Local Radial Basis Function Collocation Research Articles

Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique and the partitioned Runge–Kutta method. Specific numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schrödinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.

Meshless symplectic and multi-symplectic algorithm for Klein–Gordon–Schrödinger system with local RBF collocation

This work proposes a novel meshless symplectic and multi-symplectic approximation for Klein–Gordon–Schrödinger system. The main features of this method are: (I) the application of local radial basis function (RBF) collocation method (LRBFCM) in spatial discretization; (II) the application of symplectic integrator in time discretization. LRBFCM method can deal with the ill-conditioned problem and shape-parameter sensitivity of global RBF method, and is simple and effective. The conservatism of this method is discussed and its accuracy is evaluated. Numerical simulations are given for one dimension (1D)/2D with regular and irregular domains to verify the effectiveness of our new method.

Local knot method for solving inverse Cauchy problems of Helmholtz equations on complicated two‐ and three‐dimensional domains

AbstractThis article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore–Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi‐analytical and local radial basis function collocation method using the non‐singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy‐to‐program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

A meshless multi-symplectic local radial basis function collocation scheme for the “good” Boussinesq equation

A novel meshless multi-symplectic scheme is proposed for the “good” Boussinesq equation. The scheme consists of a local radial basis function (RBF) collocation method (LRBFCS) in space and a symplectic integrator in time. The LRBFCS is simple and efficient, since only a sparse banded linear system has to be solved. Moreover, it can avoid the ill-conditioned problem and shape-parameter-sensitivity of the global RBF method. The multi-symplectic LRBFCS (MLRBFCS) is more accurate than traditional methods. Numerical experiments with uniform knots and random knots are designed to verify the effectiveness of the method.

Meshless approach to the large-eddy simulation of the continuous casting process

The paper discusses the numerical solution of the large-eddy formulation for modelling the turbulent fluid flow with solidification in the continuous casting of steel. This industrially relevant problem is solved by a meshless method for the first time. The solid-liquid system is formulated within the continuum mixture formulation of the governing, mass momentum and energy equations. The Darcy porous media model describes the mushy region. The large-eddy formulation is based on the Smagorinsky model for sub-grid scale viscosity and the van Driest damping function. The synthetic turbulence is prescribed at the molten steel inlet. The generated turbulent fluctuations are isotopic and correlated in time through an asymmetric time filter. Explicit time discretisation, combined with meshless local radial basis function collocation method, is employed. Five-noded subdomains, non-uniform node arrangement and multiquadrics shape functions are used for solving a two-dimensional model. The sensitivity to different node arrangements, time steps, inlet temperature, and velocity boundary conditions is elaborated and successfully verified by comparison with our previously published meshless k − ε turbulence model of the same process.

Simulation of electromagnetic wave propagations in negative index materials by the localized RBF-collocation method

In this paper, a fast simulation of the electromagnetic wave propagation in negative index materials is developed by the localized meshfree radial basis function (RBF) collocation method. The perfect match layer is considered on the boundary in a unit cell to avoid the wave reflection. The experienced shape parameter is validated by simple test functions. The CPU time costs are compared with those of FEM. Numerical examples of different scatterers are presented to demonstrate the efficiency of the proposed method.

Numerical Simulation of 3d Double-Nozzles Printing by Considering a Stabilized Localized Radial Basis Function Collocation Method

Numerical Simulation of 3d Double-Nozzles Printing by Considering a Stabilized Localized Radial Basis Function Collocation Method

Three Kinds of Novel Multi-Symplectic Methods for Stochastic Hamiltonian Partial Differential Equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique, and the partitioned Runge-Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schr\"odinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.

An accurate and efficient numerical method for neural field models with transmission delays

Based on the recently developed localized radial basis function collocation method (LRBFCM) and exponential time differencing (ETD) method, we derive in this paper an accurate and efficient numerical method for solving neural field models with space-dependent delays formulated as system of integro-differential equations. This combined LRBFCM-ETD method inherits the advantages of these two methods in space and time, thereby offering high accuracy and efficiency for both uniform and non-uniform discretizations, extending the applicability of the method to multi-dimensions easily, and maintaining a good behaviour in a large time-step and long-time integration. Some numerical experiments are presented and numerical results show that the obtained simulations via the LRBFCM-ETD method are reliable and effective for approximating the solution of models investigated in the current paper.

Meshless symplectic and multi-symplectic local RBF collocation methods for nonlinear Schrödinger equation

This study formulates a novel meshless symplectic and multi-symplectic local radial basis function (RBF) collocation method (LRBFCM) for the nonlinear Schrödiner equation. The LRBFCM is employed for spatial discretization and then symplectic integrator is conducted for time discretization. The properties of the space differentiation matrix of LRBFCM are discussed in detail. The conservativeness of the proposed method is discussed and the accuracy is assessed. The LRBFCM is simple and efficient, since it can avoid the ill-conditioned problem and shape-parameter-sensitivity of global RBF method. Numerical tests with uniform knots and random knots are designed to show the effectiveness of the method.

A novel local radial basis function collocation method for multi-dimensional piezoelectric problems

The local radial basis function collocation method (LRBFCM), a strong-form formulation of the meshless numerical method, is proposed for solving piezoelectric medium problems. The proposed numerical algorithm is based on the local Kansa method using variable shape parameter. We introduce a novel technique for the determination of shape parameter in the LRBFCM, which leads to greater accuracy, and simplicity. The implemented algorithm is first verified with a 2D Poisson equation. Then, we employed LRBFCM in a numerical simulation for 2D and 3D piezoelectric problems involving mutual coupling of the electric field and elastodynamic equations for mechanical field. The presented meshless method is verified using corresponding results obtained from the finite element method and moving least squares meshless local Petrov–Galerkin method. In particular, the 2D piezoelectric problem is verified with an exact solution.

3D elastic dental analysis by a local RBF collocation method

In this paper, the local radial basis function collocation method (LRBFCM) is presented to analyze 3D elastic problems of complex geometries. In particular, a real tooth domain built by cone beam computer tomography (CBCT) data is considered. The reconstruction process of a tooth shape, the choice of the local influence domain in the LRBFCM, as well as the treatments of the boundary conditions are detailed described in this paper. Linear elastic analyses of five different examples are carried out to validate the effectiveness of the proposed approach. The stress difference between a tooth with the dental caries and a healthy tooth is investigated and discussed by the 3D elastic deformation and stress analysis.

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d -dimensional problem in space into d + 1 -dimensional one in the space-time domain by combining the d -dimensional vector space variable and 1 -dimensional time variable in one d + 1 -dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ -method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d -dimensional problem as a d + 1 -dimensional space-time one are discussed based on the type of PDEs considered. The localized radial basis function meshless method is applied to seek for the numerical solution. Different examples in two and three-dimensional space are solved to show the accuracy of such method. Different types of boundary conditions, Neumann and Dirichlet, are also considered for parabolic and hyperbolic equations to show the sensibility of the method in respect to boundary conditions. A comparison to the fourth-order Runge-Kutta method is also investigated.

An improved local radial basis function collocation method based on the domain decomposition for composite wall

In this study, the idea of the domain decomposition is applied to further enhance the computational efficiency of the local radial basis function collocation method (LRBFCM). Unlike the traditional LRBFCM, the computational domain is decomposed into small unit square domains. Then the LRBFCM is further applied to each unit square domain, and the whole computational domain is later analyzed by considering interface conditions. The direct method is proposed to deal with the instability caused by the derivative calculation at the interface. Then the proposed LRBFCM is further extended to the composite walls. Different cases are carried out to test the efficiency of the proposed method. The accuracy of the numerical results is validated by comparing with the finite element method (FEM).

Local radial basis function collocation method for stokes equations with interface conditions

In the present work, local radial basis function (RBF) collocation method is used to solve the Stokes problem numerically. Benefits of the local meshless method is its flexibility of handling complex computational domain and interface. Local meshless method produces sparse coefficient matrix, though its structure is depending on the placement of nodes, and type of boundary conditions. The Stokes equations are collocated in a rectangular region with different interface conditions and direct solvers are used instead of iterative algorithm. Due to the unknown boundary conditions for the pressure term in the Stokes equations, value of the pressure is obtained by integrating the Stokes equations and eliminating the constant of integration. In the numerical implementation, circular, elliptic and Deltoid like interfaces are taken into consideration to test performance of the proposed approach for complex problems. Meshless solution of the Stokes equations for the first two problems are compared with the published results in the literature. The last problem is different form the first two problems in the sense that the continuity equation is also discontinuous and the velocity has nonzero value in both the inner and outer sub-domains.

Noniterative Localized and Space-Time Localized RBF Meshless Method to Solve the Ill-Posed and Inverse Problem

In many references, both the ill-posed and inverse boundary value problems are solved iteratively. The iterative procedures are based on firstly converting the problem into a well-posed one by assuming the missing boundary values. Then, the problem is solved by using either a developed numerical algorithm or a conventional optimization scheme. The convergence of the technique is achieved when the approximated solution is well compared to the unused data. In the present paper, we present a different way to solve an ill-posed problem by applying the localized and space-time localized radial basis function collocation method depending on the problem considered and avoiding the iterative procedure. We demonstrate that the solution of certain ill-posed and inverse problems can be accomplished without iterations. Three different problems have been investigated: problems with missing boundary condition and internal data, problems with overspecified boundary condition, and backward heat conduction problem (BHCP). It has been demonstrated that the presented method is efficient and accurate and overcomes the stability analysis that is required in iterative techniques.

A local meshless method for the numerical solution of space‐dependent inverse heat problems

In this paper, a local radial basis function collocation method is proposed for the numerical solution of inverse space‐wise dependent heat source problems. Multiquadric radial basis function is used for spatial discretization. The method accuracy is tested in terms of absolute root mean square and relative root mean square error norms. Numerical tests on a noisy data are performed on both regular domain and irregular domain. To test the efficiency and accuracy of the proposed method, numerical experiments for one‐, two‐, and three‐dimensional cases are performed. Both regular and irregular geometries with uniform and nonuniform points are taken into consideration, and the numerical results are also compared with the existing methods reported in literature.

Application of the local RBF collocation method to natural convection in a 3D cavity influenced by a magnetic field

This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a multi-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Rem ≪ 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic three-dimensional steady natural convection benchmark for Rayleigh numbers from 105 to 107 and Prandtl number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.

A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory

In this paper, the band structures of nanoscale phononic crystals based on the nonlocal elasticity theory are calculated by using a meshfree local radial basis function collocation method (LRBFCM). The direct method is applied to enhance the stability of the derivative calculations in the LRBFCM. A simple summation in the LRBFCM is proposed to deal with the integration related to the nonlocal stresses or tractions. The LRBFCM for the band structure calculations is validated by the results obtained with the first-principle and the transfer matrix (TM) method for one-dimensional (1D) phononic crystals, as well as the comparison of the frequency responses of the two-dimensional (2D) periodic structures.

A semi-Lagrangian meshless framework for numerical solutions of two-dimensional sloshing phenomenon

This paper presents a semi-Lagrangian meshless framework based on the localized radial basis function collocation method and the second-order explicit Runge–Kutta method (RK2-LRBFCM) for analyzing the sloshing phenomenon in a 2D tank. In the present method, the semi-Lagrangian approach is adopted to constrain the lateral movements of inner collocation nodes. And the LRBFCM is used to obtain the velocity potentials at each time instant governed by Laplace equation, then second-order explicit Runge–Kutta method is introduced to calculate the free-surface elevation and velocity potential at the next time instant. In comparison with the reference results, the efficiency and accuracy of the proposed RK2-LRBFCM are demonstrated in the solutions of several benchmark examples including horizontally excited tanks, vertically excited tanks, horizontally and vertically excited tanks. Moreover, the effect of the protrusion (shape and size) of the bottom wall on the sloshing phenomenon in horizontally excited tank is investigated.