Python Implementation of the Positional Formulation of the Finite Element Method for Static Analysis Using Two-Dimensional Elements

Resumo

This work presents a computational implementation, using the Python programming language, of the positional formulation of the Finite Element Method for static analysis of planar structures modeled by triangular finite elements with cubic approximation. In the positional formulation of the Finite Element Method, the nodal positions are considered as the problem’s variables, relative to a fixed coordinate system in space, to describe the kinematics of the finite elements (total Lagrangian description). Due to the nonlinearity between the strain energy and its derivatives with respect to the nodal positions, the equilibrium analysis of the structure in the deformed configuration is performed using the Newton-Raphson method, characterizing an intrinsically nonlinear geometric formulation. Thus, this formulation is suitable for analyzing structures with large displacements. In this work, the formulation is computationally implemented using the Python programming language. The use of Python is due to its widespread use in the scientific community and because it is a language with simple and readable syntax, dynamic typing, object-oriented features, and a wide availability of libraries such as numpy, pandas, and matplotlib, which assist in the implementation of the formulation, solving the system of equations, and graphical presentation of the results. The formulation implemented in Python is then used to analyze Plane Stress and Plane Strain problems. The results obtained are compared with results from the literature to evaluate the consistency of the formulation and the computational implementation developed in Python.