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DOE OSTI2025

Multiscale Modeling Framework Using Element‐Based Galerkin Methods for Moist Atmospheric Limited‐Area Simulations

This paper presents a multiscale modeling framework (MMF) to model moist atmospheric limited-area weather. The MMF resolves large-scale convection using a coarse grid while simultaneously resolving local features through numerous fine local grids and coupling them seamlessly. Both large- and small-scale processes are modeled using the compressible Navier-Stokes equations within the Nonhydrostatic Unified Model of the Atmosphere (NUMA), and are discretized using a continuous element-based Galerkin method (spectral elements) with high-order basis functions. Consequently, the large-scale and small-scale models share the same dynamical core but have the flexibility to be adjusted individually. The proposed MMF method is tested in 2D and 3D idealized limited-area weather problems involving storm clouds produced by squall line and supercell simulations. Numerical results from the MMF showed enhanced representation of cloud processes compared to the coarse model.

Kang, Soonpil [Naval Postgraduate School, Monterey↗

DOE OSTI2024

Verification and validation of detonation-shock-dynamics relations for explosives described by general equation of state and chemical reaction models

Detonation shock dynamics is a powerful method to model the behaviour of High Explosives (HE). However in order to use this method, the underlying relationship between the local radius of curvature and the detonation speed must be known. Previous work has developed methods to calculate this effect using simple, single-step Arrhenius and polytropic gas, models for the chemical reaction and the equation of state, respectively. In recent years, more complex models for both reaction rates and equations of state have been developed which show better agreement with experimental data than these simple models, especially when considering condensed phase explosives.. This work presents the governing equations for solving these problems in a way that is generalised to use arbitrary equations of state as well as reaction models which may have more than a single step and multiple product species. This implementation is verified against exact solutions, demonstrating that the equations were implemented properly. The verified algorithm is then validated against experimental data and high fidelity simulations, showing that it is able to make accurate predictions in a regime where the underlying assumptions of the governing equations are valid. Importantly, this approach has many applications: from creating equivalent detonation shock dynamics models for existing reactive burn calibrations for HE; to developing new functional forms and calibrations of reactive burn models for condensed phase high explosives.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRSDate not supplied

Efficient Preconditioning of a High-Order Solver for Multiple Physics

This work addresses preconditioning approaches for an implicit high-order solver frame-work applied to multiple physics. The solver is based on a space-time spectral element method and matrix-free Newton-Krylov solver developed at NASA over the recent years. Within this context, most preconditioning methods are impractical, as the computational time and memory requirements scale poorly with increasing polynomial orders. To improve computational efficiency, we first describe a novel entity-based Block Jacobi preconditioner for the continuous-Galerkin solution of the linear-elasticity and linear-shell equations. Second, we introduce a multigrid algorithm to further reduce time-to-solution on stiff cases arising from continuous-and discontinuous-Galerkin discretizations. Results obtained on relevant single-physics reference solutions, demonstrate the feasibility of the methods, paving the way for high-order solutions of fully coupled multi-physics problems.

STMD↗

NASA NTRS1976

Multistep methods of numerical integration using back-corrections

A class of linear multistep methods is proposed for the solution of the equations of motion of certain dynamical systems encountered in celestial mechanics and astrodynamics. These methods are distinguished from the classical predictor-corrector methods in that they permit 'back-corrections' of the solution to be made. As the integration advances in time, the numerical solution is corrected or improved at certain points in the past. The enhanced numerical stability of these methods allows the meaningful application of high-order algorithms. Consequently, step sizes larger than those attainable with the classical methods may be adopted, and greater overall efficiency may be realized. These methods are applied to the problem of determining the orbit of an artificial satellite, and the results are compared with those obtained using classical methods.

Feagin, T.↗

NASA NTRS2019

Numerical Predictions of Dust-Induced Heat Flux Augmentation in Hypersonic Blunt-Body Flows Using a Discontinuous Galerkin Multiphase Flow Solver

Recent interest in human-scale missions to Mars has motivated the need for high-fidelity simulations of reentry flows. During a dust storm, there can be high levels of suspended dust in the Martian atmosphere, which cannot only enhance erosion of thermal protection systems but also transfer energy and momentum to the shock layer, thereby significantly augmenting the surface heat flux. Second-order finite-volume schemes are typically employed for hypersonic flow simulations, but such schemes suffer from a number of disadvantages. An attractive alternative is discontinuous Galerkin methods, which benefit from arbitrarily high spatial order of accuracy, geometric flexibility, and other properties. To enable accurate computations of high-speed particle-laden flows, an Euler-Lagrange methodology was developed in which the Eulerian field of the carrier gas is calculated using a discontinuous Galerkin scheme while the disperse phase is treated with Lagrangian particle tracking. We discuss challenges associated with coupling these two formulations and how to handle them. Momentum and energy transfer between the carrier gas and the particle phase is considered, and the importance of accounting for interparticle collisions is assessed. In addition, we describe the physical model of the particle phase and examine effects of its uncertainties on the numerical solution. We demonstrate the performance of the Euler-Lagrange method in representative testcases, with focus on the accurate prediction of particle trajectories and heating augmentation. Quantitative comparisons with experiments are provided.

Ching, Eric J.↗

NASA NTRS1990

Least-squares finite element methods for compressible Euler equations

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L-sq-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L-sq method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

Jiang, Bo-Nan↗

DOE OSTI2024

A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods

In Dzanic (2024), a limiting approach for high-order discontinuous Galerkin schemes was introduced which allowed for imposing constraints on the solution continuously (i.e., everywhere within the element). While exact for linear constraint functionals, this approach only imposed a sufficient (but not the minimum necessary) amount of limiting for nonlinear constraint functionals. This short note shows how this limiting approach can be extended to allow exactness for general nonlinear quasiconcave constraint functionals through a nonlinear limiting procedure, reducing unnecessary numerical dissipation. Finally, some examples are shown for nonlinear pressure and entropy constraints in the compressible gas dynamics equations, where both analytic and iterative approaches are used.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1992

High-order wide-band frequency domain identification using composite curve fitting

A method is presented for curve fitting nonparametric frequency domain data so as to identify a parametric model composed of two models in parallel, where each model has dynamics in a specified portion of the frequency band. This decomposition overcomes the problem of numerical sensitivity since lower order polynomials can be used compared to existing methods which estimate the model as a single entity. Consequently, composite curve fitting is useful for frequency domain identification of high-order systems and/or systems whose dynamics are spread over a large bandwidth. The approach can be extended to identify an arbitrary number of parallel subsystems in specified frequency regimes.

Bayard, D. S.↗

NASA NTRS2021

Evaluation of Higher-order Quadrature Schemes in Improving Computational Efficiency for Orientation-averaged Single-Scattering Properties of Nonspherical Ice Particles

We evaluate several high-order quadrature schemes for accuracy and efficacy in obtaining orientation-averaged single-scattering properties (SSPs). We use the highly efficient MIDAS to perform electromagnetic scattering calculations to evaluate the gain in efficiency from these schemes. MIDAS is shown to be superior to DDSCAT, a popular discrete dipole approximation (DDA) method. This study is motivated by the fact that quality physical precipitation retrievals rely on using accurate orientation-averaged SSPs derived from realistic hydrometeors as input to radiative transfer Models (RTMs). The DDA has been a popular choice for single-scattering calculations, due to its versatility with respect to target geometry. However, being iterative-solver-based (ISB), the most used DDA codes, e.g. DDSCAT and ADDA, must solve the scattering problem for each orientation of the target separately. As the size parameter and geometric anisotropy of the hydrometeor increase, the number of orientations needed to obtain accurate orientation-averages can increase drastically and so does the computation cost incurred by the ISB-DDA methods. MIDAS is a Direct-Solver-Based (DSB) code, its decomposition of the original large matrix with a high rank into multiple more manageable smaller matrices of lower ranks makes it much more computationally efficient while maintaining excellent accuracy. In addition, direct solvers consider all requested orientations at once, giving MIDAS further advantage over popular ISB-DDA methods. MIDAS, when combined with high-order quadrature for orientation averaging, can be greater than three orders of magnitude more efficient in obtaining RTM-ready SSPs of complex-shaped hydrometeors than existing ISB-DDA methods, with the native quadrature schemes they offer.

Ines Fenni↗

NASA NTRS1991

A hierarchy for modeling high speed propulsion systems

General research efforts on reduced order propulsion models for control systems design are overviewed. Methods for modeling high speed propulsion systems are discussed including internal flow propulsion systems that do not contain rotating machinery, such as inlets, ramjets, and scramjets. The discussion is separated into four areas: (1) computational fluid dynamics models for the entire nonlinear system or high order nonlinear models; (2) high order linearized models derived from fundamental physics; (3) low order linear models obtained from the other high order models; and (4) low order nonlinear models (order here refers to the number of dynamic states). Included in the discussion are any special considerations based on the relevant control system designs. The methods discussed are for the quasi-one-dimensional Euler equations of gasdynamic flow. The essential nonlinear features represented are large amplitude nonlinear waves, including moving normal shocks, hammershocks, simple subsonic combustion via heat addition, temperature dependent gases, detonations, and thermal choking. The report also contains a comprehensive list of papers and theses generated by this grant.

Hartley, Tom T.↗

NASA NTRS1987

Uniformly high order accurate essentially non-oscillatory schemes. III

In the present continuation of development and analysis efforts towards essentially nonoscillatory shock-capturing approximations of hyperbolic conservation laws, the Godunov (1959) scheme and its MUSCL second-order-accurate extension are generalized to an arbitrary order of accuracy by means of a hierarchy of uniformly high-order-accurate schemes. The design method, which involves the solution's essentially nonoscillatory piecewise polynomial reconstruction on the basis of its cell averages, followed by time-evolution through an approximate solution of the resulting initial value problem and its averaging over each cell, employs an adaptive stencil of grid points to yield schemes that are highly nonlinear.

Harten, Ami↗

NASA NTRS1991

Improved large perturbation propulsion models for control system design (1988-1989) and large perturbation models of high velocity propulsion systems (1989-1990) and reduced order propulsion models for control system design (1990-1991)

Methods for modeling high speed propulsion systems will be discussed. Included in this category are internal flow propulsion systems without rotating machinery, such as inlets, ramjets, and scramjets. Among the modeling topics discussed are modeling of linear isentropic flow, heat exchange, gasdynamics, lumped parameter systems, and infinite dimensional systems. Furthermore, a generalized overview of modeling high speed propulsion systems is presented in this collection of papers.

Hartley, Tom T.↗

NASA NTRS1997

Direct Numerical Simulations of Boundary Layer Transition on a Flat Plate

In recent years the techniques of computational fluid dynamics (CFD) have been used to compute flows associated with geometrically complex configurations. However, success in terms of accuracy and reliability has been limited to cases where the effects of turbulence and transition could be modeled in a straightforward manner. Even in simple flows, the accurate computation of skin friction and heat transfer using existing turbulence models has proved to be a difficult task, one that has required extensive fine-tuning of the turbulence models used. In more complex flows (for example, in turbomachinery flows in which vortices and wakes impinge on airfoil surfaces causing periodic transitions from laminar to turbulent flow) the development of a model that accounts for all scales of turbulence and predicts the onset of transition is an extremely difficult task. Fortunately, current trends in computing suggest that it may be possible to perform direct simulations of turbulence and transition at moderate Reynolds numbers in some complex cases in the near future. This presentation will focus on direct simulations of transition and turbulence using high-order accurate finite-difference methods. The advantage of the finite-difference approach over spectral methods is that complex geometries can be treated in a straightforward manner. Additionally, finite-difference techniques are the prevailing methods in existing application codes. An application of accurate finite-difference methods to direct simulations of transition and turbulence in a spatially evolving boundary layer subjected to high levels of freestream turbulence will be presented.

Rai, Man Mohan↗

NASA NTRS2016

Development of a Perfectly Matched Layer Technique for a Discontinuous-Galerkin Spectral-Element Method

The perfectly matched layer (PML) technique is developed in the context of a high- order spectral-element Discontinuous-Galerkin (DG) method. The technique is applied to a range of test cases and is shown to be superior compared to other approaches, such as those based on using characteristic boundary conditions and sponge layers, for treating the inflow and outflow boundaries of computational domains. In general, the PML technique improves the quality of the numerical results for simulations of practical flow configurations, but it also exhibits some instabilities for large perturbations. A preliminary analysis that attempts to understand the source of these instabilities is discussed.

Spectral-Element↗

DOE OSTI2025

Consistent Second Moment Methods with Scalable Linear Solvers for Radiation Transport

Second moment methods (SMMs) are developed that are consistent with the discontinuous Galerkin spatial discretization of the discrete ordinates (or S\(_N\)) transport equations. The low-order (LO) diffusion system of equations is discretized with fully consistent P\(_1\), local discontinuous Galerkin (LDG), and interior penalty (IP) methods. A discrete residual approach is used to derive SMM correction terms that make each of the LO systems consistent with the high-order discretization. We show that the consistent methods are more accurate and have better solution quality than independently discretized LO systems, that they preserve the diffusion limit, and that the LDG and IP consistent SMMs can be scalably solved in parallel on a challenging, multimaterial benchmark problem.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

Understanding transition and turbulence through direct simulations

Direct simulations consist in solving the full Navier-Stokes equations, without any turbulence model, and describing all the detailed features of the flow. Usually the flows are three-dimensional and time-dependent and contain both coarse and fine structures, which makes the numerical task very challenging in terms of both the algorithm and the computational effort. Most of the work until now has involved spectral methods, which are highly accurate but not very flexible in terms of geometry or complex equations. For that reason, future work will also rely on high-order finite-difference or other methods. Direct simulations complement experimental work, and both contribute to the theory and the empirical knowledge of turbulence. Once such a simulation has been shown to be accurate, the flow field is completely known in three dimensions and time, including the pressure, the vorticity and any other quantity. On the other hand, most simulations to date solved the incompressible equations in rather simple geometries, and direct simulations will always be limited to moderate Reynolds numbers. Extensive simulations have been conducted in homogeneous turbulence, channel flows, boundary layers, and mixing layers. Much effort is devoted to addressing flows with compressibility and chemical reactions, and to new geometries such as a backward-facing step.

Spalart, P. R.↗

NASA NTRS2015

Higher-Order Methods for Compressible Turbulent Flows Using Entropy Variables

Turbulent flows have a large range of spatial and temporal scales which need to be resolved in order to obtain accurate predictions. Higher-order methods can provide greater efficiency for simulations requiring high spatial and temporal resolution, allowing for solutions with fewer degrees of freedom and lower computational cost than traditional second-order computational fluid dynamics (CFD) methods.1 Higher-order methods have been widely used for turbulent flows. However, the reduced numerical stabilization present in higher-order schemes implies that special care needs to be taken in the development of numerical methods to suppress nonlinear instabilities.2–6 In this work we present the development of a higher-order space-time discontinuous Galerkin method with a focus on the aspects of our numerical scheme required for ensuring nonlinear stability for turbulent simulations at high Reynolds numbers.

Diosady, Laslo T.↗

NASA NTRS2001

On the Accuracy of Probabilistic Bucking Load Prediction

The buckling strength of thin-walled stiffened or unstiffened, metallic or composite shells is of major concern in aeronautical and space applications. The difficulty to predict the behavior of axially compressed thin-walled cylindrical shells continues to worry design engineers as we enter the third millennium. Thanks to extensive research programs in the late sixties and early seventies and the contributions of many eminent scientists, it is known that buckling strength calculations are affected by the uncertainties in the definition of the parameters of the problem such as definition of loads, material properties, geometric variables, edge support conditions, and the accuracy of the engineering models and analysis tools used in the design phase. The NASA design criteria monographs from the late sixties account for these design uncertainties by the use of a lump sum safety factor. This so-called 'empirical knockdown factor gamma' usually results in overly conservative design. Recently new reliability based probabilistic design procedure for buckling critical imperfect shells have been proposed. It essentially consists of a stochastic approach which introduces an improved 'scientific knockdown factor lambda(sub a)', that is not as conservative as the traditional empirical one. In order to incorporate probabilistic methods into a High Fidelity Analysis Approach one must be able to assess the accuracy of the various steps that must be executed to complete a reliability calculation. In the present paper the effect of size of the experimental input sample on the predicted value of the scientific knockdown factor lambda(sub a) calculated by the First-Order, Second-Moment Method is investigated.

Arbocz, Johann↗