Spectral/hp Element Methods for Computational Fluid Dynamics

By G. Em Karniadakis, Spencer J. Sherwin. 2005

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In Computational Mechanics

Citations in the corpus

Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations (2017, methodology): “We integrate equation (6) using a third-order stiffly stable scheme [15] until the system reaches a periodic steady state, as depicted in figure 2(a).”
A unified approach for a posteriori high-order curved mesh generation using solid mechanics (2016, methodology): “The performance of high-order discretisation methods for the simulation of various problems in science and engineering has been the object of intensive research during the last twodecades [32,41,44,67].”
Direct optimal growth analysis for timesteppers (2008, methodology): “While our implementations have employed spectral/hp element formulations [18], the methods we present are generally applicable, for any mesh-based discretization.”
A comparative study of the performance of high resolution advection schemes in the context of data assimilation (2006, neutral): “Spatial discretization methods for solving partial differential equations (PDEs) can be broadly classified as finite difference (FD) [1, 2], finite volume (FV) [3], finite element (FE) [4, 5, 6] and spectral methods [7] (including the discontinuous Galerkin (DG) methods [8]).”
Multifidelity Information Fusion Algorithms for High-Dimensional Systems and Massive Data sets (2016, methodology): “whereNdof is the total number of degrees of freedom, M = (P+1) 2 is the total number of modes in each quadrilateral spectral element, ξ defines a mapping from the physical space to the standard element, and φp(xe(ξ)) are local to each element polynomials of order P , which when assembled together un”
Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (2012, methodology): “FIAT uses the so-called collapsed-coordinate polynomials (Karniadakis and Sherwin, 2005) on the triangle and tetrahedra.”
CAM-SE: A scalable spectral element dynamical core for the Community Atmosphere Model (2012, methodology): “Owingtothetensorproductdecom-position,theSEMdivergence,gradientandcurlcanallbe evaluatedatthe ð d þ 1 Þ 2 GLL nodes within each element in O ð d Þ operations per node using the tensor-product property of these points (Deville et al. 2002; Karniadakis and Sherwin 2005).”
Fast formation and assembly of finite element matrices with application to isogeometric linear elasticity (2019, methodology): “Sum factorization 70 Sum factorization was first introduced in [40] and is now considered the technique of choice to attain efficient formation of local element matrices in hp-finite elements [1, 22, 35, 49].”
Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data (2018, methodology): “To generate high-resolution datasets for different benchmark problems studied in the current work, we have employed the spectral/hp-element solver NekTar in which the Navier-Stokes equations (3) along with the transport equation (2) are approximated using high-order semi-orthogonal Jacobi polynomial”
A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems (2014, background): “The reader is referred to Orszag [19] and Karniadakis and Sher102 win [20] for the original works and a recent reformulation on spectral and 103 pseudo-spectral methods.”
An isoparametric approach to high-order curvilinear boundary-layer meshing (2015, methodology): “and the additional modified hierarchical basis function ψc is defined in [1].”