Showing papers on "Meshfree methods published in 2023"

A strong-form meshfree computational method for plane elastostatic equations of anisotropic functionally graded materials via multiple-scale Pascal polynomials

Abstract: A strong-form meshfree method is proposed for solving plane elastostatic equations of anisotropic functionally graded materials. Any general function may be the grading function and it is changing smoothly from location to location in the material. The proposed method is based on Pascal polynomial basis and multiple-scale technique and it is a genuinely meshfree method since no numerical integrations over domains and meshing processes are required for considered problems. Implementation of the proposed method is straightforward and the method gives very accurate results. Stability of the solutions are examined numerically in occurrence of random noise. Some certain test problems with known exact solutions are solved both on regular and irregular geometries. Acquired solutions by the suggested method are compared with the exact solutions as well as with solutions of some existing numerical techniques in literature, such as boundary element, meshless local Petrov–Galerkin and radial basis function based meshless methods, to show accuracy of the proposed method.

A meshless method coupling peridynamics with corrective smoothed particle method for predicting material failure

Abstract: In this paper, we developed a meshless framework that couples peridynamics (PD) and the corrective smoothed particle method (CSPM) to simulate the complex damage and fracturing process of structures. This framework retains the advantages of PD in simulating material failure, and solves the problem that PD cannot simulate structures with irregular grids, and it also reduces the interface effect. Furthermore, the coupled PD-CSPM overcomes the challenge that stress boundary cannot be directly applied on CSPM-based models. In addition, PD-CSPM is capable of eliminating the limitation of Poisson's ratio imposed by the bond-based PD method. By simulating the deformation and failure behaviors of structures, the performance of PD-CSPM is verified and validated by the experimental observation and the numerical results obtained using other methods (the classic finite element method (FEM), extended finite element method (XFEM), and phase field method). Using the developed method, we successfully simulated fracture and failure behaviors of structures with irregular grids.

Meshless Galerkin analysis of the generalized Stokes problem

Abstract: This paper proposes and analyzes a stabilized element-free Galerkin (EFG) method for meshless Galerkin analysis of the generalized Stokes problem. The Nitsche-type weak form of the generalized Stokes problem is derived by adopting Nitsche's technique to deal with the lack of interpolation property of meshless shape functions. By introducing the residual-based stabilization technique to establish a stabilized Nitsche-type EFG weak form, the proposed stabilized EFG method not only allows equal-order approximation spaces for velocity and pressure, but also applies to the generalized Stokes problem with small viscosity and large reaction coefficient. Both the inf-sup stability and the error estimation of the stabilized EFG method are analyzed theoretically. Some numerical results are provided to demonstrate the efficiency of the method.

Comparing Several Different Numerical Approaches for Large Deformation Modeling with Application in Soil Dynamic Compaction

Abstract: Abstract Dynamic compaction (DC) is vastly utilized to improve the strength characteristics of the soils. To predict the soil deformations derived from the DC operations, usually numerical simulation analysis is applied. For the conduction of such simulations, several numerical approaches with different elemental formulations can be used. From the perspective of finite element analysis (FEA), there are four main formulations including the Lagrangian, Arbitrary Lagrangian-Eulerian (ALE), Coupled Lagrangian-Eulerian (CEL), and Smoothed Particle Hydrodynamic (SPH). In this research, a comparative study has been conducted to evaluate the computational efficiency of those four approaches in the prediction of soil large deformations during the DC operations. To do this, for a DC operation executed in a road embankment construction project in China, the real field data was compared to the results obtained from the numerical simulations via the ABAQUS program. The findings demonstrate that of all those approaches, the Lagrangian approach delivers the minimum accuracy of the predicted results, albeit with the least running time. In contrast, the ALE formulation predicted closer estimations of soil deformations although it was found to be less time-efficient. Interestingly, the CEL and SPH approaches predicted the soil deformations with the maximum degree of accuracy whereas they were not as time-efficient as the Lagrangian approach. To address this issue, a hybrid model of Lagrangian and SPH formulations was constituted to satisfy the maximum accuracy with the minimum running time. Such a hybrid model is highly applicable for the accurate prediction of soil large deformations during the DC operations.

A meshless multiscale method for simulating hemodynamics

Abstract: Simulating hemodynamic quantities such as pressure and velocity are of great interest to clinicians to aid in surgical planning. To accurately simulate modifications to a region of vasculature, the entire system must be modeled. To facilitate this, a localized Radial-Basis Function (RBF) collocation Meshless flow solver is developed and tightly coupled to a 0D Lumped-Parameter Model (LPM) for solution of the peripheral circulation. The Meshless solver uses localized RBF collocations at data points that are automatically generated according to the geometry. The incompressible flow equations are updated by an explicit time-marching scheme based on a pressure-velocity correction algorithm. The inlets and outlets of the domain are tightly coupled with the LPM which contains elements that draw from a fluid-electrical analogy such as resistors, capacitors, and inductors that represent the viscous resistance, vessel compliance, and flow inertia, respectively. The localized RBF Meshless approach is well-suited for modeling complicated non-Newtonian hemodynamics due to ease of spatial discretization, ease of addition of multi-physics interactions such as fluid-structure interaction of the vessel wall, and ease of parallelization for fast computations. This work introduces the tight coupling of Meshless methods and LPMs for fast and accurate hemodynamic simulations.

Meshfree Methods for PDEs on Surfaces

Abstract: This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The dissertation is subdivided into two papers and software. In the first paper, we examine the performance and accuracy of two popular meshfree methods for surface PDEs:generalized moving least squares (GMLS) and radial basis function-finite differences (RBF-FD). While these methods are computationally efficient and can give high orders of accuracy for smooth problems, there are no published works that have systematically compared their benefits and shortcomings. We perform such a comparison by examining their convergence rates for approximating the surface gradient, divergence, and Laplacian on the sphere and a torus as the resolution of the discretization increases. We investigate these convergence rates also as the various parameters of the methods are changed. We also compare the overall efficiencies of the methods in terms of accuracy per computation cost. The second paper is focused on developing a novel meshfree geometric multilevel (MGM) method for solving linear systems associated with meshfree discretizations of elliptic PDEs on surfaces represented by point clouds. Multilevel (or multigrid) methods are efficient iterative methods for solving linear systems that arise in numerical PDEs. The key components for multilevel methods: \grid" coarsening, restriction/ interpolation operators coarsening, and smoothing. The first three components present challenges for meshfree methods since there are no grids or mesh structures, only point clouds. To overcome these challenges, we develop a geometric point cloud coarsening method based on Poisson disk sampling, interpolation/ restriction operators based on RBF-FD, and apply Galerkin projections to coarsen the operator. We test MGM as a standalone solver and preconditioner for Krylov subspace methods on various test problems using RBF-FD and GMLS discretizations, and numerically analyze convergence rates, scaling, and efficiency with increasing point cloud resolution. We finish with several application problems. We conclude the dissertation with a description of two new software packages. The first one is our MGM framework for solving elliptic surface PDEs. This package is built in Python and utilizes NumPy and SciPy for the data structures (arrays and sparse matrices), solvers (Krylov subspace methods, Sparse LU), and C++ for the smoothers and point cloud coarsening. The other package is the RBFToolkit which has a Python version and a C++ version. The latter uses the performance library Kokkos, which allows for the abstraction of parallelism and data management for shared memory computing architectures. The code utilizes OpenMP for CPU parallelism and can be extended to GPU architectures.

A coupled RKPM and dynamic infinite element approach for solving static and transient heat conduction problems

Abstract: A new accurate and efficient coupled method RKPM-DIEM is proposed. This is a stable and efficient meshfree nodally-integrated reproducing kernel particle method (RKPM) coupled with a dynamic infinite element method (DIEM) for solving half-space problems. The half-space domain is defined as the near field (bounded) and the far field (unbounded) analyzed by the RKPM and DIEM, respectively. Unlike the element-based methods, RKPM is constructed using only nodal data in the global Cartesian coordinates directly to avoid mesh issues such as mesh distortion and entanglement. Also, it provides flexible control of the local smoothness and order of basis, as well as easy construction for a higher-order gradient by changing the kernel function directly. DIEM is first used to show that this approach could solve not only dynamic but also static problems by setting the wave number and the decay coefficient properly. Furthermore, various meshfree integration methods, such as the Gaussian integration, the direct nodal integration, and the natural stabilized nodal integration, are tested to show accuracy and stability. Several benchmark problems are investigated to verify the effectiveness of the proposed method. It has been found that numerical results can achieve high accuracy and stability.

Numerical Methods

Abstract: Chapter 3 cover the conventional and advanced numerical methods, which are essential for explaining the multiscale concepts and techniques. It begins by briefly reviewing the finite difference method (FDM), followed by an introduction to the finite volume method (FVM) and the finite element method (FEM). Then, the extended finite element method is described for accurate and efficient analysis of a wide variety of discontinuity and singularity problems. Then, an introduction is provided for the extended isogeometeric analysis (IGA and XIGA). The principles of the powerful meshless methods are described and some of its main approaches, the element free Galerkin (EFG), the meshless local Petrov-Galerkin (MLPG), the finite point method (FPM) and the smoothed particle hydrodynamics (SPH), are discussed. This Chapter is concluded by examining the variable node element method, which is a combination of finite element and meshless principles to provide a very flexible new element for multiscale purposes.

Support Vector Machine Guided Reproducing Kernel Particle Method for Image-Based Modeling of Microstructures

Abstract: This work presents an approach for automating the discretization and approximation procedures in constructing digital representations of composites from Micro-CT images featuring intricate microstructures. The proposed method is guided by the Support Vector Machine (SVM) classification, offering an effective approach for discretizing microstructural images. An SVM soft margin training process is introduced as a classification of heterogeneous material points, and image segmentation is accomplished by identifying support vectors through a local regularized optimization problem. In addition, an Interface-Modified Reproducing Kernel Particle Method (IM-RKPM) is proposed for appropriate approximations of weak discontinuities across material interfaces. The proposed method modifies the smooth kernel functions with a regularized heavy-side function concerning the material interfaces to alleviate Gibb's oscillations. This IM-RKPM is formulated without introducing duplicated degrees of freedom associated with the interface nodes commonly needed in the conventional treatments of weak discontinuities in the meshfree methods. Moreover, IM-RKPM can be implemented with various domain integration techniques, such as Stabilized Conforming Nodal Integration (SCNI). The extension of the proposed method to 3-dimension is straightforward, and the effectiveness of the proposed method is validated through the image-based modeling of polymer-ceramic composite microstructures.

Groundwater solute transport simulation using global collocation radial basis function based meshfree method

Abstract: We develop a global collocation based meshfree method for simulation of solute transport in the saturated aquifer formations. We prove that the projected model is free from the limitations of mesh or grid which are used in finite element method (FEM) and finite difference method (FDM). We apply this model on different one and two-dimensional synthetic aquifer problems with advection-dispersion phenomenon. The results of these problems demonstrate that the proposed model is free from the limitations of unstable convergence and has a fixed range (i.e., 2-3) of shape parameter hence it requires less number of simulations to calibrate the model. More importantly, the proposed model is simple to code and shows a higher degree of unanimity with the analytical solution, making it a viable alternative to grid-based traditional methods.  

Introduction

Abstract: AbstractAs the range of physical phenomena to be simulated in engineering practice broadens, numerical methods play an important role in various fields. Numerical methods are the main tool to find the approximate solutions of physical problems, where the exact solutions are unavailable in most cases. Since various boundary conditions can be taken into account, the numerical method has advantages over the experimental method due to its low cost and more detailed information of the model. In terms of whether the mesh is used or not, numerical methods can be generally divided into two categories: mesh-based method and meshless method. Mesh-based methods include finite element method (Feng 1965; Zienkiewicz et al. 1977), finite volume method (Versteeg and Malalasekera 2007) and so on. In the field of meshless methods, there are Smoothed Particle Hydrodynamics (SPH) (Lucy 1977) finite difference method (Gürlebeck and Hommel 2003; Taflove and Hagness 2005) Moving Least Square (MLS) approximations (Lancaster and Salkauskas 1981) Element Free Galerkin method (EFG) (Belytschko et al. 1994), Reproducing Kernel Particle Method (RKPM) (Liu et al. 1995), Partition of Unity (PU) (Babuška and Melenk 1997) Hp-cloud method (Duarte and Oden 1996) Optimal Transportation Method (OTM) (Li et al. 2010) and generalized finite difference method (Liszka and Orkisz 1980; Perrone and Kao 1975), to name a few.

A Novel “Finite Element-Meshfree” Triangular Element Based on Partition of Unity for Acoustic Propagation Problems

Abstract: The accuracy of the conventional finite element (FE) approximation for the analysis of acoustic propagation is always characterized by an intractable numerical dispersion error. With the aim of enhancing the performance of the FE approximation for acoustics, a coupled FE-Meshfree numerical method based on triangular elements is proposed in this work. In the proposed new triangular element, the required local numerical approximation is built using point interpolation mesh-free techniques with polynomial-radial basis functions, and the original linear shape functions from the classical FE approximation are employed to satisfy the condition of partition of unity. Consequently, this coupled FE-Meshfree numerical method possesses simultaneously the strengths of the conventional FE approximation and the meshfree numerical techniques. From a number of representative numerical experiments of acoustic propagation, it is shown that in acoustic analysis, better numerical performance can be achieved by suppressing the numerical dispersion error by the proposed FE-Meshfree approximation in comparison with the FE approximation. More importantly, it also shows better numerical features in terms of convergence rate and computational efficiency than the original FE approach; hence, it is a very good alternative numerical approach to the existing methods in computational acoustics fields.

A rotation-free Hellinger-Reissner meshfree thin plate formulation naturally accommodating essential boundary conditions

Abstract: A rotation-free Hellinger-Reissner meshfree thin plate formulation is proposed to naturally accommodate the essential boundary conditions in a variationally consistent way. In this approach, the Galerkin weak form is established based upon the Hellinger-Reissner variational principle, where the bending moment is expressed as the second order smoothed gradients which inherently embed the integration constraint and fulfill the variational consistency condition. Owing to the Hellinger-Reissner variational principle, the essential boundary conditions naturally arise in the weak form. Accordingly, the enforcement of essential boundary conditions has a similar form as that of the Nitsche's method, i.e., both have standard consistent and stabilized terms. Compared with the Nitsche's method, the costly second and higher order derivatives of traditional meshfree shape functions are replaced by the fast-evaluated second order smoothed gradients and their derivatives. Meanwhile, the stabilized term in proposed method does not involve any artificial parameter and thus eliminates the stabilization parameter-dependent issue in the Nitsche's formulation. Several examples are presented to illustrate the convergence, accuracy and efficiency of the proposed rotation-free Hellinger-Reissner meshfree thin plate formulation.

Smooth particle hydrodynamics: a meshless approach for structural mechanics

Abstract: This article provides an overview of the smooth particle hydrodynamics (SPH) approach and its mathematical modeling. SPH is a numerical technique based on a mesh-free Lagrangian scheme for evaluating the continuum mechanics problems. This method is suitable in the case of continuum objects undergoing large deformation, as conventional finite element methods are unreliable due to mesh failure and convergence issues. It is a widely used approach in the field of astrophysics, fluid mechanics, structural mechanics, soil mechanics, automobiles, and so on. A numerical example is also considered in this research paper to demonstrate the applicability of the method. The simulation process was achieved using LS-Dyna explicit solver software, and plots related to cutting and thrust forces, von Mises stress, plastic strain, temperature distribution, and so on, were obtained. Also, the effect of Time-Scaling Factor (TSSFAC) on SPH simulations was observed in this research.

A nodal integration based two level local projection meshfree stabilization method for convection diffusion problems

Abstract: This paper presents a new two level local projection meshfree stabilization (LPMS) method under the classical stabilized conforming nodal integration (SCNI) framework to solve convection diffusion problems suitable to a linear order approximation. Dropping the higher order terms in the residual based Petrov-Galerkin stabilization methods, usual in the Gauss integration, produces erroneous results if the classical SCNI is used because of the product of a quantity and its smoothed gradient (SCNI divergence operator) in the convective term. LPMS method is an alternative way to achieve a stable solution with the linear order approximation. In the present method, the design of the stabilization of convective terms stems from the fine scales defined on the subcells of Voronoi cells of SCNI. This LPMS with SCNI combination excludes the computational need for derivatives. First order mesh basis functions are sufficient as the SCNI replaces lower order derivatives with smoothed gradients, while local projection stabilization (LPS) excludes the terms with higher order derivatives. The present LMPS method is tested against standard benchmark problems with different distorted grids and compared with three existing stabilization methods. It is found to be in good agreement with the classical numerical methods for problems with thin layers.

On Interpolative Meshless Analysis of Orthotropic Elasticity

Abstract: As one possible alternative to the finite element method, the interpolation characteristic is a key property that meshless shape functions aspire to. Meanwhile, the interpolation meshless method can directly impose essential boundary conditions, which is undoubtedly an advantage over other meshless methods. In this paper, the establishment, implementation, and horizontal comparison of interpolative meshless analyses of orthotropic elasticity were studied. In addition, the radial point interpolation method, the improved interpolative element-free Galerkin method and the interpolative element-free Galerkin method based on the non-singular weight function were applied to solve orthotropic beams and ring problems. Meanwhile, the direct method is used to apply the displacement boundary conditions for orthotropic elastic problems. Finally, a detailed convergence study of the numerical parameters and horizontal comparison of numerical accuracy and efficiency were carried out. The results indicate that the three kinds of interpolative meshless methods showed good numerical accuracy in modelling orthotropic elastic problems, and the accuracy of the radial point interpolation method is the highest.

A Novel Coupled Meshless Model for Simulation of Acoustic Wave Propagation in Infinite Domain Containing Multiple Heterogeneous Media

Abstract: This study presents a novel coupled meshless model for simulating acoustic wave propagation in heterogeneous media, based on the singular boundary method (SBM) and Kansa’s method (KS). In the proposed approach, the SBM was used to model the homogeneous part of the propagation domain, while KS was employed to model a heterogeneity. The interface compatibility conditions associated with velocities and pressures were imposed to couple the two methods. The proposed SBM–KS coupled approach combines the respective advantages of the SBM and KS. The SBM is especially suitable for solving external sound field problems, while KS is attractive for nonlinear problems in bounded non-homogeneous media. Moreover, the new methodology completely avoids grid generation and numerical integration compared with the finite element method and boundary element method. Numerical experiments verified the accuracy and effectiveness of the proposed scheme.