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NASA NTRS2006

Accuracy Quantification of the Loci-CHEM Code for Chamber Wall Heat Fluxes in a G02/GH2 Single Element Injector Model Problem

A robust rocket engine combustor design and development process must include tools which can accurately predict the multi-dimensional thermal environments imposed on solid surfaces by the hot combustion products. Currently, empirical methods used in the design process are typically one dimensional and do not adequately account for the heat flux rise rate in the near-injector region of the chamber. Computational Fluid Dynamics holds promise to meet the design tool requirement, but requires accuracy quantification, or validation, before it can be confidently applied in the design process. This effort presents the beginning of such a validation process for the Loci- CHEM CPD code. The model problem examined here is a gaseous oxygen (GO2)/gaseous hydrogen (GH2) shear coaxial single element injector operating at a chamber pressure of 5.42 MPa. The GO2/GH2 propellant combination in this geometry represents one the simplest rocket model problems and is thus foundational to subsequent validation efforts for more complex injectors. Multiple steady state solutions have been produced with Loci-CHEM employing different hybrid grids and two-equation turbulence models. Iterative convergence for each solution is demonstrated via mass conservation, flow variable monitoring at discrete flow field locations as a function of solution iteration and overall residual performance. A baseline hybrid grid was used and then locally refined to demonstrate grid convergence. Solutions were also obtained with three variations of the k-omega turbulence model.

West, Jeff↗

NASA NTRS2014

Multiscale Analysis of Structurally-Graded Microstructures Using Molecular Dynamics, Discrete Dislocation Dynamics and Continuum Crystal Plasticity

A multiscale modeling methodology is developed for structurally-graded material microstructures. Molecular dynamic (MD) simulations are performed at the nanoscale to determine fundamental failure mechanisms and quantify material constitutive parameters. These parameters are used to calibrate material processes at the mesoscale using discrete dislocation dynamics (DD). Different grain boundary interactions with dislocations are analyzed using DD to predict grain-size dependent stress-strain behavior. These relationships are mapped into crystal plasticity (CP) parameters to develop a computationally efficient finite element-based DD/CP model for continuum-level simulations and complete the multiscale analysis by predicting the behavior of macroscopic physical specimens. The present analysis is focused on simulating the behavior of a graded microstructure in which grain sizes are on the order of nanometers in the exterior region and transition to larger, multi-micron size in the interior domain. This microstructural configuration has been shown to offer improved mechanical properties over homogeneous coarse-grained materials by increasing yield stress while maintaining ductility. Various mesoscopic polycrystal models of structurally-graded microstructures are generated, analyzed and used as a benchmark for comparison between multiscale DD/CP model and DD predictions. A final series of simulations utilize the DD/CP analysis method exclusively to study macroscopic models that cannot be analyzed by MD or DD methods alone due to the model size.

Saether, Erik↗

NASA NTRS2006

Accuracy Quantification of the Loci-CHEM Code for Chamber Wall Heat Transfer in a GO2/GH2 Single Element Injector Model Problem

A robust rocket engine combustor design and development process must include tools which can accurately predict the multi-dimensional thermal environments imposed on solid surfaces by the hot combustion products. Currently, empirical methods used in the design process are typically one dimensional and do not adequately account for the heat flux rise rate in the near-injector region of the chamber. Computational Fluid Dynamics holds promise to meet the design tool requirement, but requires accuracy quantification, or validation, before it can be confidently applied in the design process. This effort presents the beginning of such a validation process for the Loci-CHEM CFD code. The model problem examined here is a gaseous oxygen (GO2)/gaseous hydrogen (GH2) shear coaxial single element injector operating at a chamber pressure of 5.42 MPa. The GO2/GH2 propellant combination in this geometry represents one the simplest rocket model problems and is thus foundational to subsequent validation efforts for more complex injectors. Multiple steady state solutions have been produced with Loci-CHEM employing different hybrid grids and two-equation turbulence models. Iterative convergence for each solution is demonstrated via mass conservation, flow variable monitoring at discrete flow field locations as a function of solution iteration and overall residual performance. A baseline hybrid was used and then locally refined to demonstrate grid convergence. Solutions were obtained with three variations of the k-omega turbulence model.

West, Jeff↗

DOE OSTI2025

Multiphysics Meshfree Degradation Modeling of Energy Storage Materials with Kernel Enrichment

Energy storage materials exhibit strong electro-chemo-mechanical coupling and highly anisotropic material properties, contributing to the formation and propagation of micro-cracking during charge/discharge cycling and ultimately diminishing performance and service life. With microstructural images supplied by the National Renewable Energy Laboratory (NREL), pixel-based meshfree model construction by the reproducing kernel particle method (RKPM) is used to represent the complex material microstructures that dictate the coupled physics of these systems. Traditional electro-chemo-mechanical models rely on mesh-based finite element methods, which can lead to difficulties in meshing such complex geometries and capturing crack propagation due to mesh dependency. The first kernel enrichment discussed will be the interface modified reproducing kernel (IM-RK) [1, 2], constructed by scaling a smooth kernel function with an interface-distance function to achieve strategic discontinuity types (i.e. weak discontinuities for strain discontinuities and strong discontinuities for cracks) and alleviate Gibbs oscillations near these transition zones. The IM-RK is especially useful for areas in which a known discontinuity-type is expected a priori. The second kernel enrichment to be discussed is a neural network-enhanced reproducing kernel (NN-RK) [3, 4], which is introduced to effectively model non-obvious damage and crack propagation in the material microstructures; the location, orientation, and solution transition near a localization are automatically captured by superimposed block-level NN optimizations. This NN enrichment approach allows for effective modeling of localizations via a fixed background discretization, relieving tedious efforts for adaptive refinement in traditional mesh-based methods. Applications to the heterogeneous microstructures of Li-ion battery cathodes will be presented to demonstrate the effectiveness of the proposed methods. NN-RK is additionally used to inform how crack opening and closure in turn affect the electro-chemo-mechanical responses in the material microstructure. Reference: [1] Wang, Y., Baek, J., Tang, Y. et al. "Support vector machine guided reproducing kernel particle method for image-based modeling of microstructures," Comput Mech 73, 907-942 (2024). https://doi.org/10.1007/s00466-023-02394-9. [2] Susuki, K., Allen, J. & Chen, J. S.. "Image-based modeling of coupled electro-chemo-mechanical behavior of Li-ion battery cathode using an interface-modified reproducing kernel particle method," Engineering with Computers (2024). https://doi.org/10.1007/s00366-024-02016-9. [3] Baek, J., Chen, J. S., Susuki, K., "Neural Network enhanced Reproducing Kernel Particle Method for Modeling Localizations," International Journal for Numerical Methods in Engineering, Vol. 123, 4422-4454 (2022). https://doi.org/10.1002/nme.7040.

25 ENERGY STORAGE↗

DOE OSTI2026

Kernel Enriched Meshfree Multiphysics Degradation Modeling of Energy Storage Materials

Energy storage materials exhibit strong electro-chemo-mechanical coupling and highly anisotropic material properties, contributing to the formation and propagation of micro-cracking during charge/discharge cycling and ultimately diminishing performance and service life. With microstructural images supplied by the National Laboratory of the Rockies (NLR), pixel-based meshfree model construction by the reproducing kernel particle method (RKPM) is used to represent the complex material microstructures that dictate the coupled physics of these systems. Traditional electro-chemo-mechanical models rely on mesh-based finite element methods, which can lead to difficulties in meshing such complex geometries and capturing crack propagation due to mesh dependency. The first kernel enrichment discussed will be the interface modified reproducing kernel (IM-RK) [1, 2], constructed by scaling a smooth kernel function with an interface-distance function to achieve strategic discontinuity types (i.e. weak discontinuities for strain discontinuities and strong discontinuities for cracks) and alleviate Gibbs oscillations near these transition zones. The IM-RK is especially useful for areas in which a known discontinuity-type is expected a priori. The second kernel enrichment to be discussed is a neural network-enhanced reproducing kernel (NN-RK) [3, 4], which is introduced to effectively model non-obvious damage and crack propagation in the material microstructures; the location, orientation, and solution transition near a localization are automatically captured by superimposed block-level NN optimizations. This NN enrichment approach allows for effective modeling of localizations via a fixed background discretization, relieving tedious efforts for adaptive refinement in traditional mesh-based methods. Applications to the heterogeneous microstructures of Li-ion battery cathodes will be presented to demonstrate the effectiveness of the proposed methods. NN-RK is additionally used to inform how crack opening and closure in turn affect the electro-chemo-mechanical responses in the material microstructure. References: [1] Wang, Y., Baek, J., Tang, Y. et al. "Support vector machine guided reproducing kernel particle method for image-based modeling of microstructures," Comput Mech 73, 907-942 (2024). https://doi.org/10.1007/s00466-023-02394-9. [2] Susuki, K., Allen, J. & Chen, J. S.. "Image-based modeling of coupled electro-chemo-mechanical behavior of Li-ion battery cathode using an interface-modified reproducing kernel particle method," Engineering with Computers (2024). https://doi.org/10.1007/s00366-024-02016-9. [3] Baek, J., Chen, J. S., Susuki, K., "Neural Network enhanced Reproducing Kernel Particle Method for Modeling Localizations," International Journal for Numerical Methods in Engineering, Vol. 123, 4422-4454 (2022). https://doi.org/10.1002/nme.7040.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

Hanging drop crystal growth apparatus and method

An apparatus (10) is constructed having a cylindrical enclosure (16) within which a disc-shaped wicking element (18) is positioned. A well or recess (22) is cut into an upper side (24) of this wicking element, and a glass cover plate or slip (28) having a protein drop disposed thereon is sealably positioned on the wicking element (18), with drop (12) being positioned over well or recess (22). A flow of control fluid is generated by a programmable gradient former (16), with this control fluid having a vapor pressure that is selectively variable. This flow of control fluid is coupled to the wicking element (18) where control fluid vapor diffusing from walls (26) of the recess (22) is exposed to the drop (12), forming a vapor pressure gradient between the drop (12) and the control fluid vapor. Initially, this gradient is adjusted to draw solvent from the drop (12) at a relatively high rate, and as the critical supersaturation point is approached (the point at which crystal nucleation occurs), the gradient is reduced to more slowly draw solvent from the drop (12). This allows discrete protein molecules more time to orient themselves into an ordered crystalline lattice, producing protein crystals which, when processed by X-ray crystallography, possess a high degree of resolution.

Carter, Daniel C.↗

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↗

NASA NTRS2007

A Statistical Approach for the Concurrent Coupling of Molecular Dynamics and Finite Element Methods

Molecular dynamics (MD) methods are opening new opportunities for simulating the fundamental processes of material behavior at the atomistic level. However, increasing the size of the MD domain quickly presents intractable computational demands. A robust approach to surmount this computational limitation has been to unite continuum modeling procedures such as the finite element method (FEM) with MD analyses thereby reducing the region of atomic scale refinement. The challenging problem is to seamlessly connect the two inherently different simulation techniques at their interface. In the present work, a new approach to MD-FEM coupling is developed based on a restatement of the typical boundary value problem used to define a coupled domain. The method uses statistical averaging of the atomistic MD domain to provide displacement interface boundary conditions to the surrounding continuum FEM region, which, in return, generates interface reaction forces applied as piecewise constant traction boundary conditions to the MD domain. The two systems are computationally disconnected and communicate only through a continuous update of their boundary conditions. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM) as opposed to a direct coupling method where interface atoms and FEM nodes are individually related. The methodology is inherently applicable to three-dimensional domains, avoids discretization of the continuum model down to atomic scales, and permits arbitrary temperatures to be applied.

Saether, E.↗

DOE OSTI2023

Modelling and simulation of brinicle formation

Below the Arctic sea ice, under the right conditions, a flux of icy brine flows down into the sea. The icy brine has a much lower fusion point and is denser than normal seawater. As a result, it sinks while freezing everything around it, forming an ice channel called a brinicle (also known as ice stalactite). In this paper, we develop a mathematical model for this phenomenon, assuming cylindrical symmetry. The fluid is considered to be viscous and quasi-stationary. The heat and salt transport are weakly coupled to the fluid motion and are modelled with the corresponding conservation equations, accounting for diffusive and convective effects. Finite-element discretization is employed to solve the coupled system of partial differential equations. We find that the model can capture the general behaviour of the physical system and generate brinicle-like structures while also recovering dendrite composition, which is a physically expected feature aligned with previous experimental results. This represents, to our knowledge, the first complete model proposed that captures the global structure of the physical phenomenon even though it has some discrepancies, such as brine accumulation.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2000

Finite Element Analysis of Reverberation Chambers

The primary motivating factor behind the initiation of this work was to provide a deterministic means of establishing the validity of the statistical methods that are recommended for the determination of fields that interact in -an avionics system. The application of finite element analysis to reverberation chambers is the initial step required to establish a reasonable course of inquiry in this particularly data-intensive study. The use of computational electromagnetics provides a high degree of control of the "experimental" parameters that can be utilized in a simulation of reverberating structures. As the work evolved there were four primary focus areas they are: 1. The eigenvalue problem for the source free problem. 2. The development of a complex efficient eigensolver. 3. The application of a source for the TE and TM fields for statistical characterization. 4. The examination of shielding effectiveness in a reverberating environment. One early purpose of this work was to establish the utility of finite element techniques in the development of an extended low frequency statistical model for reverberation phenomena. By employing finite element techniques, structures of arbitrary complexity can be analyzed due to the use of triangular shape functions in the spatial discretization. The effects of both frequency stirring and mechanical stirring are presented. It is suggested that for the low frequency operation the typical tuner size is inadequate to provide a sufficiently random field and that frequency stirring should be used. The results of the finite element analysis of the reverberation chamber illustrate io-W the potential utility of a 2D representation for enhancing the basic statistical characteristics of the chamber when operating in a low frequency regime. The basic field statistics are verified for frequency stirring over a wide range of frequencies. Mechanical stirring is shown to provide an effective frequency deviation.

Bunting, Charles F.↗

NASA NTRS2005

Path Planning Algorithms for the Adaptive Sensor Fleet

The Adaptive Sensor Fleet (ASF) is a general purpose fleet management and planning system being developed by NASA in coordination with NOAA. The current mission of ASF is to provide the capability for autonomous cooperative survey and sampling of dynamic oceanographic phenomena such as current systems and algae blooms. Each ASF vessel is a software model that represents a real world platform that carries a variety of sensors. The OASIS platform will provide the first physical vessel, outfitted with the systems and payloads necessary to execute the oceanographic observations described in this paper. The ASF architecture is being designed for extensibility to accommodate heterogenous fleet elements, and is not limited to using the OASIS platform to acquire data. This paper describes the path planning algorithms developed for the acquisition phase of a typical ASF task. Given a polygonal target region to be surveyed, the region is subdivided according to the number of vessels in the fleet. The subdivision algorithm seeks a solution in which all subregions have equal area and minimum mean radius. Once the subregions are defined, a dynamic programming method is used to find a minimum-time path for each vessel from its initial position to its assigned region. This path plan includes the effects of water currents as well as avoidance of known obstacles. A fleet-level planning algorithm then shuffles the individual vessel assignments to find the overall solution which puts all vessels in their assigned regions in the minimum time. This shuffle algorithm may be described as a process of elimination on the sorted list of permutations of a cost matrix. All these path planning algorithms are facilitated by discretizing the region of interest onto a hexagonal tiling.

Stoneking, Eric↗

NASA NTRS1995

High-performance parallel analysis of coupled problems for aircraft propulsion

This research program deals with the application of high-performance computing methods to the numerical simulation of complete jet engines. The program was initiated in 1993 by applying two-dimensional parallel aeroelastic codes to the interior gas flow problem of a by-pass 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 an 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 3-component problem were developed in 1994. These procedures involved delayed corrections and subcycling, and have been successfully tested on several massively parallel computers. 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 for parallel versions of ENG10 has been developed. It is planned to use the steady-state global solution provided by ENG10 as input to a localized three-dimensional FSI analysis for engine regions where aeroelastic effects may be important.

Felippa, C. A.↗

NASA NTRS1997

RNS Applications for Interacting Sub- and Supersonic Flows

A solution based grid adaptation method that combines elements of the multigrid method for solution acceleration and the domain decomposition philosophy for grid optimization is described. Unlike other solution based adaptive gridding schemes, wherein the overhead of recomputing the grid and re-evaluating the solution on the adapted grid leads to higher computational costs compared to a non-adapted calculation, the present methodology reduces the computational time required to obtain the solution. The computational effort involved in the present calculation is significantly lower than a non-adapted calculation that utilizes the multigrid method purely as a convergence acceleration tool. In addition to convergence acceleration, the multigrid framework provides a mechanism of information transfer from regions wherein grid refinement is specified to unrefined coarse grid regions. The basis for domain decomposition in the current procedure is the variation in grid refinement requirements for each coordinate direction in different portions of the flow field. The method is demonstrated herein on an efficient set of governing equations termed the reduced Navier Stokes equations, applied in conjunction with a set of physical boundary conditions. The governing equations are discretized through a pressure based flux splitting procedure that is uniformly applicable from incompressible to supersonic Mach numbers.

Rubin, Stanley G.↗

NASA NTRS2009

NASA Tech Briefs, November 2009

Topics covered include: Cryogenic Chamber for Servo-Hydraulic Materials Testing; Apparatus Measures Thermal Conductance Through a Thin Sample from Cryogenic to Room Temperature; Rover Attitude and Pointing System Simulation Testbed; Desktop Application Program to Simulate Cargo-Air-Drop Tests; Multimodal Friction Ignition Tester; Small-Bolt Torque-Tension Tester; Integrated Spacesuit Audio System Enhances Speech Quality and Reduces Noise; Hardware Implementation of a Bilateral Subtraction Filter; Simple Optoelectronic Feedback in Microwave Oscillators; Small X-Band Oscillator Antennas; Free-Space Optical Interconnect Employing VCSEL Diodes; Discrete Fourier Transform Analysis in a Complex Vector Space; Miniature Scroll Pumps Fabricated by LIGA; Self-Assembling, Flexible, Pre-Ceramic Composite Preforms; Flight-speed Integral Image Analysis Toolkit; Work Coordination Engine; Multi-Mission Automated Task Invocation Subsystem; Autonomously Calibrating a Quadrupole Mass Spectrometer; Determining Spacecraft Reaction Wheel Friction Parameters; Composite Silica Aerogels Opacified with Titania; Multiplexed Colorimetric Solid-Phase Extraction; Detecting Airborne Mercury by Use of Polymer/Carbon Films; Lattice-Matched Semiconductor Layers on Single Crystalline Sapphire Substrate; Pressure-Energized Seal Rings to Better Withstand Flows; Rollerjaw Rock Crusher; Microwave Sterilization and Depyrogenation System; Quantifying Therapeutic and Diagnostic Efficacy in 2D Microvascular Images; NiF2/NaF:CaF2/Ca Solid-State High-Temperature Battery Cells; Critical Coupling Between Optical Fibers and WGM Resonators; Microwave Temperature Profiler Mounted in a Standard Airborne Research Canister; Alternative Determination of Density of the Titan Atmosphere; Solar Rejection Filter for Large Telescopes; Automated CFD for Generation of Airfoil Performance Tables; Progressive Classification Using Support Vector Machines; Active Learning with Irrelevant Examples; A Data Matrix Method for Improving the Quantification of Element Percentages of SEM/EDX Analysis; Deployable Shroud for the International X-Ray Observatory; Improved Model of a Mercury Ring Damper; Optoelectronic pH Meter: Further Details; X-38 Advanced Sublimator; and Solar Simulator Represents the Mars Surface Solar Environment.

Source record↗

NASA NTRS2020

Entropy Stable h/p-Nonconforming Discretization with the Summation-by-Parts Property for the Compressible Euler and Navier–Stokes Equations

In this paper, we extend the entropy conservative/stable algorithms presented by Del Rey Fernandez and coauthors for the compressible Euler and Navier-Stokes equations on nonconforming p-refined/coarsened curvilinear grids to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, we utilize a computationally simple and efficient approach based upon using decoupled interpolation operators. The resulting scheme is entropy conservative/stable and elementwise conservative. Numerical simulations of the isentropic vortex and viscous shock propagation con firm the entropy conservation/stability and accuracy properties of the method (achieving ~ p + 1 convergence), which are comparable to those of the original conforming scheme. Simulations of the Taylor{Green vortex at R(e) = 1,600 and turbulent flow past a sphere at R(e(infinity)) = 2,000 show the robustness and stability properties of the overall spatial discretization for unstructured grids. Finally, to demonstrate the entropy conservation property of a fully-discrete explicit entropy stable algorithm with h=p refinement/coarsening, we present the time evolution of the entropy function obtained by simulating the propagation of the isentropic vortex using a relaxation Runge-Kutta scheme.

Nonconforming interfaces↗

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.↗

DOE OSTI2021

MFIX-Exa: A path toward exascale CFD-DEM simulations

MFIX-Exa is a computational fluid dynamics–discrete element model (CFD-DEM) code designed to run efficiently on current and next-generation supercomputing architectures. MFIX-Exa combines the CFD-DEM expertise embodied in the MFIX code—which was developed at NETL and is used widely in academia and industry—with the modern software framework, AMReX, developed at LBNL. The fundamental physics models follow those of the original MFIX, but the combination of new algorithmic approaches and a new software infrastructure will enable MFIX-Exa to leverage future exascale machines to optimize the modeling and design of multiphase chemical reactors.

97 MATHEMATICS AND COMPUTING↗

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↗