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At least 109 records · Page 6

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DOE OSTI2024

A robust spectral element implementation of the $k - τ$ RANS model in Nek5000/NekRS

The $k - ω$ Reynolds Averaged Navier Stokes (RANS) model is one of the industry standard approaches for modeling of turbulent flows. It performs better than the $k - ϵ$ model for low Reynolds number flows and is also more suitable for boundary layers with adverse pressure gradients. Major drawback of the model, however, is that the asymptotic value of $ω$ at the walls is singular, necessitating the use of a contrived “sufficiently” large value for $ω$ as the boundary condition for its transport equation. Here, this invariably leads to the solution being sensitive to near wall grid spacing. While an acceptable solution for low order (finite volume) methods, the excessive near wall gradients lead to persistent numerical stability issues in high order codes. To alleviate the problem, specifically in the context of the high order spectral element code Nek5000, a regularized $k - ω$ approach was formulated in our prior work (Tomboulides et al., 2018). The formulation, however, relies on the use of wall distance and its gradients for modeling the closure terms and can pose problems for simulations in complex geometries. This work presents a novel implementation of the $k - τ$ RANS model in Nek5000, where $τ = 1/ω$, eliminating the need for regularization, owing to the asymptotically bounded behavior of the source terms in the $τ$ transport equation, and also eliminating dependence on wall distance. Robustness and stability of the $k - τ$ model is ensured through implicit treatment of the source terms and their careful numerical implementation and demonstrated through several cases aimed at verification and validation. Studies include both canonical and engineering relevant problems, viz., turbulent channel flow, pipe flow, backward facing step, flow over NACA 0012 airfoil and flow in a T-junction. Results from the $k - τ$ model are shown to be consistent with regularized $k - ω$ model and also with the $k - ω$ SST model in OpenFOAM (for select studies). Comparison with experimental data is also shown, where available, to bolster validation efforts for the $k - τ$ model implementation through prediction of key turbulent quantities of interest.

Nek5000↗

NASA NTRS1983

Spectral methods for the Euler equations

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of both Fourier and Chebyshev spectral collocation methods are provided. Filtering strategies for both shock-fitting and shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one-dimensional, periodic astrophysical 'nozzle' problem. Examples of shock-fitting approaches include a shock/acoustic wave interaction, shock/vortex interaction, and the classical blunt body problem. While the shock capturing spectral method does not yet show a clear advantage over second-order finite differences, equivalent accuracy can be obtained using shock fitting with far fewer grid points.

Hussaini, M. Y.↗

NASA NTRS1984

Spectral methods for the Euler equations: Fourier methods and shock-capturing

Hussaini, M. Y.↗

NASA NTRS1985

Spectral methods for the Euler equations. I - Fourier methods and shock capturing

Hussaini, M. Y.↗

NASA NTRS1988

The analysis of spectra of novae taken near maximum

A project to analyze ultraviolet spectra of novae obtained at or near maximum optical light is presented. These spectra are characterized by a relatively cool continuum with superimposed permitted emission lines from ions such as Fe II, Mg II, and Si II. Spectra obtained late in the outburst show only emission lines from highly ionized species and in many cases these are forbidden lines. The ultraviolet data will be used with calculations of spherical, expanding, stellar atmospheres for novae to determine elemental abundances by spectral line synthesis. This method is extremely sensitive to the abundances and completely independent of the nebular analyses usually used to obtain novae abundances.

Stryker, L. L.↗

NASA NTRS1984

Spectral methods for exterior elliptic problems

Spectral approximations for exterior elliptic problems in two dimensions are discussed. As in the conventional finite difference or finite element methods, the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. A spectral boundary treatment is introduced at infinity which is compatible with the infinite order interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although a simple Laplace problem is examined, the analysis covers more complex and general cases.

Canuto, C.↗

NASA NTRS1988

Computational structural mechanics and fluid dynamics: Advances and trends; Proceedings of the Symposium, Washington, DC, Oct. 17-19, 1988

Recent advances in computational structural and fluid dynamics are discussed in reviews and reports. Topics addressed include fluid-structure interaction and aeroelasticity, CFD techniques for reacting flows, micromechanics, stability and eigenproblems, probabilistic methods and chaotic dynamics, and perturbation and spectral methods. Consideration is given to finite-element, finite-volume, and boundary-element methods; adaptive methods; parallel processing machines and applications; and visualization, mesh generation, and AI interfaces.

Noor, Ahmed K.↗

NASA NTRS1997

High precision computing with charge domain devices and a pseudo-spectral method therefor

The present invention enhances the bit resolution of a CCD/CID MVM processor by storing each bit of each matrix element as a separate CCD charge packet. The bits of each input vector are separately multiplied by each bit of each matrix element in massive parallelism and the resulting products are combined appropriately to synthesize the correct product. In another aspect of the invention, such arrays are employed in a pseudo-spectral method of the invention, in which partial differential equations are solved by expressing each derivative analytically as matrices, and the state function is updated at each computation cycle by multiplying it by the matrices. The matrices are treated as synaptic arrays of a neural network and the state function vector elements are treated as neurons. In a further aspect of the invention, moving target detection is performed by driving the soliton equation with a vector of detector outputs. The neural architecture consists of two synaptic arrays corresponding to the two differential terms of the soliton-equation and an adder connected to the output thereof and to the output of the detector array to drive the soliton equation.

Barhen, Jacob↗

NASA NTRS1992

A spectral overlay method for dissimilar materials

The spectral overlay method is applied to examine the stress distribution along the interface between the two different materials. The essential feature of the spectral overlay method is that the high resolution of localized steep gradients can be achieved by overlaying a spectral interpolant on a standard finite element mesh. In order to evaluate this method and compare it with conventional finite element method, a test problem with high gradients in Poisson equation has been devised. The results show that this method is very powerful in capturing the shape of the gradient field and its peak value. Solutions for two problems with interfaces between dissimilar materials are then given to illustrate the effectiveness of this method.

Belytschko, T.↗

NASA NTRS1988

Semi-discrete approximations to nonlinear systems of conservation laws; consistency and L(infinity)-stability imply convergence

A convergence theory for semi-discrete approximations to nonlinear systems of conservation laws is developed. It is shown, by a series of scalar counter-examples, that consistency with the conservation law alone does not guarantee convergence. Instead, a notion of consistency which takes into account both the conservation law and its augmenting entropy condition is introduced. In this context it is concluded that consistency and L(infinity)-stability guarantee for a relevant class of admissible entropy functions, that their entropy production rate belongs to a compact subset of H(loc)sup -1 (x,t). One can now use compensated compactness arguments in order to turn this conclusion into a convergence proof. The current state of the art for these arguments includes the scalar and a wide class of 2 x 2 systems of conservation laws. The general framework of the vanishing viscosity method is studied as an effective way to meet the consistency and L(infinity)-stability requirements. How this method is utilized to enforce consistency and stability for scalar conservation laws is shown. In this context we prove, under the appropriate assumptions, the convergence of finite difference approximations (e.g., the high resolution TVD and UNO methods), finite element approximations (e.g., the Streamline-Diffusion methods) and spectral and pseudospectral approximations (e.g., the Spectral Viscosity methods).

Tadmor, Eitan↗

NASA NTRS2004

Parallel 3D Mortar Element Method for Adaptive Nonconforming Meshes

High order methods are frequently used in computational simulation for their high accuracy. An efficient way to avoid unnecessary computation in smooth regions of the solution is to use adaptive meshes which employ fine grids only in areas where they are needed. Nonconforming spectral elements allow the grid to be flexibly adjusted to satisfy the computational accuracy requirements. The method is suitable for computational simulations of unsteady problems with very disparate length scales or unsteady moving features, such as heat transfer, fluid dynamics or flame combustion. In this work, we select the Mark Element Method (MEM) to handle the non-conforming interfaces between elements. A new technique is introduced to efficiently implement MEM in 3-D nonconforming meshes. By introducing an "intermediate mortar", the proposed method decomposes the projection between 3-D elements and mortars into two steps. In each step, projection matrices derived in 2-D are used. The two-step method avoids explicitly forming/deriving large projection matrices for 3-D meshes, and also helps to simplify the implementation. This new technique can be used for both h- and p-type adaptation. This method is applied to an unsteady 3-D moving heat source problem. With our new MEM implementation, mesh adaptation is able to efficiently refine the grid near the heat source and coarsen the grid once the heat source passes. The savings in computational work resulting from the dynamic mesh adaptation is demonstrated by the reduction of the the number of elements used and CPU time spent. MEM and mesh adaptation, respectively, bring irregularity and dynamics to the computer memory access pattern. Hence, they provide a good way to gauge the performance of computer systems when running scientific applications whose memory access patterns are irregular and unpredictable. We select a 3-D moving heat source problem as the Unstructured Adaptive (UA) grid benchmark, a new component of the NAS Parallel Benchmarks (NPB). In this paper, we present some interesting performance results of ow OpenMP parallel implementation on different architectures such as the SGI Origin2000, SGI Altix, and Cray MTA-2.

Feng, Huiyu↗

NASA NTRS1992

Helicopter response to atmospheric turbulence

A new time-domain method for simulating cyclostationary turbulence as seen by a translating and rotating blade element has recently been developed for the case of one-dimensional spectral distribution. This paper extends the simulation method to the cases of two- and three-dimensional spectral distributions and presents validation results for the two-dimensional case. The statistics of an isolated rigid blade flapping response to turbulence are computed using a two-dimensional spectral representation of the von Karman turbulence model, and the results are compared with those obtained using the conventional space-fixed turbulence analysis. The new turbulence simulation method is used for predicting the Black Hawk helicopter response to atmospheric turbulence.

Riaz, J.↗

NASA NTRS1992

Theory and implementation of high-order adaptive hp methods for analysis of incompressible viscous flows

An account is given of 'smart' algorithms for CFD which change in structure and performance with during flow calculations to accommodate changing properties of the solution. Such algorithms prominently include adaptive FEM methods, which are designed to adjust mesh parameters for the control of numerical error; attention is presently given to those which change the mesh size h and the local spectral order p in order to achieve high accuracies with minimal numbers of degrees of freedom. These 'hp methods' produce exponentially convergent approximations through which flow features are resolved by automatically distributing element sizes and spectral orders. This leads to calculation of the local (elementwise) error in the approximation.

Oden, J. T.↗

NASA NTRS1974

Basic elements of power spectral analysis

The basic elements of power spectral analysis with emphasis on the Blackman-Tukey method are presented. Short discussions are included on the topics of pre-whitening, frequency and spectral windows, and statistical reliability. Examples are included whenever possible, and a FORTRAN subroutine for calculating a power spectrum is presented.

Sentman, D. D.↗

NASA NTRS2018

A Higher-Order Space-Time Finite-Element Method for Moving-Body and Fluid-Structure Interaction Problems

We present a high-order finite-element method for moving body and fluid/structure interaction problems. Our solution strategy is based on a space-time discontinuous Galerkin (DG) spectral-element discretization which extends to arbitrary order of accuracy. The space-time DG discretization is a natural choice for moving body and fluid-structure interaction problems as moving surfaces are incorporated simply by considering curved space-time elements whose space-time faces align with the moving body. We present a discontinuous-Galerkin in time discretization for six-degree of motion modeling of rigid bodies, and a continuous-Galerkin discretization for equations of linear elasticity to generate curved space-time meshes. Numerical results for several simple 2D test cases are presented in order to verify the implementation of the different models. Finally we present a preliminary dynamic simulation of a parachute.

High-Order↗

NASA NTRS1994

A spectral boundary integral equation method for the 2-D Helmholtz equation

In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the nonsmoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the nonsmoothness of the integral kernels in the spectral implementation. The present method is robust for a general boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. A numerical example of wave scattering is given in which the exponential accuracy of the present numerical method is demonstrated.

Hu, Fang Q.↗

NASA NTRS1989

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

A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.

Ho, Lee-Wing↗

NASA NTRS1992

A spectral element approach to wave motion in layered solids

A matrix methodology similar to that of the finite element method is developed for the analysis of stress waves in layered solids. Because the mass distribution is modeled exactly, the approach gives the exact frequency response of each layer. The fast Fourier transform and Fourier series are used for inversion to the time/space domain. The impact of a structured medium with multiple layers is used to demonstrate the method. Comparison with existing propagator and direct global matrix methods show the present approach to be computationally more efficient.

Rizzi, S. A.↗