Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems

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.     

Comparison of local weak and strong form meshless methods for 2-D diffusion equation

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.     

Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions

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.     

Analysis of the bullet nose of an aero-engine for bird impact through virtual and rapid prototyping

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.