On the usage of the Riemannian geometry framework for PDE constrained shape optimization Volker Schulz TEAM: H. Zorn, K. Welker, M. Siebenborn University of Trier Stephan Schmidt, Universität Würzburg Nicolas Gauger, TU Kaiserslautern Caslav Ilic, German Aerospace Center, Braunschweig Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Outline • Motivating applications • shape Newton methods • shape SQP methods Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Aerodynamic Shape Optimization Joint work with DLR and Airbus • VELA: Very Efficient Large • Design study for blended Aircraft wing-body configurations Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Framework • Collaborations with German Aerospace Center in Braunschweig and Airbus Germany within the projects: MEGADESIGN, MUNA, SPP1253, ComFliTe, DGHPOPT • Flow solvers (Flower, Tau) are matrix-free and based on multigrid methods featuring polynomial smoothers (aka RK) [for Euler-eq. cf. van Leer/Tai/Powell AIAA CFD 1989] Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Gradient-based optimization for design compute gradient w.r.t. p compute new p compute consistent y Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Primal-dual SQP c > r y L ( y , p , λ ) H yy H yp ∆ y y = � c p > r p L ( y , p , λ ) ∆ p H py H pp c ( y , p , λ ) ∆ λ c y c p 0 equivalent linear-quadratic program ◆ > ✓ ✓ ◆ 1 ∆ y ∆ y + f > y ∆ y + f > H p ∆ p min ∆ p ∆ p 2 ∆ y , ∆ p c y ∆ y + c p ∆ p + c = 0 s.t. Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
One-shot can be interpreted as an approximate reduced SQP method of the form: where A C y and B is a consistent approximation of the reduced Hessian. Convergence results for quadratic models in (Ito/Kunisch/ Schulz/Gherman, SIMAX 2010) Comparison with Griewank (ISNM 165, 2014) • Can be easily generalized to finitely many state constraints (lift) • Overall computational effort is less than 10 x effort for CFD solution. Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Modular one-shot approach to • Do few iterations in the forward solver • Do few iterations in the adjoint solver, i.e. to the equation • Use the resulting gradient for some design step, change grid accordingly Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
2D high lift configuration • 3 part wing, M=0.17146 • Shape and position of flap is to be optimized • compr. Navier Stokes solver TAU (DLR) • Reynolds number 14.7 * 10^6 • Spalart-Almaras turbulence model • Grid with 90 000 points • 10 design variables • n=20 • L-BFGS for B with 3 stages Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
design variables Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
2D High Lift results Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
… zoom Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Performance • 33 Iterations until convergence (criterion: norm of the reduced gradient) • From • To • Computing time: 4.5h compared to 65min for one simulation only (=factor 4). Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Trouble with fine parameterizations Sensitivities: solve #p linearized problems Adjoints: solve one adjoint problem by usage of an adjoint solver A supposedly trivial matrix vector product becomes a computational bottleneck Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Parametric versus nonparametric • Finite shape parametrization in many industrial shape optimizations – Pro: vector space setting, fits in CAD framework – Con: complexity inevitably increases with number of parameters, mesh sensitivities can become expensive, set of reachable shapes is restricted • Nonparametric approach built on shape calculus – Pro: avoids cons of parametric approach, can be very efficient – Con: no longer vector space setting, theoretically more challenging Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Shape gradients for free node parametrizations Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Example: simple objective, no PDE constraint Z Ω t = T t , V ( Ω o ) f ( Ω t ) = g ( x ) dx T t , V ( x ) = x + t · V ( x ) Ω t directional derivative in direction : V d f ( Ω t )[ V ] : = d f ( Ω t ) = d Z g ( x ) dx dt t = 0 dt t = 0 Ω t = d Z g ( T t , V ( x )) | det ( DT t , V ( x )) | dx dt t = 0 Ω o Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
d Z = g ( T t , V ( x )) | det ( DT t , V ( x )) | dx dt t = 0 Ω o Z ⇥ g ( x ) � V ( x ) + g ( x ) · tr ( DV ( x )) dx = | {z } Ω o divV ( x ) Z = div ( g ( x ) V ( x )) dx Ω o Z V ( x ) � = n ( x ) · g ( x ) dx ~ (Gauss) ∂ Ω o Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Shape gradients for free node parametrizations (Hadamard) Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Important shape derivatives Z Z d ( g ( x ) dx )[ W ] = g ( s )( W , n ) ds Ω ∂ Ω ( ∂ g Z Z d ( g ( s ) ds )[ W ] = n ( s ) + κ ( s ) g ( s ))( W , ~ n ) ds ∂ ~ ∂ Ω ∂ Ω (cf. Delfour/Zolesio, 2001) Z Z h ( s ) > n ( s ) ds )[ W ] = d ( div ( h ( s ))( W , n ) ds ∂ Ω ∂ Ω (cf. Stephan Schmidt, 2010) Volker Schulz ESI wor ESI w orkshop shop , February 16, 2015
Further names to be mentioned in the field of shape calculus: Zolesio, Haslinger, Sokolowski, Pironneau, Mohammadi, Delfour, Neittanmäki, Berggren, Hintermüller, Ring, Eppler, Harbrecht, Zuazua, Sturm … Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Central observation for PDE constraints (Correa-Seeger) d f ( y ( Ω ) , Ω )[ V ] = d L ( y , λ , Ω )[ V ] Thus, we just have to build the Lagrangian and perform the (partial) shape differentiation Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Example: drag in volume formulation (dissipation of kinetic energy into heat) Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Shape derivatives (Schmidt/S.: Control and Cybernetics, 2010) Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Potential problem: too much freedom • Usage of shape derivatives alone may lead to unphysical geometries • Shape Hessian approximations help to – „smooth“ gradients – Speed up convergence in the fashion of a Newton-like method – Give potentially mesh independence Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Shape Hessians d 2 J ( u , Ω )[ V 1 , V 2 ] • Can become rather cumbersome and are difficult to interprete the operators (Eppler/S./Schmidt JOTA 2008) • Fourier mode analysis gives an idea of the general structure (S./Schmidt, SICON 2009) Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Performance for Navier-Stokes speed 12 vs. 200: 96% less iterations Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Optimal non-parametric design for Euler flow in TAU (DLR) pressure From NACA0012 to Haack Ogive Mach 2.0 strong detached bow shock transformed to weak Drag reduction 45% Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Overall performance Wall clock time reduced by 99% (2.77h versus 100s) Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Vela aircraft • VELA: Very efficient large • 115,673 surface nodes to aircraft be optimized • Design study for blended • Planform constant wing-body configurations [Schmidt/Ilic/Gauger/S. AIAA Journal 2013] Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
A closer look at the shape Hessian • Symmetry No ∂ Ω ( ∂ g Z ? d ( d f ( Ω )[ W ])[ V ] = n + κ c g ) h W , ~ n i h V , ~ n i + g h DW V , ~ n i ds ∂ ~ • Taylor series ⇒ Sufficient conditions ⇒ Quadratic convergence of Newton method Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
• Distance concepts „Morphing“: Riemannian length of shortest connecting - Hausdorff distance path Ω 1 Ω 2 Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
The Riemannian metric of Michor and Mumford (2006) Shape set B e ( S 1 , 2 ) = Emb ( S 1 , 2 ) /Diff ( S 1 ) is a manifold with tangent space n , α ∈ C ∞ ( S 1 , T c B e ∼ = { h | h = α ~ ) } Scalar product Z S A ( h , k ) = c αβ + A α 0 β 0 ds Z G A ( h , k ) = c ( 1 + A κ 2 c ) αβ ds , A > 0 defines a Riemannian manifold Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
A Riemannian view on shape optimization (arXiv:1203.1493) • Defining the action of a vector field as the shape derivative, we can unleash the Riemannian structure on shape optimization h ( f )( c ) = d f ( Ω )[ V ] , V = h ~ n , c = ∂ Ω • Optimization on manifolds can be performed as in [Absil 2008] for matrix manifolds. Volker Schulz ESI w ESI wor orkshop shop , February 16, 2015
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