Space-Time Methods and Shape OptimizationShape Optimization
Michael Reichelt
Space-Time methods
Michael Reichelt
Shape Optimization
2 Space-Time-Formulation for the heat equation
3 Shape Optimization
Shape Optimization
Targets
Derivation of a Space-Time-Formulation for the heat equation as a model problem
Comparison to a conventional method
Space-Time methods
Michael Reichelt
Shape Optimization
Dirichlet problem for the heat equation
Let ⊂ Rd be a bounded Lipschitz-Domain and (0,T ) the time intervall of interest, then the Dirichlet problem for the heat equation is given as (α > 0) α∂tu(t, x)−xu(t, x) = f (t, x), (t, x) ∈ Q := × (0,T )
u(t, x) = g(t, x), (t, x) ∈ Σ := ∂× (0,T )
u(0, x) = u0(x) x ∈
For the variational formulation (later) it is possible to choose g ≡ 0 and u0 ≡ 0, which only affects f (homogenization).
Space-Time methods
Michael Reichelt
Shape Optimization
Variational formulation
The following variational formulation can be derived (blackboard). Find u ∈ L2(0,T ;H1
0 ()) ∩ H1(0,T ;H−1()) such that
α
∫ Q
∂tuvdq +
∫ Q
where
f ∈ L2(0,T ;H−10 ()).
Note that different ansatz and test spaces are used! (inf-sup condition)
Space-Time methods
Michael Reichelt
Shape Optimization
Heat equation: FEM
Let Th be a regular triangulation of . For the ansatz and test space S1
0 (Th) is used (which is a stable pairing). Its nodal basis shall be given by {i (t, x)}Ni=1. Then the approximation can be written as
uh = N∑ i=1
α
Shape Optimization
αThb + Khb = f ,
Shape Optimization
uh(t, x) = N∑ i=1
bi (t)i (x)
αMh∂tb + Khb = f
Where Mh and Kh are the classical spatial mass and stiffness matrices. This system system is then usually integrated by an ODE solver.
Space-Time methods
Michael Reichelt
Shape Optimization
u0(x) = sin(2πx) in (0, 1)
g(t, x) ≡ 0
0 ≤t ≤ 10
Space-Time methods
Michael Reichelt
Shape Optimization
Shape Optimization
Space-Time methods
Michael Reichelt
Shape Optimization
Space-Time methods
Michael Reichelt
Shape Optimization
ToDo
Sources
Problem
Let be two-dimensional a polynomial domain definded by its corners {ck}Kk=1, which can be represented by a vector c ∈ R2K .
w : L2(0,T ;H1 0 ()) ∩ H1(0,T ;H−1())→ R,
then the minimization problem is given by
min c∈R2K
subject to
u solves the homogenius heat equation in (c) for a given f .
Space-Time methods
Michael Reichelt
Shape Optimization
ToDo
Sources
Discretization
The homogenius heat equation is discretized for a given (c) which leads to a solution operator
R : R2K → S1 0 (Th) ⊂ L2(0,T ;H1
0 ()) ∩ H1(0,T ;H−1())
ck 7→ uh,
Space-Time methods
Michael Reichelt
Shape Optimization
min c∈R2K
Space-Time methods
Michael Reichelt
Shape Optimization
Algorithm scheme
Let an initial c be given, then the algorith scheme is as follows
Triangulate (c)
Perform step of Newton Method
Update c
Shape Optimization
Triangulation: gmsh
Shape Optimization
ToDo
Sources
Todo
Implementation of space time FEM in C++ keeping the template overloading for the geometry in mind
Newton method using FADBAD++ (straightforward)
Paralellization
Shape Optimization
Time, just another variable
Shape Optimization