Normalized SPH without boundary deficiency and its application to transient solid mechanics problems
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ORIGINAL PAPERS
Normalized SPH without boundary deficiency and its application to transient solid mechanics problems Yihua Xiao
. Xiangfu Hong . Ziqiang Tang
Received: 8 February 2020 / Accepted: 1 October 2020 Ó Springer Nature B.V. 2020
Abstract Smoothed particle hydrodynamics (SPH) method is a powerful tool for modeling solid mechanics problems, especially for large deformation problems. However, it suffers from boundary deficiency and difficulty of boundary condition treatment. In this work, a normalized SPH method is proposed to overcome these problems. The method is based on a newly developed normalized particle approximation. To derive this particle approximation, a normalized kernel approximation which is accurate for derivatives of linear functions everywhere in a problem domain is constructed, and all integral terms of the normalized kernel approximation including boundary terms are discretized by particle summations. The normalized particle approximation is free of matrix inversion, consequently attractive in computational stability and simplicity compared with other corrective particle approximations. Its approximation accuracy is demonstrated by calculating derivatives of test functions. Based on this particle approximation, the formulation of the normalized SPH method for transient solid mechanics problems is derived. Moreover, a direct method of treating traction boundary conditions is presented by making use of the boundary term of the normalized particle approximation. The accuracy and capability of the normalized SPH method are validated Y. Xiao (&) X. Hong Z. Tang School of Mechatronics and Vehicle Engineering, East China Jiaotong University, Nanchang 330013, China e-mail: [email protected]
by the calculation of elastic wave propagation in solids and compared with commonly used SPH method. Keywords SPH Solid mechanics Normalized particle approximation Boundary deficiency Treatment of boundary condition
1 Introduction Smoothed particle hydrodynamics (SPH) [1, 2] is a meshfree Lagrangian particle method. It is capable of modeling large deformation, fragmentation and material mixing. Currently, it has been extensively used to study engineering and science problems, such as high velocity impacts [3–6], fluid flows [7–9] and fluid– structure interactions [10–12]. The recent developments and applications of SPH were comprehensively reviewed by Ye et al. [13]. Though SPH achieved a series of successes in practical applications, it still encounters some fundamental problems that require further investigation. One of such problems is the boundary deficiency of particle approximation. In commonly used SPH (called as standard SPH later) method, the particle approximation is a discrete form of kernel approximation which estimates a field function at a given particle by an integral over a support domain surrounding the particle. It gives poor approximation
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Meccanica
accuracy for particles near boundaries of a problem domain, since t
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