Analytical expression of RKPM shape functions
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ORIGINAL PAPER
Analytical expression of RKPM shape functions Lei Zhang1 · Shaoqiang Tang1 · Wing Kam Liu2 Received: 12 April 2020 / Accepted: 3 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we derive an analytical expression for reproducing kernel particle method (RKPM) shape functions. Based on this, we propose a necessary and sufficient stability condition for general RKPM in arbitrary function space, and illustrate with degenerate cases. By selecting proper basis vectors and the support of the kernel functions, we demonstrate that the RKPM framework allows generating many kinds of shape functions, including the Lagrangian, B-spline and NURBS shape functions. Keywords Reproducing kernel particle method (RKPM) · Stability condition · Lagrangian shape functions · B-spline and NURBS
1 Introduction Reproducing Kernel Particle Method (RKPM) [1,2] represents an important class of meshfree methods, avoiding the difficulties of mesh generation, refinement and distortion in large deformations. By selecting appropriate kernel functions and basis vector, one can construct RKPM shape functions with any desired regularity. Besides large deformations [3–5], RKPM has been adopted in structural acoustics, computational fluid dynamics, peridynamics [6], micromechanics [7,8] and fractional order diffusive problems [9], among many other applications. RKPM begins the formulation with the kernel estimate of a function, and then modifies the kernel function to impose reproducing conditions. In this way, RKPM shape functions attain the consistency conditions over the whole domain under consideration. Complexity in the construction of the shape functions may arise in the inversion of a moment
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Shaoqiang Tang [email protected] Lei Zhang [email protected] Wing Kam Liu [email protected]
1
HEDPS and LTCS, College of Engineering, Peking University, Beijing 100871, China
2
Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd., Evanston, IL 60208-3111, USA
matrix, which reduces the computational efficiency and hinders further analysis such as stability analysis and derivative calculations. Analytical expression for the matrix inversion is known only for linear RKPM shape functions, in one space dimension by Liu et al. [1], and multiple space dimensions by Zhou et al. [10,11]. In this paper, we derive a general analytical expression of RKPM shape functions. This allows us to analyze the stability, i.e., the positive definitiveness of the moment matrix readily. Note that necessary stability conditions were reported in [7,12,13]. In the polynomial function space, [14] proposed a necessary condition by the quadratic form of the moment matrix. It was claimed to be also a sufficient condition if the space dimension is one. In this work, we formulate a necessary and sufficient stability condition, which is valid for arbitrary function space, and in arbitrary space dimensions. Furthermore, we may show explicitly that many kinds of shape functions are u
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