SAROD Conference, Hyderabad, december 2005 CONTINUOUS MESH ADAPTATION MODELS FOR CFD

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1 1 SAROD Conference, Hyderabad, december 2005 CONTINUOUS MESH ADAPTATION MODELS FOR CFD Bruno Koobus, 1,2 Laurent Hascoët, 1 Frédéric Alauzet, 3 Adrien Loseille, 3 Youssef Mesri, 1 Alain Dervieux 1 1 INRIA Sophia-Antipolis, France 2 Université de Montpellier II, France 3 INRIA Rocquencourt, France

2 2 (P): Find the design parameter γ opt for which a functional j is minimum. γ:aircraft shape parameter j: performances via CFD approximate model M:mesh parameterization j(m,γ) = J(M,γ,U(M,γ)), Ψ State (M,γ,U(M,γ)) = 0. - minimize the functional with respect to shape γ, - minimize the error on functional with respect to a mesh parameter, M.

3 3 Adaptation for a functional - Best ideal mesh density minimizing a continuous model of the error, see Babuska-Strouboulis - Adjoint based:. superconvergence for a functional,giles. a posteriori error for a functional,becker-rannacher

4 4 Mesh optimization strategy Minimize an error functional j with respect to mesh parameter, M. with Ψ ST AT E (γ,u) = 0 M = ArgMin J(M,γ,U exact U) with Ψ ERROR (M,γ,U exact U) = 0. J(U exact U(M)) = J exacte (U exact ).(U exact U(M)) 2

5 5 Mesh parameterization: Riemannian metric We modelize a mesh as a continuous medium, with an anisotropic property, the local metric (*): M x,y = R 1 M ( (mζ ) (m θ ) 2 ) R M, (*)(George, Hecht,..., Fortin, Habashi, Vallet,...) - interpolation error, - approximation error: elliptic theory, extension to CFD.

6 6 P1-Interpolation error E M = ( ) 2 2 u ξ 2.m2 ξ + 2 u η 2.m2 η dxdy where ξ and η are directions of diagonalization of the Hessian of u. Discontinuous case: (u(ξ + δ,η) 2u(ξ,η) + u(ξ δ,η))/δ 2 bounded in L 1 2. min M E M under the constraint N M = N.

7 7 Optimal Metric construction. M = R Λ R ( 1, where ( Λ = diag( ) λ i ) ) and λ c λi 1 1 i = min max, ε h 2, max h 2. min M opt = C N R 1 2 u 5/6 η 2 2 u 1/6 0 ξ u 5/6 ξ 2 2 u 1/6 η 2 avec: C = ( )2 2 u u 6 ξ 2. 2 η 2 dxdy. R. Erreur minimale: E opt = 4C2 N 2 ( ) 2 1/3 u ξ 2 2 1/3 u η 2 dxdy

8 8 Anisotropic mesh reconstruction Purpose: to generate a mesh such that all edges have a length of (or close to) one in the prescribed metric. For P a vertex M(P ) the metric at P, an edge P X has a local parametrization P X = P + t P X.Its average length is: l M ( P X) = 1 0 t P X M(P + t P X ) P X dt. (1) - add nodes in edges longer than 1+...

9 9 Supersonic business jet Final mesh (iteration 9) contains 798,756 vertices, 38,492 boundary triangles and 4,714,162 tetrahedra. Total CPU time 44 hours on a 600MHz workstation with 1Gb of memory. The remeshing time is 2% of total time time.

10 10 Supersonic business jet

11 11

12 12 Isotropic mesh and asymptotics How does behave the approximation error? m ζ 1 2 = m θ 1 2 = d: local mesh density M = d d h = h 1 d h > 0.

13 13 A First model: Dirichlet model, Galerkin Babuska-Rheinbolt: local problems + pollution error Mesh quality and element location effects A priori: Projection/Aubin-Nitsche Our option: extract a global smooth behavior from second-order a priori estimates.

14 14 Approximation-error splitting - P 1 -continuous, vertex-centered approximations, Π M : H k (Ω) V h = {v, continue, P 1 by element } U exact U(M) = U exact Π M U exact + Π M U exact U(M) U exact Π M U exact is interpolation error Π M U exact U(M) is the implicit error, solution of a discrete system.

15 15 Implicit error : elliptic case < (u), φ > = fφ dx. < (u h ), Π h φ > = fφ dx. < (u h Π h u), Π h φ > = < (u Π h u), Π h φ >

16 16 Lemma : We assume that the continuous solution u is in C 3 ( Ω) and that the continuous mesh size m is in C 2 ( Ω). Then, for any function φ of D 3 (Ω), we have Ω (u Π h u) Π h φ x x dm + = h 2 Ω Ω (u Π h u) Π h φ y y dm g (m)φdm + O φ (h 3 ) (2)

17 17 g (m) = g 1(m) + g 2(m) : (g 1(m),φ) = 3 48 (g 2(m),φ) = φ Ω y φ x Ω ( m 2 ( m 2 3 u x 3 ) 3 u x y 2 dm ) dm φ ( m 2 3 ) u Ω y 6 x 2 + φ 2 y y 2 φ ( ) m 2 3 u Ω y y 3 + φ 2 y 2 ( m 2 2 u x 2 ( m 2 2 u y 2 Remark: H 2 pivot space for gradient with respect to m. ) dm ) dm.

18 18 Numerical implementation Given u and its derivatives, the continuous optimization problem is discretized on a background mesh. Its adjoint and gradient are obtained by reverse-mode Automated Differentiation with TAPENADE. Optimization : One-Shot SQP and multi-level preconditioner. Example: Dirichlet problem, Minimization of L 2 error on whole domain, on left half domain.

19 19 case 1: Min 1 0 Optimization of mesh for a Dirichlet problem 1 0 u u h 2 dxdy, case 2: Min ( 1 0 u u h 2 dy)dx,

20 20 Case 1: Node-density initial(line)/final(dashes)

21 21 Case 1: Truncation error initial(line)/final(dashes)

22 22 Case 1: Approximation error initial(line)/final(dashes)

23 23 Case 2: Node-density initial(line)/final(dashes)

24 24 Case 2: truncation error initial(line)/final(dashes)

25 25 Case 2: approximation error initial(line)/final(dashes)

26 26 Towards the anisotropic case: Step 1: Pointwise optimization Given at a point a mesh density d = (m ζ m θ ) 1, find numerically at each node the: - optimal stretching direction, - optimal stretching strength. Build the reduced error model based on these outputs. Step 2: constrained global optimization Minimize the reduced error model with respect to mesh density.

27 27 A VARIATIONAL MODEL FOR EULER EQUATIONS A simplifying assumption: pure Galerkin formulation W and φ ( H 1 (Ω) ) 5,Vh is included in V. Ω φ.f(w )dω Ω φ F(W ).nd Ω = 0. Ω φ h.f h (W h )dω Ω φ h Fh (W h ).nd Ω = 0 where F h (W ) is the interpolate of F, i.e. F h (W h ) = Π h F(W h ), and same for Fh (W h ).

28 28 Replace φ by φ h = Π h φ: Ω φ h.f h (W h )dω φ h Fh (W h ).nd Ω = Ω φ h.f(w )dω φ h F(W ).nd Ω Ω Boundary data are involved inside F(W ) and F h (W ) Assumption: these terms can be split in W -dependant terms, denoted respectively by F out (W ) and F h out (Wh ), and constant terms, denoted F in and F h in. Ω

29 29 Therefore: φ h.f h (W h )dω out φ h Fh (Wh ).nd Ω = Ω Ω φ h.f(w )dω φ h F out (W ).nd Ω Ω Ω

30 30 Error estimate Assumption: both W and φ are several time continuously differentiable. φ h.(f h (W h ) Π h F(W ))dω Ω φ h ( F out h (Wh ) Π h F out (W )).nd Ω = Ω φ h.(f(w ) Π h F(W ))dω Ω φ h ( F out (W ) Π h F out (W )).nd Ω. Ω The left-hand side will be inverted and the right-hand side will be expanded to get the error estimate.

31 31 Interpolation errors We recall that Π h F(W ) = F h (W ). The left-hand side writes: LHS = Ω Ω φ h.(f h (W h ) F h (W ))dω φ h ( F h out (Wh ) out F h (W )).nd Ω.

32 32 We linearize it as follows: F LHS = φ h.(π h Ω W (W h W ))dω φ h (Π F out h h W (W h W )).nd Ω. Ω Where the derivatives F values of W. F h out W and W are evaluated from vertex LHS = A h (W )(W h Π h W )

33 33 LHS = A h (W )(W h Π h W ) Assumption:the Jacobian A h (W ) of the discretized Euler system is invertible. The implicit error W h Π h W is the unique solution of: W h Π h W = (A h (W )) 1 RHS.

34 34 In the right-hand side: RHS = φ h.(f(w ) Π h F(W ))dω Ω φ h ( F out (W ) Π h F out (W )).nd Ω Ω we recall that φ h = Π h φ and we add and substract a φ term: RHS = RHS 1 + RHS 2 with: RHS 1 = Ω Ω (Π h φ φ).(f(w ) Π h F(W ))dω (Π h φ φ)( F out (W ) Π h F out (W )).nd Ω

35 35 Assuming smoothness of φ and F(W ), we deduce that on Ω, interpolation errors are of order two and their gradients are of order one, same on boundary, and RHS 1 is thus of order three. RHS 1 const.h 3 The second term writes: RHS 2 = Ω Ω φ.(f(w ) Π h F(W ))dω φ( F out (W ) Π h F out (W )).nd Ω

36 36 and we transform it as follows: RHS 2 = ( φ).(f(w ) Π h F(W ))dω Ω + φ(f(w ) Π h F(W )).nd Ω Ω φ( F out (W ) Π h F out (W )).nd Ω Ω Same as for elliptic (easier): RHS 2 const.h 2. RHS 2 = h 2 (G(W,d),φ) + R, R = o(h 2 ).

37 37 Provisional conclusion The above study shows that the implicit error W h Π h W is a linear function of the interpolation error W Π h W. A first option consists in reducing the interpolation error. This option is studied in Sections 4 and 5. In Section 6 we shall define how to minimize with the implicit error by introducing an adjoint.

38 38 Implicit error reduction, general case Ψ W (W,d)Y = G(W,d), W is the unknown, d mesh density, Y implicit error. ( Ψ j = Ψ (W,d)Y G(W,d) = 0 ( W ) Ψ W (W,d) Π J Y (Y,d) = 0 W Y G) d (W,d) which can be solved as for elliptic case. Π + J d (Y,d) = 0

39 39 CONCLUSION This talk proposes rather sophisticated algorithms involving: flow solver, local error evaluation, sensitivity analysis by Automated Differentiation, adjoint and gradient based optimization, controlled generation of next mesh, interpolation of solution, till a convergence of a global loop is attained. These sophisticated algorithms will not bring only robustness, but also efficiency.