In this chapter, we will first develop the machinery associated with variational principles and the Rayleigh-Ritz procedure. This will be used to derive the finite element method for a simple one-dimensional Laplace problem. FEM algorithms will then be developed for two-dimensional (2-D) Helmholtz problems, including modal eigenvalue problems and radiation and scattering. FEM will then be combined with a surface integral equation to develop the FEM-BEM algorithm.

Chapter Contents:

• 8.1 Variational Principles in Mathematical Physics
• 8.1.1 Operators and Functionals
• 8.1.2 Variational Principles
• 8.1.3 Variational Calculus
• 8.1.4 Euler–Lagrange Equation
• 8.1.5 Variational Principles for PDEs
• 8.1.6 Variational Principles for Self-Adjoint, Positive Definite Operators
• 8.1.7 Functionals in Mathematical Physics
• 8.1.8 Rayleigh–Ritz Method
• 8.2 Overview of the Finite Element Method
• 8.2.1 Mesh Types and Mesh Generation
• 8.2.2 Basis Functions
• 8.2.3 Variational Principle and Rayleigh–Ritz Procedure
• 8.2.4 Linear System Solution
• 8.3 Laplace's Equation: 1-D FEM
• 8.3.1 Functional Form of the Generalized Laplace Equation
• 8.3.2 Mesh Representation
• 8.3.3 Rayleigh–Ritz Procedure
• 8.3.4 Element Stiffness Matrix
• 8.3.5 Basis Functions and Shape Functions
• 8.3.6 Evaluating the Element Stiffness Matrix
• 8.3.7 Assembly of the Global Stiffness Matrix
• 8.3.8 Example: Five-Element Mesh
• 8.3.9 Comparison of FEM and FD
• 8.3.10 Sparse Matrix and Dense Matrix Methods
• 8.4 Helmholtz Equation: 2-D FEM
• 8.4.1 Boundary Conditions for FEM
• 8.4.1.1 Boundary Terms in the Functional
• 8.4.2 Rayleigh–Ritz Method for the Helmholtz Functional
• 8.4.3 Eigenvalue Problems (Unknown k)
• 8.4.4 Scattering Problems (Known k)
• 8.4.5 FEM Formulation of the 2-D Scattering Problem
• 8.4.6 Triangular Mesh
• 8.4.7 Basis Functions and Shape Functions
• 8.4.8 Evaluating Element Matrices
• 8.4.9 Matrix Assembly
• 8.5 Finite Element Method–Boundary Element Method
• 8.5.1 Boundary Element Method
• 8.5.1.1 Extinction Theorem
• 8.5.1.2 FEM–BEM Linear System
• 8.5.2 Implementation
• 8.6 Numerical Results
• Problems
• References

Inspec keywords: finite element analysis; Helmholtz equations; boundary-elements methods; Rayleigh-Ritz methods; eigenvalues and eigenfunctions; variational techniques

Other keywords: 2D Helmholtz problem; variational principles; scattering; surface integral equation; radiation; FEM algorithms; finite element method; one-dimensional Laplace problem; Rayleigh-Ritz procedure; modal eigenvalue problem; two-dimensional Helmholtz problem

Subjects: Numerical approximation and analysis; Numerical analysis; Linear algebra (numerical analysis); Optimisation techniques; Algebra, set theory, and graph theory; Differential equations (numerical analysis); Linear algebra (numerical analysis); Differential equations (numerical analysis); Finite element analysis; Optimisation techniques; Optimisation; Finite element analysis