To scale these base scripts into production environments, integrate the following modifications. 1. Vectorized Boundary Condition Injection
% Heat source Q_dot = 10000; % Internal heat generation [W/m³]
Keep your compute_element_matrix.m separate from your main.m to make it easier to switch between 1D, 2D, or 3D elements.
This comprehensive guide breaks down the structure of high-performance MATLAB FEA codes, provides fully functional M-file scripts, and explains how to optimize them for speed. 1. The Core Architecture of a MATLAB FEA Script
% Temperature at element nodes T_elem = T(nodes);
%% Apply Boundary Conditions [K_modified, F_modified] = apply_boundary_conditions(... K_global, F_global, coordinates, T_left, T_right, h_conv, T_inf);
n_nodes = size(coordinates, 1); n_elements = size(elements, 1);
% Elements: [ElemID, Node1, Node2, E, A] elements = [1 1 2 200e9 0.001; 2 2 3 200e9 0.001; 3 1 3 200e9 0.001];
In the 2D truss script above, we use a for loop to step through elements sequentially. While suitable for simple, 1D, and 2D sparse structures, executing deep nested iterative loops for millions of 3D solid continuum elements degrades MATLAB's performance. For large-scale problems, optimize assembly using coordinate indexing arrays via the sparse() command:
c = c + x(ely,elx)^penal * Ue'*KE*Ue; dc(ely,elx) = -penal * x(ely,elx)^(penal-1) * Ue'*KE*Ue; end end
% Solve for next temperature T_solution(:,step+1) = U \ (L \ b);
% assemble.m function [K, F, freeDOF, fixedDOF, nodeMap] = assemble(nodes, elems, dirichlet, traction, C) nnode = size(nodes,1); ndof = 2 nnode; K = zeros(ndof); F = zeros(ndof,1); % assemble element stiffness for e=1:size(elems,1) enodes = elems(e,:); xy = nodes(enodes,:); ke = element_stiffness(xy, C); dof = reshape([2 enodes-1; 2 enodes],1,[]); K(dof,dof) = K(dof,dof) + ke; end % apply point tractions for i=1:size(traction,1) n = traction(i,1); F(2 n-1) = F(2 n-1) + traction(i,2); F(2 n) = F(2 n) + traction(i,3); end % Dirichlet dofs fixedDOF = []; for i=1:size(dirichlet,1) n = dirichlet(i,1); ux = dirichlet(i,2); uy = dirichlet(i,3); if ~isnan(ux); fixedDOF(end+1)=2 n-1; F = apply_prescribed(K,F,2 n-1,ux); end if ~isnan(uy); fixedDOF(end+1)=2 n; F = apply_prescribed(K,F,2 n,uy); end end allDOF = 1:ndof; freeDOF = setdiff(allDOF,fixedDOF); nodeMap = @(n) [2 n-1 2*n]; end
For a standard 1D linear elastic bar or truss element, the local stiffness matrix kebold k to the e-th power
To scale these base scripts into production environments, integrate the following modifications. 1. Vectorized Boundary Condition Injection
% Heat source Q_dot = 10000; % Internal heat generation [W/m³]
Keep your compute_element_matrix.m separate from your main.m to make it easier to switch between 1D, 2D, or 3D elements.
This comprehensive guide breaks down the structure of high-performance MATLAB FEA codes, provides fully functional M-file scripts, and explains how to optimize them for speed. 1. The Core Architecture of a MATLAB FEA Script
% Temperature at element nodes T_elem = T(nodes);
%% Apply Boundary Conditions [K_modified, F_modified] = apply_boundary_conditions(... K_global, F_global, coordinates, T_left, T_right, h_conv, T_inf);
n_nodes = size(coordinates, 1); n_elements = size(elements, 1);
% Elements: [ElemID, Node1, Node2, E, A] elements = [1 1 2 200e9 0.001; 2 2 3 200e9 0.001; 3 1 3 200e9 0.001];
In the 2D truss script above, we use a for loop to step through elements sequentially. While suitable for simple, 1D, and 2D sparse structures, executing deep nested iterative loops for millions of 3D solid continuum elements degrades MATLAB's performance. For large-scale problems, optimize assembly using coordinate indexing arrays via the sparse() command:
c = c + x(ely,elx)^penal * Ue'*KE*Ue; dc(ely,elx) = -penal * x(ely,elx)^(penal-1) * Ue'*KE*Ue; end end
% Solve for next temperature T_solution(:,step+1) = U \ (L \ b);
% assemble.m function [K, F, freeDOF, fixedDOF, nodeMap] = assemble(nodes, elems, dirichlet, traction, C) nnode = size(nodes,1); ndof = 2 nnode; K = zeros(ndof); F = zeros(ndof,1); % assemble element stiffness for e=1:size(elems,1) enodes = elems(e,:); xy = nodes(enodes,:); ke = element_stiffness(xy, C); dof = reshape([2 enodes-1; 2 enodes],1,[]); K(dof,dof) = K(dof,dof) + ke; end % apply point tractions for i=1:size(traction,1) n = traction(i,1); F(2 n-1) = F(2 n-1) + traction(i,2); F(2 n) = F(2 n) + traction(i,3); end % Dirichlet dofs fixedDOF = []; for i=1:size(dirichlet,1) n = dirichlet(i,1); ux = dirichlet(i,2); uy = dirichlet(i,3); if ~isnan(ux); fixedDOF(end+1)=2 n-1; F = apply_prescribed(K,F,2 n-1,ux); end if ~isnan(uy); fixedDOF(end+1)=2 n; F = apply_prescribed(K,F,2 n,uy); end end allDOF = 1:ndof; freeDOF = setdiff(allDOF,fixedDOF); nodeMap = @(n) [2 n-1 2*n]; end
For a standard 1D linear elastic bar or truss element, the local stiffness matrix kebold k to the e-th power
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