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

Zhang

2007

Comput Mech

Self Cite

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 L 2 -error norm in the displacements while the collocation method employing the strong formulation of the boundary-value problem has the largest value of the L 2 -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 L 2 -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,

Section: Introductionmentioning

confidence: 99%

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

Zhang

2007

Comput Mech

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“…The MLPG method has been employed in [51,52] to analyze stresses near a crack tip in a plate loaded on the cracked side by surface tractions parallel to the crack. A transient linear elastodynamic problem for a cracked plate has been studied by the MSPH method in [53].…”

Section: Crack Problemmentioning

confidence: 99%

Analysis of rubber‐like materials using meshless local Petrov–Galerkin (MLPG) method

Commun. Numer. Meth. Engng.

Porfiri²

2007

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SUMMARYLarge deformations of rubber-like materials are analyzed by the meshless local Petrov-Galerkin (MLPG) method. The method does not require shadow elements or a background mesh and therefore avoids mesh distortion difficulties in large deformation problems. Basis functions for approximating the trial solution and test functions are generated by the moving least-squares (MLS) method. A local mixed total Lagrangian weak formulation of non-linear elastic problems is presented. The deformation gradient is split into deviatoric and dilatational parts. The strain energy density is expressed as the sum of two functions: one is a function of deviatoric strains and the other is a function of dilatational strains. The incompressibility or near incompressibility constraint is accounted for by introducing the pressure field and penalizing the part of the strain energy density depending upon the dilatational strains. Unlike in the mixed finite element formulation, in the MLPG method there is no need for different sets of basis functions for displacement and pressure fields. Results computed with the MLPG method for a few sample problems are found to compare very well with the corresponding analytical solutions.

“…The numerical results clearly show that there is no tensile instability. The MSPH method was used in [34] to study wave propagation in a functionally graded bar with material properties varying continuously, in [35] to analyze crack propagation in a linear elastic plate; and in [36] to investigate the Taylor impact test. We anticipate that the SSPH method will give equally good results for these problems.…”

Section: Elastodynamic Problemmentioning

confidence: 99%

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

Zhang

2008

Comput Mech

Self Cite

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.