Matlab Codes — For Finite Element Analysis M Files

% 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)

% --- Post-processing --- disp('Nodal displacements (m):'); disp(U);

% 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. matlab codes for finite element analysis m files

% --- Apply Boundary Conditions --- % Penalty method (or elimination method) penalty = 1e12; K_global(fixed_dof, fixed_dof) = K_global(fixed_dof, fixed_dof) + penalty; F_global(fixed_dof) = penalty * 0; % zero displacement

% --- Solve --- U = K \ F;

% Elements (triangle connectivity: node1, node2, node3) elements = [1, 2, 3; 1, 3, 4];

% Boundary conditions: fix left edge (nodes 1 and 4) fixed_dofs = [1, 2; % Node 1: DOF 1 (ux), DOF 2 (uy) 4, 2]; % Node 4: DOF 2? Actually Node 4 DOF 7 and 8 % Convert to global DOF numbering (2 DOF per node) % Global DOF: (node-1)*2 + 1 for ux, +2 for uy fixed_global = []; for i = 1:size(fixed_dofs,1) node = fixed_dofs(i,1); dof_type = fixed_dofs(i,2); % 1=ux, 2=uy fixed_global = [fixed_global, (node-1)*2 + dof_type]; end % Deformed plot scale = 10; % deformation

% 3. Apply Boundary Conditions % - Modify K and F to enforce Dirichlet (displacement) BCs