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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 → free-form deformations⋆ 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

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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 → free-form deformations⋆ 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

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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 = p0 − 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

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Free-Form Deformations for Shape Parameterization

T(x;µ) Ω(0) P0

i,j

Pi,j Ω(µ) D

Choose a lattice of control points P0

i,j around the reference shape

Introduce parameters µij as perturbations of each control point Perturbed control points Pi,j = P0

i,j + µi,j define a parametric domain map

T(x;µ) =

L

∑

i=0 M

∑

j=0

  • P0

i,j + µi,j

  • bi,j(x)

Tensor product Bernstein polynomials bi,j(x1,x2) = L i M J

  • (1−x1)L−ixi

1(1−x2)M−jxj 2

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 6 / 23

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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

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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 Ω0. Parametric PDE a(u(µ),v;µ) = f(v) for all v ∈ X(Ω0) (exact) a(uN (µ),v;µ) = f(v) for all v ∈ XN (Ω0) (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

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The Idea of Reduced Basis Methods

Problem: FE solution uN ∈ XN 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

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The Idea of Reduced Basis Methods

Problem: FE solution uN ∈ XN too expensive to compute for many different values of µ. Observation: Dependence of the bilinear form a(·,·;µ) on µ is smooth ⇒ parametric manifold

  • f 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

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The Idea of Reduced Basis Methods

Problem: FE solution uN ∈ XN too expensive to compute for many different values of µ. Observation: Dependence of the bilinear form a(·,·;µ) on µ is smooth ⇒ parametric manifold

  • f solutions in X is smooth

Solution: Choose a representative set of parameter values µ1,...,µN with N ≪ N

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 9 / 23

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The Idea of Reduced Basis Methods

Problem: FE solution uN ∈ XN too expensive to compute for many different values of µ. Observation: Dependence of the bilinear form a(·,·;µ) on µ is smooth ⇒ parametric manifold

  • f solutions in X is smooth

Solution: Choose a representative set of parameter values µ1,...,µN with N ≪ N Snapshot solutions uN

1 ,...,uN N span a

subspace XN

N

with orthogonal basis {ξn}N

j=1

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 9 / 23

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The Idea of Reduced Basis Methods

Problem: FE solution uN ∈ XN too expensive to compute for many different values of µ. Observation: Dependence of the bilinear form a(·,·;µ) on µ is smooth ⇒ parametric manifold

  • f solutions in X is smooth

Solution: Choose a representative set of parameter values µ1,...,µN with N ≪ N Snapshot solutions uN

1 ,...,uN N span a

subspace XN

N

with orthogonal basis {ξn}N

j=1

Galerkin reduced basis formulation a(uN

N (µ),ξn;µ) = f(ξn)

for all n = 1,...,N.

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 9 / 23

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Comparison Between Finite Element and Reduced Basis Methods

FE basis functions Local support only Generic, work for many problems A priori estimates readily available RB basis functions Globally supported Constructed for specific problem Error estimation expensive

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 10 / 23

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Affine Decomposition of the Bilinear Form

Assumption: Tensor A affinely parameterized [A ]i,j =

Q

∑

q=1

Θq

i,j(µ)ζ q i,j (x)

for i,j = 1,2 Weak form can be written

Q

∑

q=1 2

∑

i=1 2

∑

j=1

Θq

i,j(µ)aq i,j(uN ,v) = f(v)

for all v ∈ XN , with continuous bilinear forms aq

i,j(w,v) =

  • Ω0

ζ q

i,j (x)∂w

∂xi ∂v ∂xj dΩ.

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 11 / 23

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Online/Offline Procedure for Reduced Basis Solution

Galerkin reduced basis formulation with affine parameterization:

N

∑

m=1

um

2

∑

i=1 2

∑

j=1 Q

∑

q=1

Θq

i,j(µ)aq i,j(ξm,ξn) = f(ξn) for all n = 1,...,N

Offline Greedy basis selection for the ξn’s, assembly of aq

i,j(ξm,ξn)

Computational time several hours Performed once Online Solution of N ×N reduced basis system for any µ Computational time < 0.1 s rbMIT library for Matlab at http://augustine.mit.edu

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 12 / 23

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Empirical Interpolation Method

Empirical interpolation method gives approximation for the transformation tensor that is affinely parameterized: [A ]i,j =

Q

∑

q=1

Θq

i,j(µ)ζ q i,j (x)+εi,j(x;µ).

Typically 10 ≤ Q ≤ 50

References

  • M. Barrault, Y

. Maday, N.C. Nguyen, and A.T. Patera. An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R.

  • Math. Acad. Sci. Paris, 339(9):667–672, 2004.

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 13 / 23

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Airfoil Inverse Design Problem

Let’s put everything together and solve the airfoil inverse design problem!

(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 14 / 23

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Some Reduced Basis Functions Obtained for This Problem

Figure: Pressure field around the airfoil for the first six basis shapes.

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 15 / 23

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Discrete Optimization Problem

Parametric cost functional min

µ∈D

1

0 |p(s,µ)−ptarget(s)|2 ds

1/2 +λ [α(µ)−5◦]2 To obtain the cost functional: Scale airfoil to unit length and take norm of pressure defect Add penalty term to enforce constraint AOA = 5◦ Evaluate pressure as p = p0 − 1

2|∇uN N |2, where uN N is the reduced basis

approximation for the flow potential u

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 16 / 23

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Results of the Inverse Design Procedure

(Loading optimization example.avi) Figure: Pressure distributions on the airfoil top (blue) and bottom (red) surfaces on the target airfoil (solid line) and the inverse design (×).

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 17 / 23

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Results of the Inverse Design Procedure

(a) Inverse design (b) Target airfoil NACA4412

Figure: Pressure distributions around the inverse design and target airfoil, Lassila and Rozza (2009).

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 18 / 23

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Estimated Computational Savings

2 4 6 8 10 x 10

4

10

−2

10

−1

10 10

1

10

2

10

3

10

4

CPU time (seconds) Number of mesh nodes Real−time barrier (response time < 1 s) Finite element method Reduced basis method

Figure: Computational cost for one online solution of the parametric PDE. Dimension of reduced basis space was chosen to be N = N 1/2.

200 400 600 800 1000 2 4 6 8 10 12 14 CPU time (hours) Number of PDE solves Finite element method Reduced basis method

Figure: Total computational cost for offline and online stages with N = 8043.

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 19 / 23

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How Far Can This Technique Go?

Free-form deformations independent of geometry, mesh, and underlying PDE model

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 20 / 23

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How Far Can This Technique Go?

Free-form deformations independent of geometry, mesh, and underlying PDE model Need a posteriori estimates for reduced basis solutions

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 20 / 23

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How Far Can This Technique Go?

Free-form deformations independent of geometry, mesh, and underlying PDE model Need a posteriori estimates for reduced basis solutions Reduced basis method also applicable to problems that are

unsteady (parabolic problems, e.g. Grepl and Patera (2005)) nonsymmetric (advection-diffusion, e.g. Tonn and Urban (2006)) non-Hermitian (Helmholtz, e.g. Sen (2007)) noncoercive (Stokes, e.g. Rozza and Veroy (2007)) nonlinear (Navier-Stokes, e.g. Veroy and Patera (2005))

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 20 / 23

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Summary

Our approach to parametric shape design combined:

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 21 / 23

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Summary

Our approach to parametric shape design combined:

Free-form deformations for flexible shape parameterizations

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 21 / 23

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SLIDE 40

Summary

Our approach to parametric shape design combined:

Free-form deformations for flexible shape parameterizations Empirical interpolation method to obtain affinely parameterized PDEs

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 21 / 23

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SLIDE 41

Summary

Our approach to parametric shape design combined:

Free-form deformations for flexible shape parameterizations Empirical interpolation method to obtain affinely parameterized PDEs Reduced basis methods for solution of parametric PDEs

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 21 / 23

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SLIDE 42

Summary

Our approach to parametric shape design combined:

Free-form deformations for flexible shape parameterizations Empirical interpolation method to obtain affinely parameterized PDEs Reduced basis methods for solution of parametric PDEs

Computational savings of reduced basis realized when need to solve the PDEs in real-time for many different design geometries (e.g. shape

  • ptimization)

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 21 / 23

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SLIDE 43

Summary

Our approach to parametric shape design combined:

Free-form deformations for flexible shape parameterizations Empirical interpolation method to obtain affinely parameterized PDEs Reduced basis methods for solution of parametric PDEs

Computational savings of reduced basis realized when need to solve the PDEs in real-time for many different design geometries (e.g. shape

  • ptimization)

Approach extensible to more realistic and interesting problems. Our interest is to work with viscous flows in cardiovascular modelling.

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 21 / 23

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Thank you

Acknowledgements Gianluigi Rozza and Alfio Quarteroni (EPFL) Anthony Patera and rbMIT contributors (MIT) Organizers of SIAM CS&E 2009

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 22 / 23

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References

  • M. Grepl, A.T. Patera. A posteriori error bounds for reduced-basis approximations of parametrized

parabolic partial differential equations. ESAIM: Math. Model. Numer. Anal., 39(1):157–181, 2005.

  • T. Lassila and G. Rozza. Parametric free-form shape design with PDE models and reduced basis
  • method. Preprint, 2009.
  • G. Rozza and K. Veroy. On the stability of the reduced basis method for Stokes equations in

parametrized domains. Comput. Methods Appl. Mech. Engrg., 196(7):1244–1260, 2007.

  • K. Veroy and A.T. Patera. Certified real-time solution of the parametrized steady incompressible

Navier-Stokes equations; rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47(89), 773-788, 2005. T.W. Sederberg and S.R. Parry. Free-form deformation of solid geometric models. Comput. Graph., 20(4), 1986.

  • S. Sen. Reduced basis approximation and a posteriori error estimation for non-coercive elliptic

problems: application to acoustics. PhD thesis, MIT, 2007.

  • T. Tonn and K. Urban. A reduced-basis method for solving parameter-dependent

convection-diffusion problems around rigid bodies. Proc. ECCOMAS CFD, 2006.

Toni Lassila toni.lassila@epfl.ch How to Get in Better Shape (Mathematically) SIAM CS&E 2009, Miami 23 / 23