Keynote AddressThe Analog Equation Method - A PowerfulBEM-based Solution Technique For SolvingLinear And Nonlinear Engineering Problems

A novel BEM-based method for solving boundary value problems arising in engineering discipline is presented. The actual problem governed by a linear differential operator, which is difficult to invert, or by a non-linear one is converted into an equivalent problem described by a linear differential operator whose integral representation can be readily established. The equation of the substitute problem is designated as the Analog Equation. Hence the name Analog Equation Method (AEM). The method incorporates features and advantages of FDM, FEM and BEM. Several numerical examples are presented which illustrate the range of applicability of the proposed method as well as its efficiency, accuracy and reliability. CONCEPT AND INGREDIENTS OF THE ANALOG EQUATION METHOD Numerical methods are the only methods to solve realistic problems in engineering practice. These days, the most popular computational methods are FDM, FEM and BEM. Each of them has its own advantages and disadvantages. Coupling of them can increase the computational efficiency. With the advent of modern computers and cheap computational power (memory, storage, speed) certain restrictions, which in the near past influenced enormously the development of computational techniques, have been released and the effort is focused on the reliability of the method. Transactions on Modelling and Simulation vol 7, © 1994 WIT Press, www.witpress. com, ISSN 1743-355X 168 Boundary Element Method XVI In this paper a novel method is presented based on a simple concept. We designate this method as the Analog Equation Method. According to this method the solution of a problem described by a non-linear differential operator or by a linear one, whose fundamental solution is practically not possible to establish or has a complicate expression, is obtained by solving a fictitious equivalent problem described by a linear operator having a known simple fundamental solution. In the equivalent problem the geometry of the domain and the boundary conditions are conserved, whereas the nonhomogeneous term of the equation is an unknown "appropriate" fictitious source density function which must be established. Schematically, the concept of AEM is shown in the Example of Figure 1. In this case the solution of a simply supported plate with a variable flexural rigidity resting on non-linear elastic foundation and subjected to a distributed load g(x,y) is obtained as the solution of a simply supported plate with constant flexural rigidity (D = 1) not resting on elastic foundation and subjected to a fictitious loadq(x,y). The unknown source density function is established numerically by adhering to the following steps. a. The integral representation of the field function is established from the equivalent fictitious linear problem which involves the unknown source density in the domain integral. b. Direct differentiation of this integral representation yields the derivatives involved in the operator of the real problem. c. Use of BEM technique for the boundary integrals and FEM technique for the domain integrals yields the discretized expressions for the field function and its derivatives . d. Collocation of the field function at the boundary and domain nodal points, collocation of the derivatives at the domain nodal points and elimination of the boundary quantities making use of the boundary conditions, yield the nodal values of the field function and its derivatives in terms of the values of the fictitious source density function at the nodal points inside the domain. e. Application of the governing equation of the real problem at the nodal points inside the domain and substitution of the relevant values of the field function and its derivatives yields a system of algebraic equations (linear or non-linear, depending on the operator of the real problem) from which the nodal values of the fictitious source density function are established. f. The field function and its derivative at any point inside the domain are obtained from their integral representation of the fictitious problem. Transactions on Modelling and Simulation vol 7, © 1994 WIT Press, www.witpress. com, ISSN 1743-355X Boundary Element Method XVI 169 In differentiating the integral representation of the field function singular and hypersingular domain integrals arise which are evaluated efficiently by converting them to regular boundary integrals.