Basic Principles of the Finite Element Method

Basic Principles of the Finite Element Method

Basic Principles of the Finite Element Method

The Finite Element Method (FEM) is a numerical technique for solving problems in continuum mechanics. In many engineering scenarios, due to the complexity of geometric shapes and applied loads, obtaining closed-form mathematical solutions is impractical. This necessitates numerical solutions, and one of the most flexible approaches for providing such solutions is the Finite Element Method. FEM is a numerical method in structural analysis that has become a powerful tool for analyzing continuous media. However, its application is not limited to structural mechanics; it has also been successfully used to solve problems in other fields such as heat conduction, electromagnetic fields, seepage, and fluid dynamics.

The Finite Element Method is a synthesis and refinement of classical approximate calculation methods. It incorporates the core of discrete processing from the finite difference method while inheriting the rational approach of selecting trial functions and integrating over regions from variational calculations. The fundamental idea of FEM is to discretize the continuous solution domain into an assembly of a finite number of elements that are interconnected in a specific manner. Since elements can be combined in various configurations and can themselves have different shapes, FEM can model solution domains with complex geometries.

Another key characteristic of FEM as a numerical analysis method is the use of approximate functions assumed within each element to piecewise represent the unknown field function to be solved over the entire domain. The approximate function within an element is typically expressed using the values of the unknown field function (or its derivatives) at the element’s nodes and interpolation functions. This transforms the continuous, infinite-degree-of-freedom problem into a problem where the unknowns are the values of the field function (or its derivatives) at the nodes, thereby obtaining an approximate solution over the entire domain.

Obviously, as the number of elements increases (i.e., as element sizes decrease), or as the number of element degrees of freedom and the accuracy of interpolation functions improve, the approximation of the solution will continuously improve. If the elements meet convergence requirements, the approximate solution will eventually converge to the exact solution.

The theoretical foundation of FEM lies in the variational principle or the weighted residual method. The weighted residual method is based on the formulation of equivalent integrals for differential equations and serves as an effective technique for solving approximate solutions to linear and nonlinear differential equations. In finite element analysis, the weighted residual method can be used to establish finite element equations, but it also functions as an independent numerical solution method in its own right.

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