Meshless Finite Difference Method Research Articles

Coupling finite element method with meshless finite difference method by means of approximation constraints

Coupling finite element method with meshless finite difference method by means of approximation constraints

Error bounds for a least squares meshless finite difference method on closed manifolds

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation -Delta _{mathcal {M}} u + u = f on the 2- and 3-spheres, where Delta _{mathcal {M}} is the Laplace-Beltrami operator.

Computational aspects of nonlinear and multiscale analyses by the multipoint meshless FDM

The paper focuses on the recently developed higher order extension of the MFDM – the multipoint meshless finite difference method and introduces the computational schemes and algorithms of the method application to the boundary value problems, especially to the nonlinear and multiscale analyses. There are two basic versions of the multipoint approach that allow accurate and effective solving of engineering problems. The main advantage of the multipoint general version is its generality – the basic relations of derivatives from the unknown function depend on the domain discretization only and are independent of the type of problem being solved. This feature allows to divide the multipoint computational strategy into two stages and is advantageous from the calculation efficiency point of view. The multipoint method algorithms, applied to such engineering problems as two-scale analysis of heterogeneous materials and nonlinear analysis, are developed and briefly presented. This manuscript is an extension of the paper Multipoint Meshless FD Schemes Applied to Nonlinear and Multiscale Analysis by I. Jaworska (2022) published in Lecture Notes in Computer Science [1]. In this extended version computational aspects of the algorithms are deepened, the subsections explaining specific features of the multipoint meshless approach application to the numerical homogenization and nonlinear analysis are included, and finally, the advantages and disadvantages of the method are discussed. The paper is illustrated with several examples (completed in the extended version) of the multipoint numerical analysis.

A new meshless method to solve the two-phase thermo-hydro-mechanical multi-physical field coupling problems in shale reservoirs

The efficient development of shale reservoirs relies on multi-stage hydraulic fracturing technology in horizontal wells involving the difficulty of multi-scale flow for either gas or water phases, which is characterized by the changes in the stress and temperature fields. The original simulation methods are unable to accurately solve the multi-physics field-coupled problems under such complex mechanical conditions or have low stability for two-phase gas-water problems. For this purpose, this study conducted the shale gas production simulation by the meshless finite difference method (GFDM) for the first time to ensure the stability and accuracy of calculations. Bases on the previous multi-sector coupled physical model for shale gas development and proposes a coupled thermo-hydro-mechanical multi-physical field mathematical model considering two-phase gas-water flow. The model is postulated based on different sectors considering various characteristics of gas/water flow, namely desorption attributes, deformation degree and heat transfer characteristics. This method was used to calculate the variation of the multi-physical field and compare the properties of gas-water flow within different reservoir conditions without any non-convergence situation during the calculation. The simulation results indicate that the stress field has a crucial effect on the yield, such that discard of its coupling causes the yield to be overestimated by 12.5%. In contrast, the temperature field has a marginal effect (less than 1%) and can be neglected according to the operating conditions. Furthermore, the stress field significantly characterizes the micro-fracture sector, which is also the major flow-contributor, such that the yield is overestimated by 11% in absence of its coupling, while the effect of stress field in the matrix sector is lower than 1%.

Higher order meshless approximation applied to Finite Difference and Finite Element methods in selected thermomechanical problems

This paper is focused on the application of higher order meshless schemes in a numerical analysis of selected thermomechanical problems. Numerical investigation is based upon meshless Finite Difference method as well as its combinations with Finite Element method, performed at two different levels of analysis. The first variant assumes conjugation of element and meshless approximation schemes, while in the second one, the problem domain is divided into several disjoint subdomains, with a parallel and independent operating of coupled methods in each subdomain.The most important advantage of the applied approximation technique is no requirements of modification or enhancement of the existing discretization model. Therefore, high approximation orders may be assumed without providing new nodes, elements and degrees of freedom, maintaining the entire numerical model as simple as possible. This approach is especially convenient in coupled multi-field 2D and 3D problems, for instance stationary and non-stationary thermoelastic ones. In those problems, standard higher order approximation techniques may lead to complex numerical models caused by the rapid growth of a number of degrees of freedom and ill-conditioned schemes.The proposed approach is derived for three dimensional non-stationary thermoelastic problems. Moreover, it is examined on variety of 2D and 3D benchmark examples and engineering applications. Both solution accuracy and convergence rate are taken into account. Obtained results are very promising as they reflect the competitiveness of the approach comparing to other commonly applied higher order approaches.

Higher order schemes introduced to the meshless FDM in elliptic problems

The research is focused on the development of the Meshless Finite Difference Method with higher order approximation schemes and its application in elliptic problems. On the contrary to the Finite Element Method and other meshless methods, the approximation order may be raised without introducing any new nodes or degrees of freedom. The main concept is based upon the consideration of additional correction terms of difference operators, generated by means of the Moving Weighted Least Squares technique. Higher order derivatives, included in those terms, are evaluated using basic discretization and approximation models, namely by the appropriate formulas composition and the primary numerical solution. Correction terms modify only right–hand sides of algebraic equations, which are solved iteratively. In such a manner, problems with ill–conditioned and singular finite difference schemes are avoided. This technique may be applied to elliptic problems posed in both local (strong) and global (weak variational) formulations.The paper is illustrated with results of selected one and two dimensional benchmark elliptic problems, with various geometrical shapes as well as several engineering applications. The attention is laid upon the accuracy of solution and its derivatives as well as convergence rates estimated on the set of regular meshes and irregular clouds of nodes. Moreover, a posteriori error estimates, based upon higher order solution, are taken into account.

Research using the Wendland RBF for the RBF-FD method to solve the Poisson equation in 3D

In recent years, the meshless finite difference method based on radial basis functions (RBF-FD) of solving partial differential equation in 3D has been studied by many scientists. To find the RBF-FD weight vector, the authors used the Power RBF interpolation, which does not depend on the shape parameter. This paper presents the research results of using Wendland RBF interpolation to find the weight vectors for the RBF-FD method to solve Poisson equations in 3D. The numerical experiments showed that the solution of the RBF-FD method using the Wendland function has good accuracy compared to the solution of the finite element method (FEM - Finite Element Method).

Meshless finite difference method with B-splines for numerical solution of coupled advection-diffusion-reaction problems

In this paper, a meshless finite difference (FD) method with B-splines is presented for numerical solution of coupled advection-diffusion-reaction (ADR) problems. The proposed method is mathematically simple to program and truly meshless. It combines the approximation capability and high resolution of B-splines and the ease of implementation of differential quadrature technique to discretize the system of equations. The presence of mesh is replaced by overlapping local domains containing of regular or scattered nodes, in which the solution inside is approximated by B-spline basis functions fashioned in local collocation. The fourth order Runge-Kutta (RK) method is employed for time integration. The effectiveness of the proposed method is shown by solving several coupled ADR problems, including cross reaction-diffusion, chemotaxis, pattern formation and tumor invasion into surrounding healthy tissue. Numerical results demonstrate high resolution, stability and robustness of the proposed method. It captures the emergence and evolution of sharp fronts in the problems well, thus depicting dynamics of the problems accurately. The method also places no restriction on the shape of computational domains. The present method's convergence rate is also elucidated empirically in this study and shown to be high. It is shown that the proposed method is an accurate and effective solver for coupled advection-diffusion-reaction problems in two-dimensional domains.

Generalization of the Multipoint meshless FDM application to the nonlinear analysis

The paper focuses on the new Multipoint meshless finite difference method, following the original Collatz higher order multipoint concept and the essential ideas of the Meshless FDM. The method was formulated, developed, and tested for various boundary value problems. Generalization of the multipoint method application to nonlinear analysis is the purpose of this research.The first attempt of the multipoint technique application to the geometrically nonlinear problems was successfully done recently. The case of physically nonlinear problem is considered in this paper. Several benefits of the proposed approach are highlighted, numerical algorithm and selected results are presented, and application of the multipoint method to nonlinear analysis is summarized.

A meshless finite difference method for elliptic interface problems based on pivoted QR decomposition

We propose to solve elliptic interface problems by a meshless finite difference method, where the second order elliptic operator and jump conditions are discretized with the help of the QR decomposition of an appropriately rescaled multivariate Vandermonde matrix with partial pivoting. A prescribed consistency order is achieved on irregular nodes with small influence sets, which allows to place the nodes directly on the unfitted interface and leads to sparse system matrices with the density of nonzero entries comparable to the density of the system matrices arising from the mesh-based finite difference or finite element methods. Numerical experiments on a number of standard test problems with known solutions demonstrate convergence orders up to O(h6) for both the approximate solution and its gradient, and a robust performance of the method in the case when the interface is known inaccurately.

Solving Boussinesq equations with a meshless finite difference method

This paper mainly focus on presenting a newly-developed meshless numerical scheme, named the generalized finite difference method (GFDM), to efficiently and accurately solve the improved Boussinesq-type equations (BTEs). Based on the improved BTEs, the wave propagated over a flat or irregular bottom topography is described as a two-dimensional horizontal problem with nonlinear water waves. The GFDM and the 2nd-order Runge-Kutta method (RKM) were employed for spatial and temporal discretizations for this problem, respectively. The ramping function and the sponge layer, combing in this proposed scheme, were adopted for incident and outgoing waves, respectively. As one of domain-type meshless methods, GFDM can improve the numerical efficiency due to avoiding time-consuming meshing generation and numerical quadrature. Furthermore, the partial derivatives of Boussinesq equations can be transformed as linear combinations of nearby function values by the moving-least-squares method of the GFDM, simplifying the numerical procedures. Specifically, GFDM is suitable for complex fluid field with some irregular boundaries because of the flexible distribution of nodes. Four numerical examples were selected to verify the accuracy and applicability in the improved BTEs of the proposed meshless scheme. The results were compared with other numerical predictions and experimental observations and good agreements were depicted.

New local radial point interpolation-FD methods for solving fractional diffusion and damped-wave problems

A new approach is proposed for developing numerical schemes for the solution of fractional diffusion and diffusion-wave equations. The novelty lies in the construction of analytical shape functions in a three-point local radial point interpolation method. The gain is that numerical computations of matrix entries are not required and as in the finite difference setting, the availability of matrix coefficients in closed-form facilitates stability and convergence analyses. For the fractional diffusion equation, a weak-form formulation is used to develop an implicit meshless finite difference method. The unconditional stability and convergence of the scheme is theoretically established and numerical examples are given to illustrate its second-order accuracy in space. The expected time convergence and the dependence of the convergence on the multiquadric shape parameter are also investigated. The technique is applied to develop numerical schemes for the fractional diffusion-wave equation. For this problem, the new scheme is shown to perform well in terms of both temporal and spatial accuracies when compared to some existing methods.

Octant-Based Stencil Selection for Meshless Finite Difference Methods in 3D

We study a simple meshless stencil selection algorithm in 3D for supporting the meshless finite difference method based on radial basis functions (RBF-FD) to solve the Dirichlet problem for the Poisson equation. Our numerical experiments show that the proposed method produces solutions that differ very little from the solutions by the finite element method with Courant’s piecewise linear basis functions, when the domain is discretized by the vertices of the respective tetrahedral triangulation.

A Meshless Finite Difference Method Based on Polynomial Interpolation

The finite difference (FD) formula plays an important role in the meshless methods for the numerical solution of partial differential equations. It can be created by polynomial interpolation, however, this idea has not been widely used due to the complexity of multivariate polynomial interpolation. Instead, radial basis functions interpolation is widely used to generate the FD formula. In this paper, we first propose a simple and practicable node distribution, which makes it convenient for one to choose the interpolation node set that guarantee the unique solvability of multivariate polynomial interpolation. The greatest advantage of the interpolation node set is that we can only face triangular matrix in order to obtain the Lagrange basis polynomials by the constructed basis polynomials. We then use Taylor’s formula to establish error estimates of the FD formula based on polynomial interpolation. We finally give some numerical experiments for the numerical solutions of the Poisson equation and the heat equation.

Higher order meshless schemes applied to the finite element method in elliptic problems

This paper presents selected approximation techniques, typical for the meshless finite difference method (MFDM), although applied to the finite element method (FEM). Finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes, based upon the correction terms of a simple difference operator. Those terms consist of higher order derivatives, which are evaluated by means of the appropriate formulas composition as well as a numerical solution, which corresponds to the primary interpolation order, assigned to element shape functions. Correction terms modify the right-hand sides of algebraic FE equations only, yielding an iterative procedure. Therefore, neither re-generation of the stiffness matrix nor introduction of any additional nodes and/or degrees of freedom is required. Such improved FE-MFD solution approach allows for the optimal application of advantages of both methods, for instance, a high accuracy of the nodal FE solution and a derivatives’ super-convergence phenomenon at arbitrary domain points, typical for the meshless FDM. Existing and proposed higher order techniques, applied in the FEM, are compared with each other in terms of the solution accuracy, algorithm efficiency and computational complexity.In order to examine the considered algorithms, numerical results of several two-dimensional benchmark elliptic problems are presented. Both the accuracy of a solution and the solution’s derivatives as well as their convergence rates, evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes, are taken into account.

A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

SummaryThis paper presents a numerical framework for the highly accurate solutions of transient heat conduction problems. The numerical framework discretizes the temporal direction of the problems by introducing the Krylov deferred correction (KDC) approach, which is arbitrarily high order of accuracy while remaining the computational complexity same as in the time‐marching of first‐order methods. The discretization by employing the KDC method yields a boundary value problem of the inhomogeneous modified Helmholtz equation at each time step. The meshless generalized finite difference method (GFDM) or meshless finite difference method (MFDM), a meshless method, is then applied to the solution of resulting boundary value problems at each time step. Six numerical experiments in one‐, two‐, and three‐dimensional cases show that the proposed hybrid KDC‐GFDM scheme allows big time step size for a long‐time dynamic simulation and has a great potential for the problems with complex boundaries. In addition, some comparisons are also presented between the present method, the COMSOL software, and the GFDM with implicit Euler method.

Determination of Overhead Power Line Cables Configuration by FEM and Meshless FDM

The paper addresses numerical modeling of overhead power line cables subjected to static loads. Cable extensibility, their large 3D displacements as well as an interaction with strings of insulators and elastic towers are considered. A 2D exact closed solution available for certain type of loading was generalized by us to a 3D displacement case and applied as one of the benchmarks, that were used in convergence, efficiency and accuracy studies of numerical methods to select the most reliable one. Various formulations of the problem (strong and weak in both the Bubnov and local Petrov–Galerkin MLPG-5 versions) as well as various solution approximation approaches were examined, namely the finite element method (FEM) with the higher order shape functions as well as for the first time the meshless finite difference method (MFDM). Validation done by comparison of the numerical results with in situ measurements confirmed that the assumed models are reliable.

ON SOME ASPECTS OF THE MESHLESS FDM APPLICATION FOR THE HETEROGENEOUS MATERIALS

The most commonly used engineering tool for numerical analysis of a variety of the heterogeneous materials is the finite element method. However, this paper focuses on the alternative approach based upon the meshless finite difference method (MFDM). The purpose of the work is to present some key features, as well as the selected results of the MFDM application for the heterogeneous material. The MFDM solution approach and its higher order extensions, e.g., the multipoint meshless method, may be used at both the macro and the micro levels in a two-scale analysis. At the macro level the heterogeneous material with inclusions spaced periodically was assumed. The values of effective material constants were determined by calculation at the micro level for a single representative volume element (RVE). The paper focuses on the numerical analysis, particularly on the examination of the influence of some factors, e.g., stencil configuration on solution quality based on the MFDM.

Estimation of A Posteriori Computational Error by the Higher Order Multipoint Meshless FDM

The main objective of this paper is to present a possibility of high quality a posteriori error evaluation based on reference solutions obtained by means of the new multipoint meshless finite difference method. Due to its higher order approximation, the multipoint results may be used as improved reference solution instead of the true analytical one to estimate the errors. Several types of a posteriori error estimators which can be used to evaluate the calculation error are described here. The results of selected numerical benchmarks are considered.

Coupling finite element method with meshless finite difference method in thermomechanical problems

This paper focuses on coupling two different computational approaches, namely finite element method (FEM) and meshless finite difference method (MFDM), in one domain. The coupled approach is applie...