Symmetric Smoothed Particle Hydrodynamics Method Research Articles

Efficient thermomechanical analysis of functionally graded structures using the symmetric SPH method

Tailoring the thermal and mechanical properties plays an important role in manufacturing and designing functionally graded material (FGM) which is often applied in high temperature gradient environment. The present investigation will address the basics of thermoelastic problem solved by means of a specific meshless smoothed particle hydrodynamics (SPH) method. The inconsistency of the conventional SPH method was improved by introducing the symmetric SPH (SSPH) technique, where function derivatives can be approximated by the function values in the support domain. Hence, the heat conduction and balance equations become a series of algebraic equations, which can be solved efficiently. The result accuracy achieved by the proposed approach was demonstrated in solving several numerical examples in comparison to available results in the literature. The effect of the gradation indexes on the thermoelastic behavior was also considered.

Solving 2D Poisson-type equations using meshless SPH method

Abstract In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Compared with other kernel functions, the cubic B-spline kernel function shows good capacity to reproduce a complicated function value and its first and second derivatives, when the smoothing length is chosen as 1.1 times the particles’ distance. Several types of Poisson equations are solved by SSPH method, and the numerical results exhibit a very good accuracy when comparing with the analytical solutions.

Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity

To eliminate the numerical oscillations appearing in the first-order symmetric smoothed particle hydrodynamics (FO-SSPH) method for simulating transient heat conduction problems with discontinuous initial distribution, this paper presents a second-order symmetric smoothed particle hydrodynamics (SO-SSPH) method. Numerical properties of both SO-SSPH and FO-SSPH are analyzed, including truncation error, numerical accuracy, convergence rate, and stability. Experimental results show that for transient heat conduction with initial smooth distribution, both FO-SSPH and SO-SSPH can achieve second order convergence rate, which is consistent with the theoretical analysis. However, for one- and two-dimensional conduction with initial discontinuity, the FO-SSPH method suffers from serious unphysical oscillations, which do not disappear over time, and hence it only achieves a first-order convergence rate; while the present SO-SSPH method can avoid unphysical oscillations and has second-order convergence rate. Therefore, the SO-SSPH method is a feasible tool for solving transient heat conduction problems with both smooth and discontinuous distributions, and it is easy to be extended to high dimensional cases.

Bending Analysis of Two Directional Functionally Graded Beams Using A Four-Unknown Shear and Normal Deformation Theory

The bending behaviour of two-directional functionally graded beams (FGBs) subjected to various sets of boundary conditions is investigated by using a shear and normal deformation theory and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. A simply supported conventional FGB problem is studied to validate the developed code. The comparison studies are performed along with the analytical solutions and the results from previous studies. The numerical calculations in terms of maximum dimensionless transverse deflections, dimensionless axial and transverse shear stresses are performed for various gradation exponents, aspect ratios (L/h) and sets of boundary conditions. The effects of the gradation exponents on the accuracy and the robustness of the SSPH method are also investigated for the two directional functionally graded beams which are having clamped-free boundary condition..

Simulation of three-dimensional transient heat conduction problem with variable coefficients based on the improved parallel smoothed particle hydrodynamics method

In this work, an improved parallel SPH method is proposed to accurately solve the three-dimensional (3D) transient heat conduction equation with variable coefficients. The improvements are described as follows. Firstly, the first-order symmetric smoothed particle hydrodynamics (SPH) method is extended to the simulating of the 3D problem based on Taylor expansion. Secondly, the concept of stabilized up-wind technique is introduced into the convection term. Thirdly, the MPI parallel technique based on the neighboring particle mark method is introduced into the above improved SPH method, and named the corrected parallel SPH method for 3D problems (CPSPH-3D). Subsequently, the accuracy, convergence and the computational efficiency of the proposed CPSPH-3D method are tested by simulating the 3D transient heat conduction problems with constant/variable coefficient, and compared with the analytical solution. Meanwhile, the capacity of the proposed CPSPH-3D for solving the 3D heat conduction problems with the Dirichlet and Newmann boundaries is illustrated, in which the change of temperature with time under the complex cylindrical area is also considered. The numerical results show that:1) the proposed CPSPH-3D method has the better stability, higher accuracy and computational efficiency than the conventional SPH method no matter whether the particle distribution is uniform; 2) the calculating time can be well reduced by increasing the number of CPUs when the particle number is refined in the simulations of CPSPH-3D. Finally, the temperature variation in the 3D functionally gradient material is predicted by the corrected parallel SPH method, and compared with the other numerical results. The process of temperature variation in the functionally gradient material can be shown accurately.

Analysis of isotropic tapered beams by using symmetric smoothed particle hydrodynamics method

Analysis of isotropic tapered beams by using symmetric smoothed particle hydrodynamics method

Elastostatic analysis of two-directional functionally graded beams using various beam theories and Symmetric Smoothed Particle Hydrodynamics method

The elastostatic behaviour of two-directional functionally graded beams (FGBs) subjected to various sets of boundary conditions are investigated for the first time by using the Euler-Bernoulli, Timoshenko and Reddy-Bickford beam theories and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. To validate the developed code, a simply supported conventional FGB problem is studied and the comparison studies are performed along with the analytical solutions and the results from previous studies. The numerical calculations in terms of maximum dimensionless transverse deflections, dimensionless axial and transverse shear stresses are performed based on different beam theories with varying gradation exponents (power-law index), different aspect ratios (L/h) and sets of boundary conditions.

Elastostatic Deformation Analysis of Thick Isotropic Beams by Using Different Beam Theories and a Meshless Method

The elastostatic deformations of thick isotropic beams subjected to various sets of boundary conditions are presented by using different beam theories and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The analysis is based on the Euler-Bernoulli, Timoshenko and Reddy-Bickford beam theories. The performance of the SSPH method is investigated for the comparison of the different beam theories for the first time. For the numerical results, various numbers of nodes are used in the problem domain. Regarding to the computed results for RBT, various number of terms in the Taylor Series Expansions (TSEs) is employed. To validate the performance of the SSPH method, comparison studies in terms of transverse deflections are carried out with the analytical solutions. It is found that the SSPH method has provided satisfactory convergence rate and smaller L2 error.

Analysis of Bending Deflections of Functionally Graded Beams by Using Different Beam Theories and Symmetric Smoothed Particle Hydrodynamics

The elastostatic deformations of functionally graded beams under various boundary conditions are investigated by using different beam theories and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The numerical calculations are performed based on the Euler-Bernoulli, Timoshenko and Reddy-Bickford beam theories. The performance of the SSPH method is investigated for the comparison of the different beam theories where the beams are composed of two different materials for the first time. For the numerical results various numbers of nodes are used in the problem domain. Regarding to the computed results for Reddy-Bickford beam theory various numbers of terms in the Taylor Series Expansions (TSEs) are employed to improve the accuracy. To validate the performance of the SSPH method, comparison studies in terms of transverse deflections are carried out with the analytical solutions by using the global L 2 error norm.

Bending Deflection Analysis of a Semi-Trailer Chassis by Using Symmetric Smoothed Particle Hydrodynamics

In this paper, a simple approach is presented for the calculation of bending deflection of a semi trailer chassis. The 3D model of the chassis is used to determine the moment of inertia of the cross section function and then the mathematical model of the chassis is presented as a Euler Bernoulli Beam which has the variable cross section. Different loading conditions coming from the semi trailer test procedures are applied. The bending deflections of the semi trailer chassis are numerically calculated by using the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The first time the performance of the SSPH method for the fourth order non-homogeneous variable coefficents linear boundary value problems is evaluated. For the calculations different number of terms in the TSEs are employed when the number of nodes in the problem domain increases. The comparisons are made with the results of experiments. It is observed that the SSPH method has the conventional convergence properties yields smaller L 2 error. Finally, the approach presented here may be used and found reliable for the calculation of deflection of the semi trailer chassis before the release of detail design.

Different Implementation Approaches of the Strong Form Meshless Implementation of Taylor Series Method

Based on the Taylor series expansion (TSE) and employing the technique of differential transform method (DTM), three new meshless approaches which are called Meshless Implementation of Taylor Series Methods (MITSM) are presented. In particular, Strong Form Meshless Implementation of Taylor Series Methods (SMITSM) are studied in this paper. Then, the basis functions are used to solve a 1D second-order ordinary differential equation and 2D Laplace equation by using the SMITSM. Comparisons are made with the analytical solutions and results of Symmetric Smoothed Particle Hydrodynamics (SSPH) method. We also compared the effectiveness of the SMITSM and SSPH method by considering various particle distributions, nonhomogeneous terms and number of terms in the basis functions. It is observed that the MITSM has the conventional convergence properties and, at the expense of CPU time, yields smaller L 2 error norms than the SSPH method, especially in the existence of nonsmooth nonhomogeneous problems.

Strong form Meshless Implementation of Taylor Series Method

A new meshless approach is investigated by using Taylor series expansion and technique of differential transform method, which is called Meshless Implementation of Taylor Series Method (MITSM). In particular, Strong form Meshless Implementation of Taylor Series Method (SMITSM) is studied in this paper. Then, the basis functions are used to solve a 1D second-order ordinary differential equation and 2D Laplace equation by using the SMITSM. Comparisons are made with the analytical solutions and results of Symmetric Smoothed Particle Hydrodynamics (SSPH) method. We also compared the effectiveness of the SMITSM and SSPH method by considering various particle distributions, nonhomogeneous terms and number of terms in the basis functions. It is observed that the MITSM has the conventional convergence properties and, at the expense of CPU time, yields smaller L2 error norms than the SSPH method, especially in the existence of nonsmooth nonhomogeneous problems.

Solutions of Two-Dimensional Heat Transfer Problems by Using Symmetric Smoothed Particle Hydrodynamics Method

The symmetric smoothed particle hydrodynamics (SSPH) method is used to generate the basis functions to solve 2D homogeneous and non-homogeneous steady-state heat transfer problems. The SSPH basis functions together with the collocation method (i.e, the strong formulation of the problem) are used to solve sample problems. Comparisons are made with the results obtained by using different weight functions and particle numbers. The error norms for three sample problems are computed by the use of two different kernel functions such as the revised Gauss function and revised super Gauss function, among which the revised super Gauss function yields the smallest error norm. It is observed that the SSPH method yields large errors for non-homogenous problems, especially if the forcing term is not smooth.

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

We discuss the symmetric smoothed particle hydrodynamics (SSPH) method for generating basis func- tions 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 parti- cle hydrodynamics (MSPH), the reproducing kernel particle method (RKPM), and the moving least squares (MLS) meth- ods. For finding kernel estimates of derivatives of a function, the SSPH method does not use derivatives of the kernel func- tion while other methods do, instead the SSPH method uses basis functions different from those employed to approxi- mate 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 deriv- atives of the function. Results computed for a sine function defined on a one-dimensional domain reveal that the L 2 ,t he H 1 and the H 2 error norms of the kernel estimates of a func- tion 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 sam- ple 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 magni- tudes 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 ker- nel function in the SSPH method. We use the revised super

SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations

We propose a new and simple technique called the Symmetric Smoothed Particle Hydrodynamics (SSPH) method to construct basis functions for meshless methods that use only locations of particles. These basis functions are found to be similar to those in the Finite Element Method (FEM) except that the basis for the derivatives of a function need not be obtained by differentiating those for the function. Of course, the basis for the derivatives of a function can be obtained by differentiating the basis for the function as in the FEM and meshless methods. These basis functions are used to numerically solve two plane stress/strain elasto-static problems by using either the collocation method or a weak formulation of the problem defined over a subregion of the region occupied by the body; the latter is usually called the Meshless Local Petrov–Galerkin (MLPG) method. For the two boundary-value problems studied, it is found that the weak formulation in which the basis for the first order derivatives of the trial solution are derived directly in the SSPH method (i.e., not obtained by differentiating the basis function for the trial solution) gives the least value of the L2-error norm in the displacements while the collocation method employing the strong formulation of the boundary-value problem has the largest value of the L2-error norm. The numerical solution using the weak formulation requires more CPU time than the solution with the strong formulation of the problem. We have also computed the L2-error norm of displacements by varying the number of particles, the number of Gauss points used to numerically evaluate domain integrals appearing in the weak formulation of the problem, the radius of the compact support of the kernel function used to generate the SSPH basis, the order of complete monomials employed for constructing the SSPH basis, and boundary conditions used at a point on a corner of the rectangular problem domain. It is recommended that for solving two-dimensional elasto-static problems, the MLPG formulation in which derivatives of the trial solution are found without differentiating the SSPH basis function be adopted.