The application of the complex variable boundary element method to the solution of heat conduction problems in multiply connected domains

The complex-variable boundary element method (CVBEM) is extended to the solution of potential problems in multiply connected domains. A doubly connected domain is taken for analysis, and a finite width cut is introduced in the domain. Linear basis functions are used to derive the CVBEM nodal equations through a limiting procedure. It was found that the stream functions along the cut do not cancel out but result in an additional term in the nodal equations. For the complex variable methods, the Cauchy-Riemann conditions must be used to generate additional equations relating the stream functions and heat fluxes when Neumann and Robin conditions are specified on the boundaries. The CVBEM equations for interior points are also derived, and three methods of solution of the resulting nodal equations are described. The analysis is shown to be reducible to the available simply connected formulation by introducing a new node numbering system.;The CVBEM equations are successfully tested by solving example problems with available analytical solutions. Dirichlet, Neumann, and Robin boundary conditions are tested using the implicit method of solution. The CVBEM is shown to converge as the boundary discretization scheme is refined. An example comparing the CVBEM to the real variable boundary element method is also provided. The three solution methods are compared, and the efficacy of these methods is discussed.;The CVBEM is extended to triply and generalized to multiply connected domains by using the development for doubly connected domains. The mechanism leading to the formation of double-valued stream functions is critically analyzed and applied to the formulation of the stream functions along multiple cuts. General nodal equations are also derived for an extension of the CVBEM formulation.