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.     

Modified smoothed particle hydrodynamics method and its application to transient problems

A modification to the smoothed particle hydrodynamics method is proposed that improves the accuracy of the approximation especially at points near the boundary of the domain. The modified method is used to study one-dimensional wave propagation and two-dimensional transient heat conduction problems.This work was supported by the ONR grant N00014-98-1-0300 and the ARO grant DAAD19-01-1-0657 to Virginia Polytechnic Institute and State University, and the AFOSR MURI grant to Georgia Tech that awarded a subcontract to Virginia Polytechnic Institute and State University. Opinions expressed in the paper are those of authors and not of the funding agencies.     

Search algorithm,and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method

We first present a nonuniform box search algorithm with length of each side of the square box dependent on the local smoothing length, and show that it can save up to 70% CPU time as compared to the uniform box search algorithm. This is especially relevant for transient problems in which, if we enlarge the sides of boxes, we can apply the search algorithm fewer times during the solution process, and improve the computational efficiency of a numerical scheme. We illustrate the application of the search algorithm and the modified smoothed particle hydrodynamics (MSPH) method by studying the propagation of cracks in elastostatic and elastodynamic problems. The dynamic stress intensity factor computed with the MSPH method either from the stress field near the crack tip or from the J-integral agrees very well with that computed by using the finite element method. Three problems are analyzed. One of these involves a plate with a centrally located crack, and the other with three cracks on plates’s horizontal centroidal axis. In each case the plate edges parallel to the crack are loaded in a direction perpendicular to the crack surface. It is found that, at low strain rates, the presence of other cracks will delay the propagation of the central crack. However, at high strain rates, the speed of propagation of the central crack is unaffected by the presence of the other two cracks. In the third problem dealing with the simulation of crack propagation in a functionally graded plate, the crack speed is found to be close to the experimental one.     

A corrective smoothed particle method for boundary value problems in heat conduction

Combining the kernel estimate with the Taylor series expansion is proposed to develop a Corrective Smoothed Particle Method (CSPM). This algorithm resolves the general problem of particle deficiency at boundaries, which is a shortcoming in Standard Smoothed Particle Hydrodynamics (SSPH). In addition, the method’s ability to model derivatives of any order could make it applicable for any time‐dependent boundary value problems. An example of the applications studied in this paper is unsteady heat conduction, which is governed by second‐order derivatives. Numerical results demonstrate that besides the capability of directly imposing boundary conditions, the present method enhances the solution accuracy not only near or on the boundary but also inside the domain. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States.     

Smoothed particle hydrodynamics for numerical simulation of underwater explosion

 Underwater explosion arising from high explosive detonation consists of a complicated sequence of energetic processes. It is generally very difficult to simulate underwater explosion phenomena by using traditional grid-based numerical methods due to the inherent features such as large deformations, large inhomogeneities, moving interface and so on. In this paper, a meshless, Lagrangian particle method, smoothed particle hydrodynamics (SPH) is applied to simulate underwater explosion problems. As a free Lagrangian method, SPH can track the moving interface between the gas produced by the explosion and the surrounding water effectively. The meshless nature of SPH overcomes the difficulty resulted from large deformations. Some modifications are made in the SPH code to suit the needs of underwater explosion simulation in evolving the smoothing length, treating solid boundary and material interface. The work is mainly focused on the detonation of the high explosive, the interaction of the explosive gas with the surrounding water, and the propagation of the underwater shock. Comparisons of the numerical results for three examples with those from other sources are quite good. Major features of underwater explosion such as the magnitude and location of the underwater explosion shock can be well captured. Received: 2 April 2002 / Accepted: 20 September 2002     

Development of smoothed particle hydrodynamics method and its application in the hydrodynamics of condensed matter

An improved smoothed particle hydrodynamics (SPH) method is described; in this method, the solution to the Riemann problem in strength media is described. Generalization of this approach to solving heat conduction problems is performed. The improved SPH method is used to solve a wide range of problems. Problems of heat conduction and volume energy release accompanied by spallation effects, simulation of high speed perforation, and propagation of failure waves in brittle materials are considered. Shock wave compression of porous materials and diffraction of detonation waves in heterogeneous explosives are simulated on the mesostructure scale.     

On the random differential quadrature (RDQ) method: consistency analysis and application in elasticity problems

Differential Quadrature (DQ) is an efficient derivative approximation technique but it requires a regular domain with uniformly arranged nodes. This restricts its application for a regular domain only discretized by the field nodes in a fixed pattern. In the presented random differential quadrature (RDQ) method however this restriction of the DQ method is removed and its applicability is extended for a regular domain discretized by randomly distributed field nodes and for an irregular domain discretized by uniform or randomly distributed field nodes. The consistency analysis of the locally applied DQ method is carried out, based on it approaches are suggested to obtain the fast convergence of function value by the RDQ method. The convergence studies are carried out by solving 1D, 2D and elasticity problems and it is concluded that the RDQ method can effectively handle regular as well as irregular domains discretized by random or uniformly distributed field nodes.     

The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems

In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.     

A boundary element-free method (BEFM) for three-dimensional elasticity problems

This study combines the boundary integral equation (BIE) method and improved moving least-squares (IMLS) approximation to present a direct meshless boundary integral equation method, the boundary element-free method (BEFM) for three-dimensional elasticity. Based on the improved moving least-squares approximation and the boundary integral equation for three-dimensional elasticity, the formulae of the boundary element-free method are given, and the numerical procedure is also shown. Unlike other meshless boundary integral equation methods, the BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus giving it a greater computational precision. Three selected numerical examples are presented to demonstrate the method.Aknowledgement The work in this project was fully supported by a grant from the Research Grants Council (RGC) of the Hong Kong Special Administrative Region, China (Project No. CityU 1011/02E).The work that is described in this paper was supported by Project No. CityU 1011/02E, which was awarded by the Research Grants Council of the Hong Kong Special Administrative Region, China. The authors are grateful for the financial support.     

SPH方法在剪切式碰撞能量吸收器中的应用

剪切式碰撞能量吸收器是提高汽车被动安全性的一种实用新型设计。建立了剪切式碰撞能量吸收器数值分析模型,采用SPH方法数值模拟了碰撞吸能器的碰撞过程,研究了其碰撞吸能特性,并通过不同碰撞速度的台车碰撞试验进行了试验验证研究。数值模拟结果同试验结果相符,表明SPH方法在碰撞吸能器性能研究中是行之有效的数值计算方法。同时分析了在碰撞吸能器的复杂碰撞情况下,SPH方法相比有限元法的优势。     

Analysis of functionally graded plates using an edge-based smoothed finite element method

An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap (DSG) technique using triangular meshes (ES-DSG) was recently proposed to enhance the accuracy of the existing FEM with the DSG for analysis of isotropic Reissner/Mindlin plates. In this paper, the ES-DSG is further formulated for static, free vibration and buckling analyses of functionally graded material (FGM) plates. The thermal and mechanical properties of FGM plates are assumed to vary across the thickness of the plate by a simple power rule of the volume fractions of the constituents. In the ES-DSG, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains associated with the edges of the elements. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to demonstrate the performance of the present formulation for FGM plates.     

A meshfree weak-strong (MWS) form method for time dependent problems

A meshfree weak-strong (MWS) form method, which is based on a combination of both the strong form and the local weak form, is formulated for time dependent problems. In the MWS method, the problem domain and its boundary are represented by a set of distributed field nodes. The strong form or the collocation method is used to discretize the time-dependent governing equations for all nodes whose local quadrature domains do not intersect with natural (derivative or Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The natural boundary conditions can then be easily imposed to produce stable and accurate solutions. The moving least squares (MLS) approximation is used to construct the meshfree shape functions in this study. Numerical examples of the free vibration and dynamic analyses of two-dimensional structures as well as a typical microelectromechanical system (MEMS) device are presented to demonstrate the effectivity, stability and accuracy of the present MWS formulation.     

光学玻璃材料动态冲击试验与鸟撞研究

针对多光谱硫化锌(zinc sulfide,ZnS)光学玻璃材料用于飞行器时的鸟撞问题进行了研究.对多光谱硫化锌玻璃进行了中高应变率下的压缩试验获得其材料属性.鸟体采用光滑粒子流体动力学(smoothed particle hydrodynamics,SPH)方法建模,引入Gruneisen状态方程定义鸟体本构模型.对...     

Element-free Galerkin method using directed graph and its application to creep problems

The element-free Galerkin method (EFGM) is one of the meshless methods proposed by Belytschko et al. Since node-element connectivities used in the finite element method (FEM) are not needed in the EFGM, the EFGM is expected to be applied to many problems of the continuum mechanics and to be utilized for a tool in a CAE system instead of the FEM. However the EFGM requires more CPU time to search nodes of the MLSM than the FEM. In this paper, the method of the directed graph and the Delaunay triangulation are respectively used for searching nodes and the division of the integral domain respectively. These techniques are useful for saving the CPU time and the simplification of the analysis for the EFGM. Furthermore, the EFGM has not been applied to nonlinear problems such as creep problems under elevated temperature. In this paper, the EFGM using the method of the directed graph and the Delaunay triangulation is applied to several creep problems. The CPU times for the analyses are reduced by the proposed EFGM. The results obtained from the EFGM analyses agree well with those of the FEM.     

SGBEM analysis of crack-particle(s) interactions due to elastic constants mismatch

The effects of elastic constants mismatch on the interaction between a propagating crack and single or multiple inclusions in brittle matrix materials are investigated using numerical simulations. The simulations employ a quasi-static crack-growth prediction tool based upon the symmetric-Galerkin boundary element method (SGBEM) for multiregions, a modified quarter-point crack-tip element, the displacement correlation technique for evaluating stress intensity factors (SIFs), and the maximum principal stress criterion for crack-growth direction. It is shown that, even with this simple method for calculating SIF, the crack-growth prediction tool is both highly accurate and computationally effective. This is evidenced by results for the case of a single inclusion in an infinite plate, where the SGBEM results for the SIFs show excellent agreement with known analytical solutions. The simulation results for crack growth and stress intensity behaviors in particulate media are very stable. The crack-tip shielding and amplification behaviors, as seen in similar studies using other numerical approaches, can be clearly observed.     

Analyzing three-dimensional potential problems with the improved element-free Galerkin method

The potential problem is one of the most important partial differential equations in engineering mathematics. A potential problem is a function that satisfies a given partial differential equation and particular boundary conditions. It is independent of time and involves only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neumann, or mixed conditions. When potential problems are very complex, both in their field variable variation and boundary conditions, they usually cannot be solved by analytical solutions. The element-free Galerkin (EFG) method is a promising method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by employing improved moving least-squares (IMLS) approximation, we derive the formulas for an improved element-free Galerkin (IEFG) method for three-dimensional potential problems. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed.     

The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particle methods

In the previous papers (Kim et al. Submitted for publication, Oh et al. in press), for uniformly or locally non-uniformly distributed particles, we constructed highly regular piecewise polynomial particle shape functions that have the polynomial reproducing property of order k for any given integer k ≥ 0 and satisfy the Kronecker Delta Property. In this paper, in order to make these particle shape functions more useful in dealing with problems on complex geometries, we introduce smooth-piecewise-polynomial Reproducing Polynomial Particle shape functions, corresponding to the particles that are patch-wise non-uniformly distributed in a polygonal domain. In order to make these shape functions with compact supports, smooth flat-top partition of unity shape functions are constructed and multiplied to the shape functions. An error estimate of the interpolation associated with such flexible piecewise polynomial particle shape functions is proven. The one-dimensional and the two-dimensional numerical results that support the theory are resented. June G. Kim is Visiting Professor of the University of North Carolina at Charlotte.     

Explicit model of dual programming and solving method for a class of separable convex programming problems

An objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.