!-------------------------------------------------------------------------------------------------- !> @author Pratheek Shanthraj, Max-Planck-Institut für Eisenforschung GmbH !> @brief Interpolation data used by the FEM solver !-------------------------------------------------------------------------------------------------- module FEM_quadrature use prec implicit none private integer, parameter :: & maxOrder = 5 !< maximum integration order real(pReal), dimension(2,3), parameter :: & triangle = reshape([-1.0_pReal, -1.0_pReal, & 1.0_pReal, -1.0_pReal, & -1.0_pReal, 1.0_pReal], shape=[2,3]) real(pReal), dimension(3,4), parameter :: & tetrahedron = reshape([-1.0_pReal, -1.0_pReal, -1.0_pReal, & 1.0_pReal, -1.0_pReal, -1.0_pReal, & -1.0_pReal, 1.0_pReal, -1.0_pReal, & -1.0_pReal, -1.0_pReal, 1.0_pReal], shape=[3,4]) type :: group_float !< variable length datatype real(pReal), dimension(:), allocatable :: p end type group_float integer, dimension(2:3,maxOrder), public, protected :: & FEM_nQuadrature !< number of quadrature points for spatial dimension(2-3) and interpolation order (1-maxOrder) type(group_float), dimension(2:3,maxOrder), public, protected :: & FEM_quadrature_weights, & !< quadrature weights for each quadrature rule FEM_quadrature_points !< quadrature point coordinates (in simplical system) for each quadrature rule public :: & FEM_quadrature_init contains !-------------------------------------------------------------------------------------------------- !> @brief initializes FEM interpolation data !-------------------------------------------------------------------------------------------------- subroutine FEM_quadrature_init() print'(/,a)', ' <<<+- FEM_quadrature init -+>>>'; flush(6) print*, 'L. Zhang et al., Journal of Computational Mathematics 27(1):89-96, 2009' print*, 'https://www.jstor.org/stable/43693493' !-------------------------------------------------------------------------------------------------- ! 2D linear FEM_nQuadrature(2,1) = 1 allocate(FEM_quadrature_weights(2,1)%p(FEM_nQuadrature(2,1))) FEM_quadrature_weights(2,1)%p(1) = 1._pReal FEM_quadrature_points (2,1)%p = permutationStar3([1._pReal/3._pReal]) !-------------------------------------------------------------------------------------------------- ! 2D quadratic FEM_nQuadrature(2,2) = 3 allocate(FEM_quadrature_weights(2,2)%p(FEM_nQuadrature(2,2))) FEM_quadrature_weights(2,2)%p(1:3) = 1._pReal/3._pReal FEM_quadrature_points (2,2)%p = permutationStar21([1._pReal/6._pReal]) !-------------------------------------------------------------------------------------------------- ! 2D cubic FEM_nQuadrature(2,3) = 6 allocate(FEM_quadrature_weights(2,3)%p(FEM_nQuadrature(2,3))) FEM_quadrature_weights(2,3)%p(1:3) = 2.2338158967801147e-1_pReal FEM_quadrature_weights(2,3)%p(4:6) = 1.0995174365532187e-1_pReal FEM_quadrature_points (2,3)%p = [ & permutationStar21([4.4594849091596489e-1_pReal]), & permutationStar21([9.157621350977074e-2_pReal]) ] !-------------------------------------------------------------------------------------------------- ! 2D quartic FEM_nQuadrature(2,4) = 12 allocate(FEM_quadrature_weights(2,4)%p(FEM_nQuadrature(2,4))) FEM_quadrature_weights(2,4)%p(1:3) = 1.1678627572637937e-1_pReal FEM_quadrature_weights(2,4)%p(4:6) = 5.0844906370206817e-2_pReal FEM_quadrature_weights(2,4)%p(7:12) = 8.285107561837358e-2_pReal FEM_quadrature_points (2,4)%p = [ & permutationStar21([2.4928674517091042e-1_pReal]), & permutationStar21([6.308901449150223e-2_pReal]), & permutationStar111([3.1035245103378440e-1_pReal, 5.3145049844816947e-2_pReal]) ] !-------------------------------------------------------------------------------------------------- ! 2D quintic FEM_nQuadrature(2,5) = 16 allocate(FEM_quadrature_weights(2,5)%p(FEM_nQuadrature(2,5))) FEM_quadrature_weights(2,5)%p(1:1) = 1.4431560767778717e-1_pReal FEM_quadrature_weights(2,5)%p(2:4) = 9.509163426728463e-2_pReal FEM_quadrature_weights(2,5)%p(5:7) = 1.0321737053471825e-1_pReal FEM_quadrature_weights(2,5)%p(8:10) = 3.2458497623198080e-2_pReal FEM_quadrature_weights(2,5)%p(11:16) = 2.7230314174434994e-2_pReal FEM_quadrature_points (2,5)%p = [ & permutationStar3([1._pReal/3._pReal]), & permutationStar21([4.5929258829272316e-1_pReal]), & permutationStar21([1.705693077517602e-1_pReal]), & permutationStar21([5.0547228317030975e-2_pReal]), & permutationStar111([2.631128296346381e-1_pReal, 8.3947774099576053e-2_pReal]) ] !-------------------------------------------------------------------------------------------------- ! 3D linear FEM_nQuadrature(3,1) = 1 allocate(FEM_quadrature_weights(3,1)%p(FEM_nQuadrature(3,1))) FEM_quadrature_weights(3,1)%p(1) = 1.0_pReal FEM_quadrature_points (3,1)%p = permutationStar4([0.25_pReal]) !-------------------------------------------------------------------------------------------------- ! 3D quadratic FEM_nQuadrature(3,2) = 4 allocate(FEM_quadrature_weights(3,2)%p(FEM_nQuadrature(3,2))) FEM_quadrature_weights(3,2)%p(1:4) = 0.25_pReal FEM_quadrature_points (3,2)%p = permutationStar31([0.1381966011250105_pReal]) !-------------------------------------------------------------------------------------------------- ! 3D cubic FEM_nQuadrature(3,3) = 14 allocate(FEM_quadrature_weights(3,3)%p(FEM_nQuadrature(3,3))) FEM_quadrature_weights(3,3)%p(1:4) = 7.3493043116361949e-2_pReal FEM_quadrature_weights(3,3)%p(5:8) = 1.1268792571801585e-1_pReal FEM_quadrature_weights(3,3)%p(9:14) = 4.2546020777081467e-2_pReal FEM_quadrature_points (3,3)%p = [ & permutationStar31([9.273525031089123e-2_pReal]), & permutationStar31([3.108859192633006e-1_pReal]), & permutationStar22([4.5503704125649649e-2_pReal]) ] !-------------------------------------------------------------------------------------------------- ! 3D quartic FEM_nQuadrature(3,4) = 35 allocate(FEM_quadrature_weights(3,4)%p(FEM_nQuadrature(3,4))) FEM_quadrature_weights(3,4)%p(1:4) = 0.0021900463965388_pReal FEM_quadrature_weights(3,4)%p(5:16) = 0.0143395670177665_pReal FEM_quadrature_weights(3,4)%p(17:22) = 0.0250305395686746_pReal FEM_quadrature_weights(3,4)%p(23:34) = 0.0479839333057554_pReal FEM_quadrature_weights(3,4)%p(35) = 0.0931745731195340_pReal FEM_quadrature_points (3,4)%p = [ & permutationStar31([0.0267367755543735_pReal]), & permutationStar211([0.0391022406356488_pReal, 0.7477598884818090_pReal]), & permutationStar22([0.4547545999844830_pReal]), & permutationStar211([0.2232010379623150_pReal, 0.0504792790607720_pReal]), & permutationStar4([0.25_pReal]) ] !-------------------------------------------------------------------------------------------------- ! 3D quintic FEM_nQuadrature(3,5) = 56 allocate(FEM_quadrature_weights(3,5)%p(FEM_nQuadrature(3,5))) FEM_quadrature_weights(3,5)%p(1:4) = 0.0010373112336140_pReal FEM_quadrature_weights(3,5)%p(5:16) = 0.0096016645399480_pReal FEM_quadrature_weights(3,5)%p(17:28) = 0.0164493976798232_pReal FEM_quadrature_weights(3,5)%p(29:40) = 0.0153747766513310_pReal FEM_quadrature_weights(3,5)%p(41:52) = 0.0293520118375230_pReal FEM_quadrature_weights(3,5)%p(53:56) = 0.0366291366405108_pReal FEM_quadrature_points (3,5)%p = [ & permutationStar31([0.0149520651530592_pReal]), & permutationStar211([0.0340960211962615_pReal, 0.1518319491659370_pReal]), & permutationStar211([0.0462051504150017_pReal, 0.3549340560639790_pReal]), & permutationStar211([0.2281904610687610_pReal, 0.0055147549744775_pReal]), & permutationStar211([0.3523052600879940_pReal, 0.0992057202494530_pReal]), & permutationStar31([0.1344783347929940_pReal]) ] call selfTest end subroutine FEM_quadrature_init !-------------------------------------------------------------------------------------------------- !> @brief star 3 permutation of input !-------------------------------------------------------------------------------------------------- pure function permutationStar3(point) result(qPt) real(pReal), dimension(2) :: qPt real(pReal), dimension(1), intent(in) :: point qPt = pack(matmul(triangle,reshape([ & point(1), point(1), point(1)],[3,1])),.true.) end function permutationStar3 !-------------------------------------------------------------------------------------------------- !> @brief star 21 permutation of input !-------------------------------------------------------------------------------------------------- pure function permutationStar21(point) result(qPt) real(pReal), dimension(6) :: qPt real(pReal), dimension(1), intent(in) :: point qPt = pack(matmul(triangle,reshape([ & point(1), point(1), 1.0_pReal - 2.0_pReal*point(1), & point(1), 1.0_pReal - 2.0_pReal*point(1), point(1), & 1.0_pReal - 2.0_pReal*point(1), point(1), point(1)],[3,3])),.true.) end function permutationStar21 !-------------------------------------------------------------------------------------------------- !> @brief star 111 permutation of input !-------------------------------------------------------------------------------------------------- pure function permutationStar111(point) result(qPt) real(pReal), dimension(12) :: qPt real(pReal), dimension(2), intent(in) :: point qPt = pack(matmul(triangle,reshape([ & point(1), point(2), 1.0_pReal - point(1) - point(2), & point(1), 1.0_pReal - point(1) - point(2), point(2), & point(2), point(1), 1.0_pReal - point(1) - point(2), & point(2), 1.0_pReal - point(1) - point(2), point(1), & 1.0_pReal - point(1) - point(2), point(2), point(1), & 1.0_pReal - point(1) - point(2), point(1), point(2)],[3,6])),.true.) end function permutationStar111 !-------------------------------------------------------------------------------------------------- !> @brief star 4 permutation of input !-------------------------------------------------------------------------------------------------- pure function permutationStar4(point) result(qPt) real(pReal), dimension(3) :: qPt real(pReal), dimension(1), intent(in) :: point qPt = pack(matmul(tetrahedron,reshape([ & point(1), point(1), point(1), point(1)],[4,1])),.true.) end function permutationStar4 !-------------------------------------------------------------------------------------------------- !> @brief star 31 permutation of input !-------------------------------------------------------------------------------------------------- pure function permutationStar31(point) result(qPt) real(pReal), dimension(12) :: qPt real(pReal), dimension(1), intent(in) :: point qPt = pack(matmul(tetrahedron,reshape([ & point(1), point(1), point(1), 1.0_pReal - 3.0_pReal*point(1), & point(1), point(1), 1.0_pReal - 3.0_pReal*point(1), point(1), & point(1), 1.0_pReal - 3.0_pReal*point(1), point(1), point(1), & 1.0_pReal - 3.0_pReal*point(1), point(1), point(1), point(1)],[4,4])),.true.) end function permutationStar31 !-------------------------------------------------------------------------------------------------- !> @brief star 22 permutation of input !-------------------------------------------------------------------------------------------------- function permutationStar22(point) result(qPt) real(pReal), dimension(18) :: qPt real(pReal), dimension(1), intent(in) :: point qPt = pack(matmul(tetrahedron,reshape([ & point(1), point(1), 0.5_pReal - point(1), 0.5_pReal - point(1), & point(1), 0.5_pReal - point(1), point(1), 0.5_pReal - point(1), & 0.5_pReal - point(1), point(1), point(1), 0.5_pReal - point(1), & 0.5_pReal - point(1), point(1), 0.5_pReal - point(1), point(1), & 0.5_pReal - point(1), 0.5_pReal - point(1), point(1), point(1), & point(1), 0.5_pReal - point(1), 0.5_pReal - point(1), point(1)],[4,6])),.true.) end function permutationStar22 !-------------------------------------------------------------------------------------------------- !> @brief star 211 permutation of input !-------------------------------------------------------------------------------------------------- pure function permutationStar211(point) result(qPt) real(pReal), dimension(36) :: qPt real(pReal), dimension(2), intent(in) :: point qPt = pack(matmul(tetrahedron,reshape([ & point(1), point(1), point(2), 1.0_pReal - 2.0_pReal*point(1) - point(2), & point(1), point(1), 1.0_pReal - 2.0_pReal*point(1) - point(2), point(2), & point(1), point(2), point(1), 1.0_pReal - 2.0_pReal*point(1) - point(2), & point(1), point(2), 1.0_pReal - 2.0_pReal*point(1) - point(2), point(1), & point(1), 1.0_pReal - 2.0_pReal*point(1) - point(2), point(1), point(2), & point(1), 1.0_pReal - 2.0_pReal*point(1) - point(2), point(2), point(1), & point(2), point(1), point(1), 1.0_pReal - 2.0_pReal*point(1) - point(2), & point(2), point(1), 1.0_pReal - 2.0_pReal*point(1) - point(2), point(1), & point(2), 1.0_pReal - 2.0_pReal*point(1) - point(2), point(1), point(1), & 1.0_pReal - 2.0_pReal*point(1) - point(2), point(1), point(1), point(2), & 1.0_pReal - 2.0_pReal*point(1) - point(2), point(1), point(2), point(1), & 1.0_pReal - 2.0_pReal*point(1) - point(2), point(2), point(1), point(1)],[4,12])),.true.) end function permutationStar211 !-------------------------------------------------------------------------------------------------- !> @brief star 1111 permutation of input !-------------------------------------------------------------------------------------------------- pure function permutationStar1111(point) result(qPt) real(pReal), dimension(72) :: qPt real(pReal), dimension(3), intent(in) :: point qPt = pack(matmul(tetrahedron,reshape([ & point(1), point(2), point(3), 1.0_pReal - point(1) - point(2)- point(3), & point(1), point(2), 1.0_pReal - point(1) - point(2)- point(3), point(3), & point(1), point(3), point(2), 1.0_pReal - point(1) - point(2)- point(3), & point(1), point(3), 1.0_pReal - point(1) - point(2)- point(3), point(2), & point(1), 1.0_pReal - point(1) - point(2)- point(3), point(2), point(3), & point(1), 1.0_pReal - point(1) - point(2)- point(3), point(3), point(2), & point(2), point(1), point(3), 1.0_pReal - point(1) - point(2)- point(3), & point(2), point(1), 1.0_pReal - point(1) - point(2)- point(3), point(3), & point(2), point(3), point(1), 1.0_pReal - point(1) - point(2)- point(3), & point(2), point(3), 1.0_pReal - point(1) - point(2)- point(3), point(1), & point(2), 1.0_pReal - point(1) - point(2)- point(3), point(1), point(3), & point(2), 1.0_pReal - point(1) - point(2)- point(3), point(3), point(1), & point(3), point(1), point(2), 1.0_pReal - point(1) - point(2)- point(3), & point(3), point(1), 1.0_pReal - point(1) - point(2)- point(3), point(2), & point(3), point(2), point(1), 1.0_pReal - point(1) - point(2)- point(3), & point(3), point(2), 1.0_pReal - point(1) - point(2)- point(3), point(1), & point(3), 1.0_pReal - point(1) - point(2)- point(3), point(1), point(2), & point(3), 1.0_pReal - point(1) - point(2)- point(3), point(2), point(1), & 1.0_pReal - point(1) - point(2)- point(3), point(1), point(2), point(3), & 1.0_pReal - point(1) - point(2)- point(3), point(1), point(3), point(2), & 1.0_pReal - point(1) - point(2)- point(3), point(2), point(1), point(3), & 1.0_pReal - point(1) - point(2)- point(3), point(2), point(3), point(1), & 1.0_pReal - point(1) - point(2)- point(3), point(3), point(1), point(2), & 1.0_pReal - point(1) - point(2)- point(3), point(3), point(2), point(1)],[4,24])),.true.) end function permutationStar1111 !-------------------------------------------------------------------------------------------------- !> @brief Check correctness of quadrature weights and points. !-------------------------------------------------------------------------------------------------- subroutine selfTest integer :: o, d, n real(pReal), dimension(2:3), parameter :: w = [3.0_pReal,2.0_pReal] do d = lbound(FEM_quadrature_weights,1), ubound(FEM_quadrature_weights,1) do o = lbound(FEM_quadrature_weights(d,:),1), ubound(FEM_quadrature_weights(d,:),1) if (dNeq(sum(FEM_quadrature_weights(d,o)%p),1.0_pReal,5e-15_pReal)) & error stop 'quadrature weights' enddo enddo do d = lbound(FEM_quadrature_points,1), ubound(FEM_quadrature_points,1) do o = lbound(FEM_quadrature_points(d,:),1), ubound(FEM_quadrature_points(d,:),1) n = size(FEM_quadrature_points(d,o)%p,1)/d if (any(dNeq(sum(reshape(FEM_quadrature_points(d,o)%p,[d,n]),2),-real(n,pReal)/w(d),1.e-14_pReal))) & error stop 'quadrature points' enddo enddo end subroutine selfTest end module FEM_quadrature