Meshfree Methods for Partial Differential Equations

Frontmatter

Meshless and Generalized Finite Element Methods: A Survey of Some Major Results

Abstract

In this lecture, we discuss Meshless and Generalized Finite Element Methods. We survey major results in this area with a unified approach.

I. Babuška, U. Banerjee, J. E. Osborn

Adaptive Meshfree Method of Backward Characteristics for Nonlinear Transport Equations

Abstract

In previous work, a new adaptive meshfree advection scheme for numerically solving linear transport equations has been proposed. The scheme, being a combination of an adaptive semi-Lagrangian method and local radial basis function interpolation, is essentially a method of backward characteristics. The adaptivity of the meshfree advection scheme relies on customized rules for the refinement and coarsening of scattered nodes. In this paper, the method is extended to nonlinear transport equations. To this end, in order to be able to model shock propagation, an artificial viscosity term is added to the scheme. Moreover, the local interpolation method and the node adaption rules are modified accordingly. The good performance of the resulting method is finally shown in the numerical examples by using two specific nonlinear model problems: Burgers equation and the Buckley-Leverett equation, the latter describing a two-phase fluid flow in a porous medium.

Jörn Behrens, Armin Iske, Martin Käser

New Methods for Discontinuity and Crack Modeling in EFG

Abstract

A new method for modeling discontinuities, such as cracks, in the element free Galerkin method is presented. A jump function is used for the displacement discontinuity along the crack faces and the Westergard’s solution enrichment near the crack tip. These enrichments, being extrinsic, can be limited only to the nodes surrounding the crack. The method is coupled to a new vector level set method [1] so with this approach only nodal data are used to describe the crack, no geometrical entity is introduced for the crack trajectory, and no partial differential equations need be solved to update the level sets.

Ted Belytschko, Giulio Ventura, Jingxiao Xu

SPH Simulations of MHD Shocks Using a Piecewise Constant Smoothing Length Profile

Abstract

Concerns regarding efficiency and accuracy of Smoothed Particle Hydrodynamics (SPH) compared to modern grid-based methods have been raised. Likewise, the extension of SPH to MHD problems has proven to be a challenge. In an attempt to improve the ability of SPH to treat shocks in general, and MHD shocks in particular, a modified version of SPH called Regularized Smoothed Particle Hydrodynamics (RSPH) has been presented [1]. This method allows a piecewise constant smoothing length profile to be used. Furthermore, the smoothing length profile is optimized at temporal intervals using a mass, momentum and internal erergy conserving regularization process. In this paper, we examine more closely the abilities of the RSPH method to treat MHD shocks. We present a simple stability analysis, as well as results from MHD shock tests in one and two dimensions.

Steinar Børve, Marianne Omang, Jan Trulsen

On the Numerical Solution of Linear Advection-Diffusion Equation using Compactly Supported Radial Basis Functions

Abstract

We present a meshless method based on compactly supported radial basis functions to find a stable and accurate solution for the numerical solution of linear advection-diffusion equation. The efficiency of the method in terms of computational processing time, accuracy and stability is also discussed. The results are compared with the findings from the thin plate spline radial basis function and finite difference methods. Our analysis shows that compactly supported radial basis functions 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.

Ismail Boztosun, Abdellatif Charafi, Dervis Boztosun

New RBF Collocation Methods and Kernel RBF with Applications

Abstract

A few novel radial basis function (RBF) discretization schemes for partial differential equations are developed in this study. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods. Based on the multiple reciprocity principle, the boundary particle method is introduced for general inhomogeneous problems without using inner nodes. For domain-type schemes, by using the Green integral we develop a novel Hermite RBF scheme called the modified Kansa method, which significantly reduces calculation errors at close-to-boundary nodes. To avoid Gibbs phenomenon, we present the least square RBF collocation scheme. Finally, five types of the kernel RBF are also briefly presented.

Wen Chen

Tuned Local Regression Estimators for the Numerical Solution of Differential Equations

Abstract

The first meshfree method was SPH. Central to its formulation is the statistical “kernel density estimator” which generalizes the histogram and was introduced by Rosenblatt in 1956. Stone introduced the improved “local regression estimator” in 1977, which generalizes the least-squares linear fit. Given a set of data, local regression finds a locally-weighted least-squares fit for a Taylor series expansion from a given evaluation point to nearby data points. The coefficients in the Taylor series fit are estimates for all derivatives of the data, computed simultaneously. The moving-least-squares estimator used in EFG, RKPM, and MLSPH is algebraically identical to the zeroth-derivative estimate of local regression. As with finite element and smoothing kernel interpolations, local regression estimates can be expressed as an expansion in shape functions. In the finite-element and meshfree methods, derivatives of the data are estimated by taking explicit derivatives of the shape function expansion. With local regression, the estimate of the derivative is equal to the derivative of the estimate only in the limit of small smoothing length. In familiar finite element and meshfree methods for solving differential equations, a Galerkin variational approach leads to a finite-dimensional system involving derivatives of the shape functions. Since local regression estimates all derivatives of the data simultaneously, constraining those estimates to satisfy a differential equation results in an estimate of a solution near the data. A constrained least-squares approach is the result. The paper proposes a formal setting for these ideas and supplies some very simple examples. Indications are made as to how the technique might be expanded.

Gary A. Dilts, Aamer Haque, John Wallin

Approximate Moving Least-Squares Approximation with Compactly Supported Radial Weights

Abstract

We use Maz’ya and Schmidt’s theory of approximate approximation to devise a fast and accurate approximate moving least-squares approximation method which does not require the solution of any linear systems. Since we use compactly supported weight functions, the remaining summation is also efficient. We compare our new algorithm with three other approximation methods based on compactly supported radial functions: multilevel interpolation, the standard moving least-squares approximation method, and a multilevel moving least-squares algorithm. A multilevel approximate moving least-squares approximation algorithm is also included.

Gregory E. Fasshauer

Coupling Finite Elements and Particles for Adaptivity: An Application to Consistently Stabilized Convection-Diffusion

Abstract

A mixed approximation coupling finite elements and mesh-less methods is presented. It allows selective refinement of the finite element solution without remeshing cost. The distribution of particles can be arbitrary. Continuity and consistency is preserved. The behaviour of the mixed interpolation in the resolution of the convection-diffusion equation is analyzed.

Sonia Fernández-Méndez, Antonio Huerta

A Hamiltonian Particle-Mesh Method for the Rotating Shallow-Water Equations

Abstract

A new particle-mesh method is proposed for the rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy a Kelvin circulation theorem. The generation of non-smooth components in the layer-depth is avoided by applying a smoothing operator similar to what has recently been discussed in the context of α-Euler models.

Jason Frank, Georg Gottwald, Sebastian Reich

Fast Multi-Level Meshless Methods Based on the Implicit Use of Radial Basis Functions

Abstract

A meshless technique is presented based on a special scattered data interpolation method which converts the original problem to a higher order differential equation, typically to an iterated Laplace or Helmholtz equation. The conditions of the original problem (interpolation conditions, boundary conditions and also the differential equation) are taken into account as special, non-usual boundary conditions taken on a finite set of collocation points. For the new problem, existence and uniqueness theorems are proved based on variational principles. Approximation properties are also analysed in Sobolev spaces. To solve the resulting higher order differential equation, robust quadtree/octtree-based multi-level techniques are used, which do not need any spatial and/or boundary discretisation and are completely independent of the original problem and its domain. This approach ca be considered as a special version of the method of radial basis functions (based on the fundamental solution of the applied differential operator) but avoids the solution of large and poorly conditioned systems, which significantly reduces the memory requirements and the computational cost as well.

Csaba Gáspár

A Particle-Partition of Unity Method-Part IV: Parallelization

Abstract

In this sequel to [7‐9] we focus on the parallelization of our multilevel partition of unity method for distributed memory computers. The presented parallelization is based on a data decomposition approach which utilizes a key-based tree implementation and a weighted space filling curve ordering scheme for the load balancing problem. We present numerical results in two and three dimensions with up to 128 processors and 42 million degrees of freedom. These results show the optimal scaling behavior of our algorithm in the discretization as well as the solution phase.

Michael Griebel, Marc Alexander Schweitzer

Some Studies of the Reproducing Kernel Particle Method

Abstract

Interests in meshfree (or meshless) methods have grown rapidly in the recent years in solving boundary value problems arising in mechanics, especially in dealing with difficult problems involving large deformation, moving discontinuities, etc. Rigorous error estimates of a meshfree method, the reproducing kernel particle method (RKPM), have been theoretically derived and experimentally tested in [13,14]. In this paper, we provide some further studies of the meshfree method. First, improved local meshfree interpolation error estimates are derived. Second, a new and efficient technique is proposed to implement Dirichlet boundary conditions. Numerical experiments indicate that optimal convergence orders are maintained for Dirichlet problems over higher dimensional domains. Finally, the meshfree method is applied to solve 4th-order equations. Since the smoothness of meshfree functions is the same as that of the window function, the meshfree method is a natural choice for conforming approximation of higher-order differential equations.

Weimin Han, Xueping Meng

Consistency by Coefficient-Correction in the Finite-Volume-Particle Method

Abstract

In the Finite-Volume-Particle Method, the weak formulation of a hyperbolic conservation law is discretized by restricting it to a discrete set of test functions. In contrast to the usual Finite-Volume approach, the test functions are chosen from a partition of unity with smooth and overlapping partition functions, which may even move along prescribed velocity fields. The information exchange between particles is based on standard numerical flux functions. Geometrical information, similar to the surface area of the cell faces in the Finite-Volume Method and the corresponding normal directions are given as integral quantities of the partition functions.

These quantities fulfill certain properties, which are heavily used in showing Lax-Wendroff consistency and stability estimates. We present a method which enforces the properties to be fulfilled in numerical computations. Moreover, we show a coupling among the coefficients and finally consistency of the method in space.

Dietmar Hietel, Rainer Keck

Do Finite Volume Methods Need a Mesh?

Abstract

In this article, finite volume discretizations of hyperbolic conservation laws are considered, where the usual triangulation is replaced by a partition of unity on the computational domain. In some sense, the finite volumes in this approach are not disjoint but are overlapping with their neighbors. This property can be useful in problems with time dependent geometries: while the movement of grid nodes can have unpleasant effects on the grid topology, the meshfree partition of unity approach is more flexible since the finite volumes can arbitrarily move on top of each other. In the presented approach, the algorithms of classical and meshfree finite volume method are identical - only the geometrical coefficients (cell volumes, cell surfaces, cell normal vectors) have to be defined differently. We will discuss two such definitions which satisfy certain stability conditions.

Michael Junk

An Upwind Finite Pointset Method (FPM) for Compressible Euler and Navier-Stokes Equations

Abstract

A Lagrangian scheme for compressible fluid flows is presented. The method can be viewed as a generalized finite difference upwind scheme. The scheme is based on the classical Euler equations in fluid mechanics, which concerns mainly non viscous problems. However, it can easily be extended to viscous problems as well. For the approximation of the spatial derivatives in the Euler equations, a modified moving least squares (MLS) method is used.

Jörg Kuhnert

Adaptive Galerkin Particle Method

Abstract

Adaptive procedures of a Galerkin particle method are presented. In the Galerkin particle method, the construction of shape functions and the domain integration of the weak form are entirely particle based. The stabilized conforming nodal integration is employed in the domain integration of the weak form. The Voronoi diagram used as the nodal representative domain in the stabilization of nodal integration is employed as the hierarchy for adaptive refinement. A recovery based error indicator is introduced in the adaptive analysis. Owing to the smooth shape functions in the Galerkin particle method, the stress recovery error indicator does not require any projection as was needed in the C° finite element methods.

Hongsheng Lu, Jiun-Shyan Chen

An Adaptivity Procedure Based on the Gradient of Strain Energy Density and its Application in Meshless Methods

Abstract

A gradient-based adaptation procedure is proposed in this paper. The relative error in total strain energy from two adjacent adaptation stages is used as a stop-criterion. The refinement-coarsening is guided by the gradient of strain energy density. The procedure is then implemented in Element-Free Galerkin method. Numerical examples are presented to show the performance of the proposed procedure.

Yunhua Luo, Ulrich Häussler-Combe

New Developments in Smoothed Particle Hydrodynamics

Abstract

Smoothed particle hydrodynamics (SPH) is a Lagrangian particle method which is said to be the first of the meshless methods. The characteristic of these methods is that the interpolation uses a set of disordered points and the equations of motion appear similar to the equations of motion of a set of particles. The generic name, Smoothed Particle methods seems to capture these features nicely. A useful review of SPH (Monaghan (1992)) gives the basic technique, and how it can be applied to numerous problems relevant to astrophysics. There are some useful SPH programs on the Web one of which is Gadget. This code was written by astrophysicists but it is of general interest.

Joseph J. Monaghan

The Distinct Element Method — Application to Structures in Jointed Rock

Abstract

This paper presents a brief review of the distinct element method (DEM) with particular emphasis on techniques for handling contact detection. In addition, various approaches for parallelization are considered. Our primary focus is on applying the DEM to simulations of the attack and defense of buried facilities. Some continuum approaches to this problem are discussed along with results from underground explosions. Finally, our DEM code is used to simulate dynamic loading of a tunnel in jointed rock and preliminary results are presented demonstrating the suitability of the DEM for this application.

Joseph Morris, Lew Glenn, Stephen Blair

Advance Diffraction Method as a Tool for Solution of Complex Non–Convex Boundary Problems—Implementation and Practical Application

Abstract

Varieties of meshless computational methods have been developed in recent years for the numerical solution of different problems of science and technology. In particular, these methods have found extensive application in fracture mechanics, where it is difficult to solve problems with standard finite elements.

Accurate solution of non-convex boundary problems, such as bodies that contains various types of flaws by meshless methods requires the construction of smooth continuous weight functions near discontinuities. Several methods have been recently

developed in order to handle problems with single or non-interacting discontinuities. In this work we extend the application of one of these methods, diffraction method, for the case of complex boundary problems, when a number of discontinuities can lie in the domain of influence of a single node.

The advanced diffraction method is useful for many applications, especially for the case of strongly interacting flaws which can be caused by stress corrosion cracking, creep fatigue, crack development, and cracking problems in composite materials. The improved method is able to significantly reduce computational efforts for this class of problems and so eliminate the need for extensive nodal refinement.

The general algorithm based on the Element Free Galerkin (EFG) approximation will be presented in the article. The weight functions will be constructed by the advanced diffraction method. Several numerical examples involving various crack interactions will be computed and analyzed.

Boris Muravin, Eli Turkel

On the Stochastic Weighted Particle Method

Abstract

The stochastic weighted particle method is one of the particle methods recently developed to approximate the solution of the Boltzmann equation, one of the well known kinetic equations. The main idea is to use random weight transfer between particles during collisions. In order to reduce the stochastic fluctuations, this method provides a way to increase the number of particles. But if the additional particles cannot be compensated in some natural way, then the number of particles should be reduced. To improve the method for long time intervals, two reduction procedures are proposed. One of them is based on an appropriate clustering of the particle system in the velocity space, and the other one is a recently developed method based on the weight selection of the particles. Some theoretical and numerical aspects are presented.

Endar H. Nugrahani, Sergej Rjasanow

The SPH/MLSPH Method for the Simulation of High Velocity Concrete Fragmentation

Abstract

The topic of this paper is the application of the SPH/MLSPH-method to high velocity concrete fragmentation. After a short review of the SPH/MLSPH method, a beam under static concentric loading and linear elastic material behavior is considered to show the advantage of MLSPH in opposite to SPH. A constitutive law for concrete taking into account the dynamic strength increase under high velocity loading is briefly proposed. Finally, the application of the SPH/MLSPH-Code in conjunction with this constitutive law for concrete onto two concrete slabs under contact detonation is discussed. The SPH and MLSPH-results obtained with different particle number and smoothing lengths are compared with the experimental results.

Timon Rabczuk, Josef Eibl, Lothar Stempniewski

Stability of DPD and SPH

Abstract

It is shown that DPD (Dual Particle Dynamics) and SPH (Smoothed Particle Hydrodynamics) are conditionally stable for Eulerian kernels and linear fields. This result is important because it is highly desirable to move and change neighbors where the material deformation is large. For higher dimensions (than 1D), stability for general neighborhoods is shown to require a two-step update, such a predictor-corrector. Co-locational methods (all field variables calculated on every particle) benefit from the completeness property also. We show that SPH with corrected derivatives is conditionally stable. Linear completeness of interpolations is shown to assert itself as a powerful ally with respect to stability as well as accuracy.

Philip W. Randles, Albert G. Petschek, Larry D. Libersky, Carl T. Dyka

A New Meshless Method — Finite-Cover Based Element Free Method

Abstract

Presented in the paper is a so-called finite-cover-based element-free method that is aimed to solve both continuous and discontinuous deformation problems in a mathematically consistent framework as manifold method, but no requiring mesh generation. The method is mathematically based on finite circular cover numerical technique and multiple weighted moving least square approximation. In this method, overall volume of materials is overlaid by a series of overlapped circular mathematical covers. While cut by joints, interfaces of different media and physical boundaries, a mathematical cover may be divided into two or more completely disconnected parts that are defined as physical covers. Discontinuity of materials is characterized by discontinuity of physical cover instead of disconnection of influence support. Hence, influence domain, i.e. mathematical cover can be kept regular even in a discontinuous problem. On a set of physical covers containing unknown point under consideration, the multiple weighted moving least square approximation in conjunction with cover weighting functions defined on each mathematical cover is used to determine shape functions of the unknown point for variational principle. Afterwards, discrete equations of the boundary-value problem with discontinuity can be established using variational principle. Through numerical analyses, it is shown that the proposed method that shares successfully advantages of both the manifold method and mesh-free methods is theoretically rational and practically applicable.

Rong Tian, Maotian Luan, Qing Yang

Finite Pointset Method Based on the Projection Method for Simulations of the Incompressible Navier-Stokes Equations

Abstract

A Lagrangian particle scheme is applied to the projection method for the incompressible Navier-Stokes equations. The approximation of spatial derivatives is obtained by the weighted least squares method. The pressure Poisson equation is solved by a local iterative procedure with the help of the least squares method. Numerical tests are performed for two dimensional cases. The Couette flow, Poisseulle flow and the driven cavity flow are presented. The numerical solutions are obtained for stationary as well as instationary cases and are compared with the analyatical solutions for channel flows. Finally, the driven cavity flow in a unit square is considered and the stationary solution obtained from this scheme is compared with that from the finite element method!

Sudarshan Tiwari, Jörg Kuhnert

LPRH — Local Polynomial Regression Hydrodynamics

Abstract

Local Polynomial Regression (LPR) is a weighted local least-squares method for fitting a curve to data. LPR provides a local Taylor series fit of the data at any spatial location. LPR provides estimates not only to the data, but also to derivatives of the data. This method is contrasted to the method of Moving Least Squares (MLS) which only provides a functional fit for the data. To obtain derivatives using MLS, one would be required to take analytic derivatives of the MLS functional fit. Since differentiation is known to be an unsmoothing operation, the derivatives obtained in MLS are thus less smooth than LPR derivatives. This fact has great implications for the stability of numerical methods based on MLS and LPR.

MLS and LPR can be directly used in a differential equation to provide a numerical scheme that mimics finite-differences. LPR was found to be much more stable than MLS in such a setting. However, these numerical methods cannot accurately solve nonlinear PDE’s in this fashion.

Particle or mesh-free methods for hydrodynamics typically use artificial viscosity to stabilize themselves when shocks are present. LPR can be used to solve the equations of hydrodynamics (Euler equations) without artificial viscosity. The Van Leer flux splitting scheme is used in conjunction with LPR to provide a stable and robust solution to the Euler equations. Numerical solutions are computed on both fixed and moving particle distributions.

John F. Wallin, Aamer Haque

On Multigrid Methods for Generalized Finite Element Methods

Abstract

This paper reports investigations on how multigrid methods can be applied for the solution of some generalized finite element methods based on the partition of unity technique. One feature of the generalized finite element method is that the underlying algebraic system is often singular due to the overlapping from the partition of unity. While standard iterative methods such as the conjugate gradient method, Jacobi, Gauss-Seidel methods, multigrid methods and domain decomposition methods are still convergent for this type of singular systems, we observe that a standard multigrid method does not converge uniformly with respect to mesh parameters. Using a simple model problem, we will carefully investigate why these method do not work. We will then propose a multigrid method that does converge uniformly as in the standard finite element method.

Jinchao Xu, Ludmil T. Zikatanov

The Convergence of the Finite Mass Method for Flows in Given Force and Velocity Fields

Abstract

The finite mass method is a new Lagrangian method to solve problems in continuum mechanics, primarily to simulate compressible flows. It is directly founded on a discretization of mass, not of space as with classical discretization schemes. Mass is subdivided into small mass packets of finite extension each of which is equipped with finitely many internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can change their shape to follow the motion of the fluid. In the present paper, second order convergence of the method is shown for the case of flows in a given velocity field or solely driven by external forces.

Harry Yserentant

Survey of Multi-Scale Meshfree Particle Methods

Abstract

A multiscale meshfree particle method is developed, which includes recent advances in SPH and other meshfree research efforts. Key features will include linear consistency, stability, and both local and global conservation properties. In addition, through the incorporation of Reproducing Kernal Particle Method (RKPM), standard moving least squares (MLS) enhancement and wavelet techniques, the method have the flexibility of resolving multiple scales in the solution of complex, multiple physics processes. We present the application of this approach in the following areas: 1) simulations on propagation of dynamic fracture and shear band; 2) impact and penetration; 3) fluid dynamics and 4) nano-mechanics.

Lucy T. Zhang, Wing K. Liu, Shao F. Li, Dong Qian, Su Hao

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