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Course Contents

For computing effective properties of heterogeneous materials with complex mi-
crostructures, modern computational techniques are imperative. The course provides
an introduction to modern numerical discretization and solution methods which
are based on the fast Fourier transform (FFT) and enable treating industrial-scale
microstructures and nonlinear mechanical material behavior in an efficient manner.
The course acquaints its participants to topics of current research, and is offered
exclusively in Essen at master’s level. The goal of the accompanying exercise sessions
is implementing a prototypical FFT-based micromechanics solver.

Syllabus

• Basic equations for computing effective elastic material properties
Asymptotic homogenization of linear elasticity for periodic microstructures; the
elastic cell problem to determine the effective stiffness tensor; properties of the
effective stiffness tensor; Lippmann-Schwinger formulation of the cell problem
of elasticity
• The FFT-based computational homogenization method of Moulinec-Suquet
The Lippmann-Schwinger equation as numerical solution method (basic
scheme); optimal choice of the reference material; voxel structure of micro-
computed tomography images and challenges for classical finite element solvers;
Fourier series representation of solution fields and the fast Fourier transform;
discretization of the Lippmann-Schwinger equation by trigonometric collocation;
mixed strain-stress boundary conditions for direct comparison with experiments;
problems and limits of the Moulinec-Suquet method
• Procedure for the treatment of materials with high contrast, pores or
imperfections
Eyre-Milton formulation of the cell problem of elasticity; associated solution
method; optimal choice of the reference material; the conjugate gradient method
for the Lippmann-Schwinger equation; finite differences and finite element
discretizations; optimal choice of discretization scheme and solution method for
selected examples
• Nonlinear and time-dependent mechanical problems
Formulation of time-dependent mechanical homogenization problems; time
discretization; the basic scheme in the nonlinear case - interpretation as gradient
descent method; the Newton-Raphson method for the Lippmann-Schwinger
equation

see also https://www.uni-due.de/ingmath/courses.php

[1] Milton, G. W.: The Theory of Composites. Springer, New York, 2002.