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NASA NTRS2002

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

This talk describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Yee, H. C.↗

NASA NTRS2004

Reliability of Complex Nonlinear Numerical Simulations

This work describes some of the procedure to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Yee, H. C.↗

NASA NTRS2005

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

This chapter describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations.

Yee, H. C.↗

DOE OSTI2025

Time-domain all-frequency stable formulation for low-frequency electromagnetic simulation with Newmark-β time integration

An implicitly Coulomb-gauged A-ϕ formulation has previously been proposed and validated for finite ele- ment simulations of low-frequency and multiscale electromag- netic problems in the frequency domain. This formulation has demonstrated numerical stability across all frequencies, with its accuracy, efficiency, and iterative convergence established in various frequency-domain scenarios. However, direct time- domain computation is often preferable for wideband electro- magnetic problems and is typically indispensable in nonlinear and multiphysics simulations. In this work, the A-ϕ formulation is extended to the time domain. By incorporating the well-known Newmark-β time integration scheme, the proposed formulation is validated through capacitive and inductive test cases. The results confirm the solution’s accuracy and demonstrate the formulation’s stability in the time domain.

Mekonnen, Minyechil↗

NASA NTRS2002

Dynamical Approach Study of Spurious Numerics in Nonlinear Computations

The last two decades have been an era when computation is ahead of analysis and when very large scale practical computations are increasingly used in poorly understood multiscale complex nonlinear physical problems and non-traditional fields. Ensuring a higher level of confidence in the predictability and reliability (PAR) of these numerical simulations could play a major role in furthering the design, understanding, affordability and safety of our next generation air and space transportation systems, and systems for planetary and atmospheric sciences, and in understanding the evolution and origin of life. The need to guarantee PAR becomes acute when computations offer the ONLY way of solving these types of data limited problems. Employing theory from nonlinear dynamical systems, some building blocks to ensure a higher level of confidence in PAR of numerical simulations have been revealed by the author and world expert collaborators in relevant fields. Five building blocks with supporting numerical examples were discussed. The next step is to utilize knowledge gained by including nonlinear dynamics, bifurcation and chaos theories as an integral part of the numerical process. The third step is to design integrated criteria for reliable and accurate algorithms that cater to the different multiscale nonlinear physics. This includes but is not limited to the construction of appropriate adaptive spatial and temporal discretizations that are suitable for the underlying governing equations. In addition, a multiresolution wavelets approach for adaptive numerical dissipation/filter controls for high speed turbulence, acoustics and combustion simulations will be sought. These steps are corner stones for guarding against spurious numerical solutions that are solutions of the discretized counterparts but are not solutions of the underlying governing equations.

Yee, H. C.↗

NASA NTRS1996

Multilevel Algorithm for Atmospheric Data Assimilation

A multiscale algorithm for the problem of optimal statistical interpolation of observed data has been developed. This problem includes the calculation of the vector of the 'analyzed' (best estimated) atmosphere flow field w(sup a) by the formula: w(sup a) = w(sup f) + P(sup f) H(sup T) y, where the quantity y is defined by the equation (H P(sup f) H(sup T) + R)y = w(sup o) - H w(sup f), using the given model forecast first guess w(sup f) and the vector of observations w(sup o); H is an interpolation operator from the regular grid to the observation network, P(sup f) is the forecast error covariance matrix, and R is the observation error covariance matrix. At this initial stage the case of univariate analysis of single level radiosonde height data is considered. The matrix R is assumed to be diagonal, and the matrix P(sup f) is assumed to be given by the formula P(sub ij)(sup f) = sigma(sub i)(sup f) mu(sub ij) sigma(sub j)(sub f), where mu(sub ij) is a smooth, decreasing function of the distance between the i-th and the j-th points. In this paper we describe a multiscale iterative process based on a multiresolution, simultaneous displacement technique and a localized variational calculation of iteration parameters.

Brandt, Achi↗

NASA NTRS2002

Designing Adaptive Low Dissipative High Order Schemes

Proper control of the numerical dissipation/filter to accurately resolve all relevant multiscales of complex flow problems while still maintaining nonlinear stability and efficiency for long-time numerical integrations poses a great challenge to the design of numerical methods. The required type and amount of numerical dissipation/filter are not only physical problem dependent, but also vary from one flow region to another. This is particularly true for unsteady high-speed shock/shear/boundary-layer/turbulence/acoustics interactions and/or combustion problems since the dynamics of the nonlinear effect of these flows are not well-understood. Even with extensive grid refinement, it is of paramount importance to have proper control on the type and amount of numerical dissipation/filter in regions where it is needed.

Yee, H. C.↗

DOE OSTI2025

Generative learning of densities on manifolds

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an Itô stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps , a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

Double diffusion maps↗

DOE OSTI2025

Cross-Scale Catalyst Modeling Applied to H 2 Storage and Release via Formic Acid

Here, we propose the Systems-to-Atoms (S2A) modeling framework that integrates the kinetics of reaction chemistry and structural configurations across various length scales with the aim of establishing a versatile template for multiscale modeling of reactive flow problems and to predict the operando activity of catalyst materials. The approach encompasses a microkinetic model to analyze surface reactions on individual facets of catalyst nanoparticles coupled with the computation of average surface reaction rates for catalyst nanoparticles of specific size distributions. Macro-homogeneous surface reaction kinetics are derived as a function of catalyst loading and used as input parameters for the continuum-scale reactor model. The cross-scale framework enables the optimization of catalyst utilization through reactor design and operating strategy. To demonstrate the framework, we studied the storage and release of hydrogen from formic acid, a promising liquid organic hydrogen carrier (LOHC), over Pd, Pt, and Cu catalysts. The framework predicts observed trends in formic acid dehydrogenation activity for catalysts with comparable weight loadings and metal particle diameters, demonstrating satisfactory quantitative alignment. Finally, the seamless transmission of parameter uncertainties between scales is also discussed.

08 HYDROGEN↗

NASA NTRS1990

Recognition and characterization of hierarchical interstellar structure. I - Correlation function

The problem of the quantitative description of multiscale structure in interstellar cloud complexes and gravitational collapse calculations is considered, emphasizing the recognition and characterization of hierarchical fragmentation structure. The response of the two-point correlation function to a variety of analytical models for density structure is discussed for simple clustering of pointlike clouds to more complex models involving clouds with a distribution of sizes and densities and hierarchical substructure. By expressing the density distribution as the superposition of individual clouds, it is shown that the correlation function generates two types of terms: those involving each cloud's density convolved with itself and those involving pairs of different clouds. Major distortion of the correlation function are introduced by the presence of any image features with size scales a significant fraction of the image size.

Houlahan, Padraig↗

NASA NTRS1994

High-performance parallel analysis of coupled problems for aircraft propulsion

Applications are described of high-performance parallel, computation for the analysis of complete jet engines, considering its multi-discipline coupled problem. The coupled problem involves interaction of structures with gas dynamics, heat conduction and heat transfer in aircraft engines. The methodology issues addressed include: consistent discrete formulation of coupled problems with emphasis on coupling phenomena; effect of partitioning strategies, augmentation and temporal solution procedures; sensitivity of response to problem parameters; and methods for interfacing multiscale discretizations in different single fields. The computer implementation issues addressed include: parallel treatment of coupled systems; domain decomposition and mesh partitioning strategies; data representation in object-oriented form and mapping to hardware driven representation, and tradeoff studies between partitioning schemes and fully coupled treatment.

Felippa, C. A.↗

NASA NTRSDate not supplied

Computationally Guided Development of Components for High Energy Density Solid-State Lithium-Sulfur Batteries

All electric vertical take-off and landing vehicles (eVTOL) for urban air mobility (UAM) concepts face numerous challenging technical barriers before their introduction into the consumer marketplace. The most challenging of these technical barriers to overcome is developing an energy storage system capable of meeting the rigorous aerospace safety and performance criteria1. The performance metrics for eVTOL craft, such as specific energy, specific power, and safety, exceed those of electric automobiles by a factor of two to four. Current state-of-the-art (SOA) lithium-ion batteries are incapable of meeting the key performance criteria of energy and safety for eVTOL. Therefore, next generation advanced chemistries and designs must be developed to meet required performance metrics for electric aviation2. Beyond lithium-ion chemistries, such as lithium-sulfur, show promise in their high energy, while limitations exist in their power and cyclability due to low electrical conductivity and high intermediate solubility in organic liquid electrolytes. Several strategies to overcome the low electrical conductivity involve the use of selenium as a dopant in the active sulfur material, along with the incorporation of 2-dimensional electron-conducting holey-graphene to improve the composite cathodes electronic conductivity. Furthermore, combining this chemistry with a solid-electrolyte avoids the components’ dissolution issues3. However, the development of composite solid-state cathodes is non-trivial as several components must be intimately mixed so that the active component has sufficient access to both electrons and lithium ions to undergo full electrochemical conversion. Mathematical modeling of battery components can assist experimental design through a robust and rigorous combination of computational modeling techniques covering multiple length scales. The objective is to leverage modern computational materials methods combined with battery multiphysics tools to develop radically advanced compatible cathode and electrolyte materials, build and test solid state lithium-sulfur cells and packs. A NASA-based cross-organizational team of high-powered experts combined integrated computational predictive modeling, fundamental chemistry analysis, advanced material science, and battery cell development to tackle this very challenging, multidisciplinary problem. This presentation will show a multiscale computational modeling approach that has produced a novel particle dynamics method called Solid Electrolyte Sphere Approximation Model (SESAM). SESAM modeling targets the 1-10 µm scale structures and provides electromechanical and grain interactions for predictive design guidelines for the manufacturing of solid-state components. Parameters such as particle size and volume fraction of the constituent materials were modeled and experimentally fabricated to optimize electrochemical performance through improved microstructure design. Experimental feedback was provided through ionic and electronic conductivity assessment and structural analysis of developed materials and cell components.

battery↗

NASA NTRSDate not supplied

Stiffness and Fatigue Life Estimator for Polymer Composite Laminates Using Machine Learning

Machine learning (ML) models are increasingly being used in many engineering fields due to the advancements in ML algorithms and availability of high-speed computing power. One of the most popular ML class of models is artificial neural networks (ANN). ML is increasingly being used in the design and analysis of composite materials and structures, specifically in the constitutive modeling of composite materials with the focus on greatly accelerating multiscale analyses of composite materials and structures through development of surrogate models. Towards that end, Python-based neural nets have been developed to predict initial stiffness and fatigue life of an eight-ply symmetric polymer matrix composite laminate. Two types of neural networks, a Multilayer Perceptron (MLP) and a Recurrent Neural Network (RNN), have been established. Results show that both neural net type algorithms can provide an excellent estimate of initial laminate stiffness as well as fatigue life of eight-ply symmetric polymer matrix composite laminates (PMCs). RNNs are better able to capture the shape of the fatigue curve of a laminate. The resulting tool and GUI can be very useful for system level studies to obtain an estimate of desired properties and life of PMC composite laminates. Further, the associated surrogate models can also be used in composite multiscale analyses to replace the actual physics-based calculations at lower scales and thereby significantly increase the computational efficiency of such analyses and thus make micromechanics-based multiscale analyses a viable industrial tool for large scale structural problems.

multiscale analysis↗

NASA NTRS2023

Stiffness and Fatigue Life Estimator for Polymer Composite Laminates Using Machine Learning

multiscale analysis↗

NASA NTRS2021

Micromechanics-based Modeling of Laminated SiC/SiC Ceramic Matrix Composites

The behavior and response of ceramic matrix composites (CMCs), in particular silicon carbide fiber reinforced silicon carbide matrix (SiC/SiC), is affected by many factors such as variation of fiber volume fraction, residual stresses resulting from processing of the composites at high temperature, random microstructures, and the presence of matrix flaws (e.g., voids, pores, cracks etc.) as well as general material nonlinearity and heterogeneity that occurs randomly in a composite. Residual stresses arising from the phase change of constituents are evaluated in this paper and it is shown that they do influence composite strength and need to be properly accounted for. Additionally, the microstructures (location of fiber centers, coating thickness etc.) of advanced CMCs are usually disordered (or random) and fiber diameter and strength typically have a distribution. They rarely resemble the ordered fiber packing (square, rectangular, or hexagonal) that is generally assumed in micromechanics-based models with periodic boundary conditions for computational expediency. These issues raise the question of how should one model such systems effectively? Can an ordered hexagonal packed repeating unit cell (RUC) accurately represent the random microstructure behavior? How many fibers need to be included to enable accurate representation? Clearly, the number of fibers within an RUC must be limited to insure a balance between accuracy and efficiency. NASA’s in-house micromechanics-based code MAC/GMC provides a framework to analyze such RUCs for the overall composite behavior and the FEAMAC computer code provides linkage of MAC/GMC to the commercial FEA code, ABAQUS. The appropriate level of discretization of the RUC as well as the analysis method employed, i.e., Generalized Method of Cells (GMC) or High Fidelity Generalized Method of Cells (HFGMC), is investigated in this paper in the context of a unidirectional as well as a cross-ply laminated CMC. Results including effective composite properties, proportional limit stress (an important design parameter) and fatigue are shown utilizing both GMC as well as HFGMC. Finally, a few multiscale analyses are performed on smooth bar test coupons as well as test coupons with features such as open-hole and double notches using FEAMAC. Best practices and guidance are provided to take these phenomena into account and keep a proper balance between fidelity (accuracy) and efficiency. Following these guidelines can account for important physics of the problem and provide significant advantages when performing large multiscale composite structural analyses.

Ceramic Matrix Composites↗

DOE OSTI2024

Predicting Critical Transitions in Multiscale Data

Predicting the dynamics of complex nonlinear systems remains a challenging problem both in dynamical systems theory as well as real world science and engineering applications. Data-driven methods utilizing the latest advances in machine learning (ML) provide a promising new paradigm for this task. Our work centered on Reservoir Computing (RC), which has shown itself to be capable of skillfully predicting chaotic dynamics in multiscale systems. In the first part of the work, the focus is on how to improve predictions of critical transitions in a class of slow-fast metastable systems in which the equations are known. An additional goal was to determine whether a relationship exists between RC and Koopman operator theory, to improve the efficiency and broaden the applicability of the approach. In the second part of this work, a variation on the RC model known as Reconstructive Reservoir Computing (RRC) is applied to real-world data to identify anomalies.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1997

Adaptation of a Fast Optimal Interpolation Algorithm to the Mapping of Oceangraphic Data

A fast, recently developed, multiscale optimal interpolation algorithm has been adapted to the mapping of hydrographic and other oceanographic data. This algorithm produces solution and error estimates which are consistent with those obtained from exact least squares methods, but at a small fraction of the computational cost. Problems whose solution would be completely impractical using exact least squares, that is, problems with tens or hundreds of thousands of measurements and estimation grid points, can easily be solved on a small workstation using the multiscale algorithm. In contrast to methods previously proposed for solving large least squares problems, our approach provides estimation error statistics while permitting long-range correlations, using all measurements, and permitting arbitrary measurement locations. The multiscale algorithm itself, published elsewhere, is not the focus of this paper. However, the algorithm requires statistical models having a very particular multiscale structure; it is the development of a class of multiscale statistical models, appropriate for oceanographic mapping problems, with which we concern ourselves in this paper. The approach is illustrated by mapping temperature in the northeastern Pacific. The number of hydrographic stations is kept deliberately small to show that multiscale and exact least squares results are comparable. A portion of the data were not used in the analysis; these data serve to test the multiscale estimates. A major advantage of the present approach is the ability to repeat the estimation procedure a large number of times for sensitivity studies, parameter estimation, and model testing. We have made available by anonymous Ftp a set of MATLAB-callable routines which implement the multiscale algorithm and the statistical models developed in this paper.

Menemenlis, Dimitris↗

DOE OSTI2025

Application of a temporal multiscale method for efficient simulation of degradation in PEM Water Electrolysis under dynamic operating conditions

Hydrogen is emerging as a vital energy carrier, driven by the need to reduce carbon emissions. Proton Electrolyte Membrane Water Electrolysis (PEMWE) enables hydrogen production under fluctuating renewable power conditions but requires improved understanding and stability of the anode catalyst layer under dynamic operating conditions, especially with low noble metal loadings. Long-term degradation experiments are both time-consuming and costly; therefore, a systematic, model-aided approach is essential. In the present work, a temporal multiscale method is applied to reduce the computational effort of simulating long-term degradation processes in PEMWE, with an exemplary focus on catalyst dissolution. A mechanistic model incorporating the oxygen evolution reaction, catalyst dissolution, and hydrogen permeation from the cathode to the anode was hypothesized and implemented. In this way, the local periodicity of transport and reaction processes in dynamic PEMWE operation, which influence the gradual degradation of the catalyst layer, is captured. The temporal multiscale method significantly reduces the computational effort of simulation, decreasing processing time from hours to mere minutes. This efficiency gain is attributed to the limited evolution of Slow-Scale variables during each period of time P of the Fast-Scale variables. Consequently, simulation is required only until local periodicity is achieved within each Slow-Scale time step. Hence, the fully resolved dynamic problem is decoupled into these two scales, employing a heterogeneous multiscale technique. The developed approach effectively accelerates parameter estimation and predictive simulations, supporting systematic modeling of PEMWE degradation under dynamic conditions.

08 HYDROGEN↗