Reduced order model and simultaneous computation for thermomechanical multidimensional models

Transcription

1 Reduced order model and simultaneous computation for thermomechanical multidimensional models Fatima Daïm, D. Ryckelynck, D. M. Benziane Centre des Matériaux Mines-ParisTech 17 juin 2011 D. Ryckelynck () 17 juin / 29

2 Plan 1 The extension of the concept of symmetry 2 Multi-dimensional thermo-mechanical problem 3 Reduced-order model 4 Simultaneous resolution 5 Numerical results D. Ryckelynck () 17 juin / 29

3 The extension of the concept of symmetry A postulate for the extension of the concept of symmetry. If u U ROM U, there might be explanatory variables in mechanics of materials that should permit to reduced the extent of the domain over which equilibrium conditions are considered. explanatory variables defined over Ω e 3 Ω explained variables outside Ω e 2 e 1 θ e r Canonical analysis and the PLS (Partial Least Square) method [Wold 1966] can help to find the best explanatory variables. These methods are convenient for Data Mining. But they don t take into account equilibrium equations or global conditions. We don t have data, but we have equations. D. Ryckelynck () 17 juin / 29

4 The extension of the concept of symmetry A simple theoretical example of Hyper-reduction Let s consider the following elliptic PDE (E > 0), with Dirichlet conditions : ( E(X,p) u ) = 0 x [0,L] u(0,p) = 0 u(l,p) = 0 x x u = 0 u = x=0 x= L Ω Π x ζ 2 ζ 1 ψ 2 ψ 1 Truncated test functions u c node index ψ 1 (x) = x ψ 2 (x) = x (2 L x) If u(x,p) = ψ 1 (x) a 1 (p) + ψ 2 (x) a 2 (p) + u c (x) then, a 1 and a 2 are such that : (ζ 1 (x) a 1 + ζ 2 (x) a2) ( E(X,p) u ) dx = 0 a 1 Ω Π x x a 2 D. Ryckelynck () 17 juin / 29

5 Multi-dimensional thermo-mechanical problem Plan 1 The extension of the concept of symmetry 2 Multi-dimensional thermo-mechanical problem 3 Reduced-order model 4 Simultaneous resolution 5 Numerical results D. Ryckelynck () 17 juin / 29

6 Multi-dimensional thermo-mechanical problem Parameters := p X in (x,t) {Σ p } p X out (x,t) Objective : The computation of the response in respect to parameters. Tools : Given the limits of classical discretization methods for the multi-dimensional problems, the use of the reduced order models is necessary. D. Ryckelynck () 17 juin / 29

7 Multi-dimensional thermo-mechanical problem Material for mecatronic application D. Ryckelynck () 17 juin / 29

8 Multi-dimensional thermo-mechanical problem Material for mecatronic application D. Ryckelynck () 17 juin / 29

9 Multi-dimensional thermo-mechanical problem Thermal and mechanical perfomance indicators Conductivity : 1 Φ(x)dx Γ P T = 2 Γ e ( 2 1 T (x,t max )dx 1 ) T (x,t max )dx Γ 2 Γ 2 Γ 1 Γ 1 D. Ryckelynck () 17 juin / 29

10 Multi-dimensional thermo-mechanical problem Thermal and mechanical perfomance indicators Homogenized dilatation : P E = (u(a,t max,t (A,t max )) u(b,t max,t (B,t max )) ( x ) 1 L T (x,t max )dx T 0 Γ 2 Γ 2 D. Ryckelynck () 17 juin / 29

11 Multi-dimensional thermo-mechanical problem Behaviour law (M E ) divσ(u(x,t,t ),T ) = 0 in Ω (0,t max ] ε(u(x, t, T )) = 1 + ν ν σ(u(x,t,t ),T ) E(x,T ) E(x,T ) tr(σ(u(x,t,t ),T ))I + α(t (x,t) T D)I ε(u(x,t,t)) = 1 2 ( u(x,t,t ) + u(x,t,t )T ), σ(u).n = 0 on Ω (0,t max ], u = 0 on Γ 2 (0,t max ]. Nonlinear thermal loading (M k ) ρ(x)c(x,t) T t (x,t) div(k(x,t,t ) T (x,t)) = 0 in Ω (0,t max ] T(x,0) = T D in Ω T.n = h(t (x,t) T D ) on Γ 1 (0,t max ] T.n = Φ on Γ 2 (0,t max ] T.n = 0 on Γ 3 Γ 4 (0,t max ]. D. Ryckelynck () 17 juin / 29

12 Multi-dimensional thermo-mechanical problem Correction with damage parameters { kωi = (1 p T ω i )k(x,t,t ), E ωi = (1 p E ω i )E(x,t,T ). With p = (p T ω 1,p T ω 2,p T ω 3,p E ω 1,p E ω 2,p E ω 3 ) [0,1] 6, we obtain { (M E p,m k p )} p D. Ryckelynck () 17 juin / 29

13 Multi-dimensional thermo-mechanical problem Correction with damage parameters { kωi = (1 p T ω i )k(x,t,t ), E ωi = (1 p E ω i )E(x,t,T ). With p = (p T ω 1,p T ω 2,p T ω 3,p E ω 1,p E ω 2,p E ω 3 ) [0,1] 6, we obtain { (M E p,m k p )} p Thermal performance function P T : [0,1] 6 R + p e 1 Γ 2 Φ(x)dx Γ 2 ( 1 T (x,p,t max )dx 1 Γ 2 Γ 2 Γ 1 Γ 1 T (x,p,t max )dx ), Mechanical performance function P E : [0,1] 6 R + p (u(a,p,t max,t (A,p,t max )) u(b,p,t max,t (B,p,t max )) ( x ) 1 L T (x,p,t max )dx T 0 Γ 2 Γ 2 D. Ryckelynck () 17 juin / 29

14 Reduced-order model Plan 1 The extension of the concept of symmetry 2 Multi-dimensional thermo-mechanical problem 3 Reduced-order model 4 Simultaneous resolution 5 Numerical results D. Ryckelynck () 17 juin / 29

15 Reduced-order model Weak formulation of PDE : find u(x, t) such that Ω L(x,t,u(x,t))v(x)dx = 0, v V, t (0,t max ] Finite element method (FEM) : find {a j (t)} N N dof j=1 such that u(x,t) = dof a j (t)n j (x) and Ω j=1 L(x,t n,u(x,t n ))v h (x)dx = 0, v h V h, n = 1,,N t V h = span{n j } N dof j=1 shape functions related to a given mesh of Ω, N t n=1 [ t n 1,t n] = (0,t max ]. D. Ryckelynck () 17 juin / 29

16 Reduced-order model Proper Orthogonal Decomposition (POD) : find {ϕ j (t)} N N rom j=1 such that u(x,t) = rom ϕ j (t)ψ j (x) Ω L(x,t n,u(x,t n ))v rom (x)dx = 0, v rom V rom, n = 1,,N t j=1 V rom = span{ψ j } N rom j=1. {ψ j } N rom j=1 the eigenvectors corresponding to the N rom largest eigenvalues of the correlation matrix related to the snapshots (with N rom << N dof ). D. Ryckelynck () 17 juin / 29

17 Reduced-order model Proper Orthogonal Decomposition (POD) : find {ϕ j (t)} N N rom j=1 such that u(x,t) = rom ϕ j (t)ψ j (x) Ω L(x,t n,u(x,t n ))v rom (x)dx = 0, v rom V rom, n = 1,,N t j=1 V rom = span{ψ j } N rom j=1. {ψ j } N rom j=1 the eigenvectors corresponding to the N rom largest eigenvalues of the correlation matrix related to the snapshots (with N rom << N dof ). POD needs a priori knowing of the solutions and does not offer the possibility to adapt the reduced basis. D. Ryckelynck () 17 juin / 29

18 Reduced-order model A priori hyper reduction method (APHR) Prediction Step : For V (m) rom = span{ψ (m) j (x)} N rom j=1, find {ϕ j(t n )} N rom j=1 such that N rom u(x,t n ) = j=1 ϕ j (t n )ψ (m) j (x) Ω Π Ω L(x,t n,u(x,t n ))ˆv rom dx = 0, v rom V (m) truncated rom Correction Step (using eventually a FETI solver for // computing) : N dof Find δu(x) = j=1 a j N j (x) such that Ω L(x,t n,δu + u(x,t n ))v h dx = 0, v h V h Enrich the reduced basis V (m+1) rom D. Ryckelynck () 17 juin / 29

19 Reduced-order model The construction of the RID Ω Π Ω Π F Ω Ω u = 0 u Ω Π is built by aggregating balls whose the radius is h/2 or 2h. Each vector of reduced bases provide a center of a ball. for each mode Υ A k related to an internal variable A the center of the ball is : x = arg max Υ A k() (1) Ω for each displacement mode ψ k the center of the ball is : x = arg max Ω ( ε(ψ k)() : ε(ψ k )() ) (2) D. Ryckelynck () 17 juin / 29

20 Simultaneous resolution Plan 1 The extension of the concept of symmetry 2 Multi-dimensional thermo-mechanical problem 3 Reduced-order model 4 Simultaneous resolution 5 Numerical results D. Ryckelynck () 17 juin / 29

21 Simultaneous resolution Multidimensional model : Simultaneous resolution : Weak formulation : L(x,p,t,u(x,p,t)) = 0 on Ω P ]0,t max ]. L(x,t,u(x,t)) = 0 on Ω ]0,t max ]. Ω L(x,t n,u(x,t n ))v (x )dx = 0, v L 2 (P;V ) { } Where L 2 (P;V ) = v : P V, v(.,p) 2 V dp < P D. Ryckelynck () 17 juin / 29

22 Simultaneous resolution Multidimensional problems should have more pseudo-symmetries. FIGURE: The virtual domain Ω = Ω P. D. Ryckelynck () 17 juin / 29

23 Simultaneous resolution Mechanical problem Ω P σ(u(x,p,t)) : ε(v(x,p))dxdp = α (T (x,p,t) T D ) divv(x,p)dxdp, Ω P t (0,t max ], v V = { v L 2 ( P;H 1 (Ω) ),v(.,t) Γ2 P = 0 }. Thermal problem + Ω P ρ(x)c(x,t) T t Ω P (x,p,t)w(x,p)dxdp + k(x,t,t ) T (x,p,t) w(x,p)dxdp+ h (T (x,p,t) T D )w(x,p)dxdp = Φw(x, p)dx dp Γ 1 P t (0,t max ], w L 2 ( P;H 1 (Ω) ). Γ 2 P D. Ryckelynck () 17 juin / 29

24 Simultaneous resolution Is there an optimal strategy for parallel computing? Each case of damage simulation can be run on a processor of a parallel computer. Does the hyper-reduction of the multidimensional problem is efficient when using parallel computing? D. Ryckelynck () 17 juin / 29

25 Numerical results Plan 1 The extension of the concept of symmetry 2 Multi-dimensional thermo-mechanical problem 3 Reduced-order model 4 Simultaneous resolution 5 Numerical results D. Ryckelynck () 17 juin / 29

26 Numerical results Objective : Variability of the thermal and the mechanical performances for p T ω 1,p T ω 2,p T ω 3,p E ω 1,p E ω 2,p E ω 3 {0.8,0.9} (64 cases) such that 1 Φ(x)dx Γ P T (T,p) = 2 Γ e ( 2 1 T (x,p,t max )dx 1 ), T (x,p,t max )dx Γ 2 Γ 2 Γ 1 Γ 1 P E (u,t,p) = (u(a,p,t max,t (A,p,t max )) u(b,p,t max,t (B,p,t max )) ( x ) 1 L T (x,p,t max )dx T 0 Γ 2 Γ 2 D. Ryckelynck () 17 juin / 29

27 Numerical results Domain and mesh FIGURE: The composite Ω (m 1 m 2 ) and damage zones ω 1,ω 2 and ω 3. D. Ryckelynck () 17 juin / 29

28 Numerical results Simultaneous mesh Ω = Ω P (64 simulations) D. Ryckelynck () 17 juin / 29

29 Numerical results Damage rates D. Ryckelynck () 17 juin / 29

30 Numerical results The relative errors FIGURE: The relative error between APHR computation and FEM for respectively thermal and mechanical performances D. Ryckelynck () 17 juin / 29

31 Numerical results Reduced integration domains (RID) FIGURE: In the left hand side : the reduced integration domain Ω Π t Ω P of the thermal problem and in the right hand side : the reduced integration domain Ω Π m Ω P of the mechanical problem. D. Ryckelynck () 17 juin / 29

32 Numerical results Scalability Saving on CPU time (4 procs, 32 cases) (8 procs, 64 cases) Thermal problem % % Mechanical problem % % Saving on time of computation of the internal variables (4 procs, 32 cases) (8 procs, 64 cases) Thermal problem % % Mechanical problem 88 % % D. Ryckelynck () 17 juin / 29

33 Numerical results Conclusion If multidimensional problems have more symmetries or pseudo symmetries, the hyper-reduction is more efficient in these cases than in classical dimensions. An optimal parallel strategy should exist for the simulations based on hyper-reduced multidimensionnal model. D. Ryckelynck () 17 juin / 29