Veröffentlichungen

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

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

Explizite Zeitintegrationsverfahren besitzen nur bedingte Stabilität. Diese hängt von der kritischen Zeitschrittweite ab, welche über die höchste Eigenkreisfrequenz des Systems bestimmt wird und nachweislich mit der kleinsten Elementabmessung in Verbindung steht. Speziell für dünnwandige Elemente wird ein neuer Ansatz zur selektiven Massenskalierung vorgestellt, der auf der Discrete-Shear-Gap-Methode nach Bletzinger u. a. (2000) basiert. Die eingeführte Massenskalierungsmethode beeinflusst in erster Linie die Dickenrichtung der Elemente, welche in dünnwandigen Strukturen eine deutlich geringere Abmessung als die in-ebenen-Dimensionen besitzt und reduziert damit lediglich die höchsten Frequenzen des Systems. Dadurch bleiben die niedrigen und strukturrelevanten Moden, die den hauptsächlichen Energieanteil besitzen, nahezu unbeeinflusst. Eine neue künstliche Massenmatrix mit anisotropem Aufbau und den gewünschten Eigenschaften kann erzeugt werden. Für nichtlineare Analysen, in denen sich die Steifigkeitsmatrix durch große Rotationen beständig ändert, wurde eine isotrope Version entwickelt. Diese beeinflusst auch einige niedrigere Eigenmoden des Systems, jedoch lassen sich damit die höchsten Eigenkreisfrequenzen des Systems weiter skalieren, was in expliziten Verfahren einen größeren Zeitschritt erlaubt. Für beide Versionen wurde eine Variante ermittelt, die das Frequenzspektrum noch besser abbildet, jedoch die maximal mögliche Reduktion der höchsten Frequenz partiell begrenzt. Die Methode und ihr Verhalten in Bezug auf Genauigkeit und Effektivität wird in numerischen Untersuchungen charakterisiert und anschließend diskutiert. Im Vergleich zu einigen bereits in der Literatur bekannten Methoden zur selektiven Massenskalierung liefert der hier vorgestellte Ansatz vergleichbare und unter großen Schlankheitswerten zum Teil bessere Ergebnisse. Zudem ist er auch für nahezu inkompressible Materialien in biegedominanten Problemen gut geeignet.

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

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

Hammering tools uses damping elements of various geometries and hardness levels not only to reduce the reaction forces and vibrations, but also to protect the structural parts from failure. This makes damping elements a key component during the design phase of any tool. Since many design quantities initially rely on the results obtained via simulations for optimization and improvement purposes, it makes it essential to have an appropriate material model which could capture the behaviour of the damping elements accurately. Damping elements are made up of elastomeric compounds showcasing highly non-linear behaviour especially when subjected to very high strains and strain rates when mounted in a tool. This aim of this work is to study various phenomena regarding the rubber material behaviour and develop a user-defined material model in LS-DYNA which provides error-free stress updates at a given strain level for elastomers. The fundamental concepts of hyperelastic and viscoelastic constitutive theories are emphasized in the beginning as these theories are more suitable to model elastomeric behaviour. Material parameter identification procedure is also highlighted for both hyperelastic and linear viscoleastic material models. An Ogden-based linear viscoelastic model is programmed which is extended with strain-level based non-linearity included in the material model and later validated with the experimental results obtained with a gas-gun test fixture and O-ring specimens. Discussions regarding energy dissipation is focussed towards the end as it plays a crucial role in understanding the behaviour of the damping element.

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

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

Zusammenfassung

Bei der Simulation von Blechumformprozessen werden nach derzeitigem Stand der Technik in der Regel Schalenelemente basierend auf dem Reissner-Mindlin-Modell in Kombination mit vereinfachten Konstitutivgesetzen verwendet. Diesem Modellierungsansatz liegen einige vereinfachende Annahmen zugrunde. Auf Seiten der Strukturmodellierung werden ebenbleibende Querschnittsfasern sowie vernachlässigbare Normalspannungen in Blechdickenrichtung vorausgesetzt. Bei der Materialmodellierung wird vor dem Hintergrund der Vernachlässigung transversaler Normalspannungen das richtungsabhängige Materialverhalten ausschließlich in der Blechebene beschrieben und somit anisotrope Effekte außerhalb dieser vernachlässigt. Bei bestimmten Blechumformprozessen stößt dieser Modellierungsansatz an seine Grenzen, da einige der getroffenen Annahmen nicht mehr gültig sind. Im Rahmen dieses Forschungsvorhabens wurde ein alternativer Ansatz verfolgt. Bei diesem als „3D-Blechmodellierung" bezeichneten Ansatz werden Schalenelemente höherer Ordnung, die nicht den Restriktionen des Reissner-Mindlin-Modells unterliegen, zur Simulation von Blechumformprozessen genutzt. Diese werden mit 3D-Werkstoffmodellen kombiniert, welche den vollständig dreidimensionalen Verzerrungs- und Spannungszustand berücksichtigen. Um diesen Ansatz verfolgen zu können, wurden im Rahmen des Forschungsprojektes entsprechende Schalenelemente höherer Ordnung entwickelt. Zudem wurde die Methode der virtuellen Kennwertermittlung weiterentwickelt und damit Parameter für 3D-Fließortmodelle für einen Aluminium- und einen Stahlwerkstoff bestimmt. Anhand von numerischen Benchmarks und Vergleichen mit Versuchsergebnissen werden die prinzipielle Anwendbarkeit und die Verbesserung der Ergebnisqualität durch die 3D-Blechmodellierung demonstriert. Das IGF-Vorhaben „Verbesserte Blechumformsimulation durch 3D-Werkstoffmodelle und erweiterte Schalenformulierungen" der Forschungsvereinigung EFB e.V. wurde unter der Fördernummer AiF 19707N über die Arbeitsgemeinschaft industrieller Forschungsvereinigungen (AiF) im Rahmen des Programms zur Förderung der Industriellen Gemeinschaftsforschung (IGF) vom Bundesministerium für Wirtschaft und Energie aufgrund eines Beschlusses des Deutschen Bundestages gefördert. Der Abschlussbericht ist als EFB-Forschungsbericht Nr. 532 erschienen und bei der EFB-Geschäftsstelle und im Buchhandel erhältlich.

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

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

Zusammenfassung

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

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

Die wissenschaftliche Auseinandersetzung mit der Stabilität von Tragwerken reicht in der Geschichte weit zurück. Leonhard Euler befasste sich bereits im Jahr 1744 mit dem Stabilitätsversagen druckbelasteter Stäbe. Bis heute ist die Frage nach der Stabilität in vielen Bereichen, vom Bauwesen bis zur Luft- und Raumfahrt, relevant für die Bemessung und Konstruktion einzelner Bauteile sowie gesamter Tragstrukturen. Die Analyse eines Tragwerks hinsichtlich seiner Stabilität kann dabei in drei Abschnitte untergliedert werden: erstens die Untersuchung des Systems vor dem Stabilitätsversagen, zweitens die Eigenschaften des Systems an einem kritischen Punkt sowie drittens die Untersuchung des Verhaltens im Nachbeulbereich. Für eine geometrisch nichtlineare Analyse des Vor- und Nachbeulbereichs existiert eine Vielzahl inkrementell-iterativer Lösungsverfahren, die seit den 1970er Jahren entwickelt wurden. Sie ermöglichen die zuverlässige Berechnung nichtlinearer Gleichgewichtspfade einer Struktur, auch wenn diese in Last oder Verschiebung rückläufig sind. In den vergangen Jahrzehnten wurden diese Methoden stetig weiterentwickelt und verbessert, etwa durch die Erweiterung auf mehrere Parameter. So lassen sich für Systeme mit mehreren Veränderlichen die resultierenden Gleichgewichtspfade zu Gleichgewichtsflächen erweitern. Mithilfe begleitender Maßnahmen kann zudem an jedem Gleichgewichtspunkt eine Aussage über die Stabilität des Systems getroffen werden. Der Fokus der vorliegenden Arbeit liegt auf dem zweiten der oben genannten Punkte. Es werden Methoden präsentiert, die eine direkte Berechnung von Durchschlags- und Verzweigungspunkten für Systeme ermöglichen, die durch eine von außen aufgebrachte Knotenverschiebung belastet werden. Hierfür wird die für Kraftlastfälle bereits verfügbare Methode der erweiterten Systeme, die in der Literatur meistens mit dem englischen Begriff Extended Systems bezeichnet wird, modifiziert und ergänzt. Für die direkte Berechnung von Durchschlagspunkten wird die Residuumsgleichung für das Gleichgewicht um eine Stabilitätsbedingung erweitert. Diese ist ausschließlich an kritischen Punkten erfüllt und lässt sich aus dem Indifferenzkriterium nach Euler herleiten. Eine Nebenbedingung, die eine triviale Lösung vermeidet und die Anzahl der Gleichungen und die Anzahl der Unbekannten ausgleicht, ergibt sich aus der Normierung der Länge des Eigenvektors. Für die direkte Ermittlung von Verzweigungspunkten muss zusätzlich das Kriterium zur Unterscheidung von Durchschlags- und Verzweigungspunkten für den Fall inhomogener Dirichlet-Randbedingungen neu hergeleitet werden. Nach einer konsistenten Linearisierung des nichtlinearen Gleichungssystems lässt sich die Lösung mittels inkrementell-iterativer Methoden ermitteln. Mit den präsentierten Methoden wird das Anfangsnachbeulverhalten für Systeme mit inhomogenen Dirichlet-Randbedingungen numerisch analysiert, wodurch die bisherige Kategorisierung nach Koiter (1967) erweitert wird. Zur Linearisierung des nichtlinearen Gleichungssystems ist die Ermittlung einer Richtungsableitung der Steifigkeitsmatrix in Richtung des Beulvektors erforderlich. In der bisherigen Literatur wird die Ableitung mittels Differenzenquotienten angenähert. Diese Methoden benötigen eine geringe Rechenzeit, ihre Eigenschaften hängen jedoch von einem Parameter, der Größe des Differenzenschritts, ab. Wird dieser zu groß oder zu klein gewählt, ist der resultierende Fehler zu groß und die iterative Berechnung konvergiert nur langsam oder gar nicht zur gesuchten Lösung. In dieser Arbeit werden erstmals Methoden zur Ermittlung totaler und partieller Ableitungen, die auf hyperkomplexen Zahlen basieren, mit der Methode der Extended Systems kombiniert. Diese Methoden sind unabhängig von der Parameterwahl und ermöglichen eine exakte Berechnung erster und zweiter Ableitungen. In der vorliegenden Arbeit werden die beschriebenen Methoden auf die Ermittlung der Richtungsableitung übertragen. Bei der Untersuchung der Tragfähigkeit einer stabilitätsgefährdeten Struktur haben Imperfektionen einen großen Einfluss auf die Traglast. Bereits kleinste Abweichungen in der Geometrie eines Tragwerks oder Eigenspannungen im Material können, besonders bei schlanken Tragwerken, zu einer deutlichen Abminderung der Traglast führen. Um den Einfluss geometrischer Imperfektionen genauer untersuchen zu können, werden in dieser Arbeit zwei Methoden präsentiert, die sich in ihrem Umgang mit Imperfektionen grundlegend voneinander unterscheiden. Bei der ersten Methode ist die Imperfektionsform eine gegebene Größe. Durch eine effiziente, sich wiederholende Berechnung kritischer Punkte für eine Vielzahl an Imperfektionsformen oder -amplituden lassen sich damit kritische Pfade ermitteln. Dies geschieht in der vorliegenden Arbeit durch eine geeignete Wahl der Prädiktoren für die unterschiedlichen Imperfektionsformen bzw. über eine zusätzliche Erweiterung der Gleichungssysteme zur direkten Berechnung kritischer Punkte um eine Zusatzgleichung zur Pfadverfolgung. Bei der zweiten Methode werden die Imperfektionen als Unbekannte in das Gleichungssystem eingebracht. Durch eine zusätzliche Bedingung, die sich aus der Variation des Potentials nach den Imperfektionen ergibt, sowie das Einbinden der Nebenbedingung über eine Straffunktion lassen sich mit dieser Methode ungünstigste Imperfektionsformen finden. Diese Methode ist eine Weiterentwicklung der von Deml und Wunderlich (1997) vorgschlagenen Vorgehensweise. Abschließend werden die hier vorgestellten Methoden und Algorithmen an einer Auswahl an numerischen Experimenten demonstriert. Anhand der Beispiele wird u. a. gezeigt, dass die Richtungsableitung für eine Vielzahl an Elementformulierungen effizient ermittelt werden kann. Hierbei kommen vom einfachen ebenen Stabelement über isogeometrische Schalenelemente bis hin zu Volumenschalenelementen, die durch eine Erweiterung des Verzerrungsansatzes künstliche Versteifungseffekte reduzieren, zum Einsatz. Zudem wird die Anwendbarkeit der in der vorliegenden Arbeit entwickelten Methoden auf nichttriviale Systeme im Rahmen der Finite-Elemente-Methode mit vielen Freiheitsgraden nachgewiesen.

Zusammenfassung

In this thesis, a novel approach to support the design of motions for adaptive structures is presented and gradually developed: the so-called method of motion design. It is based on the observation that, depending on the control of the actuation, the same deformation state of a structure can be reached through various motion processes. The method of motion design allows to calculate optimal deformation paths with defined properties between the initial geometry and a given deformed end geometry of a structure in a formalized way. In order to motivate the efficiency of a movement and to make it mathematically quantifiable, the so-called cost of deformation is introduced as an exemplary target value based on the strain energy. By integration over the deformation path, the motion process is considered in its entirety in this optimization problem. The method of motion design is developed based on a variational formulation using the cost of deformation as underlying functional and the displacement field as the unknown function. One of the decisive features in this work is the discretization of the motion path, i.e., the deformation process. Due to the special structure of the functional with the integration of the strain energy, analytical sensitivities can be calculated by using quantities that are generally available in finite element software. The presented basic method is particularly well suited for the identification and design of kinematic and energy-minimal motion mechanisms, which emphasizes the potential for application to deployable shape changing structures. The motion design method is extended by the use of constraints such that the actuation can be prescribed, e.g., by actuator elements, or the entire motion can be stabilized. Finally, possibilities for enhancement of the motion design method and combinations with other methods to increase the efficiency of adaptive structures are investigated. They include a combination with shape optimization of the initial geometry, an integration within an actuator placement algorithm and variations of the underlying objective function.

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.

Zusammenfassung

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

Zusammenfassung

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.

Zusammenfassung

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.

Zusammenfassung

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.