Computer Methods in Applied Mechanics and Engineering
A meshfree contact-detection algorithm
Introduction
The numerical simulation of contact between two different objects, or different parts of an object is a serious challenge in many engineering applications such as sheet metal forming, vehicle crash-worthiness, impact, penetration and fragmentation, large deformation of shell-type structures, and machining in manufactures, etc. In some situations, for instance, the large deformation of thin shell structures, a so-called self-contact phenomenon may also occur, which is very difficult to model in numerical simulations because of the complex nature of the problem. If the computation is performed without considering self-contact; unrealistic inter-penetration of different parts of the shell can be observed in the numerical computation. Fig. 1 shows such an example – a deformed cylindrical shell under axial compression.
There have been several contact-detection algorithms proposed in literature, such as Benson–Hallquist algorithm proposed by Benson & Hallquist [10], [14], Pinball algorithm proposed by Belytschko et al. [7], [8], Point-in-box search algorithm by Heinstein et al. [15], and the recent bounding box algorithms by Malone & Johnson [28], [29] and Attaway et al. [3], which are designated for parallel computation. Most of the these algorithms are complicated in implementation, especially in parallel computations, because they are involved with multiple vector operations and multiple conditional statements even for a single surface check, or single bounding box check.
Recently, a new generation of numerical methods called “meshfree” methods has emerged, and it has a profound impact on computational mechanics. Three special issues of research papers have been devoted to the subject: the special issue in Computer Methods in Applied Mechanics and Engineering (Vol. 139, 1996) Edited by Liu, Belytschko and Oden, the special issue in Int. J. Numer. Methods Engrg. (Vol. 49 No. 5, 2000) edited by Liu, Idelsohn and Onate; and the special issue in Comput. Mech. (Vol. 2/3, 2000) edited by Chen and Liu. In many meshfree methods, the construction of the approximation requires moment matrix; these include the element-free Galerkin (EFG) method by Belytschko et al. [5], [6], the reproducing kernel particle method (RKPM) by Liu et al. [20], [21], [22], [23], [24], [25], [26], [27], h-p Cloud method by Duarte and Oden [12], [13], and the meshfree local Petrov–Galerkin (MLPG) method by Atluri and Zhu [1], [2].
It has been found that this moment matrix provides a natural indicator to track the surface, or interface of any continuum object. In this paper, we address the fundamental issue of contact algorithm, i.e., contact-detection problem. This problem is deeply rooted in the question on how to accurately represent a geometric object with simple rules, and to detect the contact region by simple criterion. Based on the principle of moment of meshfree interpolation, a new contact-detection algorithm is proposed. Most remarkably, the proposed meshfree contact-detection criterion is a scalar criterion, which is easy to implement, and accurate.
The presentation is organized as follows. In Section 2, the meshfree contact-detection algorithm – a moment method is presented. In Section 3, the mathematical principle of meshfree representation of a geometric object is discussed. A brief outline of constitutive model, time integration and discretization is given in Section 4. The impenetrability and frictional conditions at the contact interface are dealt with in Section 5. Numerical results obtained by the proposed meshfree contact algorithm are reported in Section 6.
Section snippets
The meshfree contact-detection algorithm: a moment method
Before describing the meshfree contact-detection algorithm, it is expedient to recapitulate the basic procedure on how to construct moment matrix in one of the meshfree methods – reproducing kernel particle method [26], [27]. Assume that the window function is given such that for fixed ,where ρ is the dilation parameter, which is the characteristic length of the support size, and in this paper it is sometimes referred to as the radius of the
The Mathematical principle
In this section, the moment method is analyzed in details as we are trying to prove Proposition 2.1. To do so, we first define the continuous moment matrixand its discrete counterpartDefineIt is almost trivial to show 〈,〉 is an inner product in . However, its discrete counterpartis not automatically an inner product, unless certain conditions on
Weak form, discretization, and constitutive model
A total Lagrangian formulation is adopted in the numerical computation. Followed the standard convention that x denotes the spatial coordinate of a material point and X denotes the referential coordinate of the same material point, the displacement of the material point is defined asand the deformation gradient is given by,In large inelastic deformation, the deformation gradient, F, can be decomposed aswhere Fe describes the elastic deformation and rigid body rotation and Fp
Implementation of friction/impenetrability conditions
Contact problem are characterized by the so-called impenetrability condition that needs to be enforced during the computation. In other words, no inter-penetration is allowed between two contacting objects. There are two types of approaches to enforce the impenetrability condition: one is the Lagrangian multiplier method and penalty method (e.g., [14], [17]); the other is the exact enforcement of the penetration condition in a single time step (e.g., [11], [16]). The first approach is more
Numerical examples
To validate the contact-detection algorithm, numerical computations have been carried out testing its viability in computation. We consider the impact problem of the Taylor bar described in Section 2, and the exact problem statement is described in Fig. 13. The projectile is a cylindrical rod with radius, Rs=5 mm, and height, L=60 mm and the rigid block has radius Rm=12 and 10 mm in height. The material's constitutive model is chosen as the viscoplastic solid described in Section 4. The
Concluding remarks
In this paper, a new contact-detection algorithm has been proposed based on the moment method in meshfree discretization. In fact, this moment method is essential to meshfree surface fitting, and meshfree volume representation. The mathematical principle of such contact-detection algorithm is that the determinant of the moment matrix can automatically determine Lagrangian movement of the continuum, and can be used to detect a spatial point on whether or not it belongs to the object of interest.
Acknowledgements
This work is supported by grants from the the Army Research Office, and National Science Foundations. It is also sponsored in part by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAH04-95-2-003/contract number DAAH04-95-C-0008, the content of which does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred.
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