Towards an augmented domain decomposition method for nonsmooth contact dynamics models
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Abstract
This paper explores the numerical performances of algorithms enriched by an augmented interface problem in a domain decomposition method dedicated to nonsmooth dynamic systems. Starting from simulations on a single time step, different algorithms are tested on moderate size samples. The analysis of the results leads to an incomplete resolution strategy for solving a timeevolution problem.
Keywords
Discrete elements Multibody Parallel computing Augmented algorithm Distributed memory Granular material LMGC90Mathematics Subject Classification
65M55 70F35 70F40 70E551 Introduction
We are concerned herein in dense discrete systems with a potentialy large number of bodies and of nonsmooth interactions between them (mainly frictional contact), for which granular materials are the main application. The difficulties for experiments to gather data at the microscale level (the scale of the grains), and for the comprehension of the involved phenomena, lead to extensive use of numerical simulations as virtual tests. Simulation for models describing individually all the bodies (modeled for instance as rigid bodies with large displacements and rotations, with a dynamic evolution of movements and of the interaction network between bodies) leads to costly models with a large volume of results. As a first step, the examples in this article will deal with medium sized 2D granular packings.
Section 2 is dedicated to the domain decomposition in the context of granular media and to the formulation of the generic solver. Section 3 presents the augmented interface problem for enriching one of the two stages of the algorithm. Several algorithmic strategies are developed and tested in Sect. 4. Finally a fully multiscale resolution is proposed in Sect. 5. For an evolution granular problem, this amounts to an incomplete resolution at each time step according to a separation of the scales, macro scale for the interface, micro scale inside the subdomains. After the conclusions in the last section, some technical aspects on the parallel solver are developed in “Appendix”.
2 Domain decomposition
2.1 Domain partitioning
Domain partitioning of a discrete element collection is, at each time step or at a userdefined frequency, a partitioning of an interaction graph. The interaction graph consists in nodes associated to grains and edges associated to interactions.
2.2 FETIlike domain decomposition: NSCDD algorithm
The present DDM considers a non overlapping partition of the sample.
Here, the formulation described in Algorithm 1 has been implemented into the LMGC90 platform [8] for timeevolution problems. At each new time step of the incremental solving procedure, the mapping \(H\) and the contact graph have to be updated within a contact detection phase. Eventually, the domain could also be repartitioned according to the new contact graph.
3 Augmented interface problem
The NSCDD method exhibits good parallel efficiency for dense granular dynamics problems [24]. We exemplified in [25] that the global behavior and the micromechanical structure of largescale dense granular systems under biaxial loading are not disturbed by NSCDD substructuring. Moreover, extensibility is recovered when the number of particle dof is large compared to the number of interface dof. Nevertheless the number of iterations increases with the number of particles (both for sequential and parallel algorithms). This phenomenon is related to the nonsmoothness of the considered interactions. Indeed, because of the nonsmooth relations between the internal dof of a subdomain, no condensation process on the interface is allowed. The NSCDD method defines a quasidiagonal linear interface problem without coupling the different interfaces of a subdomain.
Herein, we study the properties of the augmented, or enriched, interface problem with respect to the chosen compatibility operator for the tangent search direction. A generic compatibility operator is \(G_E=H_E\), i.e. the compatibility operator used for solving the nonsmooth dynamics inside the subdomains given by the contact detection phase. The asymptotic study done in [3] shows that—at least without friction—the optimal compatibility operator is the restriction of \(H_E\) to the only active (\(r>0\)) contacts, but this optimal operator is a priori unknown.
The NSCDD enrichment leads to a coupled interface problem. Solving the NSCDD enriched interface problem is time consuming as it requires to solve a global linear problem on the whole domain viewed as a lattice structure with the same connectivity as the contact graph. Due to the distribution of the database per subdomain, and to avoid a costly direct solve, we choose to design a parallel conjugate gradient algorithm close to the one used in classical distributed parallel approach (cf. Appendix).
4 Algorithmic strategies for the enriched NSCDD

\(G_E\) such that \(G_Er_E=H_Er_E\); therefore, only active contacts are taken into account in building \(K_E=G_EG^T_E\),

\(G'_E\) such that \(G'^T_EV_E=0\); with taking into account only normal components of active contacts in building \(K'_E=G'_EG'^T_E\).
The second studied algorithm, named as ‘Relaxed enriched algorithm’—REA in the following, consists at a first step in iterating on the contacts (stage 1) then on the interface (stage 2) until convergence is reached on the interface. The convergence test is thus restricted to the interface, so the convergence criterium is relaxed. The second step consists in interating only on contacts (independently for each subdomain), until convergence within the body. Thus this second step refines the solution at the micro scale. This is summarized on Algorithm 3. It allows to focus on the convergence rate of the interface problem depending on the enrichment parameter \(\ell _E\), more precisely the dimensionless parameter \(\ell = \frac{\ell _E}{m_E}\), where \(m_E\) is a reference mass.
4.1 Granular test case
In order to illustrate the study on a single time step, a limited size example is proposed.
Setup phase The sample is constituted with 730 disks previously packed in a rigid box whose walls are clusters of disks. The final state is obtained after a vertical gravity load \(g\) is prescribed until the sample is stabilized.

is constant for \(\ell < 1\) (even with \(\ell =0\), i.e. a standard NSCDD interface problem), with \(It_2 < It_1\),

decreases for \(1 < \ell < 100\),

diverges for \(\ell > 500\) (not depicted).
For \(\ell < 10\), the compatibility operator \(G'_E\) leads to similar results as for \(K_E=G_EG^T_E\), but the number of iterations is more stable for larger values of \(\ell \). In the following, \(G'_E\) is therefore selected.
5 A fully multiscale resolution
5.1 Test on a full process (with or without friction)
This trend is similar for cases without and with friction, Figs. 5 and 6, for a substructuring in two subdomains with a partitioning grid (\(1\times 2\)). Due to the additional cost of the augmented interface problem, the algorithms FEA and REA are inefficient for such a granular evolution process problem. Such a numerical behavior may be explained by the rigid nature of the particles and the non smoothness of the interactions. In other words a largescale nonsmooth problem with exact steric exclusions cannot be correctly enough predicted by a linear problem because the local non smooth corrections strongly perturb the global dynamics relayed by the interfaces. Except if we accept to solve coarsely the global interface problem at each time step, before correcting, once and for all, the non smooth local interactions. Hence the motivation of the following section.
5.2 Incomplete resolution
The proposal in this section is to combine (i) an explicit resolution of the (linear) interface problem at the subdomain scale (macro scale), based on the active contact network as stated at the beginning of the time step, with (ii) an implicit resolution of nonsmooth problems within each subdomain, for each contact (micro scale).
This strategy relies on the assumption that interface forces traducing the global behavior of the media evolve slower than local impulses ruled by nonsmooth dynamics. The works in [22] on bimodality of the contact network exemplify that the strong network is ruled by normal impulses in the contacts hardly involving tangential sliding.
It is then possible to choose a different compatibility operator \(G_E\) for determining (\(\widetilde{r}_E,\widetilde{v}_E\)) for the different stages of the augmented algorithms; a first selection for this operator is to choose \(G_E\) such that \(G^T_EV_E=0\), by selecting normal components of active contacts. The Algorithm 4, named as ‘Incomplete enriched algorithm’—IEA, is a proposal for the implementation of such a scheme, with an update stage of active contacts at the beginning of each time step, using a single NLGS iteration. In other words the IEA algorithm consists in restricting the first step of the REA algorithm to a single iteration (\(It_{max2}=1\)).
5.3 Slow dynamic test
In order to test the algorithm IEA, the same problem of granular sample with 730 disks and rotation of gravity vector is reused. This test indeed belongs to the category of problems where the contact network is relatively persistent though the contact force distribution notably evolves. Therefore it suits the assumptions favorable to the incomplete solve strategy previously described. This incomplete solve requires also to assess the quality of the obtained solution, by checking a quality control indicator. This indicator is the mean or maximal interpenetration.
Figure 7a compares the number of iterations for algorithm IEA with respect to the references (algorithms NSCD and NSCDD). A moderate reduction is obtained, with a non monotonous dependence on parameter \(\ell \). For readability reasons, only cases \(\ell =0,10,100\) are depicted.
Concerning mean and maximal interpenetrations, Fig. 7b, c depict a series of curves corresponding to \(\ell \in [0,10^5]\). These interpenetrations largely decrease with \(\ell \). For \(\ell \in [0,10^3]\) they significantly evolve with an increasing trend as the time steps are progressing, whereas for \(\ell \in [10^4,10^5]\) they are stabilizing after a reduced number of time steps. Nevertheless, these interpenetrations are larger than their counterparts for the algorithms NSCD and NSCDD (Fig. 6) but remain acceptable with respect to the mean radius of grains that was selected to 1.
5.4 Dynamic flow test
We now consider a granular flow with an horizontal main velocity, and with a periodic boundary condition in the same direction. Thus the sample is sloped with an angle \(\theta \) equal to \(\pi /6\). Two subdomains are defined as previously, Fig. 10. This problem exhibits a large modification of the contact graph along the evolution process and a convection from the right to the left direction.
For this test case, we moreover use a parallelization of the interface problem, as described in the following section.
5.5 Parallel resolution of the interface problem

global iterations of NSCDD approach,

parallel conjugate gradient on the augmented interface gluing problem.
CPU time; sample of 730 disks with rotating gravity and \(\mu =0.3\)
Algorithm  CPU (s) 

NSCD  117 
NSCDD  63 
EA3—\(\ell =0\)  46 
EA3—\(\ell =10^6\)  51 
6 Conclusions
Domain decomposition methods are usually very well suited to implementations on distributed memory architectures, since the data locality is ensured with the geometrical domain substructuring, and is mapped to the local memories of the different processors. Therefore, favorite message passing librairies such as MPI are useful for this kind of implementation. The OpenMP paradigm is more suited to shared memory parallelization, with minimal intrusivity in the parallelized code. Nevertheless, an organization with data locality such as domain decomposition usually exhibits better performances on this kind of architecture as well (though the efficient use of parallel architecture lead usually to a smaller number of processors than for the previous approach). Load balancing is an issue for each kind of parallelization strategies, and recent advances in this study are available, see [23] for instance. With the use of coarse space (or augmented algorithms), the parallel part of these algorithms are decreasing, due to the advent of a global coarse problem on the whole physical domain (though it may also be parallelized, as done in this article).
This first attempt to enrich a domain decomposition strategy coupled with the contact dynamics underlines the difficulty to improve the convergence of a nonsmooth solver with an enriched linear predictor. Indeed the convergence rate should be significantly increased for compensating for the cost of the solution of the augmented interface problem. Such a goal cannot be reached with a complete resolution at all the scales and at each time step as proved in Sect. 4.1. With the present approach, the gain is not in the scalability performance that the algorithm enrichment may produce, but on the possibility to add a dedicated computational strategy based on a multiscale sequential strategy (using the coarse problem as a macroscopic scale): the incomplete resolution strategy. This strategy leads to admissible solutions if the contact network is stable enough to limit the interpenetration errors. This topic remains an open question for dynamic flow problems, specially if the granular medium is confined, restricting the contact releases.
The present approach has first to be tested on largescale 3D examples with several subdomains as presented in [24]. But the main improvement concerns the correction of the interpenetration during the process. The velocity formulation of the unilateral contact law used in the standard NSCD approach [20] leads to local interpenetrations which may be large if we use an averaged criterion and they are not corrected in the following time steps because no elastic restoring force is introduced. Without changing the contact law we propose to investigate the enrichment of the linear numerical step with an elastic contribution. Such an approach joins the conclusions in [1] for a related investigation.
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