% 2. Boundary conditions fixed_dofs = [...]; forces = [...];
A crucial step in any FEA is creating a good mesh. The package is a revolutionary tool that generates finite element meshes directly from 2D or 3D multi-phase images, such as microstructures of engineering materials. It supports multiple mesh generators and can export to Abaqus, Nastran, and other formats.
For large-scale industrial 3D configurations where system degree of freedom counts outpace physical RAM capacity ( matlab codes for finite element analysis m files hot
This is the fundamental building block of FEA. This script solves a simple 2D truss system using the Direct Stiffness Method. It is excellent for understanding how local element matrices are assembled into a global stiffness matrix.
% truss_fea.m - Linear Elastic 2D Truss Solver clear; clc; %% 1. PRE-PROCESSING % Node Coordinates [X, Y] nodes = [0, 0; % Node 1 90, 0; % Node 2 0, 90]; % Node 3 % Element Connectivity [Node_i, Node_j, E, A] % E = Modulus of Elasticity (ksi), A = Area (in^2) elements = [1, 2, 30000, 2.0; 3, 2, 30000, 1.5; 1, 3, 30000, 1.0]; numNodes = size(nodes, 1); numElems = size(elements, 1); GDof = 2 * numNodes; % 2 Degrees of Freedom per node % Initialize Global Matrices K_global = zeros(GDof, GDof); F_global = zeros(GDof, 1); % Apply External External Loads [Node, DOF(1=X, 2=Y), Value] loads = [2, 2, -50]; % 50 kips downward at node 2 for i = 1:size(loads, 1) nodeIdx = loads(i,1); dofIdx = loads(i,2); F_global(2 * (nodeIdx - 1) + dofIdx) = loads(i,3); end % Boundary Conditions: Prescribed Displacements [Node, DOF] % Fixed nodes: Node 1 (X, Y) and Node 3 (X, Y) fixedDofs = [1, 2, 5, 6]; activeDofs = setdiff(1:GDof, fixedDofs); %% 2. PROCESSING (ASSEMBLY) for e = 1:numElems n1 = elements(e, 1); n2 = elements(e, 2); E = elements(e, 3); A = elements(e, 4); % Element geometry x1 = nodes(n1, 1); y1 = nodes(n1, 2); x2 = nodes(n2, 1); y2 = nodes(n2, 2); L = sqrt((x2 - x1)^2 + (y2 - y1)^2); % Direction cosines C = (x2 - x1) / L; S = (y2 - y1) / L; % Local to Global Stiffness Matrix transformation k_local = (A * E / L) * [ C^2, C*S, -C^2, -C*S; C*S, S^2, -C*S, -S^2; -C^2, -C*S, C^2, C*S; -C*S, -S^2, C*S, S^2]; % Element degrees of freedom map eDof = [2*n1-1, 2*n1, 2*n2-1, 2*n2]; % Direct stiffness assembly K_global(eDof, eDof) = K_global(eDof, eDof) + k_local; end %% 3. SOLVING SYSTEM U_global = zeros(GDof, 1); U_global(activeDofs) = K_global(activeDofs, activeDofs) \ F_global(activeDofs); %% 4. POST-PROCESSING (STRESS & VISUALIZATION) fprintf('--- NODE DISPLACEMENTS ---\n'); for n = 1:numNodes fprintf('Node %d: U_x = %8.5f, U_y = %8.5f\n', n, U_global(2*n-1), U_global(2*n)); end % Plotting Configurations figure; hold on; grid on; scaleFactor = 100; % Scale up displacements for visual clarity for e = 1:numElems n1 = elements(e, 1); n2 = elements(e, 2); % Original coordinates plot([nodes(n1,1), nodes(n2,1)], [nodes(n1,2), nodes(n2,2)], 'k--', 'LineWidth', 1.2); % Deformed coordinates x1_def = nodes(n1,1) + scaleFactor * U_global(2*n1-1); y1_def = nodes(n1,2) + scaleFactor * U_global(2*n1); x2_def = nodes(n2,1) + scaleFactor * U_global(2*n2-1); y2_def = nodes(n2,2) + scaleFactor * U_global(2*n2); plot([x1_def, x2_def], [y1_def, y2_def], 'r-^', 'LineWidth', 2); end title('2D Truss Analysis: Original (Black) vs Deformed (Red)'); xlabel('X Coordinate'); ylabel('Y Coordinate'); axis equal; Use code with caution. 3. 2D Plane Stress FEM Code (Constant Strain Triangle) It supports multiple mesh generators and can export
Convection adds an additional term to the conductance matrix for nodes on the boundary:
The following M-file implements a simple 2D beam element using the finite element method: It is excellent for understanding how local element
For large systems, use sparse() to assemble the global stiffness matrix ( K ) to save memory and improve speed.
end
% Conduction matrix Ke = Ke + weight * detJ * (dN_dx * k * dN_dx');
% Efficient Sparse Matrix Assembly Matrix Formula % K = sparse(I, J, S, nDOF, nDOF) Use code with caution.