Hesch, Christian
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Hesch, Christian
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15050680100
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Publication Access status: Open Access , Dynamics of continua with interfaces(2013)Continuum mechanics is nowadays widely used to describe the material behavior of systems, which occupies a specific area in space. In contrast to atomistic and molecular models, which can be solved using molecular dynamics or Monte-Carlo simulations, we consider the system to exist as a continuum. Due to their different material behavior we distinguish between solid mechanics and fluid mechanics. The typical deformation of the former one allows us to follow the movement of each particle in space, whereas we can not do this for the latter one. This leads to different formulations, which will be presented here. Typically, we want to achieve a solution for the balance of linear momentum for the continuum mechanical system. Additionally, we will derive a pure mass transport problem to demonstrate the capabilities of the numerical framework we have developed to solve these kind of problems. Once we have introduced the continuum mechanical framework, we can extent this to include further physical effects. Moreover, we extent the solid mechanical system to include thermal contributions and apply an additional pressure field to enforce the incompressibility of the fluids in the case of low Mach numbers. Within the continuous setting, we can define various interfaces. Internal interfaces can be used to decompose bodies into different subsets, e.g. to define areas with different physical properties or, on a more technical level, to enable parallelization on modern cluster architectures. External interfaces on solids can be used to include contact between multiple bodies. Additionally we could establish an interface at the external boundary to transfer momentum between a solid and the surrounding fluid. To avoid technical problems in the case of large deformations of solids, embedded within a fluid, we employ continuum immersed strategies to include the effects of fluid-structure interaction. Finally, we want to use phase field models for the simulation of phase separation and coarsening in solder alloys. We obtain sharp interfaces between the phases using the well known Cahn-Hilliard model to represent the free energy of the phases as well as of the interface. Similar to the immersed strategies, we aim at the simulation of the whole domain, avoiding the explicit representation of interfaces. To solve the arising initial boundary value problem in space, we first apply the finite element method for all problems at hand. In particular, we introduce Lagrangian as well as NURBS based shape functions for the underlying approximation of the field equations, written in weak form. Furthermore, we show how to incorporate discrete interface models in an optimal sense with regard to the consistency error at the interface using the Mortar method. The application of Mortar methods to NURBS will be shown as well. Due to the higher continuity requirement of the Cahn-Hilliard equation, the use of NURBS seems to be natural for this kind of problems. Since we deal with initial boundary value problems, suitable time integration schemes have to be developed as well. In general, we use a common implicit integration scheme for all problems at hand, such that we could use various fields simultaneously in a consistent framework. If possible, we aim at the development of structure preserving integrators, since they provide enhanced numerical stability for large time steps. For the explicit interface representation we apply additional augmentation techniques to simplify the algebraic constraints and verify the underlying conservation properties.Source Type:HabilitationSeries:Schriftenreihe des Lehrstuhls für Numerische MechanikVolume number:71 11 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Open Access , Mechanische Integratoren für Kontaktvorgänge deformierbarer Körper unter großen Verzerrungen(2008)Forschungsziel dieser Arbeit ist die Simulation von Kontakt-Vorgängen. Ein beispielhafter und konkreter Anwendungsfall ist die computerunterstützte Crash-Simulation von Fahrzeugen der Automobilindustrie. Die begriffliche Bestimmung von Kontakt wird hierbei weitest definiert mit dem Ziel, das umfassende Feld der Gebietszerlegungsverfahren einbringen zu können. Diese notwendige Voraussetzung ergibt sich aus der Tatsache, dass ein Teil der Interface-Strukturen zur Beschreibung von Kontakt-Vorgängen ursprünglich für Gebietszerlegungsverfahren entwickelt wurde. Die Entwicklung numerischer Methoden für die Zeitintegration ist damit das substanzielle Ziel der vorliegenden Untersuchung. Die Integratoren müssen für die algorithmische Erhaltung grundlegende Symmetrie-Eigenschaften des mechanischen Systems gewährleisten. Diese Zeitschrittverfahren sind unter der Bezeichnung Mechanische Integratoren bekannt und wurden bislang ohne Erfolg auf die zur Beschreibung von Kontakt-Vorgängen notwendigen Zwangsbedingungen angewandt. Die mechanischen Integratoren bieten eine einheitliche Vorgehensweise zur Gewährleistung der fundamentalen physikalischen Eigenschaften sowohl für die nichtlineare Elastostatik als auch für allgemeine nichtlineare Zwangsbedingungen. Die algorithmische Erhaltung zentraler Eigenschaften wie der Gesamtenergie und der Impulsabbildungen führen in der Praxis zu einer herausragenden numerischen Stabilität; insbesondere bei Kontakt-Vorgängen ist dies von qualitativ hohem Interesse. Auch die hieraus resultierende Möglichkeit der Wahl beliebiger Zeitschritte zeichnet dieses Verfahren aus. Jene Eigenschaften sowie die Konzeption der mechanischen Integratoren setzen ein exaktes Einfordern der holonomen Zwangsbedingungen voraus. Dies kann mithilfe zusätzlicher Freiheitsgrade, den Lagrangeschen Multiplikatoren, erfolgen. Die zugehörigen Bewegungsgleichungen können als ein differential-algebraisches Gleichungssystem geschrieben werden. Eine spezifische Besonderheit in der Beschreibung von Kontakt-Vorgängen bei der Simulation großer Deformationen von vollständig nichtlinearen Systemen unter Einbindung beliebiger konstitutiver Gesetze liegt in der Verwendung der Karush-Kuhn-Tucker Bedingungen. Sie bestimmen, ob und wann eine Zwangsbedingung aktiv ist oder nicht. Diese Klasse der Ungleichheits-Zwangsbedingungen erfordert ein entsprechendes Anpassen der mechanischen Integratoren. In der vorliegenden Untersuchung werden alle notwendigen Formulierungen sowohl für die Elasto- als auch für die Kontaktmechanik detailliert hergeleitet. Das resultierende nichtlineare Gleichungssystem kann mit einem Newton-Raphson Verfahren gelöst werden. Nicht zuletzt werden mithilfe verschiedener numerischer Beispiele die Stärken der neu entwickelten Formulierungen nachvollziehbar demonstriert.Source Type:Doctoral ThesisSeries:Schriftenreihe des Lehrstuhls für Numerische MechanikVolume number:1722 208 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , Non‐linear space‐time elasticity(2023); ; Summary In this contribution we introduce a novel space‐time formulation for non‐linear elasticity, able to calculate large deformations and displacements with high efficiency using structured and unstructured meshes in the space‐time cylinder without changing the required regularity and thus, enabling the use of Lagrangian shape functions including tetrahedron and hypertetrahedron or tesseract elements. The common and indiscriminate treatment of spatial and temporal directions allows us to remove one of the major bottlenecks in parallel computations: The design of time‐stepping schemes, which can neither been parallelized efficiently in time nor allow for local refinements as no information transfer backwards in time is possible. Moreover, stability of time‐stepping schemes depend on the accumulation of local approximation errors in each time‐step, in contrast to space‐time formulation, characterized by enhanced stability and robustness. Eventually, we will demonstrate superiority in the convergence against classical time‐stepping schemes using various examples including highly sensitive systems in the context of non‐linear elasticity.Source Type:ArticleDOI:10.1002/nme.7194 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , Space-time rigid multibody dynamics(2023); ; In this contribution, we apply space-time formulation on constrained rigid body dynamics. In particular, we discretize directly Hamilton’s principle using appropriate space-time approximation spaces for the variational problem. Moreover, we make use of a rotationless formulation for the rigid bodies, and thus we have to define appropriate approximation spaces for the Lagrange multipliers as well. Moreover, we make use of Livens’ principle, introducing independent quantities for the position, velocity, and momentum, where the latter can be considered as Lagrange multipliers, and we apply this concept to the space-time rigid body formulation. Finally, we demonstrate the convergence of the different approaches and the superiority in terms of computational effort, and thus total energy consumption of dynamical simulations.Source Type:Article2 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , Variational formulation and monolithic solution of computational homogenization methods(2024); ; In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE and IGA approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null‐space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE and IGA formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general nonlinear problems.Source Type:ArticleDOI:10.1002/nme.75671
