Error estimation and adaptivity for model reduction in mechanics of materials

Transcription

1 Error estimation and adaptivity for model reduction in mechanics of materials David Ryckelynck Centre des Matériaux Mines ParisTech, UMR CNRS février 2017

2 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method MotivationsError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

3 Motivations To find a way to efficiently assess the accuracy of reduced basis approximation without computing "truth" finite element solutions. To find a way to efficiently adapt the reduced basis approximation with few relevant modes, or a way to sample parameter spaces. MotivationsError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

4 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method Local enrichment of a reduced basiserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

5 Local enrichment of reduced basis Prior any adaptation, local enrichment can improve the local prediction of stresses in mechanics. In [Ryckelynck 2016], the derivative extended POD (DEPOD) 1 have been used to generate additional local empirical modes. The DEPOD reads : with β 1/2. [Q, β Q Q Q] = V S WT + R Local modes are generated by considering a numerical inclusion that perturb the displacement field. 1. Derivative-Extended POD Reduced-Order Modeling for Parameter Estimation, Schmidt A., Potschka A., Koerkel S., Bock, H. G.,SIAM J. Sci. Comput., Vol. 35, 6, pp. A2696-A2717, Local enrichment of a reduced basiserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

6 Local enrichment of an hyper-reduced order model Numerical example : life duration prediction of a turbine disk. ω Virtual inclusion Zone of interest A numerical inclusion perturbs the solution locally 2. DEPOD modes. 2. Ryckelynck D., Lampoh K., Quilici, S., Hyper-reduced predictions for lifetime assessment of elasto-plastic structures, Meccanica, Vol. 51, 2, (2016), pp Local enrichment of a reduced basiserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

7 Local enrichment of an hyper-reduced order model Ω Z Γ The inclusion is added to the RID EF HR and POD 100 EF HR and DEPOD Improvement of local stress-strain prediction. Local enrichment of a reduced basiserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

8 Work in progress : bubble modes generated by image segmentation techniques Local reduced bases are proposed in 3. When POD reduced bases are large, they involve modes with many local fluctuations. The segmentation of these flucturations gives local bubble modes 4. These modes make simulation more stable and accurate. Bubble modes extracted by a region growing algorithm 3. D. Amsallem, M. J. Zahr, and Farhat C. Nonlinear model order reduction based on local reduced-order bases. International Journal for Numerical Methods in Engineering, 92 : , Bubble extension of hyper-reduced models for temperature gradient assessment in casting solidification simulation, Yang Zhang, David Ryckelynck, Michel Boussuge, International Journal of Thermal Sciences, submitted Local enrichment of a reduced basiserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

9 Hybrid approximation Full-order/hyper-reduced-order model As proposed in [Ammar 2009] [Kerfriden 2012] [Baiges 2013] [Radermacher 2014] empirical modes can be coupled to finite element approximations for nonlinear structural problems involving heat conduction, plasticity or damage. Ω = Ω FOM Ω R, Q[R, :] = V R S W T + R Ω FOM i H supp(ϕ i ), R = {1,..., N }\H where N = N R + N F. ψ k (x) = i R ϕ i (x) v R ik, x Ω, k = 1,..., N R ψ k (x) = ϕ j (x), x Ω, j = Card(R) N R + k, k = N R + 1,... N [ VR 0 V = 0 I Hyper-reduction in this context is very versatile. ] R N N (1) Local enrichment of a reduced basiserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

10 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method a priori and a posteriori approches in reduced-order modelingerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

11 a priori and a posteriori approches in reduced-order modeling In a posteriori model reduction methods, full order simulations and reduced simulation are not performed simultaneously. First, during an offline step full order predictions and a reduced basis are generated. After, the reduced-order predictions are performed during the online step. In a priori 5 model reduction methods, both reduced-basis construction and reduced-order predictions are preformed simultaneously. These methods involve an on-fly adaptive scheme An a priori model reduction method for thermomechanical problems, Réduction a priori de modèles thermomécaniques, D. Ryckelynck, Comptes Rendus Mécanique, Volume 330, Issue 7, Pages , (2002). 6. B.O. Almroth, P. Stern, F. A. Brogan, Automatic Choice of Global Shape Functions in Structural Analysis, AIAA, Vol. 16, 5, 1978, pp a priori and a posteriori approches in reduced-order modelingerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

12 Few remarks about the Reduced Basis Method and its greedy algorithm [Patera 2007] 8 [Haasdonk 2008] 9 The Reduced-Basis method is an a posteriori model reduction method. In case of elliptic problems with affine parameter dependence [Nguyen 2005] 7 : Q a(w, u; µ) = f (w) w V, a(w, v; µ) = Θ j (µ) a j (w, v) j=1 a sharp a posteriori error upper bound is available : (µ). The residual, denoted by L, reads : (L(u ROM ; µ), w) = f (w) a(w, u; µ) w V (µ) = L β where β is the Babuska "Inf-Sup" stability constant related to a. The key point is : the computational complexity of the error estimator is independent of N (FE dimension). 7. Nguyen NC, Veroy K, Patera AT. Certified real-time solution of parametrized partial differential equations. Handbook of Materials Modeling. Springer : Berlin, 2005 ; A.T. Patera, G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, http : //augustine.mit.edu/methodology/methodology_book.htm. 9. Haasdonk B., Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, M2AN, 42(2) : , 2008 a priori and a posteriori approches in reduced-order modelingerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

13 Few remarks about the Reduced Basis Method and its greedy algorithm Greedy algorithms are quasi-exhaustive sampling procedures, meant to perform the offline step. Let s consider a set of training points in the parameter space : Σ train = {µ i D, i = 1,...n train >> 1} A set of snapshots {u(µ i1,..., u(µ in )} can be supplemented by u(µ in+1 ) until (µ in+1 ) < ε o : µ in+1 = arg max µ Σ train (µ) In a posteriori approaches one need to evaluate the reduced approximation prior to the error estimation. An alternative approach to this are the a priori model reduction approaches. a priori and a posteriori approches in reduced-order modelingerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

14 a priori model reduction [Ryckelynck 2002] 10 For time dependent problems, new snapshots (full order solutions), can be incorporated on the fly, during the time integration of the equations. No snapshot selection is required when starting such reduced simulations. A (0) t i t i+1 time Just a HROM prediction if η(t i+1) > ε R, then find δu in Vh and expand the RB (n) (n+1) A A We obtain a non-incremental approach when restarting the full predictions after each reduced basis expansion. The PGD is an a priori model reduction method. 10. An a priori model reduction method for thermomechanical problems, Réduction a priori de modèles thermomécaniques, D. Ryckelynck, Comptes Rendus Mécanique, Volume 330, Issue 7, Pages , (2002). a priori and a posteriori approches in reduced-order modelingerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

15 Mesh and time discretization can be refined during the adaptation of the reduced approximation [Pelle 2000] 11 Fig. 22. Initial mesh M1. Fig. 23. Error controls during the iterations. Global error estimator at each adaptation step 4 meshes M1 M2 M3 M4 were generated during the adaptative procedure. e time ˆ 16%. Plasticity is signi cant (e time > 0:5e 0 ). Then, New modes are incorporated an initial at each iteration adaptation is performed step. to The take better meshaccount is adapted of if the related error indicator is too high. plasticity in the admissible solution (Rule R1). Thus, the error computation gives: e ref ˆ 20%, e h ˆ 16%, e time ˆ 12%. At the end of this rst iteration, space error are too high in comparison with the other error sources (e h > 11. An Efficient Adaptive Strategy e time to). Master Thanks the to Global the Quality local contribution of ViscoplasticiAnalysis, h, a newj-p. mesh Pelle et D. Ryckelynck, Computers & Structures, Ed. Elsevier Science, vol. M2 78, (94N elements) 1-3,pp , is automatically (2000) built to reduce the a priori and a posteriori approches in reduced-order modelingerror estimation and adptivity Doctoral courses, Mines ParisTech, january, 2016 space error. Data are transferred from the mesh M1 to 15/43

16 The A Priori Hyper-Reduction method (APHR) [Ryckelynck 2005] 13 The APHR method is an incremental a priori model reduction method. A (0) t i t i+1 time Just a HROM prediction if η(t i+1) > ε R, then find δu in Vh and expand the RB (n) (n+1) A A The main component of the method are : the hyper-reduction method, error estimators related to the reduced-basis approximation, having a computational complexity independent of N, an incremental POD algorithm that removes insignificant contributions to the reduced basis. When the hyper-reduction is canceled, we obtain the APR method Reduced-Order Modelling for solving linear and non-linear equations, N. Verdon, C. Allery, C. Béghein, A. Hamdouni, D. Ryckelynck, Communications in Numerical Methods in Engineering, en ligne (2010). 13. A priori hypereduction method : an adaptive approach, D. Ryckelynck, Journal of Computational Physics, Vol. 202, N 1, pp , (2005) a priori and a posteriori approches in reduced-order modelingerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

17 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method Incremental PODError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

18 Incremental singular value decomposition [Brand 2002] 14 Let s consider an existing SVD of a matrix of snapshots : Q = V diag(s) W T + E, V T V = I N, W T W = I N, E < ɛ tol The reduced-basis extension is performed by adding a new snapshot : [Q, q]. The reduced coordinate of q reads : β = V T q The orthogonal residual of the projection on V reads : δq = (I V V T ) q, p = δq T δq So : [ ] [ Vdiag(s)W T diag(s) β q = [V, δq/p] 0 p ] [ W ] T 14. Brand M., Incremental singular value decomposition of uncertain data with missing values. Computer Vision-ECCV 2002, Springer, pp , Incremental PODError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

19 Incremental singular value decomposition [Brand 2002] 14 Let s consider an existing SVD of a matrix of snapshots : Q = V diag(s) W T + E, V T V = I N, W T W = I N, E < ɛ tol The reduced-basis extension is performed by adding a new snapshot : [Q, q]. The reduced coordinate of q reads : β = V T q The orthogonal residual of the projection on V reads : So : δq = (I V V T ) q, p = δq T δq [ ] [ Vdiag(s)W T diag(s) β q = [V, δq/p] 0 p ] [ W Then, if p > ɛ R a new SVD is performed on a (N + 1) (N + 1) matrix : [ ] B diag(s ) G T diag(s) β = + E 0 p, E < ɛ tol The updated SVD reads : [Q, q] = Ṽ diag(s ) W T + Ẽ Ṽ = [V, δq/p] B [ ] W 0 W = G Brand M., Incremental singular value decomposition of uncertain data with missing values. Computer Vision-ECCV 2002, Springer, pp , Incremental PODError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 ] T

20 Incremental POD [Ryckelynck 2006] 15 Let s consider an existing POD : Q = V γ + E, V T V = I N, E < ɛ tol The reduced-basis extension is performed by adding a new snapshot : [Q, q]. The reduced coordinate of q reads : β = (V T V) 1 V T q The orthogonal residual of the projection on V reads : δq = (I V (V T V) 1 V T ) q, p = δq T δq So : [Vγ, q] = [V, δq/p] [ γ β 0 p ] 15. On the A Priori Model Reduction : Overview and Recent Developments. D. Ryckelynck, F. Chinesta, E. Cueto et A. Ammar, Archives of Computational Methods in Engineering, State of the Art Reviews, Special issue, Vol. 13, n 1, pp (2006). Incremental PODError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

21 Incremental POD [Ryckelynck 2006] 15 Let s consider an existing POD : Q = V γ + E, V T V = I N, E < ɛ tol The reduced-basis extension is performed by adding a new snapshot : [Q, q]. The reduced coordinate of q reads : β = (V T V) 1 V T q The orthogonal residual of the projection on V reads : So : δq = (I V (V T V) 1 V T ) q, p = [Vγ, q] = [V, δq/p] [ γ β 0 p δq T δq Then, if p > ɛ R a new eigendecomposition is performed on a (N + 1) (N + 1) matrix : [ ] [ ] T γ β γ β B Λ = B + E 0 p 0 p, E < ɛ 2 tol The updated POD reads : [Q, q] = Ṽ γ + Ẽ Ṽ = [V, δq/p] B [ ] γ = B T γ β 0 p 15. On the A Priori Model Reduction : Overview and Recent Developments. D. Ryckelynck, F. Chinesta, E. Cueto et A. Ammar, Archives of Computational Methods in Engineering, State of the Art Reviews, Special issue, Vol. 13, n 1, pp (2006). Incremental PODError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 ]

22 Incremental Gram Schmidt-based downdating technique A modified Gram Schmidt-based downdating technique for ULV decompositions with applications to recursive TLS problems, Hasan Erbaya, Jesse L. Barlowa, Zhenyue Zhang, Computational Statistics & Data Analysis 41 (2002) Incremental PODError estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

23 Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method

24 Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 Hyper-reduced predictions in mechanics of materials Let s consider an a posteriori approach applied to a static equilibrium coupled to the evolution of internal variables α. The output of the offline procedure are : the reduced basis V R N N, a reduced mesh of Ω Z, a set of DOF denoted by F that are not connected to the interface between Ω Z and Ω\Ω Z and a selection matrix of these DOF denoted by Z R Card(F) N.

25 Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 Hyper-reduced predictions (online procedure) The setting of the hyper-reduced equation reads : find γ(t; µ) R N such that ( N N ) u HROM = ϕ i (x) V ik γ k (t; µ) x Ω t [0, T ] i=1 k=1 V T Z T Z r(v γ, α) = 0 t [0, T ] α = g( V γ, α; x; µ) x Ω Z t [0, T ] y = l(v γ) where r is the finite element residual related to the residual L of the continuous equations : N (L(u HROM, α), w) = q T r(v γ, α), w = ϕ i (x) qi i=1 and y are the outputs of the simulation. Can we have access to an error estimator by using equations restricted to Ω Z?

26 Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 Example of an elastoviscoplastic problem. The numerical simulation aims to check the sensitivity of measured displacements to variations of constitutive coefficients. F(t) F(t) u=0 t The simulation outputs y are the transverse displacements at black points. The sensitivity is checked by this equation : y = Π u Π u(µ = µ 1 ), µ = µ 1 ± 30% The plastic flow follows a viscoplastic Norton law : ṗ =< J 2(σ X) K The vector of parameter reads µ = {K, C}. > n, X = C ε p Let s consider the metamodel µ ỹ(µ) = l(u HROM ; µ) obtained by the hyper-reduced model.

27 POD modes related to one sampling point µ = µ 1 in the parameter space Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 FIGURE: POD modes related to the decomposition of u(x, t; µ 1 )

28 The RID y z x The stress σ 11 over Ω (3400 elements) y z x sig11 map: time:0.1 min: max: The σ 11 over the RID Ω Z (700 elements) Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, 2016 sig11 map: time:0.1 min: max: /43

29 The simulation outputs are Lipschitz functions with respect to u 16. Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity, D. Ryckelynck, L. Gallimard, S. Jules, AMSES, (2015), 2 :6 doi : /s Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 l is a Lipschitz function : y = l(u HROM ), y FE = l(u FE ) c l R +, y y FE c l u HROM u FE where u HROM u FE is the approximation error related to the hyper-reduction. Can we have access to an upper bound of the approximation error by introducing an error estimator denoted by η ΩZ (u HROM ; µ)? u HROM u FE? β η ΩZ (u HROM ; µ)

30 The simulation outputs are Lipschitz functions with respect to u 16. Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity, D. Ryckelynck, L. Gallimard, S. Jules, AMSES, (2015), 2 :6 doi : /s Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 l is a Lipschitz function : y = l(u HROM ), y FE = l(u FE ) c l R +, y y FE c l u HROM u FE where u HROM u FE is the approximation error related to the hyper-reduction. Can we have access to an upper bound of the approximation error by introducing an error estimator denoted by η ΩZ (u HROM ; µ)? u HROM u FE? β η ΩZ (u HROM ; µ) Under this assumption we could have upper and lower bounds for the simulation outputs : l(u HROM ; µ) β η ΩZ (u HROM ; µ) l(u FE ; µ) l(u HROM ; µ) + β η ΩZ (u HROM ; µ), β = β cl where β can be estimated by using various truncated HROM for the prediction of u(x, t; µ 1 ) [Ryckelynck 2015] 16.

31 Numerical results Hyper-reduced predictions in mechanics of materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43 Exact metamodel y FE and estimated bounds predicted by the hyper-reduced solution and the error estimator. Output y HROM + β η ΩΖ y FE y HROM - β η ΩΖ C K Here u ROM, y HROM and η ΩZ (u ROM ) have been obtained by computation restricted to Ω Z : y z x

32 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method Error estimation by the recourse to the constitutive relation errorerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

33 Equilibrium residual When considering linear static equilibrium, the residual of the PDEs is connected to the stress tensor denoted by σ : (L(u HROM ), w) = σ : ε(w) dω F w dγ Ω F Ω where ":" is the inner product related to stress and strain tensors. The linear elasticity gives access to a statically admissible stress denoted by σ e : σ e : ε(w) dω F w dγ = 0 w V h Ω Moreover, the POD applied on stresses computed for µ = µ 1 gives access to a subspace of statically admissible stresses (for F = 0) [Kerfriden 2014] 17 : {ψ σ 1,..., ψσ N σ } = POD(σ(µ 1 ) σe ) ψ σ k : ε(w) dω = 0 w V h, k = 1,..., N σ Ω So, we can have a reduced approximation of statically admissible stress, denoted by σ : F Ω N σ γ σ σ = σ e + ψ σ k γσ k k=1 17. Kerfriden P, Ródenas JJ, Bordas SP-A (2014) Certification of projection-based reduced order modelling in computational homogenisation by the constitutive relation error. Int J Numer Meth Engng 97 : Error estimation by the recourse to the constitutive relation errorerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

34 Statically admissible stresses Usually the hyper-reduction does not provide admissible stress, then the residual is not zero. But we can define the closest statically admissible stress : T γ σ = arg min γ 0 Ω Z (σ (γ ) σ) : (σ (γ ) σ) dω dt N σ σ (γ ) = σ e + ψ σ k γ k k=1 here we have a well-posed problem if the matrix M kp = Ω ψ σ Z k : ψσ p dω is positive definite. This enters into consideration when constructing the RID. Then, the residual fulfills the following property : (L(u HROM ), w) = (σ σ) : ε(w) dω w V h Ω A constitutive relation error [Ladevèze 1983] is a norm of the discrepancy between σ obtained by the approximate solution and its statically admissible projection σ : η Ω = σ σ Ω,η where η is a convenient norm related to the constitutive equations involved in the mechanical problem. In the sequel η Ω is restricted to Ω Z and we introduce a tailored norm that establishes a connection between η Ω and u HROM u FE u. Error estimation by the recourse to the constitutive relation errorerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

35 Modes for statically admissible stresses Linear elastic solution for stresses : σ e Q σ e. Q σ Q σ e = V σ S σ W σt + R σ σ N ψ σ k (x) = ϕ σ i (x) vik σ i=1 y Ψ 1 y σ Ψ 1 z x z x z y x Ψ 2 z y x σ Ψ 2 σ = 0 11 z y x... Ψ 9 z y x... σ Ψ 17 The Gappy POD applied on stresses, gives access to the following statically admissible stresse : N σ σ = σ e + ψ σ k γσ k k=1 Error estimation by the recourse to the constitutive relation errorerror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

36 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

37 Numerical results : an estimation of the validity domain 4 x 104 N t Σ n=1 2 s(u ) - s(u ) FE HR F N t Σ n=1 n 2 u exp 4 x 104 c η η( (u HR n ) N, t (σ ^n ) Nt ) N t n 2 Σ u n=1 exp < 5% 3 < 5% C 2 C h On the left : the exact validity domain (error less than 5%) over the parameter space D R 2. On the right : the estimated validity domain obtained by η Ω. The hyper-reduced estimation of the metamodel and the constitutive relation error estimation have been performed according to a speedup of 6 compared to the usual FE approach. h Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

38 Standard materials An incremental variational principle [Ortiz 1999], [Miehe 2002], [Lahellec 2007] is available for standard materials. A time stepping is introduced : t 1,..., t i, t i+1,...t m. It exists a convex incremental potential denoted by w (ε) such that : σ i+1 = w ε (ε(u i+1/hrom)) Time indexes are removed from equations in the sequel. By recourse to the Legendre transformation, the dual potential of w, denoted by w, reads : w ( σ) = Sup ε (ε : σ w (ε )) σ The following properties are fulfilled : w (ε) + w ( σ) ε : σ 0 ε, σ σ = w ε (ε(u HROM)) w (ε(u HROM )) + w (σ) ε(u HROM) : σ = 0 ε(u HROM ) = w σ (σ) Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

39 Incremental setting of the constitutive relation error Following the method proposed by [Ladeveze, Moës], a constitutive relation error, denoted by η, can be introduced as follows : η ΩZ (u HROM, σ) = m i=1 Ω Z w (ε(u ROM )) + w ( σ) ε(u HROM) : σ dω 0, u ROM u d + V, σ σ e + Span(ψ σ 1,..., ψσ N σ ) Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

40 Incremental setting of the constitutive relation error Following the method proposed by [Ladeveze, Moës], a constitutive relation error, denoted by η, can be introduced as follows : η ΩZ (u HROM, σ) = m i=1 Ω Z w (ε(u ROM )) + w ( σ) ε(u HROM) : σ dω 0, u ROM u d + V, σ σ e + Span(ψ σ 1,..., ψσ N σ ) η ΩZ estimate the distance between σ and σ : η ΩZ (u HROM, σ) = m i=1 w ( σ) (w (σ) + w Ω Z σ (σ) : ( σ σ)) dω Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

41 Schematic view of the constitutive relation error We assume that w is a strictly convex potential : η ΩZ (u HROM, σ) = m i=1 w ( σ) (w (σ) + w Ω Z σ η ΩZ (u HROM, σ) = 0 σ = σ in Ω Z (σ) : ( σ σ)) dω 0 W Δ ^ W Δ ( σ ) W Δ ( σ ) σ σ^ d σ (σ ^, σ) W ( σ ) + σ W ( σ ^ ) : ( σ σ ) Δ σ Δ Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

42 Schematic view of the constitutive relation error We assume that w is a strictly convex potential : η ΩZ (u HROM, σ) = m i=1 w ( σ) (w (σ) + w Ω Z σ η ΩZ (u HROM, σ) = 0 σ = σ in Ω Z (σ) : ( σ σ)) dω 0 W Δ ^ W Δ ( σ ) W Δ ( σ ) σ σ^ d σ (σ ^, σ) W ( σ ) + σ W ( σ ^ ) : ( σ σ ) Δ σ Δ Similarly, one can define an estimator of strain discrepancies : u HROM u FE 2 u m i=1 Ω Z w (ε(u HROM ) w (ε(u FE )) w ε (ε(u FE )) : (ε(u HROM ) ε(u FE ))dω Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

43 The constitutive relation error and its partition [Ryckelynck 2015] 18 The following property holds : where e = u HROM u FE. N t η ΩZ (u HR, σ) = u i/hrom u i/fe 2 u + m σ i σ i/fe + δσ i 2 σ i=1 i=1 + m i=1 + Ω Z ε(e i ) : ( σ i σ i/fe ) dω m i=1 Ω Z ε(e i ) : δσ i dω The last term of the sum is a coupling term between the error e i and the error committed before the discrete time t i. Property : when Ω = Ω Z then, because of the static admissibility : ε(e i ) : ( σ i σ i/fe ) dω = 0, e i V h Ω Z Since we can t prove that this term is positive, η ΩZ is not a certified upper bound of the approximation error. 18. Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity, D. Ryckelynck, L. Gallimard, S. Jules, AMSES, (2015), 2 :6 doi : /s Constitutive relation error for standard materialserror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

44 Outline 1 Motivations 2 Local enrichment of a reduced basis 3 a priori and a posteriori approches in reduced-order modeling 4 Incremental POD 5 Hyper-reduced predictions in mechanics of materials 6 Error estimation by the recourse to the constitutive relation error 7 Constitutive relation error for standard materials 8 APHR method APHR methoderror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

45 Mathematical setting of APHR (A Priori Hyper-Reduction) method Prediction step (Petrov-Galerkin formulation) : V (n) rom = span{ψ (n) j N (n) such that u HROM (x, µ, t i ) = γ k (t i, µ)ψ (n) j (x) k=1 where V (m) Z rom (X)} N(n) L(u HROM (x, µ, t i )) v romdx = 0, v rom V (n) Ω (n) Z Ω Z rom j=1, Find {γ k (t i, µ)} N(n) k=1 is the subspace of test modes whose support is restricted to Ω(n) Z \(Ω Ω(n) Z ). Error estimation : η ΩZ (u HROM ) Correction step : N Find δu(x, µ, t i ) = δq j (t i, µ)φ j (x) such that j=1 L(δu + u HROM (x, µ, t i )) v h dx = 0, Ω v h V h Update the reduced basis by using δu V (m+1) rom and update the RID Ω (n+1) Z. APHR methoderror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

46 Example in cristal plasticity APHR methoderror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

47 Example in cristal plasticity EF Newton Raphson APHR U z Ω 100 Π Temps CPU time : FE = 1440s, APHR = 1176s APHR methoderror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43

48 Conclusion Key points for adaptivity are : an incremental scheme for the POD, an error estimator having a low computational complexity and the ability to find the worst point in the parameter space, in case of APHR the construction of the RID is performed on the fly. It must have a low computational complexity as well. Outlook : High Performance Computing, high dimensional parameter space, other tensor decompositions, less model reduction. APHR methoderror estimation and adptivity Doctoral courses, Mines ParisTech, january, /43