Fundamental solution based numerical methods for three dimensional problems: efficient treatments of inhomogeneous terms and hypersingular integrals
Date
2016
Authors
Lee, Cheuk Yu
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Abstract
In recent years, fundamental solution based numerical methods
including the meshless method of fundamental solutions (MFS), the
boundary element method (BEM) and the hybrid fundamental solution
based finite element method (HFS-FEM) have become popular for
solving complex engineering problems. The application of such
fundamental solutions is capable of reducing computation
requirements by simplifying the domain integral to the boundary
integral for the homogeneous partial differential equations. The
resulting weak formulations, which are of lower dimensions, are
often more computationally competitive than conventional
domain-type numerical methods such as the finite element method
(FEM) and the finite difference method (FDM).
In the case of inhomogeneous partial differential equations
arising from transient problems or problems involving body
forces, the domain integral related to the inhomogeneous
solutions term will need to be integrated over the interior
domain, which risks losing the competitive edge over the FEM or
FDM. To overcome this, a particular treatment to the
inhomogeneous term is needed in the solution procedure so that
the integral equation can be defined for the boundary. In
practice, particular solutions in approximated form are usually
applied rather than the closed form solutions, due to their
robustness and readiness. Moreover, special numerical treatment
may be required when evaluating stress directly on the domain
surface which may give rise to hypersingular integral
formulation. This thesis will discuss how the MFS and the BEM can
be applied to the three-dimensional elastic problems subjected to
body forces by introducing the compactly supported radial basis
functions in addition to the efficient treatment of hypersingular
surface integrals. The present meshless approach with the MFS and
the compactly supported radial basis functions is later extended
to solve transient and coupled problems for three-dimensional
porous media simulation.
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Three-dimensional linear elasticity, method of fundamental solutions, particular solutions, compactly supported radial basis functions, dual reciprocity method, boundary element method, Hypersingular integral, Hadamard finite part integral, Cauchy principal value integral, complex-step derivative
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