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In this paper, we present a characteristic-based numerical procedure for simulating incompressible flows in domains with moving boundaries. Our approach utilizes an operator-integration-factor splitting technique to help produce an efficient and stable numerical scheme. Using the spectral element method and an arbitrary Lagrangian–Eulerian formulation, we investigate flows where the convective acceleration effects are non-negligible. Several examples, ranging from laminar to turbulent flows, are considered. Comparisons with a standard, semi-implicit time-stepping procedure illustrate the improved performance of our scheme.

We investigate efficient algorithms and a practical implementation of an explicit-type high-order timestepping method based on Krylov subspace approximations, for possible application to large-scale engineering problems in electromagnetics. We consider a semi-discrete form of the Maxwell’s equations resulting from a high-order spectral-element discontinuous Galerkin discretization in space whose solution can be expressed analytically by a large matrix exponential of dimension κ × κ . We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram–Schmidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension m × m ( m ≪ κ ). For computing the matrix exponential, we obtain eigenvalues of the m × m matrix using available library packages and compute an ordinary exponential function for the eigenvalues. The scheme involves mainly matrix-vector multiplications, and its convergence rate is generally O ( Δ t m - 1 ) in time so that it allows taking a larger timestep size as m increases. We demonstrate CPU time reduction compared with results from the five-stage fourth-order Runge–Kutta method for a certain accuracy. We also demonstrate error behaviors for long-time simulations. Case studies are also presented, showing loss of orthogonality that can be recovered by adding a low-cost reorthogonalization technique.

We present a high-order spectral element method for solving layered media scattering problems featuring an operator that can be used to transparently enforce the far-field boundary condition. The incorporation of this Dirichlet-to-Neumann (DtN) map into the spectral element framework is a novel aspect of this work, and the resulting method can accommodate plane-wave radiation of arbitrary angle of incidence. In order to achieve this, the governing Helmholtz equations subject to quasi-periodic boundary conditions are rewritten in terms of periodic unknowns. We construct a spectral element operator to approximate the DtN map, thus ensuring nonreflecting outgoing waves on the artificial boundaries introduced to truncate the computational domain. We present an explicit formula that accurately computes the Fourier coefficients of the solution in the spectral element discretization space projected onto the boundary which is required by the DtN map. Our solutions are represented by the tensor product basis of one-dimensional Legendre–Lagrange interpolation polynomials based on the Gauss–Lobatto–Legendre grids. We study the scattered field in singly and doubly layered media with smooth and nonsmooth interfaces. We consider rectangular, triangular, and sawtooth interfaces that are accurately represented by the body-fitted quadrilateral elements. We use GMRES iteration to solve the resulting linear system, and we validate our results by demonstrating spectral convergence in comparison with exact solutions and the results of an alternative computational method.

Here, we introduce a characteristic-based numerical procedure for simulating incompressible flows in domains with moving boundaries. Our approach utilizes an operator-integration-factor splitting technique to help produce an efficient and stable numerical scheme. Using the spectral element method and an arbitrary Lagrangian-Eulerian formulation, we explore flows where the convective acceleration effects are non-negligible. Several examples, ranging from laminar to turbulent flows, are considered. Comparisons with a standard, semi-implicit time-stepping procedure illustrate the improved performance of our scheme.

This paper presents large-scale computations and theoretical or computational aspects of the spectral element methods for solving Maxwell's equations that have potential applications in nanoscience for surface-enhanced Raman scattering (SERS) and solar cell devices. We study the surface-enhanced electromagnetic fields near the surface of metallic nanoparticles using spectral element discontinuous Galerkin method. We solve Maxwell's equations in time-domain and provide accuracy and efficiency of our method compared to the conventional finite difference method. We demonstrate light transmission properties for nanoslab and nanoslits, and time-averaged electric fields over the cross sections of nanoholes in a hexagonal array.

We develop hybrid RANS-LES strategies within the spectral element code Nek5000 based on the \\(k-\\tau\\) class of turbulence models. We chose airfoil sections at small flight configurations as our target problem to comprehensively test the solver accuracy and performance. We present verification and validation results of an unconfined NACA0012 wing section in a pure RANS and in a hybrid RANS-LES setup for an angle of attack ranging from 0 to 90 degrees. The RANS results shows 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 a RANS formulation. On the other hand, DDES captures both the attached and separated flow dynamics well when compared with available numerical datasets. We demonstrate that for the hybrid turbulence modeling 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 finite-volume and finite-difference approaches. We also revise some of the guidelines on sample size requirements for statistics convergence. Furthermore, we analyze some of the observed discrepancies of our unconfined DDES at higher angles with the experiments by evaluating the side wall \"blocking\" effect. We carry out additional simulations in a confined 'numerical wind tunnel' and assess the observed differences as a function of Reynolds number.

In the realm of Computational Fluid Dynamics (CFD), the demand for memory and computation resources is extreme, necessitating the use of leadership-scale computing platforms for practical domain sizes. This intensive requirement renders traditional checkpointing methods ineffective due to the significant slowdown in simulations while saving state data to disk. As we progress towards exascale and GPU-driven High-Performance Computing (HPC) and confront larger problem sizes, the choice becomes increasingly stark: to compromise data fidelity or to reduce resolution. To navigate this challenge, this study advocates for the use of in situ analysis and visualization techniques. These allow more frequent data \"snapshots\" to be taken directly from memory, thus avoiding the need for disruptive checkpointing. We detail our approach of instrumenting NekRS, a GPU-focused thermal-fluid simulation code employing the spectral element method (SEM), and describe varied in situ and in transit strategies for data rendering. Additionally, we provide concrete scientific use-cases and report on runs performed on Polaris, Argonne Leadership Computing Facility's (ALCF) 44 Petaflop supercomputer and J\"ulich Wizard for European Leadership Science (JUWELS) Booster, J\"ulich Supercomputing Centre's (JSC) 71 Petaflop High Performance Computing (HPC) system, offering practical insight into the implications of our methodology.

Electrostatic gating and optical pumping schemes enable efficient time modulation of graphene's free carrier density, or Drude weight. We develop a theory for plasmon propagation in graphene under temporal modulation. When the modulation is on the timescale of the plasmonic period, we show that it is possible to create a backwards-propagating or standing plasmon wave and to amplify plasmons. The theoretical models show very good agreement with direct Maxwell simulations.

We demonstrate NekRS performance results on various advanced GPU architectures. NekRS is a GPU-accelerated version of Nek5000 that targets high performance on exascale platforms. It is being developed in DOE's Center of Efficient Exascale Discretizations, which is one of the co-design centers under the Exascale Computing Project. In this paper, we consider Frontier, Crusher, Spock, Polaris, Perlmutter, ThetaGPU, and Summit. Simulations are performed with 17x17 rod-bundle geometries from small modular reactor applications. We discuss strong-scaling performance and analysis.

We develop an all-hex meshing strategy for the interstitial space in beds of densely packed spheres that is tailored to turbulent flow simulations based on the spectral element method (SEM). The SEM achieves resolution through elevated polynomial order N and requires two to three orders of magnitude fewer elements than standard finite element approaches do. These reduced element counts place stringent requirements on mesh quality and conformity. Our meshing algorithm is based on a Voronoi decomposition of the sphere centers. Facets of the Voronoi cells are tessellated into quads that are swept to the sphere surface to generate a high-quality base mesh. Refinements to the algorithm include edge collapse to remove slivers, node insertion to balance resolution, localized refinement in the radial direction about each sphere, and mesh optimization. We demonstrate geometries with 10^2-10^5 spheres using approximately 300 elements per sphere (for three radial layers), along with mesh quality metrics, timings, flow simulations, and solver performance.