12. Finite Element Methods#
The following chapter covers the Finite Element Method, a comprehensive exploration into a powerful numerical technique for solving complex engineering and scientific problems. In this lecture, we delve into the Finite Element Method (FEM), a versatile approach used to approximate solutions for differential equations over complex geometries.
Introduction to Finite Element Method (FEM): An explanation of FEM as a numerical technique to solve partial differential equations over complex domains. We will also provide an overview of scenarios where FEM is used, such as structural analysis, heat transfer, and fluid dynamics.
Shape Functions and Interpolation: We will explain shape functions and how they define the behavior of the solution within each element and additionally, discuss interpolation and how it allows for the approximation of the solution within elements.
Assembly and Global Equations: Conduct an overview of the assembly process to combine element equations into a global system and show an explanation of how boundary conditions and constraints are applied within the global equations.
Finite Element Discretization: We will explore the process of discretizing complex domains into smaller elements and introduce the concept of nodes, elements, and their connectivity within the FEM framework.
Applications of FEM: Demonstrate the applications of FEM in various engineering fields, from structural analysis to fluid dynamics and explore how FEM provides insights into stress distributions, heat transfer, and more.
By the end of this chapter, you will have a solid understanding of how FEM breaks down intricate problems into manageable elements, enabling you to tackle challenges that were once considered computationally infeasible.