finite elements in matlab

Finite element method is a numerical technique used to solve partial differential equations. In Matlab, the Partial Differential Equation Toolbox provides a platform to implement this technique. The following steps can be followed to implement FEM in Matlab:

1. Create a 2D or 3D geometry model using the PDE Toolbox.

main.m
model = createpde(2);
geometryFromEdges(model,@circleg)
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2. Define the PDE equation to be solved. For example, consider the Poisson equation in 2D:

main.m
specifyCoefficients(model,'m',0,'d',1,'c',1,'a',0,'f',@(x,y) -4*pi^2*sin(2*pi*x)*sin(2*pi*y));
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3. Create a mesh structure using the generateMesh function.

main.m
mesh = generateMesh(model,'Hmax',0.1);
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4. Define the boundary conditions on the edges using the applyBoundaryCondition function.

main.m
applyBoundaryCondition(model,'edge',1:4,'u',0);
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5. Assemble the stiffness matrix and load vector using the assempde function.

main.m
[stiffnessMatrix,loadVector] = assempde(model,c,b,f);
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6. Apply the boundary conditions to the stiffness matrix and load vector.

main.m
[stiffnessMatrix,loadVector] = applyBoundaryCondition(model,stiffnessMatrix,loadVector);
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7. Solve the system of linear equations using the backslash operator.

main.m
solution = stiffnessMatrix\loadVector;
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8. Interpolate the solution onto the mesh using the interpolateSolution function.

main.m
u = interpolateSolution(model,solution,mesh);
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These are the basic steps required to implement FEM in Matlab. The PDE Toolbox provides several other functions to further refine the mesh, visualize the solution, and perform transient simulations.