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Bosch Sensortec

Matlab Codes For Finite Element Analysis M Files Hot Here

function [K_global, M_global, F_global] = assemble_thermal_matrices(...
    coordinates, elements, k, rho, cp, Q_dot)
% Assemble global stiffness, mass, and force matrices
% Inputs:
%   coordinates - nodal coordinates
%   elements - element connectivity
%   k - thermal conductivity
%   rho - density
%   cp - specific heat
%   Q_dot - internal heat generation

n_nodes = size(coordinates, 1);
n_elements = size(elements, 1);


% Initialize global matrices
K_global = sparse(n_nodes, n_nodes);
M_global = sparse(n_nodes, n_nodes);
F_global = zeros(n_nodes, 1);


% Gauss quadrature points and weights (4-point for quadrilateral)
gauss_points = [-1/sqrt(3), 1/sqrt(3)];
gauss_weights = [1, 1];


% Loop through all elements
for elem = 1:n_elements
nodes = elements(elem, :);
elem_coords = coordinates(nodes, :);


% Initialize element matrices
Ke = zeros(4, 4);
Me = zeros(4, 4);
Fe = zeros(4, 1);
% Numerical integration
for i = 1:2
    for j = 1:2
        xi = gauss_points(i);
        eta = gauss_points(j);
        weight = gauss_weights(i) * gauss_weights(j);
% Shape functions and derivatives
        [N, dN_dxi] = shape_functions_quad4(xi, eta);
% Jacobian matrix
        J = dN_dxi' * elem_coords;
        detJ = abs(det(J));
        invJ = inv(J);
% Derivatives in physical coordinates
        dN_dx = dN_dxi * invJ;
% Conduction matrix
        Ke = Ke + weight * detJ * (dN_dx * k * dN_dx');
% Mass matrix (consistent)
        Me = Me + weight * detJ * (rho * cp) * (N * N');
% Force vector (heat generation)
        Fe = Fe + weight * detJ * Q_dot * N;
    end
end
% Assemble into global matrices
K_global(nodes, nodes) = K_global(nodes, nodes) + Ke;
M_global(nodes, nodes) = M_global(nodes, nodes) + Me;
F_global(nodes) = F_global(nodes) + Fe;



end
end


function [N, dN_dxi] = shape_functions_quad4(xi, eta)
% Shape functions for 4-node quadrilateral element
N = 1/4 * [(1-xi)(1-eta);
(1+xi)(1-eta);
(1+xi)(1+eta);
(1-xi)(1+eta)];


dN_dxi = 1/4 * [-(1-eta), -(1-xi);
(1-eta), -(1+xi);
(1+eta),  (1+xi);
-(1+eta),  (1-xi)];
end


%% Convergence study for mesh refinement
function error_analysis()
% Perform convergence study by refining mesh

% Mesh sizes to test
mesh_sizes = [5, 10, 15, 20, 30];
n_refinements = length(mesh_sizes);
errors = zeros(n_refinements, 1);
h_values = zeros(n_refinements, 1);


% Reference solution (very fine mesh)
nx_fine = 100;
ny_fine = 100;
[coord_fine, elem_fine] = generate_mesh_2D(0.1, 0.1, nx_fine, ny_fine);
[K_fine, M_fine, F_fine] = assemble_thermal_matrices(coord_fine, elem_fine, ...
15, 2700, 900, 10000);
[K_mod, F_mod] = apply_boundary_conditions(K_fine, F_fine, coord_fine, ...
100, 25, 50, 25);
T_ref = K_mod \ F_mod;


% Compute error for each mesh
for i = 1:n_refinements
nx = mesh_sizes(i);
ny = mesh_sizes(i);
h_values(i) = 0.1 / nx;


% Solve on current mesh
[coord, elem] = generate_mesh_2D(0.1, 0.1, nx, ny);
[K, M, F] = assemble_thermal_matrices(coord, elem, 15, 2700, 900, 10000);
[K_mod, F_mod] = apply_boundary_conditions(K, F, coord, 100, 25, 50, 25);
T_current = K_mod \ F_mod;
% Interpolate to fine mesh for error calculation
T_current_interp = griddata(coord(:,1), coord(:,2), T_current, ...
                             coord_fine(:,1), coord_fine(:,2), 'linear');
% L2 error norm
errors(i) = sqrt(sum((T_current_interp - T_ref).^2) / length(T_ref));
fprintf('Mesh: %dx%d, h = %.4f, Error = %.2e\n', nx, ny, h_values(i), errors(i));



end


% Plot convergence
figure;
loglog(h_values, errors, 'bo-', 'LineWidth', 2);
hold on;
% Theoretical convergence rate (linear elements)
h_ref = logspace(log10(min(h_values)), log10(max(h_values)), 100);
plot(h_ref, errors(1) * (h_ref/h_values(1)).^2, 'r--', 'LineWidth', 1.5);
xlabel('Element size h [m]');
ylabel('L2 Error Norm');
title('Convergence Study');
legend('FEA Solution', 'Theoretical O(h²)', 'Location', 'best');
grid on;


% Calculate convergence rate
p = polyfit(log(h_values), log(errors), 1);
fprintf('\nConvergence rate: %.2f (theoretical: 2.00)\n', p(1));
end

FEA combined with time-stepping (Backward Euler, Newmark-Beta).

Features:

Why popular: Foundation for linear elastic continuum FEA.

Here’s a complete, minimal M-file that assembles and solves a 2D truss bridge:

% hot_truss_solver.m
% A hot M-file for 2D truss analysis
clear; clc; close all;

% Nodal coordinates (x, y)
nodes = [0 0; 4 0; 8 0; 2 3; 6 3];
% Connectivity (element: node1 node2 A E)
elements = [1 2 0.01 200e9; 2 3 0.01 200e9; 1 4 0.015 200e9;
2 4 0.01 200e9; 2 5 0.015 200e9; 3 5 0.01 200e9;
4 5 0.01 200e9; 4 2 0.02 200e9];


% Boundary conditions (fixed nodes 1 and 3)
fixed = [1 1 0; 3 1 0]; % node, x-dof fixed (1), y-dof free (0)
loads = [3 1 -10000]; % node 3, x-direction, -10 kN


% Assemble and solve
[K, F] = assembleTruss(nodes, elements, fixed, loads);
U = K \ F;  % Nodal displacements matlab codes for finite element analysis m files hot


% Plot deformed shape (exaggerated)
plotDeformedTruss(nodes, elements, U, 100);
title('Hot Truss FEA: Deformed Shape (100x scale)');

(Note: The functions assembleTruss and plotDeformedTruss are separate M-files available in the hot FEA library.)

If your M-files run slowly, they aren’t hot—they’re cold. Apply these optimizations:

What’s next for "hot" MATLAB FEA codes?

The hottest emerging trend is coupling MATLAB’s FEA codes with machine learning. Researchers are creating M-files that: