Comparison of local weak and strong form meshless methods for 2-D diffusion equation

Abstract

A comparison between weak form meshless local Petrov–Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.

Introduction

In recent years, a number of meshless approaches have been developed for the numerical solution of partial differential equations (PDEs) to circumvent the problem of polygonisation encountered in the classical, mesh-based numerical methods, such as the finite difference (FDM) [1] and finite volume methods (FVM) [2], finite element method (FEM) [3] or boundary domain integral method (BDIM) [4]. The “mesh” denotes the connectivity between the corresponding neighboring nodes obtained by some sort of spatial discretization. For non-structured 3-D geometries from real world the mesh construction is one of the most cumbersome step in the entire numerical solution process [5].

Instead of using a mesh, a set of geometrically unconnected nodes can be used for the global domain discretization, resulting in the meshless methods (MLM) (also referred to as mesh-free methods). Initial concepts of MLM were published in the seventies with the smoothed particle hydrodynamics (SPH) [6], based on the combination of weak-form and collocation techniques. Many variants of the meshless methods have been developed later, based on different strong/weak formulations and approximation/interpolation techniques. Diffuse element method (DEM) [7], element-free Galerkin method (EFG) [8], reproducing kernel particle method (RKPM) [9], hp-cloud method [10], partition of unity FEM (PUFEM) [11], meshless Galerkin method using radial basis functions (MGRBF) [12], [13], [14], and others are based on the global weak-form. These methods became widely used in different areas of application. General finite difference method (GFDM) [15], [16] with arbitrary mesh, radial basis function collocation methods (RBFCM) [17] and finite point method (FPM) [18], which are based on collocation, are meshless methods devised from the global strong-form. In spite of the simple formulation, additional strategies are needed for their stabilization [19]. All above MLM methods do not need the mesh for the construction of the trial function but by weak-form methods a simple background mesh is still needed for the weighted residual integration. Detailed information on a variety of meshless methods can be found in the review papers [20], [21].

An alternative formulation of the meshless methods is based on the local weak-form. The integration can now be performed in independent quadrature subdomains, therefore no background mesh is needed. Interpolation [22], [23] or moving least squares (MLS) approximation of nodal parameters from subdomains can be used for the construction of shape functions. Even though the interpolation simplifies the implementation of the essential boundary conditions, there is no general method known to solve the possible singularities of the interpolation coefficient matrix in case of degenerated distribution of nodes or to ensure the continuity of the solution. On the other hand, MLS is less influenced by inappropriate nodal distribution in the support domain because it is over-determined by default. It is an essential approach that guarantees the locality of many meshless methods. The shape functions are built from the weighted contributions of a certain number of the nearest nodes [24], [25]. However, the approximative nature of the MLS trial functions makes it more difficult to impose essential boundary conditions because the trial functions do not pass through the nodal values.

The meshless local Petrov–Galerkin method (MLPG) [26], [27] is the earliest representative of the “truly” meshless methods. Several variants of MLPG differ in the construction of trial functions and the use of different test functions. Note that both of them are local. We will further test the MLPG1, a variant of MLPG [28], which uses the MLS weight function as a test function over a square quadrature subdomain. New methods that combine previous knowledge from FEM and BEM with the MLPG approach have been proposed in [29]. Several convergence issues have been addressed [28], [12], [30], mostly in connection with the structural analysis, and often on smaller systems with a uniform distribution of nodes. The convergence of MLPG methods depends on the MLS approximation accuracy and stability and on the integration accuracy of the local weak-form, all three depending on the distribution of discretization nodes [31], [32]. Further studies have confirmed that MPLG is a general concept that can be applied in various fields of application [33], [34], [35], [36], [37].

An alternative, much simpler local meshless approach, is Diffuse Approximate Method (DAM) [38]. Similar to the MLPG, in the DAM, the local trial functions are constructed through the MLS approach over the local support domain. In the DAM “local” refers to the locality of the MLS trial functions. The conceptual difference is in the treatment of partial differential operations. The DAM uses strong-form and thus no integration is required. All differential operations are performed by straightforward application of the differential operator on the trial function.

To our knowledge this is the first work that compares errors of two meshless local methods MLPG and DAM, with most of method-specific parameters the same, e.g. local subdomain, base functions, nodal distribution. Thus the comparison of the weak and strong solution principles remains the primary focus of the present paper. As a reference, we add also the errors of the two mesh-based methods, FDM and FEM, obtained under the same circumstances. The meshless methods are claimed to perform well in situations with complicated geometry and non-uniform node arrangement resulting in smoother solution than mesh-based methods. However, since these methods are in development, they are in the great majority of numerical simulations [39], [40], [14] demonstrated only on simple geometries with regular node arrangements. Also direct comparisons between different meshless methods are very rarely found in the literature. The main incitement of this paper is in contributing towards a better understanding of the meshless approaches. Respectively, we compare the well known meshless local Petrov–Galerkin method (MLPG), which is based on the week formulation, and the local diffuse approximate method (DAM), based on the strong formulation. The methods are tested in solving the dimensionless diffusion equation. We use uniform and non-uniform node arrangements and simple as well as more complicated geometry for assessment. Errors were evaluated in discretization nodes on a square domain and on a domain with a hole with Dirichlet boundary conditions. Our analysis has taken into account different spatial domain discretization ranging from nodes on a simple orthogonal mesh, through randomized nodes with an approximately even density, to nodes on a fully optimized triangular meshes.

The structure of the paper is as follows. In the next section a short background on the meshless approaches used for the solution of a diffusion equation is given. Then the test conditions are described in more details. In Section 4, the obtained results are presented and analyzed. The paper concludes with the discussion on the obtained results and future work.