Multiquadric Radial Basis Function Method Research Articles

A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory

The free vibration analysis of functionally graded plates is performed by the multiquadric radial basis function method and a higher-order shear deformation theory. The multiquadric method is a truly meshless method, allowing a fast and simple domain and boundary discretization. We use an homogenization technique for material properties based on the Mori–Tanaka scheme. Numerical tests show that the method is reliable and produces good results.

Static and free vibration analysis of composite shells by radial basis functions

The higher-order shear deformation theory of laminated orthotropic elastic shells of Reddy accounts for parabolic distribution of the transverse shear strains through the thickness of the shell. The Reddy shell theory allows the fulfillment of homogeneous conditions (zero values) at the top and bottom surfaces of the shell. This paper deals with a meshless solution of the Reddy higher order shell theory in static and free vibration analysis. The meshless technique is based on the asymmetric global multiquadric radial basis function method proposed by Kansa. This paper demonstrates that this truly meshless method is very successful in the static and free vibration analysis of laminated composite shells.

Modelling cross-ply laminated elastic shells by a higher-order theory and multiquadrics

The higher-order shear-deformation theory of laminated orthotropic elastic shells of Vlasov–Reddy is a modification of Sanders’ theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the shell. The Vlasov–Reddy shell theory allows the fulfillment of homogeneous conditions (zero values) at the top and bottom surfaces of the shell. This paper deals with a meshless solution of the Vlasov–Reddy higher-order shell theory. The meshless technique is based on the asymmetric global multiquadric radial basis function method proposed by Hardy and Kansa. This paper demonstrates that this truly meshless method is successful in the analysis of laminated composite shells.

A COMPARISON OF THE EFFECTIVENESS OF USING THE MESHLESS METHOD AND THE FINITE DIFFERENCE METHOD IN GEOSTATISTICAL ANALYSIS OF TRANSPORT MODELING

Disposal of radioactive waste in geological formations is a great concern with regards to nuclear safety. The general reliability and accuracy of transport modeling depends predominantly on input data such as hydraulic conductivity, water velocity, radioactive inventory, and hydrodynamic dispersion. The most important input data are obtained from field measurements, but they are not always available. One way to study the spatial variability of hydraulic conductivity is geostatistics. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). These methods require a mesh to support the localized approximations. The multiquadric (MQ) radial basis function method is a recent meshless collocation method with global basis functions. Solving PDEs using radial basis function (RBF) collocations is an attractive alternative to these traditional methods because no tedious mesh generation is required. We compare the meshless method, which uses radial basis functions, with the traditional finite difference scheme. In our case we determine the average and standard deviation of radionuclide concentration with regard to spatial variability of hydraulic conductivity that was modeled by a geostatistical approach.

Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions

In this paper, we use the first-order shear deformation theory in the multiquadric radial basis function (MQRBF) procedure for predicting the free vibration behavior of moderately thick symmetrically laminated composite plates. The transverse deflection and two rotations of the laminate are independently approximated with the MQRBF approximation. The natural frequencies of vibration are computed for various laminated plates and compared with some available published results. Through numerical experiments, the capability and efficiency of the MQRBF method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.

Solving time-dependent engineering problems with multiquadrics

This paper deals with the solution of time-dependent problems. The multiquadric radial basis function method is formulated, with a new approach for transient problems. One- and two-dimensional problems are considered. The forward difference and the Crank–Nicolson time-marching schemes for parabolic cases are considered. The central difference integration method of the Newmark family is considered for hyperbolic problems. The method proves its accuracy in four numerical examples.

Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates

In the present study the third-order theory of Reddy for composite laminated beams and plates is discretized using a meshless method 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

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.

A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates

The multiquadric radial basis functions (MQ) method is a recent mesh-free collocation method with global basis functions. The MQ method for the solution of partial differential equations was first introduced by Kansa in the early 1990s and showed exponential convergence for interpolation problems. In this paper we formulate and discuss the MQ method to the solution of moderately thick rectangular laminated composite plates. Numerical results show the high accuracy of the MQ method in the analysis of this class of structural problems.

Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary

The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by Kansa in the early 1990s. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems. In [1], we have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as auto. We have observed high accuracy with a small number of unknowns, as compared with most known results from the literature. In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.

CONTINUATION FOR NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS DISCRETIZED BY THE MULTIQUADRIC METHOD

The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest-size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.