submodule(domain_element) domain_element_triangle_first implicit none contains !----------------------------------------------------------------------! ! triangle_first_Construct: !----------------------------------------------------------------------! ! This function constructs a triangle_first element object based on the ! given element index, global nodal coordinates, connectivity, and ! spatial dimension type. ! ! Arguments: ! id : Element index (int32). ! Identifies the target element. ! ! global_coordinate : type_dp_3d type pointer containing the global coordinates ! of all nodes in the mesh. ! ! Connectivity : Integer array (size 3) specifying the indices of ! nodes that form the triangular element. ! ! Return Value: ! element : Allocated polymorphic object of type ! triangle_first (extends Abstract_ElementType). ! ! Function Details: ! - Allocates a new triangle_first element object. ! - Stores element ID and connectivity information. ! - Links to the corresponding global coordinates for each node. ! - Initializes Gauss point and weight for integration. ! !----------------------------------------------------------------------! module function construct_triangle_first(id, global_coordinate, cell_info, integration) result(element) implicit none integer(int32), intent(in) :: id type(type_dp_3d), pointer, intent(in) :: global_coordinate type(type_vtk_cell), intent(in) :: cell_info type(type_geometry_settings), intent(in) :: integration class(abst_element), allocatable :: element integer(int32) :: i if (allocated(element)) deallocate (element) allocate (type_triangle_first :: element) element%id = id element%type = cell_info%cell_type element%group = cell_info%cell_entity_id element%dimension = cell_info%get_dimension() element%order = cell_info%get_order() element%num_nodes = cell_info%num_nodes_in_cell allocate (element%connectivity(element%num_nodes)) element%connectivity(:) = cell_info%connectivity(1:element%num_nodes) allocate (element%x(element%num_nodes)) allocate (element%y(element%num_nodes)) allocate (element%z(element%num_nodes)) do i = 1, element%num_nodes nullify (element%x(i)%val) nullify (element%y(i)%val) nullify (element%z(i)%val) element%x(i)%val => global_coordinate%x(element%connectivity(i)) element%y(i)%val => global_coordinate%y(element%connectivity(i)) element%z(i)%val => global_coordinate%z(element%connectivity(i)) end do select case (integration%integration_type) case ("full") element%num_gauss = 1_int32 call allocate_array(element%weight, element%num_gauss) call allocate_array(element%gauss, element%dimension, element%num_gauss) element%weight(:) = [0.5d0] element%gauss(:, 1) = [1.0d0 / 3.0d0, 1.0d0 / 3.0d0] case ("reduced") call global_logger%log_warning(message="Reduced-type integration is not implemented for first order triangles.") element%num_gauss = 1_int32 call allocate_array(element%weight, element%num_gauss) call allocate_array(element%gauss, element%dimension, element%num_gauss) element%weight(:) = [0.5d0] element%gauss(:, 1) = [1.0d0 / 3.0d0, 1.0d0 / 3.0d0] case ("free") call global_logger%log_warning(message="Free-type integration is not implemented for first order triangles.") element%num_gauss = 1_int32 call allocate_array(element%weight, element%num_gauss) call allocate_array(element%gauss, element%dimension, element%num_gauss) element%weight(:) = [0.5d0] element%gauss(:, 1) = [1.0d0 / 3.0d0, 1.0d0 / 3.0d0] end select ! Initialize the interpolation function pointer if (associated(element%interpolate)) nullify (element%interpolate) element%interpolate => interpolate if (associated(element%get_connectivity)) nullify (element%get_connectivity) element%get_connectivity => get_connectivity end function construct_triangle_first !----------------------------------------------------------------------! ! getNumNodes_triangle_first: !----------------------------------------------------------------------! ! This function returns the number of nodes associated with a ! triangle_first element. ! ! Arguments: ! self : triangle_first type object. ! Represents the current triangular element instance. ! ! Return Value: ! n : Integer (int32) indicating the number of nodes used by the ! element. This is typically 3 for a linear triangle. ! ! Function Details: ! - Retrieves the value stored in `self%size`, which represents ! the number of nodes for the element. ! !----------------------------------------------------------------------! module function get_id_triangle_first(self) result(id) implicit none class(type_triangle_first), intent(in) :: self integer(int32) :: id id = self%id end function get_id_triangle_first module function get_type_triangle_first(self) result(type) implicit none class(type_triangle_first), intent(in) :: self integer(int32) :: type type = self%type end function get_type_triangle_first module function get_num_nodes_triangle_first(self) result(num_nodes) implicit none class(type_triangle_first), intent(in) :: self integer(int32) :: num_nodes num_nodes = self%num_nodes end function get_num_nodes_triangle_first module function get_group_triangle_first(self) result(group) implicit none class(type_triangle_first), intent(in) :: self integer(int32) :: group group = self%group end function get_group_triangle_first module function get_dimension_triangle_first(self) result(dimension) implicit none class(type_triangle_first), intent(in) :: self integer(int32) :: dimension dimension = self%dimension end function get_dimension_triangle_first module function get_order_triangle_first(self) result(order) implicit none class(type_triangle_first), intent(in) :: self integer(int32) :: order order = self%order end function get_order_triangle_first module function get_num_gauss_triangle_first(self) result(num_gauss) implicit none class(type_triangle_first), intent(in) :: self integer(int32) :: num_gauss num_gauss = self%num_gauss end function get_num_gauss_triangle_first !----------------------------------------------------------------------! ! psi_triangle_first: !----------------------------------------------------------------------! ! This function evaluates the shape function ψ_i(ξ, η) for a linear ! triangular element at the given natural coordinates (ξ, η). ! ! Arguments: ! self : triangle_first type object. ! Represents the triangular element for which the shape ! function is evaluated. ! ! i : Integer (int32), index of the shape function (i = 1 ~ 3). ! Each index corresponds to a vertex of the triangle. ! ! xi : Real(real64), the ξ coordinate in the natural coordinate ! system. ! ! eta : Real(real64), the η coordinate in the natural coordinate ! system. ! ! Return Value: ! psi : Real(real64), value of the i-th shape function ψ_i at (ξ, η). ! ! Function Details: ! - For a linear triangular element, the shape functions are: ! ψ₁(ξ, η) = ξ ! ψ₂(ξ, η) = η ! ψ₃(ξ, η) = 1 - ξ - η ! - Returns 0.0d0 for indices outside the range [1, 3]. ! !----------------------------------------------------------------------! module function psi_triangle_first(self, i, xi, eta) result(psi) implicit none class(type_triangle_first), intent(in) :: self integer(int32), intent(in) :: i real(real64), intent(in) :: xi, eta real(real64) :: psi select case (i) case (1) psi = xi case (2) psi = eta case (3) psi = 1.0d0 - xi - eta case default psi = 0.0d0 end select end function psi_triangle_first !----------------------------------------------------------------------! ! dpsi_dxi_triangle_first: !----------------------------------------------------------------------! ! This function evaluates the partial derivative ∂ψ_i/∂ξ of the i-th ! shape function for a linear triangular element with respect to ξ ! at a given η coordinate. ! ! Arguments: ! self : triangle_first type object. ! Represents the triangular element for which the derivative ! is being evaluated. ! ! i : Integer (int32), index of the shape function (i = 1 ~ 3). ! ! ! xi : Real(real64), the ξ coordinate in the natural coordinate ! system (not used in linear case, but included for interface). ! ! eta : Real(real64), the η coordinate in the natural coordinate ! system (not used in linear case, but included for interface). ! ! Return Value: ! dpsi : Real(real64), value of ∂ψ_i/∂ξ evaluated at (ξ, η). ! ! Function Details: ! - For a linear triangle element: ! ∂ψ₁/∂ξ = 1.0 ! ∂ψ₂/∂ξ = 0.0 ! ∂ψ₃/∂ξ = -1.0 ! - Returns 0.0d0 for indices outside [1, 3]. ! !----------------------------------------------------------------------! module function dpsi_dxi_triangle_first(self, i, xi, eta) result(dpsi) implicit none class(type_triangle_first), intent(in) :: self integer(int32), intent(in) :: i real(real64), intent(in) :: xi, eta real(real64) :: dpsi select case (i) case (1) dpsi = 1.0d0 case (2) dpsi = 0.0d0 case (3) dpsi = -1.0d0 case default dpsi = 0.0d0 end select end function dpsi_dxi_triangle_first !----------------------------------------------------------------------! ! dpsi_deta_triangle_first: !----------------------------------------------------------------------! ! This function evaluates the partial derivative ∂ψ_i/∂η of the i-th ! shape function for a linear triangular element with respect to η ! at a given ξ coordinate. ! ! Arguments: ! self : triangle_first type object. ! Represents the triangular element for which the derivative ! is being evaluated. ! ! i : Integer (int32), index of the shape function (i = 1, 2, 3). ! ! xi : Real(real64), the ξ coordinate in the natural coordinate ! system (not used in linear case, but included for interface). ! ! eta : Real(real64), the η coordinate in the natural coordinate ! system (not used in linear case, but included for interface). ! ! Return Value: ! dpsi : Real(real64), value of ∂ψ_i/∂η evaluated at (ξ, η). ! ! Function Details: ! - For a linear triangle element: ! ∂ψ₁/∂η = 0.0 ! ∂ψ₂/∂η = 1.0 ! ∂ψ₃/∂η = -1.0 ! - Returns 0.0d0 for indices outside [1, 3]. ! !----------------------------------------------------------------------! module function dpsi_deta_triangle_first(self, i, xi, eta) result(dpsi) implicit none class(type_triangle_first), intent(in) :: self integer(int32), intent(in) :: i real(real64), intent(in) :: xi, eta real(real64) :: dpsi select case (i) case (1) dpsi = 0.0d0 case (2) dpsi = 1.0d0 case (3) dpsi = -1.0d0 case default dpsi = 0.0d0 end select end function dpsi_deta_triangle_first !----------------------------------------------------------------------! ! jacobian_triangle_first: !----------------------------------------------------------------------! ! This function computes the (i,j) component of the Jacobian matrix J ! for a linear triangular finite element at a given natural coordinate ! (ξ, η). The Jacobian maps natural coordinates (ξ, η) to physical ! coordinates (x, y). ! ! Arguments: ! self : triangle_first type object. ! Represents the element whose Jacobian is being evaluated. ! ! i : Integer (int32), the row index of the Jacobian component. ! i = 1 → corresponds to x-component (dx/dξ or dx/dη), ! i = 2 → corresponds to y-component (dy/dξ or dy/dη). ! ! j : Integer (int32), the column index of the Jacobian component. ! j = 1 → partial derivative w.r.t ξ, ! j = 2 → partial derivative w.r.t η. ! ! xi : Real(real64), ξ coordinate in natural coordinate system. ! ! eta : Real(real64), η coordinate in natural coordinate system. ! ! Return Value: ! Jval : Real(real64), the (i,j) component of the Jacobian matrix. ! ! Function Details: ! - The Jacobian matrix J is a 2×2 matrix defined as: ! [ ∂x/∂ξ ∂x/∂η ] ! [ ∂y/∂ξ ∂y/∂η ] ! ! - Each entry is computed as a weighted sum over the shape function ! derivatives with respect to ξ or η, multiplied by the physical ! coordinates (X or Y) of the element's nodes. ! ! - The derivatives of shape functions are accessed via: ! self%dpsi_dxi(ii, eta) ! self%dpsi_deta(ii, xi) ! ! - For example: ! ∂x/∂ξ = Σ (∂ψ_i/∂ξ) * x_i ! ∂y/∂η = Σ (∂ψ_i/∂η) * y_i ! ! - This function supports 2D problems. ! !----------------------------------------------------------------------! module function jacobian_triangle_first(self, i, j, xi, eta) result(Jval) implicit none class(type_triangle_first), intent(in) :: self integer(int32), intent(in) :: i, j real(real64), intent(in) :: xi, eta real(real64) :: Jval integer(int32) :: ii, jlocal Jval = 0 !! dx select case (i) case (1) select case (j) case (1) !! dx_dxi do ii = 1, self%num_nodes Jval = Jval + self%dpsi_dxi(ii, xi, eta) * self%x(ii)%val end do case (2) !! dx_deta do ii = 1, self%num_nodes Jval = Jval + self%dpsi_deta(ii, xi, eta) * self%x(ii)%val end do end select !! dy case (2) select case (j) case (1) !! dy_dxi do ii = 1, self%num_nodes Jval = Jval + self%dpsi_dxi(ii, xi, eta) * self%y(ii)%val end do case (2) !! dy_deta do ii = 1, self%num_nodes Jval = Jval + self%dpsi_deta(ii, xi, eta) * self%y(ii)%val end do end select end select end function jacobian_triangle_first !----------------------------------------------------------------------! ! jacobian_det_triangle_first: !----------------------------------------------------------------------! ! This function computes the determinant of the Jacobian matrix J ! for a linear triangular element at a specified point (ξ, η) in ! the natural coordinate system. ! ! Arguments: ! self : triangle_first type object. ! Represents the finite element whose Jacobian is evaluated. ! ! xi : Real(real64), ξ coordinate in the natural coordinate system. ! ! eta : Real(real64), η coordinate in the natural coordinate system. ! ! Return Value: ! J_Det : Real(real64), the determinant of the Jacobian matrix J. ! ! Function Details: ! - The Jacobian matrix J is a 2×2 matrix defined as: ! [ ∂x/∂ξ ∂x/∂η ] ! [ ∂y/∂ξ ∂y/∂η ] ! ! - The determinant is calculated using: ! det(J) = (∂x/∂ξ)(∂y/∂η) - (∂x/∂η)(∂y/∂ξ) ! ! - This determinant gives the area scaling factor for transformation ! from natural to physical coordinates and is used in numerical ! integration (e.g., Gauss quadrature) on the element. ! ! - A zero or negative determinant typically indicates a problem ! with the element geometry (e.g., inverted element). ! !----------------------------------------------------------------------! module function jacobian_det_triangle_first(self, xi, eta) result(J_Det) implicit none class(type_triangle_first), intent(in) :: self real(real64), intent(in) :: xi, eta real(real64) :: J_Det real(real64) :: dx_xi, dx_eta real(real64) :: dy_xi, dy_eta integer(int32) :: i dx_xi = self%jacobian(1, 1, xi, eta) dx_eta = self%jacobian(1, 2, xi, eta) dy_xi = self%jacobian(2, 1, xi, eta) dy_eta = self%jacobian(2, 2, xi, eta) J_Det = dx_xi * dy_eta - dx_eta * dy_xi end function jacobian_det_triangle_first !----------------------------------------------------------------------! ! is_in_triangle_first: !----------------------------------------------------------------------! ! This function checks if the given physical coordinates (px, py) lie ! within the boundaries of a square element. ! The function uses a reverse mapping (Newton-Raphson method) to map ! the physical coordinates to natural coordinates (ξ, η) and then ! checks if the point lies within the square element. ! ! Arguments: ! self : triangle_first type object. ! ! px : x-coordinate (real64 type) in the physical coordinate system. ! This coordinate is checked to see if it lies inside the square element. ! ! py : y-coordinate (real64 type) in the physical coordinate system. ! This coordinate is checked to see if it lies inside the square element. ! ! Return Value: ! is_in : .true. if the point lies within the square element, ! .false. otherwise. ! The function also returns .false. if the Newton-Raphson method ! does not converge or if the natural coordinates fall outside ! the square element's domain. ! ! Algorithm: ! - The function uses the Newton-Raphson method to map the physical ! coordinates (px, py) to the natural coordinates (ξ, η). ! - The function then checks if the natural coordinates (ξ, η) are ! within the valid range [-1, 1]. If they are, the point is inside ! the square element. ! - If the method does not converge, or the natural coordinates fall ! outside the valid range, the function returns .false. ! !----------------------------------------------------------------------! module subroutine is_in_triangle_first(self, px, py, pxi, peta, is_in) class(type_triangle_first), intent(in) :: self real(real64), intent(in) :: px, py real(real64), intent(inout) :: pxi, peta logical, intent(inout) :: is_in real(real64) :: xi, eta real(real64) :: x0, y0 real(real64) :: dx_xi, dx_eta, dy_xi, dy_eta real(real64) :: detJ real(real64) :: dx, dy integer(int32) :: iter, max_iter real(real64) :: tol integer(int32) :: i logical :: converged ! 初期化 xi = 0.0d0 eta = 0.0d0 tol = 1.0d-15 max_iter = 100 converged = .false. ! Newton-Raphson 法による逆写像 do iter = 1, max_iter x0 = 0.0d0 y0 = 0.0d0 do i = 1, self%num_nodes x0 = x0 + self%psi(i, xi, eta) * self%x(i)%val y0 = y0 + self%psi(i, xi, eta) * self%y(i)%val end do dx = px - x0 dy = py - y0 if (sqrt(dx * dx + dy * dy) < tol) then converged = .true. exit end if dx_xi = self%jacobian(1, 1, xi, eta) dx_eta = self%jacobian(1, 2, xi, eta) dy_xi = self%jacobian(2, 1, xi, eta) dy_eta = self%jacobian(2, 2, xi, eta) detJ = self%jacobian_det(xi, eta) if (abs(detJ) < 1.0d-20) exit ! ヤコビ行列の特異性チェック ! Newton-Raphson 更新 xi = xi + (dy_eta * dx - dx_eta * dy) / detJ eta = eta + (-dy_xi * dx + dx_xi * dy) / detJ end do ! 最終判定：収束かつ自然座標が範囲内 is_in = converged .and. (xi >= 0.0d0) .and. (eta >= 0.0d0) .and. (xi + eta <= 1.0d0) if (is_in) then pxi = xi peta = eta end if end subroutine is_in_triangle_first module function interpolate_triangle_first(self, xi, eta, value) result(interpolated_value) implicit none class(abst_element), intent(in) :: self real(real64), intent(in) :: xi, eta real(real64), intent(in) :: value(:) real(real64) :: interpolated_value integer(int32) :: i interpolated_value = 0.0d0 do i = 1, self%num_nodes interpolated_value = interpolated_value + self%psi(i, xi, eta) * value(self%connectivity(i)) end do end function interpolate_triangle_first module function interpolate_reordered_triangle_first(self, xi, eta, value) result(interpolated_value) implicit none class(abst_element), intent(in) :: self real(real64), intent(in) :: xi, eta real(real64), intent(in) :: value(:) real(real64) :: interpolated_value integer(int32) :: i interpolated_value = 0.0d0 do i = 1, self%num_nodes interpolated_value = interpolated_value + self%psi(i, xi, eta) * value(self%connectivity_reordered(i)) end do end function interpolate_reordered_triangle_first end submodule domain_element_triangle_first