Numerical Solution of PDEs: Bounds for Functional Outputs and Certificates

Transcription

1 Numerical Solution of PDEs: Bounds for Functional Outputs and Certificates J. Peraire Massachusetts Institute of Technology, USA ASC Workshop, August 2005, Stanford University

2 People MIT: A.T. Patera, A.M. Budge, H. Ciria, and J. Wong UCS: J. Bonet UPC: N. Pares, A. Huerta NUS: Z.C. Xuan ASC Workshop, August

3 Acknowledgments Singapore-MIT Alliance Sandia National Laboratories DARPA/AFOSR ASC Workshop, August

4 Sleipner Platform Failure Sank in August 1991, causing an event registering 3.0 on the Richter scale and leaving nothing but a pile of debris at a depth of 220m Sinking traced to a failure of a concrete tricell FEM performed with NASTRAN underestimated shear stresses by 47% More precise simulation of underdesigned component predicted failure at 62m Actually sank at 65m ASC Workshop, August

5 Basic Question How do we know if the answer computed with a FE code is correct 1? given that: the solution may not be well behaved we may not have similar solutions to compare we may not have access to the source code the code may no longer exist!! 1 i.e. consistent with the mathematical model ASC Workshop, August

6 Basic Question How do we know if the answer computed with a FE code is correct? Provide a Certificate ASC Workshop, August

7 Certificates What is a Certificate? A data set that documents a given claim Can be used to rigorously proof correctness Simple to exercise Stand alone - access to the code used to compute it not required The stronger the claim the longer the certificate (usually) ASC Workshop, August

8 Certificates Current Paradigm COMPUTATIONAL CODE SOLUTION UNCERTAINTY... - ALGORITHM - SOFTWARE ASC Workshop, August

9 Certificates Proposed Paradigm COMPUTATIONAL CODE SOLUTION + CERTIFICATE + A-POSTERIORI BOUNDS ASC Workshop, August

10 Certificates Proposed Paradigm COMPUTATIONAL CODE SOLUTION + CERTIFICATE CERTAINTY GUARANTEED!! + A-POSTERIORI BOUNDS VERIFICATION TEST ASC Workshop, August

11 Certificates Disclaimer Physical Phenomenon Modelling Uncertainty Continuous Mathematical Model Discretization Uncertainty Discrete Mathematical Model Software Uncertainty Prediction ASC Workshop, August

12 Certificates Disclaimer Physical Phenomenon Modelling Uncertainty Continuous Mathematical Model Prediction with certified error bounds ASC Workshop, August

13 Certificates Examples Polynomial Bounds Given a polynomial F (x), x IR n Claim : F (x) γ, x Certificate : Polynomials f 1 (x),..., f m (x) s.t. or ( F (x) γ = n i=1 f 2 i m i=1 (x)) (F (x) γ) = m f 2 i (x) (SOS) i=n+1 f 2 i (x) ASC Workshop, August

14 Certificates Examples Bounds for solutions of IVP... Given ẋ = f(x, t), x(0) = x 0, (f(x, t) polynomial) Claim : x(t ) γ Certificate : Polynomial function B(x, t) s.t. B t (x, t) + B x (x, t)f(x, t) 0, x, t B(x T, T ) > B(x 0, 0), x T γ Parrilo, Doyle,... ASC Workshop, August

15 Certificates Examples...Bounds for solutions of IVP... B t (x, t) + B x (x, t)f(x, t) 0, x, t B(x T, T ) > B(x 0, 0), x T γ ASC Workshop, August

16 Certificates Examples...Bounds for solutions of IVP... B t (x, t) + B x (x, t)f(x, t) 0, x, t B(x T, T ) > B(x 0, 0), x T γ ASC Workshop, August

17 Certificates Examples...Bounds for solutions of IVP Given: ẋ = px 3 Barrier function B(x,t) 3 Separating level set x(0) [0.85, 0.95] x p [0.05, 0.2] x t t? x(2) [2.0, 2.5] x(2) / [2.0, 2.5] ASC Workshop, August

18 Objective Compute Certificates for Bounds of Outputs of PDE s Work with quantities of interest Work with equations of interest Guarantee certainty even for low cost Cost effective ASC Workshop, August

19 Objective Examples Elasticity Non-regular solution (Plane Stress) OUTPUTS l(u) = l(u) = Γ O u ds Γ 5 t x ds ASC Workshop, August

20 What can we do Today? Linear Functional Outputs for: - Convection-Diffusion-Reaction Equation (high Pe) - Linear Elasticity Equations - Stokes Equations (DG) Collapse Loads in Limit Analysis (SOCP) Energy Release Rates in Linear Elasticity ASC Workshop, August

21 Outline Problem Description Function Minimization/Duality in IR n Method Overview (for CG) 1.- Bounds for Energy 2.- Bounds for Arbitrary Outputs 3.- Bounds for Arbitrary Equations 4.- Domain Decomposition (Hybridization) Method Summary and Examples Extension to a non-linear Convex Problem: Limit Analysis ASC Workshop, August

22 Problem Description Let u(x) X, x IR d, be the solution of a PDE A u = f. e.g. A 2, 2 + U, etc. We are typically interested in outputs of the form s = l(u) IR, e.g. l(v) v(x 0 ), l(v) = v x dx,... ASC Workshop, August

23 Problem Description u(x) is not computable ( dimensional) In practice, we compute approximation ū(x), such that u ū = C( 0) (as cost increases ). - For a given ū, C is unknown, and, any output approximation s = l(ū), is uncertain. Existing error estimates are either, - certain but uncomputable, or, - computable but uncertain. ASC Workshop, August

24 Problem Description Approach Compute Strict upper and lower bounds for functional outputs of the Exact solutions of PDE s... and give Certificates ASC Workshop, August

25 Function Minimization f : Y IR Unconstrained e.g. Y IR n s = min v Y f(v) f(v ) ASC Workshop, August

26 Function Minimization Unconstrained Upper Bound s = min v Y f(v) f(v ) f( v) }{{} upper bound EASY! s +, v Y... we expect a reasonable bound if v v ASC Workshop, August

27 Function Minimization Unconstrained Upper Bound: Example... Linear-Quadratic Problem f(v) = 1 2 vt Av v T b, A IR n IR n SPD, b, v IR n Stationarity condition Av = b s = min v Y f(v) = f(v ) = 1 2 v T Av ASC Workshop, August

28 Function Minimization Unconstrained...Upper Bound: Example Let v = v + ɛ s + = f( v) = f(v ) ɛt Aɛ Error is O(ɛ 2 ) ASC Workshop, August

29 Function Minimization Unconstrained Lower Bound s = min v Y f(v) f(v ) It appears that we need to know v explicitly Not generally available?? ASC Workshop, August

30 Function Minimization f : Y IR, g : Y Λ Equality Constrained objective constraints e.g. Y IR n, Λ IR m s = min v Y g(v) = 0 f(v) ASC Workshop, August

31 Function Minimization Equality Constrained Lagrangian L(v, λ) : Y Λ IR L(v, λ) = f(v) + λ T g(v) It follows that λ : Lagrange multipliers s = min v Y max λ Λ L(v, λ) ASC Workshop, August

32 Function Minimization Equality Constrained Duality min v Y max λ Λ L(v, λ) max λ Λ min v Y L(v, λ) Proof: Let Then, L(v, λ (v)) max λ Λ L(v (λ), λ) min v Y L(v, λ) L(v, λ) L(v, λ (v)) L(v, λ) L(v (λ), λ), v, λ. ASC Workshop, August

33 Function Minimization Equality Constrained Lower Bound Since by duality, s = min v Y Then, s max λ Λ max λ Λ L(v, λ) max λ Λ G(λ) G( λ) }{{} lower bound min L(v, λ) } v Y {{} L(v (λ), λ) G(λ) Dual Function s, λ Λ... reasonable bounds to be expected when λ λ ASC Workshop, August

34 Function Minimization Unconstrained Lower Bound... s = min v Y f(v) Assume f(v) F (v, g(v)) and consider s = min v Y, w Λ g(v) w = 0 F (v, w) ASC Workshop, August

35 Function Minimization Unconstrained...Lower Bound Lagrangian L(v, w, λ) = F (v, w) + λ T (g(v) w) Dual function G(λ) = min L(v, w, λ) = v Y,w Λ L(v (λ), w (λ), λ) s = min v Y f(v) max λ Λ G(λ) G( λ) s, λ ASC Workshop, August

36 Function Minimization Unconstrained Lower Bound: Example... Linear-Quadratic Problem f(v) = v T K T Kv + v T b, K IR n IR n, b, v IR n f(v) F (v, g(v)) = g T g + v T b, g(v) = Kv IR n L(v, w, λ) = w T w + b T v + λ T (Kv w) ASC Workshop, August

37 Function Minimization Unconstrained...Lower Bound: Example Dual Function G(λ) = min v Y,w Λ L(v, w, λ) = { 1 2 λt λ if K T λ = b otherwise s = min v Y f(v) max G(λ) λ Λ 1 2 λ T λ s, λ s.t. Kλ = b ASC Workshop, August

38 Method Overview 1.- Energy s = J(u) Poisson s Equation: Find u X() 2 u = f(x), x, (+ b.c. s) Energy functional: J(v) : X IR J(v) = v v dx 2 fv dx ASC Workshop, August

39 Method Overview 1.- Energy s = J(u) Minimization Minimization formulation min v X J(v) = J(u) = uf dx ASC Workshop, August

40 Method Overview 1.- Energy s = J(u) Upper Bound Upper bound s + J(u h ), u h X h X ( trivial ) ASC Workshop, August

41 Method Overview 1.- Energy s = J(u) Lower Bound... Lower bound s ( harder ) Construct dual problem (J(u) =) J c (p) = max q Q f J c (q), ASC Workshop, August

42 Method Overview 1.- Energy s = J(u)...Lower Bound... s = min v X = min max v X q Q max min q Q v X = max q Q f Q f = {q Q ( v v 2vf) dx (q = v) ( q q + 2q v 2vf) dx ( q q + 2q v 2vf) dx q q dx q v dx = q=f fv dx, v X }{{} } ASC Workshop, August

43 Method Overview 1.- Energy s = J(u)...Lower Bound... s = min v X = min max v X q Q max min q Q v X = max q Q f Q f = {q Q J(v) J c (q) ( q q + 2q v 2vf) dx ( q q + 2q v 2vf) dx q v dx = q=f fv dx, v X }{{} } ASC Workshop, August

44 Method Overview 1.- Energy s = J(u)...Lower Bound... or, in a different way... q q dx 2 q q dx 2 q v dx + fv dx + }{{}}{{} (q v) 2 dx 0, v X, q Q v v dx 0, v X, q Q v v dx 0, v X, q Q f J c (q) + J(v) 0, v X, q Q f Q f = {q Q q v dx = fv dx, v X} ( q = f) J(v) J c (q), v X, q Q f ASC Workshop, August

45 Method Overview 1.- Energy s = J(u)...Lower Bound... Duality ASC Workshop, August

46 Method Overview 1.- Energy s = J(u)...Lower Bound... Then, s J c (p h ), p h (Q f ) h Q f. ASC Workshop, August

47 Method Overview 1.- Energy s = J(u)...Lower Bound Idea : We can exchange an infinite dimensional minimization problem by a finite dimensional feasibility problem while retaining the bounding property ASC Workshop, August

48 Method Overview 1.- Energy s = J(u) Lower Bound - Summary Claim : s = J(u) = Given 2 u = f(x) uf dx s Certificate : Any p h (Q f ) h Q f s.t. s J c (p h ) Recall: Q f = {q Q q v dx = fv dx, v X} ( q = f) ASC Workshop, August

49 Method Overview 2.- General Outputs s = l(u) Find s = l(u), where u X() (l(v) = or, 2 u = f(x), x, (+ b.c. s) ( u v fv) dx = 0, Modified Energy : E(v) : X IR E(v) v v dx v X f O v dx) fv dx E(u) = 0 ASC Workshop, August

50 Method Overview 2.- General Outputs s = l(u) Lagrangian s = l(u) = min v X ( v ψ fψ) dx = 0, ψ X l(v) +E(v) Lagrangian : L(v, ψ) : X X IR L(v, ψ) = E(v) + l(v) + ( v ψ fψ) dx s = l(u) = min v max ψ L(v, ψ) ASC Workshop, August

51 Method Overview 2.- General Outputs s = l(u) Lower Bound... Weak duality + Relaxation s = l(u) = min v max ψ L(v, ψ) max ψ min v L(v, ψ) min v L(v, ψ), ψ X ASC Workshop, August

52 Method Overview L(v, ψ) = 2.- General Outputs s = l(u) v v dx + l(v) +...Lower Bound... fv dx ( v ψ f ψ) dx For a given ψ, L(v, ψ), contains quadratic and linear terms in v identical to J(v) (for an appropriate f ψ). L(v, ψ) = v v dx 2 f ψv dx f ψ dx ASC Workshop, August

53 Method Overview 2.- General Outputs s = l(u)...lower Bound Idea : Write output as a constrained minimization problem. Relax constraint to obtain an energy-like minimization problem. Obtain lower bound by finding a feasible solution of the dual problem. ASC Workshop, August

54 Method Overview 2.- General Outputs s = l(u) Upper Bound Define l (v) = l(v) and compute, s l (u) Idea: s + s l (u) = l(u) Upper Bound for l(v) Lower Bound for l(v) ASC Workshop, August

55 Method Overview 2.- General Outputs s = l(u) Summary Given 2 u = f(x) Claim : s + s = l(u) s Certificate : ψ Xh X, p + h (Q f +) h Q f +, p h (Q f ) h Q f ASC Workshop, August

56 Method Overview 3.- Non-symmetric equations or, 2 u + U u = f(x), x, (+ b.c. s) ( u v + (U u)v fv) dx = 0, Modified Energy : E(v) : X IR E(v) v v dx v X fv dx E(u) = 0 ASC Workshop, August

57 Method Overview 3.- Non-symmetric equations Lagrangian... s = l(u) = min l(v) +E(v) v X ( v ψ+(u v)ψ fψ) dx = 0, ψ X Lagrangian : L(v, ψ) : X X IR L(v, ψ) = E(v) + l(v) + ( v ψ + (U v)ψ fψ) dx s = l(u) = min v max ψ L(v, ψ)... ASC Workshop, August

58 Method Overview 3.- Non-symmetric equations...lagrangian Idea : Non-symmetric terms do not contribute to the energy and only enter in the Lagrangian linearly. After relaxation, minimization problem retains convex structure. ASC Workshop, August

59 Method Overview 4.- Domain Decomposition Recall that a lower bound for s = J(u), is given by s = J c (q), q Q f Q f = {q Q q v dx = fv dx, v X} i.e. find q Q s.t. q = f, in... not trivial ASC Workshop, August

60 Method Overview 4.- Domain Decomposition v X() continuous ˆv ˆX() discontinuous X ˆX ASC Workshop, August

61 Method Overview 4.- Domain Decomposition v X() continuous ˆv ˆX() discontinuous Idea : solve local minimization problems use piecewise polynomial certificates ASC Workshop, August

62 Method Overview 4.- Domain Decomposition Continuity Form Jump bilinear form b : ˆX() Λ(Γ) IR b(ˆv, λ) = γ γ [ˆv] γ λ γ ds, X {ˆv ˆX b(ˆv, λ) = 0, λ Λ} ASC Workshop, August

63 Method Overview 4.- Domain Decomposition Relaxation... J(u) = min v X J(v) = min ˆv ˆX b(ˆv, λ) = 0, λ Λ J(ˆv) = min ˆv ˆX max λ Λ J(ˆv) + b(ˆv, λ) min ˆv ˆX J(ˆv) + b(ˆv, λ), λ Λ (equil.) } {{ } J λ(ˆv) Solve local minimization problems ASC Workshop, August

64 Method Overview 4.- Domain Decomposition...Relaxation Feasibility problem over each triangle q = f, in T e q n = λ, on T e has an explicit solution provided f and λ are polynomial Ladeveze... ASC Workshop, August

65 Method Overview Summary 1. Primal problem: u h X h Au h = f 2. Dual problem: ψ Xh A ψ = f O, (l(v) = f O v dx) ASC Workshop, August

66 Method Overview Summary 3. Domain decomposition (Equilibration) λ Global Solution Equilibrated Solution ASC Workshop, August

67 Method Overview Summary 4. Obtain lower bounds for local minimization problems s + s... and piecewise polynomial certificates 5. It can be shown that the bound gap can be written as s + s = with e 0 T e T H e... Adaptivity ASC Workshop, August

68 Examples Convection-Diffusion f =10 3 ν 2 u + U u = f f O =1 s = l(u) = f O u dx ASC Workshop, August

69 Examples Convection-Diffusion u: ψ: -7.1x x x10-04 Solution Adjoint ASC Workshop, August

70 Examples Convection-Diffusion Adaptive Solution gap = s = ± Gap: Elements Uniform refinement would require 6356 elements ASC Workshop, August

71 Examples Stokes Flows 1 1/2 1 1/2 σ + (p + p) = f u = 0 U X 2 X 1 w l(u) = p = 1 x 1 cylinder t x ds ASC Workshop, August

72 Examples Stokes Flows DG Results 1 Pressure h S S + 1/ / / Special attention paid to incompressibility constraint!! ASC Workshop, August

73 Examples Linear Elasticity Energy Release Rates... p Total Potential Energy Π(v) = 1 a(v, v) (f, v) g, v 2 Displacement solution u minimizes Π(v) 5 Crack tip 60 Π(u) = 1 2 a(u, u) = 1 2 u 30 Energy Release Rate J(u) p 20 δπ(u) = J (u) δl... l crack length ASC Workshop, August

74 Examples Linear Elasticity...Energy Release Rates... Mixed mode crack problem (Plane Strain, ν = 0.3) p Crac k tip O Crac k tip p ASC Workshop, August

75 Examples Linear Elasticity...Energy Release Rates... ASC Workshop, August

76 Examples Linear Elasticity...Energy Release Rates Mesh size H H/2 H/4 H/8 H/16 J (u H ) η χ e J J ASC Workshop, August

77 Nonlinear Extension Limit Analysis Compute Bounds on the Collapse Load under the assumption of rigid-plastic material behavior Linear vs. Rigid-Plastic ASC Workshop, August

78 Nonlinear Extension Limit Analysis Limit Analysis vs Non-linear Analysis CONS Limited Physics PROS Existence of solution Uniqueness of solution Computable Certifiable... ASC Workshop, August

79 Nonlinear Extension a(σ, v) = F (v) = fv dx + σ : ε(v) dx gv ds X F = {v X F (v) = 1} Σ = {σ f(σ) σ Y } ε(v) = { 0 if f(σ) < σy κ f σ if f(σ) = σ Y Limit Analysis ϕ = max σ Σ Formulation max σ Σ a(σ, v) = ϕf (v), v X = min max v X F σ Σ = max σ Σ a(σ, v) min a(σ, v) v X F ϕ a(σ, v) Upper Bound min a( σ, v) Lower Bound v X F ASC Workshop, August

80 Nonlinear Extension Limit Analysis Outline By choosing appropriate piecewise polynomial interpolations for v and σ we can obtain strict upper and lower bounds on ϕ Discrete minimization/maximization problems are convex (SOCP) and solved (globally) with an IPM ϕ + ϕ can be decomposed into elemental contributions Adaptivity ASC Workshop, August

81 Nonlinear Extension Limit Analysis Examples... Cantilever Beam in Plane Stress 1.6 g = 0 1 x x ASC Workshop, August

82 Nonlinear Extension Limit Analysis...Examples... UNIFORM ADAPTED ASC Workshop, August

83 Nonlinear Extension Limit Analysis...Examples... Uniform Mesh Number Number Low. Bound Upp. Bound Bound Low. Bound Upp. Bound of refin. of elem. λ LB h λ UB h Gap h Error (%) Error (%) Adaptive Mesh Number Number Low. Bound Upp. Bound Bound Low. Bound Upp. Bound of refin. of elem. λ LB h λ UB h Gap h Error (%) Error (%) ASC Workshop, August

84 Conclusions Uniform bounds on Relevant engineering outputs (linear functionals) of Exact weak solutions of linear PDEs, with a Stand-alone certificate of precision, including Non-symmetric operators, using Standard FE solutions and purely local subproblems. ASC Workshop, August

85 Conclusions Certificates allow to Standardize the use of more accurate and safer mathematical models (e.g. construction codes) Eliminate costlier-than-necessary computations Allow for true black boxes that can be used by non-experts in numerical analysis Document computations Address software error issues... ASC Workshop, August

86 Current Work Exploit Discontinuous Galerkin Discretizations Time dependent parabolic problems µ PDE s Non-coercive operators with positivity constraints on the solution Deformation theory of plasticity Recent papers can be found at: ASC Workshop, August