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 Element Method

The Analytic Element Method provides a foundation to solve boundary value problems commonly encountered in engineering and science. The goals are: to introduce readers to the basic principles of the AEM, to provide a template for those interested in pursuing these methods, and to empower readers to extend the AEM paradigm to an even broader range of problems. A comprehensive paradigm: 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 with nearly exact precision. Collectively, sets of elements interact to transform their environment, and these synergistic interactions are expanded upon for three common types of problems. The first problem studies a vector field that is directed from high to low values of a function, and applications include: groundwater flow, vadose zone seepage, incompressible fluid flow, thermal conduction and electrostatics. A second type of problem studies the interactions of elements with waves, with applications including water waves and acoustics. A third type of problem studies the interactions of elements with stresses and displacements, with applications in elasticity for structures and geomechanics. The Analytic Element Method paradigm comprehensively employs a background of existing methodology using complex functions, separation of variables and singular integral equations. This text puts forth new methods to solving important problems across engineering and science, and has a tremendous potential to broaden perspective and change the way problems are formulated.

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.

Application of analytic element method in hydrogeology

The analytic element method (AEM) has been successfully used in practice worldwide for many years. This method provides the possibility of fast preliminary quantitative analysis of the hydrogeological systems or boundary conditions of the numerical models, as it is shown in the case study of groundwater source of the city of Vrbas. The AEM is also applicable for the initial analysis of a hydrogeological system, which is of particular importance in case of excess pollution that cannot be predicted where it could happen. One example of the application of the AEM is presented in this article. The analytical model is calibrated based on the measured data from several drilled monitoring wells, and this was the base for the numerical model of the contaminant transport. In this case, the AEM enabled the quick access to information on the hydrogeological system and effective response to excess pollution.