Simulations with adaptive modeling

Transcription

1 Simulations with adaptive modeling Alexandre Ern CERMICS Ecole des Ponts, ParisTech INdAM Workshop, Cortona, 21/09/06

2 Introduction Engineering problems often involve models of different complexity, scales and accuracy Most accurate and validated model cannot be chosen because of computational costs Simpler models are computationally efficient but may only capture a few features of original problem

3 Adaptive model simulation Apply detailed model only where needed Balance model and discretization errors Estimate modeling errors together with discretization errors Main tool: A posteriori error indicators of modeling and of discretization errors Reliable (upper bound) Efficient (lower bound) Localized to mesh cells

4 Literature overview Computational solid and fluid mechanics Fatone, Gervasio & Quarteroni 01 (NS/Oseen) Perotto 04 (Shallow-water eqs.) Amara, Capatina & Trujillo 04 (Free-surface NS 3D/2D/1D) Oden & Vemaganti 00, Oden & Prudhomme 02 (elasticity) elliptic problems on thin plates Vogelius & Babuška 81, Babuška & Schwab 96, Ainsworth 98 Heterogeneous Multiscale Methods E & Engquist 03

5 Outline 1 error estimates & computational results coll. with M. Braack [Univ. Heidelberg] 03 2 no error estimates, promising computational results coll. with T. Lelièvre [ENPC] & M. Braack 3 adaptive modeling with functional space conformity error estimates, implementation in progress coll. with S. Perotto & A. Veneziani [MOX]

6 linear model error nonlinear model error nonlinear model and discretization errors selecting and evaluating the error indicators example on a toy problem application: adaptive diffusion modeling in flames

7 Linear model error Exact solution u V : a(u, φ) }{{} + d(u, φ) }{{} simple model complexities = (f, φ) }{{} data φ V Approximate solution (same functional space) u m V : a(u m, φ) = (f, φ) φ V Both problems are assumed to be well posed Given linear output functional j, estimate modeling error j(e u ) = j(u u m ) = j(u) j(u m )

8 Main tool: Duality-based a posteriori error estimates discretization error estimates [Johnson 95, Becker & Rannacher, 96] Dual problem { z V : a(φ, z) + d(φ, z) = j(φ) φ V j(e u ) = d(u m, z) Reduced dual problem { zm V : a(φ, z m ) = j(φ) φ V j(e u ) = d(u m, z m ) + R where R = 1 2 [(d(u m, e z ) + d(e u, z m )] is a h.o.t. in modeling errors e u := u u m and e z := z z m

9 Nonlinear model error Semilinear forms a(u; φ) and d(u; φ) Nonlinear output functional j(u) Directional (Gateaux)-derivative a 1 ( ) (u; v, φ) = lim a(u + ɛv; φ) a(u; φ) ɛ 0 ɛ

10 Full and reduced models u V : a(u; φ) + d(u; φ) = (f, φ) φ V u m V : a(u m ; φ) = (f, φ) φ V Full and reduced dual problems z V : a (u; φ, z) + d (u; φ, z) = j (u; φ) φ V z m V : a (u m ; φ, z m ) = j (u m ; φ) φ V Dual problems are linear

11 Error representation j(u) j(u m ) = d(u m ; z m ) (order 0) 1 2 [d(u m; e z ) + d (u m ; e u, z m )] (order 1) R (order 3) with e u := u u m and e z := z z m

12 Nonlinear model and discretization error Discretization subspace V h V Residuals ρ(u hm )(φ) := (f, φ) a(u hm ; φ) ρ (u hm, z hm )(φ) := j (u hm ; φ) a (u hm ; φ, z hm ) Discrete reduced problems u hm V h : ρ(u hm )(φ) = 0 φ V h z hm V h : ρ (u hm, z hm )(φ) = 0 φ V h

13 Error representation j(u) j(u hm ) = d(u hm ; z hm ) (order 0) 1 2 [ρ(u hm)(z i h z) + ρ (u hm, z hm )(u i h u)] (order 0) 1 2 [d(u hm; e z ) + d (u hm ; e u, z hm ) (order 1) R (order 3) for arbitrary interpolation operator i h : V V h

14 Recall: j(u) j(u hm ) = d(u hm ; z hm ) 1 2 [ρ(u hm)(z i h z) + ρ (u hm, z hm )(u i h u)] 1 2 [d(u hm; e z ) + d (u hm ; e u, z hm )] R Model error indicator η m = d(u hm ; z hm ) Mesh error indicator η h = 1 2 [ρ(u hm)(z i h z) + ρ (u hm, z hm )(u i h u)] Neglect higher-order terms some can be estimated by solving local problems

15 Evaluating the error indicators η m = d(u hm ; z hm ) can be readily evaluated Many techniques available to evaluate η h V h : pcw. linears on locally refined, tensor-product meshes V (2) 2h : pcw. quadratics on coarser mesh interpolation operator i (2) 2h : V h V (2) 2h

16 Approximate u i h u i (2) 2h u hm u hm z i h z i (2) 2h z hm z hm Mesh error indicator η h = 1 2 [ρ(u hm)(i (2) 2h z hm) + ρ (u hm, z hm )(i (2) 2h u hm)]

17 Localize error indicators η m and η h no theoretical justification Nodal localization of model error indicator Lagrangian nodal basis {φ i } V h U, Z : nodal components of u hm and z hm nodal model error estimator η m,i = Z i d(u hm ; φ i )

18 Nodal localization of mesh error indicator quadratic basis functions φ (2) i = i (2) 2h φ i residual contributions Ψ i = ρ(u hm )(φ (2) i ) Ψ i = ρ (u hm, z hm )(φ (2) i ) P nodal mesh error estimator η h = 1 2 i (Ψ iz i + Ψ i U i ) Filter to remove node to node oscillation of residual filter operator π h φ = φ i2hφ h filtered estimator η h = 1 2 Pi (Ψπ i Z i + Ψ π i U i ) PSfrag replacements i (2) 2h φ φ π h φ

19 Adapting the model and the mesh Adaptive step i Discrete subspace V i depends on i because of mesh refinement Semilinear form a i depends on i because of model modification Solve u i V i : a i (u i ; φ) = (f, φ) φ V i z i V i : a i (u i; φ, z i ) = j (u i ; φ) φ V i

20 Transfer nodal error indicators to mesh cells Error balancing to select cells where modifications occur η i > αη α (0, 1) Either the model or the mesh is modified locally check locally which error indicator is larger

21 Example on toy problem Model problem: (µ u) = f on Ω = (0, 1) 2 Adaptivly choosing between detailed model: µ oscillates on upper right quadrant simplified model: µ 1 everywhere Very fine mesh is used (no discretization error)

22 j(u) = u dx (0.8,1) (0.4,0.6) iter % exact m. η m j(e) I eff e e e e e e e e e e e e

23 Areas with exact model (light) and crude model (dark) 2 iterations 4 iterations

24 Application: adaptive diffusion models in flames Systems of nonlinear coupled PDE s Ozone flame (6 PDE s) moderate impact of detailed model Hydrogen flame (12 PDE s) sizeable impact of detailed model

25 Ozone flame Ozone flame: 2O 3 + O 2 4O 2 Three chemical species participating in 6 elementary reactions with Arrhenius type rates O 3, O 2 and O Reaction A β E a O+O+M O 2 +M O 2 +M O+O+M O 2 +O+M O 3 +M O 3 +M O 2 +O+M O+O 3 O 2 +O O 2 +O 2 O+O

26 2D premixed flame in a slot with heated lateral walls Temperature and O-radical profiles

27 Governing equations compressible NS equations for 2D flow velocities energy balance for temperature 3 mass balance equations for chemical species 6 coupled, strongly nonlinear PDE s Species balance equations v(y) y + F(y) = f (y) y = (y O3, y O2, y O ) T : species mass fractions F = (F O3, F O2, F O ) T : species diffusion fluxes

28 Simplified model Fick type diffusion F i = D i y i Detailed model multicomponent diffusion and thermal diffusion (Soret effect) F i = X j D ij (y) y j θ i (y) T D ij and θ i can be evaluated from the kinetic theory of gas mixtures need to solve constrained linear systems of size the number of species at each mesh node use transport algorithms of Ern & Giovangigli 94

29 Output functional j(u) = c y O Ω Reference solution with detailed model on very fine mesh Impact of detailed model is moderate #nodes % of multi. η h η m η j(u u hm ) I eff e e e e e e e e e e e e e e e e e e e e e e

30 Convergence history as a function of number of nodes 10 1 eta mesh eta model eta error # nodes

31 Local model and mesh adaption after 2, 3 and 4 iterations

32 Hydrogen flame 9 chemical species, 19 elementary reactions Underventilated diffusion flame flown from a slot in periodic setting Simplified model: Fick type diffusion Detailed model: Multicomponent diffusion with Soret effect Sizeable impact of detailed model simplified detailed

33 Output functional j(u) = Γ y OH along a line Γ cutting the flame #nodes % of multi. j(u h ) η h η m η e e e e e e e e e e e e e e e-03

34 Comparison of OH mass fraction along a line Profile well captured after only 2/3 adaptive steps FL MC/FK 2.Adap. step MC/FK 3.Adap. step MC mass fraction OH y-coordinate

35 Dilute polymeric fluids non-interacting polymer chains diluted in a Newtonian solvent Rheology of the fluid influenced by polymer chains: Non-Newtonian behavior Stress tensor depends in a complicated manner on the history of the deformation of the fluid

36 Micro-macro models microscopic description of the dynamics of polymer chains stochastic PDE modeling solvent-polymer interactions coupled to macroscopic description of solvent through stress tensor compare favorably with experiments computationally intensive CONNFFESSIT approach (Monte Carlo+FE) [Laso & Öttinger 93; Hulsen et al. 97; Bonvin & Picasso 99, Keunings 00] Macro models full macroscopic description can be derived from micro-macro models via closure relations linking stress and strain (e.g. Oldroyd-B) less favorable comparison with experiments computationally cheaper than micro-macro models

37 Micro-macro model Dumbbell model polymer chain modeled by two beads linked by a spring end-to-end vector: random variable X t Brownian configuration field: X t depends on x Governing equations Re ( t u + u u) = (1 ɛ) u p + τ u = 0 τ = ɛ We (E (X t Π(X t )) Id ) ( dx t + u x X t dt = x ux t 1 ) 2We Π(X t) dt + 1 dw t We dw t : d-dimensional standard Brownian motion F (X t ) := Π(X t ) models the force between the two beads

38 Non-dimensional numbers Re : Reynolds number We : Weissenberg number (polymer chain time to fluid time ratio) ɛ (0, 1): polymer chain viscosity to total viscosity ratio Finite extensible nonlinear elastic (FENE) dumbbells [Warner 72] Π(X ) = b 2 ln ( 1 ) X 2 b Polymer contribution to stresses ( τ = E ɛ We ( X t X t 1 X t 2 /b F (X ) = ) ) Id X 1 X 2 /b

39 Macro model FENE-P model [Peterlin 66; Bird et al. 81] F (X t ) = X t 1 E( X t 2 )/b Covariance tensor obtained from nonlinear macroscopic PDE da dt + u A ua + A u T = 1 We A 1 tr(a)/b + 1 We Id and recover τ = ɛ We ( ) A 1 tr(a)/b Id FENE-P model can be simulated by deterministic methods

40 Adaptive micro-macro approach Use elementwise either detailed (FENE, micro-macro) or simplified (FENE-P, macro) models Solvers FENE: CONNFFESSIT method plus VR FENE-P: P 1 (velocity)/p 0 (stresses) FE Adaptive algorithm For each time step, at each mesh cell, 1 compute stress tensor and velocity 2 estimate alternative stress tensor and velocity (which would have been obtained with the other model) 3 based on these estimates, possibly switch model

41 If detailed (FENE, micro-macro) model is used in a cell, Monte Carlo method to solve stochastic PDE for M dumbbells switching to simplified model simply involves ensemble averages If simplified (FENE-P, macro) model is used in a cell, evolve 1 M, δ 1, dumbbells to compute estimates δ switch to detailed model by replicating the ensemble of 1 M δ dumbbells δ times

42 Results Start-up of shear flow (1D problem) 1.5 y 1 u=0 1 inflow outflow 0 u= Initially, flow is at rest and dumbbells at equilibrium X 0 follows the law b+2 2πb (1 X 2 /b) b/2 1 X 2 <b dx Simplified model initially used in each cell Parameters: Re = 0.1, We = 0.5, ɛ = 0.9, b = 20, M = 10 4, δ = 10 2, I = 10 mesh cells, N = 2000 timesteps

43 Speedups w.r. to execution time of fully detailed model η speedup η: error control parameter Number of cells where detailed model is used 10 η = 0.1 η = 0.01 η = PSfrag replacements

44 Accuracy assessment: time evolution of relative velocity error η = 0.1 η = 0.01 η = g replacements

45 Elliptic problems on thin plates [Babuška & Vogelius 81] problem dimension reduced by 1 (3D 2D; 2D 1D) reduced space spanned by modal basis functions reduced solution: energy projection of exact solution onto reduced space careful selection of modal basis for asymptotic optimality w.r. to plate thickness A posteriori error analysis residual-type modeling error estimates [Babuška & Schwab 96] modeling and discretization error estimates by solving local problems [Ainsworth 98]

46 The present setting Fibre domain Ω = x Ω 1 γ x with 1D generatrix Ω 1 and fibers γ x PSfrag replacementsω γ x x Ω 1 Geometric maps ψ x : γ x S d 1 R d 1 C 1 -diffeomorphisms with smooth dependence on x

47 Model reduction: 3D 1D Targetted applications haemodynamics (blood flows in vessels) hydraulics (river flows in estuaries) Simpler setting: elliptic linear PDE with energy space V u V : a(u, v) = f (v) v V

48 Reduced energy space: Two ingredients energy space for 1D problem on Ω 1 : V 1D modal basis functions {ϕ k } k N, ϕ k : S d 1 R V m = n v(x, y) = mx v k (x) ϕ k (ψ x(y)), with v k V 1Do k=1 Conformity: V m V Spectral approximability v V, lim m + ( ) inf v v m V v m V m = 0

49 Reduced problem: energy projection of exact solution onto V m u m V m : a(u m, v m ) = f (v m ) v m V m Spectral optimality u u m V inf v m V m u v m V Reduced problem amounts to solving m coupled 1D problems

50 Illustrations Transverse effects induced by source term mode modes Transverse effects induced by shape of domain A. Ern INdAM Workshop, Cortona 09/06 Simulations with adaptive modeling 20 25

51 FE approximation of reduced problem FE approximation for 1D problem, mesh T h V 1D h V 1D Spatially variable modal expansions m := {m T } T Th Vhm { T = m T } T h, v T = v kh (x) ϕ k (ψ x (y)), with v kh Vh 1D k=1 Conforming subspace V hm = V hm V

52 Discrete reduced problem u hm V hm : a(u hm, v hm ) = f (v hm ) v hm V hm Practical implementation discrete reduced problem posed on V hm matching conditions at interfaces between different modal expansions enforced by DN-type iterations

53 A posteriori error analysis Hierarchical setting Spectrally enriched space m V + hm { T = + T } T h, v T = v kh (x) ϕ k (ψ x (y)), with v kh Vh 1D k=1 with m + T > m T for all T T h Auxiliary problem to be solved u + hm V + hm : a(u + hm, v + hm ) = f (v + hm ) v + hm V + hm

54 Output functional J : V R Saturation assumption J(u u + m) β J(u u m ) (β (0, 1)) yields J(u u hm ) J(u m + u+ hm }{{ ) + J(u m u hm ) + J(u + }}{{} hm u hm) }{{} A B C A + B: discretization errors estimated by standard techniques C: modeling error readily computable Global lower bounds can be established under additional assumptions Implementation in progress

55 Conclusions Adaptive model/mesh simulations driven by a posteriori error estimates are feasible Many interesting mathematical issues remain open, including error analysis for coupled stochastic/determinsitic PDE s well-posedness of dimensionally reduced problems beyond coercivity paradigm [Amara et al. 04] Is it computationally worthwhile? moderate speedups in flame simulation: model and mesh refinement strongly correlated significant speedups for polymeric fluid flow simulations speedups must be weighted against implementing complex code

56 References M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul., 1(2), (2003). M. Braack and A. Ern, Coupling multimodeling with local mesh refinement for the numerical computation of laminar flames, Combust. Theory Model., 8(4), (2004). M. Braack, A. Ern and T. Lelièvre, Adaptive models for the simulation of polymeric fluid flows, In preparation (2006). A. Ern, S. Perotto and A. Veneziani, Hierarchical model dimension reduction, In progress (2006).