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Spatial Interpolation Schemes in OpenFOAM

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Spatial Interpolation Schemes in OpenFOAM

  1. 1. OpenFOAM Spatial Interpolation Fumiya Nozaki Last Updated: 16 June 2014 English
  2. 2. 2 FVM discretization ๐›ป๐œ“ ๐‘‘๐‘‰ ๐‘‰ = ๐‘‘๐‘บ โˆ™ ๐œ“ ๐‘† = ๐‘บ ๐‘“ โˆ™ ๐œ“ ๐‘“ ๐‘“ Values at face centers ๏ƒ˜ Most spatial derivative terms are first integrated over a cell volume ๐‘‰ and then converted to integrals over the cell surface bounding the volume using Gaussโ€™s theorem Gaussโ€™s theorem Spatial discretization
  3. 3. 3 Interpolation from cells to faces Owner cell center Neighbour cell center Face center ๐œ“ ๐‘ƒ ๐œ“ ๐‘ ๐œ“ ๐‘“ ๏ƒ˜ What we need is some algebraic relational expressions ๐œ“ ๐‘“ = ๐‘“ ๐œ“ ๐‘ƒ, ๐œ“ ๐‘
  4. 4. ๏ƒ˜ Typical interpolation schemes in OpenFOAM ๏ƒผ upwind ๏ƒผ linearUpwind ๏ƒผ linear ๏ƒผ limitedLinear etc. Many other schemes are available for use in the spatial interpolation [1]. 4 Available interpolation schemes in OpenFOAM ๐œ“ ๐‘“ = ๐‘“ ๐œ“ ๐‘ƒ, ๐œ“ ๐‘
  5. 5. 5 Specification of interpolation schemes gradSchemes { default Gauss linear; } divSchemes { default none; div(phi,U) bounded Gauss linearUpwind grad(U); div((nuEff*(T(grad(U))))) Gauss linear; } laplacianSchemes { default Gauss linear corrected; } interpolationSchemes { default linear; } ๏ƒ˜ Interpolation schemes are chosen on a term-by-term basis
  6. 6. gradSchemes { default Gauss linear; } divSchemes { default none; div(phi,U) bounded Gauss linearUpwind grad(U); div((nuEff*dev(T(grad(U))))) Gauss linear; } laplacianSchemes { default Gauss linear corrected; } interpolationSchemes { default linear; } 6 Specification of interpolation schemes ๏ƒ˜ Interpolation schemes are chosen on a term-by-term basis ๐›ป๐‘ ๐‘‘๐‘‰ ๐‘‰ = ๐‘‘๐‘บ โˆ™ ๐‘ ๐‘† = ๐‘บ ๐‘“ โˆ™ ๐‘ ๐‘“ ๐‘“
  7. 7. 7 Specification of interpolation schemes gradSchemes { default Gauss linear; } divSchemes { default none; div(phi,U) bounded Gauss linearUpwind grad(U); div((nuEff*(T(grad(U))))) Gauss linear; } laplacianSchemes { default Gauss linear corrected; } interpolationSchemes { default linear; } ๏ƒ˜ Interpolation schemes are chosen on a term-by-term basis ๐›ป โˆ™ ๐œˆ ๐›ป๐’– ๐‘‡ ๐‘‘๐‘‰ ๐‘‰ = ๐‘‘๐‘บ โˆ™ ๐œˆ ๐›ป๐’– ๐‘‡ ๐‘† = ๐‘บ ๐‘“ โˆ™ ๐œˆ ๐›ป๐’– ๐‘‡ ๐‘“ ๐‘“ ๐›ป โˆ™ ๐‘ผ๐‘ผ ๐‘‘๐‘‰ ๐‘‰ = ๐‘‘๐‘บ โˆ™ ๐‘ผ๐‘ผ ๐‘† = ๐‘บ ๐‘“ โˆ™ ๐‘ผ ๐‘“ ๐‘ผ ๐‘“ ๐‘“ = ๐น๐‘ผ ๐‘“ ๐‘“
  8. 8. 8 Specification of interpolation schemes gradSchemes { default Gauss linear; } divSchemes { default none; div(phi,U) bounded Gauss linearUpwind grad(U); div((nuEff*dev(T(grad(U))))) Gauss linear; } laplacianSchemes { default Gauss linear corrected; } interpolationSchemes { default linear; } ๏ƒ˜ Interpolation schemes are chosen on a term-by-term basis ๐›ป โˆ™ ฮ“๐›ป๐œ™ ๐‘‘๐‘‰ ๐‘‰ = ๐‘‘๐‘บ โˆ™ ฮ“๐›ป๐œ™ ๐‘† = ฮ“๐‘“ ๐‘บ ๐‘“ โˆ™ ๐›ป๐œ™ ๐‘“ ๐‘“
  9. 9. 9 Letโ€™s look into the code!
  10. 10. 10 surfaceInterpolationScheme.C 0322 //- Return the face-interpolate of the given cell field 0323 // with explicit correction 0324 template<class Type> 0325 tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > 0326 surfaceInterpolationScheme<Type>::interpolate 0327 ( 0328 const GeometricField<Type, fvPatchField, volMesh>& vf 0329 ) const 0330 { 0331 if (surfaceInterpolation::debug) 0332 { 0333 Info<< "surfaceInterpolationScheme<Type>::interpolate" 0334 "(const GeometricField<Type, fvPatchField, volMesh>&) : " 0335 "interpolating " 0336 << vf.type() << " " 0337 << vf.name() 0338 << " from cells to faces" 0339 << endl; 0340 } 0341 0342 tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > tsf 0343 = interpolate(vf, weights(vf)); 0344 0345 if (corrected()) 0346 { 0347 tsf() += correction(vf); 0348 } 0349 0350 return tsf; 0351 } surfaceInterpolationScheme.C Without explicit correction Addition of explicit correction
  11. 11. 11 surfaceInterpolationScheme.C 0322 //- Return the face-interpolate of the given cell field 0323 // with explicit correction 0324 template<class Type> 0325 tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > 0326 surfaceInterpolationScheme<Type>::interpolate 0327 ( 0328 const GeometricField<Type, fvPatchField, volMesh>& vf 0329 ) const 0330 { 0331 if (surfaceInterpolation::debug) 0332 { 0333 Info<< "surfaceInterpolationScheme<Type>::interpolate" 0334 "(const GeometricField<Type, fvPatchField, volMesh>&) : " 0335 "interpolating " 0336 << vf.type() << " " 0337 << vf.name() 0338 << " from cells to faces" 0339 << endl; 0340 } 0341 0342 tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > tsf 0343 = interpolate(vf, weights(vf)); 0344 0345 if (corrected()) 0346 { 0347 tsf() += correction(vf); 0348 } 0349 0350 return tsf; 0351 } surfaceInterpolationScheme.C โ€œweights()โ€ are different depending on the interpolation scheme.
  12. 12. 12 surfaceInterpolationScheme.C 0322 //- Return the face-interpolate of the given cell field 0323 // with explicit correction 0324 template<class Type> 0325 tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > 0326 surfaceInterpolationScheme<Type>::interpolate 0327 ( 0328 const GeometricField<Type, fvPatchField, volMesh>& vf 0329 ) const 0330 { 0331 if (surfaceInterpolation::debug) 0332 { 0333 Info<< "surfaceInterpolationScheme<Type>::interpolate" 0334 "(const GeometricField<Type, fvPatchField, volMesh>&) : " 0335 "interpolating " 0336 << vf.type() << " " 0337 << vf.name() 0338 << " from cells to faces" 0339 << endl; 0340 } 0341 0342 tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > tsf 0343 = interpolate(vf, weights(vf)); 0344 0345 if (corrected()) 0346 { 0347 tsf() += correction(vf); 0348 } 0349 0350 return tsf; 0351 } surfaceInterpolationScheme.C โ€œinterpolate()โ€ calculates interpolated face value without explicit correction. Turn to the next page.
  13. 13. 13 surfaceInterpolationScheme.C 0266 const surfaceScalarField& lambdas = tlambdas(); 0267 0268 const Field<Type>& vfi = vf.internalField(); 0269 const scalarField& lambda = lambdas.internalField(); 0270 0271 const fvMesh& mesh = vf.mesh(); 0272 const labelUList& P = mesh.owner(); 0273 const labelUList& N = mesh.neighbour(); 0274 0275 tmp<GeometricField<Type, fvsPatchField, surfaceMesh> > tsf 0276 ( 0277 new GeometricField<Type, fvsPatchField, surfaceMesh> 0278 ( 0279 IOobject 0280 ( 0281 "interpolate("+vf.name()+')', 0282 vf.instance(), 0283 vf.db() 0284 ), 0285 mesh, 0286 vf.dimensions() 0287 ) 0288 ); 0289 GeometricField<Type, fvsPatchField, surfaceMesh>& sf = tsf(); 0290 0291 Field<Type>& sfi = sf.internalField(); 0292 0293 for (register label fi=0; fi<P.size(); fi++) 0294 { 0295 sfi[fi] = lambda[fi]*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]]; 0296 } surfaceInterpolationScheme.C sfi[fi] represents the interpolated value at the fi-th face center.
  14. 14. 14 First look into โ€œupwindโ€ scheme Table 4.6: Interpolation schemes [1] upwind Flow directions are considered.
  15. 15. 15 Calculation of weights 0128 //- Return the interpolation weighting factors 0129 tmp<surfaceScalarField> weights() const 0130 { 0131 return pos(this->faceFlux_); 0132 } 0133 0134 //- Return the interpolation weighting factors 0135 virtual tmp<surfaceScalarField> weights 0136 ( 0137 const GeometricField<Type, fvPatchField, volMesh>& 0138 ) const 0139 { 0140 return weights(); 0141 } upwind.H ๏ƒ˜ l. 0131 ๐‘ค๐‘’๐‘–๐‘”โ„Ž๐‘ก๐‘ () facei = 1 0 if ๐‘“๐‘Ž๐‘๐‘’๐น๐‘™๐‘ข๐‘ฅ facei > 0 if ๐‘“๐‘Ž๐‘๐‘’๐น๐‘™๐‘ข๐‘ฅ facei โ‰ค 0 upwind
  16. 16. 16 Explicit correction 0175 //- Return true if this scheme uses an explicit correction 0176 virtual bool corrected() const 0177 { 0178 return false; 0179 } surfaceInterpolationScheme.H ๏ƒ˜ For โ€œupwindโ€ scheme, corrected() returns โ€œfalseโ€ So, โ€œupwindโ€ interpolation has no explicit correction. ๏ƒ˜ Evaluation of the interpolated value โ€ข If phi[facei] > 0 sfi[fi] = lambda[fi]*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = weights()[fi]*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = 1*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = vfi[P[fi]] = owner cell value โ€ข If phi[facei] โ‰ฆ 0 sfi[fi] = lambda[fi]*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = 0*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = vfi[N[fi]] = neighbour cell value upwind
  17. 17. 17 Interpolated valueupwind phi[facei] > 0 phi[facei] โ‰ฆ 0 owner neighbour owner neighbour owner is the upstream cell neighbour is the upstream cell ๐ ๐’‡ ๐Ÿ๐š๐œ๐ž๐ข = ๐ ๐’๐’˜๐’๐’†๐’“ ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข ๐ ๐’‡ ๐Ÿ๐š๐œ๐ž๐ข = ๐ ๐’๐’†๐’Š๐’ˆ๐’‰๐’ƒ๐’๐’–๐’“ ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข
  18. 18. 18 Linear upwind differencing Table 4.6: Interpolation schemes [1] linearUpwind Flow directions are considered.
  19. 19. 19 Explicit correctionlinearUpwind ๏ƒ˜ โ€œweightsโ€ are same as that of โ€œupwindโ€ scheme ๏ƒ˜ For โ€œlinearUpwindโ€ scheme, corrected() returns โ€œtrueโ€ ๏ƒ˜ Calculation of explicit correction term 0139 //- Return true if this scheme uses an explicit correction 0140 virtual bool corrected() const 0141 { 0142 return true; 0143 } linearUpwind.H 0085 forAll(faceFlux, facei) 0086 { 0087 label celli = (faceFlux[facei] > 0) ? owner[facei] : neighbour[facei]; 0088 sfCorr[facei] = (Cf[facei] - C[celli]) & gradVf[celli]; 0089 } linearUpwind.C ๐’“ โˆ™ ๐›ป๐œ“ ๐‘ข๐‘๐‘ ๐‘ก๐‘Ÿ๐‘’๐‘Ž๐‘š ๐‘๐‘’๐‘™๐‘™ label of an upstream cell โ€ข Upstream cell is judged by the sign of the face flux field โ€œphiโ€ (faceFlux means phi in the above code)
  20. 20. 20 Interpolated value phi[facei] > 0 phi[facei] โ‰ฆ 0 owner neighbour owner neighbour upstream cell ๐ ๐’‡ ๐Ÿ๐š๐œ๐ž๐ข = ๐ ๐’–๐’‘๐’”๐’•๐’“๐’†๐’‚๐’Ž ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข + ๐’“ โˆ™ ๐œต๐ ๐’–๐’‘๐’”๐’•๐’“๐’†๐’‚๐’Ž ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข linearUpwind Center of upstream cell Face center ๐’“
  21. 21. ๐ ๐’‡ ๐Ÿ๐š๐œ๐ž๐ข = ๐ ๐’–๐’‘๐’”๐’•๐’“๐’†๐’‚๐’Ž ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข + ๐’“ โˆ™ ๐œต๐ ๐’–๐’‘๐’”๐’•๐’“๐’†๐’‚๐’Ž ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข 21 Discretization of gradient gradSchemes { grad(psi) Gauss linear; } interpolationSchemes { interpolate(psi) linearUpwind phi grad(psi); } ๏ƒ˜ Specification of the discretization scheme of gradient term ๐›ป๐‘ The same strings have to be specified. โ€œfvSchemesโ€ file linearUpwind
  22. 22. 22 Linear interpolation with symmetric weighting Table 4.6: Interpolation schemes [1] midPoint Flow directions are not considered.
  23. 23. 23 Calculation of weightsmidPoint 0095 //- Return the interpolation weighting factors 0096 tmp<surfaceScalarField> weights 0097 ( 0098 const GeometricField<Type, fvPatchField, volMesh>& 0099 ) const 0100 { 0101 tmp<surfaceScalarField> taw 0102 ( 0103 new surfaceScalarField 0104 ( 0105 IOobject 0106 ( 0107 "midPointWeights", 0108 this->mesh().time().timeName(), 0109 this->mesh() 0110 ), 0111 this->mesh(), 0112 dimensionedScalar("0.5", dimless, 0.5) 0113 ) 0114 ); 0115 0116 surfaceScalarField::GeometricBoundaryField& awbf = 0117 taw().boundaryField(); 0118 0119 forAll(awbf, patchi) 0120 { 0121 if (!awbf[patchi].coupled()) 0122 { 0123 awbf[patchi] = 1.0; 0124 } 0125 } 0126 0127 return taw; 0128 } midPoint.H weights on the internal faces are 0.5
  24. 24. 24 Explicit correction ๏ƒ˜ For โ€œmidPointโ€ scheme, corrected() returns โ€œfalseโ€ So, โ€œmidPointโ€ interpolation has no explicit correction. ๏ƒ˜ Evaluation of the interpolated value sfi[fi] = lambda[fi]*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = weights()[fi]*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = 0.5*(vfi[P[fi]] - vfi[N[fi]]) + vfi[N[fi]] = 0.5*(vfi[P[fi]] + vfi[N[fi]]) = 0.5*(owner cell value + neighbour cell value) Arithmetic mean midPoint 0175 //- Return true if this scheme uses an explicit correction 0176 virtual bool corrected() const 0177 { 0178 return false; 0179 } surfaceInterpolationScheme.H
  25. 25. 25 Interpolated valuemidPoint owner neighbour ๐Ÿ ๐Ÿ ๐ ๐’๐’˜๐’๐’†๐’“ ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข + ๐ ๐’๐’†๐’Š๐’ˆ๐’‰๐’ƒ๐’๐’–๐’“ ๐’„๐’†๐’๐’ ๐’๐’‡ ๐Ÿ๐š๐œ๐ž๐ข ๐ ๐’‡ ๐Ÿ๐š๐œ๐ž๐ข = Arithmetic mean
  26. 26. 26 References [1] User Guide http://foam.sourceforge.net/docs/Guides-a4/UserGuide.pdf (accessed 06/15/2014)
  27. 27. 27 Thank You!

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