maximize

Numerical Simulation (V4E2)

Course numbers 611500602 (Lecture) and 611700602 (Tutorial)

Lecture: Prof. Dr. Martin Rumpf, Tutorial: Stefan W. von Deylen, Benedict Geihe


Date

Lecture Tuesday, 10 (c.t.) - 12 Seminar room 0.011, Endenicher Allee 60
  Thursday, 8:30 - 10:00 Seminar room 0.011, Endenicher Allee 60
 
Tutorial Monday 16 (c.t.) - 18 Seminar room 0.007, Endenicher Allee 60

Tutorial

Problem sheets Solutions
Notes on partial integration
Sheet 01: 19.04.2010 Solutions
Sheet 02: 26.04.2010 Solutions
Sheet 03: 03.05.2010 Solutions
Sheet 04: 10.05.2010 Solutions
Sheet 05: 17.05.2010 Solutions
Sheet 06: 31.05.2010 Solutions
Sheet 07: 14.06.2010 Solutions
Notes on differential geometry
Sheet 08: 21.06.2010 Solutions
Sheet 09: 28.06.2010 Solutions
Sheet 10: 12.07.2010 Solutions

Lab

Documentation (doxygen)
Reference
Sheet 1: 28.06.2010
Sheet 2: 05.07.2010
Sheet 3: 12.07.2010

Content

This course together with the accompanying exercises will give an introduction to optimization with PDE constraints.
As a guiding example elastic shape optimization will be investigated in detail starting from modeling issues and the related modern foundation in analysis to the numerical discretization and the convergence of the approximating models.
A couple of numerical approaches will be derived and in detail discussed ranging from elastic domains described by a finite number of shape parameters to the implicit description of shapes as level sets of a scalar function or as one phase in a diffusive interface model.


Optimal shapes for different loading scenarios characterized by fine scale geometric details and
optimal shaping of holes in a perforated elastic plate with a color coded stress distribution.

A particular emphasize will be on the multi-scale characteristics of optimal shapes. Typically, during the optimizing fine scale geometric structures show up and the actual solution of the shape optimization problem is known to exist only in a generalized sense taking into account macroscopic elastic objects with a spatially varying microstructure.


Aluminum foam in a polymer matrix and microstructure of bones
with corresponding stress distribution under a specific load

In the lectures efficient methods for the numerical treatment of such multi-scale problems will be presented and then used in the context of shape optimization.
There will be an exercise course linked to the lecture which will allow to deepen the knowledge and to practice the discussed tools in analysis, numerical analysis, and algorithm development.