arXiv subjects

Peter Munch

Publications and source records attributed to Peter Munch.

24 records · Page 2Linked to original sources

arXiv2022

Efficient distributed matrix-free multigrid methods on locally refined meshes for FEM computations

This work studies three multigrid variants for matrix-free finite-element computations on locally refined meshes: geometric local smoothing, geometric global coarsening, and polynomial global coarsening. We have integrated the algorithms into the same framework-the open-source finite-element library deal.II-, which allows us to make fair comparisons regarding their implementation complexity, computational efficiency, and parallel scalability as well as to compare the measurements with theoretically derived performance models. Serial simulations and parallel weak and strong scaling on up to 147,456 CPU cores on 3,072 compute nodes are presented. The results obtained indicate that global coarsening algorithms show a better parallel behavior for comparable smoothers due to the better load balance particularly on the expensive fine levels. In the serial case, the costs of applying hanging-node constraints might be significant, leading to advantages of local smoothing, even though the number of solver iterations needed is slightly higher.

math.NA↗

arXiv2021

Lethe-DEM : An open-source parallel discrete element solver with load balancing

Approximately $75 \%$ of the raw material and $50 \%$ of the products in the chemical industry are granular materials. The Discrete Element Method (DEM) provides detailed insights of phenomena at particle scale and it is therefore often used for modeling granular materials. However, because DEM tracks the motion and contact of individual particles separately, its computational cost increases non-linearly $O(n_p\log(n_p))$ -- $O(n_p^2)$ depending on the algorithm) with the number of particles ($n_p$). In this article, we introduce a new open-source parallel DEM software with load balancing: Lethe-DEM. Lethe-DEM, a module of Lethe, consists of solvers for two-dimensional and three-dimensional DEM simulations. Load-balancing allows Lethe-DEM to significantly increase the parallel efficiency by $\approx 25 - 70 \%$ depending on the granular simulation. We explain the fundamental modules of Lethe-DEM, its software architecture, and the governing equations. Furthermore, we verify Lethe-DEM with several tests including analytical solutions and comparison with other software. Comparisons with experiments in a flat-bottomed silo, wedge-shaped silo, and rotating drum validate Lethe-DEM. We investigate the strong and weak scaling of Lethe-DEM with $1 \leq n_c \leq 192$ and $32 \leq n_c \leq 320$ processes, respectively, with and without load-balancing. The strong-scaling analysis is performed on the wedge-shaped silo and rotating drum simulations, while for the weak-scaling analysis, we use a dam break simulation. The best scalability of Lethe-DEM is obtained in the range of $5000 \leq n_p/n_c \leq 15000$. Finally, we demonstrate that large scale simulations can be carried out with Lethe-DEM using the simulation of a three-dimensional cylindrical silo with $n_p=4.3 \times 10^6$ on 320 cores.

cond-mat.mtrl-sci↗

arXiv2020

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: Towards grid-converged results for the inviscid Taylor-Green problem

Providing evidence of finite-time singularities of the incompressible Euler equations in three space dimensions is still an unsolved problem. Likewise, the zeroth law of turbulence has not been proven to date by numerical experiments. We address this issue by high-resolution numerical simulations of the inviscid three-dimensional Taylor-Green vortex problem using a novel high-order discontinuous Galerkin discretization approach. Our main finding is that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but instead converges to a solution with nonzero kinetic energy dissipation rate. This implies an energy dissipation anomaly in the absense of viscous dissipation according to Onsager's conjecture, and serves as an indication of finite-time singularities in incompressible inviscid flows. We demonstrate convergence to a dissipative solution for the three-dimensional inviscid Taylor-Green problem with a measured relative $L_2$-error of $0.27 \%$ for the temporal evolution of the kinetic energy and $3.52 \%$ for the kinetic energy dissipation rate.

physics.flu-dyn↗

arXiv2020

hyper.deal: An efficient, matrix-free finite-element library for high-dimensional partial differential equations

This work presents the efficient, matrix-free finite-element library hyper.deal for solving partial differential equations in two to six dimensions with high-order discontinuous Galerkin methods. It builds upon the low-dimensional finite-element library deal.II to create complex low-dimensional meshes and to operate on them individually. These meshes are combined via a tensor product on the fly and the library provides new special-purpose highly optimized matrix-free functions exploiting domain decomposition as well as shared memory via MPI-3.0 features. Both node-level performance analyses and strong/weak-scaling studies on up to 147,456 CPU cores confirm the efficiency of the implementation. Results of the library hyper.deal are reported for high-dimensional advection problems and for the solution of the Vlasov--Poisson equation in up to 6D phase space.

cs.MS↗

arXiv2019

Hybrid multigrid methods for high-order discontinuous Galerkin discretizations

The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE solver. The multigrid hierarchy exploits all possibilities of geometric, polynomial, and algebraic coarsening, targeting engineering applications on complex geometries. Additionally, a transfer from discontinuous to continuous function spaces is performed within the multigrid hierarchy. This does not only further reduce the problem size of the coarse-grid problem, but also leads to a discretization most suitable for state-of-the-art algebraic multigrid methods applied as coarse-grid solver. The relevant design choices regarding the selection of optimal multigrid coarsening strategies among the various possibilities are discussed with the metric of computational costs as the driving force for algorithmic selections. We find that a transfer to a continuous function space at highest polynomial degree (or on the finest mesh), followed by polynomial and geometric coarsening, shows the best overall performance. The success of this particular multigrid strategy is due to a significant reduction in iteration counts as compared to a transfer from discontinuous to continuous function spaces at lowest polynomial degree (or on the coarsest mesh). The coarsening strategy with transfer to a continuous function space on the finest level leads to a multigrid algorithm that is robust with respect to the penalty parameter of the SIPG method. Detailed numerical investigations are conducted for a series of examples ranging from academic test cases to more complex, practically relevant geometries. Performance comparisons to state-of-the-art methods from the literature demonstrate the versatility and computational efficiency of the proposed multigrid algorithms.

physics.comp-ph↗

arXiv2019

A Hermite-like basis for faster matrix-free evaluation of interior penalty discontinuous Galerkin operators

This work proposes a basis for improved throughput of matrix-free evaluation of discontinuous Galerkin symmetric interior penalty discretizations on hexahedral elements. The basis relies on ideas of Hermite polynomials. It is used in a fully discontinuous setting not for higher order continuity but to minimize the effective stencil width, namely to limit the neighbor access of an element to one data point for the function value and one for the derivative. The basis is extended to higher orders with nodal contributions derived from roots of Jacobi polynomials and extended to multiple dimensions with tensor products, which enable the use of sum factorization. The beneficial effect of the reduced data access on modern processors is shown. Furthermore, the viability of the basis in the context of multigrid solvers is analyzed. While a plain point-Jacobi approach is less efficient than with the best nodal polynomials, a basis change via sum-factorization techniques enables the combination of the fast matrix-vector products with effective multigrid constituents. The basis change is essentially for free on modern hardware because these computations can be hidden behind the cost of the data access.

math.NA↗