The code evaluates elemental equations and maps local interactions into a collective system.
: Visualize the results, such as deformed shapes or stress distributions. Specialized Toolboxes iFEM: an integrated finite element method package in MATLAB matlab codes for finite element analysis m files
% Efficient sparse assembly syntax K_global = sparse(i_index, j_index, k_values, total_dof, total_dof); Use code with caution. Numerical Integration (Gauss Quadrature) The code evaluates elemental equations and maps local
%% 2D Truss Finite Element Analysis Solver clc; clear; close all; % --- 1. Pre-processing --- % Nodal Coordinates matrix [X, Y] nodes = [0.0, 0.0; % Node 1 1.0, 0.0; % Node 2 0.5, 1.0]; % Node 3 num_nodes = size(nodes, 1); num_dofs = num_nodes * 2; % Element Connectivity matrix [Node_i, Node_j] elements = [1, 2; 2, 3; 3, 1]; num_elems = size(elements, 1); E = 200e9; % Pa A = 1e-4; % m^2 K_global = zeros(num_dofs, num_dofs); F_global = zeros(num_dofs, 1); % Define External Forces (Node 3, negative Y direction) F_global(6) = -10000; % 10 kN downward load at Node 3 Y-DOF % --- 2 & 3. Local Stiffness and Global Assembly --- for e = 1:num_elems n1 = elements(e, 1); n2 = elements(e, 2); dx = nodes(n2, 1) - nodes(n1, 1); dy = nodes(n2, 2) - nodes(n1, 2); L = sqrt(dx^2 + dy^2); % Direction cosines c = dx / L; s = dy / L; % Transformation matrix components for 2D truss k_local_global = (E * A / L) * [ c*c, c*s, -c*c, -c*s; c*s, s*s, -c*s, -s*s; -c*c, -c*s, c*c, c*s; -c*s, -s*s, c*s, s*s ]; % Global DOFs for this element elem_dofs = [2*n1-1, 2*n1, 2*n2-1, 2*n2]; % Assemble K_global(elem_dofs, elem_dofs) = K_global(elem_dofs, elem_dofs) + k_local_global; end % --- 4. Boundary Conditions --- % Fixed nodes: Node 1 (X, Y) and Node 2 (Y only) constrained_dofs = [1, 2, 4]; free_dofs = setdiff(1:num_dofs, constrained_dofs); % --- 5. Solve --- U = zeros(num_dofs, 1); U(free_dofs) = K_global(free_dofs, free_dofs) \ F_global(free_dofs); % --- 6. Post-processing Visualization --- scale_factor = 50; % Exaggerate displacement for visibility figure; hold on; grid on; % Plot Original and Deformed Structure for e = 1:num_elems n1 = elements(e,1); n2 = elements(e,2); % Original coordinates plot([nodes(n1,1), nodes(n2,1)], [nodes(n1,2), nodes(n2,2)], 'k--', 'LineWidth', 1.5); % Deformed coordinates x1_def = nodes(n1,1) + scale_factor * U(2*n1-1); y1_def = nodes(n1,2) + scale_factor * U(2*n1); x2_def = nodes(n2,1) + scale_factor * U(2*n2-1); y2_def = nodes(n2,2) + scale_factor * U(2*n2); plot([x1_def, x2_def], [y1_def, y2_def], 'r-生', 'LineWidth', 2); end title('2D Truss Deflection (Red) vs Undeformed Shape (Black)'); xlabel('X Position (m)'); ylabel('Y Position (m)'); legend('Undeformed', 'Deformed'); Use code with caution. 4. 2D Plane Stress Continuum Element (CST) Numerical Integration (Gauss Quadrature) %% 2D Truss Finite
: For time-dependent problems, such as a 1D heat equation, you can create animated plots to show how a profile evolves over time. Coding Best Practices for FEA
By using well-structured scripts and functions, you can create a clear and logical representation of the finite element method itself. For instance, a typical script will define the mesh and material properties, call a function to assemble the global stiffness matrix, another function to apply constraints and loads, a solver function to compute displacements, and finally, a post-processing function to visualize results. Furthermore, the extensive ecosystem of community-contributed toolboxes and functions can significantly accelerate development, allowing you to focus on the unique aspects of your problem rather than re-coding fundamental methods.
A matrix mapping local element nodes to global node numbers.