Finite Element Implementation with MATLAB Code

Implementing finite element method in MATLAB is a preferred approach for many engineers and researchers, not only due to its efficient matrix computation capabilities but also because of its concise syntax and extensive toolbox support. The core principle of finite element method involves discretizing complex continuous domains into finite small elements (such as triangles or quadrilaterals), then numerically solving local equations for these elements, and finally combining them into a global solution. Writing finite element programs in MATLAB typically involves these key implementation steps: Mesh Generation: Divide the computational domain into finite elements using built-in tools or third-party libraries like DistMesh for structured/unstructured grids. For example, the DistMesh library can be called using `pdistmesh` function to generate quality triangular meshes. Stiffness Matrix Assembly: Each element contributes a local stiffness matrix, which are superimposed to form the global stiffness matrix. MATLAB's efficient sparse matrix operations (using `sparse` function) make this assembly process computationally efficient through vectorized operations. Boundary Condition Handling: Modify the stiffness matrix or introduce constraint conditions to correctly represent the system's physical properties. Dirichlet conditions can be implemented by modifying corresponding rows/columns in the stiffness matrix, while Neumann conditions are typically incorporated into the force vector. Linear System Solution: Use MATLAB's solvers such as the backslash operator `` for direct solving or `pcg` function for iterative conjugate gradient methods to compute nodal displacements or other physical quantities. For large systems, preconditioned iterative methods often provide better performance. Post-processing and Visualization: Utilize functions like `contour` for stress contours, `quiver` for vector fields, and `patch` for deformation plots to visualize results. The `pdeplot` function from PDE Toolbox offers specialized finite element visualization capabilities. MATLAB's flexibility enables easy extension of finite element methods to various domains like heat transfer, fluid dynamics, or electromagnetic field analysis without relying on commercial software. For beginners, starting with simple elasticity problems (such as cantilever beams or plane stress problems) serves as the best practice for understanding core finite element concepts. A typical implementation might involve defining shape functions using isoparametric formulation and performing Gaussian quadrature for element matrix integration.