Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Comparison of Slope Stability Using Smoothed Particle Hydrodynamics, Finite Element Method, and Limit Equilibrium Method

Abstract: The principles of slope stability analysis embrace the mechanics of slope failure to develop methods of stability analysis, to predict factors of safety and corresponding slope movements. By equating the disturbing and resisting forces on potential slip surfaces, limit equilibrium method computes the corresponding factor of safety. These static stability methods can be used to calculate the minimum factor of safety, but they cannot predict pre- and post-failure movements of the system in case of failure. As an advanced numerical method, finite element method and the strength reduction technique take into consideration the stress distribution within the slope. This stress distribution controls the deformations, movements, and development of failure zones for a given slope. Along with the factor of safety, FEM also predicts these expected deformations/movements. These deformations in FEM are dependent heavily on the meshing of the whole system and suffer from mesh distortion and low accuracy in case of run-out/flow analysis where mesh deformations are relatively more significant. Run-out/flow failures (landslides, landfill failures), where the debris flows over a great distance, have been a prominent research topic for many years. Numerical methods have been developed such as the center of the mass method, FEM with updated mesh techniques, DAN/W, and meshless methods (SPH, DEM, and DDM). Smoothed particle hydrodynamics (SPH) is a meshless technique which discretizes continuous material into discrete particles. When combined with yield strength, SPH solves Navier–Stokes equations to predict the slip surfaces as well as run-out/flow of the slope. In this study, a comparison has been made between LEM, FEM, and SPH on slope failures taken from literature. The slip surfaces as obtained from these slope failures have been compared, and the deformations have been studied between FEM and SPH. The advantage of SPH in predicting flow failures has been highlighted.

Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF's Shape Parameter.

Abstract: New engineering materials exhibit a complex internal structure that determines their properties. For thermal metamaterials, it is essential to shape their thermophysical parameters’ spatial variability to ensure unique properties of heat flux control. Modeling heterogeneous materials such as thermal metamaterials is a current research problem, and meshless methods are currently quite popular for simulation. The main problem when using new modeling methods is the selection of their optimal parameters. The Kansa method is currently a well-established method of solving problems described by partial differential equations. However, one unsolved problem associated with this method that hinders its popularization is choosing the optimal shape parameter value of the radial basis functions. The algorithm proposed by Fasshauer and Zhang is, as of today, one of the most popular and the best-established algorithms for finding a good shape parameter value for the Kansa method. However, it turns out that it is not suitable for all classes of computational problems, e.g., for modeling the 1D heat conduction in non-homogeneous materials, as in the present paper. The work proposes two new algorithms for finding a good shape parameter value, one based on the analysis of the condition number of the matrix obtained by performing specific operations on interpolation matrix and the other being a modification of the Fasshauer algorithm. According to the error measures used in work, the proposed algorithms for the considered class of problem provide shape parameter values that lead to better results than the classic Fasshauer algorithm.

Moving mesh finite volume method and its applications

Abstract: Moving mesh method has become an important numerical tool for computing singular or nearly singular problems arising from a variety of physical and engineering areas. This thesis is mainly concerned with the application of moving mesh method to nonlinear conservation laws, convection-diffusion equations, phase-field equations and magneto-hydrodynamics (MHD) model problems. Our moving mesh algorithm is an extension of Tang’s recent work which focuses on hyperbolic conservation laws. We have written the underlying PDEs in the computational domain via a coordinate transformation, and then the transformed PDEs are solved in computational domain equipped with fixed uniform mesh. Effectiveness and robustness of the proposed algorithm are demonstrated by numerical experiments. It is well known that the time steps associated with the moving mesh methods are proportional to the smallest mesh size in space in order to guarantee stability. Since the moving mesh method is useful for problems whose solutions are singular in fairly localized regions, it reduces the allowable time step only in small part of the solution domain. It is then natural to use locally varying time steps to enhance the efficiency of the moving mesh methods. We design an efficient local time stepping scheme for the nonlinear hyperbolic conservation laws and the convection-dominated problems. Finally we apply moving mesh method to the phase-field equations and magnetohydrodynamics (MHD) model. Numerical results demonstrate the advantages of our moving mesh method in solving these problems.

Numerical integration by using local-node gauss-hermite cubature

Abstract: A local–node numerical integration scheme for meshless methods is presented in this work. The distinguishing characteristic of the introduced scheme is that not support mesh or grid to perform numerical integration is needed, besides the fact that gauss cubature points for each node are generated in a properly fashion and that the extension of the methodology to high dimensions is straightforward. The numerical integration is computed with the Gauss–Hermite cubature formulas, and the partition of unity is employed to introduce the Gaussian weight in a natural way. Selected numerical tests in two-dimensions are used to illustrate the validity of the proposed methodology. Although the obtained results are encouraging, the behavior of the integration error is not still well understood when the dimensionless parameter which control the width of the Gaussian kernel varies.

Meshless methods applied to linear and nonlinear structural analysis [CD-ROM]

Abstract: The Element-free Galerkin, EFGM, and hp-clouds meshless methods are ap- plied to linear and nonlinear structural analysis. The aim is to evaluate the numerical response of the method on different kind of structural problems. On the linear side, the case of tall buildings structures formed by plane association of wall and frame panels, submitted to lateral loading is presented. The mechanical behavior of the association is modeled by the continuous medium technique. The fundamental concepts of the technique are presented. Then, both the meshless methods studied are applied to solve numerically the boundary value problem. The numerical solutions are compared with the exact re- sponses and with numerical ones obtained by the FEM. An investigation of the rate of convergence is included. On the nonlinear side, a reinforced concrete beam is considered. The behavior of the concrete is accounted for the Mazars'damage model. The beam is considered as a layered system. General details about the numerical implementation of both methods are outlined but only the hp-cloud method is employed to perform static and dynamic simulations. In the static case, the numerical results obtained are compared with experimental measures and the good performance of the method is verified. For the dy- namic analysis the system is considered without damping. The results show the influence of damage on the structural behavior.