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Auto-generated structured meshes for evolving domains / vorgelegt von Camilla Hahn. Wuppertal, 2021
Inhalt
Introduction
Finite element method
Elliptic boundary value problems
Elliptic partial differential equations with boundary conditions
Weak solutions of elliptic PDEs
Introduction to finite elements
Error bounds
Discretization of the linear elasticity equation via finite elements
Objective functionals
Shape optimization
Survival probabilities
Ceramic material
Low cycle fatigue
Discretization of the objective functionals
Discretization of the two-dimensional functional for ceramic material
Adjoint equation
Derivative of the objective functional
Derivative with respect to Uloc
Derivative with respect to local Xloc
Structured meshing
Introduction to mesh generation
Mesh morphing and re-meshing
Structured meshing
Two-dimensional meshing
How to treat kinks
Shape optimization on the structured mesh
Higher order finite elements
Adapting in three dimensions
Krylov subspace recycling
Krylov subspace methods
MINRES
Krylov subspace recycling
Recycling MINRES for evolving geometries
Mapping on structured meshes
Implementation and numerical results
Details of the implementation
Assembling the linear system of equations
Simple test case
Updating the mesh
The objective functionals
The functional for low cycle fatigue
Ceramic material
Numerical results
Three Dimensional Examples
Krylov subspace recycling
Test of mapping an approximate invariant subspace
Poisson equation on a turbine blade
Gradient based shape optimization with linear elasticity as governing PDE
Conclusion