% Connectivity (Element e connects Node i to Node i+1) % For 1D linear elements, this is implicit, but we define it for clarity connectivity = [1:nNode-1; 2:nNode]';
% truss2d_analysis.m % A clean, modular 2D Truss Finite Element Solver clearvars; clc; %% 1. INPUT DATA (Pre-Processing) % Nodal Coordinates [x, y] nodes = [0, 0; % Node 1 10, 0; % Node 2 5, 5]; % Node 3 % Element Connectivity [Node_Start, Node_End, E, A] % E = Young's Modulus (Pa), A = Cross-sectional Area (m^2) elements = [1, 2, 210e9, 0.01; 2, 3, 210e9, 0.01; 3, 1, 210e9, 0.01]; % Boundary Conditions: [Node_ID, DOF (1=X, 2=Y), Prescribed_Value] BCs = [1, 1, 0; % Node 1 fixed in X 1, 2, 0; % Node 1 fixed in Y 2, 2, 0]; % Node 2 fixed in Y (Roller) % Nodal Loads: [Node_ID, DOF, Force_Value] loads = [3, 2, -50000]; % 50kN downward load at Node 3 %% 2. INITIALIZATION numNodes = size(nodes, 1); numElements = size(elements, 1); GDof = 2 * numNodes; % 2 Degrees of Freedom per node K_global = zeros(GDof, GDof); F_global = zeros(GDof, 1); %% 3. GLOBAL STIFFNESS MATRIX ASSEMBLY for e = 1:numElements % Extract element properties n1 = elements(e, 1); n2 = elements(e, 2); E = elements(e, 3); A = elements(e, 4); % Geometry calculations x1 = nodes(n1, 1); y1 = nodes(n1, 2); x2 = nodes(n2, 1); y2 = nodes(n2, 2); L = hypot(x2 - x1, y2 - y1); % Direction cosines c = (x2 - x1) / L; s = (y2 - y1) / L; % Local stiffness matrix for a 2D truss element k_local = (E * A / L) * [ c^2, c*s, -c^2, -c*s; c*s, s^2, -c*s, -s^2; -c^2, -c*s, c^2, c*s; -c*s, -s^2, c*s, s^2]; % Element global degree of freedom mapping elementDof = [2*n1-1, 2*n1, 2*n2-1, 2*n2]; % Assembly into global stiffness matrix K_global(elementDof, elementDof) = K_global(elementDof, elementDof) + k_local; end %% 4. APPLY LOADS for i = 1:size(loads, 1) nodeID = loads(i, 1); dof = loads(i, 2); val = loads(i, 3); globalDof = 2 * (nodeID - 1) + dof; F_global(globalDof) = val; end %% 5. APPLY BOUNDARY CONDITIONS (Penalty Method) K_constrained = K_global; F_constrained = F_global; penalty = 1e15; % Large stiffness factor to enforce zero displacement for i = 1:size(BCs, 1) nodeID = BCs(i, 1); dof = BCs(i, 2); val = BCs(i, 3); globalDof = 2 * (nodeID - 1) + dof; K_constrained(globalDof, globalDof) = K_constrained(globalDof, globalDof) + penalty; F_constrained(globalDof) = F_constrained(globalDof) + penalty * val; end %% 6. SOLVE SYSTEM U_global = K_constrained \ F_constrained; %% 7. POST-PROCESSING (Stress & Strain) fprintf('\n--- NODAL DISPLACEMENTS ---\n'); for n = 1:numNodes fprintf('Node %d: U_x = %12.4e m, U_y = %12.4e m\n', n, U_global(2*n-1), U_global(2*n)); end fprintf('\n--- ELEMENT STRESSES ---\n'); for e = 1:numElements n1 = elements(e, 1); n2 = elements(e, 2); E = elements(e, 3); x1 = nodes(n1, 1); y1 = nodes(n1, 2); x2 = nodes(n2, 1); y2 = nodes(n2, 2); L = hypot(x2 - x1, y2 - y1); c = (x2 - x1) / L; s = (y2 - y1) / L; % Extract displacements for this element u = [U_global(2*n1-1); U_global(2*n1); U_global(2*n2-1); U_global(2*n2)]; % Transformation matrix row to get axial strain strain = (1/L) * [-c, -s, c, s] * u; stress = E * strain; fprintf('Element %d: Stress = %12.2f MPa\n', e, stress / 1e6); end Use code with caution. 3. Advanced Hot Topics in Modern FEA Coding
% Efficient Sparse Matrix Assembly Matrix Formula % K = sparse(I, J, S, nDOF, nDOF) Use code with caution.
: Calculate secondary variables such as element stresses, strains, and reaction forces. Production-Ready MATLAB Code: 2D Truss FEA Solver
Instantiate the global matrix instantaneously using K_global = sparse(I, J, V, GDof, GDof) . Non-Linear Solver Implementation matlab codes for finite element analysis m files hot
Global stiffness matrices contain mostly zeros. Use MATLAB’s built-in sparse storage system ( sparse and spalloc ) to save memory and drastically speed up the processing phase:
This script models planar heat transfers using 4-node isoparametric bilinear quadrilateral elements and 2x2 Gauss-Legendre quadrature integration.
function Thermal2D_Quad4() % Element Conductivity Matrix Tensor D = [1.5, 0; 0, 1.5]; % Thermal conductivity kx = ky = 1.5 W/mK % Four-Node Element Local Coordinates (Normalized Square) % Order: (-1,-1), (1,-1), (1,1), (-1,1) nodes_e = [0.0, 0.0; 1.0, 0.0; 1.0, 1.0; 0.0, 1.0]; % Gauss Quadrature Points and Weights (2x2 Rule) gp = [-1/sqrt(3), 1/sqrt(3)]; gw = [1.0, 1.0]; ke = zeros(4, 4); % Numerical Integration Loop for i = 1:2 for j = 1:2 xi = gp(i); eta = gp(j); % Shape Function Derivatives in Natural Coordinates dN_dxi = 0.25 * [-(1-eta), (1-eta), (1+eta), -(1+eta)]; dN_deta = 0.25 * [-(1-xi), -(1+xi), (1+xi), (1-xi)]; Jacobian = [dN_dxi; dN_deta] * nodes_e; detJ = det(Jacobian); invJ = inv(Jacobian); % Transform Shape Function Gradients to Physical Space B = invJ * [dN_dxi; dN_deta]; % Accumulate Gauss Point Contributions ke = ke + (B' * D * B) * detJ * gw(i) * gw(j); end end fprintf('\n--- Element Conductivity Matrix (Ke) ---\n'); disp(ke); end Use code with caution. 4. Advanced Enhancements: Solver Performance and UI
Before diving into M-files, let's review some basic MATLAB commands used in FEA: % Connectivity (Element e connects Node i to
: This widely used resource provides scripts for beams, plates, and shells. Ferreira’s book is popular because it bridges the gap between theory and numerical implementation.
For detailed stress distributions, 4-node isoparametric elements utilizing Gauss-Legendre quadrature integration are preferred.
%% --- 1. Input Parameters & Mesh Generation --- L = 1.0; % Length of the rod k = 10.0; % Thermal conductivity Q = 5.0; % Internal heat generation rate (source term)
% Nodes at boundaries bc_node_left = 1; bc_node_right = nNode; GLOBAL STIFFNESS MATRIX ASSEMBLY for e = 1:numElements
% element_stiffness.m function ke = element_stiffness(xy, C) [B, area] = shape(xy); ke = B' C B*area; end
Notes:
If you need help adapting this code for a specific project, please let me know:
A typical thermal FEA in MATLAB follows a structured path from geometry to results visualization.
top_opt_88.m
% Get coordinates x1 = node_coords(node1); x2 = node_coords(node2);