A robust contact detection algorithm based on the Contact Theory in the three-dimensional discontinuous deformation analysis

Abstract

The two-dimensional discontinuous deformation analysis (DDA), as a discrete numerical method, has been successfully used to simulate many rock mechanics and rock engineering problems. However, for the development of the three-dimensional DDA (3D-DDA), the contact detection between polyhedral blocks has long been one of the key difficulties awaited to be solved. A universal Contact Theory was proposed by Shi in 2015, in which the complicated contact relationship of two any-shaped blocks is represented by a simple relationship between a reference point and an entrance block, along with a concept of the contact cover. In the present study, a robust contact detection algorithm in the 3D-DDA for convex polyhedral blocks based on the Contact Theory is proposed. In this algorithm, all the six basic contact types of polyhedral blocks are functionally treated as two contact types, namely, the vertex-to-face contact and the non-parallel edge-to-edge contact; meanwhile, the vertex-to-vertex and vertex-to-edge contacts are specially handled to guarantee the effectiveness and efficiency of the algorithm. Two special contact cases with the parallel edge-to-edge contact and the vertex-to-vertex contact, respectively, and two multi-block cases, in which various contact types and the transition between them are involved, are simulated to verify the effectiveness and robustness of the proposed contact detection algorithm. Moreover, failure simulations of sliding and toppling slopes with massive rock blocks are conducted and in the sliding case the effect of a retaining wall is also simulated. Results indicate that, the proposed algorithm can handle the contact detections of convex polyhedral blocks effectively under critical and complex conditions, which builds a good precondition for the successful development and practical application of the 3D-DDA method.

Introduction

The discontinuous deformation analysis (DDA)¹ as a numerical simulation method has widely attracted the attentions of researchers and engineers, since it was put forward more than 30 years ago. It is based on the mechanics of discontinuous medium and calculates the static or dynamic behaviors of discontinuous systems of two-dimensional polygonal or three-dimensional polyhedral blocks. It is an implicit displacement-based method theoretically parallel to the finite element method (FEM) and a discrete method similar to the distinct element method (DEM) to simulate discontinuous mediums. With the development of its theory and the improvement of its calculation accuracy and capability,2, 3, 4 the two-dimensional DDA (2D-DDA) has been broadly used to simulate rock mechanics and rock engineering problems.5, 6, 7, 8, 9, 10, 11 In terms of solving dynamic problems,¹² the 2D-DDA has been successfully employed to simulate stress wave propagations and the induced rock failures,13, 14, 15, 16 the seismic response of rock mass structures,17, 18, 19, 20, 21, 22 the drilling blasting of rock,23, 24, 25, 26 and rock burst,^27,28 etc.. However, the theoretical development and application of the three-dimensional DDA (3D-DDA)²⁹ is far behind, mainly restricted by the complexity of the contact detection between three-dimensional polyhedral blocks, the efficiency of contact open-close iteration³⁰ and equation solving, and some other factors, among which the contact detection is a primary issue to be faced.

Contact extensively exists in rock mechanics and rock engineering problems where the compression, sliding, and impact take place between discrete bodies or along discontinuous surfaces. Proper treatment of the contact, namely, to find out where, when and what kind of contact occurs, and to apply contact springs accordingly, e.g., by the penalty method,³⁰ is the basis of discontinuous medium simulation. Obviously, a correct and robust contact detection algorithm is the prerequisite for the correct application of contact springs and the effective handling of contact problems of block systems. At present, the contact detection algorithms generally used for 3D blocks are the direct method and the common plane (CP) method,³¹ however, both with certain deficiencies. The direct method, which is commonly used in the DDA, encounters problems of low efficiency and possibilities to miss or misjudge contacts under critical and complex conditions, while the CP method, which is mainly used in the DEM, shows poor performance in dealing with non-face contact. Nezami et al. proposed two methods to obtain the common plane between polygonal particles, namely, the fast common plane (FCP)³² and the shortest link method (SLM).³³ Both methods dramatically reduced the search space for the common plane and the SLM is faster than the FCP. The SLM even runs up to 17 times faster than the conventional CP method. As for the 3D-DDA, the CP method also has been tried as a complementary approach.³⁴ Furthermore, some researchers coupled and modified the direct and the CP methods to improve the contact detection efficiency of 3D blocks. For example, a penetration edge method³⁵ and an incision body method³⁶ were developed, however, being limited to be applied to simple convex polyhedral blocks, while Liu et al. further introduced a probe to the movement of block corners to identify the correctness of contact, and improved the ability to simulate a large number of blocks with the 3D-DDA.³⁷

In fact, the direct method has been greatly developed and improved by itself for contact treatment in the 3D-DDA. Jiang and Yeung derived the contact formulas for the vertex-to-face contact and expounded the open-close iteration criteria and operations for different changes of the contact state.³⁸ Independently, Wu et al. presented a vertex-to-face contact model based on the topology, and derived the formulas for the normal contact spring by minimizing the total potential energy.³⁹ Thereafter, Yeung et al. enriched the contact form with the edge-to-edge contact via vector analysis and simulated three test cases to demonstrate the applicability of the contact model.⁴⁰ Wu presented a new edge-to-edge contact calculating algorithm that can reduce the complexity of 3D contact calculations and improve the efficiency of calculation procedure.⁴¹ Beyabanaki et al. transformed all the six basic contact types (vertex-to-vertex, vertex-to-edge, vertex-to-face, parallel/non-parallel edge-to-edge, edge-to-face, and face-to-face) of 3D polyhedral blocks into the form of vertex-to-face contact, and derived the formulas for this contact model in detail.⁴² Subsequently, some researchers tried to modify the contact detection algorithm by the direct method and improve its efficiency. Keneti et al. improved the approaching face method of contact detection for convex blocks, and made the contact patterns easy to identify based on the theory of segment intersection.⁴³ Ahn and Song used virtual spheres inserted into the contact vertices to stabilize the process of contact detection, and this algorithm shows a better performance in reducing the open-close iteration times.⁴⁴ Mousakhani and Jafari provided an efficient algorithm of edge-to-edge contact model to judge the type and location of contacts that can handle different kinds of contact.⁴⁵ The above studies are all for convex blocks, whereas Zhang et al. further extended the capacity of the 3D-DDA for concave blocks.⁴⁶ In general, the direct method is a logically simple and intuitional method. It is more prone to errors in the actual implementation because of its lack of completeness in theory to detect all possible contacts under critical and complex conditions.

Shi⁴⁷ put forward the Contact Theory, and this new theory is expected to solve the contact detection problems of all kinds of geometries completely.⁵⁷ Afterwards, Zheng et al. presented a fast direct search algorithm in the 3D-DDA by classifying the contact types of vertex-to-face and crossing edge-to-edge into four contact forms based on the Contact Theory. This improved algorithm was compared with the direct method and the fast common plane method to show its efficiency of contact detection by modeling different contact types.⁴⁸ In 2018, Zhang et al. reviewed the work of many researchers on the contact detection algorithm for the 3D-DDA during the past decade, and used the first entrance approach from the Contact Theory to determine the suitable state and force of each contact.⁴⁹ In addition, Lin et al. tried to interpret the Contact Theory in detail and suggested rough strategies for the computer programming of this theory.⁵⁰ So far, the research and application of the Contact Theory in the 3D-DDA are far to demonstrate the robustness of this theory and to make the 3D-DDA practically used.

In the present paper, a contact detection algorithm for convex polyhedral blocks based on the Contact Theory is introduced within the 3D-DDA framework. First, the contact detection algorithm is described in detail in Section 2. Then, in Section 3, two special cases of critical contact conditions and two multi-block examples are simulated to verify the effectiveness and robustness of the proposed contact detection algorithm. Thereafter, failure simulations of slopes with massive rock blocks are conducted in Section 4. Finally, conclusions are drawn in Section 5.