Finite Element Plane Problem Solving Framework

This article provides an in-depth exploration of finite element plane problem solving programs. We begin by detailing the computational methodologies for mass and stiffness matrices, including numerical integration techniques like Gaussian quadrature for element matrix assembly and shape function implementation for displacement interpolation. The discussion covers their critical roles in solving plane stress/strain problems, with particular attention to consistent vs. lumped mass matrix formulations. Subsequently, we examine the main solver program architecture, explaining how to implement efficient solution procedures using direct methods (like Cholesky decomposition) or iterative solvers, along with practical debugging strategies using conditional breakpoints and matrix condition number checks. Finally, we demonstrate visualization procedures that transform numerical results into interpretable contour plots and deformation animations using vector graphics libraries, enabling clearer understanding of solution behaviors. Through this guide, you'll acquire comprehensive skills for developing, implementing, and troubleshooting finite element programs for plane problems.