A mixed finite element method for deformed cubic meshesInria Project Lab C2S@Exa, annual meeting,
Bordeaux
Nabil Birgle
POMDAPI project-teamInria Paris-Rocquencourt, UPMC
With :Jérôme Jaffré and Martin Vohralík
July 10, 2014
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 1 / 26
Goals
Numerical methodDefine a mixed finite element method for deformed cubes1 pressure per cell1 flux per face
Mixed finite element methods Deformed cube
High performance computingImplementation in Traces (ANDRA)Parallelism and optimization
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 2 / 26
Mixed finite element methods
RTN0 for tetrahedra and cubes (Raviart-Thomas-Nédélec [1])
Tetrahedron Cube
Composite element for hexahedron (Sboui-Jaffré-Roberts [2])
Hexahedron Hexahedron split into 5 tetrahedra
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 3 / 26
Composite mixed finite element method
Composite element with 5 tetrahedra
Assume the faces are planarChoose the splitting
Cube split into 5 tetrahedra
Composite element with 24 tetrahedra
Works with curved facesSingle splittingSymmetryConforming tetrahedral submesh Cube split into 24 tetrahedra
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 4 / 26
Composite mixed finite element method
Incompressible Darcy flowFind u P Hpdiv ; Ωq and p P L2pΩq such that
u “ ´K ∇ p in Ω∇¨u “ f in Ωp “ p0 on BΩ
Weak formulationFind uh P Wh and ph P Mh such that
ż
ΩK´1uh ¨ vh ´
ż
Ωph ∇¨vh “ ´
ż
BΩp0vh ¨ n @vh P Wh
´
ż
Ωqh ∇¨uh “ ´
ż
Ωfqh @qh P Mh
Define the approximation spaces Wh and MhNabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 5 / 26
Composite mixed finite element method
Conditions to meet for the approximation spaces Wh and Mh
uh P Wh and ph P Mh
ph must be constant on each hexahedron E of the mesh Th
∇¨uh must be constant on Euh must be in RTN0 inside the tetrahedral submesh TE of Euh must be uniquely defined by this value on each face F of the mesh
Definition of the approximation spaces Wh and Mh
Mh “ tq P L2pEq : q|E is constant on E ,@E P Thu
Fh is the set of faces of the mesh Th
Wh is defined as a vectorial space with the basis functions wF , F P Fh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 6 / 26
Composite mixed finite element method
Definition of the approximation spaces Wh and Mh
Wh “ Vect twF , F P Fh : wF |E is solution of (PE,F )u
A local problem (PE,F ) is defined to meet the conditions for each wF
The basis function wF will solve the local problem (PE,F ) inside E
The local approximation spaces ĂWE and ĂME
TE is the tetrahedral mesh of EĂWE and ĂME are the mixed finite element spaces
ĂME “ tq P L2pEq : q|T is constant on T ,@T P TEu
ĂWE “ tv P Hpdiv ; Eq : v|T P RTN0pT q,@T P TEu
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 7 / 26
Composite mixed finite element method
The local problem (PE,F) inside the composite element
Find wF P ĂWE and rpF P ĂME such thatż
EK´1wF ¨ rv´
ż
ErpF ∇¨ rv “ 0 @rv P ĂWE
´
ż
Erq ∇¨wF “ ´
ż
E
1|E |rq @rq P ĂME
(PE,F )
Explicit solution with 5 tetrahedra
Neumann boundary conditionnF normal of the face F
wF ¨ nF 1 “
#
1|F | if F “ F 1
0 elseCube split into 24 tetrahedra
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 8 / 26
Composite mixed finite element method with curved faces
Problem with curved facesConstant velocities are not inside the approximation space Wh
Proof (Nordbotten-Hægland [3])F is the union of 4 sub-faces Fi
nF “ nFi on the triangular sub-face Fi
u is a constant velocity
u ¨ nFi ‰ u ¨ nFj
Deformed cube Constant velocity u
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 9 / 26
Composite mixed finite element method with curved faces
Neumann boundary condition with curved facewF |E solves a local problem inside EwF ¨ nF 1 “ 0 if F ‰ F 1
Constant velocity u Approximated velocity uh
The error between u and uh depends on the mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 10 / 26
Composite mixed finite element method with curved faces
Neumann boundary condition with curved faceu ¨ nF is not constant if F is a curved face
u ¨ nFi ‰ u ¨ nFj
Adapt the Neumann boundary condition
wF ¨ nF “1
|F |Deformed cube
nF is the mean of the normal nFi
nF “
ř4i“1 |Fi | nFi
›
›
›
ř4i“1 |Fi | nFi
›
›
›
L2
Neumann boundary condition
wF ¨ nF “nF ¨ nF
ş
F nF ¨ nF
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 11 / 26
Numerical Experiment
Exact solutionConvergence error inside the domain Ω “ r0 ; 1s3 with different meshes.
p “ 2xz ` y2
2 ` z u “ ´
¨
˝
2zy
2x ` 1
˛
‚
RTN0 on tetrahedron Composite RTN0 RTN0 on hexahedron
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 12 / 26
Numerical Experiment
Regular mesh Convergence error on the regular mesh
10−1 10010−3
10−2
10−1
h‖p−
ph‖ L
2
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 13 / 26
Numerical Experiment
Regular mesh Convergence error on the regular mesh
10−1 10010−3
10−2
10−1
h‖u−
u h‖ L
2
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 14 / 26
Numerical Experiment
Deformed mesh Convergence error on the deformed mesh
10−1 10010−3
10−2
10−1
100
h‖p−
ph‖ L
2
y = 0.2y = 0.8
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 15 / 26
Numerical Experiment
Deformed mesh Convergence error on the deformed mesh
10−1 10010−3
10−2
10−1
h‖u−
u h‖ L
2
y = 0.2y = 0.8
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 16 / 26
Numerical Experiment (Fixed aspect ratio)
Hexahedral mesh Convergence error on the hexahedral mesh
10−1 10010−3
10−2
10−1
100
h‖p−
ph‖ L
2
1.6h
0.4h RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 17 / 26
Numerical Experiment (Fixed aspect ratio)
Hexahedral mesh Convergence error on the hexahedral mesh
10−1 10010−3
10−2
10−1
100
h‖u−
u h‖ L
2
1.6h
0.4h RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 18 / 26
Numerical Experiment (Fixed aspect ratio)
Random mesh Convergence error on the random mesh
10−1 10010−3
10−2
10−1
100
h‖p−
ph‖ L
2
Shift randomlythe vertices˘0.3h
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 19 / 26
Numerical Experiment (Fixed aspect ratio)
Random mesh Convergence error on the random mesh
10−1 100
10−2
100
h‖u−
u h‖ L
2
Shift randomlythe vertices˘0.3h
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 20 / 26
Numerical Experiment (Hybrid form - pcg - Traces)
Deformed mesh CPU time to build the linear equation
10−1 100
10−2
100
hcp
utim
e(s
)
32768 hexahedra786432 tetrahedra
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 21 / 26
Numerical Experiment (Hybrid form - pcg - Traces)
Deformed mesh CPU time to solve the linear equation
10−1 100
10−2
100
102
hcp
utim
e(s
)
32768 hexahedra786432 tetrahedra
RTN0 on tetraedral meshComposite RTN0RTN0 on hexaedral mesh
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 22 / 26
Conclusion
Convergence errorThe convergence is optimal for planar facesIf the curved faces are fixed, the velocity convergesThe composite error is between the two RTN0 errors
Prisms and pyramidsThe same methodology can be apply for prisms and pyramidsLocal and conforming refinement
Projection of the solution into the RTN0 tetrahedral spaceWith rpF in (PE,F ) the pressure is defined on the tetrahedral submeshA posteriori error estimation (Vohralík [4])
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 23 / 26
Traces
Implementation in TracesPut the composite method in Traces (hexahedron, prism, pyramid)Check the matrixCheck the method with basic test cases
PerspectiveBuild a specific test caseStudy the parallelismTry another solverDo a posteriori error estimation
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 24 / 26
References I
Pierre-Arnaud Raviart and Jean-Marie Thomas.A mixed finite element method for 2-nd order elliptic problems.In Mathematical aspects of finite element methods, pages 292–315.Springer, 1977.
Amel Sboui, Jérôme Jaffré, and Jean Roberts.A composite mixed finite element for hexahedral grids.SIAM Journal on Scientific Computing, 31(4) :2623–2645, 2009.
J.M. Nordbotten and H. Hægland.On reproducing uniform flow exactly on general hexahedral cells usingone degree of freedom per surface.Advances in Water Resources, 32(2) :264–267, Feb 2009.
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 25 / 26
References II
Martin Vohralík.Unified primal formulation-based a priori and a posteriori error analysisof mixed finite element methods.Mathematics of Computation, 79(272) :2001–2032, 2010.
Nabil Birgle ( Inria - UPMC ) C2S@Exa, ANDRA July 10, 2014 26 / 26