context. This additive manufacturing process, using fiber-reinforced plastics, exhibits great potential for efficient structures when its intrinsic parameter variations can be minimized. The presented method aims to make data sets comparable by identifying the steps each data set needs to undergo (acquisition, pre-processing, mapping, post-processing, analysis, and evaluation). These processes are imperative to provide the means to find domain interrelations, which in the future can provide quantitative results that will help to inform the design process, making it more reliable, and allowing for the reduction of safety factors. The results of the case study demonstrate the data set processes, proving the necessity of these methods for the comprehensive inter-domain data comparison.

Zusammenfassung

We investigate the variational formulation and corresponding minimizing energies for the detection of energetically favorable magnetization states of thin cylindrical magnetic nanodots. Opposed to frequently used heuristic procedures found in the literature, we revisit the underlying governing equations and construct a rigorous variational approach that takes both exchange and demagnetization energy into account. Based on a combination of Ritz’s method and a Fourier series expansion of the solution field, we are able to pinpoint the precision of solutions, which are given by vortex modes or single-domain states, down to an arbitrary degree of precision. Furthermore, our model allows to derive an expression for the demagnetization energy in closed form for the in-plane single-domain state, which we compare to results from the literature. A key outcome of the present investigation is an accurate phase diagram, within the problem class of constant magnetization through the thickness and rotational symmetry, which we obtain by comparing the vortex mode’s energy minimizers with those of the single-domain states. This phase diagram is validated with data of two- and three-dimensional models from literature. By means of the phase diagram, we particularly find the critical radius at which the vortex mode becomes unfavorable with machine precision. All relevant data and codes related to the present contribution are available at Müller (2023).

Zusammenfassung

Floor systems are typically designed to satisfy tight deflection limits under out-of-plane loading. Although the use of concrete flat slabs is common in the built environment due to the ease of construction, the load-bearing performance is inefficient because the material is not optimally distributed within the cross section to take the bending caused by external loads. This typically results in significant oversizing. Floor slabs account for more than 50% of the material mass and associated emissions embodied in typical low-rise reinforced concrete buildings. In addition, the volume of carbon-intensive cement production has tripled in the last three decades. Therefore, lightweight floor systems that use minimum material resources causing low emissions can have a significant impact on reducing adverse environmental impacts of new constructions. Recent work has shown that rib-stiffened slabs offer significant potential for material savings compared with flat slabs. This work investigates adaptive rib-stiffened slabs equipped with an adaptive tensioning system. The adaptive tensioning system comprises cables embedded within the concrete rib through a duct that enables varying the cable tension as required to counteract the effect of different loading conditions without applying permanent prestress that might cause unwanted long-term effects including tension loss and amplified deflection. The cables are positioned following a profile so that the tension force is applied eccentrically to the neutral axis of the slab-ribs assembly. The resulting system of forces causes a bending moment that counteracts the effect of the external load. The rib placement is optimized through a greedy algorithm with a heuristic based on the direction of the principal stresses. The deflection of the slab is reduced by adjusting the cable tensile forces computed by a quasi-static controller. Benchmark studies comparing different cable profiles and active rib layouts are carried out to determine an efficient control configuration. A case study of an 8x8 m adaptive rib-stiffened slab is implemented to evaluate material savings potential. Results show that the adaptive slab solution can achieve up to 67% of material savings compared with an equivalent passive flat slab.

Zusammenfassung

Large statically indeterminate truss and frame structures exhibit complex load-bearing behavior, and redundancy matrices are helpful for their analysis and design. Depending on the task, the full redundancy matrix or only its diagonal entries are required. The standard computation procedure has a high computational effort. Many structures fall in the category of moderately redundant, i.e., the ratio of the statical indeterminacy to the number of all load-carrying modes of all elements is less one half. This paper proposes a closed-form expression for redundancy contributions that is computationally efficient for moderately redundant systems. The expression is derived via a factorization of the redundancy matrix that is based on singular value decomposition. Several examples illustrate the behavior of the method for increasing size of systems and, where applicable, for increasing degree of statical indeterminacy.

Zusammenfassung

This paper discusses the role that structural stiffness plays in the context of designing adaptive structures. The focus is on load-bearing structures with adaptive displacement control. A design methodology is implemented to minimize the control effort by making the structure as stiff as possible against external loads and as flexible as possible against the effect of actuation. This rationale is tested using simple analytical and numerical case studies.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

Redundancy matrices provide insights into the load carrying behavior of statically indeterminate structures. This information can be employed for the design and analysis of structures with regard to certain objectives, for example reliability, robustness, or adaptability. In this context, the structure is often iteratively examined with the help of slight adjustments. However, this procedure generally requires a high computational effort for the recalculation of the redundancy matrix due to the necessity of costly matrix operations. This paper addresses this problem by providing generic algebraic formulations for efficiently updating the redundancy matrix (and related matrices). The formulations include various modifications like adding, removing, and exchanging elements and are applicable to truss and frame structures. With several examples, we demonstrate the interaction between the formulas and their mechanical interpretation. Finally, a performance test for a scaleable structure is presented.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

(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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

In this paper, we study the accuracy and robustness of the computational time reversal approach based on the explicit finite element method for application in nondestructive testing in solids. The main goal of this paper is to find a methodology for correct and accurate reconstruction of the original source time history. For numerical modeling of frontal (forward) and reverse (backward) problems of elastic wave propagation, we use the finite element method and explicit time integration with the lumped mass matrix. The suggested methodology is applicable in each finite element open source or commercial software. A special attention is paid to prescription of boundary conditions/loading for the reverse problem for accurate reconstruction of time history of the original source. For evaluation of the reconstruction quality, we suggest certain cost functions. Based on several numerical tests, we show effects of prescription of boundary conditions/loading in time reversal, effect of mesh size and time step size, an unknown obstacle, a number of sources, and environmental disturbance (noise) on the correctness of reconstruction of the original source.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

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Customization of finite elements for low dispersion error through grid dispersion analysis requires a symbolic expansion of a determinant of a representative dynamic stiffness matrix. Such an expansion turns out to be a bottleneck for many practical cases with the size of the representative matrix greater than eight or ten even if the modern computer algebra systems are applied. In this contribution, we propose an alternative approach for low-dispersion customization that avoids explicit determinant expansion. This approach reduces the customization problem to a series of quadratic programming problems and consist of two main steps. Firstly, the customization problem is reformulated as a rank minimization problem for the representative dynamic stiffness matrix evaluated at several discrete pairs of wavenumbers and frequencies. Secondly, the rank minimization problem is solved approximately via log-det heuristic. Examples for customization of reciprocal mass matrices illustrate capabilities of the proposed approach.

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Finite elements with Allman’s rotations provide good computational efficiency for explicit codes exhibiting less locking than linear elements and lower computational cost than quadratic finite elements. One way to further raise their efficiency is to increase the feasible time step or increase the accuracy of the lowest eigenfrequencies via reciprocal mass matrices. This paper presents a formulation for variationally scaled reciprocal mass matrices and an efficient estimator for the feasible time step for finite elements with Allman’s rotations. These developments take special care of two core features of such elements: existence of spurious zero‐energy rotation modes implying the incompleteness of the ansatz spaces, and the presence of mixeddimensional degrees of freedom. The former feature excludes construction of dual bases used in the standard variational derivation of reciprocal mass matrices. The latter feature destroys the efficiency of the existing nodal‐based time step estimators stemming from the Gershgorin’s eigenvalue bound. Finally, the developments are tested for standard benchmarks and triangular, quadrilateral and tetrahedral finite elements with Allman’s rotations.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

Zusammenfassung

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.

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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.

Zusammenfassung

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.

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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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

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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.

Zusammenfassung

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.

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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.

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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.

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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.

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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.

Zusammenfassung

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.

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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.

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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.

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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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.