Meshfree Methods for Partial Differential Equations III

We present a new approach to construct approximation schemes from scattered data on a node set, i.e. in the spirit of meshfree methods. The rational procedure behind these methods is to harmonize the locality of the shape functions and the information-theoretical optimality (entropy maximization) of the scheme, in a sense to be made precise in the paper. As a result, a one-parameter family of methods is defined, which smoothly and seamlessly bridges meshfree-style approximants and Delaunay approximants. Besides an appealing theoretical foundation, the method presents a number of practical advantages when it comes to solving partial differential equations. The non-negativity introduces the well-known monotonicity and variation-diminishing properties of the approximation scheme. Also, these methods satisfy ab initio a weak version of the Kronecker-delta property, which makes essential boundary conditions straightforward. The calculation of the shape functions is both efficient and robust in any spacial dimension. The implementation of a Galerkin method based on local maximum entropy approximants is illustrated by examples.

In this paper we present the application of the meshfree method of finite spheres to the solution of thin and thick plates composed of isotropic as well as functionally graded materials. For the solution of such problems it is observed that using Gaussian and adaptive quadrature schemes are computationally inefficient. In this paper a new technique, presented in [26, 21], in which the integration points and weights are generated using genetic algorithms and stored in a lookup table using normalized coordinates as part of an offline preprocessing step, is shown to provide significant reduction of computational time without sacrificing accuracy.

Recently, very intensive efforts have been devoted to develop meshless or element free methods that eliminate the need of element connectivity in the solution of PDEs. The motivation is to cut down modelling costs in industrial applications by avoiding the labor intensive step of mesh generation. In addition, these methods are particularly attractive in problems with moving interfaces since no remeshing is necessary.

In this paper we introduce various forms of strain smoothing for stabilization and regularization of two types of instability: (1) numerical instability resulting from nodal domain integration of weak form, and (2) material instability due to material strain softening and localization behavior. For numerical spatial instability, we show that the conforming strain smoothing in stabilized conforming nodal integration only suppresses zero energy modes resulting from nodal domain integration. When the spurious nonzero energy modes are excited, additional stabilization is proposed. For problems involving strain softening and localization, regularization of the ill-posed problem is needed. We show that the gradient type regularization method for strain softening and localization can be formulated implicitly by introducing a gradient reproducing kernel strain smoothing. It is also demonstrated that the gradient reproducing kernel strain smoothing also provides a stabilization to the nodally integrated stiffness matrix. For application to modeling of fragment penetration processes, a nonconforming strain smoothing as a simplification of conforming strain smoothing is also introduced.

This paper addresses the numerical solution of the system of the Lagrangian gas dynamics equation. Usually Finite Difference Methods are applied for the simulation of fluid dynamics. However, for problems with large deformations as the application under consideration they encounter problems with intercrossed numerical grids. This disadvantage can be overcome by meshfree methods which do not require a numerical grid like the Smoothed Particle Hydrodynamics method in principle [1], [2]. We show, however, that this approach does not work properly for the solution of the Lagrangian system for hydrodynamic flow, except for special cases, because of conservation violations.

A method is proposed for arbitrary discontinuities, without the need for a mesh that aligns with the interfaces, and without introducing additional unknowns as in the extended finite element method. The approximation space is built by special shape functions that are able to represent the discontinuity, which is described by the level-set method. The shape functions are constructed by means of the moving least-squares technique. This technique employs special mesh-based weight functions such that the resulting shape functions are discontinuous along the interface. The new shape functions are used only near the interface, and are coupled with standard finite elements, which are employed in the rest of the domain for efficiency. The coupled set of shape functions builds a linear partition of unity that represents the discontinuity. The method is illustrated for linear elastic examples involving strong and weak discontinuities.

A meshless boundary element method (BEM) for stress analysis in two-dimensional (2-D), isotropic, continuously non-homogeneous, and linear elastic functionally graded materials (FGMs) is presented in this paper. It is assumed that Young’s modulus has an exponential variation, while Poisson’s ratio is taken to be constant. Since no fundamental solutions are yet available for general FGMs, fundamental solutions for isotropic, homogeneous, and linear elastic solids are applied, which results in a boundary-domain integral formulation. Normalized displacements are introduced in the formulation, which avoids displacement gradients in the domain-integrals. The radial integration method (RIM) is used to transform the domain-integrals into boundary integrals along the global boundary. The normalized displacements appearing in the domain-integrals are approximated by a series of prescribed basis functions, which are taken as a combination of radial basis functions and polynomials in terms of global coordinates. Numerical examples are presented to verify the accuracy and the efficiency of the present meshless BEM.

This paper is concerned with the adaptive multilevel solution of elliptic partial differential equations using the partition of unity method. While much of the work on meshfree methods is concerned with convergence-studies, the issues of fast solution techniques for the discrete system of equations and the construction of optimal order algorithms are rarely addressed. However, the treatment of large scale real-world problems by meshfree techniques will become feasible only with the availability of fast adaptive solvers.

In this paper we propose a new approximation technique within the context of meshless methods able to reproduce functions with discontinuous derivatives. This approach involves some concepts of the reproducing kernel particle method (RKPM), which have been extended in order to reproduce functions with discontinuous derivatives. This strategy will be referred as Enriched Reproducing Kernel Particle Approximation (E-RKPA). The accuracy of the proposed technique will be compared with standard RKP approximations (which only reproduces polynomials).

We consider a mathematical model for polymeric liquids which requires the solution of high-dimensional Fokker-Planck equations related to stochastic differential equations. While Monte-Carlo (MC) methods are classically used to construct approximate solutions in this context, we consider an approach based on Quasi- Monte-Carlo (QMC) approximations. Although QMC has proved to be superior to MC in certain integration problems, the advantages are not as pronounced when dealing with stochastic differential equations. In this article, we illustrate the basic difficulty which is related to the construction of QMC product measures.

This paper is a review of the bridging scale method, which was recently proposed to couple atomistic and continuum simulation methods. The theory will be shown in a fully generalized three-dimensional setting, including the numerical calculation of the time history kernel in multiple dimensions, such that a two-way coarse/fine coupled non-reflecting molecular dynamics boundary condition can be found. We present numerical examples in three dimensions validating the bridging scale methodology. The bridging scale method is tested on highly nonlinear dynamic fracture examples, and the ability of the numerically calculated time history kernel in eliminating high frequency wave reflection at the MD/FE interface is shown. All results are compared to benchmark full MD simulations for verification and validation.

A new stabilized nodal integration scheme is proposed and implemented. In this work, focus is on the natural neighbor meshless interpolation schemes. The approach is a modification of the stabilized conforming nodal integration (SCNI) scheme and is shown to perform well in several benchmark problems.

The Finite Pointset Method (FPM) is a meshfree Lagrangian particle method for flow problems. We focus on incompressible, viscous flow equations, which are solved using the Chorin projection method. In the classical FPM second order derivatives are approximated by a least squares approximation. In general this approach yields stencils with both positive and negative entries. We present how optimization routines can force the stencils to have only positive weights aside from the central point, and investigate conditions on the particle geometry. This approach yields an M-matrix structure, which is of interest for various linear solvers, for instance multigrid. We solve the arising linear systems using algebraic multigrid.

This paper presents two different formulations for in-plane generalized finite elements for geometrical non-linear analysis. The results from analyses which employ the proposed elements are also presented. One of the proposed elements has one additional degree of freedom at each node and shows good performance for analysis in which bending deformation is dominant. The other can reproduce quadratic deformation mode with only corner nodes and it has no linear dependency, which is a well known problem of generalized finite elements. The formulation is based on the rate form of the virtual work principle and obtained by a simple extension of standard FEM. The convergence of analytical solutions and the robustness against element distortion are investigated and the results are compared with those of standard displacement based first and second order elements. In most cases, the proposed elements provide good solution convergence which is similar to, if not better than, those of conventional second order elements. Additionally, it is also shown that high-precision solutions can be obtained even if the mesh is strongly distorted.

We present the application of a meshfree method for simulations of interaction between fluids and flexible structures. As a flexible structure we consider a sheet of paper. In a two-dimensional framework this sheet can be modeled as curve by the dynamical Kirchhoff-Love theory. The external forces taken into account are gravitation and the pressure difference between upper and lower surface of the sheet. This pressure difference is computed using the Finite Pointset Method (FPM) for the incompressible Navier-Stokes equations. FPM is a meshfree, Lagrangian particle method. The dynamics of the sheet are computed by a finite difference method. We show the suitability of the meshfree method for simulations of fluid-structure interaction in several applications.

A novel approach for implicit residual-type error estimation in meshfree methods is presented. This allows to compute upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods.

In this paper, new natural element approximations are proposed, in order to address issues associated with incompressibility as well as to increase the accuracy in the Natural Element Method (NEM). The NEM exhibits attractive features such as interpolant shape functions or auto-adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well known inf-sup, or LBB condition. The first proposed approach constructs richer NEM approximation schemes by means of bubbles associated with the topological entities of the underlying Delaunay tessellation, allowing to pass the LBB and to remove pressure oscillations in the incompressible limit. Despite of its simplicity, this approach does not construct approximation with higher order consistency. The second part of the paper deals with a discussion on the construction of second-order accurate NEM approximations. For this purpose, two techniques are investigated : (a) the enrichment in the MLS framework of the bubbles with higher-order polynomials and (b) the use of a new Hermite-NEM formulation.