6,436 research outputs found

    VoroCrust: Voronoi Meshing Without Clipping

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    Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

    The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows

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    The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a `sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart

    FullSWOF_Paral: Comparison of two parallelization strategies (MPI and SKELGIS) on a software designed for hydrology applications

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    In this paper, we perform a comparison of two approaches for the parallelization of an existing, free software, FullSWOF 2D (http://www. univ-orleans.fr/mapmo/soft/FullSWOF/ that solves shallow water equations for applications in hydrology) based on a domain decomposition strategy. The first approach is based on the classical MPI library while the second approach uses Parallel Algorithmic Skeletons and more precisely a library named SkelGIS (Skeletons for Geographical Information Systems). The first results presented in this article show that the two approaches are similar in terms of performance and scalability. The two implementation strategies are however very different and we discuss the advantages of each one.Comment: 27 page

    Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption

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    In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation −Δu−(k2+iΔ)u=f-\Delta u - (k^2+ {\rm i} \varepsilon)u = f, with absorption parameter Δ∈R\varepsilon \in \mathbb{R}. Multigrid approximations of this equation with Δ=Ìž0\varepsilon \not= 0 are commonly used as preconditioners for the pure Helmholtz case (Δ=0\varepsilon = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (Δ=Ìž0\varepsilon \not=0), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a kk- and Δ\varepsilon-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if âˆŁÎ”âˆŁâˆŒk2|\varepsilon|\sim k^2, then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case Δ=0\varepsilon = 0. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about O(n4/3)\mathcal{O}(n^{4/3}) for solving finite element systems of size n=O(k3)n=\mathcal{O}(k^3), where we have chosen the mesh diameter h∌k−3/2h \sim k^{-3/2} to avoid the pollution effect. Experiments on problems with h∌k−1h\sim k^{-1}, i.e. a fixed number of grid points per wavelength, are also given
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