Filters

Settings

Now showing 1 - 7 of 7
  • Some of the metrics are blocked by your 
    consent settings
    PublicationAccess status: Open Access ,
    Dynamics of continua with interfaces
    (2013)
    Hesch, Christianorcid-logo
    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:Habilitation
    Series:Schriftenreihe des Lehrstuhls für Numerische Mechanik
    Volume number:7
      1  11
  • Some of the metrics are blocked by your 
    consent settings
    PublicationAccess status: Open Access ,
    Energie-Entropie-konsistente Zeitintegratoren für die nichtlineare Thermoviskoelastodynamik
    (2012)
    Krüger, Melanie
    This work deals with an energy-entropy-consistent simulation of a thermoviscoelastic model problem and continuum. Both systems are described by the Poissonian variables - linear momentum, configuration, entropy and internal variable. The linear momentum and the configuration as independent variables lead to the equations of motion as differential equations of first order. The thermal evolution equation is derived by the second law of thermodynamics. The heat flux is described by Fourier's law. The equations of motion and the thermal evolution equation are linked through the constitutive equation of the internal energy. A viscous evolution equation, as fourth equation, is necessary to describe the viscous deformation behavior. This equation is based on deformation-like internal variables and a fourth order compliance tensor, which is restricted to the one-dimensional case for the model problem. The internal dissipation is given by a quadratic form of the viscous Mandel stress. The four differential equations of first order are transformed by the refined General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) format into a matrix-vector notation. This format is called in the following enhanced GENERIC format. The enhanced GENERIC format yields with the related degeneracy conditions structure preservation properties for an isolated system. An isolated system is defined as an adiabatic system, which does not do mechanical work. These properties are in addition to a constant linear and angular momentum, the constant total energy, an increasing total entropy and a decreasing Lyapunov function. The last one is a stability criterion for thermoviscoelastic systems. New is furthermore the consistent embedding of external mechanical and thermal loads. These external loads affect the aforementioned preservation properties of isolated systems. In this case the related balance equations of the system (consistency properties) are considered. The discretization in time is done for the model problem and the continuum with two different integrators. On the one hand the midpoint-rule and on the other hand the enhanced TC (Thermodynamically Consistent) integrator is used. The enhanced TC integrator is constructed such, that the underlying enhanced GENERIC format reflects the algorithmic properties after the discretization in time, for an isolated system. For a continuum a discretization in space is necessary, which is given by the Finite-Element-Method. A projection of the test function of the thermal evolution equation is necessary for an energy consistent discretization. Furthermore the energy consistency can only be guaranteed using the enhanced TC integrator, which leads to enhanced stability. The external mechanical and thermal loads are included with fixed bearings, external mechanical loads, thermal constraints and external thermal loads. The Lagrangian multipliers are used to fulfill the thermal constraint. These Lagrangian multipliers are well known for constraining the motion of a system. The enhanced GENERIC format will be extended for the model problem after the discretization with the external loads. In contrast to that, the enhanced GENERIC format for the continuum, which is here given in the strong evolution equations, contains the external loads. This yields the necessary weak evolution equations for the solution of the system. The consistency properties are shown for representative numerical examples with different boundary conditions. The coupled mechanical system under consideration is formulated in terms of the Poissonian variables. This leads to a monolithic solution with the Newton-Raphson method. Therefore, two Newton-Raphson methods are necessary, one to resolve the viscous internal variable on local (element) level and another one to resolve the equations of motion, the thermal evolution equation and if applicable the equation of projection on global level. This is called a multi-level Newton-Raphson method.
    Source Type:Doctoral Thesis
    Series:Schriftenreihe des Lehrstuhls für Numerische Mechanik
    Volume number:6
      642  245
  • Some of the metrics are blocked by your 
    consent settings
    PublicationAccess status: Open Access ,
    Mechanical integrators for the optimal control in multibody dynamics
    (2012)
    Siebert, Ralf
    The present thesis deals with inverse multibody dynamics problems. In particular, optimal control problems will be treated, which are governed by differential-algebraic equations. The main goal therein is the minimization of the control effort, which is necessary for moving a multibody system from one configuration to another. A basic task therefore is the formulation of the underlying equations of motion. Main focus will be on the formulation of the equations of motion with natural coordinates, which facilitates the design of structure-preserving time-stepping schemes. It is well known that schemes preserving basic properties of the mechanical system algorithmically exhibit superior stability properties in comparison to standard integrators. The application of these schemes within optimal control problems requires a consistent incorporation of the control torques. A convincing way for the incorporation of the control torques will be proposed in this contribution. In addition to the schemes based on the rotationless formulation, also an energy-momentum conserving time-stepping scheme based on quaternions will be presented. Although the quaternion-based scheme turns out to be competitive in the forward dynamics of rigid bodies, the extension to multibody problems is not as straightforward and easy to handle as with the rotationless formulation in terms of natural coordinates. Therefore quaternions will not be applied within optimal control problems in this thesis. Regarding optimal control of multibody systems, new energy-momentum consistent direct transcription methods in terms of natural coordinates will be presented. In a first step, the equations of motion obtained by a reduction process via the discrete null space method will be applied. The arising results will be compared with those achieved by a formulation of the equations of motion with the widely used minimal coordinates. In a second step, the original equations of motion in form of differential-algebraic equations will serve as basis for the formulation of the optimal control problem. In addition to the direct transcription methods mentioned before, a novel optimal control method based on indirect transcription will be presented. The newly proposed method conserves the Hamiltonian corresponding to the optimal control problem. It is worth mentioning that this method is directly related to previously developed energy consistent schemes in forward dynamics.
    Source Type:Doctoral Thesis
    Series:Schriftenreihe des Lehrstuhls für Numerische Mechanik
    Volume number:5
      1  7
  • Some of the metrics are blocked by your 
    consent settings
    PublicationAccess status: Open Access ,
    Elemente für die Dynamik flexibler Mehrkörpersysteme
    (2011)
    Sänger, Nicolas
    Die vorliegende Arbeit behandelt die Erstellung energie-, impuls- und drehimpulskonsistenter Zeitschrittverfahren für die Dynamik flexibler Mehrkörpersysteme mit großen Verformungen. Der vorgestellte Ansatz basiert auf nichtlinearen Finite-Elemente-Methoden für die räumliche Diskretisierung flexibler Körper. Es wird gezeigt, daß sich die Bewegungsgleichungen in Form von Algebrodifferenzialgleichungen (differential algebraic equations, DAEs) schreiben lassen. Diese DAEs erlauben eine einheitliche Systematik für eine rotationsfreie Formulierung flexibler Mehrkörpersysteme, d.h. die Verwendung von Rotationsfreiheitsgraden wird während des kompletten Diskretisierungsprozesses in Raum und Zeit vermieden. Die rotationsfreie Formulierung wiederum erleichtert die direkte Einbeziehung geometrisch exakter Schalen und Balken. Zusätzlich erlaubt sie einheitliche energie- und drehimpulskonsistente Zeitschrittverfahren für jegliche Arten von Mehrkörpersystemen. Neben der Beschreibung der Zeitschrittverfahren erfolgt die der zugrundeliegenden Strukturelemente: einer Starrkörper-, einer Balken- und einer Schalenformulierung. Bei ersterer handelt es sich um einen Spezialfall der sogenannten "natürlichen Koordinaten", zweitere ist eine rotationsfreie Balkenformulierung mit Simo-Reissner-Kinematik und letztere eine Schalenformulierung mit Reissner-Mindlin-Kinematik. Die Einbindung von Gelenken für die rotationsfreie Starrkörperformulierung wird auf eine systematische Basis gestellt, die insbesondere für eine effiziente Implementierung, aber auch für eine Bewahrung der obengenannten Erhaltungseigenschaften geeignet ist. Linienförmige Verbindungen betreffen im Rahmen der flexiblen Mehrkörpersysteme insbesondere Balken und Schalenränder. Die Behandlung von Schalenverschneidungen als flexible Mehrkörpersysteme zeigt die Universalität des gewählten Ansatzes, insbesondere da sich durch die integrale Betrachtung mit Mortar-ähnlichen Ansätzen selbst nicht-konforme Netze mühelos verbinden lassen, ohne künstliche Versteifungseffekte hervorzurufen. Auch hier wird die Wahrung der Konsistenzeigenschaften nicht beeinträchtigt. Numerische Beispiele dienen der Demonstration der hervorragenden numerischen Stabilitätseigenschaften des vorgestellten Zeitschrittverfahrens sowie der Eignung der vorliegenden Systematik für reale Problemstellungen.
    Source Type:Doctoral Thesis
    Series:Schriftenreihe des Lehrstuhls für Numerische Mechanik
    Volume number:4
      5  4
  • Some of the metrics are blocked by your 
    consent settings
    PublicationAccess status: Open Access ,
    Energy consistent time-integration of hybrid multibody systems
    (2009)
    Uhlar, Stefan
    In the present contribution we address the design of energy-consistent time stepping schemes for hybrid multibody systems, which consist of rigid and flexible parts. First focusing on rigid body dynamics, we will present a specific rotationless formulation to describe rigid body rotations. This kinematics alleviates the design of energy-consistent integration schemes which facilitate a stable numerical integration of the corresponding set of differential algebraic equations. Furthermore we will mainly focus on vital modeling features concerning rigid bodies. In this connection we present the coordinate augmentation technique, the design of explicit null space matrices for closed loop systems, the incorporation of control constraints, the modeling of dissipation and its consistent time integration, the incorporation of kinematic constraints as well as the coupling of flexible and rigid structures. The performance of the final hybrid scheme will be demonstrated with an example incorporating all modeling features listed above.
    Source Type:Doctoral Thesis
    Series:Schriftenreihe des Lehrstuhls für Numerische Mechanik
    Volume number:3
      1  4
  • Some of the metrics are blocked by your 
    consent settings
    PublicationAccess status: Open Access ,
    Higher-order accurate and energy-momentum consistent discretisation of dynamic finite deformation thermo-viscoelasticity
    (2009)
    Groß, Michael
    This paper is concerned with the energy consistent simulation of motions of a viscoelastic continuum body, under inclusion of the coupling of thermal and mechanical fields. The corresponding algorithm is based on a four-field formulation in the Lagrangian description, in which the deformation mapping, the velocity field, the temperature field and a strain-like viscous internal variable field are independent unknowns. Hence, the equations of motion are formulated in first-order form. The Lagrangian temperature field is determined by the first-order entropy evolution equation, associated with Fourier's law of heat conduction. The first-order viscous evolution equation is derived from an internal dissipation being quadratic in a nonlinear viscous strain-rate tensor. This coupled system of nonlinear differential equations is discretised by a new space-time finite element method, consisting of continuous as well as discontinuous finite element approximations in time. Owing to particular time approximations in the constitutive laws, beside the total linear momentum as well as the total angular momentum balance, a nonlinear stability estimate with respect to a relative energy function is exactly fulfilled in the fully discrete case as well. Hence, the resulting time integration algorithm is long-time nonlinear stable also when changing the time step size. The obtained coupled system of nonlinear algebraic equations is solved by a monolithic solution strategy. The corresponding Newton-Raphson methods on the global and the element level are based on a consistent linearisation. The new convergence criteria used for these iterative solution procedures take the energy consistency into account, and is free of the scaling in the independent variables. Representative numerical simulations with various boundary conditions show the higher-order accuracy and the superior stability of the new time integration algorithm.
    Source Type:Habilitation
    Series:Schriftenreihe des Lehrstuhls für Numerische Mechanik
    Volume number:2
      5  6
  • Some of the metrics are blocked by your 
    consent settings
    PublicationAccess status: Open Access ,
    Mechanische Integratoren für Kontaktvorgänge deformierbarer Körper unter großen Verzerrungen
    (2008)
    Hesch, Christianorcid-logo
    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 Thesis
    Series:Schriftenreihe des Lehrstuhls für Numerische Mechanik
    Volume number:1
      722  208