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Full Description
"Analytic Element Method" (AEM) assembles a broad range of mathematical and computational approaches to solve important problems in engineering and science. As the subtitle "Complex Interactions of Boundaries and Interfaces" suggests, problems are partitioned into sets of elements and methods are formulated to solve conditions along their boundaries and interfaces. Presentation will place an element within its landscape, formulate its interactions with other elements using linear series of influence functions, and then solve for its coefficients to match its boundary and interface conditions. Computational methods enable boundary and interface conditions of closely interacting elements to be matched with nearly exact precision, commonly to within 8-12 significant digits. Comprehensive solutions provide elements that collectively interact and shape the environment within which they exist.
This work is grounded in a wide range of foundational studies, using exact solutions for important boundary value problems. However, the computational capacity of their times limited solutions to idealized problems, commonly involving a single isolated element within a uniform regional background. With the advent of modern computers, such mathematically based methods were passed over by many, in the pursuit of discretized domain solutions using finite element and finite difference methods. Yet, the elegance of the mathematical foundational studies remains, and the rationale for the Analytic Element Method was inspired by the realization that computational advances could also lead to advances in the mathematical methods that were unforeseeable in the past.
Contents
Analytic Element Method across Fields of Study
1: Philosophical Perspective
2: Studies of Flow and Conduction
3: Studies of Periodic Waves
4: Studies of Deformation by Forces
Further Reading
Foundation of the Analytic Element Method
5: The Analytical Element Method Paradigm
6: Solving Systems of Equations to Match Boundary Conditions
7: Consistent Notation for Boundary Value Problems
Further Reading
Analytic Elements from Complex Functions
8: Point Elements in a Uniform Vector Field
9: Domains with Circular Boundaries
10: Ellipse Elements with Continuity Conditions
11: Slit Element Formulation: Courant's Sewing Theorem with Circle Elements
12: Circular Arcs and Joukowsky's Wing
13: Complex Vector Fields with Divergence and Curl
14: Biharmonic Equation and the Kolosov Formulas
Further Reading
Analytic Elements from Separation of Variables
15: Overview
16: Separation for One-Dimensional Problems
17: Separation in Cartesian Coordinates
18: Separation in Circular-Cylindrical Coordinates
19: Separation in Spherical Coordinates
20: Separation in Spheroidal Coordinates
Further Reading
Analytic Elements from Singular Integral Equations
21: Formulation of Singular Integral Equations
22: Double Layer Elements
23: Single Layer Elements
24: Simpler Far-Field Representation
25: Polygon Elements
26: Curvilinear Elements
27: Three-Dimensional Vector Fields
Further Reading
A List of Symbols
B Solutions to Selected Problem Sets
References
Index
