% Apply force F_global(force_dof) = applied_force;
for e = 1:size(elements,1) % Element nodes n1 = elements(e,1); n2 = elements(e,2); n3 = elements(e,3);
% B matrix for CST B = zeros(3, 6); for i = 1:3 j = mod(i,3)+1; k = mod(i+1,3)+1; B(1, 2*i-1) = (y(j)-y(k)) / (2*area); B(2, 2*i) = (x(k)-x(j)) / (2*area); B(3, 2*i-1) = (x(k)-x(j)) / (2*area); B(3, 2*i) = (y(j)-y(k)) / (2*area); end
% Element length L = nodes(n2) - nodes(n1); matlab codes for finite element analysis m files
% --- Post-processing --- disp('Nodal displacements (m):'); disp(U);
% Deformed plot scale = 10; % deformation scale factor deformed = nodes + scale * U_nodes; figure; patch('Faces', elements, 'Vertices', deformed, 'FaceColor', 'cyan', 'EdgeColor', 'red'); hold on; patch('Faces', elements, 'Vertices', nodes, 'FaceColor', 'none', 'EdgeColor', 'black', 'LineStyle', '--'); axis equal; grid on; xlabel('X (m)'); ylabel('Y (m)'); title('Deformed (cyan) vs Undeformed (dashed) Shape'); legend('Deformed', 'Undeformed'); | Tip | Description | |------|-------------| | Vectorization | Avoid loops when possible; use reshape , repmat , and index vectors | | Sparse Matrices | For large problems, use sparse() to store global K matrix | | Modular Programming | Write functions for elem_stiffness , elem_mass , post_process | | Input Files | Store mesh, BCs, and loads in separate .mat or .txt files | | Visualization | Use patch , trisurf , quiver for 2D/3D results | | Verification | Compare with analytical solutions for simple cases | 6. Example Function Library (Modular Approach) File: bar2e.m (2-node bar element)
% 5. Post-processing % - Compute stresses, strains, reaction forces % - Visualize results Problem: Axially loaded bar with fixed-free boundary conditions. M-file: truss_1d.m % Apply force F_global(force_dof) = applied_force; for e
% Load: tension on right edge (nodes 2 and 3) force_val = 1000; % N/m % Node 2: Fx = force_val * area? For simplicity, point load F_applied = zeros(size(nodes,1)*2, 1); F_applied((2-1)*2 + 1) = force_val * 0.05 * thickness; % Node 2, ux F_applied((3-1)*2 + 1) = force_val * 0.05 * thickness; % Node 3, ux
% Plot deformed shape plot(nodes, U, 'ro-', 'LineWidth', 2); xlabel('X (m)'); ylabel('Displacement (m)'); title('1D Truss Deformation'); grid on; Problem: Thin plate with a hole under tension (simplified mesh). M-file: cst_plate.m
% Nodes (x, y) nodes = [0, 0; % Node 1 0.1, 0; % Node 2 0.1, 0.1; % Node 3 0, 0.1]; % Node 4 M-file: truss_1d
% Assembly dof_list = [(n1-1)*2+1, (n1-1)*2+2, ... (n2-1)*2+1, (n2-1)*2+2, ... (n3-1)*2+1, (n3-1)*2+2]; K(dof_list, dof_list) = K(dof_list, dof_list) + ke; end
% Element stiffness matrix ke = thickness * area * (B' * D * B);
% main_bar_assembly.m clear; clc; % ... define nodes, elements, E, A ... K_global = zeros(n_dof); for e = 1:ne n1 = elements(e,1); n2 = elements(e,2); L = nodes(n2) - nodes(n1); ke = bar2e(E, A, L); dof = [n1, n2]; K_global(dof, dof) = K_global(dof, dof) + ke; end % ... apply BCs, solve, post-process ... | Element Type | MATLAB Implementation Key Points | |---------------|----------------------------------| | 2D Quadrilateral (Q4) | Gauss quadrature, shape functions in natural coordinates | | Beam (2D Euler-Bernoulli) | 4 DOF per element (u1, theta1, u2, theta2) | | 3D Tetrahedron (TET4) | Volume coordinates, B matrix size 6x12 | | Heat Transfer (2D) | Same structure, but D becomes thermal conductivity matrix | 8. Conclusion MATLAB M-files provide a transparent, educational, and flexible environment for implementing Finite Element Analysis. The step-by-step approach—pre-processing, assembly, BC application, solving, and post-processing—remains consistent across problem types. While not as efficient as commercial FEA packages for large-scale problems, MATLAB FEA codes are invaluable for learning, prototyping, and research.
% 3. Apply Boundary Conditions % - Modify K and F to enforce Dirichlet (displacement) BCs