Overview of Modules

Continuum-mechanical knowledge is the fundamental basis for the computation of deformation processes of solid materials. Based on the methods of tensor calculus, the lecture offers the following content:

• Vector and Tensor Algebra: symbols, spaces, products, specific tensors and definitions 
• Vector and Tensor Analysis: functions of scalar-, vector- and tensor-valued variables, integral theorem (e. g., after Gauss or Stokes
• Foundations of Continuum Mechanics: kinematics and deformation, forces and stress concepts; Cauchy’s lemma and theorem, Cauchy, Kirchhoff and Piola-Kirchhoff stress tensors
• Fundamental Balance Laws: master balance, axiomatic balance relations of mechanics (mass balance, momentum and angular momentum balances)
• Related Balance Laws and Concepts: balance of mechanical energy, stress power and the concept of conjugate variables, d’Alembert’s principle and the principle of virtual work
• Numerical Aspects of Continuum Mechanics: strong and weak formulation of the boundary value problem
• The Closure Problem of Mechanics: finite elasticity of solid mechanics (as an example), linearization of the field equations

This core course focuses on the theory and numerics of material models. Important classes of material models are investigated both for the one-dimensional and the three-dimensional context.

• Introduction to discrete and continuous modeling of materials (microstructures, homogenization techniques and multi-scale approaches)
• Fundamental theoretical concepts (basic rheology, classification of the phenomeno- logical material response, elements of continuum thermodynamics)
• Fundamental numerical concepts (discretization techniques for evolution systems, linearization techniques, and iterative solution of nonlinear systems)
• Linear and nonlinear elasticity, damage mechanics, viscoelasticity (linear and nonlinear models, stress update algorithms and consistent linearization)
• Rate-independent plasticity (theoretical formulations, return mapping schemes, incremental variational formulations, consistent elastic-plastic tangent moduli)
  Viscoplasticity (classical approaches and overstress models)
• Material stability, failure analysis (nonlocal modes, discrete failure mechanisms) 

• direct stiffness method
• isoparametric concept
• variational formulation of finite elements, mixed variational principles, shape functions, approximation spaces and mathematical convergence requirements
• finite elements for trusses, beams, plates and solids
• locking, reduced integration, mixed and hybrid finite element methods
• modelling in structural mechanic, mathematical model and numerical model (discretization)
• interpretation of numerical results

• Dr.-Ing. Martin Bernreuther
• Dr.-Ing. André Schmidt

Discretization Methods : 

The lecture deals with the numerical treatment of differential equations 
which arise from different mechanical and thermodynamical problems. 
Contents are: 

• deduction of differential equations based on the principles of 
  mechanics and thermodynamics and their classification 
• the Finite Difference Method 
• the method of weighted residuals: method of subdomains, collocation 
  method, least squares method, and Galerkin's method 
• the Finite Element Method 
• different time integration schemes 
• convergence and stability

Introduction to Scientific Programming:

part I: layout of a computer 

• von Neumann architecture 
• design of modern microprocessors, memory hierarchy , parallelism 
• programming languages 

part II: algorithms and data structures 

• complexity, Bachmann-Landau notation 
• example sorting algorithms (sub-topic: recursion)
• arrays,lists, hashtables, trees (binary, KD, Quadtree, Octree), Heap 
• graphs (exemplary algorithms Cuthill McKee, Dijkstra) 

part III: numerics 

• number representation intager and floating point 
• rounding and rounding errors 
• condition and stability 
• matrices and linear mappings (demonstrative meaning, effect on 
  geometric objects) 
• solving of linear systems of equations (Gaussian elimination, LU- 
  decomposition, Pivoting, Cholesky decomposition) 
• polynomial interpolation (different bases and algorithms: Lagrange, 
  Newton, Aitken-Neville, divided differences, error estimation, 
  condition) 
• spline interpolation & parametric curves (Beacute;zier)

• Formulation of the optimization problem: optimization criteria, scalar optimization problem, multicriteria optimization
• Sensitivity Analysis: Numerical differentiation, semianalytical methods, automatic differentiation
• Unconstrained parameter optimization: theoretical basics, strategies, Quasi-Newton methods, stochastic methods
• Constrained parameter optimization: theoretical basics, strategies, Lagrange-Newton methods

• Prof. Dr. rer. nat. Dr. h.c. Siegfried Schmauder 
• Prof. Dr.-Ing. Jan Hofmann
• Prof. Dr.-Ing. habil. Christian Moormann

Metals:

• Fundamentals of dislocation theory
• Plastic deformation of metals
• Possibilities of strengthening
• Influences on behaviour of material

Concrete:

• Properties of concrete
• The behaviour of concrete under compressive loading
• The behaviour of concrete under tensile loading
• Time dependent behaviour
• Special concretes

• Soils:
• Stresses in soils
• Stiffness of soils
• Strength of soils

The course covers variational formulations, various locking phenomena and alternative formulations for finite elements and advanced discretiza-tion schemes.

• variational formulation of finite elements, mixed variational principles
•  geometrical and material locking effects in structural and solid mechanics
• hybrid-mixed and enhanced assumed strain finite element for-mulations, reduced integration and stabilization, DSG method, u-p formulations
• patch test, stability, convergence
• linear and non-linear analyses
• locking effects and their avoidance in advanced discretization schemes, like isogeometric analysis 

• Provide overview of fracture mechanics
• Show link between fracture mechanics, strength of materials and damage mechanics
• Create interest in the diverse world of fracture phenomena

• Dynamic crack branching
• Anticrack initiation
• Crack pattern analysis
• Mixed mode crack growth and among others

• Thermodynamics of a general internal variable formulation of inelasticity at small strains
• Linear and nonlinear elasticity (isotopic spectral forms, anisotropic models based on structural tensors)
• Finite Element implementation of nonlinear elasticity
• Viscoelasticity (linear and nonlinear models, stress update algorithms and consistent linearization)
• Rate-independent and rate-dependent plasticity (theoretical formulations, stress update algorithms and local variational formulations, consistent linearization)
• Micromechanically-based models of plasticity for crystalline solids
• Introduction to homogenization methods and micro-to-macro transitions
• General internal variable formulation of inelasticity at large strains
• Approaches to the modelling and numerics finite elasticity and finite viscoelasticity

Discretization methods:

• Knowledge of the common methods (finite differences, finite elements, finite volume) and the differences between them
• Advantages and disadvantages and of the methods and thus of their applicability
• Derivation of the various methods
• Use and choice of the correct boundary conditions for the various methods

Time discretization:

• Knowledge of the various possibilities
• Assessment of stability, computational effort, precision
• Courant number, CFL criterion

Transport equation:

• Various discretisation possibilities
• Physical background
• Stabiity criteria of the methods (Peclet number)

Choice of a grid 
Overview of discretazation techniques on the basis of the stationary groundwater equation:

• Finite differences
• Finite volume (integral finite differences)
• Finite elements

Time discretisation on the basis of the instationary groundwater equation:

• Explicit and implicit methods

Discretisation of the transport equation:

• Central difference methods
• Upwinding

Introduction to stability analysis, convergence 
Clarification of concepts: model, simulation 
Derivation of the finite element method 
Application of the finite element method to the stationary groundwater equation 
Setting-up of a simulation programme for modeling groundwater:

• Programme requirements
• Programming individual routines

Fundamentals of programming in C:

• Control structures
• Functions
• Arrays
• Debugging

Visualisation of the simulation results

The course covers the theory of non-linear structural mechanics and 
corresponding discretization methods and algorithms with a focus on the 
finite element methods.

• basic principles, phenomena and concepts of structural mechanics
• non-linear strain measures and stress measures
• large deformations, stability problems
• methods and algorithms of non-linear structural mechanics
• iteration methods and path following techniques
• stability analysis, buckling problems 

This course provides an introduction to intrinsically nonlinear phenomena in dynamical systems. The main focus of this course is on differential geometric methods. Applications will include problems from nonlinear control, optimization and mechanics. Some of the topics covered in this course are:

• Nonlinear vector fields and flows
• Stability and bifurcations
• Lie brackets and commutativity of flows
• Manifolds, vector fields on manifolds and integrability
• Center manifolds and advanced stability theory
• Applications in Control, Mechanics and Optimization
• Extremum Seeking Control

The course Nonsmooth Dynamics is concerned with the mathematical description and numerical simulation of mechanical systems with unilateral constraints.

Nonsmooth mechanical systems are characterized by jumps in the velocities or accelerations of the bodies in contact. Examples are systems with impact and friction. Mathematical concepts from convex and nonsmooth analysis, which are essential for the description of nonsmooth systems, are discussed in the first part of the course. Topics such as convex sets and functions, the subdifferential and set-valued functions are introduced. The second part of the course discusses set-valued force laws. Special attention is paid to Coulomb friction, both in the planar and spatial case, and to Newtonian impact laws. The third part of the course is concerned with nonsmooth dynamical systems. Numerical methods for the simulation of nonsmooth systems are explained and illustrated with example problems.

Topcis:

• Different types of differential equations (ordinary differential equations, partial differential equations)
• Existence of (unique) solution
• Algorithms for solving differential equations
• Theoretical analysis of numerical schemes
• Implementation of numerical schemes within MATLAB

Goal:

• knowledge about the theory of ordinary and partial differential equations
• overview about numerical solution methods for these problems
• ability to link a given problem and the appropriate method
• ability to program solution methods in MATLAB

The lecture's goal is to improve understanding of general soil behaviour and its numerical treatment. The course will cover the fundamentals of elasticity and plasticity, as well as more advanced constitutive models used not only in typical industrial applications but also in research. The course will not only provide a theoretical framework for the constitutive laws, but will also demonstrate their implementation in a simple Finite Element software written in FORTRAN. In addition to the lectures, tutorials on the commercial finite element software Plaxis™ will be provided. Candidates will be expected to work independently to simulate geotechnical structures and present their findings in a 10-minute presentation. The course will cover the following topics:

• Introduction to Plasticity
• Different yield criterions used in soil mechanics : Drucker-Prager, Matsuoka-Nakai, etc.,
• Modified Cam Clay model
• Hardening soil model
• Hypoplastic soil model
• Programming of constitutive laws in Fortran
• Simulating geotechnical structures using Plaxis™

• Prof. Dr.-Ing. Marc-André Keip
• Prof. Dr.-Ing. Holger Steeb

The ultimate goal of the lecture to foster the understanding of general inelastic material behavior with regard to the theoretical modeling and the numerical treatment based on selected model problems. For example, the selected material models under consideration may cover (i) micromechanically motivated approaches to inelastic material response such as crystal plasticity or (ii) purely phenomenological formulations of an inelastic material response such as viscoelasticity. Contents:

• Introduction to inelastic material behavior
• Micromechanical structure of solids
• Kinematics of inelastic deformations at finite strains
• Foundations of continuum-based material modeling for selected problems, e.g. finite crystal plasticity and viscoelasticity
• Integration algorithms of evolution systems, stress-update algorithms and consistent linearization of updating schemes

Selected Topics in the Theories of Plasticity and Viscoelasticity [16100] from Chair of Material Theory

Selected Topics in the Theories of Plasticity and Viscoelasticity [16100] from Chair of Continuum Mechanics

• Prof. Dr.-Ing. Marc-André Keip

The lecture provides an in-depth perspective on the formulation and algorithmic implementation of material models for the description of physically and geometrically nonlinear deformation and failure mechanisms of solids. Material models of elasticity, visco-elasticity, plasticity as well as damage and fracture at finite deformations are covered in the course. This also includes non-mechanical effects such as thermomechanical or electromechanical coupling. In addition to continuum mechanical models, discrete modeling approaches on different space and time scales are presented, and fundmental concepts of multi-scale models and mathematical homogenization techniques are addressed. The lecture covers theoretical and numerical aspects in a integrated sense. For example, model-specific algorithms for time integration, global solvers for coupled nonlinear field equations as well as different finite element formulations for the spatial discretization of nonlinear material models and discontinuities are considered. Many of the presented developments and methods are current topics in research. A specification and orientation of this broad subject based on the interests of the audience is possible. Contents:

• Direct variational methods of finite elasticity and uniqueness
• Anisotropic finite elasticity and isotropic tensor functions
• Damage models and elements of fracture mechanics
• Finite elasto-visco-plasticity of metals and polymers
• Discrete models: particle methods and dislocation dynamics
• Multi-scale models and numerical homogenization techniques
• Material instabilities, phase transitions and microstructures

Jun.-Prof., Ph.D. Carina Bringedal

The course covers examples of multiscale models, in particular from porous media, together with analytical and numerical multiscale methods to simplify and simulate such models. The course is structured into three parts; first part is on multiscale models, where the focus is to describe their multiscale characteristics. The second part is on (analytical) multiscale methods, where matched asymptotic expansions and averaging techniques such as volume averaging and homogenization are covered. The theoretical justification of homogenization through two-scale convergence is also included. The third part is on multiscale numerical methods, where discretization methods such as multiscale finite element methods and finite volume methods are covered, and the idea behind heterogeneous multiscale methods is described. The course is mathematically oriented, but the main part of the content is covered through applying these methods to examples, including implementing multiscale discretizations. The following topics will be covered:

    • Multiscale models in porous media
    • Phase-field models
    • Averaging techniques
    • Homogenization Ansatz
    • Use of asymptotic expansions
    • Two-scale convergence
    • Multiscale FEM
    • Multiscale FV
    • Heterogeneous multiscale methods
    • Adaptive two-scale methods 

Lecturer

Dr.-Ing. Albrecht Eiber 

Content

Inhalte der Vorlesung sind: Einführung, Problemstellung, Aufgaben; Modelle und Modellbildungsverfahren (Gewebe, Muskeln, Knochen); Sensorik; Motorik; Messverfahren (zur Parameteridentifikation und Diagnose); Simulation von Bewegungsabläufen sowie natürlicher und pathologischer Funktionen; Rekonstruktion gestörter Funktionen (Gelenke, Körperteile); Beispiel: das natürliche, pathologische und rekonstruierte Gehör (siehe auch unten im Abschnitt Inhalt der Vorlesung ).

ECTS points

3

• Introduction to kinematics of contact, Signorini conditions
• Weak and strong forms of a contact problem
• Spatial discretization
• Global and local contact search
• Global solution algorithms: active set and complementarity algorithms
• Treatment of contact for explicit time integration
• Treatment of contact for implicit time integration
• Mesh tying techniques

Lecturer

Prof. Dr. C. David Remy

Description

This course covers the description of kinematics and dynamics of multibody systems as they are typical for applications in robotics, mechatronics, and biomechanics.

The course provides the theoretical background to describe such systems in a precise mathematical way, while it also pays attention to an intuitive physical understanding of the underlying dynamics. It discusses the tools and methods necessary to create the governing differential equations analytically and it covers a range of computational algorithms that do so in a numerically efficient way. Special attention is paid to the handling of closed loops, collisions, and variying structure.

As part of the exercises accompanying this course, the students will implement their own multibody dynamics engine in MATLAB, using advanced programming techniques that include recursive formulations and object oriented programming. The gained knoweldege will enable a creative approach to the design and control of robotic systems. It will enable the students to debug their own solutions more intuitively and understand what is going on when using off-the-shelf software for design or analysis.

ECTS points

6

Lecturer

Prof. Dr. rer. nat. Dr. h.c. Siegfried Schmauder 

Content

The theoretical foundations of Monte Carlo (MC), Molecular Dynamics (MD) and other advanced simulation techniques with respect to atomistic phenomena in computational materials science, such as, e.g., precipitation strengthening in steels. Another focus is put on dislocation theory including the dislocation dynamics and the applications for the understanding of the local deformation processes in metallic materials. Finite-Element-methods, crystal plasticity and damage mechanical modelling are further essential topics in this course.

ECTS points

6

The course covers design and analysis of shells, membrane and shell theory as well as mathematical and computational models for analysis of shells. The theoretical contents is supplemented and exemplified with applications of commercial finite element software.

• Historical overview of shell theory
• Geometrical basics and load carrying behavior
• Shell models, Prerequisites and assumptions
• Membrane theory, basic equations and analytical solutions for shells of revolution
• Computation of stress resultants and displacements
• Bending theory, analytical solutions for cylindrical shells
• Computational models for shells with arbitrary geometry, shear deformable (Reissenr/Mindlin) shell finite elements
• Non-linear analysis and stability
• Application of shell elements using commercial codes

• Biological tissue as a porous medium: the intervertebral disc as a porous medium, basic concepts and fundamental equations of the Theory of Porous Media.
• Material modelling: basic concepts and principles of material modelling, material symmetry, symmetry groups, invariants.
• Strain energy functions for selected material types: mechanical characteristics of soft tissues, rubber-like materials, Fung-type material laws, passive and active behaviour of the heart muscle.
• Student presentations on recent research studies related to the modelling of biological tissues.

Motivation
Data is omnipresent in engineering and science. The processing of field data and simulation results is an essential tool in knowledge building. In recent years the focus on data-oriented modeling an simulation has gained significant momentum and it has led to the establishment of this discipline as a field in its own right.

Scope
The course teaches basic elements in data acquisition, preparation, analysis, visualization and data-based/-assisted modeling. The content is particularly designed for the use in engineering and in science (mathematics, physics, chemistry). A part of the lecture is designated to the construction of data-driven surrogate models using kernel interpolation and regression and artificial neural networks.

Selected topics
denoising • filtering • statistical characterization • image processing • dimensionality reduction • Fourier Transform • Kernel interpolation and regression • introduction to artificial neural networks

The lecture deals with flow in natural hydrosystems with particular emphasis on groundwater / seepage flow and on flow in surface water / open channels. Groundwater hydraulics includes flow in confined, semi-confined and unconfined groundwater aquifers, wells, pumping tests and other hydraulic investigation methods for exploring groundwater aquifers. In addition, questions concerning regional groundwater management (z.B. recharge, unsaturated zone, saltwater intrusion) are discussed. Using the example of groundwater flow, fundamentals of CFD (Computational Fluid Dynamics) are explained, particularly the numerical discretisation techniques finite volume und finite difference. The hydraulics of surface water deals with shallow water equations / Saint Venant equations, unstationary channel flow, turbulence und layered systems. Calculation methods such as the methods of characteisitcs are explained. The contents are:

• Potential flow and groundwater flow
• Computational Fluid Dynamics
• Shallow water equations for surface water
• Charakteristikenmethode
• Examples from civil and environmental engineering

• Fundamentals of Fuzzy theory
• Fuzzy logic
• Fuzzy systems
• Fuzzy control
• Fuzzy arithmetic
• Fuzzy cluster analysis

The knowledge of continuum mechanics and continuum thermodynamics is the fundamental requirement for the theoretical and algorithmic understanding of geometrically and physically nonlinear deformation, failure and transport processes in solids consisting of metallic, polymer or geological materials. This course offers a presentation of fundamental concepts of continuum mechanics and constitutive theory at finite elastic and inelastic strains. The chosen formulation accentuates geometrical aspects based on the modern terminology of differential geometry, which also includes the description of multi-field theories with thermo- and electromechanical coupling. Algorithmic aspects of the computer implementation of nonlinear continuum mechanics models are covered in parallel to the theoretical description.

Contents:

• Tensor algebra and -analysis on manifold
• Differential geometry of finite deformations
• Balance principles of nonlinear continuum thermodynamics
• Phenomenological material theory at finite strains
• Uniqueness of boundary value problems and stability theory

• principal structure of a finite element code
• pre- and post-processing, software engineering in the context of finite element programs
• integration of element stiffness matrices and load vectors, implementation of boundary conditions
• assembly of stiffness matrices
• solution of linear systems of equations
• storage formats for sparse matrices

Fundamental knowledge of nonlinear continuum thermodynamics is a crucial prerequisite for the description of large deformations of arbitrary materials with nonlinear constitutive laws. The lecture provides a systematic representation of nonlinear continuum mechanics and the basics of thermodynamics (energy balance, entropy inequality). Proceeding from the fundamental principles of constitutive theory and the 2nd law of thermodynamics, the procedure for the derivation of thermodynamic consistent and admissible material models is described. All methods are exemplarily applied for the description of a nonlinear deformable, thermoelastic solid. Moreover, some aspects of the numerical treatment of nonlinear processes in space and time are discussed. In particular, the lecture comprises the following topics:

• Motivation and introduction of the problem
• Nonlinear continuum mechanics: kinematics, transport theorems, nonlinear deformation and strain measures in absolute and convective notation
• Stress tensors of Cauchy, Kirchhoff, Piola-Kirchhoff, Biot,Mandel and Green-Naghdi
• Mechanical balance relations: balances of mass, linear momentum and angular momentum
• Thermodynamic balance relations: energy balance and entropy inequality (1st and 2nd law of thermodynamics)
• Elements of classical thermodynamics: internal energy and caloric state variables, thermodynamic potentials, Legendre transformations
• Thermodynamic materials theory: thermodynamic principles and process variables, material symmetry
• Thermoelastic solid: evaluation of the entropy principle, isotropy, the coupled problem of thermomechanics, thermoelasticity in nominal form, energy and entropy elasticity
• Numerical aspects: weak form of the boundary-value problem, time integration of coupled problems, linearization of the field equations, stability criteria

•  Motivation: necessity for model order reduction in numerical studies; properties of parameterized mechanical systems (with examples)
• Continuum mechanical foundations: Heat conduction (stationary; instationary), Discrete mechanical systems (spring-mass-systems) elasto statics
• matrix algebra (eigenproblems/SVD, ...); formal definitions of function spaces
• substructuring techniques
• definition of local and global measures of the approximation error
• proper orthogonal decomposition (POD)
• reduced basis methods for linear time invariant problems (LTI)
• reduced basis methods for linear time dependent problems
• introduction to model order reduction of nonlinear systems
• numerical aspects of model order reduction for nonlinear problems

The modeling approaches are rooted in micromechanics, mostly phenomenological, and build on the framework of continuum mechanics and the thermodynamically-consistent formulation of constitutive equations as taught in earlier courses. This framework, which accounts for thermomechanical coupling, is extended, where necessary, to include electric and magnetic coupling effects. The lecture covers the following topics:

• Introduction to continuum mechanics and basic modeling concepts
• Numerical solution of partial differential equations with the Finite Element Method in one dimension
• Maxwell Theory (electro-magnetism)
• Modeling of electro-mechanically coupled materials
• Modeling of magneto-mechanically coupled materials 

Using complex models in engineering practice requires well-founded knowledge of the characteristics of discretisation techniques as well as of the capabilities and limitations of numerical models, taking into account the respective concepts implemented and the underlying model assumptions. The contents are: 

• Theory of multiphase flow in porous media
• Derivation of the differential equations
• Constitutive relations

Numerical solution of the multiphase flow equation

• Box method
• Linearisation
• Time discretisation

Multicomponent systems

• Thermodynamic fundamentals and non-isothermal processes

Application examples:

• Thermal remediation techniques
• CO2 storage in geological formations
• Water/ oxygen transport in gas diffusion layers of fuel cells

In many practical control problems it is desired to optimize a given cost functional while satisfying constraints involving dynamical systems. These kind of problems typically fall into the area of optimal control, a centerpiece of modern control theory. This course gives an introduction to the theory and application of optimal control for linear and nonlinear systems. Topics covered in the course are:

• Nonlinear programming approach
• Dynamic programming
• Model predictive control
• Pontryagin maximum principle
• Applications

The course is intended to students having visited SGRT/ERT lectures.

• Dr.-Ing. Arndt Wagner
• Prof. Dr.-Ing. Holger Steeb

Erdbeben führen als unvermeidbare und derzeit nur schwer vorhersagbare Naturkatastrophen zu schwerwiegenden Folgen in den betroffenen Gebieten. Die Vorlesung gibt eine Einführung in die Technik des erdbebensicheren Bauens in theoretischen und konstruktiven Belangen. Insbesondere soll der Blick für den erdbebengerechten Entwurf von  Hochbauten geschärft werden. Der Inhalt der Veranstaltung gliedert sich hierbei wie folgt:

• Erdbebenentstehung, seismische Grundlagen (Plattentektonik, seismische Wellen, Erdbebenskalen), Erdbebenfolgen und Erdbebenbeanspruchung
• Schwingungen mit einem Freiheitsgrad, freie ungedämpfte und gedämpfte Schwingung, erzwungene Schwingungen, Resonanz, Faltungsintegral
• Schwingungen mit mehreren Freiheitsgraden, modale Koordinaten, Modalanalyse
• Antwortspektren der Relativverschiebung, Relativgeschwindigkeit und Absolutbeschleunigung, Bemessungsgrundlagen nach DIN 4149 bzw. EC 8
• Bauliche Aspekte, erdbebengerechter Entwurf, typische Schadensmuster, konstruktive Manahmen für erdbebensicheres Bauen (Grundriss, Aufriss, Gründung,Massenverteilung)
• Modellbildung, Ersatzstabmodell, Modell der starren Stockwerksscheiben
• Zeitverlaufsverfahren, numerische Integration der Schwingungsdifferential- gleichungen, Newmark-Verfahren
• Ausblick: weitere Methoden zur Erdbebensimulation

• History of Computers
• Finite-Element-Method
• Molecular Dynamics (MD)
• Integrators
• Different Ensembles: Thermostats, Barostats
• Observables
• Simulation of quantum mechanical problems
• Solving the Schrödinger equation
• Lattice models, Lattice gauge theory
• Monte-Carlo-Simulations (MC)
• Spin Systems, Critical Phenomena, Finite Size Scaling
• Statistical Errors, Autocorrelation

In addition to purely mechanical questions, the Finite Element Method (FEM) can also be used to address more complex questions with coupled field equations. Examples include: thermo-mechanical couplings, electro-mechanical couplings, chemical-mechanical couplings or combinations thereof. The treatment of these problems requires the development of coupled material equations, which do not contradict the thermodynamic principles. Furthermore, the system of equations may be extended by an additional process variable, e.g. the temperature, the electric field or a chemical state variable, which negatively influence the numerical solution properties in the context of the finite element approximation. For a stable solution of coupled problems using the finite element method, it is necessary to formulate thermodynamically consistent material equations, to develop advanced finite element formulations and to use suitable numerical solution methods. The lecture will be complemented with computer pool exercises.

As a conceptual approach for the treatment of discrete multicomponent materials, the Theory of Porous Media (TPM) is presented. The conceptual procedure for the development of thermodynamically consistent material equations is discussed. The resulting system of equations is solved numerically using the finite element method (FEM). Due to the mostly strongly coupled and non-linear nature of the system of equations to be solved, special element formulations are presented.

• Motivation and overview
• Introduction to the Theory of Porous Media (TPM)
• Development of thermodynamically consistent material equations
• Continuum mechanical treatment
• Example: fluid-saturated porous solid
• Preparation of the coupled model equations for numerical implementation
• Verification of the model concept with sample calculation

The first part of the lecture communicates the fundamentals of vehicle dynamics via selected contributions in research. In doing so, the mechanical modeling and the mathematical description are concerned with vehicle systems for ground transportation that contain vehicle superstructures, support and guidance systems and tracks. In the second part, occupant protection systems in vehicles are introduced from industrial practice such as airbags and belt restraint systems, with all steps from modeling via simulation to experimental verification being treated.

Part I of the lecture is based on a modularization of the vehicle substructures with standardized interfaces: vehicle superstructures, support and guidance systems and track descriptions are the elements of complete vehicle-track-systems, which are supplemented by assessment criteria for the human vibrational feeling. The theoretical methods are clarified by examples of simple longitudinal, lateral and vertical motions.

Part II of the lecture gives attention to the fundamentals of occupant protection systems and the modeling of airbags and belt systems in a full vehicle. The experiments from subsystem testing towards crash tests are introduced. They are followed by the design of restraint systems. The excursion to TRW Automotive in Alfdorf delivers insight into the industrial practice.

Contents:

• Cluster Methods
• Discretization of Newtonian mechanics
• Molecular dynamics: linked lists, parallelization
• Lattice Boltzmann
• Advanced simulation methods
• Partial differential equations: Schrödinger equation