Sixth-order finite volume approximations for smooth curved boundary domains. Mathematical Center, School of Sciences, Minho University – Portugal
St´ephane Clain, Ricardo Costa, Abderhamanne Boularas This research was financed by FEDER Funds through Programa Operacional Factores de Competitividade,COMPETE and by Portuguese Funds through FCT, Funda¸c˜ ao para a Ciˆ encia e a Tecnologia, within the Project PTDC/MAT/121185/2010, the project FCT-ANR/MAT-NAN/0122/2012 and the strategic programme PEst-OE/MAT/UI0013/2014.
Ofir, 2015 May, 18th -22th
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Position of the problem
+ Substitution of smooth boundary domains with the polygonal domain associated to the mesh provides at most a second-order scheme. â Very high-order approximations (third-order or more) require a specific treatment of the boundary conditions. â Techniques developed for finite difference, finite elements, Discontinuous Galerkin (isoparametric elements). 6 Very few things in the finite volume context (larger than third-order).
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F. V. techniques for the boundary treatment
¬ Extra rows of ghost cells which are added beyond the geometric boundary of the computational domain (thick condition). Enforce the boundary conditions by constraining the least-squares reconstruction: - C. F. Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, Journal of Computational Physics, (2002).
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What we propose â Similar to C. F. Ollivier-Gooch and M. Van Altena: boundary condition prescription via the Polynomial Reconstruction (PR). The main difference â O-V use the Gauss points on the local curve to build the PR. â We use the Gauss points on the edge to build the PR and introduce a free parameter to prescribe the BC condition on the Gauss points of the curve ”in the integral sense”.
b
pr b
b
v2
b b b
b
b
v1
qr c
+ Main advantage: PL independent of the curve, use the free parameter to recover the H-O: a problem independent blackbox procedure (we just need BC). Trento, 2015 March, 16th -20th
Generic H-O F.V. for convection diffusion ż
ż pV.nφ ´ k∇φ.nqds ´
qij,r
cj
nij ci
eij R ÿ
ÿ |eij | jPνpiq
f dx “ 0. (divergence theorem) ci
Bci
ci cell, eij “ ci X cj edge. nij normal vector i Ñ j. qij,r Gauss points on edge. νpiq adjacent cells subset to ci .
” ı ζr V pqij,r q.nij φpqij,r q ´ kpqij,r q∇φpqij,r q.nij
r“1
´|ci |fi “ Oph2R i q. (Gauss Points)
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Fluxes and residual formulation
Based on the previous expression: the residual formulation is Gi “
R ÿ |eij | ÿ ζr Fij,r ´ fi , |ci | r“1
jPνpiq
where Fij,r « V pqij,r q.nij φpqij,r q ´ kpqij,r q∇φpqij,r q.nij , approximation of the flux at the Gauss point qij,r . + Sixth-order approximation for Fij,r .
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P.R. for Boundary Condition Prescribe Dirichlet on ΓD “ BΩ, eiD : edge on the boundary , d: the polynomial degree, SpeiD , dq: the associated stencil, φiD : the free parameter, φpiD px; d, φiD q “ φiD `
ÿ
! ) α α Rd,α px ´ m q ´ M iD iD , iD
1ď|α|ďd
α “ pα1 , α2 q, miD the centroid of edge eiD ż 1 α + Set MiD “ px ´ miD qα ds to provide the |eiD | eiD conservation.
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Coefficients for φpiD RdiD vector gathering coefficients Rd,α iD Assume mean values φ` on cells c` , ` P SpeiD , dq are known, p d minimizes the functional R iD EiD pRdiD ; d, φiD q “
ÿ
ωiD,`
`PSpeiD ,dq
” 1 ż ı2 φpiD px; d, φiD q dx´φ` , |c` | c`
with ωiD,` positive weights. + Coefficients ωiD,` are very important to provide ”good”properties (M-matrix).
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Polynomial reconstruction machinery and fluxes for vector Φ “ pφi qi“1,...,I given + Several polynomial reconstructions Conservative for cell ci : φpi pΦq, Non conservative for edge eij : φrij pΦq, Conservative for boundary edge eiD : φpiD pΦ; φiD q. + Flux evaluations Inner edges eij : Fij,r “ pi pqij,r q`rV pqij,r q.nij s´ φ pj pqij,r q´kpqij,r q∇φ rij pqij,r q.nij rV pqij,r q.nij s` φ
Boundary edge eiD : FiD,r “
pi pqiD,r q` rV pqiD,r q.niD s` φ piD pqiD,r q´kpqiD,r q∇φ piD pqiD,r q.niD . rV pqiD,r q.niD s´ φ
+ We use the geometry of the mesh! Not of the curve. Trento, 2015 March, 16th -20th
Resolution
¬ The polynomial reconstruction operators are linear. The flux computations are linear. ® The residual expression is linear: Φ Ñ Gi pΦq. We get a linear operator Φ P RI Ñ GpΦq “ pG1 pΦq, ..., GI pΦqq P RI . + Problem: Find Φ such that GpΦq “ 0. * Matrix-free problem: use GMRES method.
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Curved boundary treatment
‚ qiD,r Gauss points on eiD , ‚ piD,r Gauss points on the curve,
‚ viD,1 viD,2 vertices of the edge, ‚ uiD,1 uiD,2 vertices of arc. + Edges and curves may not have common vertices
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Free parameter evaluation Define the boundary integral default on the piece of curve. BiD pφiD q “
R ÿ
´ ¯ ζr φD ppiD,r q ´ φpiD ppiD,r ; φiD q .
r“1
Notice that φiD Ñ BiD pφiD q is affine: unique solution BiD pφ‹iD q 1
Compute Hpφ¯iD q and Hpφ¯iD ` 1q
2
Compute φ‹iD with φ‹iD “ ´
BiD pφ¯iD ` 1qφ¯iD ´ BiD pφ¯iD qpφ¯iD ` 1q . BiD pφ¯iD ` 1q ´ BiD pφ¯iD q
+ Just need BC + Gauss Points and very fast.
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Numerical Tests
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The annulus
´∆φ “ 0 Constant Dirichlet condition Rin “ 1{10, Rout “ 1. Solution φpx, yq “ a lnpx2 ` y 2 q ` b.
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The annulus:
convergence tables without (top) and with (bottom) corrections
Cell 804 2178 8226 24502
P1 9.26E-02 3.09E-02 7.17E-03 2.92E-03
Cell 804 2178 8226 24502
P1 6.40E-02 2.20E-02 5.19E-03 1.20E-03
— 2.0 2.2 1.7
P3 3,80E-02 9.46E-03 2.09E-03 8.05E-04
— 2.2 2.2 2.1
P3 1.98E-02 1.00E-03 4.87E-05 3.60E-06
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— 2.8 2.3 1.4
P5 5.10E-02 9.60E-03 1.98E-03 7.79E-04
— 3.0 2.4 1.3
— 5.1 4.6 3.7
P5 2.40E-02 8.84E-04 4.99E-06 9.09E-08
— 6.6 7.8 5.9
The 3D annulus
´∆φ “ 0 Constant Dirichlet conditions Rin “ 1{2, Rout “ 1. Solution φpx, yq “ a a ` b. 2 x ` y2 ` z2
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The 3D annulus:
convergence tables with (top) and without (bottom)
Cell 2924 7917 17901 31976
P1 5.19E-01 3.02E-01 1.96E-01 1.50E-01
Cell 2924 7917 17901 31976
P1 4.17E-01 2.06E-01 1.21E-01 8.11E-02
— 1.6 1.5 1.3
P3 6.41E-02 2.52E-02 1.27E-02 7.47E-03
— 2,1 1,9 2,1
P3 3.94E-02 1.07E-02 3.50E-03 1.72E-03
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— 2.8 2.5 2.7
P5 2.24E-01 8.77E-02 4.39E-02 2.33E-02
— 2.8 2.5 3.2
— 3.9 4.1 3.7
P5 1.55E-01 7.33E-03 1.17E-03 4.11E-04
— 9.2 6.8 5.4
The annulus: non matching mesh with the boundary b
b
b b
h b
%h
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Non matching mesh test: convergence tables comparison P1 case with and without conrrections Cell 1557 3257 7083 13771 30569
% 42 40 38 36 33
P1 with 9.42e-04 5.13e-04 2.13e-04 1.18e-04 5.23e-05
— 1.6 2.3 1.8 2.0
P1 without 4.03e-03 2.83e-03 1.91e-03 1.42e-03 9.00e-04
— 0.96 1.00 0.91 1.14
P2 , P3 , P5 cases Cell 1557 3257 7083 13771 30569
% 21 20 18 18 18
P2 1.43e-04 5.36e-05 1.73e-05 8.61e-06 2.15e-06
— 2.7 2.9 2.1 3.5
% 19 17 15 15 13
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P3 4.06e-05 6.43e-06 1.44e-06 4.39e-07 7.61e-08
— 5.0 3.9 3.6 4.4
% 18 13 12 11 10
P5 1.25e-05 1.03e-06 7.23e-08 1.23e-08 1.08e-09
— 6.8 6.8 5.3 6.0
Other Non matching mesh test
-100% of h radius: 1 Ñ 0.86 Cell 1460 3120 7082 13506
h 1.4e-1 9.7e-2 6.5e-2 4.7E-2
P2 6.26e-04 3.31e-04 1.50e-04 8.15e-05
— 1.7 1.9 1.9
P3 6.48e-05 1.56e-05 2.87e-06 7.54e-07
— 3.8 4.1 4.1
P5 5.71e-05 3.35e-06 2.51e-07 3.83e-08
— 7.5 6.3 5.8
P5 9.98e-06 9.55e-07 6.89e-08 2.41e-08
— 6.2 6.4 —
+100% of h radius: 1 Ñ 1.14 Cell 1460 3120 7082 13506
h 1.4e-1 9.7e-2 6.5e-2 4.7E-2
P2 6.91e-04 3.66e-04 1.56e-04 8.44e-05
— 1.7 2.1 1.9
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P3 3.43e-05 8.09e-06 1.48e-06 4.82e-07
— 3.8 4.1 3.5
Find the Gauss points on the curve e “ v1 v2 boundary edge of length |v1 v2 | vŊ 1 v2 boundary arc of length |vŊ 1 v2 | equal-distance property characterizes the Gauss points |vŊ |vŊ |vŊ 1 pr | 2 pr | 1 v2 | “ “ |v1 qr | |v2 qr | |v1 v2 |
âQuadrature rule for the numerical integration over the arc ż vŊ 1 v2
φD ppqdp “ |vŊ 1 v2 |
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R2 ÿ r“1
ξr φD ppr q.
Conclusions and perspectives
A simple ”blackbox”procedure with one free parameter Reconstruction matrix structure independent of the curve Enable to handle boundaries which not fit with the mesh Preserve the optimal order Work for more complex problem (Stokes + ∇ ¨ U “ 0)
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Questions How treat the 3D case? do we have the equal-area property for surface? How to treat the Neuman condition (flux and condition are not at the same place)? What about the hyperbolic case with (partial) Dirichlet condition? Uniform meshes with complex boundaries can be tackled? Fracture, crack, operator with coefficient discontinuities across a smooth line? Tracking discontinuities or interfaces up to the sixth-order?
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