A posteriori error estimators for FFT-based numerical techniques - PowerPoint PPT Presentation

A posteriori error estimators for FFT-based numerical techniques Sébastien Brisard 1 Ludovic Chamoin 2 1 Université Paris-Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, F-77455 Marne-la-Vallée 2 LMT (ENS Cachan/CNRS/Université Paris Saclay), 61 av. Président Wilson, F-94235 Cachan Sept. 14, 2015

Outline of this talk Introduction constitutive relation error (CRE) uniform grid, periodic Lippmann–Schwinger (UGPLS) solvers Overall strategy combining CRE and UGPLS solvers reconstructing (kinematically admissible) displacements Example: a square inclusion S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 1

A non-specialist’s presentation of CRE The setting Elastic equilibrium of a structure Ω : div σ + b =  σ statically admissible with b and T ∂Ω N : σ · n = T ∂Ω D : u = u u kinematically admissible with u Ω : σ = C : ε (local) constitutive relation Energy norms � � � σ � 2 � σ 1 , σ 2 � S = σ 1 : S : σ 2 d V S = σ : S : σ d V Ω Ω � � � ε � 2 � ε 1 , ε 2 � C = ε 1 : C : ε 2 d V C = ε : C : ε d V Ω Ω S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 2

A non-specialist’s presentation of CRE The Prager–Synge theorem The Prager–Synge theorem u , σ : true solution ε � 2 σ − σ � 2 ε − ε � 2 � ˆ σ − C : ˆ S = � ˆ S + � ˆ u : kinematically admissible ˆ C ˆ σ : statically admissible Consequence � σ − σ � 2 � ˆ S ε � 2 ≤ � ˆ σ − C : ˆ S ε − ε � 2 � ˆ � �� � C Constitutive relation error, � �� � computable “True” error, unknown Prager & Synge (1947), Quart. Appl. Math. 5 :261-269 S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 3

A non-specialist’s presentation of CRE Application to FEM Construction of ˆ u Trivial with displacement-based FEM! Construction of ˆ σ : the EET method (Equilibrating Element Tractions) Postprocessing of the FEM nodal displacements. Succession of local linear problems step 1. construction of tractions: node-wise, step 2. construction of stresses: element-wise. Ladevèze & Pelle (2005), Mastering Calculations in Linear and Nonlinear Mechanics , Springer Evaluation of constitutive relation error Simple integration of standard FE fields. Provides an upper-bound on the “true” error. S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 4

How about Uniform Grid Periodic Lippmann–Schwinger Solvers (UGPLS)?

UGPLS in a nutshell The corrector problem and the Green operator for strains Periodic homogenization: Ω = ( 0 , L 1 ) × · · · × ( 0 , L d ) (unit-cell) The corrector problem The effective stiffness C eff : E def ∇ · ( C : ( E + ∇ s u )) =  . = C : ( E + ∇ s u ) u : periodic fluctuation of total displacement Intermezzo: the Green operator for strains ∇ · ( C 0 : ∇ s u + ̟ ) =  ⇐ ⇒ ∇ s u = − Γ 0 [ ̟ ] u : periodic displacement ̟ : heterogeneous prestress C 0 : homogeneous, reference material S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 6

UGPLS in a nutshell The Lippmann–Schwinger equation � ∇ · ( C 0 : ∇ s u + τ ) =  ∇ · ( C : ( E + ∇ s u )) =  ⇐ ⇒ τ = ( C − C 0 ) : ( E + ∇ s u ) � ∇ s u = − Γ 0 [ τ ] ⇐ ⇒ τ = ( C − C 0 ) : ( E + ∇ s u ) � ( C − C 0 ) − 1 : τ + Γ 0 [ τ ] = E ⇐ ⇒ τ = ( C − C 0 ) : ( E + ∇ s u ) S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 7

UGPLS in a nutshell Discretization of the Lippmann–Schwinger equation Discretization of the Lippmann–Schwinger equation ( C − C 0 ) − 1 : τ + Γ 0 [ τ ] = E ( C − C 0 ) − 1 : τ N + Γ N 0 [ τ N ] = E � Various discretization strategies Point collocation (Moulinec & Suquet, 1994, 1998) Galerkin, piece-wise constant (Brisard & Dormieux, 2010, 2012) Galerkin, trigonometric polynomials (Vondˇ rejc, Zeman & Marek, 2014) Structure of resulting linear system Block-diagonal + block-circulant ⇒ Matrix-free implementation, using FFT S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 8

Combining CRE and UGPLS solvers Proposed strategy ( C − C 0 ) − 1 : τ + Γ 0 [ τ ] = E Kinematically admissible displacements? Statically admissible stresses? Remember: true u retrieved from true τ ∇ · ( C 0 : ∇ s u + τ ) =  u derived from approximate τ N The idea: kinematically admissible ˆ C 0 : ∇ s ˆ � u + τ N � ∇ · =  τ N : output of the UGPLS solver Use finite elements on a uniform grid! S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 9

Combining CRE and UGPLS solvers Anticipated workflow 1. Compute approximate solution τ N to LS (use UGPLS solvers) ( C − C 0 ) − 1 : τ N + Γ N 0 [ τ N ] = E . 2. Compute kinematically admissible u N as FEM approximation of solution to � C 0 : ∇ s u + τ N � ∇ · =  S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 10

Combining CRE and UGPLS solvers Now, wait a minute. . . Question Why not use FEM to solve the corrector problem ∇ · ( C : ( E + ∇ s u )) =  and be done with it? S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 11

Combining CRE and UGPLS solvers Now, wait a minute. . . Question Why not use FEM to solve the corrector problem ∇ · ( C : ( E + ∇ s u )) =  and be done with it? Hint: compare � C 0 : ∇ s u + τ N � ∇ · ( C : ( E + ∇ s u )) =  vs. ∇ · =  S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 11

Combining CRE and UGPLS solvers Now, wait a minute. . . Question Why not use FEM to solve the corrector problem ∇ · ( C : ( E + ∇ s u )) =  and be done with it? Hint: compare � C 0 : ∇ s u + τ N � ∇ · ( C : ( E + ∇ s u )) =  vs. ∇ · =  Answer The unit-cell is now homogeneous, which makes a HUGE difference. . . S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 11

Efficient reconstruction of displacements The canonical problem x 3 L 1 The canonical problem ∇ · ( C 0 : ∇ s u + ̟ ) =  L 3 Uniform grid x 2 L = ( L 1 , . . . , L d ) : size of unit-cell, x 1 N = ( N 1 , . . . , N d ) : size of grid, N : set of node multi-indices. L 2 S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 12

Efficient reconstruction of displacements Handy element–by–element vector operations d � def def = ( x 1 y 1 , . . . , x d y d ) f ( x ) = f ( x j ) xy x = ( x 1 , . . . , x d j = 1 def ) y y 1 y d f : scalar function Examples | x | = | x 1 | · · · | x d | = | x 1 · · · x d | � � � � d 2 i π kx k j x j � = exp exp 2 i π L L j j = 1 In the following formulas, think of vectors as mere scalars! S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 13

Efficient reconstruction of displacements Discretization Periodic shape function P1 approx. of displacement � Φ N � � u N ( x ) = x − x N u N n n n ∈N Φ N : shape function x 1 x N n : coordinates of nodes x 2 L 2 L 1 S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 14

Efficient reconstruction of displacements Exact Fourier series expansion of periodic displacement The periodic P1 shape function � π k � � � 2 i π kx 1 � Φ N ( x ) = sinc 2 exp | N | N L k ∈ Z d The periodic displacement � π k � � � 2 i π kx 1 � u N ( x ) = sinc 2 u N ˆ exp k | N | N L k ∈ Z d DFT of nodal displacements � � − 2 i π kn � u N u N ˆ k = exp n N n ∈N S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 15

Efficient reconstruction of displacements The FE system The total potential energy � 1 � Π = | L | � k · ˆ k · ˆ K N u N ̟ N 2 ˆ u N k · ˆ k + ˆ u N k · ˆ B N , k | N | 2 k ∈N Notes 1. Finite sum (no approximation)! 2. Closed-form expressions for ˆ k and ˆ K N B N k . Optimization of potential energy: finite set of d × d systems ˆ k · ˆ K N u N ̟ N k · ˆ k = − ˆ B N k Brisard (2015), IJNME , submitted S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 16

Efficient reconstruction of displacements Outline of reconstruction 1. Compute DFT of prestress (use FFT!). 2. Solve d × d linear system for each frequency. 3. Compute inverse DFT of displacement (use FFT!). Notes 1. Very simple code, soon available (meanwhile, contact me!). 2. Applies to any variant of UGPLS! S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 17

A very simple example The problem L Material properties x 2 µ i = 100 µ m µ m , ν m ν m = 0 . 3 ν i = 0 . 2 L L / 2 µ i , ν i Loading x 1 E = E 12 ( e 1 ⊗ e 2 + e 2 ⊗ e 1 ) L / 2 S. Brisard, L. Chamoin MAI Workshop – Micromechanics of cementitious materials Sept. 14, 2015 18