Symmetric Smoothed Particle Hydrodynamics Research Articles

Dual-Point Symmetric Smoothed Particle Hydrodynamics for modelling of Fourier and non-Fourier heat conduction phenomena

Dual-Point Symmetric Smoothed Particle Hydrodynamics for modelling of Fourier and non-Fourier heat conduction phenomena

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.

Meshless SPH analysis for transient heat conduction in the functionally graded structures

Transient heat conduction analysis of the functionally graded structures is of great importance for the structure design and manufacturing. In this study, 2D heat conduction problem is solved by the meshless smoothed particle hydrodynamics (SPH) method. To improve the computational accuracy and efficiency of traditional SPH method, a symmetric SPH approach is adopted, which can calculate a function and its derivatives of higher order using one symmetric matrix. The heat conduction function is discretized by the improved SPH method, as well as the boundary conditions. The SPH analysis of the heat conduction in several functionally graded structures with different geometries is performed and validated by the analytical values and the FEM results in literature. The effect of the graded index on the temperature distribution in the structure domain is also studied. The present paper provides an alternative potential method for the heat conduction analysis of functionally graded materials.

Meshless analysis of bi-directional functionally graded beam structures based on physical neutral surface

More and more attention has been paid on the two-directional functionally graded materials (FGMs) applied in harsh heat environment. A novel meshless model is established for the bending analysis, based on the Smoothed Particle Hydrodynamics (SPH) method. To improve the prediction accuracy, the conventional SPH method is improved by the symmetric SPH method starting from the Taylor series expansion. Moreover, the neutral surface position of the bi-directional FG beam is determined and the third order Reddy-Bickford beam theory based on this exact neutral surface is adopted. The governing equations are deduced for FGB composed by two or four different materials and then discretized by the proposed method. The present model is verified by comparison of the deflection with the solution obtained in literature and a self-programmed finite element code. The deformation of the beam with different gradient indexes and boundary conditions is also investigated.

A novel meshless particle method for nonlocal analysis of two-directional functionally graded nanobeams

Based on the Reddy–Bickford beam theory (RBBT), a comprehensive study on high-order bending, buckling and free vibration of two-directional functionally graded (FG) nanobeams is presented. The variation of the material properties is assumed to obey arbitrary power-law form in both axial and thickness directions. The equations of motion are derived using Hamilton’s principle, and small-scale effects are captured by nonlocal elasticity theory of Eringen. The RBBT formulation leads to complicated governing equations, and 2D FGM introduces additional stiffness terms. Accordingly, the governing equations cannot be solved by classical analytical schemes, and thus, symmetric smoothed particle hydrodynamics (SSPH) meshless method is adopted as an efficient numerical solution approach. The revised super-Gauss function is used as the kernel function. To validate the developed SSPH code, benchmark problems are studied and the results are compared to the analytical solutions. In this context, excellent agreement is observed. Several numerical examples are included to illustrate the effects of gradient indexes, boundary conditions, size scale parameters, aspect and elastic modulus ratios on static and dynamic responses of two-directional FG nanobeams.

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.

Size dependent buckling analysis of nano sandwich beams by two schemes

Within the framework of Timoshenko beam theory, the buckling of nano sandwich beams is developed. The material properties are assumed to vary arbitrarily in both axial and thickness directions. These types of beams are referred to as bi-directional functionally graded (BDFG) beams. Two types of nano sandwich beams with different material distribution patterns and immovable supports are considered. Since the size effects play a significant role in mechanical behavior of nanostructures, the small-scale effects are captured by Eringen’s nonlocal theory of elasticity. The governing equations are derived using the variational formulation. Symmetric smoothed particle hydrodynamics (SSPH) and the Galerkin method are adopted as numerical solution approaches. As a truly meshless method, the convergence of the SSPH technique mainly depends on the smoothing length value and distribution of particles in the compact support domain of the kernel function. The Revised Super Gauss Function is used as the kernel function and an optimum value for the smoothing length that bears the fastest convergence rate is obtained. The solution methods are verified through benchmark problems found in the literature. Numerical and illustrative results show that various parameters, including the aspect ratio, nonlocal parameter, gradient indexes, and cross-sectional types have significant effects on the buckling responses of BDFG nano sandwich beams.

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..

Flexure Analysis of Laminated Composite and Sandwich Beams Using Timoshenko Beam Theory

The static behaviour of laminated composite and sandwich beams subjected to various sets of boundary conditions is investigated by using the Timoshenko beam theory and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. In order to solve the problem, a SSPH code which consists of up to sixth order derivative terms in Taylor series expansion is developed. The validation and convergence studies are performed by solving symmetric and anti-symmetric cross-ply composite beam problems with various boundary conditions and aspect ratios. The results in terms of mid-span deflections, axial and shear stresses are compared with those from previous studies to validate the accuracy of the present method. The effects of fiber angle, lay-up and aspect ratio on mid-span displacements and stresses are studied. At the same time, the problems not only for the convergence analysis but also for the extensive analysis are also solved by using the Euler-Bernoulli beam theory for comparison purposes.

Device Simulation using Symmetric Smoothed Particle Hydrodynamics

We have applied symmetric smoothed particle hydrodynamics (SSPH) to electronic structure calculations for high electron mobility transistors (HEMTs). In layered structures such as field effect transistors (FETs), and especially HEMTs, the current density is mainly dependent on the electron mobility and the electronic field near the gate, where both can be taken to be constant. The relation between the channel current and the applied gate voltage can be obtained by a one-dimensional calculation. Then, it is easy to apply SSPH to evaluate the simple quantum properties of a device. We mainly focus on the I-V characteristics, which are typical device features. The electronic structure of a HEMT was calculated using both SSPH and finite-difference (FD) methods. The results from SSPH calculations are in good agreement with those from the FD method, and the accuracy of SSPH is similar to that of FD. In a simple example, where three particles are employed in the SSPH domain, we show there is an equivalence to the three-point method in FD.

Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory

This paper presents the static behaviour of two-directional functionally graded (FG) sandwich beams subjected to various sets of boundary conditions by using a quasi-3D shear deformation theory and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The SSPH code, which is developed based on the present formulation of the FG sandwich beam, is validated by solving a simply supported conventional functionally graded beam problem. Numerical results which are in terms of maximum dimensionless transverse deflections, dimensionless axial, normal and shear stresses are compared with the analytical solutions and the results from previous studies. Various FG sandwich beam structures are investigated by considering different aspect ratios (L/h) and sets of boundary conditions and using power-law distribution.

Numerical simulation of metal removal in laser drilling using symmetric smoothed particle hydrodynamics

A simple numerical model is proposed for predicting the penetration depth of metal laser drilling. A simplified 2D axisymmetric model for transient metal laser drilling is introduced. Strong-form of Symmetric Smoothed Particle Hydrodynamics (SSPH) method is used to harness its significant reduction in computational time. The 2D axisymmetric domain is discretized, then SSPH formulation is used to obtain shape functions. Collocation method is used to discretize governing and boundary conditions equations to construct the global stiffness matrix. Laser beam is assumed to be continuous wave with Gaussian distribution. MATLAB code is constructed for numerical simulation, and the results are compared with published work. A good agreement is shown, and thus the proposed numerical model is found to be computationally efficient and accurate standalone platform for predicting the penetration depth of metal laser drilling process.

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.

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.

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.

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.

Development of the SSPH Method for Real-Space Electronic Structure Calculations

We have applied symmetric smoothed particle hydrodynamics (SSPH) to real-space electronic structure calculations. The electronic structure of a simple atom such as hydrogen, helium and lithium is calculated. The results of SSPH are compared to that of the higher-order finite difference method (FD), and are in good agreement with that of FD. Our results indicate that SSPH can be applicable to the electronic structure calculations that require high accuracy.