Reduced order modelling for Crash Simulation

Transcription

1 Reduced order modelling for Crash Simulation Fatima Daim ESI Group 28 th january 2016 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

2 High Fidelity Model (HFM) and Model Order Reduction (ROM) X in Σ M,T (u(x,t )) X out (u(x,t )) Σ M,T := physical model discretized on Mesh M during time interval simulation (0, T ]. p := parameter vector. u := model state variable of Σ M,T such that HFM u(x, t ) N dof α i (t )N i (x) N dof = O(10 6 ) i=1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

3 High Fidelity Model (HFM) and Model Order Reduction (ROM) X in Σ M,T (u(x,t )) X out (u(x,t )) Σ M,T := physical model discretized on Mesh M during time interval simulation (0, T ]. p := parameter vector. u := model state variable of Σ M,T such that HFM u(x, t ) N dof α i (t )N i (x) N dof = O(10 6 ) i=1 N rom ROM u(x, t ) β i (t )φ i (x) N rom N dof Fatima Daim (ESI Group) ROMi=1 Crash Simulation 28 th january / 50

4 High Fidelity Model (HFM) and Model Order Reduction (ROM) Parameters := p X in Σ M,T (u(x,t,p)) X out (u(x,t,p)) Σ M,T := physical model discretized on Mesh M during time interval simulation (0, T ]. p := parameter vector. u := model state variable of Σ M,T such that HFM u(x, t, p) N dof α i (t,p)n i (x) N dof = O(10 6 ) i=1 N rom ROM u(x, t ) β i (t )φ i (x) N rom N dof Fatima Daim (ESI Group) ROMi=1 Crash Simulation 28 th january / 50

5 High Fidelity Model (HFM) and Model Order Reduction (ROM) Parameters := p X in Σ M,T (u(x,t,p)) X out (u(x,t,p)) Σ M,T := physical model discretized on Mesh M during time interval simulation (0, T ]. p := parameter vector. u := model state variable of Σ M,T such that HFM u(x, t, p) N dof α i (t,p)n i (x) N dof = O(10 6 ) i=1 N rom ROM u(x, t, p) β i (t,p)φ i (x) N rom N dof Fatima Daim (ESI Group) ROMi=1 Crash Simulation 28 th january / 50

6 Which model reduction? Construction of X out (u(x,t,p)) for a given vector parameter p Two possibilities : I. approximation of the function,p X out (u(x,t,p)) by Xout (p) Metamodels, response surfaces, kriging, linear regression. II. fast approximation of the state variable u(x, t, p) via reduced basis and computation then X out (ũ) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

7 Which model reduction? Construction of X out (u(x,t,p)) for a given vector parameter p Two possibilities : I. approximation of the function,p X out (u(x,t,p)) by Xout (p) Metamodels, response surfaces, kriging, linear regression. II. fast approximation of the state variable u(x, t, p) via reduced basis and computation then X out (ũ) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

8 Protection assessment of Car Construction "Euro NCAP" Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

9 Ratings Euro NCAP 2015 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

10 Crash Assessments Quantitative Assessment maximum global compression, maximum sectional efforts, nodal displacement Head Injury Criterion HIC HIC = max 0<t 2 t 1 <τ [ 1 t 2 t 1 t 2 t 1 ü(t)dt ] 2.5 (t 2 t 1 ), τ 15ms Qualitative Assessment Scenario (the kinematics of the components) :the order of components deformation, cabin integrity (no escape of door fillet),... Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

11 Crash Simulation 3M to 7M elements, T 200ms with time step < 1µs. 15 hours for one simulation on HPC. 12 hours for result analysis. For optimization study : 20 to 50 parameter (thickness, materials,...) and 3 to 10 crash simulation per design parameter. Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

12 Virtual Crash 5h on 13 PROCS, 200K elements, T = 150ms, t [0.2µs,1µs] Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

13 Virtual Crash Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

14 PDE : find u defined over Ω (0,T ] ρü divσ = f in Ω (0,T ] σ.n = g on Γ N (0,T ] u = 0 on Γ D (0,T ] u(.,0) = u 0 on Ω u(.,0) = u 0 on Ω u, σ and ε are respectively the displacement, the stress and strain tensors such that σ = Σ(ε, ε) ε = 1 2 ( u T + u ) + u T u f is the external body force and g is the traction on Γ N, the displacement is fixed on Γ D Ω. u 0 and u 0 are respectively the initial displacement and the velocity. Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

15 PDE : find u defined over Ω (0,T ] ρü divσ = f in Ω (0,T ] σ.n = g on Γ N (0,T ] u = 0 on Γ D (0,T ] u(.,0) = u 0 on Ω u(.,0) = u 0 on Ω u, σ and ε are respectively the displacement, the stress and strain tensors such that σ = Σ(ε, ε) Material Nonlinearity ε = 1 2 ( u T + u ) + u T u Geometrical Nonlinearity f is the external body force and g is the traction on Γ N, the displacement is fixed on Γ D Ω. u 0 and u 0 are respectively the initial displacement and the velocity. Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

16 Weak formulation (Spacial discretization) : find u (ρüϕ i + σ : ε(ϕ i )dx) = f.ϕ i dx + g.ϕ i dγ. Ω Ω Γ N {ϕ i } N dof i=1, the N dof shape functions related to the DoF of given mesh of Ω. N dof u(x,t) = i=1 ϕ i (x)u i (t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

17 Matrix form : find U = (U 1 (t),,u Ndof (t)) T MÜ(t) + F int ( U(t), U(t) ) = F ext U(0) = U 0 U(0) = U0 M R N dof N dof the mass matrix. F int the vector of the internal forces depending on U and U. F ext the external forces. Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

18 Notations : Let {t n } N n=1 be the partition of the time interval simulation with time step t n = t n t n 1. Let U n R N dof be the approximation of U(t n ) Let Un R N dof be the approximation of U(t n ) Let Ü n R N dof be the approximation of Ü(t n ) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

19 Newmark family methods MÜ n + F int (U n, Un ) = F ext U n = U n 1 + t n Un 1 + t2 [ (1 2β)Ü n 1 + 2βÜ n] 2 U n = Un 1 + [ t n (1 γ)ü n 1 + γü n] Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

20 Implicit time discretization : β = 1 4,γ = 1 2 Uncondionnally stable Efficient for slow variation phenomena (long load durations, low frequencies) Requires the assemble and the resolution of non-linear system at each time step. Not efficient with fast variations phenomena (costly convergence). Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

21 Implicit time discretization : β = 1 4,γ = 1 2 Uncondionnally stable Efficient for slow variation phenomena (long load durations, low frequencies) Requires the assemble and the resolution of non-linear system at each time step. Not efficient with fast variations phenomena (costly convergence). Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

22 Explicit time discretization : β = 0,γ = 1 2 Efficient for short loading durations (impact, blast,...) Requires a CFL stability condition (Small time step). Run through the elements of the structure to compute the internal forces F int. Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

23 Explicit time discretization : β = 0,γ = 1 2 Efficient for short loading durations (impact, blast,...) Requires a CFL stability condition (Small time step). Run through the elements of the structure to compute the internal forces F int. Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

24 Explicit time discretization :central finite difference ( Ü 0 = M 1 F ext F int (U 0, U ) 0 ) U 1 2 = U 0 + t1 2 Ü 0 n = 1 U n = U n 1 + t n U n 1 2 ) Ü n = M (F 1 ext F int (U n, U n 1 2 ) t n+ 1 2 = tn + t n+1 2 U n+ 1 2 = U n t n+ 1 2 Ü n n = n + 1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

25 Taylor Impact U 0 = 227m/s, L = 32.4mm, R = 3.2mm, T = 80µs ν = 0.33, σ e = 0.4GPa, E = 117GPa, E t = 0.1GPa Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

26 Taylor Impact Final displacement 10 and rate plasticity = 280% Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

27 Explicit Versus Implicit Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

28 Proper Orthogonal Decomposition (POD) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

29 POD principle Find the best direction ϕ such that the set of observations S = {u(.,t i ),i = 1,,N s } H provides the best projection on it max ϕ H 1 N s N s ( u(.,t i ),ϕ ) H ϕ 2 i=1 max ϕ H 1 N s N s i=1 ( u(.,t i ),ϕ ) 2 H (ϕ,ϕ) H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

30 POD principle Find the best direction ϕ such that the set of observations S = {u(.,t i ),i = 1,,N s } H provides the best projection on it max ϕ H N 1 s N s i=1 ( ) u(.,t i ϕ ϕ ), ϕ H H ϕ H 2 max ϕ H 1 N s N s i=1 ( u(.,t i ),ϕ ) 2 H (ϕ,ϕ) H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

31 POD principle Find the best direction ϕ such that the set of observations S = {u(.,t i ),i = 1,,N s } H provides the best projection on it max ϕ H N 1 s N s i=1 ( ) u(.,t i ϕ ϕ ), ϕ H H ϕ H 2 max ϕ H 1 N s N s i=1 ( u(.,t i ),ϕ ) 2 H (ϕ,ϕ) H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

32 FIGURE : ϕ 1 and ϕ 2 are the principal directions Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

33 Discrete POD We consider T = {p 1,,p Ns }, U i = U(.,p i ) and the Snapshots matrix : A = u1 1 u N s 1 u2 1 u N s u 1 N dof u N s N dof = ( ) U 1 U N s We consider the operator K such that : K : H H ϕ K ϕ = 1 N s ( U i,ϕ ) N H Ui = 1 AA T ϕ s N s i=1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

34 Discrete POD We consider T = {p 1,,p Ns }, U i = U(.,p i ) and the Snapshots matrix : A = u1 1 u N s 1 u2 1 u N s u 1 N dof u N s N dof = ( ) U 1 U N s We consider the operator K such that : K : H H ϕ K ϕ = 1 N s ( U i,ϕ ) N H Ui = 1 AA T ϕ s N s i=1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

35 Discrete POD We consider T = {p 1,,p Ns }, U i = U(.,p i ) and the Snapshots matrix : A = u1 1 u N s 1 u2 1 u N s u 1 N dof u N s N dof = ( ) U 1 U N s We consider the operator K such that : K : H H ϕ K ϕ = 1 N s ( U i,ϕ ) N H Ui = 1 AA T ϕ s N s i=1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

36 Reformulation of the maximization problem max ϕ H 1 N s ( N t U i,ϕ ) 2 H i=1 (ϕ,ϕ) H (K ϕ,ϕ) = max H ϕ H (ϕ,ϕ) H Proposition K is linear, symmetric and positive hence λ 1 λ 2 λ d > 0 and orthogonal {ϕ i } d i=1 H such that K ϕ i = λ i ϕ i ϕ 1 is a solution of the problem (K ϕ 1,ϕ 1 ) H = max (ϕ 1,ϕ 1 ) H ϕ H (K ϕ,ϕ) H (ϕ,ϕ) H = max ϕ H 1 N s ( N s U i,ϕ ) 2 H = λ 1 i=1 (ϕ,ϕ) H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

37 Reformulation of the maximization problem max ϕ H 1 N s ( N t U i,ϕ ) 2 H i=1 (ϕ,ϕ) H (K ϕ,ϕ) = max H ϕ H (ϕ,ϕ) H Proposition K is linear, symmetric and positive hence λ 1 λ 2 λ d > 0 and orthogonal {ϕ i } d i=1 H such that K ϕ i = λ i ϕ i ϕ 1 is a solution of the problem (K ϕ 1,ϕ 1 ) H = max (ϕ 1,ϕ 1 ) H ϕ H (K ϕ,ϕ) H (ϕ,ϕ) H = max ϕ H 1 N s ( N s U i,ϕ ) 2 H = λ 1 i=1 (ϕ,ϕ) H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

38 Reformulation of the maximization problem max ϕ H 1 N s ( N t U i,ϕ ) 2 H i=1 (ϕ,ϕ) H (K ϕ,ϕ) = max H ϕ H (ϕ,ϕ) H Proposition K is linear, symmetric and positive hence λ 1 λ 2 λ d > 0 and orthogonal {ϕ i } d i=1 H such that K ϕ i = λ i ϕ i ϕ 1 is a solution of the problem (K ϕ 1,ϕ 1 ) H = max (ϕ 1,ϕ 1 ) H ϕ H (K ϕ,ϕ) H (ϕ,ϕ) H = max ϕ H 1 N s ( N s U i,ϕ ) 2 H = λ 1 i=1 (ϕ,ϕ) H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

39 Proposition ϕ k (1 k d) is a solution of the maximization problem max ϕ H (ϕ,ϕ j ) H = 0,j < k (K ϕ,ϕ) H = max (ϕ,ϕ) H ϕ H (ϕ,ϕ j ) H = 0,j < k = λ k 1 N s ( N s U i,ϕ ) 2 H i=1 (ϕ,ϕ) H Optimality of POD basis : Given M d and {ψ i } M i=1 an orthonormal family 1 N s N s i=1 Ui M k=1 ( ) 2 U i,ϕ k H ϕ k H 1 N s N s Ui i=1 M k=1 ( ) 2 U i,ψ k H ψ k H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

40 Proposition ϕ k (1 k d) is a solution of the maximization problem max ϕ H (ϕ,ϕ j ) H = 0,j < k (K ϕ,ϕ) H = max (ϕ,ϕ) H ϕ H (ϕ,ϕ j ) H = 0,j < k = λ k 1 N s ( N s U i,ϕ ) 2 H i=1 (ϕ,ϕ) H Optimality of POD basis : Given M d and {ψ i } M i=1 an orthonormal family 1 N s N s i=1 Ui M k=1 ( ) 2 U i,ϕ k H ϕ k H 1 N s N s Ui i=1 M k=1 ( ) 2 U i,ψ k H ψ k H Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

41 Proposition ϕ k (1 k d) is a solution of the maximization problem max ϕ H (ϕ,ϕ j ) H = 0,j < k (K ϕ,ϕ) H = max (ϕ,ϕ) H ϕ H (ϕ,ϕ j ) H = 0,j < k = λ k 1 N s ( N s U i,ϕ ) 2 H i=1 (ϕ,ϕ) H Efficiency of POD basis 1 N s N s i=1 Ui M k=1 ( ) 2 U i,ϕ k H ϕ k H = d λ k i=m+1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

42 Reduced order POD basis : is the family of {ϕ i } N rom i=1 eigenvectors the most "energetic" modes associated to λ 1 λ 2 λ Nrom such that N rom λ i i=1 d i=1 λ i = 1 ε Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

43 Reduced order POD basis : is the family of {ϕ i } N rom i=1 eigenvectors computed via N dof N dof eigenvalue problem computation AA T ϕ i = λ i ϕ i i = 1,,N rom such that N rom λ i i=1 d i=1 λ i 1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

44 POD and Singular Value Decomposition (SVD) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

45 Let A = [U 1 U 2 U N s ] RN dof N s and d min(n dof,n s ) Singular Value Decomposition (SVD) Φ = [ϕ 1 ϕ Ndof ] R N dof N dof,ψ = [ψ 1 ψ Ns ] R N s N s and σ 1 σ 2 σ d > σ d+1 = = 0 such that D = diag(σ 1,,σ d ),Φ T Φ = I R N dof N dof,ψ T Ψ = I R Ns Ns Φ T AΨ = D Consequences AA T ϕ i = σ 2 i ϕ i, i = 1,,N dof A T Aψ i = σ 2 i ψ i, i = 1,,N s ϕ i = 1 σ 2 i Aψ i i = 1,,d Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

46 Let A = [U 1 U 2 U N s ] RN dof N s and d min(n dof,n s ) Singular Value Decomposition (SVD) Φ = [ϕ 1 ϕ Ndof ] R N dof N dof,ψ = [ψ 1 ψ Ns ] R N s N s and σ 1 σ 2 σ d > σ d+1 = = 0 such that D = diag(σ 1,,σ d ),Φ T Φ = I R N dof N dof,ψ T Ψ = I R Ns Ns Φ T AΨ = D Consequences AA T ϕ i = σ 2 i ϕ i, i = 1,,N dof A T Aψ i = σ 2 i ψ i, i = 1,,N s ϕ i = 1 σ 2 i Aψ i i = 1,,d Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

47 Snapshot POD method POD basis via N s N s eigenvalue problem computation, {ϕ N rom i=1 } computed by : A T Aψ i = λ i ψ i i = 1,,N rom such that ϕ i = 1 λ i Aψ i N rom λ i i=1 d i=1 λ i 1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

48 The essential cost for explicit : t F int (U(t), U(t)) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

49 Discrete Empirical Interpolation Method (DEIM) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

50 Let Ψ = [ψ 1 ψ Nrom ] R N dof N rom be the reduced basis such that U(t) = Ψα(t) α(t) R N rom t To determine α(t) = (α 1 (t),,α Nrom (t)) T R N rom, we have to solve Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

51 Let Ψ = [ψ 1 ψ Nrom ] R N dof N rom be the reduced basis such that U(t) = Ψα(t) α(t) R N rom t To determine α(t) = (α 1 (t),,α Nrom (t)) T R N rom, we have to solve α 1 (t)ψ 1 (x 1 ) + α 2 (t)ψ 2 (x 1 ) + + α Nrom (t)ψ Nrom (x 1 ) = U(x 1,t) α 1 (t)ψ 1 (x 2 ) + α 2 (t)ψ 2 (x 2 ) + + α Nrom (t)ψ Nrom (x 2 ) = U(x 2,t). α 1 (t)ψ 1 (x Ndof ) + α 2 (t)ψ 2 (x Ndof ) + + α Nrom (t)ψ Nrom (x Ndof ) = U(x Ndof,t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

52 Let Ψ = [ψ 1 ψ Nrom ] R N dof N rom be the reduced basis such that U(t) = Ψα(t) α(t) R N rom t To determine α(t) = (α 1 (t),,α Nrom (t)) T R N rom, we have to solve α 1 (t)ψ 1 (x 1 ) + α 2(t)ψ 2 (x 1 ) + + α N rom (t)ψ Nrom (x 1 ) = U(x 1,t) α 1 (t)ψ 1 (x 2 ) + α 2(t)ψ 2 (x 2 ) + + α N rom (t)ψ Nrom (x 2 ) = U(x 2,t). α 1 (t)ψ 1 (x N rom ) + α 2 (t)ψ 2 (x N rom ) + + α Nrom (t)ψ Nrom (x N rom ) = U(x Nrom,t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

53 Let Ψ = [ψ 1 ψ Nrom ] R N dof N rom be the reduced basis such that U(t) = Ψα(t) α(t) R N rom t To determine α(t) = (α 1 (t),,α Nrom (t)) T R N rom, we have to solve P T U(t) = P T Ψα(t) ] P = [e x e 1 x e 2 x R Nrom N dof N rom e x i = (0,,,0,1,0,,0) T ( The x i column of I Ndof N dof ) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

54 Let Ψ = [ψ 1 ψ Nrom ] R N dof N rom be the reduced basis such that U(t) = Ψα(t) α(t) R N rom t To determine α(t) = (α 1 (t),,α Nrom (t)) T R N rom, we have to solve P T U(t) = P T Ψα(t) ] P = [e x e 1 x e 2 x R Nrom N dof N rom e x i = (0,,,0,1,0,,0) T ( The x i column of I Ndof N dof ) α(t) = (P T Ψ) 1 P T U(t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

55 1: INPUT : [ψ 1,,ψ Nrom ] 2: OUTPUT : {x i } i=1,,n rom 3: INITIALIZATION : { x 1 = argmax ψ 1, Ψ = [ψ 1 ],P = [ e x 1 ] 4: for l = 2 to N rom do 5: Solve P T Ψα = P T Ψ l for α 6: R = ψ l Ψα 7: x l = argmax R 8: UPDATE : Ψ = [ Ψ,ψl ],P = [ ] P,e x l 9: end for Algorithm 1: DEIM Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

56 Hyper reduction The knowledge of the unknown field U(t) at "well selected points" i.e. P T U(t) with a given reduced basis Ψ is sufficient to approximate U(t) over all the domain. { α(t) = (P T Ψ) 1 P T U(t) U(t) = Ψα(t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

57 Hyper reduction Ingredients Reduced Basis Ψ Ω RID Ω : selected points Ω RID/ Γ N Γ B Γ D Solve on Ω RID MÜ(t) + F int ( U(t), U(t) ) = F ext U(0) = U 0 U(0) = U0 Ω But how to treat the interface Γ B? Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

58 Hyper reduction Ingredients Reduced Basis Ψ Ω RID Ω : selected points Ω RID/ Γ N Γ B Γ D Solve on Ω RID MÜ(t) + F int ( U(t), U(t) ) = F ext U(0) = U 0 U(0) = U0 Ω But how to treat the interface Γ B? Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

59 Boundary Condition over Ω RID We denote Ω RID = Ω int Γ B U ΩRID (t) = Ψ ΩRID α(t) { U Ωint (t) = Ψ Ωint α(t) U ΓB (t) = Ψ ΓB α(t) α(t) = U ΓB (t) = Ψ ΓB α(t) ( ) 1 Ψ T Ωint Ψ Ωint Ψ T Ωint U Ωint (t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

60 Boundary Condition over Ω RID We denote Ω RID = Ω int Γ B U ΩRID (t) = Ψ ΩRID α(t) { U Ωint (t) = Ψ Ωint α(t) U ΓB (t) = Ψ ΓB α(t) α(t) = U ΓB (t) = Ψ ΓB α(t) ( ) 1 Ψ T Ωint Ψ Ωint Ψ T Ωint U Ωint (t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

61 Boundary Condition over Ω RID We denote Ω RID = Ω int Γ B U ΩRID (t) = Ψ ΩRID α(t) { U Ωint (t) = Ψ Ωint α(t) U ΓB (t) = Ψ ΓB α(t) α(t) = U ΓB (t) = Ψ ΓB α(t) ( ) 1 Ψ T Ωint Ψ Ωint Ψ T Ωint U Ωint (t) Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

62 Explicit time discretization :central finite difference ( Ü 0 = M 1 F ext F int (U 0, U ) 0 ) U 1 2 = U 0 + t1 2 Ü 0 n = 1 U n = U n 1 + t n U n 1 2 ) Ü n = M (F 1 ext F int (U n, U n 1 2 ) t n+ 1 2 = tn + t n+1 2 U n+ 1 2 = U n t n+ 1 2 Ü n n = n + 1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

63 Hyper reduction for explicit scheme Ü 0 = M (F 1 ext ΩRID F int ΩRID (U 0, U ) 0 ) Ω RID U 1 2 = U 0 + t1 2 Ü 0 ( ) U ΓB = Ψ ΓB Ψ T Ψ Ω Ωint Ψ Ωint U 1 2 int n = 1 Ωint U n = U n 1 + t n U n 1 2 ) Ü n = M (F 1 ext ΩRID F int ΩRID (U n, U n 1 2 ) Ω RID t n+ 1 2 = t n + t n+1 2 U n+ 1 2 = U n t n+ 1 2 Ü n U n+ 1 2 Γ B = Ψ ΓB ( Ψ T Ω int Ψ Ωint ) 1 Ψ Ωint U n+ 1 2 Ω int n = n + 1 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

64 Taylor Impact T = 10µs N rom = 19,Ω RID = 17% of Ω, relative error : on displacement = 0.9%, on velocity = 2%, Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

65 Taylor Impact T = 10µs N rom = 19,Ω RID = 17% of Ω, relative error : on displacement = 0.9%, on velocity = 2%, Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

66 Modes FIGURE : Left : Ψ 1, Right : Ψ 2 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

67 Modes FIGURE : Left : Ψ 3, Right : Ψ 9 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

68 HROM Vs HFM Time Saving : on internal forces 75% Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

69 Taylor Impact T = 40µs, with two local bases [0,20µs] and [20µs, 40µs] N rom = (20,25),Ω RID = 30% of Ω, relative error : on displacement = 2%, on velocity = 4%, Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

70 Taylor Impact T = 40µs, with two local bases [0,20µs] and [20µs, 40µs] N rom = (20,25),Ω RID = 30% of Ω, relative error : on displacement = 2%, on velocity = 4%, Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

71 Modes of the RB [0,20µs] FIGURE : Left : Ψ 1, Right : Ψ 2 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

72 Modes of the RB [0,20µs] FIGURE : Left : Ψ 3, Right : Ψ 5 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

73 Modes of the RB [20µs,40µs] FIGURE : Left : Ψ 1, Right : Ψ 2 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

74 Modes of the RB [20µs,40µs] FIGURE : Left : Ψ 3, Right : Ψ 5 Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

75 Parametric study over 0.08GPa < Et < 0.2GPa The variation of the plasticity rate is between 4% to 16% Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

76 Error on displacement Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

77 Shell with small rotations U 0 = 56m/s, L = 300mm, T = 0.3ms ν = 0.3, σ e = 0.2GPa, E = 100GPa. Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

78 Shell impact with small rotations Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50

79 Large rotations Fatima Daim (ESI Group) ROM Crash Simulation 28 th january / 50