This article explores the rich heritage of the boundary element method (BEM) by examining its mathematical foundation from the potential theory, boundary value problems, Green's functions, Green's identities, to Fredholm integral equations. The 18th to 20th century mathematicians, whose contributions were key to the theoretical development, are honored with short biographies. The origin of the numerical implementation of boundary integral equations can be traced to the 1960s, when the electronic computers had become available. The full emergence of the numerical technique known as the boundary element method occurred in the late 1970s. This article reviews the early history of the boundary element method up to the late 1970s.

background

https://www.olemiss.edu/sciencenet/benet/05eabeb.pdf

The wind and beyond

Heritage and early history of the boundary element method

Tutorial introduction Algorithms of CM, TM, and CTM Coupling techniques Boundary element methods Other kinds of boundary methods Comparisons Part I: Collocation Trefftz method 1 - Basic algorithms and theory Notations and preliminaries Approximation problems Error estimates Debye-Huckel equation Stability analysis 2 - Motz's problem and its variants Introduction Basic algorithms of CTM Error bounds and integration approximation Variants of Motz's problem Concluding remarks 3 - Coupling techniques Introduction Description of generalized TMs Penalty TMs Simplified hybrid TMs Penalty plus hybrid TMs Lagrange multiplier TM Effective condition number Numerical experiments 4 - Biharmonic equations with singularities Introduction The Green formulas of A2u The collocation Trefftz methods Error bounds Part II: Collocation methods 5 - Collocation methods Introduction Description of collocation methods Error analysis Robin boundary conditions Inverse inequalities Final remarks 6 - Combinations of collocation and finite element methods Introduction Combinations of FEMs Linear algebraic equations of combination of FEM and CM Uniformly Vh0-elliptic inequality Uniformly Vh0-elliptic inequality involving integration approximation Final remarks 7 - Radial basis function collocation methods Introduction Radial basis functions Description of radial basis function collocation methods Inverse estimates for radial basis functions Numerical experiments Comparisons and conclusions Part III: Advanced topics 8 - Combinations with high-order FEMs Introduction Combinations of TM and Lagrange FEMs Global superconvergence Adini's elements 9 - Eigenvalue problems Introduction New numerical algorithms for eigenvalue problems Error bounds of eigenvalues Error bounds of eigenfunctions Computational models and numerical experiments Eigenvalues for the singularity problem Summaries and discussions 10 - The Helmholtz equation Introduction The Trefftz method Error analysis Summaries and discussions 11 - Explicit harmonic solutions of Laplace's equation Introduction Harmonic functions Harmonic solutions involving Neumann conditions Extensions and analysis on singularity New models of singularities for Laplace's equation Concluding remarks Appendix - Historic review of boundary methods Potential theory Existence and uniqueness Reduction in dimensions and Green's formula Integral equations Extended Green's formula Pre-electronic computer era Electronic computer era Boundary integral equation and boundary element methods

/pdf/trefftz-and-collocation-methods-1x7y3q2kmy.pdf

Trefftz and Collocation Methods

On Making Things the Best-Aeronautical Uses of Optimization

Flow-induced streamwise vibrations of structures

Why is it important? Hypersonics has had two major applications. The first has been to provide thermal protection during atmospheric entry. Success in this enterprise has supported ballistic-missile nose cones, has returned strategic reconnaissance photos from orbit and astronauts from the Moon, and has even dropped an instrument package into the atmosphere of Jupiter. The last of these approached Jupiter at four times the speed of a lunar mission returning to Earth.

/pdf/facing-the-heat-barrier-a-history-of-hypersonics-3e097uz7h3.pdf

The wind and beyond

Citations

Heritage and early history of the boundary element method

Trefftz and Collocation Methods

On Making Things the Best-Aeronautical Uses of Optimization

Flow-induced streamwise vibrations of structures

Facing the Heat Barrier: A History of Hypersonics

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