Research Group of Prof. Dr. D. Peterseim
Institute for Numerical Simulation

Research Projects

Numerical Homogenization

Numerical homogenization refers to a class of numerical methods for PDEs with multiscale data aiming at the determination of macroscopic (effective) approximations that account for the complexity of the microstructure. The possible added value of computational homogenization when compared with classical analytical techniques is its applicability, reliability, and accuracy in the absence of strong (unrealistic) assumptions such as periodicity and scale separation. We are developing new methods with the aims of efficiency and reliability for representative classes of multiscale problems in the context of high-contrast, strong anisotropy and uncertainty.

DFG Priority Programme 1748: Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis
Adaptive isogeometric modeling of propagating strong discontinuities in heterogeneous materials
Cooperation with Dr.-Ing. Markus Kästner, TU Dresden

Multi-material lightweight designs and smart devices with characteristic microscopic material structures are the key features for the development of innovative products. In this context, an adaptive isogeometric framework for the modeling and simulation of crack propagation in heterogeneous materials is to be developed, implemented, and mathematically analyzed in this project. The mechanical modeling of interface failure will be based on increasing knot multiplicities driven by cohesive zone models for crack propagation along material interfaces. In addition, a phase-field model will account for propagating cracks in the bulk material including interaction phenomena such as crack branching and coalescence. The spline-based discretization used offers higher efficiency compared to Lagrangian polynomials, control of regularity, accurate approximation of strong gradients in the phase-field order parameter, as well as the possibility to discretize higher-order phase-field equations. Local mesh adaptivity required for the resolution of material interfaces and the phase-field variables will be provided by T-splines as well as hierarchical spline approximations. In addition to the physical modeling, open mathematical problems include a practicable characterization of T-meshes suitable for IGA in 3D and clear understanding of the role of increased regularity in the approximation.

Linear and Nonlinear Eigenvalue Problems

Our new techniques for computational homogenization for linear elliptic problems yield promising results also for elliptic eigenvalue problems with possibly very rough data. Those results show that numerical upscaling may be performed in such a way that the homogenized (effective) operator preserves small eigenvalues extremely accurate. This observation has surprising applications, e.g., the computation of ground states of Bose-Einstein condensates in quantum chemistry. This research is continued in the context of non-linear Schrödinger equations and other classes of linear and nonlinear (polynomial) eigenvalue problems, for example the mechanical analysis of damped vibrating structures.

Wave Propagation and Scattering

The numerical simulation of acoustic or electromagnetic high-frequency wave propagation is still among the most challenging tasks of computational partial differential equations. The highly oscillatory nature of the solution plus a wave number dependent pollution effect put very restrictive assumptions on the smallness of the underlying mesh. Typically, this condition is much stronger than the minimal requirement for a meaningful representation of highly oscillatory functions from approximation theory. We develop pollution-free methods for homogeneous and heterogeneous frequency domain models as well as adaptive and parallel solution strategies in the time-domain. We expect that these techniques will be very useful in the context of parameter studies and inverse problems and we aim at their transfer to geophyiscal and medical applications.

Adaptive Algorithms for Partial Differential Equations

Nowadays, a posteriori error estimation is commonly used for the control of adaptive mesh refinement or, in general, the control of the adaptive enrichment of finite element spaces. We develop estimators and refinement/enrichment strategies with the aim to minimize computational complexity. We emphasize, however, that for many singularly perturbed or parameter dependent problems the condition "the mesh width has to be sufficiently small" still arises. The generation of optimal initial meshes is, hence, of utmost importance and we develop new concepts for this purpose.

DFG Research Center MATHEON
Project C33: Modeling and Simulation of Composite Materials

Composite materials are engineered materials made from two or more constituent materials with significantly different physical properties. In a typical configuration, (hard) filler particles are surrounded by a matrix of a second material which binds the filler particles together. Because the characteristics and relative volumes of both the matrix material and the various filler particles can be manipulated, these materials show an almost infinite range of physical properties. The aim of this project is to develop efficient numerical methods for the simulation of material responses. In addition, the analysis of the corresponding discrete models shall lead to a better understanding of how composites behave and how certain properties depend on the controllable variables. Both simulation and analysis provide the opportunity to develop materials with enhanced performance for the particular industrial applications.