A mixed finite element method for deformed cubic meshes

Transcription

1 A mixed finite element method for deformed cubic meshes Inria Project Lab technical meeting at ANDRA, hâtenay-malabry Nabil Birgle POMDAPI project-team Inria Paris-Rocquencourt, UPM With : Martin Vohralík and Jérôme Jaffré April 24, 2013 Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

2 Goals Numerical method Define a mixed finite element method for deformed cubes 1 pressure per cell 1 flux per face High performance computing Implementation in Traces (ANDRA) Parallelism and optimization Deformed cube Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

3 Mixed finite element methods Works well with tetrahedral and cubic meshes (Raviart-Thomas, Nédélec) Tetrahedron ube omposite element for hexahedron (Sboui-Jaffré-Roberts) Hexahedron Hexahedron split into 5 tetrahedrons Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

4 Approximation of a deformed cube Approximation of a deformed cube { Face divided into 4 triangles ell divided into 24 tetrahedrons Deformed cube approximation The alternatives : 1. Build a composite element 2. Use the static condensation In two dimensions { Face divided into 2 segments ell divided into 8 triangles Deformed square approximation Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

5 Mixed finite element formulation Incompressible Darcy flow u = K p in Ω u = g in Ω p = f on Ω Find u H(div; Ω) and p L 2 (Ω) such that (K 1 u, v) (p, v) = (f, v) ( u, φ) = (g, φ) v H(div; Ω) φ L 2 (Ω) Find u h RTN 0 (Ω h ) and p h L 2 (Ω h ) such that (K 1 u h, v h ) (p h, v h ) = (f, v h ) ( u h, φ h ) = (g, φ h ) v h RTN 0 (Ω h ) φ h L 2 (Ω h ) Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

6 Static condensation in two steps Standard mixed finite element on T h gives ( A B t B 0 ) ( ) U P = ( ) F G The mesh h and the triangular sub-mesh T h Reduction of the dimension of the system in two steps : 1. Eliminate internal d.o.f.s 1 pressure per cell h 2 fluxes per face 2. Reduction to 1 flux per face 1 pressure per cell h 1 flux per face Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

7 The global system in terms of local matrices The linear system ( ) ( ) A B t U = B 0 P ( ) F G ell divided into 8 triangles T γ s and θ s are local to global matrix extensions. A = γ T (A T ) = γ (A ) A = γ T (A T ) T T h h B = T T h γ T (B T ) = γ (B ) B = γ T (B T ) T T h h F = T T h θ T (F T ) = θ (F ) F = θ T (F T ) T T h h G = T T h θ T (G T ) = θ (G ) G = θ T (G T ) T T h h T T h Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

8 Eliminate internal d.o.f.s for a cell h Internal flux U int External flux U ext Pressure variation P 0 Pressure mean P 1 Local equations on the cell The cell A int,int U int +A int,ext U ext + (B 0,int ) t P 0 =F int (1) B 0,int U int + B 0,ext U ext =G 0 (2) A ext,int U int +A ext,ext U ext + (B 0,ext ) t P 0 + (B 1,ext ) t P 1 =F ext (3) B 1,ext U ext =G 1 (4) Eliminate U int, P0 from eqs (1),(2) to obtain local eqs for Uext, P1. Thus one obtains the global intermediary system (Ā t) (Ū P) ( B FḠ) Ū : 2 fluxes per face of = B 0 P : 1 pressure per cell Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

9 Static condensation : second step We start from the intermediary system (Ā t) (Ū P) ( B FḠ) Ū : 2 fluxes per face of = h B 0 P : 1 pressure per cell h How to reduce the system to only 1 pressure per cell h and 1 flux per face? The mesh h It is not possible when considering a single cell. We need to consider patches P of 4 cells : A patch P Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

10 Reduction to 1 flux per face On the patch P define red subface s flux Ūr P blue subface s flux Ūb P For each face e with subfaces define A patch P U + e = n e,b Ū b e + n e,r Ū r e In matrix form U + = NŪ Under some constraint on N, we will construct a linear system with unknown U + and P Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

11 Reduction to 1 flux per face On the patch P : Rectangular system (12 24) (Ār P B r P Ā b P B b P ) (Ūr ) P = Ū b P ( Fint P Ḡ P ( B P ) t ) PP Add equations from the definition of U + A patch P Square system (24 24), nonsingular when n e,r n e,b for all faces e and (n e,r, n e,b ) (n e,r, n e,b) for at least two faces e, e in the patch : Ā r P Ā b (Ūr ) P F P ( B P ) t PP B r P B b P P N r P N b Ū b = Ḡ P P P Ū + P One can eliminate Ūr P and Ūb P in terms of P P and Ū+ P and proceed as in the first step. Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

12 Numerical experiment Test case : 36 cells (288 triangles) Initial values K = I f = 0 g = 2(2π) 2 sin(2πx) sin(2πy) Exact solution p = sin(2πx) sin(2πy) ( ) cos(2πx) sin(2πy) u = 2π sin(2πx) cos(2πy) The mesh h and the triangular sub-mesh T h Exact pressure Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

13 Numerical experiment Test case : 36 cells (288 triangles) Approximation of the pressure MFE method on T h Intermediary method Final method p P L 2 = p P L 2 = p P L 2 = onditioning of the matrices cond = 20 cond = 55 cond = onditioning s mean of the local matrices cond = 49 cond = Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

14 Numerical experiment Test case : 36 cells (288 triangles) Nonzero coefficients of the matrices MFE method on T h ( ) Intermediary method ( ) Final method ( ) Stencil of the matrices s = 7 s = 16 s = 29 Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

15 onclusion and perspective onclusion 1. We succeeded at solving the problem with 1 pressure per cell and 1 flux per face on a very coarse mesh 2. Resulting matrix is lower dimension, much denser and badly conditioned Perspective Is it possible to improve dramatically the condition number by choosing a suitable N? Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16

16 References I Amel Sboui, Jérôme Jaffré, and Jean Roberts. A composite mixed finite element for hexahedral grids. SIAM Journal on Scientific omputing, 31(4) : , Martin Vohralík and Barbara I. Wohlmuth. From face to element unknowns by local static condensation with application to nonconforming finite elements. omputer Methods in Applied Mechanics and Engineering, 253(0) : , Martin Vohralík and Barbara I. Wohlmuth. Mixed finite element methods : Implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods. Mathematical Models and Methods in Applied Sciences, 23(05) : , Nabil Birgle ( Inria - UPM ) 2S@Exa, ANDRA April 24, / 16