Analytic Elements from Complex Functions

The mathematical functions associated with analytic elements may be formulated using a complex function $\Omega$ of a complex variable ${\zcomplex}$. Complex formulation of analytic elements is introduced in Section 3.1 for exact solutions obtained by embedding point elements that generate divergence, circulation, or velocity within a uniform vector field. Influence functions for analytic elements with circular geometry are obtained using Taylor and Laurent series expansions in Section 3.2, and conformal mapping extends this formulation to analytic elements with the geometry of ellipses (Section 3.3). The Courant's Sewing Theorem is employed in Section 3.4 to develop solutions for interface conditions across straight line segments, and the Joukowsky transformation extends methods to circular arcs and wings (Section 3.5), which satisfy a Kutta condition of non-singular vector field at their trailing edges. Vector fields with spatially distributed divergence and curl are formulated using the complex variable ${\zcomplex}$ with its complex conjugate $\overline{\zcomplex}$ in Section 3.6, and the complex conjugate is further employed in the Kolosov formulas (Section 3.7) to solve force deformation problems for analytic elements with traction or displacement specified boundary conditions.

Analytic Element Method across Fields of Study

This chapter introduces the philosophical perspective for solving problems with the Analytic Element Method, organized within three common types of problems: gradient driven flow and conduction, waves, and deformation by forces. These problems are illustrated by classic, well known solutions to problems with a single isolated element, along with their extension to complicated interactions occurring amongst collections of elements. Analytic elements are presented within fields of study to demonstrate their capacity to represent important processes and properties across a broad range of applications, and to provide a template for transcending solutions across the wide range of conditions occurring along boundaries and interfaces. While the mathematical and computational developments necessary to solve each problem are developed in later chapters, each figure documents where its solutions are presented.

Analytic Elements from Singular Integral Equations

Solutions to interface problems may be developed using analytic elements with mathematical solutions to the Laplace equation developed by singular integral equations. This formulation leads to solutions with discontinuities occurring across line segments, where the potential or stream function is discontinuous across double layer elements in Section 5.2, and the normal or tangential component of the vector field is discontinuous across single layer elements in Section 5.3. Examples illustrate a broad range of solutions to interface conditions possible with these elements. Series expansions are used to represent the far-field at larger distances from elements in Section 5.4, which leads to higher-order elements with nearly exact solutions and also provides a simpler representation for contiguous strings of adjacent elements. Such strings of elements are used with polygon elements in 5.5 to solve conditions along the interfaces of heterogeneities, and to provide a common series expansion to represent the far-field for a group of neighboring elements. Methods are extended to analytic elements with curvilinear geometry using conformal mappings (Section 5.6) and to three-dimensional fields in Section 5.7.

Analytic Elements from Separation of Variables

Separation of variables provides influence functions for analytic elements, which extend the solutions available with complex functions to problems involving the Helmholtz and modified Helmholtz equations. Methods are introduced for one-dimensional problems that provide the background vector field for many problems, and these solutions are extended to finite domains with interconnected rectangle elements in Section 4.3. Circular elements are developed in Section 4.4 using series of Bessel and Fourier functions to model wave propagation around and through collections of elements, and vadose zone solutions are extended to solve the nonlinear interface conditions occurring along circles. Methods are extended to three-dimensional problems for spheres (Section 4.5), and prolate and oblate spheroids in Section 4.6.

Foundation of the Analytic Element Method

The Analytic Element Method provides a foundation to solve boundary value problems commonly encountered in engineering and science, where problems are structured around elements to organize mathematical functions and methods. While this text mostly adheres to a ``just in time mathematics'' philosophy, whereby mathematical approaches are introduced when they are first needed, a comprehensive paradigm is presented in Section 2.1 as four steps necessary to achieve solutions. Likewise, Section 2.2 develops general solution methods, and Section 2.3 presents a consistent notation and concise representation to organize analytic elements across the broad range of disciplinary perspectives introduced in Chapter 1.