Smoothed Point Interpolation Methods

Liu, G R; Gui-Rong,; Gy, Zhang; Guiyong,

doi:10.1142/8742

Cited by 64 publications

(8 citation statements)

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“…Compared to MLS, RPIMp has a significant advantage, as shown in subsection II-D: it does not require any additional method to impose the essential boundary conditions since it has the Kronecker delta property [28]. However, if the method uses only radial-based functions, it does not approximate linear functions consistently.…”

Section: B the Local Weak Formmentioning

confidence: 99%

Node-to-Node Realization of Meshless Local Petrov Galerkin (MLPG) Fully in GPU

et al. 2019

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This paper presents an end-to-end massively parallelized procedure for the solution of boundary value problems on Graphics Processing Units (GPU). The proposal is an integrated strategy that not only entails the calculation of nodal contributions, and the stiffness matrix assembly using the Meshless Local Petrov Galerkin Method (MLPG) but also the iterative solution of the system of algebraic equations in combination with methods from the Conjugate Gradient (CG) family. This end-to-end solution is fully developed using the Compute Unified Device Architecture (CUDA) platform without the need for extra data movement between the device and host after the matrix assembly. This is possible thanks to the parallel nature of the MLPG; each node designates a thread on the device. The introduced solution is wholly executed in the GPU, with minimal auxiliary structures or global synchronization points. The proposed approach was applied to the solution of a simple electromagnetic problem, and a sevenfold speedup was observed.

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Section: B the Local Weak Formmentioning

confidence: 99%

Node-to-Node Realization of Meshless Local Petrov Galerkin (MLPG) Fully in GPU

et al. 2019

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“…A class of meshfree methods was developed by GR Liu's group for solids and structures, known as smoothed point interpolation methods or S‐PIM and smoothed FEMs or S‐FEM . The S‐PIM and S‐FEM are all based on the so‐called weakened weak form, and they use gradient smoothing technique or the generalized gradient smoothing technique .…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

Mao

Liu

2017

Numerical Meth Engineering

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A novel Lagrangian gradient smoothing method (L-GSM) is developed to solve "solid-flow" (flow media with material strength) problems governed by Lagrangian form of Navier-Stokes equations. It is a particle-like method, similar to the smoothed particle hydrodynamics (SPH) method but without the socalled tensile instability that exists in the SPH since its birth. The L-GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no-slip solid boundary, free-surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L-GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L-GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear-driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L-GSM performance. It shows that the L-GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testingresults demonstrate that the proposed L-GSM approach restores first-order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L-GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.

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“…Reddy found that Eringen nonlocal constitutive relation has a softening effect on the beam (Khodabakhshi and Reddy, 2015;Reddy and Srinivasa, 2015). Meanwhile, various meshfree methods have been developed and extended in recent decades, such as the reproducing kernel particle method (RKPM) (Liu et al, 1995), the meshless local Petrov-Galerkin method (MLPG) (Atluri and Shen, 2002), the element-free Galerkin method (EFG) (Belytschko et al, 1994) and smoothed point interpolation methods (S-PIM) (Liu and Zhang, 2013). S-PIM is a meshfree method that uses the pointed interpolation method (PIM) (Liu, 2002) as its nodal shape functions and the generalized gradient smoothing technique for strain field construction (Liu and Zhang, 2013).…”

Section: Introductionmentioning

confidence: 99%

“…Meanwhile, various meshfree methods have been developed and extended in recent decades, such as the reproducing kernel particle method (RKPM) (Liu et al, 1995), the meshless local Petrov-Galerkin method (MLPG) (Atluri and Shen, 2002), the element-free Galerkin method (EFG) (Belytschko et al, 1994) and smoothed point interpolation methods (S-PIM) (Liu and Zhang, 2013). S-PIM is a meshfree method that uses the pointed interpolation method (PIM) (Liu, 2002) as its nodal shape functions and the generalized gradient smoothing technique for strain field construction (Liu and Zhang, 2013). According to different types of smoothing domains used, a class of S-PIM models has been developed based on the weakened weak (w 2 ) formulation (Liu and Zhang, 2013;Liu and Zhang, 2009;Liu et al, 2005;Liu and Zhang, 2008).…”

Section: Introductionmentioning

confidence: 99%

“…S-PIM is a meshfree method that uses the pointed interpolation method (PIM) (Liu, 2002) as its nodal shape functions and the generalized gradient smoothing technique for strain field construction (Liu and Zhang, 2013). According to different types of smoothing domains used, a class of S-PIM models has been developed based on the weakened weak (w 2 ) formulation (Liu and Zhang, 2013;Liu and Zhang, 2009;Liu et al, 2005;Liu and Zhang, 2008). With the same type of linear triangular meshes, all these S-PIM models have been found to possess softer stiffness than the FEM does, which leads to the increase of computational performance (Liu and Zhang, 2013).…”

Section: Introductionmentioning

confidence: 99%

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A cell-based smoothed point interpolation method (CS-PIM) for 2D thermoelastic problems

Liu

Zhang

et al. 2017

HFF

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Purpose Due to the strong reliance on element quality, there exist some inherent shortcomings of the traditional finite element method (FEM). The model of FEM behaves overly stiff, and the solutions of automated generated linear elements are generally of poor accuracy about especially gradient results. The proposed cell-based smoothed point interpolation method (CS-PIM) aims to improve the results accuracy of the thermoelastic problems via properly softening the overly-stiff stiffness. Design/methodology/approach This novel approach is based on the newly developed G space and weakened weak (w2) formulation, and of which shape functions are created using the point interpolation method and the cell-based gradient smoothing operation is conducted based on the linear triangular background cells. Findings Owing to the property of softened stiffness, the present method can generally achieve better accuracy and higher convergence results (especially for the temperature gradient and thermal stress solutions) than the FEM does by using the simplest linear triangular background cells, which has been examined by extensive numerical studies. Practical implications The CS-PIM is capable of producing more accurate results of temperature gradients as well as thermal stresses with the automated generated and unstructured background cells, which make it a better candidate for solving practical thermoelastic problems. Originality/value It is the first time that the novel CS-PIM was further developed for solving thermoelastic problems, which shows its tremendous potential for practical implications.

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Smoothed Point Interpolation Methods

Cited by 64 publications

References 0 publications

Node-to-Node Realization of Meshless Local Petrov Galerkin (MLPG) Fully in GPU

Node-to-Node Realization of Meshless Local Petrov Galerkin (MLPG) Fully in GPU

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

A cell-based smoothed point interpolation method (CS-PIM) for 2D thermoelastic problems

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