Reproducing kernel enhanced local radial basis collocation method

Standard radial basis functions (RBFs) offer exponential convergence, however, the method is suffered from the large condition numbers due to their ‘nonlocal’ approximation. The nonlocality of RBFs also limits their applications to small‐scale problems. The reproducing kernel functions, on the other hand, provide polynomial reproducibility in a ‘local’ approximation, and the corresponding discrete systems exhibit relatively small condition numbers. Nonetheless, reproducing kernel functions produce only algebraic convergence. This work intends to combine the advantages of RBFs and reproducing kernel functions to yield a local approximation that is better conditioned than that of the RBFs, while at the same time offers a higher rate of convergence than that of reproducing kernel functions. Further, the locality in the proposed approximation allows its application to large‐scale problems. Error analysis of the proposed method is also provided. Numerical examples are given to demonstrate the improved conditioning and accuracy of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

Double grid diffuse collocation method

 In the present paper we propose a new method for constructing a second order Moving Least Squares (MLS) approximation. The method leads to shape functions which are then used for solving Partial Differential Equations (PDE) by a collocation method. This work is an extension of the Generalized Finite Difference Method originally proposed by Liszka and Orkisz (GFDM). However it differs from GFDM by using a sequence of two first order numerical derivations based on linear polynomial basis instead of a second order derivation based on a quadratic polynomial basis. This two-stage approach leads to continuous approximation coefficients using a limited number of surrounding points and results into quite a simple program structure, very similar to that of the finite elements. The method is in an early stage of development so no definitive conclusions may be drawn, however example problems exhibit good convergence properties.     

Shape variable radial basis function and its application in dual reciprocity boundary face method

The radial basis functions (RBFs) is an efficient tool in multivariate approximation, but it usually suffers from an ill-conditioned interpolation matrix when interpolation points are very dense or irregularly spaced. The RBFs with variable shape parameters can usually improve the interpolation matrix condition number. In this paper a new shape parameter variation scheme is implemented. Comparison studies with the constant shaped RBF on convergence and stability are made. Results show that under the same accuracy level, the interpolation matrix condition number by our scheme grows much slower than that of the constant shaped RBF interpolation matrix with increase in the number of interpolation points. As an application example, the dual reciprocity method equipped with the new RBF is combined with the boundary face method to solve boundary value problems governed by Poisson equations. Numerical results further demonstrate the robustness and better stability of the new RBF.     

A NURBS-based Error Reproducing Kernel Method with Applications in Solid Mechanics

A novel method for derivation of mesh-free shape functions is proposed. The first step in the method is to approximate a function and its derivatives through non-uniform-rational-B-spline (NURBS) basis functions. However since NURBS functions neither reproduce polynomials of degree higher than one nor interpolate the control points (also referred to as grid or nodal points), the approximated function leads to uncontrolled errors over the domain including the nodal points. Accordingly the error function in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Since any desired order of continuity in the approximation can be achieved through NURBS, the proposed error reproducing kernel method (ERKM) can even approximate functions with discontinuous derivatives. Moreover, thanks to the variation diminishing property of NURBS, it has advantages in representing sharp layers without the so-called Gibbs‘ or Runge’s phenomena. Since derivatives are reproduced within polynomial spaces of appropriately reduced dimensions, differentiability requirements of the kernel functions are avoided. Any compactly supported continuous function, monotonically decreasing on either side of its maximum, may be used as the weight function (unlike other mesh free approximations). As it turns out, a target function is mainly approximated via NURBS and error functions are just supposed to add corrections, whose magnitudes are typically an order less than those of the NURBS components. The proposed method is observed to be nearly insensitive to the support size of the weight function. The proposed method is next applied to some linear and nonlinear boundary value problems of typical interest in solid mechanics. Some of these results are compared with those obtained via the standard form of RKPM. In the process, the relative numerical advantages and accuracy of the new method are brought out to an extent.     

Developments of the method of difference potentials for linear elastic fracture mechanics problems

Recently, the method of difference potentials has been extended to linear elastic fracture mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the extended finite element method, were introduced to improve the approximation near the crack tip leading to near‐optimal convergence rates. Now, the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has been proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near‐optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the fast Fourier transform is provided with the aim of further increasing the efficiency of the method.     

Derivatives of maximum‐entropy basis functions on the boundary: Theory and computations

In this paper, we obtain explicit expressions to evaluate the derivatives of maximum‐entropy (max‐ent) basis function on the boundary of a convex domain. In the max‐ent formulation, the basis functions are obtained by maximizing a concave functional subjected to linear constraints (reproducing conditions). In doing so, it is found that the Lagrange multipliers blow up when x ∈ ?Ω, and the expressions for the derivatives of the max‐ent basis functions in Ω are of an indeterminate form for points on ?Ω. We appeal to l'Hôpital's rule to derive expressions to determine the derivatives of the basis functions. We consider the Shannon entropy functional and the relative entropy functional with different choices of the prior weight function. The first‐order derivatives of all basis functions are bounded. In contrast, on an irregular grid with a certain nodal spacing, some of the second derivatives of the basis functions are unbounded on the boundary. Necessary and sufficient conditions on the priors to obtain bounded Lagrange multipliers are established. Optimal convergence rates for fourth‐order problems are demonstrated for a Galerkin approach with a quadratically complete partition‐of‐unity enriched max‐ent approximation.Copyright © 2013 John Wiley & Sons, Ltd.     

Generalized boundary element method for galerkin boundary integrals

A meshless approach to the Boundary Element Method in which only a scattered set of points is used to approximate the solution is presented. Moving Least Square approximations are used to build a Partition of Unity on the boundary and then used to construct, at low cost, trial and test functions for Galerkin approximations. A particular case in which the Partition of Unity is described by linear boundary element meshes, as in the Generalized Finite Element Method, is then presented. This approximation technique is then applied to Galerkin boundary element formulations. Finally, some numerical accuracy and convergence solutions for potential problems are presented for the singular, hypersingular and symmetric approaches.     

Meshless methods in dual analysis: Theoretical and implementation issues

This paper presents a meshless implementation of dual analysis for 2D linear elasticity problems. The derivation of the governing systems of equations for the discretized compatible and equilibrated models is detailed and crucial implementation issues of the proposed algorithm are discussed: (i) arising of deficiencies associated with the independent approximation field used for the imposition of the essential boundary conditions (EBC) for the two parts of the boundary sharing a corner and (ii) determination of the Lagrange multipliers functional space used to impose EBC. An attempt to implement the latter resulted in an approximation which is nothing more than the trace on the essential boundary of the domain nodal functions. The difficulties posed by such approximation are explained using the inf–sup condition.Several examples of global (energy) and local (displacements) quantities of interest and their bounds determination are used to demonstrate the validity of the presented meshless approach to dual analysis. Numerical assessment of the convergence rates obtained for both models is made, for different polynomial basis degrees.     

关于Lagrange插值逼近中几个问题研究的新进展

本文将综述 L agrange插值逼近中几个问题研究的新进展 ,并提出几个新问题 .主要的论题是 Lagrange插值多项式序列的收敛与发散 ,用 L agrange插值多项式同时逼近可微分函数及其导数 ,以及修改的 L agrange插值多项式对函数的逼近     

Improving the modified XFEM for optimal high-order approximation

This paper investigates the accuracy of high-order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method, is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher-order contributions to the approximation space. An improved modified XFEM is presented, with basis functions “corrected” by projecting out higher-order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order p ≤ 4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity and are also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order p>1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order p ≤ 4.     

The intrinsic partition of unity method

A method is presented which enables the global enrichment of the approximation space without introducing additional unknowns. Only one shape function per node is used. The shape functions are constructed by means of the moving least-squares method with an intrinsic basis vector and weight functions based on finite element shape functions. The enrichment is achieved through the intrinsic basis. By using polynomials in the intrinsic basis, optimal rates of convergence can be achieved even on distorted elements. Special enrichment functions can be chosen to enhance accuracy for solutions that are not polynomial in character. Results are presented which show optimal convergence on randomly distorted elements and improved accuracy for the oscillatory solution of the Helmholtz equation.     

The Trefftz method for solving eigenvalue problems

For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.     

SOLVING NONLINEAR DIFFERENTIAL EQUATIONS OF MECHANICS WITH THE BOUNDARY ELEMENT METHOD AND RADIAL BASIS FUNCTIONS

The Boundary Element Method is a very effective method for solving linear differential equations. To use it also in the consideration of non-linear problems some different procedures were developed, among them the dual reciprocity method and the particular integral method. Both procedures use interpolation conditions for the approximation with radial basis functions. In this paper a method is presented which avoids problems connected with interpolation by means of quasi-interpolation. It is possible to solve differential equations of the kind Δ^mu=p(u) approximately; the application to two non-linear problems of plate theory yield good results. Hints to a theoretical examination of the method including sufficient conditions for feasibility and convergence are given. © 1997 by John Wiley & Sons, Ltd.     

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 John Wiley & Sons, Ltd.     

Perturbation and stability analysis of strong form collocation with reproducing kernel approximation

Solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence, but with the cost of a dense and ill‐conditioned linear system. In this work, the local approximation functions based on reproducing kernel approximation are introduced for strong form collocation method, called the reproducing kernel collocation method (RKCM). We perform the perturbation and stability analysis of RKCM, and estimate the condition numbers of the discrete equation. Our stability analyses, validated with numerical tests, show that this approach yields a well‐conditioned and stable linear system similar to that in the finite element method. We also introduce an effective condition number where the properties of both matrix and right‐hand side vector of a linear system are taken into consideration in the measure of conditioning. We first derive the effective condition number of the linear systems resulting from RKCM, and show that using the effective condition number offers a tighter estimation of stability of a linear system. The mathematical analysis also suggests that the effective condition number of RKPM does not grow with model refinement. The numerical results are also presented to validate the mathematical analysis. Copyright © 2011 John Wiley & Sons, Ltd.     

Towards criteria for selecting approximation functions in the Dual Reciprocity Method

Since the publication of a book about the Dual Reciprocity Method in 1992 there has been much research into the development of approximation functions employed with this method, and many new functions have been proposed. Given this wealth of new functions, it is important for engineers to be able to identify an adequate function for a given problem. Unfortunately, most authors, in proposing functions for use in DRM compare results principally with the r function used in the book mentioned above, and which is now considered to be that which has the worst performance. In view of this six examples are solved here using a wide range of functions and the results obtained with each function compared with the aim of establishing some guidelines for selecting the function.     

Generalized Multiscale Finite Element Methods with energy minimizing oversampling

In this paper, we propose a general concept for constructing multiscale basis functions within Generalized Multiscale Finite Element Method, which uses oversampling and stable decomposition. The oversampling refers to using larger regions in constructing multiscale basis functions and stable decomposition allows estimating the local errors. The analysis of multiscale methods involves decomposing the error by coarse regions, where each error contribution is estimated. In this estimate, we often use oversampling techniques to achieve a fast convergence. We demonstrate our concepts in the mixed, the Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin discretizations. One of the important features of the proposed basis functions is that they can be used in online Generalized Multiscale Finite Element Method, where one constructs multiscale basis functions using residuals. In these problems, it is important to achieve a fast convergence, which can be guaranteed if we have a stable decomposition. In our numerical results, we present examples for both offline and online multiscale basis functions. Our numerical results show that one can achieve a fast convergence when using online basis functions. Moreover, we observe that coupling using Hybridized Discontinuous Galerkin provides a better accuracy compared with Interior Penalty Discontinuous Galerkin, which is due to using multiscale glueing functions.     

An implicit time integration scheme for inelastic constitutive equations with internal state variables

In a broad class of inelastic constitutive models for the deformation of metals the inelastic strain rates are functions of the current state of stress and internal state variables only. All known models are in some regions of application mathematically stiff and therefore difficult to integrate. The unconditionally stable implicit Euler rule is used for integration. It leads to a system of highly nonlinear algebraic equations which have to be solved by an iterative process. The general Newton-Raphson method, which converges under very broad conditions, requires repeated solution of the finite element system and is infeasible for large inelastic problems. But for the inelastic strains and internal state variables the Jacobian can be computed analytically and therefore the NRI can be used. For the stresses the Jacobian cannot be computed analytically and therefore the accelerated Jacobi iteration is used. A new method for computing the relaxation parameter is introduced which increases the rate of convergence significantly. The new algorithm is applied on Hart's model. A comparison with prior computations using an approximation is made.     

Improvements for the ultra weak variational formulation

In this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two‐dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.     

The 2D large deformation analysis using Daubechies wavelet

In this paper, Daubechies (DB) wavelet is used for solution of 2D large deformation problems. Because the DB wavelet scaling functions are directly used as basis function, no meshes are needed in function approximation. Using the DB wavelet, the solution formulations based on total Lagrangian approach for two-dimensional large deformation problems are established. Due to the lack of Kroneker delta properties in wavelet scaling functions, Lagrange multipliers are used for imposition of boundary condition. Numerical examples of 2D large deformation problems illustrate that this method is effective and stable.