Domain Decomposition Techniques for Boundary Elements: Application to Fluid Flow

The sub-domain techniques in the BEM are nowadays finding its place in the toolbox of numerical modellers, especially when dealing with complex 3D problems. We see their main application in conjunction with the classical BEM approach, which is based on a single domain, when part of the domain needs to be solved using a single domain approach, the classical BEM, and part needs to be solved using a domain approach, BEM subdomain technique. This has usually been done in the past by coupling the BEM with the FEM, however, it is much more efficient to use a combination of the BEM and a BEM sub-domain technique. The advantage arises from the simplicity of coupling the single domain and multi-domain solutions, and from the fact that only one formulation needs to be developed, rather than two separate formulations based on different techniques. There are still possibilities for improving the BEM sub-domain techniques. However, considering the increased interest and research in this approach we believe that BEM sub-domain techniques will become a logical choice in the future substituting the FEM whenever an efficient solution requires coupling of the BEM with a domain technique.

Table of Contents:
Chapter 1: Diffusion-convection problems Introduction; Boundary element formulations; Numerical implementation; Numerical stability for homogeneous diffusion-convection; Numerical examples of diffusion-convection problems; Burgers' equation; Numerical formulations of Burgers' equation; Numerical examples of Burgers' equation; Conclusion Chapter 2: Viscous compressible fluid dynamics Introduction; Conservation equations; Linear gradient type of constitutive models; Primitive variables formulation; Velocity-vorticity formulation; Pressure equation; Boundary-domain integral equations; Discrete models; Test example: differentially-heated tall enclosure; Conclusions Chapter 3: Multi-domain DRM boundary element method for the numerical simulation of non-isothermal Newtonian and non-Newtonian flow problems Introduction; Thermal convection Newtonian flow problems; Non-isothermal non-Newtonian Stokes flow with viscous dissipation; Conclusion Chapter 4: Modelling flow and solute transport in fractured porous media using the DRM multidomain technique Introduction; Governing equations; Numerical method; Numerical implementation; Coupling strategy; Numerical results; Conclusion Chapter 5: Parallel domain decomposition boundary element method approach for large-scale transient and steady nonlinear heat conduction Introduction; Explicit domain decomposition; Iterative solution algorithm; Parallel implementation on a PC cluster; Applications in heat transfer; Numerical validation and examples; Conclusions Chapter 6: Computational implementation for 3D problems Introduction; Equations and DRM formulation; The dual reciprocity method multidomain approach; Schematic view of a DRM code for Poisson's problems; General aspects of DRM-MD implementation for Poisson's problems; A 3D DRM-MD implementation for Poisson's problems using discontinuous elements; A 3D DRM-MD implementation for Poisson's problems using continuous element; 3D DRM-MD implementation for advection-diffusion problems; A convenient arrangement; Examples; Transient advection-diffusion; Conclusions Chapter 7: Iterative schemes for the solution of systems of equations arising from the DRM in multidomains Introduction; Preliminary remarks on the BEM; The dual reciprocity approximation; Approximating functions f j; Iterative solution methods for sparse linear systems; Numerical analysis; Conclusions