SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations
SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations
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DOI：
10.1007/s00466-007-0209-3
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发表时间：
2007-08
影响因子：
4.1
通讯作者：
R. Batra;G. M. Zhang
中科院分区：
文献类型：
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作者：
R. Batra;G. M. Zhang
我们提出了一种新的和简单的技术，称为对称光滑粒子流体动力学（SSPH）的方法来构造基函数的无网格方法，只使用粒子的位置。这些基函数被发现是类似于那些在有限元法（FEM），除了基础上的衍生物的功能不需要获得通过微分那些功能。当然，函数的导数的基可以通过对函数的基进行微分来获得，如在FEM和无网格方法中。这些基函数用于数值求解两个平面应力/应变弹性静力学问题，通过使用配置法或弱制定的问题定义在一个子区域的区域所占据的身体，后者通常被称为无网格局部彼得罗夫-伽辽金（MLPG）方法。对于所研究的两个边值问题，发现在SSPH方法中直接导出试探解的一阶导数的基的弱公式（即，不是通过对试解的基函数求微分得到的）给出位移中的L2误差范数的最小值，而采用边界值问题的强公式化的配点法具有最大的L2误差范数值。使用弱公式的数值解比使用强公式的数值解需要更多的CPU时间。我们还计算了位移的L2-误差范数，通过改变粒子的数量，用于数值计算出现在问题的弱公式中的区域积分的高斯点的数量，用于生成SSPH基的核函数的紧支撑的半径，用于构造SSPH基的完全单项式的阶，以及在矩形问题域的角上的点处使用的边界条件。建议在求解二维弹性静力问题时，采用MLPG公式，在MLPG公式中，求试解的导数时不需要对SSPH基函数进行微分。
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.