Four-Node Rectangular Element MATLAB Finite Element Implementation

The four-node rectangular element represents one of the most fundamental element types in finite element analysis, suitable for plane stress, plane strain, and similar problems. Implementing this element in MATLAB typically involves the following key computational steps:

Element Stiffness Matrix Calculation The stiffness matrix for the four-node rectangular element is commonly derived using isoparametric transformation. In the local coordinate system (ξ, η), displacement fields are interpolated through shape functions, and the element stiffness matrix is calculated using strain-displacement matrices and material constitutive relationships. Due to the bilinear characteristics of rectangular elements, computing shape function derivatives and the Jacobian determinant is relatively straightforward. In MATLAB implementation, this involves defining shape functions N1-N4 and their derivatives with respect to ξ and η coordinates, followed by Jacobian matrix calculation using the element's nodal coordinates.

Numerical Integration Method Gauss quadrature is typically employed for numerical integration of the stiffness matrix. For four-node rectangular elements, a 2×2 Gauss integration point scheme provides sufficient accuracy. Each integration point's weight combines with shape function derivatives, and the results are superimposed to form the complete stiffness matrix. The MATLAB code implementation would include loops over integration points, calculating the B matrix (strain-displacement matrix) and D matrix (material matrix) at each Gauss point, then summing their contributions with appropriate weights.

Boundary Condition Handling General-purpose programs must support the application of displacement boundary conditions, such as fixing specific nodal degrees of freedom. This can be achieved by modifying the global stiffness matrix and load vector, or through penalty function methods for constraint implementation. In MATLAB, this typically involves identifying constrained degrees of freedom and applying appropriate modifications to the system equations before solving.

Result Post-processing After computation completion, strains and stresses need to be extracted from nodal displacement solutions. Since strains vary linearly within four-node elements, stress recovery is typically performed directly at Gauss points or nodal locations. The MATLAB implementation would include routines to calculate strain components (εxx, εyy, γxy) from displacement derivatives, then compute stresses using the material constitutive law.

The advantage of this element lies in its simple formulation and computational efficiency, making it ideal as an introductory case for finite element learning. By extending shape functions or increasing integration points, it can be further adapted to meet higher-order problem requirements. The MATLAB code structure typically includes separate functions for shape function calculation, stiffness matrix assembly, boundary condition application, and results computation, facilitating modular development and testing.