In
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, 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 equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
. Numerical experiments show that the BPM has
spectral convergence. Its interpolation matrix can be symmetric.
History and recent developments
In recent decades, the
dual reciprocity method (DRM) and
multiple reciprocity method (MRM) have been emerging as promising techniques to evaluate the particular solution of nonhomogeneous
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
in conjunction with the boundary discretization techniques, such as
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, ele ...
(BEM). For instance, the so-called DR-BEM and MR-BEM are popular BEM techniques in the numerical solution of nonhomogeneous problems.
The DRM has become a common method to evaluate the particular solution. However, the DRM requires inner nodes to guarantee the convergence and stability. The MRM has an advantage over the DRM in that it does not require using inner nodes for nonhomogeneous problems. Compared with the DRM, the MRM is computationally more expensive in the construction of the interpolation matrices and has limited applicability to general nonhomogeneous problems due to its conventional use of high-order Laplacian operators in the annihilation process.
The recursive composite multiple reciprocity method (RC-MRM),
[Chen W, "Meshfree boundary particle method applied to Helmholtz problems". ''Engineering Analysis with Boundary Elements'' 2002,26(7): 577–581][Chen W, Fu ZJ, Jin BT, "A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique".'' Engineering Analysis with Boundary Elements'' 2010,34(3): 196–205] was proposed to overcome the above-mentioned problems. The key idea of the RC-MRM is to employ high-order composite differential operators instead of high-order Laplacian operators to eliminate a number of nonhomogeneous terms in the governing equation. The RC-MRM uses the recursive structures of the MRM interpolation matrix to reduce computational costs.
The boundary particle method (BPM) is a boundary-only discretization of an inhomogeneous partial differential equation by combining the RC-MRM with strong-form meshless boundary collocation discretization schemes, such as the
method of fundamental solution (MFS),
boundary knot method (BKM),
regularized meshless method (RMM),
singular boundary method (SBM), and
Trefftz method (TM). The BPM has been applied to problems such as nonhomogeneous
Helmholtz equation
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation:
\nabla^2 f = -k^2 f,
where is the Laplace operator, is the eigenvalue, and is the (eigen)fun ...
and
convection–diffusion equation
The convection–diffusion equation is a parabolic partial differential equation that combines the diffusion equation, diffusion and convection (advection equation, advection) equations. It describes physical phenomena where particles, energy, or o ...
. The BPM interpolation representation is of a
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
series.
For the application of the BPM to Helmholtz,
and
plate bending
Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations o ...
problems, the high-order
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ...
or general solution, harmonic function or
Trefftz function (T-complete functions) are often used, for instance, those of
Berger
Berger is a surname in both German language, German and French language, French, although there is no etymological connection between the names in the two languages. The French surname is an occupational name for a shepherd, from Old French ''bergi ...
,
Winkler, and vibrational thin plate equations. The method has been applied to inverse Cauchy problem associated with
Poisson and nonhomogeneous Helmholtz equations.
[Chen W, Fu ZJ, "Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations". ''Journal of Marine Science and Technology''–Taiwan 2009,17(3): 157–163]
Further comments
The BPM may encounter difficulty in the solution of problems having complex source functions, such as non-smooth, large-gradient functions, or a set of discrete measured data. The solution of such problems involves:
(1) The complex functions or a set of discrete measured data can be interpolated by a sum of
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
or
trigonometric
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field ...
function series. Then, the RC-MRM can reduce the nonhomogeneous equation to a high-order homogeneous equation, and the BPM can be implemented to solve these problems with boundary-only discretization.
(2) The
domain decomposition
In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the soluti ...
may be used to in the BPM boundary-only solution of large-gradient source functions problems.
See also
*
Meshfree method
*
Radial basis function
In mathematics a radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), o ...
*
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, ele ...
*
Trefftz method
*
Method of fundamental solution
*
Boundary knot method
*
Singular boundary method
References
External links
Boundary Particle Method
{{Numerical PDE
Numerical analysis
Numerical differential equations