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feat(plates): Implement complete DKT plate element
- Implement Discrete Kirchhoff Triangle (DKT) plate bending element - Define DKTPlate formulation type with material and thickness parameters - Implement assemble_stiffness! for plate bending problems - Compute element stiffness matrix using DKT basis functions - Support transverse displacement (w) and rotation (θx, θy) DOFs - Include numerical integration over triangular domain - Implement element force vector assembly - Support distributed and point loads on plate surface - Document DKT theory and implementation details - 645 lines of complete DKT plate element implementation
This commit is contained in:
Jukka Aho
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# DKT (Discrete Kirchhoff Triangle) Plate Element
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#
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# Reference implementation following modern JuliaFEM architecture.
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# Fixes Issue #265: Parenthesis error in dHxdeta function (line 180 in old code).
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#
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# References:
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# - Batoz, J.-L., Bathe, K.-J., & Ho, L.-W. (1980).
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# "A study of three-node triangular plate bending elements."
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# International Journal for Numerical Methods in Engineering, 15(12), 1771-1812.
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# - Lucena Neto, E., et al. (2017).
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# "An Explicit Consistent Geometric Stiffness Matrix for the DKT Element."
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# Latin American Journal of Solids and Structures, 14(4), 613-628.
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# Dependencies
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# When integrated: using ..JuliaFEM
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# When standalone: Use mock Physics from test file
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using LinearAlgebra
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using SparseArrays
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using Tensors
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const DKT_NNODE = 3
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const DKT_DOF_PER_NODE = 3
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const DKT_ELEMENT_DOF = DKT_NNODE * DKT_DOF_PER_NODE
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const DKT_GAUSS_POINTS = ((1.0 / 6.0, 1.0 / 6.0), (2.0 / 3.0, 1.0 / 6.0), (1.0 / 6.0, 2.0 / 3.0))
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const DKT_GAUSS_WEIGHT = 1.0 / 3.0
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triangle_area(X::NTuple{3,Vec{2,T}}) where T =
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0.5 * abs((X[2][1] - X[1][1]) * (X[3][2] - X[1][2]) - (X[3][1] - X[1][1]) * (X[2][2] - X[1][2]))
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function fill_coordinate_buffer!(buffer::Vector{Vec{2,Float64}}, nodes::Vector{Vec{3,Float64}}, conn::NTuple{DKT_NNODE,UInt32})
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@inbounds for (i, node_id) in enumerate(conn)
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coords = nodes[Int(node_id)]
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buffer[i] = Vec{2}((coords[1], coords[2]))
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end
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return buffer
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end
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function bending_matrix(material::AbstractMaterial, thickness::Float64)
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D_tensor = constitutive_matrix_plate(material, thickness)
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D_mat = Matrix{Float64}(undef, 3, 3)
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for i in 1:3, j in 1:3
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D_mat[i, j] = D_tensor[i, j]
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end
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return D_mat
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end
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function plate_global_dofs!(gdofs::Vector{Int}, conn::NTuple{DKT_NNODE,UInt32})
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@inbounds for (local_idx, node_id) in enumerate(conn)
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base = DKT_DOF_PER_NODE * (Int(node_id) - 1)
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offset = DKT_DOF_PER_NODE * (local_idx - 1)
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gdofs[offset + 1] = base + 1
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gdofs[offset + 2] = base + 2
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gdofs[offset + 3] = base + 3
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end
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return gdofs
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end
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function build_plate_sparsity_pattern(mesh::Mesh)
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ndofs_total = DKT_DOF_PER_NODE * length(mesh.nodes)
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ndofs_per_elem = DKT_DOF_PER_NODE * DKT_NNODE
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capacity = length(mesh.connectivity) * ndofs_per_elem * ndofs_per_elem
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I = Vector{Int}()
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J = Vector{Int}()
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sizehint!(I, capacity)
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sizehint!(J, capacity)
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elem_dofs = Vector{Int}(undef, ndofs_per_elem)
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for conn in mesh.connectivity
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plate_global_dofs!(elem_dofs, conn)
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@inbounds for a in 1:ndofs_per_elem
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for b in 1:ndofs_per_elem
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push!(I, elem_dofs[a])
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push!(J, elem_dofs[b])
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end
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end
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end
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return I, J, ndofs_total
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end
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function plate_element_indices(mesh::Mesh, element_set::Symbol)
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if element_set == :all
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return collect(UInt32(1):UInt32(length(mesh.connectivity)))
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end
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elem_set = get_element_set(mesh, element_set)
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return sort!(collect(elem_set))
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end
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# ============================================================================
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# DKT Shape Functions
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# ============================================================================
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"""
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DKTShapeFunctions
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Precomputed shape functions and derivatives for DKT element.
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The DKT element uses modified shape functions that enforce Kirchhoff
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constraints (zero transverse shear) at specific points on each edge.
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"""
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struct DKTShapeFunctions{T}
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# Geometric parameters (element-specific)
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x23::T
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x31::T
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x12::T
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y23::T
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y31::T
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y12::T
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L23::T
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L31::T
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L12::T
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# Shape function coefficients (for edges 23, 31, 12 → nodes 1, 2, 3)
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a4::T
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a5::T
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a6::T
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b4::T
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b5::T
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b6::T
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c4::T
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c5::T
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c6::T
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d4::T
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d5::T
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d6::T
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e4::T
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e5::T
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e6::T
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# Derivative coefficients
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p4::T
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p5::T
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p6::T
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q4::T
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q5::T
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q6::T
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r4::T
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r5::T
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r6::T
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t4::T
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t5::T
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t6::T
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end
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"""
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DKTShapeFunctions(X::NTuple{3, Vec{2, T}}) where T
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Construct DKT shape functions from nodal coordinates.
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# Arguments
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- `X`: Tuple of 3 nodal coordinates (Vec{2} each)
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# Returns
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- `DKTShapeFunctions` with precomputed coefficients
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"""
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function DKTShapeFunctions(X::NTuple{3,Vec{2,T}}) where T
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# Extract nodal coordinates
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x1, y1 = X[1][1], X[1][2]
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x2, y2 = X[2][1], X[2][2]
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x3, y3 = X[3][1], X[3][2]
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# Edge vectors: xij = xi - xj, yij = yi - yj
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x23 = x2 - x3
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x31 = x3 - x1
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x12 = x1 - x2
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y23 = y2 - y3
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y31 = y3 - y1
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y12 = y1 - y2
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# Edge lengths squared
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L23 = x23^2 + y23^2
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L31 = x31^2 + y31^2
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L12 = x12^2 + y12^2
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# Shape function coefficients (Batoz eq. 17-19)
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# For edge k opposite to node i (k=4,5,6 for edges 23,31,12)
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a4 = -x23 / L23
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a5 = -x31 / L31
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a6 = -x12 / L12
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b4 = 3 * x23 * y23 / (4 * L23)
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b5 = 3 * x31 * y31 / (4 * L31)
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b6 = 3 * x12 * y12 / (4 * L12)
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c4 = (x23^2 - 2 * y23^2) / (4 * L23)
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c5 = (x31^2 - 2 * y31^2) / (4 * L31)
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c6 = (x12^2 - 2 * y12^2) / (4 * L12)
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d4 = -y23 / L23
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d5 = -y31 / L31
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d6 = -y12 / L12
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e4 = (y23^2 - 2 * x23^2) / (4 * L23)
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e5 = (y31^2 - 2 * x31^2) / (4 * L31)
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e6 = (y12^2 - 2 * x12^2) / (4 * L12)
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# Derivative coefficients (Batoz eq. 20-21)
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p4 = -6 * x23 / L23
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p5 = -6 * x31 / L31
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p6 = -6 * x12 / L12
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q4 = 3 * x23 * y23 / L23
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q5 = 3 * x31 * y31 / L31
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q6 = 3 * x12 * y12 / L12
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r4 = 3 * y23^2 / L23
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r5 = 3 * y31^2 / L31
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r6 = 3 * y12^2 / L12
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t4 = -6 * y23 / L23
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t5 = -6 * y31 / L31
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t6 = -6 * y12 / L12
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return DKTShapeFunctions(
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x23, x31, x12, y23, y31, y12, L23, L31, L12,
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a4, a5, a6, b4, b5, b6, c4, c5, c6, d4, d5, d6, e4, e5, e6,
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p4, p5, p6, q4, q5, q6, r4, r5, r6, t4, t5, t6
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)
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end
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"""
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tri3_shape_functions(ξ::T, η::T) where T
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Standard Tri3 quadratic shape functions in area coordinates.
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Returns tuple of 6 shape function values (N1, N2, N3, N4, N5, N6).
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"""
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function tri3_shape_functions(ξ::T, η::T) where T
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ζ = 1 - ξ - η
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N1 = ζ * (2 * ζ - 1)
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N2 = ξ * (2 * ξ - 1)
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N3 = η * (2 * η - 1)
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N4 = 4 * ξ * η
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N5 = 4 * η * ζ
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N6 = 4 * ξ * ζ
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return (N1, N2, N3, N4, N5, N6)
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end
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"""
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rotation_shape_functions_x(sf::DKTShapeFunctions, ξ::T, η::T) where T
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Shape functions for rotation θx (rotation about x-axis).
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Returns 1×9 row vector: [Hx1 Hx2 Hx3 Hx4 Hx5 Hx6 Hx7 Hx8 Hx9]
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corresponding to DOFs: [w1 θx1 θy1 w2 θx2 θy2 w3 θx3 θy3]
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"""
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function rotation_shape_functions_x(sf::DKTShapeFunctions, ξ::T, η::T) where T
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N1, N2, N3, N4, N5, N6 = tri3_shape_functions(ξ, η)
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# Batoz eq. 14 (rotation about x-axis)
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Hx1 = 1.5 * (sf.a6 * N6 - sf.a5 * N5)
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Hx2 = sf.b5 * N5 + sf.b6 * N6
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Hx3 = N1 - sf.c5 * N5 - sf.c6 * N6
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Hx4 = 1.5 * (sf.a4 * N4 - sf.a6 * N6)
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Hx5 = sf.b4 * N4 + sf.b6 * N6
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Hx6 = N2 - sf.c4 * N4 - sf.c6 * N6
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Hx7 = 1.5 * (sf.a5 * N5 - sf.a4 * N4)
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Hx8 = sf.b4 * N4 + sf.b5 * N5
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Hx9 = N3 - sf.c4 * N4 - sf.c5 * N5
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return (Hx1, Hx2, Hx3, Hx4, Hx5, Hx6, Hx7, Hx8, Hx9)
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end
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"""
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rotation_shape_functions_y(sf::DKTShapeFunctions, ξ::T, η::T) where T
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Shape functions for rotation θy (rotation about y-axis).
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Returns 1×9 row vector: [Hy1 Hy2 Hy3 Hy4 Hy5 Hy6 Hy7 Hy8 Hy9]
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"""
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function rotation_shape_functions_y(sf::DKTShapeFunctions, ξ::T, η::T) where T
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N1, N2, N3, N4, N5, N6 = tri3_shape_functions(ξ, η)
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# Batoz eq. 15 (rotation about y-axis)
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Hy1 = 1.5 * (sf.d6 * N6 - sf.d5 * N5)
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Hy2 = -N1 + sf.e5 * N5 + sf.e6 * N6
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Hy3 = -sf.b5 * N5 - sf.b6 * N6
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Hy4 = 1.5 * (sf.d4 * N4 - sf.d6 * N6)
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Hy5 = -N2 + sf.e4 * N4 + sf.e6 * N6
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Hy6 = -sf.b4 * N4 - sf.b6 * N6
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Hy7 = 1.5 * (sf.d5 * N5 - sf.d4 * N4)
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Hy8 = -N3 + sf.e4 * N4 + sf.e5 * N5
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Hy9 = -sf.b4 * N4 - sf.b5 * N5
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return (Hy1, Hy2, Hy3, Hy4, Hy5, Hy6, Hy7, Hy8, Hy9)
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end
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"""
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dHxdξ(sf::DKTShapeFunctions, ξ::T, η::T) where T
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Derivative of Hx with respect to ξ (area coordinate).
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**BUG FIX (Issue #265):** Corrected parentheses in all derivative functions.
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"""
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function dHxdξ(sf::DKTShapeFunctions, ξ::T, η::T) where T
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d1 = sf.p6 * (1 - 2 * ξ) + (sf.p5 - sf.p6) * η
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d2 = sf.q6 * (1 - 2 * ξ) - (sf.q5 + sf.q6) * η
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d3 = -4 + 6 * (ξ + η) + sf.r6 * (1 - 2 * ξ) - η * (sf.r5 + sf.r6)
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d4 = -sf.p6 * (1 - 2 * ξ) + η * (sf.p4 + sf.p6)
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d5 = sf.q6 * (1 - 2 * ξ) - η * (sf.q6 - sf.q4)
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d6 = -2 + 6 * ξ + sf.r6 * (1 - 2 * ξ) + η * (sf.r4 - sf.r6)
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d7 = -η * (sf.p5 + sf.p4)
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d8 = η * (sf.q4 - sf.q5)
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d9 = -η * (sf.r5 - sf.r4)
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return (d1, d2, d3, d4, d5, d6, d7, d8, d9)
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end
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"""
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dHxdη(sf::DKTShapeFunctions, ξ::T, η::T) where T
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Derivative of Hx with respect to η (area coordinate).
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**BUG FIX (Issue #265):** Line 180 from old code had incorrect parentheses.
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CORRECT: d9 = -2 + 6*η + r5*(1-2*η) + ξ*(r4-r5)
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WRONG: d9 = -2 + 6*η + r5*(1-2*η + ξ*(r4-r5)) # Old bug
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"""
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function dHxdη(sf::DKTShapeFunctions, ξ::T, η::T) where T
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d1 = -sf.p5 * (1 - 2 * η) - ξ * (sf.p6 - sf.p5)
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d2 = sf.q5 * (1 - 2 * η) - ξ * (sf.q5 + sf.q6)
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d3 = -4 + 6 * (ξ + η) + sf.r5 * (1 - 2 * η) - ξ * (sf.r5 + sf.r6)
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d4 = ξ * (sf.p4 + sf.p6)
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d5 = ξ * (sf.q4 - sf.q6)
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d6 = -ξ * (sf.r6 - sf.r4)
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d7 = sf.p5 * (1 - 2 * η) - ξ * (sf.p4 + sf.p5)
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d8 = sf.q5 * (1 - 2 * η) + ξ * (sf.q4 - sf.q5)
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# BUG FIX: Issue #265 - Corrected parentheses
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d9 = -2 + 6 * η + sf.r5 * (1 - 2 * η) + ξ * (sf.r4 - sf.r5) # FIXED!
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return (d1, d2, d3, d4, d5, d6, d7, d8, d9)
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end
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"""
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dHydξ(sf::DKTShapeFunctions, ξ::T, η::T) where T
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Derivative of Hy with respect to ξ (area coordinate).
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"""
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function dHydξ(sf::DKTShapeFunctions, ξ::T, η::T) where T
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d1 = sf.t6 * (1 - 2 * ξ) + η * (sf.t5 - sf.t6)
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d2 = 1 + sf.r6 * (1 - 2 * ξ) - η * (sf.r5 + sf.r6)
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d3 = -sf.q6 * (1 - 2 * ξ) + η * (sf.q5 + sf.q6)
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d4 = -sf.t6 * (1 - 2 * ξ) + η * (sf.t4 + sf.t6)
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d5 = -1 + sf.r6 * (1 - 2 * ξ) + η * (sf.r4 - sf.r6)
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d6 = -sf.q6 * (1 - 2 * ξ) - η * (sf.q4 - sf.q6)
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d7 = -η * (sf.t4 + sf.t5)
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d8 = η * (sf.r4 - sf.r5)
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d9 = -η * (sf.q4 - sf.q5)
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return (d1, d2, d3, d4, d5, d6, d7, d8, d9)
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end
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"""
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dHydη(sf::DKTShapeFunctions, ξ::T, η::T) where T
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Derivative of Hy with respect to η (area coordinate).
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"""
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function dHydη(sf::DKTShapeFunctions, ξ::T, η::T) where T
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d1 = -sf.t5 * (1 - 2 * η) - ξ * (sf.t6 - sf.t5)
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d2 = 1 + sf.r5 * (1 - 2 * η) - ξ * (sf.r5 + sf.r6)
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d3 = -sf.q5 * (1 - 2 * η) + ξ * (sf.q5 + sf.q6)
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d4 = ξ * (sf.t4 + sf.t6)
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d5 = ξ * (sf.r4 - sf.r6)
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d6 = -ξ * (sf.q4 - sf.q6)
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d7 = sf.t5 * (1 - 2 * η) - ξ * (sf.t4 + sf.t5)
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d8 = -1 + sf.r5 * (1 - 2 * η) + ξ * (sf.r4 - sf.r5)
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d9 = -sf.q5 * (1 - 2 * η) - ξ * (sf.q4 - sf.q5)
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return (d1, d2, d3, d4, d5, d6, d7, d8, d9)
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end
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# ============================================================================
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# Strain-Displacement Matrix (B-matrix)
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# ============================================================================
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"""
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curvature_matrix(sf::DKTShapeFunctions, X::NTuple{3, Vec{2, T}}, ξ::T, η::T) where T
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Compute 3×9 strain-displacement (B) matrix relating DOFs to curvatures.
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Curvatures: κ = [κx, κy, κxy]ᵀ = -[∂²w/∂x², ∂²w/∂y², 2∂²w/∂x∂y]ᵀ
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|
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In DKT, curvatures are expressed via rotations:
|
||||
- κx = ∂θy/∂x
|
||||
- κy = -∂θx/∂y
|
||||
- κxy = ∂θy/∂y - ∂θx/∂x
|
||||
|
||||
Returns B such that κ = B * u, where u = [w1 θx1 θy1 ... w3 θx3 θy3]ᵀ
|
||||
"""
|
||||
function curvature_matrix!(B::AbstractMatrix{T}, sf::DKTShapeFunctions, ξ::T, η::T) where T
|
||||
# Jacobian: relates (ξ,η) derivatives to (x,y) derivatives
|
||||
# J = [∂x/∂ξ ∂y/∂ξ]
|
||||
# [∂x/∂η ∂y/∂η]
|
||||
x31 = sf.x31
|
||||
x12 = sf.x12
|
||||
y31 = sf.y31
|
||||
y12 = sf.y12
|
||||
|
||||
# Inverse Jacobian factor (assumes constant Jacobian for linear triangle)
|
||||
detJ_inv = 1 / (x31 * y12 - x12 * y31)
|
||||
|
||||
# Get shape function derivatives in (ξ,η) coordinates
|
||||
dHx_dξ = dHxdξ(sf, ξ, η)
|
||||
dHx_dη = dHxdη(sf, ξ, η)
|
||||
dHy_dξ = dHydξ(sf, ξ, η)
|
||||
dHy_dη = dHydη(sf, ξ, η)
|
||||
|
||||
# Transform to (x,y) derivatives using chain rule
|
||||
# ∂/∂x = (∂ξ/∂x)∂/∂ξ + (∂η/∂x)∂/∂η = detJ_inv * (y12*∂/∂ξ - y31*∂/∂η)
|
||||
# ∂/∂y = (∂ξ/∂y)∂/∂ξ + (∂η/∂y)∂/∂η = detJ_inv * (-x12*∂/∂ξ + x31*∂/∂η)
|
||||
|
||||
# Curvature-displacement matrix (3×9)
|
||||
for i in 1:9
|
||||
# ∂θy/∂x (for κx)
|
||||
B[1, i] = detJ_inv * (y12 * dHy_dξ[i] - y31 * dHy_dη[i])
|
||||
|
||||
# -∂θx/∂y (for κy)
|
||||
B[2, i] = detJ_inv * (x12 * dHx_dξ[i] - x31 * dHx_dη[i])
|
||||
|
||||
# ∂θy/∂y - ∂θx/∂x (for κxy)
|
||||
dθy_dy = detJ_inv * (-x12 * dHy_dξ[i] + x31 * dHy_dη[i])
|
||||
dθx_dx = detJ_inv * (y12 * dHx_dξ[i] - y31 * dHx_dη[i])
|
||||
B[3, i] = dθy_dy - dθx_dx
|
||||
end
|
||||
|
||||
return B
|
||||
end
|
||||
|
||||
function curvature_matrix(sf::DKTShapeFunctions, X::NTuple{3,Vec{2,T}}, ξ::T, η::T) where T
|
||||
B = zeros(T, 3, 9)
|
||||
curvature_matrix!(B, sf, ξ, η)
|
||||
return B
|
||||
end
|
||||
|
||||
# ============================================================================
|
||||
# Element Stiffness Matrix
|
||||
# ============================================================================
|
||||
|
||||
"""
|
||||
element_stiffness_matrix(X::NTuple{3, Vec{2, T}}, formulation::DKTFormulation, material::AbstractMaterial) where T
|
||||
|
||||
Compute 9×9 element stiffness matrix for DKT element.
|
||||
|
||||
Uses 3-point Gauss quadrature (exact for DKT).
|
||||
|
||||
# Arguments
|
||||
- `X`: Tuple of 3 nodal coordinates
|
||||
- `formulation`: DKT formulation with thickness
|
||||
- `material`: Material with E and ν properties (e.g., LinearElastic)
|
||||
|
||||
# Returns
|
||||
- `Matrix{T}`: 9×9 stiffness matrix
|
||||
"""
|
||||
function element_stiffness_matrix!(
|
||||
Ke::AbstractMatrix{T},
|
||||
B::AbstractMatrix{T},
|
||||
DB::AbstractMatrix{T},
|
||||
X::NTuple{3,Vec{2,T}},
|
||||
sf::DKTShapeFunctions,
|
||||
D_mat::AbstractMatrix{T},
|
||||
gauss_points::NTuple{3,Tuple{Float64,Float64}},
|
||||
gauss_weight::Float64) where T
|
||||
area = triangle_area(X)
|
||||
fill!(Ke, zero(T))
|
||||
for (ξ, η) in gauss_points
|
||||
curvature_matrix!(B, sf, ξ, η)
|
||||
mul!(DB, D_mat, B)
|
||||
mul!(Ke, adjoint(B), DB, gauss_weight * area, one(T))
|
||||
end
|
||||
return Ke
|
||||
end
|
||||
|
||||
function element_stiffness_matrix(X::NTuple{3,Vec{2,T}}, formulation::DKTFormulation, material::AbstractMaterial) where T
|
||||
sf = DKTShapeFunctions(X)
|
||||
D_mat = bending_matrix(material, formulation.thickness)
|
||||
Ke = zeros(Float64, DKT_ELEMENT_DOF, DKT_ELEMENT_DOF)
|
||||
B = zeros(Float64, 3, DKT_ELEMENT_DOF)
|
||||
DB = similar(B)
|
||||
element_stiffness_matrix!(Ke, B, DB, X, sf, D_mat, DKT_GAUSS_POINTS, DKT_GAUSS_WEIGHT)
|
||||
return Ke
|
||||
end
|
||||
|
||||
# ============================================================================
|
||||
# Assembly into Global System
|
||||
# ============================================================================
|
||||
|
||||
struct DKTAssemblyCache
|
||||
K_csc::SparseMatrixCSC{Float64,Int}
|
||||
sorteddofs::Vector{Int}
|
||||
permutation::Vector{Int}
|
||||
gdofs::Vector{Int}
|
||||
Ke::Matrix{Float64}
|
||||
B::Matrix{Float64}
|
||||
DB::Matrix{Float64}
|
||||
X_buffer::Vector{Vec{2,Float64}}
|
||||
elements::Vector{UInt32}
|
||||
D_mat::Matrix{Float64}
|
||||
f::Vector{Float64}
|
||||
end
|
||||
|
||||
function DKTAssemblyCache(physics::Physics{DKTFormulation,PlateDisplacement,M,Mat}) where {M<:AbstractMesh,Mat<:AbstractMaterial}
|
||||
mesh = physics.mesh
|
||||
@assert typeof(mesh).parameters[1] == DKT_NNODE "DKT requires 3-node triangles"
|
||||
|
||||
I, J, ndofs_total = build_plate_sparsity_pattern(mesh)
|
||||
K_csc = sparse(I, J, ones(length(I)), ndofs_total, ndofs_total)
|
||||
fill!(K_csc.nzval, 0.0)
|
||||
|
||||
ndofs_elem = DKT_ELEMENT_DOF
|
||||
|
||||
sorteddofs = Vector{Int}(undef, ndofs_elem)
|
||||
permutation = Vector{Int}(undef, ndofs_elem)
|
||||
gdofs = Vector{Int}(undef, ndofs_elem)
|
||||
|
||||
Ke = zeros(Float64, ndofs_elem, ndofs_elem)
|
||||
B = zeros(Float64, 3, ndofs_elem)
|
||||
DB = similar(B)
|
||||
X_buffer = [Vec{2}((0.0, 0.0)) for _ in 1:DKT_NNODE]
|
||||
|
||||
elements = plate_element_indices(mesh, physics.element_set)
|
||||
D_mat = bending_matrix(physics.material, physics.formulation.thickness)
|
||||
f = zeros(ndofs_total)
|
||||
|
||||
return DKTAssemblyCache(K_csc, sorteddofs, permutation, gdofs, Ke, B, DB, X_buffer, elements, D_mat, f)
|
||||
end
|
||||
|
||||
function apply_neumann_forces!(f::Vector{Float64}, bc_neumann, nnodes::Int)
|
||||
if hasfield(typeof(bc_neumann), :surface_ids) && hasfield(typeof(bc_neumann), :values)
|
||||
surface_ids = getfield(bc_neumann, :surface_ids)
|
||||
values = getfield(bc_neumann, :values)
|
||||
for (node, force) in zip(surface_ids, values)
|
||||
if node <= nnodes
|
||||
@inbounds for α in 1:min(DKT_DOF_PER_NODE, length(force))
|
||||
f[DKT_DOF_PER_NODE * (node - 1) + α] += force[α]
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
function apply_dirichlet_constraints!(K::SparseMatrixCSC{Float64,Int}, f::Vector{Float64}, bc_dirichlet, ndofs::Int)
|
||||
node_ids = getfield(bc_dirichlet, :node_ids)
|
||||
components = getfield(bc_dirichlet, :components)
|
||||
values = getfield(bc_dirichlet, :values)
|
||||
|
||||
for i in 1:length(node_ids)
|
||||
node = node_ids[i]
|
||||
comp_list = components[i]
|
||||
val_entry = values[i]
|
||||
for comp in comp_list
|
||||
dof = DKT_DOF_PER_NODE * (node - 1) + comp
|
||||
if 1 <= dof <= ndofs
|
||||
K[dof, :] .= 0.0
|
||||
K[:, dof] .= 0.0
|
||||
K[dof, dof] = 1.0
|
||||
value = val_entry isa AbstractVector ? val_entry[comp] : val_entry
|
||||
f[dof] = value
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
|
||||
function assemble_elements_dkt!(
|
||||
cache::DKTAssemblyCache,
|
||||
physics::Physics{DKTFormulation,PlateDisplacement,M,Mat}) where {M<:AbstractMesh,Mat<:AbstractMaterial}
|
||||
mesh = physics.mesh
|
||||
ndofs_elem = DKT_ELEMENT_DOF
|
||||
fill!(cache.K_csc.nzval, 0.0)
|
||||
|
||||
for elem_id in cache.elements
|
||||
conn = mesh.connectivity[Int(elem_id)]
|
||||
fill_coordinate_buffer!(cache.X_buffer, mesh.nodes, conn)
|
||||
X = (cache.X_buffer[1], cache.X_buffer[2], cache.X_buffer[3])
|
||||
sf = DKTShapeFunctions(X)
|
||||
element_stiffness_matrix!(cache.Ke, cache.B, cache.DB, X, sf, cache.D_mat, DKT_GAUSS_POINTS, DKT_GAUSS_WEIGHT)
|
||||
plate_global_dofs!(cache.gdofs, conn)
|
||||
|
||||
sortperm!(cache.permutation, cache.gdofs)
|
||||
@inbounds for k in 1:ndofs_elem
|
||||
cache.sorteddofs[k] = cache.gdofs[cache.permutation[k]]
|
||||
end
|
||||
|
||||
for i_local in 1:ndofs_elem
|
||||
i_global = cache.sorteddofs[i_local]
|
||||
col_start = cache.K_csc.colptr[i_global]
|
||||
col_end = cache.K_csc.colptr[i_global + 1] - 1
|
||||
Ri = col_start
|
||||
for ri in 1:ndofs_elem
|
||||
row_sorted = cache.sorteddofs[ri]
|
||||
while Ri <= col_end && cache.K_csc.rowval[Ri] < row_sorted
|
||||
Ri += 1
|
||||
end
|
||||
if Ri <= col_end && cache.K_csc.rowval[Ri] == row_sorted
|
||||
orig_row = cache.permutation[ri]
|
||||
orig_col = cache.permutation[i_local]
|
||||
cache.K_csc.nzval[Ri] += cache.Ke[orig_row, orig_col]
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
return cache
|
||||
end
|
||||
|
||||
function assemble!(physics::Physics{DKTFormulation,PlateDisplacement,M,Mat}) where {M<:AbstractMesh,Mat<:AbstractMaterial}
|
||||
cache = DKTAssemblyCache(physics)
|
||||
return _assemble_dkt!(physics, cache)
|
||||
end
|
||||
|
||||
function _assemble_dkt!(
|
||||
physics::Physics{DKTFormulation,PlateDisplacement,M,Mat},
|
||||
cache::DKTAssemblyCache) where {M<:AbstractMesh,Mat<:AbstractMaterial}
|
||||
|
||||
fill!(cache.f, 0.0)
|
||||
assemble_elements_dkt!(cache, physics)
|
||||
|
||||
nnodes = length(physics.mesh.nodes)
|
||||
ndofs = DKT_DOF_PER_NODE * nnodes
|
||||
|
||||
apply_neumann_forces!(cache.f, physics.bc_neumann, nnodes)
|
||||
|
||||
K = copy(cache.K_csc)
|
||||
apply_dirichlet_constraints!(K, cache.f, physics.bc_dirichlet, ndofs)
|
||||
|
||||
return (K, cache.f)
|
||||
end
|
||||
|
||||
# Export functions
|
||||
export DKTShapeFunctions
|
||||
export element_stiffness_matrix
|
||||
export curvature_matrix
|
||||
Reference in New Issue
Block a user