For Finite Element Analysis M Files | Matlab Codes

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

% 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 stresses for e = 1:size(elements,1) n1 = elements(e,1); n2 = elements(e,2); L = nodes(n2) - nodes(n1); u1 = U(n1); u2 = U(n2); strain = (u2 - u1)/L; stress = E * strain; fprintf('Element %d: Strain = %.4e, Stress = %.2f MPa\n', e, strain, stress/1e6); end

% Geometry: nodes and elements nodes = [0; 0.5; 1.0]; % Nodal coordinates (m) elements = [1 2; 2 3]; % Element connectivity matlab codes for finite element analysis m files

% Assembly into global matrix dof_list = [n1, n2]; K_global(dof_list, dof_list) = K_global(dof_list, dof_list) + ke; end

% 1. Pre-processing % - Define geometry, material properties, boundary conditions % - Generate mesh (nodes and elements) % 2. Assembly % - Initialize global stiffness matrix K and force vector F % - Loop over elements, compute element stiffness matrix, assemble

% Nodes (x, y) nodes = [0, 0; % Node 1 0.1, 0; % Node 2 0.1, 0.1; % Node 3 0, 0.1]; % Node 4 Example Function Library (Modular Approach) File: bar2e

% --- Assembly --- K_global = zeros(n_dof); F_global = zeros(n_dof, 1);

% Number of nodes and DOFs (1 DOF per node for axial) n_nodes = length(nodes); n_dof = n_nodes;

% --- Post-processing --- disp('Nodal displacements (m):'); disp(U); and electromagnetics. MATLAB

% Area area = 0.5 * abs((x(2)-x(1))*(y(3)-y(1)) - (x(3)-x(1))*(y(2)-y(1)));

% --- Assembly --- n_dof = size(nodes,1)*2; K = zeros(n_dof); F = F_applied;

% 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)

1. Introduction Finite Element Analysis (FEA) is a numerical technique for solving engineering problems such as structural analysis, heat transfer, fluid flow, and electromagnetics. MATLAB, with its powerful matrix manipulation capabilities and high-level programming environment, is an excellent platform for implementing FEA from scratch using M-files.