Spectral element simulation of ultrafiltration

Algebraic multigrid is investigated as a solver for linear systems that arise from high-order spectral element discretizations. An algorithm is introduced that utilizes the efficiency of low-order finite elements to precondition the high-order method in a multilevel setting. In particular, the efficacy of this approach is highlighted on simplexes in two and three dimensions with nodal spectral elements up to order $n=11$. Additionally, a hybrid preconditioner is also developed for use with discontinuous spectral element methods. The latter approach is verified for the discontinuous Galerkin method on elliptic problems.

The SEL macroscopic modeling code

We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14,259 degrees of freedom. The steady-state snapshot solutions define a reduced order space, which allows to accurately evaluate the steady-state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

Semtex: A spectral element–Fourier solver for the incompressible Navier–Stokes equations in cylindrical or Cartesian coordinates

A statically condensed discontinuous Galerkin spectral element method on Gauss-Lobatto nodes for the compressible Navier-Stokes equations

Overlapping Schwarz Methods for Unstructured Spectral Elements

High-Performance Spectral Element Algorithms and Implementations

The spectral element method is a high-order technique for solution of the incompressible Navier-Stokes equations which combines spectral expansions with finite element methodology to give high accuracy in general geometries. In the spectral element discretization, the computational domain is broken up into macro-elements, and the velocity and pressure in each element are represented as high-order Lagrangian inter polants. The nonlinear terms in the equation are then treated with explicit collocation, while the pressure and viscous contributions are handled implicitly with variational projection operators. Parallel static condensation applied to the implicit equations gives an operation count commensurate with that for low-order sub-structure techniques at the same resolution. A time-splitting technique is presented for solution of the Navier-Stokes equations, and results are given for linear and (three dimensional) secondary spatial stability of plane Poiseuille flow, and for steady and unsteady separated channel flows at Reynolds numbers of several thousand.

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

The spectral and spectral element discretizations of partial differential equations rely on high degree polynomial approximation and on the use of tensorized bases of polynomials. Firstly, on a square or a cube, we describe the basic tools for spectral methods, and we prove some optimal properties of polynomial approximation and interpolation. We apply them to the analysis of the spectral discretization of the Laplace and Navier-Stokes equations and also of some hyperbolic problems. Secondly, we present their extension to more complex geometries: the mortar spectral element method allows for working on any domain which admits a decomposition, non necessarily conforming, into curved rectangles or hexahedra without overlapping, while a simple change of variables allows for handling axisymmetric geometries.KeywordsSpectral MethodPolynomial ApproximationDiscrete ProblemSpectral ElementStokes ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A priori and a posteriori error analysis of hp spectral element discretization for optimal control problems with elliptic equations

We develop hybrid RANS–LES strategies within the spectral element code Nek5000 based on the k − τ SST turbulence model. We chose airfoil sections with chord-based Reynolds number on the order of 10 5 − 10 6 , in both attached and stalled conditions, as our target problem to comprehensively test the solver accuracy and performance. Verification and validation of the k − τ SST model are performed for two reference cases: for the zero-pressure gradient boundary layer developing on a flat plate and for mild adverse-pressure gradient boundary layers developing on suction side of NACA0012. The k − τ SST model shows good grid convergence characteristics, at par or better in comparison to existing reference results. The results also show good corroboration with existing experimental and numerical datasets for low incoming flow angles. A small discrepancy appears at higher angle in comparison with the experiments, which is in line with our expectations from an RANS formulation. Building on this foundation, we construct a hybrid RANS–LES framework based on the Delayed Detached-Eddy Simulation (DDES) approach. DDES captures both the attached and separated flow dynamics well when compared with available numerical datasets. We demonstrate that for the hybrid approach a high-order spectral element discretization converges faster (i.e. with less resolution) and captures the flow dynamics more accurately than representative low-order approaches. We also revise some of the guidelines on sample size requirements for statistics convergence for massively separated flow within the current numerical framework. Finally, we analyse some of the observed discrepancies of our unconfined DDES at higher angles with the experiments by evaluating the ‘blocking’ effect of wind tunnel walls. We carry out additional simulations for confined domains and assess the observed differences as a function of Reynolds number.

An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier–Stokes Equations

Two dimensional mixed flows of buoyant and thermocapillary convection are investigated by the spectral element method (SEM). The Navier–Stokes equations are solved based on a spectral element discretization with Chebyshev and Legendre polynomials. The semi-implicit time-splitting method is employed for temporal discretization. Finally, the SEM is applied to solve two dimensional mixed flows of buoyant and thermocapillary convection. After qualitative and quantitative comparisons between the SEM and finite volume method (FVM), it is found that the numerical solutions agree closely with the reference data. In addition, exhaustive sources of errors are given. These numerical results reveal that incompressible fluid flow with free surface can be accurately solved by the SEM.

Stability Analysis for Different Formulations of the Nonlinear Term in PN−PN−2 Spectral Element Discretizations of the Navier–Stokes Equations