Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations

doi:10.1016/j.enganabound.2008.03.004

Engineering Analysis with Boundary Elements

Volume 33, Issue 1, January 2009, Pages 83-92

https://doi.org/10.1016/j.enganabound.2008.03.004 Get rights and content

Abstract

A meshfree method namely, discrete least squares meshless (DLSM) method, is presented in this paper for the solution of elliptic partial differential equations. In this method, computational domain is discretized by some nodes and the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is defined as the sum of squared residuals of the differential equation and its boundary condition calculated at a set of points called sampling points, generally different from nodal points. A moving least squares (MLS) technique is used to construct the shape functions. The proposed method automatically leads to symmetric and positive-definite system of equations. The proposed method does not need any background mesh and, therefore, it is a truly meshless method. The solutions of several one- and two-dimensional examples of elliptic partial differential equations are presented to illustrate the performance of the proposed method. Sensitivity analysis on the parameters of the method is also carried out and the results are presented.

Introduction

There is a growing interest in the development of meshless methods for numerical solution of partial differential equations (PDEs), as meshless techniques do not require the generation of a mesh, which can be a costly and time consuming for two- and three-dimensional problems with complex geometries. Instead, meshless techniques require only a scattered set of nodes representing the domain of interest. No connectivity information among the scattered set of nodes is required, unlike finite element, boundary element or classical finite difference techniques. Meshless methods have become very attractive and efficient for the development of adaptive methods for the solution of boundary value problems because nodes can be easily added and deleted without a burdensome remeshing of the entire problem domain [1]. The main advantage of meshless methods is to get rid of or at least alleviate the difficulty of meshing and remeshing the entire problem domain.

Over the last decade, meshless methods for the solution of PDEs have become increasingly popular. The main idea of these methods is to approximate the unknown field by a linear combination of shape functions built without having recourse to mesh the domain. Instead, nodes are scattered in the domain and a certain weight function with a local support is associated with each of these nodes. The shape function associated with a given node is then built considering the weight functions whose support overlaps the weight function of this node. Several meshless methods have been proposed in the literature, many of them in the last decade. The meshless methods have been developed in recent years are smoothed particle hydrodynamics (SPH) [2], the element free Galerkin (EFG) method [3], [4], the reproducing kernel particle (RKP) method [5], the finite point (FP) method [6], the hp clouds method [7], meshless local Petrov–Galerkin (MLPG) method [8], [9], [10], [11], local boundary integral equation (LBIE) method [12], [13], [14], finite cloud method [15].

A multiquadric (MQ) collocation method was suggested by Kansa [16] based on a standard collocation method in which the number of collocation points were equal to the number of nodes used to discretize the differential equations. A FP, least squares collocation meshless (LSCM) method was proposed in [17]. Except for the nodes, which were used to construct the shape functions, a number of auxiliary points were also used. The system of discretized equations was constructed by collocating the differential equation and the boundary conditions at these auxiliary points. The number of resulting equations was consequently greater than that of unknowns. Therefore, the set of equations was solved using least squares method. Numerical studies showed that LSCM is stable and efficient with high accuracy, but its coefficient matrix is asymmetric. In a point weighted least squares (PWLS) method introduced in [18], the weighted least squares technique is used to obtain the solution of the problem by minimizing a functional defined as the squared weighted summation of the residuals at nodal points constructed using radial point interpolation method (RPIM) shape functions. A meshless Galerkin least squares (MGLS) method proposed by Pan et al. [19] combines the advantage of Galerkin and collocation methods. In this approach, the Galerkin method is applied on the boundary, whereas least squares technique is applied in the interior domain. A meshless weighted least squares (MWLS) method was also proposed in [20]. Numerical studies show high accuracy, high efficiency and stability of the method on mesh of regular nodes. The method, however, lacked enough accuracy when it is applied to mesh of irregular nodes.

A discrete least squares meshless (DLSM) method with sampling points is presented in this paper. The method is based on minimizing the least squares functional defined by the sum of squared residuals of the differential equation and its boundary condition at some points, called sampling points, over the domain and its boundary. The sampling points are generally considered to be different from the nodal points used to discretize the problem domain. The method provides symmetric coefficient matrix and preserves high accuracy and high stability even for irregularly distributed nodes. The method does not need any background mesh. Therefore, it can be considered a truly meshless method. In what follows, the construction of moving least squares (MLS) shape functions is first explained. DLSM method with sampling points for discretization of the governing differential equation is then explained. And finally some numerical examples are solved and the results are presented.

Section snippets

Moving least squares shape functions

The most frequently used approximation in meshless methods is the MLS interpolant in which the unknown function φ is approximated by $φ (X) = \sum_{i = 1}^{m_{p}} p_{i} (X) a_{i} (X) = P^{T} (X) a (X)$ where P^T(X) is a polynomial basis in the space coordinate X, m_p is the total number of the terms in the basis, and a(X) is the vector of coefficients. The polynomial basis of first and second orders in one and two dimensions can be specified as follows: $P^{T} = [\begin{matrix} 1 & x \end{matrix}]$ for basis of first order in one dimension, $P^{T} = [\begin{matrix} 1 & x & y \end{matrix}]$ for basis of 1st order

Discrete least squares meshless (DLSM) method

A general form of differential equations is used to present the procedure of the DLSM method in establishing discretized system of equations. The PDE (ordinary) and its boundary condition are written in the form $I (φ) + f = 0 in Ω$ $φ - \bar{φ} = 0 on Γ_{2}$ $ℜ (φ) - \bar{t} = 0 on Γ_{1}$ where Γ₂ and Γ₁ are Dirichlet and Neumann boundaries, respectively, $I$ and $ℜ$ are (partial) differential operators, and f represents external forces or source term in the problem domain. The residual of the differential equation at a typical point k is

Numerical examples

In this section some one- and two-dimensional numerical examples are solved using regular and irregular mesh of nodes to illustrate the efficiency and accuracy of the proposed method. Following error norms are defined as error indicators in this paper: $e_{0} = \frac{\sum_{i = 1}^{n_{e}} | u_{i}^{exact} - u_{i}^{num} |}{\sum_{i = 1}^{n_{e}} | u_{i}^{exact} |}$ $e_{1 x} = \frac{\sum_{i = 1}^{n_{e}} | u_{i, x}^{exact} - u_{i, x}^{num} |}{\sum_{i = 1}^{n_{e}} | u_{i, x}^{exact} |}$ $e_{1 y} = \frac{\sum_{i = 1}^{n_{e}} | u_{i, y}^{exact} - u_{i, y}^{num} |}{\sum_{i = 1}^{n_{e}} | u_{i, y}^{exact} |}$

In which e₀, e₁_x and e₁_y are the error norms for the solution and its derivatives with respect to x and y,

Conclusion

A truly meshless method namely DLSM method is presented in this paper. The method is based on the minimization of a least squares functional defined as the sum of the squared residuals of the differential equation and its boundary conditions calculated at sampling points chosen on the problem domain and its boundaries. The minimization is carried out with respect to the nodal parameters associated with nodal points used to discretize the problem domain and its boundaries. The MLS method is used

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The method does not require any background mesh, therefore, it is considered as a truly meshless method. In cases with irregular distributions of nodes, the accuracy and stability of DLSM can be improved by providing a set of auxiliary points called sampling points [14]. The suggested method utilizes the Moving Least Squares (MLS) shape functions and minimizes least-squares functional with respect to nodal parameters.
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In this paper, an improved node moving technique was developed for adaptive solution of some flow problems. Collocated Discrete Least Squares Meshless (CDLSM) method was applied for simulation. This method is a truly meshless method and also enjoys the symmetric and positive-definite property. Then, an improved node moving technique was developed based on the spring analogy. In this mechanism, each node was assumed to be connected to its neighboring nodes via some virtual springs. Then, higher values of stiffness were allocated to the springs located at higher error areas. In the previously published works, the corresponding system of equations was assumed to be linear. In this work, first, it was shown that this assumption may lead to inaccurate results in some cases and then, an improved method was proposed considering the nonlinear nature of the system of equations. Some numerical examples were used to illustrate the ability of the proposed node moving technique for some simple examples and benchmark problems in the Computational Fluid Dynamics (CFD) context. The Results of the study showed a considerable improvement in comparison with those of previously published works.
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View full text

Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations

Abstract

Introduction

Section snippets

Moving least squares shape functions

Discrete least squares meshless (DLSM) method

Numerical examples

Conclusion

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

Comput Math Appl

Arbitrary placement of secondary nodes, and error control, in the meshless local Petrov–Galerkin (MLPG) method

CMES: Comput Modeling Eng Sci

Smoothed particle hydrodynamics: theory and applications to non-spherical stars

Mon Not R Astron Soc

Element free Galerkin methods

Int J Numer Methods Eng

Reproducing kernel particle methods

Int J Numer Methods Fluids

A finite point method in computational mechanics. Applications to convective transport and fluid flow

Int J Numer Methods Eng

The meshless local Petrov–Galerkin (MLPG) approach for solving problems in elasto-statics

Comput Mech

New concepts in meshless methods

Int J Numer Methods Eng