% ========================================================================= % MATLAB M-File: 1D Finite Element Analysis Solver for Bar Elements % ========================================================================= clear; clc; %% 1. Geometry and Material Properties % Node coordinates (x-coordinate for each node) nodes = [0.0; 0.5; 1.0]; % Element connectivity [Node_Start, Node_End] elements = [1, 2; 2, 3]; % Material properties for each element [E, A] % Element 1: E = 200e9 Pa, A = 0.01 m^2 % Element 2: E = 200e9 Pa, A = 0.005 m^2 prop = [200e9, 0.01; 200e9, 0.005]; num_nodes = size(nodes, 1); num_elements = size(elements, 1); %% 2. Initialize Global Matrices K = zeros(num_nodes, num_nodes); F = zeros(num_nodes, 1); U = zeros(num_nodes, 1); %% 3. Global Assembly Loop for e = 1:num_elements % Get node indices for current element node1 = elements(e, 1); node2 = elements(e, 2); % Calculate element length L = nodes(node2) - nodes(node1); % Extract material data E = prop(e, 1); A = prop(e, 2); % Compute local stiffness matrix k_local = (E * A / L) * [1, -1; -1, 1]; % Assembly into global stiffness matrix element_nodes = [node1, node2]; K(element_nodes, element_nodes) = K(element_nodes, element_nodes) + k_local; end %% 4. Apply Loads and Boundary Conditions % Apply external force of 10,000 N at Node 3 F(3) = 10000; % Apply boundary condition: Node 1 is fixed (U(1) = 0) fixed_nodes = [1]; all_nodes = 1:num_nodes; free_nodes = setdiff(all_nodes, fixed_nodes); %% 5. Solve for Displacements U(free_nodes) = K(free_nodes, free_nodes) \ F(free_nodes); %% 6. Display Results fprintf('--- FEA Results ---\n'); for i = 1:num_nodes fprintf('Node %d Displacement: %e m\n', i, U(i)); end Use code with caution. 2D Truss Element Assembly (.m Function)
% Define the problem parameters Lx = 1; Ly = 1; % Length of the domain Nx = 10; Ny = 10; % Number of elements g = @(x, y) sin(pi*x).*sin(pi*y); % Source term
Every standard Finite Element Analysis program written in MATLAB follows a structured, sequential pipeline. Maintaining this architecture ensures that your .m files remain modular, readable, and easy to debug.
% Apply boundary conditions K(1, :) = 0; K(1, 1) = 1; F(1) = 0; K(end, :) = 0; K(end, end) = 1; F(end) = 0; matlab codes for finite element analysis m files
2D truss elements (pin-jointed frames) handle forces in a two-dimensional plane. Each node has two degrees of freedom (horizontal displacement and vertical displacement
A well-structured FEA script typically follows a modular workflow:
A matrix mapping node IDs to spatial positions (e.g., Global Assembly Loop for e = 1:num_elements %
ke=t⋅Ae⋅(BTDB)k sub e equals t center dot cap A sub e center dot open paren cap B to the cap T-th power cap D cap B close paren is thickness and Aecap A sub e is the triangular element area.
Defining nodes, connectivity, material properties (Young's modulus), and section properties.
These frameworks often include extensive documentation, test suites, and example problems, allowing you to focus on the physics and numerics rather than low‑level coding details. Display Results fprintf('--- FEA Results ---\n'); for i
tol = 1e-6; maxit = 1000; [U_free, flag] = pcg(K_free, F_free, tol, maxit);
The core of a 2D elasticity solver is the element stiffness matrix for a quadrilateral or triangular element. A typical M‑file for a 4‑node quadrilateral might include:
: 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
function [Kmod,Fmod,freeDOF,u0] = apply_bc(K,F,bc) % bc: struct with fields .prescribed = [dof, value; ...] u0 = zeros(size(F)); pres = bc.prescribed; for i=1:size(pres,1) u0(pres(i,1)) = pres(i,2); end fixed = pres(:,1); allDOF = (1:length(F))'; freeDOF = setdiff(allDOF, fixed); Fmod = F(freeDOF) - K(freeDOF, fixed)*u0(fixed); Kmod = K(freeDOF, freeDOF); end
: Discretize the domain into nodes and elements.