+-------------------------------------------------------+ | 1. Pre-Processing | | - Define Geometry, Nodes, and Connectivity (Mesh) | | - Define Material Properties & Section Data | +-------------------------------------------------------+ | v +-------------------------------------------------------+ | 2. Stiffness Matrix Assembly | | - Loop over elements to compute local k | | - Map local k to Global Stiffness Matrix K | +-------------------------------------------------------+ | v +-------------------------------------------------------+ | 3. Boundary Conditions & Loads | | - Apply known nodal forces to Global Vector F | | - Enforce essential boundary conditions (Displacements)| +-------------------------------------------------------+ | v +-------------------------------------------------------+ | 4. Solver | | - Partition system or apply penalty/elimination | | - Solve systems of equations: U = K \ F | +-------------------------------------------------------+ | v +-------------------------------------------------------+ | 5. Post-Processing | | - Calculate element strains and stresses | | - Plot deformed shapes, stress contours, & reactions | +-------------------------------------------------------+ Pre-Processing
%% ---------- STEP 1: INPUT DATA ---------- % Nodes: [x, y] coordinates (meters) nodes = [0, 0; % Node 1 4, 0; % Node 2 4, 3; % Node 3 0, 3]; % Node 4
% Assemble global stiffness matrix (K) and force vector (F) K = zeros(numDofs, numDofs); F = zeros(numDofs, 1);
% --- Apply Boundary Conditions (Penalty Method / Matrix Partitioning) --- % We use the 'Zeroing' method: K_ff * d_f = F_f free_dofs = setdiff(1:ndof, fixed_dofs);
Modifies the system equations to account for fixed supports (e.g., using the penalty method or direct elimination). matlab codes for finite element analysis m files
– Separate functions for isotropic, orthotropic, thermal, etc.
: The Finite Element Toolbox 2.1 on MathWorks File Exchange offers basic scripts for 2D/3D problems, ideal for students and researchers. Common Workflow in FEA M-Files
If your custom FEA solver outputs unexpected results, use these systematic checks to isolate the error:
Vectorization is essential for maintaining performance when moving from small academic models to complex, high-density meshes. Boundary Conditions & Loads | | - Apply
B = (1/(2*A_e)) * [ y(2)-y(3), 0, y(3)-y(1), 0, y(1)-y(2), 0; 0, x(3)-x(2), 0, x(1)-x(3), 0, x(2)-x(1); x(3)-x(2), y(2)-y(3), x(1)-x(3), y(3)-y(1), x(2)-x(1), y(1)-y(2) ];
% --- 1D Truss FEA .m File --- function Truss1D() % 1. Preprocessing E = 200e9; % Young's Modulus (Pa) A = 0.01; % Area (m^2) L = 2; % Total Length (m) numElems = 10; nodes = linspace(0, L, numElems+1); elemLen = L/numElems; % Global Stiffness Matrix Assembly K = zeros(numElems+1, numElems+1); ke = (E*A/elemLen) * [1 -1; -1 1]; % Local Stiffness for i = 1:numElems K(i:i+1, i:i+1) = K(i:i+1, i:i+1) + ke; end % Force vector and Boundary Conditions F = zeros(numElems+1, 1); F(end) = 10000; % 10kN load at end % Fix Node 1 (U1 = 0) K_reduced = K(2:end, 2:end); F_reduced = F(2:end); % 3. Solve U_reduced = K_reduced \ F_reduced; U = [0; U_reduced]; % 4. Post-processing disp('Displacements (m):'); disp(U); plot(nodes, U, '-o'); xlabel('Position (m)'); ylabel('Displacement (m)'); title('1D Truss Deflection'); end Use code with caution. 4. Advanced: 2D Plane Stress (Triangle Elements)
%% ---------- STEP 2: SETUP GLOBAL SYSTEM ---------- numNodes = size(nodes,1); numDofs = 2 * numNodes; % 2 dofs per node (ux, uy) numElem = size(elements,1);
This public link is valid for 7 days and shares a thread, including any personal information you added. This link or copies made by others cannot be deleted. If you share with third parties, their policies apply. Can’t copy the link right now. Try again later. For an element with Young's modulus
% Plot deformed truss (red, scaled) scale = 100; X_def_scaled = X_orig + scale * U(1:2:end); Y_def_scaled = Y_orig + scale * U(2:2:end); for e = 1:numElem n1 = elements(e,1); n2 = elements(e,2); plot([X_def_scaled(n1), X_def_scaled(n2)], ... [Y_def_scaled(n1), Y_def_scaled(n2)], 'r--o', 'LineWidth',1.5); end
MATLAB is widely used in academic and industrial settings for developing and prototyping Finite Element Analysis (FEA) codes due to its powerful matrix manipulation capabilities, built-in linear algebra solvers, and easy-to-use visualization tools. While commercial FEA packages (e.g., ANSYS, Abaqus) offer robust solutions, writing MATLAB .m files from scratch provides deep insight into the mathematical and computational foundations of the finite element method.
Element 1: Stress = 150.00 MPa Element 2: Stress = 75.00 MPa
: Critically important for large models to reduce memory usage by storing only non-zero entries in Kbold cap K \ (mldivide) : Efficiently solves the sparse system
The 1D bar element is the simplest finite element. It resists axial deformation along its longitudinal axis. For an element with Young's modulus , cross-sectional area , and length , the local element stiffness matrix is defined as: