Multiquadric Research Articles

Spectral collocation method with a flexible angular discretization scheme for radiative transfer in multi-layer graded index medium

The spectral collocation method (SCM) is employed to solve the radiative transfer in multi-layer semitransparent medium with graded index. A new flexible angular discretization scheme is employed to discretize the solid angle domain freely to overcome the limit of the number of discrete radiative direction when adopting traditional SN discrete ordinate scheme. Three radial basis function interpolation approaches, named as multi-quadric (MQ), inverse multi-quadric (IMQ) and inverse quadratic (IQ) interpolation, are employed to couple the radiative intensity at the interface between two adjacent layers and numerical experiments show that MQ interpolation has the highest accuracy and best stability. Variable radiative transfer problems in double-layer semitransparent media with different thermophysical properties are investigated and the influence of these thermophysical properties on the radiative transfer procedure in double-layer semitransparent media is also analyzed. All the simulated results show that the present SCM with the new angular discretization scheme can predict the radiative transfer in multi-layer semitransparent medium with graded index efficiently and accurately.

Numerical simulation of metal removal in laser drilling using radial point interpolation method

Prediction of penetration depth in metal laser drilling is done through a simple meshfree numerical model. 2D axisymmetric simplified model of transient metal laser drilling is proposed for continuous laser beam of Gaussian distribution with strong form of Radial Point Interpolation Method (RPIM) used for approximating the temperature field. The commonly used Multi-Quadrics (MQ) and Exponential (EXP) Radial Basis Functions (RBFs) are tested and compared with each other. The model logic is constructed in MATLAB code, while the results are compared with published numerical and experimental work. The simulation results give good agreement with the previous numerical and experimental work, showing the model reliability in predicting the penetration depth in such a physically complex process.

Two-dimensional convection—diffusion problem solved using method of localized particular solutions

A meshless local method of approximated particular solutions (LMAPS) is used to analyze problem described by the convection-diffusion equation. The method solves the steady convection-diffusion equation with reaction term. The discretized system of equations is derived via interpolation procedure using integrated multiquadrics (MQ) radial basis functions (RBF). The solution of the equation is performed over simple geometry with non-uniform velocity field and results are presented in the article. The LMAPS method is capable to produce stable solutions with results comparable to the analytical solutions.

Numerical analysis for multi-group neutron-diffusion equation using Radial Point Interpolation Method (RPIM)

A mesh-free method is introduced to overcome the drawbacks (e.g., mesh generation and connectivity definition between the meshes) of mesh-based (nodal) methods such as the finite-element method and finite-difference method. In particular, the Point Interpolation Method (PIM) using a radial basis function is employed in the numerical analysis for the multi-group neutron-diffusion equation. The benchmark calculations are performed for the 2D homogeneous and heterogeneous problems, and the Multiquadrics (MQ) and Gaussian (EXP) functions are employed to analyze the effect of the radial basis function on the numerical solution. Additionally, the effect of the dimensionless shape parameter in those functions on the calculation accuracy is evaluated. According to the results, the radial PIM (RPIM) can provide a highly accurate solution for the multiplication eigenvalue and the neutron flux distribution, and the numerical solution with the MQ radial basis function exhibits the stable accuracy with respect to the reference solutions compared with the other solution. The dimensionless shape parameter directly affects the calculation accuracy and computing time. Values between 1.87 and 3.0 for the benchmark problems considered in this study lead to the most accurate solution. The difference between the analytical and numerical results for the neutron flux is significantly increased in the edge of the problem geometry, even though the maximum difference is lower than 4%. This phenomenon seems to arise from the derivative boundary condition at (x,0) and (0,y) positions, and it may be necessary to introduce additional strategy (e.g., the method using fictitious points and Hermite-type collocation method) in order to solve this problem. On the basis of these results, it is expected that the mesh-free method including the RPIM can be sufficiently employed in numerical analysis for the multi-group neutron-diffusion equation and can be considered as an alternative numerical approach to overcome the drawbacks of existing nodal methods.

A global Stokes method of approximated particular solutions for unsteady two-dimensional Navier–Stokes system of equations

ABSTRACTThe unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions (MAPS) in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric (MQ) radial basis function (RBF) as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers (Re), corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step (BFS) flows (steady-state problems) and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows (transient problems). The first two problems are solved up to Re=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices at Re=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved at Re=700. The influence of the cavity length, l, in determining the different structures of the flow pattern is studied for values of , showing that the scheme is able to reproduce the previously reported change in the flow pattern when l=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions at Re values greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature.

Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model

As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank–Nicolson scheme and another one is based on explicit Runge–Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.

A high-order finite volume method on unstructured grids using RBF reconstruction

This paper proposes a high-order finite volume method based on radial basis function (RBF) reconstruction for the solution of Euler and Navier–Stokes equations on unstructured grids. Unlike traditional polynomial K-exact method, RBF method has stronger adaptability for different reconstruction stencils and more flexibility in choosing interpolating points. We expatiate on the detailed process of flow-field reconstruction by using multiquadric (MQ) basis function for the second-order and third-order schemes on unstructured triangular grids. Subsequently, we validate the accuracy order of RBF method through the numerical test case. Furthermore, the method is used to solve several typical flow fields. Compared with traditional K-exact high-order scheme, RBF method is more accurate and has lower numerical dissipation, which can obtain more elaborate and precise results.

Computing a numerical solution of two dimensional non-linear Schrödinger equation on complexly shaped domains by RBF based differential quadrature method

In this paper, two-dimensional Schrödinger equations are solved by differential quadrature method. Key point in this method is the determination of the weight coefficients for approximation of spatial derivatives. Multiquadric (MQ) radial basis function is applied as test functions to compute these weight coefficients. Unlike traditional DQ methods, which were originally defined on meshes of node points, the RBFDQ method requires no mesh-connectivity information and allows straightforward implementation in an unstructured nodes. Moreover, the calculation of coefficients using MQ function includes a shape parameter c. A new variable shape parameter is introduced and its effect on the accuracy and stability of the method is studied. We perform an analysis for the dispersion error and different internal parameters of the algorithm are studied in order to examine the behavior of this error. Numerical examples show that MQDQ method can efficiently approximate problems in complexly shaped domains.

Application of the MQ RBF-FD method to calculating transient voltages of power transmission lines

This paper presents an application of the Radial Basis Function – Based Finite Difference Method (RBF-FD) to solving the electrical transient problems defined by the time-dependent ordinary differential equations. In this method, the finite difference approximations of first- and second-order derivatives in time domain are formalated the same as those in space domain based on the MQ (Multiquadrics) function presented in [1]. The MQ RBF-FD method are for the sake of evaluating the accuracy, effectiveness and applicability used to compute the transient voltages on the benchmark circuit and 220 kV three-phase transmission line of Viet Nam. Our numerical results are compared with those obtained by the analytical method, the traditional FD method and ATP/EMTP software. The compared results have been shown that the MQ RBF-FD method has accuracy that is higher than ones of the traditional numerical methods, especially with the optimal shape parameter.

The RBF-FDTD method for modeling the lightning-induced voltages on overhead distribution lines

This paper presents an application of the Radial Basis Function – Based Finite Difference Time Domain Method (RBF-FDTD) such as MQ (Multiquadrics), IMQ (Inverse Multiquadrics) and GA (Gaussian) is developed in [1] for modeling the lightning-induced voltages on overhead power lines in both cases of ideal ground and lossy ground. In addition, the influence of corona on the lightning-induced voltages has been considered as well. In order to increasing the accuracy of proposed method, the optimal algorithm of finding the shape parameter has been used. The accuracy, effectiveness and applicability of The MQ, IMQ and GA RBF-FDTD are evaluated through computing the lightning-induced voltages on 110kV overhead distribution lines. The solutions obtained by the RBF-FDTD are compared with those of the traditional FDTD based on the basic solution of the LIOV. The obtained results demonstrate that the RBF-FDTD is always more accurate than the traditional FDTD, in particular with the optimal shape parameter.

A robust interpolation method for constructing digital elevation models from remote sensing data

A digital elevation model (DEM) derived from remote sensing data often suffers from outliers due to various reasons such as the physical limitation of sensors and low contrast of terrain textures. In order to reduce the effect of outliers on DEM construction, a robust algorithm of multiquadric (MQ) methodology based on M-estimators (MQ-M) was proposed. MQ-M adopts an adaptive weight function with three-parts. The weight function is null for large errors, one for small errors and quadric for others. A mathematical surface was employed to comparatively analyze the robustness of MQ-M, and its performance was compared with those of the classical MQ and a recently developed robust MQ method based on least absolute deviation (MQ-L). Numerical tests show that MQ-M is comparative to the classical MQ and superior to MQ-L when sample points follow normal and Laplace distributions, and under the presence of outliers the former is more accurate than the latter. A real-world example of DEM construction using stereo images indicates that compared with the classical interpolation methods, such as natural neighbor (NN), ordinary kriging (OK), ANUDEM, MQ-L and MQ, MQ-M has a better ability of preserving subtle terrain features. MQ-M replaces thin plate spline for reference DEM construction to assess the contribution to our recently developed multiresolution hierarchical classification method (MHC). Classifying the 15 groups of benchmark datasets provided by the ISPRS Commission demonstrates that MQ-M-based MHC is more accurate than MQ-L-based and TPS-based MHCs. MQ-M has high potential for DEM construction.

RBF collocation method and stability analysis for phononic crystals

The mesh-free radial basis function (RBF) collocation method is explored to calculate band structures of periodic composite structures. The inverse multi-quadric (MQ), Gaussian, and MQ RBFs are used to test the stability of the RBF collocation method in periodic structures. Much useful information is obtained. Due to the merits of the RBF collocation method, the general form in this paper can easily be applied in the high dimensional problems analysis. The stability is fully discussed with different RBFs. The choice of the shape parameter and the effects of the knot number are presented.

A total error-based multiquadric method for surface modeling of digital elevation models

Digital elevation model (DEM) source data are subject to both horizontal and vertical errors owing to improper instrument operation, physical limitations of sensors, and bad weather conditions. These factors may bring a negative effect on some DEM-based applications requiring low levels of positional errors. Although classical smoothing interpolation methods have the ability to handle vertical errors, they are prone to omit horizontal errors. Based on the statistical concept of the total least squares method, a total error-based multiquadric (MQ-T) method is proposed in this paper to reduce the effects of both horizontal and vertical errors in the context of DEM construction. In nature, the classical multiquadric (MQ) method is a vertical error regression procedure, whereas MQ-T is an orthogonal error regression model. Two examples, including a numerical test and a real-world example, are employed in a comparative performance analysis of MQ-T for surface modeling of DEMs. The numerical test indicates that MQ-T performs better than the classical MQ in terms of root mean square error. The real-world example of DEM construction with sample points derived from a total station instrument demonstrates that regardless of the sample interval and DEM resolution, MQ-T is more accurate than classical interpolation methods including inverse distance weighting, ordinary kriging, and Australian National University DEM. Therefore, MQ-T can be considered as an alternative interpolator for surface modeling with sample points subject to both horizontal and vertical errors.

A comparative analysis of local meshless formulation for multi-asset option models

A local meshless radial basis function collocation differential quadrature (LMRBFCDQ) is proposed for the numerical solution of a single and multi-asset option pricing PDE models arising in computational finance. Spatial discretization is performed by both local and a standard global meshless collocation procedures coupled with a set of different time integrators based on the forward Euler difference formula (FEDF), the fully Implicit method (FIM), the Crank–Nicolson method (CNM), the explicit Runge–Kutta method of order two (ERK2), the Crank–Nicolson Runge–Kutta method of order two (CNRK2), the fully Implicit Runge–Kutta method of order two (IRK2), the Runge–Kutta method of order four (RK4), the Embedded Runge–Kutta method (RK23). Operator splitting techniques like the ordinary operator splitting (OOS), the Lie–Trotter splitting, the additive splitting and the Strang splitting are also tested for time integration. The proposed hybrid schemes are the amalgamation of the meshless differential quadrature procedure and the finite difference approximations. Different types of radial basis functions (RBFs) i.e. the multiquadric (MQ), the inverse quadric (IQ) and the Gaussian (GA) are utilized for the spatial discretization of the PDE models. Numerical analysis of a range of computational finance related models are shown to demonstrate accuracy, efficiency and ease of implementation of the proposed meshless-finite difference procedure.

Shape parameter selection for multi-quadrics function method in solving electromagnetic boundary value problems

Purpose – The multi-quadrics (MQ) function is a kind of radial basis function. And the MQ method has been successfully adopted as a type of meshless method in solving electromagnetic boundary value problems. However, the accuracy of MQ interpolation or solving equations is severely influenced by shape parameter. Thus the purpose of this paper is to propose a case-independent shape parameter selection strategy from the aspect of coefficient matrix condition number analysis. Design/methodology/approach – The condition number of coefficient matrix is investigated. It is shown that the condition number is only a function of shape parameter and MQ node number, and is irrelevant to the interpolated function which means case-independent. The effective condition number which takes into account the interpolated function is introduced. Then, the relation between the relative root mean square error and condition number is analyzed. Three numerical experiments as transmission line, cable channel and grounding metal box model were carried out. Findings – In the numerical experiments, there is an approximate linear relationship between the logarithm of the condition number and shape parameter, an approximate quadratic relationship with node number. And the optimal shape parameter is corresponding to the early stage of condition number oscillation. Originality/value – This paper proposed a case-independent shape parameter selection strategy. For a finite precision computation, the upper limit of the condition number is predetermined. Therefore, the shape parameter can be chosen where condition number oscillates in early stage.

Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis–Procesi equation

In this paper, the Degasperis–Procesi (DP) equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form: u−P. Based on the equivalent formulation of DP equation, a new numerical method is presented. The multi-quadric (MQ) quasi-interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order TVD method is used to approximate the time derivative. Several examples are presented to demonstrate the effectiveness of the proposed method.

An improved localized method of approximate particular solutions for solving elliptic PDEs

In this paper we improve the localized method of approximate particular solutions (LMAPS) in Yao et al. (2011) by utilizing the polyharmonic splines (PS) radial basis function (RBF) for solving elliptic partial differential equations (PDEs). LMAPS has been widely circulated since it is published in 2010. The multiquadric (MQ) has been considered as the most popular choice among all RBFs. However, adjusting the shape parameter is a critical issue when utilizing the original LMAPS. In this paper, we modified LMAPS by combining conditionally positive definite RBF-PS and an additional low degree of polynomial basis in the localization process. The accuracy of the proposed LMAPS is significantly improved. We can simply increase the order of PS to achieve even higher accuracy. Other than the unexpected high accuracy, there is no need to deal with the difficult issues of choosing optimal shape parameter. This is a huge advantage in the RBF simulations of PDEs. In the numerical experiments, we will present the pros and cons of improved LMAPS (ILMAPS) using PS and some commonly used RBFs (MQ, Matérn, and Gaussian) versus the original LMAPS (OLMAPS).

The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The Crank–Nicolson scheme and the method of lines (MOL)

MHD equations have many applications in physics and engineering. The model is coupled equations in velocity and magnetic field and has a parameter namely Hartmann. The value of Hartmann number plays an important role in the equations. When this parameter increases, using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem. In the present paper a numerical meshless method based on radial basis functions (RBFs) is provided to solve MHD equations. For approximating the spatial variable, a new approach which is introduced by Bozzini et al. (2015) is applied. The method will be used here is based on the interpolation with variably scaled kernels. The methodology of the new technique is defining the scale function c on the domain Ω⊂Rd. Then the interpolation problem from the data locations xj∈Rd transforms to the new interpolation problem in the data locations (xj,c(xj))∈Rd+1 (Bozzini et al., 2015). The radial kernels used in the current work are Multiquadrics (MQ), Inverse Quadric (IQ) and Wendland’s function. Of course the latter one is based on compactly supported functions. To discretize the time variable, two techniques are applied. One of them is the Crank–Nicolson scheme and another one is based on MOL. The numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem. Also in appendix, we provide a computational algorithm for implementing the new technique in MATLAB software.

Local geoid determination in strip area projects by using polynomials, least-squares collocation and radial basis functions

Orthometric heights are used in many engineering projects. However, the heights determined by the widely-used Global Navigation Satellite System (GNSS) are ellipsoid heights. Leveling measurements conducted with the purpose of determining the orthometric heights on points are quite arduous and time-consuming processes. To be able to use ellipsoid heights in engineering projects, their transformation to orthometric heights defined in the height datum of the region is necessary. Therefore, in terms of convenience and feasibility GNSS/levelling method is preferred in determining geoid heights. This method is based on the principle of transformation of ellipsoid heights to orthometric heights. In effect, the main purpose of the method can also be regarded as the estimation of geoid undulation values for the study area.During the estimation process, polynomial (surface, curve) models are generally used. Polynomial models produce meaningful results for points which are scattered uniformly on the study area. However, for strip areas where the points scatter along a route (road etc. projects), the accuracy of the geoid heights obtained from these models is low. Therefore, different estimation techniques have to be implemented in strip areas instead of polynomial models. In this study, interpolation methods used in determining the geoid undulation of a strip area were researched and the identification of the best suitable method for the area was examined. For this purpose, geoid undulation values were calculated with the help of least-squares collocation (LSC) and radial basis functions such as Multiquadric (MQ), Thin Plate Spline (TPS) along with polynomial models, and results were presented. According to the results, it was observed that TPS, MQ, LSC methods respectively yield better results compared to polynomial methods.

Local multiquadric RBF meshless scheme for radiative heat transfer in strongly inhomogeneous media

A local radial basis function meshless method (LRBFM) is developed to solve radiative heat transfer in participating media, in which multiquadric (MQ) radial basis functions (RBF) augmented with polynomial basis are employed to construct the trial functions, and the radiative transfer equation (RTE) is discretized directly at nodes by collocation method. The LRBFM belongs to a class of truly meshless methods which do not need any mesh, and can be implemented on a set of uniform or irregular nodes without nodes׳ connectivity. To improve numerical stability of LRBFM for the solution to radiative heat transfer in strongly inhomogeneous media, an upwind support domain scheme is introduced. The upwind scheme is implemented by moving the support domain of local radial basis function interpolation approximation to the opposite direction of each streamline, which can fully capture the information from upstream and improve the accuracy and stability of LRBFM. Performances of the LRBFM and upwind LRBFM (LRBFM_U) are compared with analytical solutions and other numerical results reported earlier in the literatures via a variety of problems in 1-D and 2-D geometries with strongly inhomogeneous media. It is demonstrated that the local radial basis function meshless method with upwind support domain scheme (LRBFM_U) provides high accuracy and great stability to solve radiative heat transfer in strongly inhomogeneous media.