Isoparametric elements are commonly used in finite element analysis for modeling complex geometries. In Matlab, the process of creating isoparametric elements involves defining the geometry of the element and then mapping it to a reference element, such as a quadrilateral or triangular element.
Here's a sample code for creating isoparametric elements for a simple twodimensional rectangular domain:
% Define the rectangular domain
xmin = 0; xmax = 1; ymin = 0; ymax = 1;
% Define the number of elements in the x and y directions
nx = 10; ny = 10;
% Define the nodes of the rectangular domain
x = linspace(xmin, xmax, nx+1); Visit: www.matlabassignmentexperts.com
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y = linspace(ymin, ymax, ny+1);
[X,Y] = meshgrid(x,y);
nodes = [X(:) Y(:)];
% Define the connectivity of the rectangular domain
n1 = 1:ny; n2 = n1 + 1; n3 = n2 + nx; n4 = n1 + nx; connectivity = [n1' n2' n3' n4'];
% Define the isoparametric mapping function
syms r s
N = 0.25 * [(1-r)*(1-s), (1+r)*(1-s), (1+r)*(1+s), (1-r)*(1+s)];
J = [diff(N,r)*X(:) diff(N,s)*X(:) diff(N,r)*Y(:) diff(N,s)*Y(:)];
invJ = simplify(inv(J));
x = N*nodes(:,1);
y = N*nodes(:,2);
% Define the element stiffness matrix
B = invJ*[diff(N,r) zeros(1,4); zeros(1,4) diff(N,s)];
D = eye(2);
K = B'*D*B*det(J);
% Assemble the global stiffness matrix
numnodes = size(nodes,1); numelem = size(connectivity,1);
K_global = sparse(numnodes*2,numnodes*2); for elem = 1:numelem
nodes_elem = connectivity(elem,:);
K_elem = K; for i = 1:4 for j = 1:4
K_global([2*nodes_elem(i)-1 2*nodes_elem(i)], [2*nodes_elem(j)-1
2*nodes_elem(j)]) = ...
K_global([2*nodes_elem(i)-1 2*nodes_elem(i)], [2*nodes_elem(j)-1
2*nodes_elem(j)]) + ...
K_elem([2*i-1 2*i], [2*j-1 2*j]); end end end
% Apply boundary conditions
fixed_nodes = find(nodes(:,1)==xmin | nodes(:,1)==xmax | nodes(:,2)==ymin);
K_global(fixed_nodes*2-1,:) = 0;
K_global(fixed_nodes*2-1,fixed_nodes*2-1) = speye(length(fixed_nodes));
K_global(fixed_nodes*2,:) = 0;
K_global(fixed_nodes*2,fixed_nodes*2) = speye(length(fixed_nodes));
% Solve for the nodal displacements and reactions
f = zeros(numnodes*2,1);
f(end-1) = -1; % apply a unit load at the top right corner
u = K_global\f;
reaction_forces = K_global*u;
% Plot the results
figure subplot(2,2,1)
trimesh(connectivity, x, y, zeros(size(x)), 'EdgeColor', 'black', 'FaceColor', 'none')
title('Undeformed mesh