submodule(domain_mesh_element) domain_mesh_element_square_second implicit none contains !----------------------------------------------------------------------! ! construct_square_second: !----------------------------------------------------------------------! ! This function constructs a square_second element object based on the ! given element index, global nodal coordinates, connectivity, and ! spatial dimension type. ! ! Arguments: ! iElem : 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 4) specifying the indices of ! nodes that form the square element. ! ! Return Value: ! element : Allocated polymorphic object of type ! square_second (extends Abstract_ElementType). ! ! Function Details: ! - Allocates a new square_second 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_square_second(id, global_coordinate, input) result(element) implicit none integer(int32), intent(in) :: id type(type_dp_3d), pointer, intent(in) :: global_coordinate type(type_input), intent(in) :: input class(abst_element), allocatable :: element integer(int32) :: i integer(int32) :: num_nodes, num_gauss real(real64), allocatable :: weight(:) real(real64), allocatable :: gauss(:, :) allocate (type_square_second :: element) num_nodes = input%geometry%vtk%cells(id)%num_nodes_in_cell select case (input%basic%geometry_settings%integration_type) case ("full") num_gauss = 9_int32 call allocate_array(weight, num_gauss) call allocate_array(gauss, 3_int32, num_gauss) weight(:) = [25.0d0 / 81.0d0, 40.0d0 / 81.0d0, 25.0d0 / 81.0d0, 40.0d0 / 81.0d0, & 64.0d0 / 81.0d0, 40.0d0 / 81.0d0, 25.0d0 / 81.0d0, 40.0d0 / 81.0d0, & 25.0d0 / 81.0d0] gauss(:, 1) = [-sqrt(3.0d0 / 5.0d0), -sqrt(3.0d0 / 5.0d0), 0.0d0] gauss(:, 2) = [0.0d0, -sqrt(3.0d0 / 5.0d0), 0.0d0] gauss(:, 3) = [sqrt(3.0d0 / 5.0d0), -sqrt(3.0d0 / 5.0d0), 0.0d0] gauss(:, 4) = [-sqrt(3.0d0 / 5.0d0), 0.0d0, 0.0d0] gauss(:, 5) = [0.0d0, 0.0d0, 0.0d0] gauss(:, 6) = [sqrt(3.0d0 / 5.0d0), 0.0d0, 0.0d0] gauss(:, 7) = [-sqrt(3.0d0 / 5.0d0), sqrt(3.0d0 / 5.0d0), 0.0d0] gauss(:, 8) = [0.0d0, sqrt(3.0d0 / 5.0d0), 0.0d0] gauss(:, 9) = [sqrt(3.0d0 / 5.0d0), sqrt(3.0d0 / 5.0d0), 0.0d0] case ("reduced") num_gauss = 4_int32 call allocate_array(weight, num_gauss) call allocate_array(gauss, 3_int32, num_gauss) weight(:) = [1.0d0, 1.0d0, 1.0d0, 1.0d0] gauss(:, 1) = [-sqrt(1.0d0 / 3.0d0), -sqrt(1.0d0 / 3.0d0), 0.0d0] gauss(:, 2) = [-sqrt(1.0d0 / 3.0d0), sqrt(1.0d0 / 3.0d0), 0.0d0] gauss(:, 3) = [sqrt(1.0d0 / 3.0d0), sqrt(1.0d0 / 3.0d0), 0.0d0] gauss(:, 4) = [sqrt(1.0d0 / 3.0d0), -sqrt(1.0d0 / 3.0d0), 0.0d0] case ("free") num_gauss = 4_int32 call allocate_array(weight, num_gauss) call allocate_array(gauss, 3_int32, num_gauss) weight(:) = [1.0d0, 1.0d0, 1.0d0, 1.0d0] gauss(:, 1) = [-input%basic%geometry_settings%integration_points, -input%basic%geometry_settings%integration_points, 0.0d0] gauss(:, 2) = [-input%basic%geometry_settings%integration_points, input%basic%geometry_settings%integration_points, 0.0d0] gauss(:, 3) = [input%basic%geometry_settings%integration_points, input%basic%geometry_settings%integration_points, 0.0d0] gauss(:, 4) = [input%basic%geometry_settings%integration_points, -input%basic%geometry_settings%integration_points, 0.0d0] end select call element%initialize(id=id, & type=input%geometry%vtk%cells(id)%cell_type, & group=input%geometry%vtk%cells(id)%cell_entity_id, & dimension=input%geometry%vtk%cells(id)%get_dimension(), & order=input%geometry%vtk%cells(id)%get_order(), & num_nodes=num_nodes, & connectivity=input%geometry%vtk%cells(id)%connectivity(1:num_nodes), & num_gauss=num_gauss, & weight=weight, & gauss=gauss, & global_coordinate=global_coordinate) end function construct_square_second pure module function get_area_square_second(self) result(area) implicit none class(type_square_second), intent(in) :: self real(real64) :: area type(type_dp_vector_3d) :: r ! 初期化 area = 0.0d0 r%z = 0.0d0 ! ガウス点 1 r%x = -sqrt(3.0d0 / 5.0d0) r%y = -sqrt(3.0d0 / 5.0d0) area = area + 25.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 2 r%x = 0.0d0 r%y = -sqrt(3.0d0 / 5.0d0) area = area + 40.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 3 r%x = sqrt(3.0d0 / 5.0d0) r%y = -sqrt(3.0d0 / 5.0d0) area = area + 25.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 4 r%x = -sqrt(3.0d0 / 5.0d0) r%y = 0.0d0 area = area + 40.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 5 r%x = 0.0d0 r%y = 0.0d0 area = area + 64.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 6 r%x = sqrt(3.0d0 / 5.0d0) r%y = 0.0d0 area = area + 40.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 7 r%x = -sqrt(3.0d0 / 5.0d0) r%y = sqrt(3.0d0 / 5.0d0) area = area + 25.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 8 r%x = 0.0d0 r%y = sqrt(3.0d0 / 5.0d0) area = area + 40.0d0 / 81.0d0 * self%jacobian_det(r) ! ガウス点 9 r%x = sqrt(3.0d0 / 5.0d0) r%y = sqrt(3.0d0 / 5.0d0) area = area + 25.0d0 / 81.0d0 * self%jacobian_det(r) end function get_area_square_second !----------------------------------------------------------------------! ! psi_square_second: !----------------------------------------------------------------------! ! This function evaluates the shape function ψ_i(ξ, η) for a linear ! square element at the given natural coordinates (ξ, η). ! ! Arguments: ! self : square_second type object. ! Represents the square element for which the shape ! function is evaluated. ! ! i : Integer (int32), index of the shape function (i = 1 ~ 8). ! Each index corresponds to a vertex of the square. ! ! 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 square element, the shape functions are: ! ψ₁(ξ, η) = 0.25 * (1 - ξ) * (1 - η) * (-ξ - η - 1) ! ψ₂(ξ, η) = 0.25 * (1 + ξ) * (1 - η) * (ξ - η - 1) ! ψ₃(ξ, η) = 0.25 * (1 + ξ) * (1 + η) * (ξ + η - 1) ! ψ₄(ξ, η) = 0.25 * (1 - ξ) * (1 + η) * (-ξ + η - 1) ! ψ₅(ξ, η) = 0.5 * (1 - ξ) * (1 + ξ) * (1 - η) ! ψ₆(ξ, η) = 0.5 * (1 + ξ) * (1 - η) * (1 + η) ! ψ₇(ξ, η) = 0.5 * (1 - ξ) * (1 + ξ) * (1 + η) ! ψ₈(ξ, η) = 0.5 * (1 - ξ) * (1 - η) * (1 + η) ! - Returns 0.0d0 for indices outside the range [1, 8]. ! !----------------------------------------------------------------------! pure elemental module function psi_square_second(self, i, r) result(psi) implicit none class(type_square_second), intent(in) :: self integer(int32), intent(in) :: i type(type_dp_vector_3d), intent(in) :: r real(real64) :: psi select case (i) case (1) psi = 0.25d0 * (1.0d0 - r%x) * (1.0d0 - r%y) * (-r%x - r%y - 1.0d0) case (2) psi = 0.25d0 * (1.0d0 + r%x) * (1.0d0 - r%y) * (r%x - r%y - 1.0d0) case (3) psi = 0.25d0 * (1.0d0 + r%x) * (1.0d0 + r%y) * (r%x + r%y - 1.0d0) case (4) psi = 0.25d0 * (1.0d0 - r%x) * (1.0d0 + r%y) * (-r%x + r%y - 1.0d0) case (5) psi = 0.5d0 * (1.0d0 - r%x) * (1.0d0 + r%x) * (1.0d0 - r%y) case (6) psi = 0.5d0 * (1.0d0 + r%x) * (1.0d0 - r%y) * (1.0d0 + r%y) case (7) psi = 0.5d0 * (1.0d0 - r%x) * (1.0d0 + r%x) * (1.0d0 + r%y) case (8) psi = 0.5d0 * (1.0d0 - r%x) * (1.0d0 - r%y) * (1.0d0 + r%y) case default psi = 0.0d0 end select end function psi_square_second !----------------------------------------------------------------------! ! dpsi_square_second: !----------------------------------------------------------------------! ! This function evaluates the partial derivative ∂ψ_i/∂ξ of the i-th ! shape function for a linear square element with respect to ξ ! at a given η coordinate. ! ! Arguments: ! self : square_second type object. ! Represents the square element for which the derivative ! is being evaluated. ! ! i : Integer (int32), index of the shape function (i = 1 ~ 8). ! ! xi : Real(real64), the ξ coordinate in the natural coordinate ! system. ! ! eta : Real(real64), the η coordinate in the natural coordinate ! system. ! ! Return Value: ! dpsi : Real(real64), value of ∂ψ_i/∂ξ evaluated at (ξ, η). ! ! Function Details: ! - For a bilinear square element: ! ∂ψ₁/∂ξ = 0.25 * (1 - η) * (2ξ + η) ! ∂ψ₂/∂ξ = 0.25 * (1 - η) * (2ξ - η) ! ∂ψ₃/∂ξ = 0.25 * (1 + η) * (2ξ + η) ! ∂ψ₄/∂ξ = 0.25 * (1 + η) * (2ξ - η) ! ∂ψ₅/∂ξ = -ξ * (1 - η) ! ∂ψ₆/∂ξ = 0.5 * (1 + ξ) * (1 - ξ) ! ∂ψ₇/∂ξ = -ξ * (1 + η) ! ∂ψ₈/∂ξ = -0.5 * (1 + η) * (1 - η) ! ∂ψ₁/∂η = 0.25 * (1 - ξ) * (2η + ξ) ! ∂ψ₂/∂η = 0.25 * (1 + ξ) * (2η - ξ) ! ∂ψ₃/∂η = 0.25 * (1 + ξ) * (2η + ξ) ! ∂ψ₄/∂η = 0.25 * (1 - ξ) * (2η - ξ) ! ∂ψ₅/∂η = -0.5 * (1 + ξ) * (1 - ξ) ! ∂ψ₆/∂η = -(1 + ξ) * η ! ∂ψ₇/∂η = 0.5 * (1 + ξ) * (1 - ξ) ! ∂ψ₈/∂η = -(1 - ξ) * η ! - Returns 0.0 for indices outside [1, 8]. ! !----------------------------------------------------------------------! pure elemental module function dpsi_square_second(self, i, j, r) result(dpsi) implicit none class(type_square_second), intent(in) :: self integer(int32), intent(in) :: i integer(int32), intent(in) :: j type(type_dp_vector_3d), intent(in) :: r real(real64) :: dpsi select case (j) case (1) select case (i) case (1) dpsi = 0.25d0 * (1.0d0 - r%y) * (2.0d0 * r%x + r%y) case (2) dpsi = 0.25d0 * (1.0d0 - r%y) * (2.0d0 * r%x - r%y) case (3) dpsi = 0.25d0 * (1.0d0 + r%y) * (2.0d0 * r%x + r%y) case (4) dpsi = 0.25d0 * (1.0d0 + r%y) * (2.0d0 * r%x - r%y) case (5) dpsi = -r%x * (1.0d0 - r%y) case (6) dpsi = 0.5d0 * (1.0d0 + r%y) * (1.0d0 - r%y) case (7) dpsi = -r%x * (1.0d0 + r%y) case (8) dpsi = -0.5d0 * (1.0d0 + r%y) * (1.0d0 - r%y) case default dpsi = 0.0d0 end select case (2) select case (i) case (1) dpsi = 0.25d0 * (1.0d0 - r%y) * (r%x + 2.0d0 * r%y) case (2) dpsi = 0.25d0 * (1.0d0 - r%y) * (-r%x + 2.0d0 * r%y) case (3) dpsi = 0.25d0 * (1.0d0 + r%y) * (r%x + 2.0d0 * r%y) case (4) dpsi = 0.25d0 * (1.0d0 + r%y) * (-r%x + 2.0d0 * r%y) case (5) dpsi = -0.5d0 * (1.0d0 + r%x) * (1.0d0 - r%x) case (6) dpsi = -(1.0d0 + r%x) * r%y case (7) dpsi = 0.5d0 * (1.0d0 + r%x) * (1.0d0 - r%x) case (8) dpsi = -(1.0d0 - r%x) * r%y case default dpsi = 0.0d0 end select end select end function dpsi_square_second !----------------------------------------------------------------------! ! jacobian_square_second: !----------------------------------------------------------------------! ! This function computes the (i,j) component of the Jacobian matrix J ! for a linear square finite element at a given natural coordinate ! (ξ, η). The Jacobian maps natural coordinates (ξ, η) to physical ! coordinates (x, y). ! ! Arguments: ! self : square_second 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: ! jacobian : 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(ii,1, eta) ! self%dpsi(ii,2, xi) ! ! - For example: ! ∂x/∂ξ = Σ (∂ψ_i/∂ξ) * x_i ! ∂y/∂η = Σ (∂ψ_i/∂η) * y_i ! ! - This function supports 2D problems. ! !----------------------------------------------------------------------! pure elemental module function jacobian_square_second(self, i, j, r) result(jacobian) implicit none class(type_square_second), intent(in) :: self integer(int32), intent(in) :: i, j type(type_dp_vector_3d), intent(in) :: r real(real64) :: jacobian integer(int32) :: ii type(type_dp_vector_3d) :: coordinate jacobian = 0 !! dx select case (i) case (1) select case (j) case (1) !! dx_dxi do ii = 1, self%get_num_nodes() coordinate = self%get_coordinate(ii) jacobian = jacobian + self%dpsi(ii, 1, r) * coordinate%x end do case (2) !! dx_deta do ii = 1, self%get_num_nodes() coordinate = self%get_coordinate(ii) jacobian = jacobian + self%dpsi(ii, 2, r) * coordinate%x end do end select !! dy case (2) select case (j) case (1) !! dy_dxi do ii = 1, self%get_num_nodes() coordinate = self%get_coordinate(ii) jacobian = jacobian + self%dpsi(ii, 1, r) * coordinate%y end do case (2) !! dy_deta do ii = 1, self%get_num_nodes() coordinate = self%get_coordinate(ii) jacobian = jacobian + self%dpsi(ii, 2, r) * coordinate%y end do end select end select end function jacobian_square_second !----------------------------------------------------------------------! ! jacobian_det_square_second: !----------------------------------------------------------------------! ! This function computes the determinant of the Jacobian matrix J ! for a linear square element at a specified point (ξ, η) in ! the natural coordinate system. ! ! Arguments: ! self : square_second 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: ! jacobian_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). ! !----------------------------------------------------------------------! pure elemental module function jacobian_det_square_second(self, r) result(jacobian_det) implicit none class(type_square_second), intent(in) :: self type(type_dp_vector_3d), intent(in) :: r real(real64) :: jacobian_det real(real64) :: dx_xi, dx_eta real(real64) :: dy_xi, dy_eta dx_xi = self%jacobian(1, 1, r) !& dx_eta = self%jacobian(1, 2, r) !& dy_xi = self%jacobian(2, 1, r) !& dy_eta = self%jacobian(2, 2, r) !& jacobian_det = dx_xi * dy_eta - dx_eta * dy_xi end function jacobian_det_square_second !-------------------------------------------------------------------------------------- ! is_in_square_second: !-------------------------------------------------------------------------------------- ! This subroutine checks if the given physical coordinates (px, py) lie ! within the boundaries of a square element. ! The subroutine 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 : square_second type object. Represents a square element. ! It contains the coordinates (X, Y, Z) and connectivity ! information (conn) of the element. ! ! 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 subroutine 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 subroutine uses the Newton-Raphson method to map the physical ! coordinates (px, py) to the natural coordinates (ξ, η). ! - The subroutine 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 subroutine returns .false. ! !-------------------------------------------------------------------------------------- module subroutine is_in_square_second(self, cartesian, normalized, is_in) class(type_square_second), intent(in) :: self type(type_dp_vector_3d), intent(in) :: cartesian type(type_dp_vector_3d), intent(inout) :: normalized logical, intent(inout) :: is_in type(type_dp_vector_3d) :: r type(type_dp_vector_3d) :: coordinate 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 call r%set(0.0d0, 0.0d0, 0.0d0) tol = 1.0d-15 max_iter = 1000 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%get_num_nodes() coordinate = self%get_coordinate(i) x0 = x0 + self%psi(i, r) * coordinate%x y0 = y0 + self%psi(i, r) * coordinate%y end do dx = cartesian%x - x0 dy = cartesian%y - y0 if (sqrt(dx * dx + dy * dy) < tol) then converged = .true. exit end if dx_xi = self%jacobian(1, 1, r) dx_eta = self%jacobian(1, 2, r) dy_xi = self%jacobian(2, 1, r) dy_eta = self%jacobian(2, 2, r) detJ = self%jacobian_det(r) if (abs(detJ) < 1.0d-20) exit ! ヤコビ行列の特異性チェック ! Newton-Raphson 更新 r%x = r%x + (dy_eta * dx - dx_eta * dy) / detJ r%y = r%y + (-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. (abs(r%x) <= 1.0d0) .and. (abs(r%y) <= 1.0d0) if (is_in) normalized = r end subroutine is_in_square_second end submodule domain_mesh_element_square_second