Goal-oriented mesh adaptation for FSI problems ´ eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ El´ ed´ ( + ) INRIA, Team Gamma3, Saclay, France ( ∗ ) INRIA, Team Ecuador, Sophia-Antipolis, France Eccomas Hersonissos, June 2016 eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 1 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Motivations Objective Combine goal-oriented mesh adaptation and simulations with moving bodies. Main numerical difficulty How to handle the geometry displacement ? eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 2 El´ ed´ Goal-oriented mesh adaptation for FSI problems
1 Scientific Context 2 ALE mesh adaptation 3 Goal-oriented mesh adaptation 4 Coupling Goal-oriented method and ALE method eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 3 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Mesh Adaptation Main idea : introduce the use of metrics field, and notion of unit mesh. [George, Hecht and Vallet., Adv Eng. Software 1991] Riemannian metric space: M : d × d symmetric definite positive matrix � 1 � u , v � M = t u M v ⇒ ℓ M ( a , b ) = � t ab M ( a + tab ) ab d t 0 √ � | K | M = det M d | K | K continuous Metric Field → discrete Mesh . eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 4 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Time-accurate Feature-based Mesh Adaptation Deriving the best mesh to compute the characteristics of a given solution w in space and time [Tam et al.,CMAME 2000], [Picasso, SIAMJSC 2003], [Formaggia et al, ANM 2004 ], [Frey and Alauzet CMAME 2005], [Gruau and Coupez, CMAME 2005 ], [Huang, JCP 2005 ], [Compere et al., 2007 ] Discrete space-time mesh adaptation problem : Find H opt L p having N st vertices such that H opt L p = Argmin H || u − Π h u || H , L p (Ω × [0 , T ]) Well-posed Continuous space-time mesh adaptation problem : Find M opt L p of complexity N st such that �� T � � Trace ( M ( x , t ) − 1 2 | H u ( x , t ) |M ( x , t ) − 1 1 E L p ( M opt 2 ) p d x d t L p ) = min M p 0 Ω ⇒ Solved by variational calculus eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 5 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Time-accurate Feature-based Mesh Adaptation Optimal Mesh �� T � 2 − 2 p − 2 2 − 1 M opt 2 p +3 K ( t ) d t 2 p +3 (det | H u ( x , t ) | ) 2 p +3 | H u ( x , t ) | 3 τ ( t ) L p = N τ ( t ) 3 st 0 p �� � 2 p +3 d x with K ( t ) = Ω (det | H u ( x , t ) | ) Global normalization term requires the whole computation A global fixed-point algorithm ⇒ to compute the space-time metric complexity ⇒ to converge the non-linear mesh adaptation problem ⇒ to predict the solution evolution Split the simulation into several time sub-intervals and set an adapted mesh for each sub-interval ⇒ to limit the number of meshes [0 , T ] = [ t 0 = 0 , t 1 ] ∪ [ t 1 , t 2 ] ∪ ... [ t kmax , T ] eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 6 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Unsteady Feature-based Mesh Adaptation : Algorithm For j=1,nptfx For i=1,nadap S j 0 , i = InterpolateSolution ( H j i − 1 , S j i − 1 , H j i ) S j i = SolveState ( S j 0 , i , H j i ) | H max | j i = ComputeHessianMetric ( H j i , {S j i ( k ) } k =1 , nk ) C j = ComputeSpaceTimeComplexity ( {| H max | j End for i } i =1 , nadap ) M j − 1 = ComputeUnsteadyLpMetrics ( C j − 1 , | H max | j − 1 ) i i H j i = GenerateAdaptedMeshes ( H j − 1 , M j − 1 ) i i End for [0 , T ] = [ t 0 = 0 , t 1 ] ∪ [ t 1 , t 2 ] ∪ ... [ t kmax , T ] eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 7 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Unsteady Feature-based mesh adaptation in ALE : Algorithm For j=1,nptfx For i=1,nadap S j 0 , i = InterpolateSolution ( H j i − 1 , S j i − 1 , H j i ) S j i = SolveState ( S j 0 , i , H j i ) | H max | j i = ComputeHessianMetric ( H j i , {S j i ( k ) } k =1 , nk ) C j = ComputeSpaceTimeComplexity ( {| H max | j i } i =1 , nadap ) End for M j − 1 = ComputeUnsteadyLpMetrics ( C j − 1 , | H max | j − 1 ) i i H j i = GenerateAdaptedMeshes ( H j − 1 , M j − 1 ) i i End for [0 , T ] = [ t 0 = 0 , t 1 ] ∪ [ t 1 , t 2 ] ∪ ... [ t kmax , T ] It is then possible to take into account the mesh motion inside the error estimate = ⇒ optimal space-time adapted mesh in ALE framework Mesh optimization for moving meshes As mesh quality tends to decrease while the mesh is moving ⇒ regular optimization phases must be performed : smoothing and edge swapping eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 8 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Unsteady Feature-based mesh adaptation in ALE : Two F117s crossing flight paths Two planes moved at Mach 0.4 inside an inert air. The planes are translated and rotating 50 sub intervals and 3 adaptation loops Total space time complexity: 36 , 000 , 000 vertices, average mesh size: 732 , 000 vertices, 80 , 000 timesteps eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 9 El´ ed´ Goal-oriented mesh adaptation for FSI problems
2D Pilot case Blast initialization : high density (10, 0 ,25) and air (1, 0 ,2.5) 2D geometry : (5 x 0.5 m 2 ) : 2395 Vertices � T 1 � 2 ( p − p air ) 2 d Γ d t . cost function j : j : j ( W ) = 0 Γ eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 10 El´ ed´ Goal-oriented mesh adaptation for FSI problems
2D Pilot case eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 11 El´ ed´ Goal-oriented mesh adaptation for FSI problems Figure:
Goal-oriented mesh adaptation : Introduction of the Adjoint State Deriving the best mesh to observe a given output scalar functional j ( w ) = � g , w � [Venditti and Darmofal, JCP 2003], [Jones et al., AIAA 2006], [Power et al, CMA 2006 ], [Wintzer et al., AIAA 2008], [Leicht and Hartmann, JCP 2010 ] Let W be the solution of the state equation Ψ( W ) = 0 Choose a scalar functional j . Minimize δ j h = | j − j h | = | ( g , W ) − ( g , W h ) | Introduce the adjoint state W ∗ � ∂ Ψ h � ϕ h , W ∗ = ( g , ϕ h ) h ∂ W h to estimate the error δ j h ≈ ( W ∗ , Ψ h ( W ) − Ψ( W )) Minimize δ j h with an a priori error estimate eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 12 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Unsteady Adjoint Resolution (Euler Equation) The continuous state model on Ω × [0 , T ] obeys to : Ψ( W ) = 0 The discrete state model writes : n = W h n − 1 + δ t n Φ h ( W n − 1 W h ) h Consider a time-dependent functional : � T � j ( W ) = j Γ ( W ( x , t ) d x d t 0 Γ The continuous adjoint state on Ω × [0 , T ] obeys to : � ∂ F − ∂ W ∗ � ∇ W ∗ = g Ψ ∗ ( W , W ∗ ) = 0 or − (1) ∂ t ∂ W The discrete adjoint state writes : ∗ , n + δ t n ∂ j n − 1 ∂ Φ h ∗ , n − 1 = W h ( W n − 1 ∗ , n ) T ( W n − 1 h ) − δ t n ( W h ) W h ∂ W n − 1 h ∂ W n − 1 h h h eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 13 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Time accurate Goal-oriented mesh adaptation Difficulties Computing W ∗ , n − 1 at time t n − 1 requires the knowledge of state W n − 1 and adjoint state W ∗ , n ⇒ the knowledge of all states W n , n = 1 · · · N is needed Solve state foreward: Ψ( W ) = 0 Solve adjoint state backward: Ψ ∗ ( W, W ∗ ) = 0 ⇒ Large memory storage effort in 3D (10 6 vertices & 10 3 iterations request 37.25 Gb) Adopted solution Solve state once to get checkpoints State interpolation between two memory storage eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 14 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Unsteady Feature-based mesh adaptation : Algorithm For j=1,nptfx For i=1,nadap S j 0 , i = ConservativeSolutionTransfer ( H j i − 1 , S j i − 1 , H j i ) S j i = SolveState ( S j 0 , i , H j i ) End for For i=nadap,1 ( S ∗ ) j i = AdjointStateTransfer ( H j 0 ) j i +1 , H j i +1 , ( S ∗ i ) {S j i ( k ) , ( S ∗ ) j i ( k ) } = SolveStateAndAdjointBackward ( S j 0 , i , ( S ∗ ) j i , H j i ) | H max | j i = ComputeGoalOrientedHessianMetric ( H j i , {S j i ( k ) , ( S ∗ ) j i ( k ) } ) End for C j = ComputeSpaceTimeComplexity ( {| H max | j i } i =1 , nadap ) M j i = ComputeUnsteadyLpMetrics ( C j − 1 , | H max | j − 1 ) i H j +1 = GenerateAdaptedMeshes ( H j i , M j i ) i End for Solve state once to get checkpoints Ψ( W ) = 0 Ψ( W ) = 0 Ψ ∗ ( W, W ∗ ) = 0 Solve state and backward adjoint state from checkpoints eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 15 El´ ed´ Goal-oriented mesh adaptation for FSI problems
Blast in the city Blast initialization : high density (10,0,0,0,25) and air (1,0,0,0,2.5) 3D town geometry (85 x 70 x 70 m 3 ) : 4187548 vertices � T 2 ( p − p air ) 2 d Γ d t . 1 � cost function j : j ( W ) = 0 Γ The observation Γ are these 2 buildings eonore Gauci ∗ + , Fr´ eric Alauzet + , Alain Dervieux ∗ ´ 16 El´ ed´ Goal-oriented mesh adaptation for FSI problems
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