Distributed-memory parallelization of the aggregated unfitted finite element method

Abstract

The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR.

Embedded Multilevel Monte Carlo for Uncertainty Quantification in Random Domains

“…AgFE spaces are not affected by the small cut cell problem and the condition number of the resulting matrix is O(h −2 ℓ ) [9]. The numerical experiments in [67] show that a standard parallel algebraic multigrid solver is effective to solve AgFE linear systems using default settings. The AgFE space approximation of (1) at a given level ℓ reads as follows: find…”

“…We represent the resulting aggregated finite element (AgFE) space with V(G ℓ , ω) ⊆ V(T D ℓ , ω). We refer the interested reader to [9] for a detailed exposition of the mesh aggregation algorithm, the computation of constraint, the implementation issues, and its numerical analysis, to [67] for its parallel implementation, and to [8] for a mixed AgFE space for the Stokes problem. AgFE spaces are not affected by the small cut cell problem and the condition number of the resulting matrix is O(h −2 ℓ ) [9].…”

Distributed and vectorized method of characteristics for fast transient simulations in water distribution systems

“…Accurate transient models are essential to estimate and forecast system failures due to sudden changes in pressure, such as pipe ruptures, which account for major economic losses (Duan et al, 2020). In practice, transient models for WDSs are based on conservation laws expressed in the form of one-dimensional © 2021 Computer-Aided Civil and Infrastructure Engineering hyperbolic partial differential equations (PDEs), whose solution can be found numerically using the method of characteristics (MOC) (Wylie et al, 1993), finite difference methods (Blanco et al, 2015;Chaudhry & Hussaini, 1985;Kiuchi, 1994;Verdugo et al, 2019), or finite volume methods (Cao et al, 2020;Castro et al, 2006;Fernández-Pato & García-Navarro, 2014;Mesgari Sohani & Ghidaoui, 2019;Zhao & Ghidaoui, 2004). Overall, MOC has been predominantly used over other numerical schemes due to its ease of implementation and accuracy.…”

High‐order cut discontinuous Galerkin methods with local time stepping for acoustics

“…Basis functions with small support can also be dropped completely by replacing them by an extension of the basis functions of a neighboring element. This approach is known as cell agglomeration or cell merging and is presented in Reference in combination with higher order finite element and DG methods for cut problems applied to elliptic problems and flow equations. The cell agglomeration can be thought of as a strong enforcement of the continuity of the solution field to larger neighboring elements, while penalizing jumps in derivatives represent a weak enforcement of the continuity.…”