How to Get in Better Shape (Mathematically) In search of efficient - PowerPoint PPT Presentation

How to Get in Better Shape (Mathematically) In search of efficient procedures for engineering shape design Toni Lassila toni.lassila@epfl.ch Modelling and Scientific Computing Institute of Mathematics Institute of Analysis and Scientific Computing Helsinki University of Technology ´ Ecole Polytechnique F´ ed´ erale de Lausanne SIAM CS&E 2009, Miami Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 1 / 23

Outline Shape optimization of PDE-modelled systems Why more efficient shape design procedures are required? Application: Airfoil inverse design Free-form deformations for parametric shape design The idea of reduced basis methods for parametric PDEs Results and conclusions Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 2 / 23

Shape Optimization of PDE-modelled Systems Ingredients Geometry State equations Optimization Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 3 / 23

Shape Optimization of PDE-modelled Systems Ingredients Geometry Computational geometry (triangular/quadrilateral mesh) State equations Optimization Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 3 / 23

Shape Optimization of PDE-modelled Systems Ingredients Geometry Computational geometry (triangular/quadrilateral mesh) Parametric shapes (splines, mesh deformations, basis shapes) State equations Optimization Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 3 / 23

Shape Optimization of PDE-modelled Systems Ingredients Geometry Computational geometry (triangular/quadrilateral mesh) Parametric shapes (splines, mesh deformations, basis shapes) State equations PDE model equation (elasticity, Stokes, Navier-Stokes, Helmholtz, Maxwell) Optimization Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 3 / 23

Shape Optimization of PDE-modelled Systems Ingredients Geometry Computational geometry (triangular/quadrilateral mesh) Parametric shapes (splines, mesh deformations, basis shapes) State equations PDE model equation (elasticity, Stokes, Navier-Stokes, Helmholtz, Maxwell) Numerical PDE solver (FEM, FVM) Optimization Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 3 / 23

Shape Optimization of PDE-modelled Systems Ingredients Geometry Computational geometry (triangular/quadrilateral mesh) Parametric shapes (splines, mesh deformations, basis shapes) State equations PDE model equation (elasticity, Stokes, Navier-Stokes, Helmholtz, Maxwell) Numerical PDE solver (FEM, FVM) Optimization Cost functional (linear/nonlinear functional of state) Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 3 / 23

Shape Optimization of PDE-modelled Systems Ingredients Geometry Computational geometry (triangular/quadrilateral mesh) Parametric shapes (splines, mesh deformations, basis shapes) State equations PDE model equation (elasticity, Stokes, Navier-Stokes, Helmholtz, Maxwell) Numerical PDE solver (FEM, FVM) Optimization Cost functional (linear/nonlinear functional of state) Numerical optimization (nonlinear programming, AD, evolutionary algorithms) Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 3 / 23

Why More Efficient Procedures Are Required? PDEs expensive to solve when solutions need to capture fine details Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 4 / 23

Why More Efficient Procedures Are Required? PDEs expensive to solve when solutions need to capture fine details Finite element assembly expensive when problem geometry keeps changing Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 4 / 23

Why More Efficient Procedures Are Required? PDEs expensive to solve when solutions need to capture fine details Finite element assembly expensive when problem geometry keeps changing Nonlinear optimization requires many evaluations of cost functional Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 4 / 23

Why More Efficient Procedures Are Required? PDEs expensive to solve when solutions need to capture fine details Finite element assembly expensive when problem geometry keeps changing Nonlinear optimization requires many evaluations of cost functional Path to Efficient Shape Design Optimal shape design problems can be solved efficiently if: Number of design parameters is low Solving the state equations is inexpensive Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 4 / 23

Why More Efficient Procedures Are Required? PDEs expensive to solve when solutions need to capture fine details Finite element assembly expensive when problem geometry keeps changing Nonlinear optimization requires many evaluations of cost functional Path to Efficient Shape Design Optimal shape design problems can be solved efficiently if: → free-form deformations ⋆ Number of design parameters is low Solving the state equations is inexpensive ⋆ Sederberg and Parry (1986) Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 4 / 23

Why More Efficient Procedures Are Required? PDEs expensive to solve when solutions need to capture fine details Finite element assembly expensive when problem geometry keeps changing Nonlinear optimization requires many evaluations of cost functional Path to Efficient Shape Design Optimal shape design problems can be solved efficiently if: → free-form deformations ⋆ Number of design parameters is low Solving the state equations is inexpensive → reduced basis methods ⋆⋆ ⋆ Sederberg and Parry (1986) ⋆⋆ http://augustine.mit.edu Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 4 / 23

Airfoil Inverse Design Problem Consider reference airfoil in exterior potential flow ( △ u = 0) Choose target airfoil and compute pressure distribution on its surface using Bernoulli equation ( p = p 0 − 1 2 | ∇ u | 2 ) Find small perturbation of reference airfoil s.t. pressure distribution on the airfoil surface close to target airfoil (a) Reference airfoil NACA0012 (b) Target airfoil NACA4412 Figure: Pressure distributions around the airfoils Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 5 / 23

Free-Form Deformations for Shape Parameterization P 0 i , j P i , j Ω( 0 ) T ( x ; µ ) Ω( µ ) D Choose a lattice of control points P 0 i , j around the reference shape Introduce parameters µ ij as perturbations of each control point Perturbed control points P i , j = P 0 i , j + µ i , j define a parametric domain map L M � � P 0 ∑ ∑ T ( x ; µ ) = i , j + µ i , j b i , j ( x ) i = 0 j = 0 Tensor product Bernstein polynomials � L �� M � 1 ( 1 − x 2 ) M − j x j ( 1 − x 1 ) L − i x i b i , j ( x 1 , x 2 ) = 2 i J Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 6 / 23

Free-Form Deformations in Action Figure: An example of the reference airfoil and a deformed configuration. Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 7 / 23

Parametric Partial Differential Equations Deforming the mesh not sensible due to degrading element quality. Instead keep mesh fixed and transform problem from Ω( µ ) back to Ω 0 . Parametric PDE a ( u ( µ ) , v ; µ ) = f ( v ) for all v ∈ X (Ω 0 ) (exact) a ( u N ( µ ) , v ; µ ) = f ( v ) for all v ∈ X N (Ω 0 ) (FEM) , where the problem coefficient matrices are now parametric: a ( u ( µ ) , v ; µ ) � � A ( · ; µ ) ∇ u , ∇ v � + B ( · ; µ ) uv . For potential flow B = 0. Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 8 / 23

The Idea of Reduced Basis Methods Problem: FE solution u N ∈ X N too expensive to compute for many different values of µ . Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 9 / 23

The Idea of Reduced Basis Methods Problem: FE solution u N ∈ X N too expensive to compute for many different values of µ . Observation: Dependence of the bilinear form a ( · , · ; µ ) on µ is smooth ⇒ parametric manifold of solutions in X is smooth Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 9 / 23

The Idea of Reduced Basis Methods Problem: FE solution u N ∈ X N too expensive to compute for many different values of µ . Observation: Dependence of the bilinear form a ( · , · ; µ ) on µ is smooth ⇒ parametric manifold of solutions in X is smooth Solution: Choose a representative set of parameter values µ 1 ,..., µ N with N ≪ N Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 9 / 23