Madych and Nelson [1] proved multiquadric (MQ) mesh-independent radial basis functions (RBFs) enjoy exponential convergence. The primary disadvantage of the MQ scheme is that it is global, hence, the coefficient matrices obtained from this discretization scheme are full. Full matrices tend to become progressively more ill-conditioned as the rank increases. In this paper, we explore several techniques, each of which improves the conditioning of the coefficient matrix and the solution accuracy. The methods that were investigated are 1. (1) replacement of global solvers by block partitioning, LU decomposition schemes, 2. (2) matrix preconditioners, 3. (3) variable MQ shape parameters based upon the local radius of curvature of the function being solved, 4. (4) a truncated MQ basis function having a finite, rather than a full band-width, 5. (5) multizone methods for large simulation problems, and 6. (6) knot adaptivity that minimizes the total number of knots required in a simulation problem. The hybrid combination of these methods contribute to very accurate solutions. Even though FEM gives rise to sparse coefficient matrices, these matrices in practice can become very ill-conditioned. We recommend using what has been learned from the FEM practitioners and combining their methods with what has been learned in RBF simulations to form a flexible, hybrid approach to solve complex multidimensional problems.

Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations

The radial basis function (RBF) collocation method uses global shape functions to interpolate and collocatethe approximate solution of PDEs. It is a truly meshless method as compared to some of the so-calledmeshless or element-free ﬁnite element methods. For the multiquadric and Gaussian RBFs, there are twoways to make the solution converge—either by reﬁning the mesh size

Exponential convergence and H‐c multiquadric collocation method for partial differential equations

We survey some recent applications of radial basis functions (rbfs) for the BEM and related algorithms such as the method of fundamental solutions. Among these are the use of alternatives to the traditional 1+ r function in the dual reciprocity method such as thin plate splines, multquadrics and the recently discovered compactly supported positive definite rbfs, and convergence proofs of the DRM for Poisson’s equation. Newly discovered particular solutions for Helmholtz-type operators are discussed and applied to give efficient mesh free algorithms for the diffusion equation. In addition, a number of proposals are given for future applications of rbfs such as the use of surface rbfs for interpolation and the solution of boundary integral equations and the application of Kansa’s method to develop new rbf based coupled domain-boundary approximation methods.

Some recent results and proposals for the use of radial basis functions in the BEM

In the present study the third-order theory of Reddy for composite laminated plates is discretized using a new type of meshless method, a finite point based on the multiquadric radial basis function method. The method allows a very accurate prediction of the field variables.

Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method

Y. M. Zhu

Papers

Multiquadric method for the numerical solution of a biphasic mixture model