Energy Exascale Computational Fluid Dynamics Simulations With the Spectral Element Method

In this paper, we present a semi-analytic approach to enhance the integration of elliptic Dirichlet-to-Neumann (DtN) boundary condition and high order spectral element method in solving scattering problem with slender scatterer. By using appropriate elemental mapping in the spectral element discretization, semi-analytic formulas are obtained for the computation of Mathieu expansion coefficients involved in the global DtN operator. Further, a semi-analytic approach is proposed for the computation of global boundary integral terms in the spectral element discretization. The proposed semi-analytic formulas can also be used to calculate Mathieu expansion coefficients for functions given values on spectral element grids. Numerical examples show that spectral element method with the proposed semi-analytic approach can produce high order numerical solution for scattering problem with slender scatterer.

CFD simulation of the stratified atmospheric boundary layer: Consistency between Monin-Obukhov similarity theory and the standard k-ε model

We investigate the applicability of a high-order Spectral Element Method (SEM) to density based topology optimization of unsteady flows in two dimensions. Direct Numerical Simulations (DNS) are conducted relying on Brinkman penalization to describe the presence of solid within the domain. The optimization procedure uses the adjoint-variable method to compute gradients and a checkpointing strategy to reduce storage requirements. A nonlinear filtering strategy is used to both enforce a minimum length scale and to provide smoothing across the fluid–solid interface, preventing Gibbs oscillations. This method has been successfully applied to the design of a channel bend and an oscillating pump, and demonstrates good agreement with body fitted meshes. The precise design of the pump is shown to depend on the initial material distribution. However, the underlying topology and pumping mechanism is the same. The effect of a minimum length scale has been studied, revealing it to be a necessary regularization constraint for the oscillating pump to produce meaningful designs. The combination of SEM and density based optimization offer some unique challenges which are addressed and discussed, namely a lack of explicit boundary tracking exacerbated by the interface smoothing. Nevertheless, SEM can achieve equivalent levels of precision to traditional finite element methods, while requiring fewer degrees of freedom. Hence, the use of SEM addresses the two major bottlenecks associated with optimizing unsteady flows: computation cost and data storage.

Numerical simulations are a powerful tool to understand the kinematic and thermodynamic behaviour of impinging jet, which is of great interest in many industrial application. However, in addition to the usual difficulties of computer simulation of incompressible fluid motion and heat exchange, this flow also presents the added issue of involving severely truncated open boundaries. The most commonly used boundary conditions either cause numerical instabilities or have an exceedingly high computational cost. A two-dimensional simulation of the confined impinging jet at Re = 2500 with a non-dimensional nozzleto-plate distance H/D = 2 has been used to test the effectiveness of alternative boundary conditions designed to correctly reproduce the physics of the phenomenon. A large eddy simulation of the three-dimensional case with the same parameters has been performed as well. In both cases, the boundary conditions prevented the divergence of the simulation and the production of non-physical reflections.

A three-dimensional computational fluid dynamics analysis of low Reynolds number [O(104)] flow over a 30P30N three-element high lift wing is carried out using a high-order spectral element method. In this article, we study the flow in the slat cove region and the slat wake/shear layer interaction. Vortical structures, identified in the computations, are very similar to those visualized in recent experiments. For Reynolds numbers below a critical interval (found in recent experiments), Görtler vortices are observed in the slat wake, while for Reynolds numbers above the critical interval, a roll-up is observed in the slat cove and both streamwise and spanwise vortices form in the slat wake. Prior to the formation of Görtler and roll-up vortices, three-dimensional tongue- or rib-like vortex shapes, similar to those found in the wake of bluff bodies, are observed in the slat cove and promote transition to three-dimensional flow. Above the critical interval, streaks and spanwise vortices are observed to dominate the slat wake and lead to the formation of hairpin vortices which contribute to the transition to turbulence. Integral flow parameters such as lift, drag, and pressure coefficients are analyzed in the range of Reynolds numbers studied.

In the frequency domain, we optimize the entries of the element stiffness matrix computed on a square element with the purpose of approximating the 2D acoustic wave equation with the second spatial order. The optimized matrices are computed from the minimization of the normalized phase velocity as a function of propagation angle and the number of grid points per wavelength, leading to a remarkable reduction in the numerical error. The optimized number of 4.1 sample points in a wavelength, with a maximum error in velocities <0.3%, offering the simulation of large-scale realistic models. The superior performance of the optimized matrices to those from the lumped, consistent and eclectic matrices, is evident from the numerical examples presented, with consequent reduction not only in the numerical dispersion/anisotropy but also in an improved resolution. It is noteworthy that optimized matrices give modeling results with better accuracy than models from the high-order spectral element method. Our methodology is easily extendable to the accurate discretizing of the general 3D elastic wave equation that optimizes the bandwidth of the impedance matrix, availing computational resources for accurately modeling the compressional and shear wave velocities, density, as well as multi-parameter inversion.

This paper concerns the design of a high-frequency rotating transformer in the vicinity of an aluminum shield. The objective is to maximize the output power within the available volume. Both axially and radially gapped transformers are considered on a wide frequency range and the final design are compared. A high ratio between the object dimensions and the skin depth exist. Consequently, meshing problems arise when applying the Finite Element Method. Therefore, the high-order Spectral Element Method is employed.

The aim of this work is to propose a novel, fast solver for the Poisson problem discretized with high-order spectral element methods (HO-SEM) in canonical geometries (rectangle in two dimensions, r...

Control of vortex breakdown in a closed cylinder with a small rotating rod

High order methods for the solution of PDEs expose a trade-off between computational cost and accuracy on a per degree of freedom basis. In many cases, the cost increases due to higher arithmetic intensity while affecting data movement minimally. As architectures tend towards wider vector instructions and expect higher arithmetic intensities, the best order for a particular simulation may change.

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derivedp-version,h-version, andhp-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.

The high order Spectral Element Method is used to solve the reaction-diffusion-advection problem as described by the premixed flame case. An h–p adaptive refinement-coarsening algorithm is developed based on a posteriori error estimators. Simulations of the wrinkling of a premixed flame front are used to illustrate how the mesh is adaptively refined. A similar, idealized heat transfer problem is used to show the adaptive refinement and coarsening of the mesh. Adaptivity efficiently provides high resolution in areas of the domain where large or rapidly varying physical changes exist, while saving unnecessary computation where the solution is smooth or physical phenomena have passed by.

Large-eddy simulation (LES) of a stable atmospheric boundary layer is performed using recently developed dynamic subgrid-scale (SGS) models. These models not only calculate the Smagorinsky coefficient and SGS Prandtl number dynamically based on the smallest resolved motions in the flow, they also allow for scale dependence of those coefficients. This dynamic calculation requires statistical averaging for numerical stability. Here, we evaluate three commonly used averaging schemes in stable atmospheric boundary-layer simulations: averaging over horizontal planes, over adjacent grid points, and following fluid particle trajectories. Particular attention is focused on assessing the effect of the different averaging methods on resolved flow statistics and SGS model coefficients. Our results indicate that averaging schemes that allow the coefficients to fluctuate locally give results that are in better agreement with boundary-layer similarity theory and previous LES studies. Even among models that are local, the averaging method is found to affect model coefficient probability density function distributions and turbulent spectra of the resolved velocity and temperature fields. Overall, averaging along fluid pathlines is found to produce the best combination of self consistent model coefficients, first- and second-order flow statistics and insensitivity to grid resolution.

In recent years, the spectral element method (SEM) has been extensively applied to computational electromagnetics for large scale problems. As a special high-order finite element method, however, the SEM also includes DC spurious modes as in the traditional finite element method. Since 2015, the mixed spectral element method has been developed to remove such spurious modes by incorporating Gauss's law into the traditional SEM (N. Liu, L. Tobon, Y. Tang, Q. H. Liu, “Mixed Spectral Element Method for 2D Maxwell Eigenvalue Problem,” Communications in Computational Physics, vol. 17, no. 2, pp. 458-486, 2015; N. Liu, L. Tobon, Y. Zhao, Y. Tang, Q. H. Liu, “Mixed Spectral Element Method for 3-D Maxwell Eigenvalue Problem,” IEEE Trans. Microwave Theory Tech., vol. 64, no. 2, pp. 317-325, 2015).

Time/wave domain analysis for axially moving pre-stressed nanobeam by wavelet-based spectral element method