How do you implement boundary conditions and loads in FEM in MATLAB?

1 What is FEM?

FEM is a method that divides a domain into smaller elements and approximates the solution of a differential equation or a system of equations by using polynomial functions. FEM can handle problems with complicated geometries, nonlinearities, and multiple physics. FEM consists of four main steps: pre-processing, assembly, solution, and post-processing. In pre-processing, you define the problem domain, the material properties, the boundary conditions, and the loads. In assembly, you construct the global system of equations from the local element equations. In solution, you solve the system of equations for the unknown variables. In post-processing, you analyze and visualize the results.

2 How to create and manipulate meshes?

A mesh is a collection of nodes and elements that discretize the problem domain. Nodes are the points where the unknown variables are calculated. Elements are the regions between the nodes that define the shape functions and the material properties. MATLAB has several built-in functions and toolboxes for creating and manipulating meshes, such as pdetool, pdegeom, initmesh, refinemesh, and pdeplot. You can also import or export meshes from other software or formats, such as Gmsh, STL, or COMSOL.

3 How to specify boundary conditions and loads?

Boundary conditions and loads are essential for defining the problem and obtaining a unique solution. Boundary conditions specify the values or the derivatives of the unknown variables at the boundaries of the domain. Loads are the external forces or sources that act on the domain. MATLAB allows you to specify boundary conditions and loads in different ways, depending on the type of problem and the format of the mesh. For example, you can use pdesetbd for 2D problems with pdetool meshes, or applyBoundaryCondition for 2D or 3D problems with PDEModel objects. You can also use functions, expressions, or matrices to define the boundary conditions and loads.

4 How to solve and visualize the results?

Once you have defined the problem, created the mesh, and specified the boundary conditions and loads, you can solve the system of equations using MATLAB functions or toolboxes. For example, you can use pdesolve for 2D problems with pdetool meshes, or solvepde for 2D or 3D problems with PDEModel objects. You can also use direct or iterative solvers, such as backslash, bicgstab, or gmres. After solving the problem, you can visualize the results using MATLAB functions or toolboxes, such as pdeplot, pdeplot3D, pdegplot, or contourf. You can also export the results to other software or formats, such as Paraview, VTK, or MATLAB figures.

5 What are some tips and tricks for FEM in MATLAB?

FEM in MATLAB is a powerful and flexible tool for numerical analysis, but it also requires some skills and knowledge to use it effectively. To get the most out of FEM in MATLAB, you should consider choosing an appropriate mesh size and shape for your problem, such as mesh refinement, adaptive meshing, or error estimation techniques. Additionally, you should select an appropriate polynomial degree and basis function for your problem. For example, higher degree polynomials can capture more details but increase complexity. Finally, it is important to choose an appropriate solver and preconditioner for your problem as well as an appropriate output and visualization format. MATLAB functions or toolboxes can help customize the solver and preconditioner options, as well as convert and export the results to different formats.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?