In
scientific computation and
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...
, the method of fundamental solutions (MFS) is a technique for solving
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
based on using the
fundamental solution as a basis function. The MFS was developed to overcome the major drawbacks in the
boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, e ...
(BEM) which also uses the fundamental solution to satisfy the governing equation. Consequently, both the MFS and the BEM are of a boundary discretization numerical technique and reduce the computational complexity by one dimensionality and have particular edge over the domain-type numerical techniques such as the
finite element
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...
and finite volume methods on the solution of infinite domain, thin-walled structures, and
inverse problems
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
.
In contrast to the BEM, the MFS avoids the numerical integration of singular fundamental solution and is an inherent
meshfree method
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, origina ...
. The method, however, is compromised by requiring a controversial fictitious boundary outside the physical domain to circumvent the singularity of fundamental solution, which has seriously restricted its applicability to real-world problems. But nevertheless the MFS has been found very competitive to some application areas such as infinite domain problems.
The MFS is also known by different names in the literature, including the charge simulation method, the superposition method, the desingularized method, the indirect boundary element method and the virtual boundary element method.
MFS formulation
Consider a partial differential equation governing certain type of problems
:
:
:
where
is the differential partial operator,
represents the computational domain,
and
denote the Dirichlet and Neumann boundary, respectively,
and
.
The MFS employs the fundamental solution of the operator as its basis function to represent the approximation of unknown function u as follows
:
where
denotes the Euclidean distance between collocation points
and source points
,
is the fundamental solution which satisfies
:
where
denotes Dirac delta function, and
are the unknown coefficients.
With the source points located outside the physical domain, the MFS avoid the fundamental solution singularity. Substituting the approximation into boundary condition yields the following matrix equation
:
where
and
denote the collocation points, respectively, on Dirichlet and Neumann boundaries. The unknown coefficients
can uniquely be determined by the above algebraic equation. And then we can evaluate numerical solution at any location in physical domain.
History and recent developments
The ideas behind the MFS were developed primarily by V. D. Kupradze and M. A. Alexidze in the late 1950s and early 1960s. However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s, followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications. The MFS then gradually became a useful tool for the solution of a large variety of physical and engineering problems.
In the 1990s, M. A. Golberg and C. S. Chen extended the MFS to deal with inhomogeneous equations and time-dependent problems, greatly expanding its applicability. Later developments indicated that the MFS can be used to solve partial differential equations with variable coefficients. The MFS has proved particularly effective for certain classes of problems such as inverse, unbounded domain, and free-boundary problems.
[A.K. G. Fairweather, The method of fundamental solutions for elliptic boundary value problems, ''Advances in Computational Mathematics''. 9 (1998) 69–95.]
Some techniques have been developed to cure the fictitious boundary problem in the MFS, such as the
boundary knot method,
singular boundary method, and
regularized meshless method In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to solve certain partial differential equations whose ...
.
See also
*
Radial basis function
*
Boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, e ...
*
Boundary knot method
*
Boundary particle method In applied mathematics, the boundary particle method (BPM) is a boundary-only meshless (meshfree) collocation technique, in the sense that none of inner nodes are required in the numerical solution of nonhomogeneous partial differential equation ...
*
Singular boundary method
*
Regularized meshless method In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to solve certain partial differential equations whose ...
References
External links
International Center for Numerical Simulation Software in Engineering & Sciences
{{DEFAULTSORT:Meshfree Methods
Numerical analysis
Numerical differential equations