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DOE OSTI2014

Numerical Evidence of Multiple Solutions for the Reynolds-Averaged Navier–Stokes Equations

In this paper, evidence is presented for the existence of multiple solutions of Reynolds-averaged Navier–Stokes equations with the one-equation Spalart–Allmaras and two-equation Wilcox $k-ω$ turbulence models on fixed grids in three dimensions and how they were obtained is described. The two major configurations considered are an “academic” extruded two-dimensional airfoil geometry and the trap wing test case. The observed appearance of the multiple solutions seems to be closely related to smooth body separation (sometimes massive) routinely observed in flows over high-lift configurations, especially near stall angles of attack. The results are obtained and cross-verified with two stabilized finite-element codes (streamwise upwind Petrov–Galerkin), which provide residual converged results for complex flows with second-order discretizations. In the paper, the ways multiple solutions have been obtained are described, including such obvious ones as providing a different initial guess to the steady-state solver as well as somewhat unexpected (in this context) techniques of using implicit residual smoothing while time marching to steady state. The phenomenon of the so-called pseudosolutions is also discussed, which is loosely defined as solutions to the discrete system of equations having “sharp” convergence behavior (sometimes up to six to seven orders of the relative residual reduction), which fail to achieve the stronger, machine-zero convergence criterion. Some numerical observations are also presented on the sensitivity of the obtained multiple solutions to the discretization and grid perturbations.

Engineering↗

NASA NTRS1984

Static shape determination and control for large space structures. I - The flexible beam. II - A large space antenna

A method for determining and controlling the shape of large, continuous space structures by means of discrete or pointwise observations and control devices is presented. The general linear boundary value problem satisfied by a one-dimensional shape function is defined, and the existence of solutions is studied. The static shape control problems for one-dimensional systems with and without rigid body modes and the static shape estimation problem are presented and solved. Eigenfunction expansions are presented which provide approximations to the algorithm terms when the associated Green's function is not known. An integral operator approach is applied to the multidimensional static problem, and the results are illustrated with a finite element model of the disk of a large space antenna which assumes no rigid body modes. It is shown that the shape control algorithm must be modified for systems with rigid body modes.

Weeks, C. J.↗

NASA NTRS1996

High-Performance Parallel Analysis of Coupled Problems for Aircraft Propulsion

This research program dealt with the application of high-performance computing methods to the numerical simulation of complete jet engines. The program was initiated in January 1993 by applying two-dimensional parallel aeroelastic codes to the interior gas flow problem of a bypass jet engine. The fluid mesh generation, domain decomposition and solution capabilities were successfully tested. Attention was then focused on methodology for the partitioned analysis of the interaction of the gas flow with a flexible structure and with the fluid mesh motion driven by these structural displacements. The latter is treated by a ALE technique that models the fluid mesh motion as that of a fictitious mechanical network laid along the edges of near-field fluid elements. New partitioned analysis procedures to treat this coupled three-component problem were developed during 1994 and 1995. These procedures involved delayed corrections and subcycling, and have been successfully tested on several massively parallel computers, including the iPSC-860, Paragon XP/S and the IBM SP2. For the global steady-state axisymmetric analysis of a complete engine we have decided to use the NASA-sponsored ENG10 program, which uses a regular FV-multiblock-grid discretization in conjunction with circumferential averaging to include effects of blade forces, loss, combustor heat addition, blockage, bleeds and convective mixing. A load-balancing preprocessor tor parallel versions of ENG10 was developed. During 1995 and 1996 we developed the capability tor the first full 3D aeroelastic simulation of a multirow engine stage. This capability was tested on the IBM SP2 parallel supercomputer at NASA Ames. Benchmark results were presented at the 1196 Computational Aeroscience meeting.

Felippa, C. A.↗

NASA NTRS2001

Characterizing Aeroelastic Systems Using Eigenanalysis, Explicitly Retaining The Aerodynamic Degrees of Freedom

Discrete time aeroelastic models with explicitly retained aerodynamic modes have been generated employing a time marching vortex lattice aerodynamic model. This paper presents analytical results from eigenanalysis of these models. The potential of these models to calculate the behavior of modes that represent damped system motion (noncritical modes) in addition to the simple harmonic modes is explored. A typical section with only structural freedom in pitch is examined. The eigenvalues are examined and compared to experimental data. Issues regarding the convergence of the solution with regard to refining the aerodynamic discretization are investigated. Eigenvector behavior is examined; the eigenvector associated with a particular eigenvalue can be viewed as the set of modal participation factors for that particular mode. For the present formulation of the equations of motion, the vorticity for each aerodynamic element appears explicitly as an element of each eigenvector in addition to the structural dynamic generalized coordinates. Thus, modal participation of the aerodynamic degrees of freedom can be assessed in M addition to participation of structural degrees of freedom.

Heeg, Jennifer↗

DOE OSTI2022

An Efficient B-Spline Lagrangian/Eulerian Method for Compressible Flow, Shock Waves, and Fracturing Solids

This study presents a new method for modeling the interaction between compressible flow, shock waves, and deformable structures, emphasizing destructive dynamics. Extending advances in time-splitting compressible flow and the Material Point Methods (MPM), we develop a hybrid Eulerian and Lagrangian/Eulerian scheme for monolithic flow-structure interactions. We adopt the second-order WENO scheme to advance the continuity equation. To stably resolve deforming boundaries with sub-cell particles, we propose a blending treatment of reflective and passable boundary conditions inspired by the theory of porous media. The strongly coupled velocity-pressure system is discretized with a new mixed-order finite element formulation employing B-spline shape functions. Shock wave propagation, temperature/density-induced buoyancy effects, and topology changes in solids are unitedly captured.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Exploration of High-Dimensional Scalar Function for Nuclear Reactor Safety Analysis and Visualization

The next generation of methodologies for nuclear reactor Probabilistic Risk Assessment (PRA) explicitly accounts for the time element in modeling the probabilistic system evolution and uses numerical simulation tools to account for possible dependencies between failure events. The Monte-Carlo (MC) and the Dynamic Event Tree (DET) approaches belong to this new class of dynamic PRA methodologies. A challenge of dynamic PRA algorithms is the large amount of data they produce which may be difficult to visualize and analyze in order to extract useful information. We present a software tool that is designed to address these goals. We model a large-scale nuclear simulation dataset as a high-dimensional scalar function defined over a discrete sample of the domain. First, we provide structural analysis of such a function at multiple scales and provide insight into the relationship between the input parameters and the output. Second, we enable exploratory analysis for users, where we help the users to differentiate features from noise through multi-scale analysis on an interactive platform, based on domain knowledge and data characterization. Our analysis is performed by exploiting the topological and geometric properties of the domain, building statistical models based on its topological segmentations and providing interactive visual interfaces to facilitate such explorations. We provide a user’s guide to our software tool by highlighting its analysis and visualization capabilities, along with a use case involving dataset from a nuclear reactor safety simulation.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Exploration of high-dimensional scalar function for nuclear reactor safety analysis and visualization

The next generation of methodologies for nuclear reactor Probabilistic Risk Assessment (PRA) explicitly accounts for the time element in modeling the probabilistic system evolution and uses numerical simulation tools to account for possible dependencies between failure events. The Monte-Carlo (MC) and the Dynamic Event Tree (DET) approaches belong to this new class of dynamic PRA methodologies. A challenge of dynamic PRA algorithms is the large amount of data they produce which may be difficult to visualize and analyze in order to extract useful information. We present a software tool that is designed to address these goals. We model a large-scale nuclear simulation dataset as a high-dimensional scalar function defined over a discrete sample of the domain. First, we provide structural analysis of such a function at multiple scales and provide insight into the relationship between the input parameters and the output. Second, we enable exploratory analysis for users, where we help the users to differentiate features from noise through multi-scale analysis on an interactive platform, based on domain knowledge and data characterization. Our analysis is performed by exploiting the topological and geometric properties of the domain, building statistical models based on its topological segmentations and providing interactive visual interfaces to facilitate such explorations. We provide a user's guide to our software tool by highlighting its analysis and visualization capabilities, along with a use case involving data from a nuclear reactor safety simulation. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1972

Analysis of a laterally loaded ring with a hinged cross section.

A ring assembly constructed by lacing together three elements into a basic channel cross section is analyzed. The ring is supported at three equidistant points and loaded by a uniform distribution of radial and transverse loads. Bulkheads may be introduced at discrete cross sections to prevent distortion. The purpose of this analysis is to determine the behavior of the deflection as the number and location of the bulkheads and the degree of rigidity of the supports are varied. The method of analysis is an application of the Principle of Virtual Work within the framework of small displacement theory. Numerical results are presented for the geometrical parameters of a model. An important result is the observation that bulkheads have virtually no effect on the deflections of the web.

Williams, H. E.↗

NASA NTRS1993

Remote sensing x ray fluorescence spectrometry for future lunar exploration missions

Measurement of discrete line X-ray emission from space can be used to obtain both qualitative and quantitative elemental surface composition information. Remote orbital X-ray elemental analysis is the measurement of characteristic X-rays following the interaction of solar X-rays with the surface of a given solid body. Thus, X-ray emission from a surface is strongly dependent on the incident solar spectrum as well as on the chemical composition of the surface. In this paper, the relationship between the solar flux and the major lunar element fluorescence and scattered X-ray flux is characterized. A model was developed to calculate lunar surface X-ray fluorescence emission spectra under a variety of solar conditions and, in particular, conditions that might be expected for a Lunar Scout mission proposed for 1995 or 1996. This time period should be during solar minimum.

Clark, Pamela E.↗

NASA NTRS1996

Analysis of Sensory/Active Piezoelectric Composite Structures in Thermal Environments

Although there has been extensive development of analytical methods for modeling the behavior of piezoelectric structures, only a limited amount of research has been performed concerning the implications of thermal effects on both the active and sensory response of smart structures. Thermal effects become important when the piezoelectric structure has to operate in either extremely hot or cold temperature environments. Consequently, the purpose of this paper is to extend the previously developed discrete layer formulation of Saravanos and Heyliger to account for the coupled mechanical, electrical, and thermal response in modern smart composite beams. The mechanics accounts for thermal effects which may arise in the elastic and piezoelectric media at the material level through the constitutive equations. The displacements, electric potentials, and temperatures are introduced as state variables, allowing them to be modeled as variable fields through the laminate thickness. This unified representation leads to an inherent capability to model both the active compensation of thermal distortions in smart structures and the resultant sensory voltage when thermal loads are applied. The corresponding finite element formulation is developed and numerical results demonstrate the ability to model both the active and sensory modes of composite beams with heterogeneous plies with attached piezoelectric layers under thermal loadings.

Lee, Ho-Jun↗

DOE OSTI2020

BISON Robustness and Performance Improvements

BISON is a modern finite-element based nuclear fuel performance code that has been under development at the Idaho National Laboratory (USA) since 2009 [1]. The code is applicable to both steady and transient fuel behavior and can be used to analyze 1D (spherically symmetric), 2D (axisymmetric and generalized plane strain) or 3D geometries. BISON is the fuel performance code used within CASL for LWR fuel under both normal operating and accident conditions. BISON is built using the INL Multiphysics ObjectOriented Simulation Environment, or MOOSE [2, 3]. MOOSE is a massively parallel, finite element-based framework to solve systems of coupled non-linear partial differential equations using the Jacobian-Free Newton Krylov (JFNK) method [4]. This enables investigation of computationally large problems, for example a full stack of discrete pellets in a LWR fuel rod, or every rod in a full reactor core. MOOSE supports the use of complex two and three-dimensional meshes and uses implicit time integration, important for the widely varied time scale in nuclear fuel simulation. An object-oriented architecture is employed which greatly minimizes the programming effort required to add new material and behavioral models. The flexibility of the implicit and fully coupled multiphysics approach comes with a need for constructing suitable approximations for the Jacobian matrix of the coupled system used for either preconditioning a Krylov solve or in a direct Newton solve. Preconditioning options for Bison problems need to be revisited with new preconditioning methods becoming available.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2021

A solution framework for linear PDE-constrained mixed-integer problems

Abstract We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Computational Evaluation of Thermal Response of Open-Cell Foam With Circular Pore (Computional Evaluation of Thermal Response of Open-Cell Foam with Circular Pore)

The evaluation of effective material properties in heterogeneous materials (e.g., composites or multicomponent structures) critically relevant to a wide spectrum of applications, including nuclear power, electronic packaging, flame retardants, hypersonics, and gas turbine power. The work described in this paper is centered around the numerical assessment of the thermal behavior of porous materials obtained from finite element thermal modeling and simulation. Here, two-dimensional, steady state analyses were performed on unit cells with centered, circular pores using a second order accurate Galerkin finite element method (FEM). The effective thermal conductivities of the porous systems were examined, encompassing a range of porosities from 4.9% to 60.1%. The geometries of the models were generated based on ordered circular pores for each modeled porosity level. The system response quantity (SRQ) under investigation was the dimensionless effective thermal conductivity across the unit cell. The dimensionless effective thermal conductivity was compared across all simulated cases, producing a trend between porosity and effective thermal conductivity. In the presented investigation, the method of manufactured solutions (MMS) was used to perform code verification, and the grid convergence index (GCI) was employed to estimate discretization uncertainty as solution verification. Code verification concluded an approximately second order accurate Galerkin FEM solver. It was found that the introduction of porosity to the unit cell material structure reduces effective thermal conductivity, as anticipated. Numerical results obtained in this study are compared to an analytical solution and to a sample of empirical data.

36 MATERIALS SCIENCE↗

DOE OSTI2021

A review of rare earth elements and yttrium in coal ash: Content, modes of occurrences, combustion behavior, and extraction methods

Rare earth elements and yttrium (REY) have attracted considerable attention over the last decade because of their vital roles in clean energy, consumer product, national defense and security applications, among other uses. Due to the retention of REY during coal burning, coal combustion ash is considered as potential alternative sources for REY. Understanding the content, speciation, retention and/or transformation behavior of REY during coal combustion not only expands our knowledge of the combustion behavior of the trace elements in coal, but also provides basis for modeling REY partitioning during coal combustion and for developing economically viable REY recovery technologies. This review makes a critical summary of recent progress in the study of REY in coal ash. The contents and the extraction potentials of REY in coal ash derived from 15 major coal-producing countries worldwide were summarized and evaluated. Various analytical methods for determining REY bulk contents and speciation, together with the solid sample pretreatment, analytical accuracy and precision, advantages and disadvantages were summarized and compared. Modern analytical approaches combined indirect methods (e.g., sequential extraction) shed light on the physical distribution, mineralogy, and the chemical state of REY in coal ash. Three types of REY occurrences in coal ash, including Si-Al glassy association, discrete minerals or compounds, and organic association (bound with unburned carbon) were defined in the review. The glassy association can be further divided into REY minerals closely bound to glass phases and dispersed throughout the glassy structure. REY partitioning in various emission streams, the size distribution, and their enrichment behavior in coal ash were discussed. Additionally, thermal behavior and transformation of various REY forms in coal during combustion process, including organic-associated REY, REY phosphates, REY carbonates, clay-bound REY and among others were summarized. Two possible retention mechanisms of REY by aluminosilicate glass at boiler temperature were proposed: the incorporation of the individual REY phases into the glass as inclusions and the diffusion of REY phases throughout Si-Al glass structures in the melting process. Feed coal mineral types, mineral-mineral associations, boiler conditions, and other factors control the retention process. After coal combustion, the speciation of REY in fly ash may be modified by the reactions of REY phases with flue gas components. Further, an overview of REY transformation mechanisms during coal combustion was deeply discussed. Finally, current extraction techniques for REY recovery from coal combustion ash were introduced. Future outlooks and research problems were also identified.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2019

A single crystal plasticity finite element formulation with embedded deformation twins

Deformation twinning is an important plastic deformation mechanism in some polycrystalline metals such as titanium and magnesium. In this paper, we present a novel crystal plasticity finite element framework that accounts for deformation twinning explicitly, in addition to crystallographic slip. Within this computational framework, deformation twins are treated as weak discontinuities embedded within individual finite elements, such that a jump in the velocity gradient field is introduced (via the discretized gradient operator) between the twinned and untwinned crystalline regions, taking into account compatibility and traction continuity conditions at the interface between these two regions. The deformation gradient is multiplicatively split into elastic and plastic parts in the untwinned region, as is customary in finite-deformation crystal plasticity formulations. A different multiplicative decomposition of the deformation gradient into elastic, plastic (slip), and twinning parts is adopted in the twinned region, allowing deformation twinning to be accounted for as an additional mode of plastic deformation. Here, a stochastic model is used to predict twin nucleation at grain boundaries, and the evolution of the length and thickness of the twinned region under subsequent deformation is taken into account.

36 MATERIALS SCIENCE↗

DOE OSTI2014

Intrinsic Uncertainties in Modeling Complex Systems.

Models are built to understand and predict the behaviors of both natural and artificial systems. Because it is always necessary to abstract away aspects of any non-trivial system being modeled, we know models can potentially leave out important, even critical elements. This reality of the modeling enterprise forces us to consider the prospective impacts of those effects completely left out of a model – either intentionally or unconsidered. Insensitivity to new structure is an indication of diminishing returns. In this work, we represent a hypothetical unknown effect on a validated model as a finite perturbation whose amplitude is constrained within a control region. We find robustly that without further constraints, no meaningful bounds can be placed on the amplitude of a perturbation outside of the control region. Thus, forecasting into unsampled regions is a very risky proposition. We also present inherent difficulties with proper time discretization of models and representing inherently discrete quantities. We point out potentially worrisome uncertainties, arising from mathematical formulation alone, which modelers can inadvertently introduce into models of complex systems.

Cooper, Curtis S.↗

NASA NTRS1998

Control System and Flexible Satellite Interaction During Orbit Transfer Maneuver

In this paper the interaction between the attitude control system and the flexible structure of an artificial satellite during orbit transfer maneuver has been investigated. The satellite was modeled by a rigid central body with one or more flexible appendages. The dynamics equations were obtained by Lagrangean approach. The flexible appendages were treated as clamped-free beam and its displacement was discretized by assumed- mode method. In order to transfer the satellite, a typical Hohmann transfer and a burn-coast-burn strategy were used and the attitude was controlled by an on-off controller. During transfer procedure a global analysis of satellite has been done, such as: performance of control system, influence of elastic response in control system, thruster firing frequency, fuel consumption and variation of orbital elements. In order to avoid the interaction with structure motion, a control system with bandwidth of one decade bellow the fundamental frequency was used. In the simulations the firing frequency was evaluated in an approximately way but kept below the fundamental frequency of the structure. The control system has kept the attitude below the specifications. As a result, the orbit transfer maneuvering has been done correctly without excessive excitation of flexible appendage.

Adenilson Roberto da Silva↗

NASA NTRS2015

Grid Convergence for Turbulent Flows(Invited)

A detailed grid convergence study has been conducted to establish accurate reference solutions corresponding to the one-equation linear eddy-viscosity Spalart-Allmaras turbulence model for two dimensional turbulent flows around the NACA 0012 airfoil and a flat plate. The study involved three widely used codes, CFL3D (NASA), FUN3D (NASA), and TAU (DLR), and families of uniformly refined structured grids that differ in the grid density patterns. Solutions computed by different codes on different grid families appear to converge to the same continuous limit, but exhibit different convergence characteristics. The grid resolution in the vicinity of geometric singularities, such as a sharp trailing edge, is found to be the major factor affecting accuracy and convergence of discrete solutions, more prominent than differences in discretization schemes and/or grid elements. The results reported for these relatively simple turbulent flows demonstrate that CFL3D, FUN3D, and TAU solutions are very accurate on the finest grids used in the study, but even those grids are not sufficient to conclusively establish an asymptotic convergence order.

Diskin, Boris↗