DFG - Schwerpunktprogram 1095 A nalysis M odeling & S imulation of M ultiscale P roblems Marie Curie Research Training Network MULTIMAT Workshop on Multiscale Numerical Methods for Advanced Materials Institute Henri Poincaré, Paris March 14th – 16th, 2005 Numerical relaxation of nonconvex functionals in phase transitions of solids and finite strain elastoplasticity Sören Bartels ‡ , Carsten Carstensen † & Antonio Orlando † ‡ Department of Mathematics, University of Maryland, College Park † Humboldt-Universität zu Berlin, Institut für Mathematik . PlechᡠThanks to: G. Dolzmann, K. Hackl, A. Mielke, P c, A. Prohl. Supported by: DFG Schwerpunktprogram 1095 AMSMP
SSSS SSSS AMSMSPAMSM SSSS SSSS PPPPP PPPPP SSSSS SSSSS PPPP PP SSSSSS SSSSSS PPPPP SSS SSS SSS SSS PPPPP AMS MSP SSS SSS PPPPP SSS SSSSS SSS PPPPP SSS SSS SSS PPPPP SSS SSS PP PPPP SSS SSS PPPPP PPPPP SSSSS SSSSS PPPPPPPPPPP MMM PAMSMSPAMSMSP MMMMM AAA AAAA MMMMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAAA MMM MMM AAAAAAAAAAAA MMM MMM AAA Computational microstructures in phase-transition solids AMSMSPAMSMSPA AAA MMM MMM AAA MMMMMM MMMMMM AAAAA & finite-strain elastoplasticity Overview Computational Microstructures in 2 D Scientific computing in 1� 0.� 8� vector nonconvex variational problem 0.� 6� h� u� 0.� 4� 17000 1.� 5� 0.� 2� 16500 1� 0� 16000 0.� 5� 0� 0.� 1� 0.� 2� 0.� 3� 0.� 4� 0.� 5� 0.� 6� 0.� 7� 0.� 8� 0� energy 15500 0.� 9� 1� 15000 14500 W(F ξ ) R 2 1 W(F ξ ) 14000 W pc d,r (F ξ ) 13500 0 0.5 1 1.5 2 2.5 ξ Concluding Remarks 2 MULTIMAT Workshop on Multiscale Numerical Methods, Paris 2005
SSSS SSSS AMSMSPAMSM SSSS SSSS PPPPP PPPPP SSSSS SSSSS PPPP PP SSSSSS SSSSSS PPPPP SSS SSS SSS SSS PPPPP AMS MSP SSS SSS PPPPP SSS SSSSS SSS PPPPP SSS SSS SSS PPPPP SSS SSS PP PPPP SSS SSS PPPPP PPPPP SSSSS SSSSS PPPPPPPPPPP MMM PAMSMSPAMSMSP MMMMM AAA AAAA A 2 D scalar benchmark problem MMMMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAAA MMM MMM AAAAAAAAAAAA MMM MMM AAA AMSMSPAMSMSPA AAA MMM MMM AAA MMMMMM MMMMMM AAAAA • Ericksen-James density energy in antiplane shear conditions ( m = 1 , n = 2) motivates √ √ 13 | 2 | F + (3 , 2) / 13 | 2 W ( F ) := | F − (3 , 2) / Ω | u − f | 2 dx over u ∈ A = u D + W 1 , 4 � � ( P ) Minimize E ( u ) := Ω W ( Du ) dx + (Ω) 0 √ with Ω = (0 , 1) × (0 , 3 / 2) , f ( x, y ) := − 3 t 5 / 128 − t 3 / 3 for t = (3( x − 1) + 2 y )( 13) • inf E ( A ) < E ( u ) for all u ∈ A • All the weakly converging infimising sequences ( u j ) of ( P ) have the same weak limit u Finite element solution u h ( x, y ) for ( P h ) 1.� 2� 1� • Oscillations mesh sensitive 0.� 8� 0.� 6� • Difficult numerics 0.� 4� ⇒ Why don’t we relax? 1.� 5� 0.� 2� 1� 0� � 0.� 2� 0.� 5� 0� 0.� 1� 0.� 2� 0.� 3� 0.� 4� 0.� 5� 0.� 6� 0.� 7� 0.� 8� 0� 0.� 9� 1� 3 MULTIMAT Workshop on Multiscale Numerical Methods, Paris 2005
SSSS SSSS AMSMSPAMSM SSSS SSSS PPPPP PPPPP SSSSS SSSSS PPPP PP SSSSSS SSSSSS PPPPP SSS SSS SSS SSS PPPPP AMS MSP SSS SSS PPPPP SSS SSSSS SSS PPPPP SSS SSS SSS PPPPP SSS SSS PP PPPP SSS SSS PPPPP PPPPP SSSSS SSSSS PPPPPPPPPPP MMM PAMSMSPAMSMSP MMMMM AAA AAAA MMMMM AAA AAA Relax FE minimization for the benchmark problem MMM MMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAAA MMM MMM AAAAAAAAAAAA MMM MMM AAA AMSMSPAMSMSPA AAA MMM MMM AAA MMMMMM MMMMMM AAAAA ( RP ) Minimize Ω | u − f | 2 dx Ω W ∗∗ ( Du ) dx + � � RE ( u ) := √ with W ∗∗ ( F ) = (( | F | 2 − 1) + ) 2 + 4( | F | 2 − ((3 , 2) · F ) 2 / 13) . • ( RP ) has a unique solution u ∈ A equals to the weak limit u • E ( u j ) → inf E ( A ) ⇒ σ j := DW ( Du j ) → σ := DW ∗∗ ( Du ) in measure Finite element solution u h ( x, y ) for ( RP h ) 1.� 2� 1� • No oscillations and interface no sharp 0.� 8� 0.� 6� • Simple numerics 0.� 4� 1.� 5� ⇒ Where is the microstructure? 0.� 2� 1� 0� 0.� 2� � 0.� 5� 0� 0.� 1� 0.� 2� 0.� 3� 0.� 4� 0.� 5� 0.� 6� 0.� 7� 0.� 8� 0� 0.� 9� 1� 4 MULTIMAT Workshop on Multiscale Numerical Methods, Paris 2005
SSSS SSSS AMSMSPAMSM SSSS SSSS PPPPP PPPPP SSSSS SSSSS PPPP PP SSSSSS SSSSSS PPPPP SSS SSS SSS SSS PPPPP AMS MSP SSS SSS PPPPP SSS SSSSS SSS PPPPP SSS SSS SSS PPPPP SSS SSS PP PPPP SSS SSS PPPPP PPPPP SSSSS SSSSS PPPPPPPPPPP MMM PAMSMSPAMSMSP MMMMM AAA AAAA GYM for 2 D scalar benchmark problem MMMMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAAA MMM MMM AAAAAAAAAAAA MMM MMM AAA AMSMSPAMSMSPA AAA MMM MMM AAA MMMMMM MMMMMM AAAAA There exists a unique gradient Young measure (C & Plechᡠc ’97) ν x = λ ( F ) δ S + ( F ) + (1 − λ ( F )) δ S − ( F ) P F ± F 2 (1 − | P F | 2 ) − 1 / 2 if | F | ≤ 1; ℓ 1 with P = I − F 2 ⊗ F 2 , λ ( F ) = ℓ 1 + ℓ 2 , and S ± ( F ) = F if 1 < | F | . Volume fraction from u h of ( RP ) on ( T 15 , N = 2485) 1.5 F 0.9 F 2 0.8 S (F) + 0.7 F +F 1 1 2 2 0.6 0.5 l 2 F 1 0.4 F 0.5 0.3 S_(F) l 1 0.2 0.1 0 5 MULTIMAT Workshop on Multiscale Numerical Methods, Paris 2005 0 0.2 0.4 0.6 0.8 1
SSSS SSSS AMSMSPAMSM SSSS SSSS PPPPP PPPPP SSSSS SSSSS PPPP PP SSSSSS SSSSSS PPPPP SSS SSS SSS SSS PPPPP AMS MSP SSS SSS PPPPP SSS SSSSS SSS PPPPP SSS SSS SSS PPPPP SSS SSS PP PPPP SSS SSS PPPPP PPPPP SSSSS SSSSS PPPPPPPPPPP MMM PAMSMSPAMSMSP MMMMM AAA AAAA MMMMM AAA AAA Convergence rate on uniform meshes for ( P ) & ( RP ) MMM MMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAAA MMM MMM AAAAAAAAAAAA MMM MMM AAA AMSMSPAMSMSPA AAA MMM MMM AAA MMMMMM MMMMMM AAAAA A priori error analysis for ( RP h ) � u − u h � L 2 + � σ − σ h � L 4 / 3 � v h ∈A h � D ( u − v h ) � L 4 (Ω) � | u − Iu | W 1 , 4 (Ω) inf 1� 10� 0� 10� 0.125� 1� 1� � 10� 0.375� 1� 2� � 10� (P) ||u-u h || 2 0.375� 1� (P) |u-u h | 1,4 �� (P) || s - s h || 4/3 (RP) ||u-u h || 2 (RP) |u-u h | 1,4 �� (RP) || s - s h || 4/3 3� � 10� 0� 1� 2� 3� 4� 5� 10� 10� 10� 10� 10� 10� N� • a priori bounds of limitate use in error control (lack of regularity for u ) ⇒ use a posteriori error estimate 6 MULTIMAT Workshop on Multiscale Numerical Methods, Paris 2005
SSSS SSSS AMSMSPAMSM SSSS SSSS PPPPP PPPPP SSSSS SSSSS PPPP PP SSSSSS SSSSSS PPPPP SSS SSS SSS SSS PPPPP AMS MSP SSS SSS PPPPP SSS SSSSS SSS PPPPP SSS SSS SSS PPPPP SSS SSS PP PPPP SSS SSS PPPPP PPPPP SSSSS SSSSS PPPPPPPPPPP MMM PAMSMSPAMSMSP MMMMM AAA AAAA MMMMM AAA AAA A posteriori error estimate and adaptivity for ( RP ) MMM MMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAAA MMM MMM AAAAAAAAAAAA MMM MMM AAA AMSMSPAMSMSPA AAA MMM MMM AAA MMMMMM MMMMMM AAAAA Averaging a posteriori error estimate for ( RP h ) η M − h.o.t. ≤ � σ − σ h � L 4 / 3 ≤ cη 1 / 2 M + h.o.t. η 4 / 3 � ) 3 / 4 , η T = � σ h − A σ h � L 4 / 3 ( T ) , A averaging operator with η M = ( T T ∈T ⇒ Efficiency-reliability gap (C & Jochimsen ’03) 1� 0.� 8� 0.� 6� blueL blueR h� u� 0.� 4� 1.� 5� 0.� 2� 1� green red 0� 0.� 5� 0� 0.� 1� 0.� 2� 0.� 3� 0.� 4� 0.� 5� 0.� 6� 0.� 7� 0.� 8� 0� 0.� 9� 1� 7 MULTIMAT Workshop on Multiscale Numerical Methods, Paris 2005
SSSS SSSS AMSMSPAMSM SSSS SSSS PPPPP PPPPP SSSSS SSSSS PPPP PP SSSSSS SSSSSS PPPPP SSS SSS SSS SSS PPPPP AMS MSP SSS SSS PPPPP SSS SSSSS SSS PPPPP SSS SSS SSS PPPPP SSS SSS PP PPPP SSS SSS PPPPP PPPPP SSSSS SSSSS PPPPPPPPPPP MMM PAMSMSPAMSMSP MMMMM AAA AAAA MMMMM AAA AAA Experimental Convergence Rates for ( RP ) MMM MMM AAA AAA MMM MMM AAA AAA MMM MMM AAA AAAA MMM MMM AAAAAAAAAAAA MMM MMM AAA AMSMSPAMSMSPA AAA MMM MMM AAA MMMMMM MMMMMM AAAAA 8 MULTIMAT Workshop on Multiscale Numerical Methods, Paris 2005
Recommend
More recommend
