Automatic particular solutions of arbitrary high-order splines associated with polyharmonic and poly-Helmholtz equations

The explicit analytical particular solutions of the splines and monomials for the polyharmonic and poly-Helmholtz operators and their products were derived in the author's previous study. These solutions enable an implementation for automatically approximating particular solutions of arbitrary high-order splines. The automatic particular solutions obtained by the proposed implementation are extremely accurate. In the case of a polyharmonic equation, these solutions are more accurate than the numerical solutions obtained using the multiquadrics within the limits of the IEEE double precision. In the case of poly-Helmholtz and product equations, this implementation can also result in very accurate solutions despite the fact that an analytical particular solution for the multiquadrics does not exist. After particular solutions are obtained, boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions and the Trefftz method, can be applied to solve the homogeneous differential equations.     

Solution of Maxwell's Equations Using the MQ Method

A meshless time domain numerical method based on the radial basis functions using multiquadrics (MQ) is employed to simulate electromagnetic field problems by directly solving the time-varying Maxwell's equations without transforming to simplified versions of the wave or Helmholtz equations. In contrast to the conventional numerical schemes used in the computational electromagnetism such as FDTD, FETD or BEM, the MQ method is a truly meshless method such that no mesh generation is required. It is also easy to deal with the appropriate partial derivatives, divergences, curls, gradients, or integrals like semi-analytic solutions. For illustration purposes, the MQ method is employed to calculate the propagations of the electric and magnetic waves in the homogeneous, isotropic, and non-lossy 2D rectangular waveguide as well as 3D cavity resonator. Good agreements are obtained as compared to analytical solutions. By directly solving the Maxwell's equations, the MQ scheme provides a very simple and effective numerical scheme for the computational electromagnetism.     

Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

In this study, the homotopy analysis method (HAM) is combined with the method of fundamental solutions (MFS) and the augmented polyharmonic spline (APS) to solve certain nonlinear partial differential equations (PDE). The method of fundamental solutions with high-order augmented polyharmonic spline (MFS–APS) is a very accurate meshless numerical method which is capable of solving inhomogeneous PDEs if the fundamental solution and the analytical particular solutions of the APS associated with the considered operator are known. In the solution procedure, the HAM is applied to convert the considered nonlinear PDEs into a hierarchy of linear inhomogeneous PDEs, which can be sequentially solved by the MFS–APS. In order to solve strongly nonlinear problems, two auxiliary parameters are introduced to ensure the convergence of the HAM. Therefore, the homotopy method of fundamental solutions can be applied to solve problems of strongly nonlinear PDEs, including even those whose governing equation and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the MFS. Several numerical experiments were carried out to validate the proposed method.     

Analysis of Thin Isotropic Rectangular and Circular Plates with Multiquadrics

A computational method based on radial basis functions has been applied to the linear solution of thin plates. This meshless numerical method gives high flexibility in the analysis of irregular geometries, due to its insensivity to spatial dimension. The multiquadrics approach is used in this paper. The numerical solution is compared with Kirchhoff theory for plates.__________Translated from Problemy Prochnosti, No. 2, pp. 72 – 84, March – April, 2005.     

DRM—MFS for two-dimensional finite elasticity

This paper presents a meshless method for the solution of problems in finite elasticity. The method is based on coupling the method of fundamental solutions (MFS) with dual reciprocity method (DRM). The solution is obtained by adding the homogeneous solution generated by MFS to the particular solution obtained by the radial basis function employed in DRM. The procedure has the advantage of eliminating domain integration. The proposed method is tested through two numerical examples that confirm its efficiency and accuracy.     

A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions

 A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties. Received: 10 November 2002 / Accepted: 5 March 2003     

Modelling of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions

In this paper we use a trigonometric layerwise deformation theory for modelling symmetric composite plates. We use a meshless discretization method based on global multiquadric radial basis functions. The results obtained are compared with solutions derived from other models and numerical techniques. The results show that the use of trigonometric layerwise deformation theory discretized with multiquadrics provides very good solutions for composite plates and excellent solutions for sandwich plates.     

The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids

This paper describes the method of fundamental solutions (MFS) to solve three-dimensional elastostatic problems of transversely isotropic solids. The desired solution is represented by a series of closed-form fundamental solutions, which are the displacement fields due to concentrated point forces acting on the transversely isotropic material. To obtain the unknown intensities of the fundamental solutions, the source points are properly located outside the computational domain and the boundary conditions are then collocated. Furthermore, the closed-form traction fields corresponding to the previously published point force solutions are reviewed and addressed explicitly in suitable forms for numerical implementations. Three numerical experiments including Dirichlet, Robin, and peanut-shaped-domain problems are carried out to validate the proposed method. It is found that the method performs well for all the three cases. Furthermore, a rescaling method is introduced to improve the accuracy of Robin problem with noisy boundary data. In the spirits of MFS, the present meshless method is free from numerical integrations as well as singularities.     

A meshless method for solving 1D time-dependent heat source problem

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) and the heat polynomials is proposed for recovering a time-dependent heat source and the boundary data simultaneously in an inverse heat conduction problem (IHCP). We will transform the problem into a homogeneous IHCP and initial value problems for the first-order ordinary differential equation. An improved method of MFS is used to solve the IHCP and a finite difference method is applied for solving the initial value problems. The advantage of applying the proposed meshless numerical scheme is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Numerical experiments for some examples are provided to show the effectiveness of the proposed algorithm.     

The Method of Fundamental Solutions for One-Dimensional Wave Equations

A meshless numerical algorithm is developed for the solutions of one-dimensional wave equations in this paper. The proposed numerical scheme is constructed by the Eulerian-Lagrangian method of fundamental solutions (ELMFS) together with the D'Alembert formulation. The D'Alembert formulation is used to avoid the difficulty to constitute the linear algebraic system by using the ELMFS in dealing with the initial conditions and time-evolution. Moreover the ELMFS based on the Eulerian-Lagrangian method (ELM) and the method of fundamental solutions (MFS) is a truly meshless and quadrature-free numerical method. In this proposed wave model, the one-dimensional wave equation is reduced to an implicit form of two advection equations by the D'Alembert formulation. Solutions of advection equations are then approximated by the ELMFS with exceptionally small diffusion effects. We will consider five numerical examples to test the capability of the wave model in finite and infinite domains. Namely, the traveling wave propagation, the time-space Cauchy problems and the problems of vibrating string, etc. Numerical validations of the robustness and the accuracy of the proposed method have demonstrated that the proposed meshless numerical model is a highly accurate and efficient scheme for solving one-dimensional wave equations.     

An Alternating Iterative MFS Algorithm for the Cauchy Problem in Two-Dimensional Anisotropic Heat Conduction

In this paper, the alternating iterative algorithm originally proposed by Kozlov, Maz'ya and Fomin (1991) is numerically implemented for the Cauchy problem in anisotropic heat conduction using a meshless method. Every iteration of the numerical procedure consists of two mixed, well-posed and direct problems which are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The iterative MFS algorithm is tested for Cauchy problems related to heat conduction in two-dimensional anisotropic solids to confirm the numerical convergence, stability and accuracy of the method.     

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.     

An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation

We investigate the numerical implementation of the alternating iterative algorithm originally proposed by Kozlov et al. (Comput Math Math Phys 31:45–52) for the Cauchy problem associated with the two-dimensional modified Helmholtz equation using a meshless method. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The iterative MFS algorithm is tested for Cauchy problems for the two-dimensional modified Helmholtz operator to confirm the numerical convergence, stability and accuracy of the method.     

基于遗传算法和复合二次径向基函数的复合材料层合板自由振动分析

为了提高复合材料层合板自由振动分析的精度,采用无网格径向基配点法分析复合材料材料层合板的自由振动问题,径向基函数的形状参数对计算精度有很大影响。利用遗传算法对复合二次径向基函数的形状参数进行优化,用优化后的形状参数的复合二次径向基函数计算复合材料层合板的固有频率,计算结果与文献中的结果具有较好的一致性。遗传算法在形状参数优化方面具有很大的潜力,所提出的方法具有较高的计算精度。     

A BEM-based meshless solution to buckling and vibration problems of orthotropicplates

In this paper a BEM-based meshless solution is presented to buckling and vibration problems of Kirchhoff orthotropic plates with arbitrary shape. The plate is subjected to compressive centrally applied load together with arbitrarily transverse distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting buckling and vibration problems are described by partial differential equations in terms of the deflection. Both problems are solved employing the Analog Equation Method (AEM). According to this method the fourth-order partial differential equation describing the response of the orthotropic plate is converted to an equivalent linear problem for an isotropic plate subjected only to a fictitious load under the same boundary conditions. The AEM is applied to the problem at hand as a boundary-only method by approximating the fictitious load with a radial basis function series. Thus, the method retains all the advantages of the pure BEM using a known simple fundamental solution. Example problems are presented for orthotropic plates, subjected to compressive or vibratory loading, to illustrate the method and demonstrate its efficiency and its accuracy.     

The method of fundamental solutions for steady-state groundwater flow problems

The method of fundamental solutions is a meshless method. Only boundary collocation points are needed during the whole solution process. It has the merits of mathematical simplicity, ease of programming, high solution accuracy, and others. In this paper, the method of fundamental solutions is applied to simulate 2D steady-state groundwater flow problems. The principle of superposition is used during the whole solution process. Numerical results are compared with the multiquadrics method and the mixed finite element method as well as analytical solutions. It is shown that the method of fundamental solutions is promising in dealing with steady groundwater flow problems.     

Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems

We investigate two algorithms involving the relaxation of either the given Dirichlet data (boundary temperatures) or the prescribed Neumann data (normal heat fluxes) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [26] applied to two-dimensional steady-state heat conduction Cauchy problems, i.e. Cauchy problems for the Laplace equation. The two mixed, well-posed and direct problems corresponding to each iteration of the numerical procedure are solved using a meshless method, namely the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the Laplace operator in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

The method of fundamental solutions for solving incompressible Navier–Stokes problems

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) is proposed to solve the primitive variables formulation of the Navier–Stokes equations. The MFS is a meshless method since it is free from the mesh generation and numerical integration. We will transform the Navier–Stokes equations into simple advection–diffusion and Poisson differential operators via the operator-splitting scheme or the so-called projection method, instead of directly using the more complicated fundamental solutions (Stokeslets) of the unsteady Stokes equations. The resultant velocity advection–diffusion equations and the pressure Poisson equation are then calculated by using the MFS together with the Eulerian–Lagrangian method (ELM) and the method of particular solutions (MPS). The proposed meshless numerical scheme is a first attempt to apply the MFS for solving the Navier–Stokes equations in the moderate-Reynolds-number flow regimes. The lid-driven cavity flows at the Reynolds numbers up to 3200 for two-dimensional (2D) and 1000 for three-dimensional (3D) are chosen to validate the present algorithm. Through further simulating the flows in the 2D circular cavity with an eccentric rotating cylinder and in the 3D cube with a fixed sphere inside, we are able to demonstrate the advantages and flexibility of the proposed meshless method in the irregular geometry and multi-dimensional flows, even though very coarse node points are used in this study as compared with other mesh-dependent numerical schemes.     

Regularized meshless method for antiplane piezoelectricity problems with multiple inclusions

In this paper, solving antiplane piezoelectricity problems with multiple inclusions are attended by using the regularized meshless method (RMM). This is made possible that the troublesome singularity in the MFS disappears by employing the subtracting and adding-back techniques. The governing equations for linearly electro-elastic medium are reduced to two uncoupled Laplace's equations. The representations of two solutions of the two uncoupled system are obtained by using the RMM. By matching interface conditions, the linear algebraic system is obtained. Finally, typical numerical examples are presented and discussed to demonstrate the accuracy of the solutions.     

Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations

In this paper, the static and free vibration analysis of composite plates are performed, using a layerwise deformation theory and multiquadrics discretization. This meshless discretization method considers radial basis functions as the approximation method for both the equations of motion and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the natural frequencies.