On the method of finite spheres in applications: towards the use with ADINA and in a surgical simulator

 In this paper we report some recent advances regarding applications using the method of finite spheres; a truly meshfree numerical technique developed for the solution of boundary value problems on geometrically complex domains. First, we present the development of a preprocessor for the generation of nodal points on two-dimensional computational domains. Then, the development of a specialized version of the method of finite spheres using point collocation and moving least squares approximation functions and singular weight functions is reported for rapid computations in virtual environments involving multi-sensory (visual and touch) interactions. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

A bridging transition technique for the combination of meshfree method with finite element method in 2D solids and structures

For certain continuum problems, it is desirable and beneficial to combine two different methods together in order to exploit their advantages while evading their disadvantages. In this paper, a bridging transition algorithm is developed for the combination of the meshfree method (MM) with the finite element method (FEM). In this coupled method, the MM is used in the sub-domain where the MM is required to obtain high accuracy, and the FEM is employed in other sub-domains where FEM is required to improve the computational efficiency. The MM domain and the FEM domain are connected by a transition (bridging) region. A modified variational formulation and the Lagrange multiplier method are used to ensure the compatibility of displacements and their gradients. To improve the computational efficiency and reduce the meshing cost in the transition region, regularly distributed transition particles, which are independent of either the meshfree nodes or the FE nodes, can be inserted into the transition region. The newly developed coupled method is applied to the stress analysis of 2D solids and structures in order to investigate its’ performance and study parameters. Numerical results show that the present coupled method is convergent, accurate and stable. The coupled method has a promising potential for practical applications, because it can take advantages of both the MM and FEM when overcome their shortcomings.     

On the evaluation of the method of finite spheres for the solution of Reissner–Mindlin plate problems using the numerical inf–sup test

In this paper we use the numerical inf–sup test to evaluate both displacement‐based and mixed discretization schemes for the solution of Reissner–Mindlin plate problems using the meshfree method of finite spheres. While an analytical proof of whether a discretization scheme passes the inf–sup condition is most desirable, such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved. The numerical inf–sup test (Int. J. Numer. Meth. Engng 1997; 40 :3639–3663), developed to test finite element discretization spaces, has therefore been adopted in this paper. Tests have been performed for both regular and irregular nodal configurations. While, like linear finite elements, pure displacement‐based approximation spaces with linear consistency do not pass the inf–sup test and exhibit shear locking, quadratic discretizations, unlike quadratic finite elements, pass the test. Pure displacement‐based and mixed approximation spaces that pass the numerical inf–sup test exhibit optimal or near optimal convergence behaviour. Copyright © 2007 John Wiley & Sons, Ltd.     

A numerical method for determining a quasi solution of a backward time-fractional diffusion equation

In this study, we present existence and uniqueness theorems of a quasi solution to backward time-fractional diffusion equation. To do that, we consider a methodology, involving minimization of a least squares cost functional, to identify the unknown initial data. Firstly, we prove the continuous dependence on the initial data for the corresponding forward problem and then we obtain a stability estimate. Based on this, we give the existence theorem of a quasi solution in an appropriate class of admissible initial data. Secondly, it is shown that the cost functional is Fréchet-differentiable and its derivative can be formulated via the solution of an adjoint problem. These results help us to prove the convexity of cost functional and subsequently the uniqueness theorem of the quasi solution. In addition, in order to approximate the quasi solution, WEB-spline finite element method is used. Since the obtained system of linear equations is ill-posed, we apply the Levenberg-Marquardt regularization. Finally, a numerical example is given to show the validation of the introduced method.     

Coupling of the meshfree and finite element methods for determination of the crack tip fields

This paper develops a new concurrent simulation technique to couple the meshfree method with the finite element method (FEM) for the analysis of crack tip fields. In the sub-domain around a crack tip, we applied a weak-form based meshfree method using the moving least squares approximation augmented with the enriched basis functions, but in the other sub-domains far away from the crack tip, we employed the finite element method. The transition from the meshfree to the finite element (FE) domains was realized by a transition (or bridge region) that can be discretized by transition particles, which are independent of both the meshfree nodes and the FE nodes. A Lagrange multiplier method was used to ensure the compatibility of displacements and their gradients in the transition region. Numerical examples showed that the present method is very accurate and stable, and has a promising potential for the analyses of more complicated cracking problems, as this numerical technique can take advantages of both the meshfree method and FEM but at the same time can overcome their shortcomings.     

A new method for the numerical solution of integral equation approximations

A new numerical technique for solving the Ornstein-Zernike equation is described. It is particularly useful in solving the Ornstein-Zernike equation for approximations and pair potentials (such as the Percus-Yevick and mean spherical approximations for finite ranged potentials) which imply a finiteranged direct correlation function since for such approximations the numerical technique is essentially exact. The only approximation involved in such cases is the discretization of direct and total correlation functions over the finite range on which the direct correlation function is nonzero. Thus, the new method avoids truncation of the total correlation function and should permit the critical point and spinodal curve to be mapped out with greater accuracy than is permitted by existing methods. Preliminary explorations on the stability and accuracy of the method are described.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

Meshless point collocation for the numerical solution of Navier-Stokes flow equations inside an evaporating sessile droplet

The Navier-Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid-gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier-Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.     

A modification of the Petrov–Galerkin method for the transient convection–diffusion equation

A variation of the Petrov–Galerkin method of solution of a partial differential equation is presented in which the weight function applied to the time derivative term of the transient convection–diffusion equation is different from the weight function applied to the special derivatives. This allows for the formulation of fourth-order explicit and centred difference schemes. Comparison with analytic solutions show that these methods are able to capture steep wave fronts. The ability of the explicit method to capture wave fronts increases as the amount of convective transport increases.     

A comparison of two methods for calculating solid angle coefficients in a BIEM numerical wave tank

By taking the simulation of nonlinear numerical wave tank using the boundary integral equation method (BIEM) as an example, direct and indirect methods are used to compute solid angle coefficients (also named as free-term coefficients). The computing precision and cost are compared between these two methods. The comparisons show that the same good numerical results can be obtained for both direct and indirect methods, and the computing cost of the indirect method is greater than that of the direct method, especially for larger calculation domain.     

Enrichment of the method of finite spheres using geometry‐independent localized scalable bubbles

In this paper, we report the development of two new enrichment techniques for the method of finite spheres, a truly meshfree method developed for the solution of boundary value problems on geometrically complex domains. In the first method, the enrichment functions are multiplied by a weight function with compact support, while in the second one a floating ‘enrichment node’ is introduced. The scalability of the enrichment bubbles offers flexibility in localizing the spatial extent to which the enrichment field is applied. The bubbles are independent of the underlying geometric discretization and therefore provide a means of achieving convergence without excessive refinement. Several numerical examples involving problems with singular stress fields are provided demonstrating the effectiveness of the enrichment schemes and contrasting them to traditional ‘geometry‐dependent’ enrichment strategies in which one or more nodes associated with the geometric discretization of the domain are enriched. An additional contribution of this paper is the use of a meshfree numerical integration technique for computing the J‐integral using the domain integral method. Copyright © 2006 John Wiley & Sons, Ltd.     

Tsallis entropy in dual homogenization of random composites using the stochastic finite element method

This work concerns an application of the Tsallis entropy to homogenization problem of the fiber‐reinforced and also of the particle‐filled composites with random material and geometrical characteristics. Calculation of the effective material parameters is done with two alternative homogenization methods—the first is based upon the deformation energy of the Representative Volume Element (RVE) subjected to the few specific deformations, while the second uses explicitly the so‐called homogenization functions determined under periodic boundary conditions imposed on this RVE. Probabilistic homogenization is made with the use of three concurrent non‐deterministic methods, namely Monte‐Carlo simulation, iterative generalized stochastic perturbation technique as well as the semi‐analytical approach. The last two approaches are based on the Least Squares Method with polynomial basis of the statistically optimized order— this basis serves for further differentiation in the 10th‐order stochastic perturbation technique, while semi‐analytical method uses it in probabilistic integrals. These three approaches are implemented all as the extensions of the traditional Finite Element Method (FEM) with contrastively different mesh sizes, and they serve in computations of Tsallis entropies of the homogenized tensor components as the functions of input coefficient of variation.     

Improved numerical method for the Traction Boundary Integral Equation by application of Stokes' theorem

This paper concerns the direct numerical evaluation of singular integrals arising in Boundary Integral Equations for displacement (BIE) and displacement gradients (BIDE), and the formulation of a Traction Boundary Integral Equation (TBIE) for solving general elastostatic crack problems. Subject to certain continuity conditions concerning displacements and tractions at the source point, singular integrals in the BIE and the BIDE corresponding to coefficients of displacement and displacement gradients at the source point are shown to be of a form that allows application of Stokes' theorem. All the singular integrals in 3-D BIE and BIDE are reduced to non-singular line integrals, and those in 2-D BIE and BIDE are evaluated in closed form. Remaining terms involve regular integrals, and no references to Cauchy or Hadamard principal values are required. Continuous isoparametric interpolations used on continuous elements local to the source point are modified to include unique displacement gradients at the source point which are compatible with all local tractions. The resulting numerical BIDE is valid for source points located arbitrarily on the boundary, including corners, and a procedure is given for constructing a TBIE from the BIDE. Some example solutions obtained using the present numerical method for the TBIE in 2-D and 3-D are presented. © British Crown Copyright 1997/DERA.     

Testing complete and compact radial basis functions for solution of eigenvalue problems using the boundary element method with direct integration

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.     

SBP-SAT方法及其在波动领域的应用

基于分部求和(Summation By Parts)方法和同时逼近项(Simultaneous Approximation Terms)技术建立的有限差分方法,具有更高的精度和稳定性。同时在介质几何不连续、参数突变条件具有较大的优势。国内对SBP-SAT方法的相关研究目前较少,论文对该方法的研究背景,方法发展过程进行了介绍并基于SBP-SAT方法和弹性波动理论,结合初边值条件,推导出曲线网格条件下的弹性波动SBP-SAT离散方程。最后,通过数值模拟实现地震波传播过程,介绍该方法在地震数值模拟领域中的应用价值和前景。     

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain

The development of a hybrid high order time domain finite element solution procedure for the simulation of two dimensional problems in computational electromagnetics is considered. The chosen application area is that of electromagnetic scattering. The spatial approximation adopted incorporates both a continuous Galerkin spectral element method and a high order discontinuous Galerkin method. Temporal discretisation is achieved by means of a fourth order Runge–Kutta procedure. An exact analytical solution is employed initially to validate the procedure and the numerical performance is then demonstrated for a number of more challenging examples.     

An analysis of the linear advection–diffusion equation using mesh-free and mesh-dependent methods

The numerical solution of advection–diffusion equations has been a long standing problem and many numerical methods that attempt to find stable and accurate solutions have to resort to artificial methods to stabilize the solution. In this paper, we present a meshless method based on thin plate radial basis functions (RBF). The efficiency of the method in terms of computational processing time, accuracy and stability is discussed. The results are compared with the findings from the dual reciprocity/boundary element and finite difference methods as well as the analytical solution. Our analysis shows that the RBFs method, with its simple implementation, generates excellent results and speeds up the computational processing time, independent of the shape of the domain and irrespective of the dimension of the problem.     

The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation

In this article, a meshless numerical method for solving the inverse source problem of the space-fractional diffusion equation is proposed. The numerical solution is approximated using the fundamental solution of the space-fractional diffusion equation as a basis function. Since the resulting matrix equation is extremely ill-conditioned, a regularized solution is obtained by adopting the Tikhonov regularization scheme, in which the choice of the regularization parameter is based on generalized cross-validation criterion. Two typical numerical examples are given to verify the efficiency and accuracy of the proposed method.     

A finite volume method for the approximation of convection–diffusion equations on general meshes

A new finite volume method is presented for approximating convection–diffusion equations. This method allows general (unstructured, non‐matching, distorted) meshes to be used without the numerical results being too much altered. The method has been tested for some well‐known benchmarks involving convection and convection–diffusion operators in two space dimensions. These numerical experiments show that it is between first and second‐order accurate, according to the type of the underlying mesh. Further numerical experiments regarding the striation equations have been carried out successfully. Copyright © 2012 John Wiley & Sons, Ltd.     

An adaptive strategy for the bivariate solution of finite element problems using multivariate nested Padé approximants

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present paper, an adaptive approach is proposed as a way to estimate the solution of such multivariate finite element problems. It is based upon the integration of so‐called nested Padé approximants within the finite element procedure. This procedure includes an effective control of the approximation error, which enables adaptive refinements of the converged intervals upon reconstruction of the solution. The main advantages lie in a potential reduction of the computational effort and the fact that the level of a priori knowledge required about the solution in order to have an accurate, efficient, and well‐sampled estimate of the solution is low. The approach is introduced for bivariate problems, for which it is validated on elasto‐poro‐acoustic problems of both academic and more industrial scale. It is argued that the methodology in general holds for more than two variables, and a discussion is opened about the truncation refinements required in order to generalize the results accordingly. Copyright © 2014 John Wiley & Sons, Ltd.