Investigation of highly efficient algorithms for solving linear equations in the discontinuous deformation analysis method

The efficiency of solving equations plays an important role in implicit-scheme discontinuous deformation analysis (DDA). A systematic investigation of six iterative methods, namely, symmetric successive over relaxation (SSOR), Jacobi (J), conjugate gradient (CG), and three preconditioned CG methods (ie, J-PCG, block J-PCG [BJ-PCG], and SSOR-PCG), for solving equations in threedimensional sphere DDA (SDDA) is conducted in this paper. Firstly, simultaneous equations of the SDDA and iterative formats of the six solvers are presented. Secondly, serial and OpenMP-based parallel computing numerical tests are done on a 16-core PC, the result of which shows that (a) for serial computing, the efficiency of the solvers is in this order: SSOR-PCG > BJ-PCG > J-PCG > SSOR>J > CG, while for parallel computing, BJ-PCG is the best solver; and (b) CG is not only the most sensitive to the ill-condition of the equations but also the most time consuming under both serial and parallel computing. Thirdly, to estimate the effects of equation solvers acting on SDDA computations, an application example with 10 000 spheres and 200 000 calculation steps is simulated on this 16-core PC using serial and parallel computing. The result shows that SSOR-PCG is about six times faster than CG for serial computing, while BJ-PCG is about four times faster than CG for parallel computing. On the other hand, the whole computation time using BJ-PCG for parallel computing is 3.37 hours (ie, 0.061 s per step), which is about 36 times faster than CG for serial computing. Finally, some suggestions are given based on this investigation result. K E Y W O R D Shigh efficiency, iterative method, linear equation solver, three-dimensional discontinuous deformation analysis, sphere DDA | INTRODUCTIONBoth the discontinuous deformation analysis (DDA) method 1 and discrete element method (DEM) 2 are proposed to simulate the mechanical behaviors of discrete block systems. The DDA and DEM have extensive prospects in engineering application since they can automatically trace and simulate block rotation, fracture opening, and complete detachments, which are often included in real jointed rock masses failure process. 3,4 Compared with the DEM, the DDA has more rigorous mathematical theory and higher computational accuracy. 5,6 Although the DDA is relatively young, it has been widely applied in various aspects of geotechnical engineering, such as tunnel engineering, 7 rock hydraulic fracturing, 8,9 rock burst analysis, 10 and landslide. 11 However, the computational efficiency of the DDA is much lower than that in the DEM, although there has been great progress within DDA researches over the past three decades. 12 The main reason for this is that the DEM uses an explicit time integration scheme and the second law of Newton is used to calculate the displacement of each block independently, while the DDA often uses an implicit solution method in which simultaneous equilibrium equations are assembled by applying the minimum potential energy principle and solved in each calcul...

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Highly efficient iterative methods for solving linear equations of three‐dimensional sphere discontinuous deformation analysis

Huang

et al. 2020

“…With increasing numbers of blocks and contact pairs, more OCI cycles may be necessary to achieve convergence (of no‐tension and no‐penetration conditions), with each cycle being associated with the inversion of a large matrix, which lead to drastic increases in the computational demands . Therefore, to enhance the computational efficiency in solving simultaneous equations, parallelization methods such as OpenMP, GPU, and MPI have been previously applied in DDA . Alternatively, the central difference (CD) approach in DEM and FDEM is an explicit and efficient way to solve large‐scale discontinuous problems, in which the size of time step and damping coefficients should be carefully chosen to ensure numerical stability.…”

Section: Introductionmentioning

confidence: 99%

“…36 Therefore, to enhance the computational efficiency in solving simultaneous equations, parallelization methods such as OpenMP, GPU, and MPI have been previously applied in DDA. [37][38][39] Alternatively, the central difference (CD) approach in DEM 1 and FDEM 40 is an explicit and efficient way to solve large-scale discontinuous problems, in which the size of time step and damping coefficients should be carefully chosen to ensure numerical stability. An explicit approach based on the velocity verlet algorithm has been implemented in DDA to improve its computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

Modified predictor‐corrector solution approach for efficient discontinuous deformation analysis of jointed rock masses

Zheng

Leung

Zhu

et al. 2018

Summary The high computational costs associated with the implicit formulation of discontinuous deformation analysis (DDA) have been one of the major obstacles for its implementation to engineering problems involving jointed rock masses with large numbers of blocks. In this paper, the Newmark‐based predictor‐corrector solution (NPC) approach was modified to improve the performance of the original DDA solution module in modeling discontinuous problems. The equation of motion for a discrete block system is first established with emphasis on the consideration of contact constraints. A family of modified Newmark‐based predictor‐corrector integration (MNPC) scheme is then proposed and implemented into a unified analysis framework. Comparisons are made between the proposed approach and the widely used constant acceleration (CA) integration approach and central difference (CD) approach, regarding the stability and numerical damping features for a single‐degree‐of‐freedom model, where the implications of the proposed approach on open‐close iteration are also discussed. The validity of the proposed approach is verified by several benchmarking examples, and it is then applied to two typical problems with different numbers of blocks. The results show that the original CA approach in DDA is efficient for the simulation of quasi‐static deformation of jointed rock masses, while the proposed MNPC approach leads to improved computational efficiency for dynamic analysis of large‐scale jointed rock masses. The MNPC approach therefore provides an additional option for efficient DDA of jointed rock masses.

A cover‐based contact detection approach for irregular convex polygons in discontinuous deformation analysis

Zhuang

Zheng

et al. 2020

Irregular polygon shapes (eg, with small edges or small angles) are usually encountered in the contact simulation of discrete block systems. Treatment of irregular polygons in contact detection process has critical effects on the robustness and efficiency of the discontinuous computation approach. The present work proposes a cover‐based strategy to detect and solve contacts of irregular convex polygons in a robust and efficient way. Contact constraints of two polygons are represented by vertex‐edge and edge‐vertex contact covers in 2D. Two loops, namely vertex‐edge loop and edge‐vertex loop, and two filter criteria, namely the entrance filter criterion and the distance filter criterion, are used to establish the potential contact cover list of two neighbor polygons. The initial active and closed contact covers are chosen based on block configuration at the beginning of the step and they are then updated in the open‐close iteration process using proposed criteria. This strategy is implemented in discontinuous deformation analysis. The robustness of the proposed cover‐based approach and the conventional type‐based approach in handling contact of irregular blocks is verified first. Then, the contact analysis efficiency of the cover‐based approach with different contact tolerances is evaluated. This cover‐based method can be extended to 3D case for efficient and robust contact analysis of irregular polyhedral blocks.