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Matlab Codes For Finite Element Analysis M Files [new]

% FEM_TrussAnalysis.m
% Finite Element Analysis of a 2D Truss Structure
% Solves for displacements, reactions, and element forces
% Units: N, m, Pa (consistent system)
clear; clc; close all;
%% ---------- 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
% Elements: [node_i, node_j, E (Pa), A (m^2)]
E = 200e9;        % Young's modulus (steel)
A = 0.0005;       % Cross-sectional area (m^2)
elements = [1, 2, E, A;
            2, 3, E, A;
            3, 4, E, A;
            4, 1, E, A;
            1, 3, E, A];   % Diagonal brace
% Boundary conditions: [node, dof (1=x,2=y), displacement]
% 0 = fixed, [] = free, value = prescribed displacement
BC = [1, 1, 0;    % Node1, x-fixed
      1, 2, 0;    % Node1, y-fixed
      4, 1, 0;    % Node4, x-fixed
      4, 2, 0];   % Node4, y-fixed
% Loads: [node, dof, force (N)]
loads = [2, 2, -10000;   % Node2, downward 10 kN
         3, 1, 5000];    % Node3, rightward 5 kN
%% ---------- STEP 2: SETUP GLOBAL SYSTEM ----------
numNodes = size(nodes,1);
numDofs = 2 * numNodes;          % 2 dofs per node (ux, uy)
numElem = size(elements,1);
% Assemble global stiffness matrix (K) and force vector (F)
K = zeros(numDofs, numDofs);
F = zeros(numDofs, 1);
% Function to get global dof indices for an element
getDofs = @(node_i, node_j) [2*node_i-1, 2*node_i, 2*node_j-1, 2*node_j];
for e = 1:numElem
    n1 = elements(e,1);
    n2 = elements(e,2);
    Ee = elements(e,3);
    Ae = elements(e,4);
% Coordinates of nodes
    x1 = nodes(n1,1); y1 = nodes(n1,2);
    x2 = nodes(n2,1); y2 = nodes(n2,2);
% Element length and direction cosines
    L = sqrt((x2-x1)^2 + (y2-y1)^2);
    C = (x2-x1)/L;
    S = (y2-y1)/L;
% Element stiffness matrix in global coordinates
    k_local = (Ee*Ae/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];
% Assembly into global matrix
    dofs = getDofs(n1, n2);
    K(dofs, dofs) = K(dofs, dofs) + k_local;
end
% Assemble global force vector
for l = 1:size(loads,1)
    node = loads(l,1);
    dof = loads(l,2);
    val = loads(l,3);
    global_dof = 2*(node-1) + dof;
    F(global_dof) = F(global_dof) + val;
end
%% ---------- STEP 3: APPLY BOUNDARY CONDITIONS ----------
% Identify fixed dofs and free dofs
fixed_dofs = [];
for bc = 1:size(BC,1)
    node = BC(bc,1);
    dof = BC(bc,2);
    global_dof = 2*(node-1) + dof;
    fixed_dofs = [fixed_dofs, global_dof];
    % Set prescribed displacement (if nonzero, handled via penalty or reduction)
    % Here we assume zero displacement for BC (can extend later)
end
free_dofs = setdiff(1:numDofs, fixed_dofs);
% Reduced system for free dofs
Kff = K(free_dofs, free_dofs);
Ff = F(free_dofs);
%% ---------- STEP 4: SOLVE FOR DISPLACEMENTS ----------
Uf = Kff \ Ff;       % Solve linear system
% Full displacement vector
U = zeros(numDofs, 1);
U(free_dofs) = Uf;
%% ---------- STEP 5: POST-PROCESSING ----------
% Compute reaction forces
R = K * U - F;
R_fixed = R(fixed_dofs);
% Compute element forces (axial)
fprintf('\n===== ELEMENT FORCES (N) =====\n');
for e = 1:numElem
    n1 = elements(e,1);
    n2 = elements(e,2);
    Ee = elements(e,3);
    Ae = elements(e,4);
x1 = nodes(n1,1); y1 = nodes(n1,2);
    x2 = nodes(n2,1); y2 = nodes(n2,2);
    L = sqrt((x2-x1)^2 + (y2-y1)^2);
    C = (x2-x1)/L;
    S = (y2-y1)/L;
dofs = getDofs(n1, n2);
    u_e = U(dofs);      % element displacements [u1, v1, u2, v2]
% Local axial displacement = u2_local - u1_local
    u_local = [C, S, 0, 0; 0, 0, C, S] * u_e;
    axial_strain = (u_local(2) - u_local(1)) / L;
    axial_force = Ee * Ae * axial_strain;
fprintf('Element %d: Force = %.2f N (%s)\n', e, axial_force, ...
        sign(axial_force)*'Compression' + (axial_force>=0)*'Tension');
end
%% ---------- STEP 6: DISPLAY RESULTS ----------
% Nodal displacements
fprintf('\n===== NODAL DISPLACEMENTS (m) =====\n');
for n = 1:numNodes
    ux = U(2*n-1);
    uy = U(2*n);
    fprintf('Node %d: ux = %.4e, uy = %.4e\n', n, ux, uy);
end
% Reaction forces
fprintf('\n===== REACTION FORCES (N) =====\n');
for i = 1:length(fixed_dofs)
    global_dof = fixed_dofs(i);
    node = ceil(global_dof/2);
    dof = mod(global_dof-1,2)+1;
    fprintf('Node %d, DOF %d: R = %.2f N\n', node, dof, R_fixed(i));
end
%% ---------- STEP 7: PLOT DEFORMED SHAPE ----------
% Original coordinates
X_orig = nodes(:,1);
Y_orig = nodes(:,2);
% Deformed coordinates
X_def = X_orig + U(1:2:end);
Y_def = Y_orig + U(2:2:end);
% Plot
figure;
hold on; grid on; axis equal;
title('Truss Deformation (Scale = 100x)');
xlabel('X (m)'); ylabel('Y (m)');
% Plot original truss (blue)
for e = 1:numElem
    n1 = elements(e,1);
    n2 = elements(e,2);
    plot([X_orig(n1), X_orig(n2)], [Y_orig(n1), Y_orig(n2)], 'b-o', 'LineWidth',1);
end
% 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
legend('Undeformed', 'Deformed (scaled)');
hold off;

Part 7: Solving Linear Systems – Beyond Backslash

MATLAB’s \ (mldivide) is excellent for small to medium problems. For larger FEA models, use:

Example iterative solver in an M-file:

tol = 1e-6;
maxit = 1000;
[U_free, flag] = pcg(K_free, F_free, tol, maxit);

If your matlab codes for finite element analysis m files become slow, profile using profile on and vectorize element loops.

References

  1. Hughes, T. J. R. (2000). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover.
  2. Reddy, J. N. (2019). Introduction to the Finite Element Method. McGraw-Hill.
  3. MATLAB Documentation: "Partial Differential Equation Toolbox" – Finite element solvers.
  4. Kwon, Y. W., & Bang, H. (2018). The Finite Element Method Using MATLAB. CRC Press.

Finite Element Analysis with MATLAB: A Comprehensive Guide to M-Files

Finite Element Analysis (FEA) is a powerful numerical method used to solve partial differential equations (PDEs) in various fields, including physics, engineering, and mathematics. MATLAB is a popular programming language used extensively in FEA due to its ease of use, flexibility, and high-performance computing capabilities. In this blog post, we will provide an overview of FEA using MATLAB and share some essential M-files for solving common FEA problems.

What is Finite Element Analysis?

Finite Element Analysis is a computational method that discretizes a complex problem into smaller, manageable parts called finite elements. Each element is a simple shape, such as a triangle or quadrilateral, with a set of nodes that define its geometry. The solution is approximated within each element using a set of basis functions, and the global solution is obtained by assembling the local solutions.

MATLAB for Finite Element Analysis

MATLAB provides an extensive range of tools and functions for FEA, including:

  1. Partial Differential Equation Toolbox: This toolbox provides a comprehensive set of tools for solving PDEs using FEA.
  2. MATLAB Coder: This tool allows you to generate C code from your MATLAB code, enabling high-performance computing and deployment.
  3. Parallel Computing Toolbox: This toolbox enables you to parallelize your FEA computations, reducing simulation time.

Basic Steps in FEA using MATLAB

To perform FEA using MATLAB, follow these basic steps: matlab codes for finite element analysis m files

  1. Mesh Generation: Create a mesh of finite elements that represents your problem domain.
  2. Element Stiffness Matrix: Compute the stiffness matrix for each element.
  3. Assemble Global Stiffness Matrix: Assemble the global stiffness matrix by combining the element stiffness matrices.
  4. Apply Boundary Conditions: Apply boundary conditions to the global stiffness matrix.
  5. Solve the System: Solve the linear system to obtain the solution.

MATLAB M-Files for FEA

Here are some essential M-files for solving common FEA problems:

2. 2D Laplace Equation

% 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
% Generate the mesh
[x, y] = meshgrid(linspace(0, Lx, Nx+1), linspace(0, Ly, Ny+1));
% Compute the stiffness matrix and load vector
K = zeros(Nx*Ny, Nx*Ny);
F = zeros(Nx*Ny, 1);
for i = 1:Nx
    for j = 1:Ny
        idx = (i-1)*Ny + j;
        K(idx, idx) = 2*(1/(x(i+1, j)-x(i, j))^2 + 1/(y(i, j+1)-y(i, j))^2);
        F(idx) = (x(i+1, j)-x(i, j))*(y(i, j+1)-y(i, j))/4*g(x(i, j), y(i, j)) + ...
                 (x(i+1, j)-x(i, j))*(y(i, j+1)-y(i, j))/4*g(x(i+1, j), y(i, j)) + ...
                 (x(i+1, j)-x(i, j))*(y(i, j+1)-y(i, j))/4*g(x(i, j), y(i, j+1)) + ...
                 (x(i+1, j)-x(i, j))*(y(i, j+1)-y(i, j))/4*g(x(i+1, j), y(i, j+1));
    end
end
% Apply boundary conditions
K(1, :) = 0; K(1, 1) = 1; F(1) = 0;
K(end, :) = 0; K(end, end) = 1; F(end) = 0;
% Solve the system
u = K\F;
% Plot the solution
surf(x, y, reshape(u, Ny+1, Nx+1));

Conclusion

In this blog post, we provided an overview of Finite Element Analysis using MATLAB and shared some essential M-files for solving common FEA problems. These examples demonstrate the power and flexibility of MATLAB for FEA. By mastering these concepts and M-files, you can efficiently solve complex problems in various fields.

References

Reviewing MATLAB codes for Finite Element Analysis (FEA) involves distinguishing between custom user-written scripts (.m files) and professional toolboxes. For educational purposes, A.J.M. Ferreira’s MATLAB Codes are the industry standard for learning the underlying mechanics. Core Components of FEA M-Files

A well-structured FEA script typically follows a modular workflow:

Preprocessing: Defining nodes, connectivity, material properties (Young's modulus), and section properties.

Assembly: Creating local element stiffness matrices (e.g., Q4elementstiffnessMatrix) and assembling them into a global sparse matrix.

Solver: Applying boundary conditions (Dirichlet/Neumann) and solving the system of equations ( % FEM_TrussAnalysis

Postprocessing: Visualizing results such as nodal displacements, stresses, and deformed shapes using MATLAB’s graphics engine. Top MATLAB FEA Code Repositories & Toolboxes

This content is structured as a standalone tutorial. It includes the main solver script, the core functions (m-files), and an explanation of how to run a sample problem (a cantilever beam).

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