Extended and Mesh Free Finite Element Methods for Boundary Value Problems with Discontinuities and Multiscale Methods for Fracture (long seminar without exam)

Over the past decades, finite element methods have been developed into one of the most general
and powerful class of techniques to solve boundary value problems governed by partial differential
equations in order to study, predict and model the behavior of structures, materials, processes and
fluids. Within this course, extended finite element and mesh free methods will be presented to handle
problems with arbitrary strong and weak discontinuities. In these methods, the approximation
space of the test and trial functions are modified such that arbitrary discontinuities can be handled.
Afterwards, details of atomistic models, continuum models and possible coupling techniques in
statics and dynamics are presented in relation to the problem of fracture of materials. Techniques
of adaptive adjustment of the atomistic region based on the propagation of defects at the nanoscale will be addressed.

In detail, the following topics will be covered:

? The partition of unity and its relation to completeness
? Lagrangian and Eulerian kernel functions
? Weak and strong form and weak and strong discontinuities
? PUFEM, EFG, RKPM and DRKPM and their shape functions
? Level sets and signed distance function
? Nodal, stress­point and cell­integration in a meshfree method
? Principles and implementation procedure of XFEM
? Principles of embedded and interface elements
? Modeling aspects of atomistic simulations in statics and dynamics
? Computer algorithms of atomistic simulations
? Modeling fracture in the continuum based on the phantom node method and virtual atom
cluster models
? Coupling techniques of continuum and atomistic regions
? Adaptive adjustment of the atomistic region
? Coarse graining and refinement techniques
? Applications to realistic materials