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1.
We discuss the symmetric smoothed particle hydrodynamics (SSPH) method for generating basis functions for a meshless method.
It admits a larger class of kernel functions than some other methods, including the smoothed particle hydrodynamics (SPH),
the modified smoothed particle hydrodynamics (MSPH), the reproducing kernel particle method (RKPM), and the moving least squares
(MLS) methods. For finding kernel estimates of derivatives of a function, the SSPH method does not use derivatives of the
kernel function while other methods do, instead the SSPH method uses basis functions different from those employed to approximate
the function. It is shown that the SSPH method and the RKPM give the same value of the kernel estimate of a function but give
different values of kernel estimates of derivatives of the function. Results computed for a sine function defined on a one-dimensional
domain reveal that the L
2, the H
1 and the H
2 error norms of the kernel estimates of a function computed with the SSPH method are less than those found with the MSPH method.
Whereas the L
2 and the H
2 norms of the error in the estimates computed with the SSPH method are less than those with the RKPM, the H
1 norm of the error in the RKPM estimate is slightly less than that found with the SSPH method. The error norms for a sample
problem computed with six kernel functions show that their rates of convergence with an increase in the number of uniformly
distributed particles are the same and their magnitudes are determined by two coefficients related to the decay rate of the
kernel function. The revised super Gauss function has the smallest error norm and is recommended as a kernel function in the
SSPH method. We use the revised super Gauss kernel function to find the displacement field in a linear elastic rectangular
plate with a circular hole at its centroid and subjected to tensile loads on two opposite edges. Results given by the SSPH
and the MSPH methods agree very well with the analytic solution of the problem. However, results computed with the SSPH method
have smaller error norms than those obtained from the MSPH method indicating that the former will give a better solution than
the latter. The SSPH method is also applied to study wave propagation in a linear elastic bar. 相似文献
2.
A comparison between weak form meshless local Petrov-Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost. 相似文献
3.
D.F. Gilhooley R.C. Batra J.R. Xiao M.A. McCarthy J.W. Gillespie Jr. 《Composite Structures》2007,80(4):539-552
Infinitesimal deformations of a functionally graded thick elastic plate are analyzed by using a meshless local Petrov–Galerkin (MLPG) method, and a higher-order shear and normal deformable plate theory (HOSNDPT). Two types of Radial basis functions RBFs, i.e. Multiquadrics and Thin Plate Splines, are employed for constructing the trial solutions, while a fourth-order Spline function is used as the weight/test function over a local subdomain. Effective material moduli of the plate, made of two isotropic constituents with volume contents varying only in the thickness direction, are computed using the Mori–Tanaka homogenization technique. Computed results for a simply supported aluminum/ceramic plate are found to agree well with those obtained analytically. Results for a plate with two opposite edges free and the other two simply supported agree very well with those obtained by analyzing three-dimensional deformations of the plate by the finite element method. The distributions of the deflection and stresses through the plate thickness are also presented for different boundary conditions. It is found that both types of basis functions give accurate values of plate deflection, but the multiquadrics give better values of stresses than the thin plate splines. 相似文献
4.
On the convergence analysis,stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations
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Elyas Shivanian 《International journal for numerical methods in engineering》2016,105(2):83-110
In this article, the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form subject to given initial and Dirichlet boundary conditions. The main difficulty of the great number of methods in full 3‐D problems is the large computational costs. In meshless local radial point interpolation method, it does not require any background integration cells, so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method with the help of radial basis functions is proposed to construct shape functions that have Kronecker delta function property. A weak formulation with the Heaviside step function converts the set of governing equations into local integral equations on local subdomains. A two‐step time discretization method is employed to evaluate the time derivatives. This suggests Crank‐Nicolson technique to be applied on the right hand side of the equation. The convergence analysis and stability of the method are fully discussed. Three illustrative examples are presented, and satisfactory agreements are achieved. It is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
5.
Deniz Can
Kolukisa Murat Ozbulut Emre Pesman Mehmet Yildiz 《International journal for numerical methods in engineering》2020,121(18):4187-4207
In Lagrangian particle-based methods such as smoothed particle hydrodynamics (SPH), computing totally divergence-free velocity field in a flow domain with the smallest error possible is the most critical issue, which might be achieved through solving pressure Poisson equation implicitly with higher particle resolutions. However, implicit solutions are computationally expensive and may be particularly challenging in the solution of multiphase flows with highly nonlinear deformations as well as fluid-structure interaction problems. Augmented Lagrangian SPH (ALSPH) method is a new alternative algorithm as a prevalent pressure solver where the divergence-free velocity field is achieved by iterative calculation of velocity and pressure fields. This study investigates the performance of the ALSPH technique by solving a challenging flow problem such as two-dimensional flow around a cylinder within the Reynolds number range of 50 to 500 in terms of improved robustness, accuracy, and computational efficiency. The same flow conditions are also simulated using the conventional weakly compressible SPH (WCSPH) method. The results of ALSPH and WCSPH solutions are not only compared in terms of numerical validation/ verification studies, but also rigorous investigations are performed for all related physical flow characteristics, namely, hydrodynamic coefficients, frequency domain analyses, and velocity divergence fields. 相似文献
6.
G. Gabard P. Gamallo T. Huttunen 《International journal for numerical methods in engineering》2011,85(3):380-402
Several numerical methods using non‐polynomial interpolation have been proposed for wave propagation problems at high frequencies. The common feature of these methods is that in each element, the solution is approximated by a set of local solutions. They can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation. There are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated. The similarities and discrepancies between several non‐polynomial numerical methods are discussed in the context of the Helmholtz equation. The present comparison is concerned with the ultra‐weak variational formulation (UWVF), the least‐squares method (LSM) and the discontinuous Galerkin method with numerical flux (DGM). An analysis in terms of Trefftz methods provides an interesting insight into the properties of these methods. Second, the UWVF and the LSM are reformulated in a similar fashion to that of the DGM. This offers a unified framework to understand the properties of several non‐polynomial methods. Numerical results are also presented to put in perspective the relative accuracy of the methods. The numerical accuracies of the methods are compared with the interpolation errors of the wave bases. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
7.
《Virtual and Physical Prototyping》2013,8(2):121-130
Flying was inspired by birds. But ironically bird strikes are a menace. Therefore, aero-engines have to be designed to survive these strikes. The design of the bullet nose of an aero-engine during a bird strike is presented in this paper. A Finite Element Analysis (FEA) model was developed for this purpose. It was then fine-tuned through experiments on bullet noses built using Laser-Engineered Net-Shaping (LENS). Through this approach, several design alternatives could be evaluated virtually and only a few physical experiments were required for validation. The outcome is not only a rapid and safe bullet nose design but also a realistic FEA model. 相似文献