High-performance dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites

Abstract

The key innovation in this paper is an open-source, high-performance iterative solver for high contrast, strongly anisotropic elliptic partial differential equations implemented within dune-pdelab. The iterative solver exploits a robust, scalable two-level additive Schwarz preconditioner, GenEO (Spillane et al. 2014). The development of this solver has been motivated by the need to overcome the limitations of commercially available modeling tools for solving structural analysis simulations in aerospace composite applications. Our software toolbox dune-composites encapsulates the mathematical complexities of the underlying packages within an efficient C++ framework, providing an application interface to our new high-performance solver. We illustrate its use on a range of industrially motivated examples, which should enable other scientists to build on and extend dune-composites and the GenEO preconditioner for use in their own applications. We demonstrate the scalability of the solver on more than 15,000 cores of the UK national supercomputer Archer, solving an aerospace composite problem with over 200 million degrees of freedom in a few minutes. This scale of computation brings composites problems that would otherwise be unthinkable into the feasible range. To demonstrate the wider applicability of the new solver, we also confirm the robustness and scalability of the solver on SPE10, a challenging benchmark in subsurface flow/reservoir simulation. Nature of problem:dune-composites is designed to solve anisotropic linear elasticity equations for anisotropic, heterogeneous materials, e.g. composite materials. To achieve this, our contribution also implements a new preconditioner in dune-pdelab. Solution method:The anisotropic elliptic partial differential equations are solved via the finite element method. The resulting linear system is solved via an iterative solver with a robust, scalable two-level overlapping Schwarz preconditioner: GenEO.

Novel design and analysis of generalized FE methods based on locally optimal spectral approximations

“…Proof of Theorem 3. 6. In what follows, we show how Lemma 3.12 leads to the desired estimate (3.20).…”

“…Whilst, in many cases, only the macroscopic properties of the solution are of interest, they are often strongly influenced by the micro-and mesoscopic details of the media, making a direct discretization at the macroscale unreliable. However, direct numerical solution on a fine mesh that resolves all the small-scale features is computationally expensive and notoriously illconditioned [6]. This motivates the development of multiscale methods which reduce the computational cost by efficiently incorporating physically important fine-scale information into a coarse-scale representation.…”

Recent advances in domain decomposition methods for large-scale saddle point problems

“…The library also provides access to the GCRODR method [24] useful when solving linear systems with multiple right-hand sides. • the FreeFem DD library ffddm [55] • a Dune solver as described in [56] It is worth noticing that the access to the variational formulation is made easy if one makes use of domain specific languages or libraries such as FreeFem [53], Dune [57] or Firedrake [58]. It facilitates the encapsulation of two-level methods.…”

Multilevel Spectral Domain Decomposition

“…We report results on modelling carbon fibre composite materials from aerospace engineering, described in detail in [26,7]. The setup is similar to the one in [26, p. 271], except that the domain is flattened out and consists of only 9 ply layers and 8 interface (resin) layers.…”