Irregular Distribution Of Nodes Research Articles

A spatio-temporal fully meshless method for hyperbolic PDEs

We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear hyperbolic PDEs. The method is based on the moving least squares method, more precisely in the Generalized Finite Difference Method which allows us to select well-conditioned stars. Several 2D and 3D examples including the time variable are shown for both regular and irregular node distributions. The results are compared with explicit GFDM both in terms of errors and execution time.

Full-waveform inversion in acoustic-elastic coupled media with irregular seafloor based on the generalized finite-difference method

The marine environment can be considered as a fluid-solid coupled media governed by the acoustic-elastic coupled equation (AECE). For complex fluid-solid coupled models with irregular seabeds, the well-known finite-difference method (FDM) often causes ripple-shaped scattering due to the rectangular grid. To overcome this problem, a meshless method (MM), called the generalized finite-difference method (GFDM), for irregular node distributions is developed and applied to AECE forward modeling by using a body-fitted node-generation method, which provides robust node distributions for GFDM. Compared with the FDM, our GFDM improves modeling accuracy and eliminates divergence. Furthermore, our MM was applied to an AECE-based elastic full-waveform inversion (EFWI). Owing to the well-matched irregular seabed, our EFWI can invert the accurate elastic model for ocean-bottom cable and pressure data. Numerical examples demonstrate that this method can efficiently improve the numerical accuracy and build an accurate velocity model even using noisy data, which is helpful for high-precision seismic exploration in the complex marine environment.

A Novel Spatio-Temporal Fully Meshless Method for Parabolic PDEs

We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear parabolic PDEs. The method is based on a moving least squares method, more precisely, the generalized finite difference method (GFDM), which allows us to select well-conditioned stars. Several 2D and 3D examples, including the time variable, are shown for both regular and irregular node distributions. The results are compared with explicit GFDM both in terms of errors and execution time.

A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

Topology optimization of continuum structures using element free Galerkin method on irregular nodal distribution

In this paper, topology optimization of structures is achieved through integrating the element free Galerkin (EFG) method and the solid isotropic method with penalization (SIMP). The strain energy is minimized as objective function. In the proposed method the density is assigned to the Gauss points to obtain an acceptable resolution. Irregular distribution of nodes has also been used in this study. Several benchmark examples were applied to demonstrate the ability of the proposed algorithm and the obtained results compared to the other previous works.

Accuracy analysis of different Laplacian models of incompressible SPH method improved by using Voronoi diagram

The purpose of this paper is to investigate the effect of employment of Voronoi diagram drawing as an alternative approach to estimate the particle or nodal volume on accuracy of three well-known Laplacian operator models of incompressible SPH method (Hu and Adams in J Comput Phys 227:264–278, 2007; Schwaiger in Int J Numer Methods Eng 75:647–671, 2008; Shao and Lo in Adv Water Resour 26:787–800, 2003). In addition to this, the numerical performance of these modified Laplacian models is compared with that of another newly developed higher-order Laplacian model proposed by Shobeyri (Iran J Sci Technol Trans Civil Eng https://doi.org/10.1007/s40996-018-0226-9 , 2019). The numerical errors of the above Laplacian models for solving different 2-D elliptic partial differential equations are analyzed on a unit square computational domain discretized with highly irregular node distributions. In addition to this, the patch test for an arbitrary function is conducted to evaluate numerical behavior of the Laplacian models. The numerical results indicate that the three Laplacian models coupled with Voronoi diagram technique can get smaller errors compared with the standard models. The most significant finding of this study is that the Laplacian model proposed by Shobeyri (Iran J Sci Technol Trans Civil Eng https://doi.org/10.1007/s40996-018-0226-9 , 2019) produces lower numerical errors even in comparison with the improved models. Voronoi diagram technique with simple implementation can significantly improve numerical accuracy of the presented higher-order Laplacian models for arbitrary and highly irregular node configurations, and it is expected that this strategy is applicable for other SPH Laplacian models to obtain higher-accurate results.

Solving the telegraph equation in 2-D and 3-D using generalized finite difference method (GFDM)

In this paper it is shown the application of the generalized finite difference method (GFDM) for solving numerically the Telegraph equation in two and three-dimensional spaces. The explicit time discretization is used and for approximating the spatial variables a GFDM is applied. The GFDM is a truly meshless method. The possibility of using a nodal method allowing irregular distribution of nodes in a natural way is one of the main advantages of the generalized finite difference method (GFDM). The stability condition using von Neumann method is done, which is provided in Theorem 1. Some numerical results have been reported and compared with the exact solutions for showing the accuracy and efficiency of the proposed numerical scheme.

Accuracy analysis of different higher-order Laplacian models of incompressible SPH method

PurposeThis paper aims to examine the accuracy of several higher-order incompressible smoothed particle hydrodynamics (ISPH) Laplacian models and compared with the classic model (Shao and Lo, 2003).Design/methodology/approachThe numerical errors in solving two-dimensional elliptic partial differential equations using the Laplacian models are investigated for regular and highly irregular node distributions over a unit square computational domain.FindingsThe numerical results show that one of the Laplacian models, which is newly developed by one of the authors (Shobeyri, 2019) can get the smallest errors for various used node distributions.Originality/valueThe newly proposed model is formulated by the hybrid of the standard ISPH Laplacian model combined with Taylor expansion and moving least squares method. The superiority of the proposed model is significant when multi-resolution irregular node distributions commonly seen in adaptive refinement strategies used to save computational cost are applied.

Improving Accuracy of Laplacian Model of Incompressible SPH Method Using Higher-Order Interpolation

In this study, a new model of Laplacian operator is formulated as a hybrid of an incompressible SPH (I-SPH) method with Taylor expansion and moving least-squares method. Accuracy of the proposed Laplacian model in solving 2-D elliptic partial differential equations for a unit square computational domain is compared with the conventional I-SPH Laplacian operator. The results show significant improvement in accuracy for the proposed model on regular, highly irregular and multi-resolution irregular node distributions employed for computational domain discretization. The proposed Laplacian model because of notable accuracy can be applied for more efficient simulation of free surface flows.

The Application of the Generalized Finite Difference Method (GFDM) for Modelling Geophysical Test

The possibility of using a nodal method allowing irregular distribution of nodes in a natural way is one of the main advantages of the generalized finite difference method (GFDM) with regard to the classical finite difference method. Moreover, this feature has made it one of the most-promising meshless methods because it also allows us to reduce the time-consuming task of mesh generation and the numerical solution of integrals. This characteristic allows us to shape geological features easily whilst maintaining accuracy in the results, which can be a source of great interest when dealing with this kind of problems. Two widespread geophysical investigation methods in civil engineering are the cross-hole method and the seismic refraction method. This paper shows the use of the GFDM to model the aforementioned geophysical investigation tests showing precision in the obtained results when comparing them with experimental data.

Physics–Dynamics Coupling with Element-Based High-Order Galerkin Methods: Quasi-Equal-Area Physics Grid

Atmospheric modeling with element-based high-order Galerkin methods presents a unique challenge to the conventional physics–dynamics coupling paradigm, due to the highly irregular distribution of nodes within an element and the distinct numerical characteristics of the Galerkin method. The conventional coupling procedure is to evaluate the physical parameterizations ( physics) on the dynamical core grid. Evaluating the physics at the nodal points exacerbates numerical noise from the Galerkin method, enabling and amplifying local extrema at element boundaries. Grid imprinting may be substantially reduced through the introduction of an entirely separate, approximately isotropic finite-volume grid for evaluating the physics forcing. Integration of the spectral basis over the control volumes provides an area-average state to the physics, which is more representative of the state in the vicinity of the nodal points rather than the nodal point itself and is more consistent with the notion of a “large-scale state” required by conventional physics packages. This study documents the implementation of a quasi-equal-area physics grid into NCAR’s Community Atmosphere Model Spectral Element and is shown to be effective at mitigating grid imprinting in the solution. The physics grid is also appropriate for coupling to other components within the Community Earth System Model, since the coupler requires component fluxes to be defined on a finite-volume grid, and one can be certain that the fluxes on the physics grid are, indeed, volume averaged.

Element-free Galerkin method for numerical simulation of sediment transport equations on regular and irregular distribution of nodes

In this paper, the numerical simulation of the bed-load sediment transport using Element-Free Galerkin (EFG) method is presented. The governing equations of this model include shallow water equations for the hydraulic behaviour, Exner equation for morphodynamic variations, and Grass model for solid discharge. The governing equations are formulated for the coupled approach. The problem is solved using EFG meshfree method, in which the problem domain is represented by a set of arbitrarily distributed nodes; there is no need to use meshes, elements, or any other node connectivity information for field variable interpolation. The EFG method is based on moving least-squares (MLS) shape functions originated in scattered data fitting. Finally, to assess the ability and the efficiency of the EFG method, several benchmark examples on regular and irregular distribution of nodes are investigated and the results are compared with those of previously published works.

Improving Accuracy of SPH Method Using Voronoi Diagram

In this paper, the mesh-less SPH formulation is modified based on Voronoi diagram to approximate region influence of computational nodal points to achieve higher accuracy. The accuracy of the proposed method is examined for a 2-D elliptical partial differential equation with known analytical solution using both regular and irregular node distributions. In addition, a comparison between the accuracy is accomplished for conventional SPH and the proposed method. The numerical results indicate the accuracy of the proposed method over SPH method especially for irregular node distributions.

Local RBF meshless scheme for coupled radiative and conductive heat transfer

ABSTRACTA local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.

Overview with details for exploring geo-located graphs on maps

Geo-located graph drawings often suffer from map visualization problems, such as overplotting of nodes as well as edges and location of parts of the graph being outside of the screen. One cause of these problems is often an irregular distribution of nodes on the map. Zooming and panning do not solve the problems, as they either only show the overview of the whole graph or only the details of a part of the graph. We present an interactive graph drawing technique that overcomes these problems without affecting the overall geographical structure of the graph. First, we introduce a method that uses insets to visualize details of small or remote areas. Second, to prevent the subgraphs within insets from overplotting and edge crossing, we introduce a local area re-arrangement. Moreover, insets are automatically drawn/hidden and repositioned in accordance with the user’s navigation. We test our technique on real-world geo-located graph data and show the effectiveness of our approach for showing overview and details at the same time. Additionally, we report on expert feedback concerning our approach.

Investigation of RPIM Shape Parameter Effects on the Solution Accuracy of 2D Elastoplastic Problems

The effects of radial point interpolation method (RPIM) shape parameters on the solution accuracy of 2D elastoplastic problems are investigated. The RPIM algorithm is adapted for the solution of 2D elastoplastic problems using the multi-quadric radial basis function. The convergence rates after yielding are investigated for perfectly plastic and hardening cases with various shape parameters. Both regular and irregular node distributions are used by considering the standard Gauss integration and nodal integration schemes. It is shown that the solutions after yielding can be improved using appropriate shape parameters. The propagation of plastic region and solution time of each case are presented against load increments.

ASSESSMENT OF RPIM SHAPE PARAMETERS FOR SOLUTION ACCURACY OF 2D GEOMETRICALLY NONLINEAR PROBLEMS

This study discussed the effects of shape parameters on the radial point interpolation method (RPIM) accuracy in 2D geometrically nonlinear problems. Four finite deformation problems with compressible Neo-Hookean material are numerically solved with the RPIM algorithm using the multi-quadric (MQ) radial basis function. Both regular and irregular node distributions are used. Their displacements and Cauchy stresses are compared for different values of shape parameters and monomial basis. It is found that the shape parameters proposed for linearly elastic problems (q = 1.03, αc = 4) can still be applicable to 2D geometrically nonlinear problems but careful selections should be made for the calculation of stress. For example, when q is used as 1.75 with irregular node distributions, stresses can be calculated more precisely.

A boundary face method for potential problems in three dimensions

AbstractThis work presents a new implementation of the boundary node method (BNM) for numerical solution of Laplace's equation. By coupling the boundary integral equations and the moving least‐squares (MLS) approximation, the BNM is a boundary‐type meshless method. However, it still uses the standard elements for boundary integration and approximation of the geometry, thus loses the advantages of the meshless methods. In our implementation, here called the boundary face method, the boundary integration is performed on boundary faces, which are represented in parametric form exactly as the boundary representation data structure in solid modeling. The integrand quantities, such as the coordinates of Gauss integration points, Jacobian and out normal are calculated directly from the faces rather than from elements. In order to deal with thin structures, a mixed variable interpolation scheme of 1‐D MLS and Lagrange Polynomial for long and narrow faces. An adaptive integration scheme for nearly singular integrals has been developed. Numerical examples show that our implementation can provide much more accurate results than the BNM, and keep reasonable accuracy in some extreme cases, such as very irregular distribution of nodes and thin shells. Copyright © 2009 John Wiley & Sons, Ltd.

A local boundary integral-based meshless method for Biot's consolidation problem

Traditional numerical techniques such as FEM and BEM have been successfully applied to the solutions of Biot's consolidation problems. However, these techniques confront some difficulties in dealing with moving boundaries. In addition, pre-designing node connectivity or element is not an easy task. Recently, developed meshless methods may overcome these difficulties. In this paper, a meshless model, based on the local Petrov–Galerkin approach with Heaviside step function as well as radial basis functions, is developed and implemented for the numerical solution of plane strain poroelastic problems. Although the proposed method is based on local boundary integral equation, it does not require any fundamental solution, thus avoiding the singularity integral. It also has no domain integral over local domain, thus largely reducing the computational cost in formulation of system stiffness. This is a truly meshless method. The solution accuracy and the code performance are evaluated through one-dimensional and two-dimensional consolidation problems. Numerical examples indicate that this meshless method is suitable for either regular or irregular node distributions with little loss of accuracy, thus being a promising numerical technique for poroelastic problems.