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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

Negative magnetic eddy diffusivities from the test-field method and multiscale stability theory

The generation of a large-scale magnetic field in the kinematic regime in the absence of an α-effect is investigated by following two different approaches: the test-field method and the multiscale stability theory relying on the homogenization technique. Our computations of the magnetic eddy diffusivity tensor of the parity-invariant flow IV of G. O. Roberts and the modified Taylor–Green flow confirm the findings of previous studies and also explain some of their apparent contradictions. The two flows have large symmetry groups; this is used to considerably simplify the eddy diffusivity tensor. Finally, a new analytic result is presented: upon expressing the eddy diffusivity tensor in terms of solutions to auxiliary problems for the adjoint operator, we derive relations between the magnetic eddy diffusivity tensors that arise for mutually reverse small-scale flows v(x) and −v(x).

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Demonstrating Computational Equivalence Between Continuous and Discrete Adjoint Methods by Calculating Time-Dependent Adjoint Solutions with Neutron Diffusion Models

The continuous adjoint method and the discrete adjoint method are two alternative approaches used to calculate adjoint solutions for adjoint systems. The continuous adjoint method derives adjoint equations analytically from continuous forward equations and then solves the adjoint equations either analytically or numerically in a discretized form whereas the discrete adjoint method calculates the adjoint solutions directly from the discretized forward equations. With regard to the methodology development and calculation procedure, distinct differences are well recognized between the two methods. For certain reasons, both methods are exclusively preferred and commonly used by different computational communities, but limited studies clarify the connections between the two adjoint methods from either of the communities. Herein, this paper demonstrates the computational equivalence between the continuous and discrete adjoint methods by investigating time-dependent adjoint solutions to the two-group neutron diffusion model in nuclear reactor analysis problems using both methods. Adjoint solutions can be used to estimate system parameters for reactor safety analysis. Appropriate final state conditions for the adjoint systems are specified in both of the methods, and the conditions are clarified with proper physical explanations. With the help of an event-based case study on neutron diffusion models, the accuracy of the time-dependent adjoint fluxes obtained from both methods is verified, and the pros and cons of both adjoint methods are examined. More importantly, the computational equivalence of both methods is demonstrated when they are applied to multigroup neutron diffusion systems. The advantage of calculating time-dependent adjoint fluxes by directly solving time-dependent adjoint systems rather than taking steady-state approximations as in common practice is also demonstrated.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2015

Inverse problems in heterogeneous and fractured media using peridynamics

The following work presents an adjoint-based methodology for solving inverse problems in heterogeneous and fractured media using state-based peridynamics. We show that the inner product involving the peridynamic operators is self-adjoint. The proposed method is illustrated for several numerical examples with constant and spatially varying material parameters as well as in the context of fractures. We also present a framework for obtaining material parameters by integrating digital image correlation (DIC) with inverse analysis. This framework is demonstrated by evaluating the bulk and shear moduli for a sample of nuclear graphite using digital photographs taken during the experiment. The resulting measured values correspond well with other results reported in the literature. Lastly, we show that this framework can be used to determine the load state given observed measurements of a crack opening. Furthermore, this type of analysis has many applications in characterizing subsurface stress-state conditions given fracture patterns in cores of geologic material.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed-point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

42 ENGINEERING↗
DOE OSTI2025

A time-parallel multiple-shooting method for large-scale quantum optimal control

Quantum optimal control plays a crucial role in quantum computing by providing the interface between compiler and hardware. Solving the optimal control problem is particularly challenging for multi-qubit gates, due to the exponential growth in computational complexity with the system's dimensionality and the deterioration of optimization convergence. To ameliorate the computational complexity of time-integration, this paper introduces a multiple-shooting approach in which the time domain is divided into multiple windows and the intermediate states at window boundaries are treated as additional optimization variables. Further, this enables parallel computation of state evolution across time-windows, significantly accelerating objective function and gradient evaluations. Since the initial state matrix in each window is only guaranteed to be unitary upon convergence of the optimization algorithm, the conventional gate trace infidelity is replaced by a generalized infidelity that is convex for non-unitary state matrices. Continuity of the state across window boundaries is enforced by equality constraints. A quadratic penalty optimization method is used to solve the constrained optimal control problem, and an efficient adjoint technique is employed to calculate the gradients in each iteration. We demonstrate the effectiveness of the proposed method through numerical experiments on quantum Fourier transform gates in systems with 2, 3, and 4 qubits, noting a speedup of 80x for evaluating the gradient in the 4-qubit case, highlighting the method's potential for optimizing control pulses in multi-qubit quantum systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires half of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Calculating Alpha Eigenvalues and Eigenfunctions with a Markov Transition Rate Matrix Monte Carlo Method

For a nuclear system in which the entire α-eigenvalue spectrum is known, eigenfunction expansion yields the time-dependent flux response to any arbitrary source. Applications in which this response is of interest include pulsed-neutron experiments, accelerator-driven subcritical systems, and fast burst reactors, where a steady-state assumption used in neutron transport is invalid for characterizing the time-dependent flux. To obtain the α-eigenvalue spectrum, the transition rate matrix method (TRMM) tallies transition rates describing neutron behavior in a discretized position-direction-energy phase space using Monte Carlo. Interpretation of the resulting Markov process transition rate matrix as the operator in the adjoint α-eigenvalue problem provides an avenue for determining a large finite set of α eigenvalues and eigenfunctions of a nuclear system. Results from the TRMM are verified using analytic solutions, time-dependent Monte Carlo simulations, and modal expansion from diffusion theory. For simplified infinite-medium and one-dimensional geometries, the TRMM accurately calculates eigenvalues, eigenfunctions, and eigenfunction expansion solutions. Applications and comparisons to measurements are made for the small fast burst reactor CALIBAN and the Fort St. Vrain high-temperature gas-cooled reactor. For large three-dimensional geometries, discretization of the large position-energy-direction phase space limits the accuracy of eigenfunction expansion solutions using the TRMM, but it can still generate a fair estimate of the fundamental eigenvalue and eigenfunction. These results show that the TRMM generates an accurate estimate of a large number of α eigenvalues. This is not possible with existing Monte Carlo–based methods.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2013

Efficient checkpointing schemes for depletion perturbation solutions on memory-limited architectures

We describe a methodology for decreasing the memory footprint and machine I/O load associated with the need to access a forward solution during an adjoint solve. Specifically, we are interested in the depletion perturbation equations, where terms in the adjoint Bateman and transport equations depend on the forward flux solution. Checkpointing is the procedure of storing snapshots of the forward solution to disk and using these snapshots to recompute the parts of the forward solution that are necessary for the adjoint solve. For large problems, however, the storage cost of just a few copies of an angular flux vector can exceed the available RAM on the host machine. We propose a methodology that does not checkpoint the angular flux vector; instead, we write and store converged source moments, which are typically of a much lower dimension than the angular flux solution. This reduces the memory footprint and I/O load of the problem, but requires that we perform single sweeps to reconstruct flux vectors on demand. We argue that this trade-off is exactly the kind of algorithm that will scale on advanced, memory-limited architectures. We analyze the cost, in terms of FLOPS and memory footprint, of five checkpointing schemes. We also provide computational results that support the analysis and show that the memory-for-work trade off does improve time to solution. (authors)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model

We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection–diffusion equation, which couples to the nonlinear Stokes ice flow equations; together they determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to render the problem well posed. We derive adjoint-based gradient and Hessian expressions for the resulting partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov–Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems – i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian – we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. Here, we show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.

15 GEOTHERMAL ENERGY↗
DOE OSTI2022

A mixed, unified forward/inverse framework for earthquake problems: fault implementation and coseismic slip estimate

SUMMARY We introduce a new finite-element (FE) based computational framework to solve forward and inverse elastic deformation problems for earthquake faulting via the adjoint method. Based on two advanced computational libraries, FEniCS and hIPPYlib for the forward and inverse problems, respectively, this framework is flexible, transparent and easily extensible. We represent a fault discontinuity through a mixed FE elasticity formulation, which approximates the stress with higher order accuracy and exposes the prescribed slip explicitly in the variational form without using conventional split node and decomposition discrete approaches. This also allows the first order optimality condition, that is the vanishing of the gradient, to be expressed in continuous form, which leads to consistent discretizations of all field variables, including the slip. We show comparisons with the standard, pure displacement formulation and a model containing an in-plane mode II crack, whose slip is prescribed via the split node technique. We demonstrate the potential of this new computational framework by performing a linear coseismic slip inversion through adjoint-based optimization methods, without requiring computation of elastic Green’s functions. Specifically, we consider a penalized least squares formulation, which in a Bayesian setting—under the assumption of Gaussian noise and prior—reflects the negative log of the posterior distribution. The comparison of the inversion results with a standard, linear inverse theory approach based on Okada’s solutions shows analogous results. Preliminary uncertainties are estimated via eigenvalue analysis of the Hessian of the penalized least squares objective function. Our implementation is fully open-source and Jupyter notebooks to reproduce our results are provided. The extension to a fully Bayesian framework for detailed uncertainty quantification and non-linear inversions, including for heterogeneous media earthquake problems, will be analysed in a forthcoming paper.

58 GEOSCIENCES↗
DOE OSTI2021

Space to Air High-Altitude Region Adjoint Neutron Transport

Neutrons from an atmospheric nuclear explosion can be detected by sensors in orbit. Current tools for characterizing the neutron energy spectrum assume a known source and use forward transport to recreate the detector response. In realistic scenarios the true source is unknown, making this an inefficient, iterative approach. In contrast, the adjoint approach directly solves for the source spectrum, enabling source reconstruction. The time–energy fluence at the satellite and adjoint transport equation allow a Monte Carlo method to characterize the neutron source’s energy spectrum directly in a new model: the Space to High-Altitude Region Adjoint (SAHARA) model. A new adjoint source event estimator was developed in SAHARA to find feasible solutions to the neutron transport problem given the constraints of the adjoint environment. This work explores SAHARA’s development and performance for mono-energetic and continuous neutron energy sources. In general, the identified spectra were shifted towards energies approximately 5% lower than the true source spectra, but SAHARA was able to capture the correct spectral shapes. Continuous energy sources, including real-world sources Fat Man and Little Boy, resulted in identifiable spectra that could have been produced by the same distribution as the true sources as demonstrated by two-dimensional (2D) Kolmogorov–Smirnov tests.

LaMere, Zachary W.↗
DOE OSTI2015

Second-order adjoint sensitivity analysis methodology (2ND-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale systems - 14356

This work presents the second-order adjoint sensitivity analysis methodology (2.-ASAM) for computing exactly and efficiently the second-order functional derivatives ('sensitivities') of system responses to the system's model parameters. For a physical system comprising N{sub α} parameters and N{sub r} responses, forward methods require a total of at least (N{sup 2}{sub α}/2+3N{sub α}/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all N{sub r} system responses. On the other hand, for one functional-type system response, the 2.-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1.-LASS) for obtaining all of the first-order sensitivities, followed by at most (2N{sub α}+1) large-scale computations using the second-level adjoint sensitivity systems (2.-LASS) for obtaining all of the second-order sensitivities. The second-order sensitivities contribute decisively to causing asymmetries in the response distribution, since they are the leading contributors to the third-order response correlations (skewness). They also cause the 'expected value of the response' to differ from the 'computed nominal value of the response'. The implementation of the 2.-ASAM requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Only the sources on the right-sides of the diffusion (differential) operator needed to be modified; the left-side of the differential equations (and hence the 'solver' in large-scale practical applications) remain unchanged. The application of the 2.-ASAM is illustrated on a benchmark heat conduction/convection problem, which makes transparent the underlying mathematical derivations. For this illustrative problem, 4 'large-scale' adjoint computations suffice for computing exactly all of the 6 first- and 21 distinct second-order derivatives. The 2.-ASAM presented in this work should enable the hitherto very difficult, if not intractable, exact computation of all of the second-order response sensitivities for large-systems involving many parameters, which is expected to affect significantly other fields that need efficiently computed second-order response sensitivities, such as optimization, data assimilation/adjustment, model calibration, and predictive modeling. (author)

42 ENGINEERING↗
DOE OSTI2017

Transmutation approximations for the application of hybrid Monte Carlo/deterministic neutron transport to shutdown dose rate analysis

In fusion energy systems (FES) neutrons born from burning plasma activate system components. The photon dose rate after shutdown from resulting radionuclides must be quantified. This shutdown dose rate (SDR) is calculated by coupling neutron transport, activation analysis, and photon transport. The size, complexity, and attenuating configuration of FES motivate the use of hybrid Monte Carlo (MC)/deterministic neutron transport. The Multi-Step Consistent Adjoint Driven Importance Sampling (MS-CADIS) method can be used to optimize MC neutron transport for coupled multiphysics problems, including SDR analysis, using deterministic estimates of adjoint flux distributions. When used for SDR analysis, MS-CADIS requires the formulation of an adjoint neutron source that approximates the transmutation process. In this work, transmutation approximations are used to derive a solution for this adjoint neutron source. It is shown that these approximations are reasonably met for typical FES neutron spectra and materials over a range of irradiation scenarios. When these approximations are met, the Groupwise Transmutation (GT)-CADIS method, proposed here, can be used effectively. GT-CADIS is an implementation of the MS-CADIS method for SDR analysis that uses a series of single-energy-group irradiations to calculate the adjoint neutron source. For a simple SDR problem, GT-CADIS provides speedups of 200 100 relative to global variance reduction with the Forward-Weighted (FW)-CADIS method and 9 ± 5 • 10 4 relative to analog. As a result, this work shows that GT-CADIS is broadly applicable to FES problems and will significantly reduce the computational resources necessary for SDR analysis.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Inverse deep learning methods and benchmarks for artificial electromagnetic material design

In this work we investigate the use of deep inverse models (DIMs) for designing artificial electromagnetic materials (AEMs) – such as metamaterials, photonic crystals, and plasmonics – to achieve some desired scattering properties (e.g., transmission or reflection spectrum). DIMs are deep neural networks (i.e., deep learning models) that are specially-designed to solve ill-posed inverse problems. There has recently been tremendous growth in the use of DIMs for solving AEM design problems however there has been little comparison of these approaches to examine their absolute and relative performance capabilities. In this work we compare eight state-of-the-art DIMs on three unique AEM design problems, including two models that are novel to the AEM community. Our results indicate that DIMs can rapidly produce accurate designs to achieve a custom desired scattering on all three problems. Although no single model always performs best, the Neural-Adjoint approach achieves the best overall performance across all problem settings. As a final contribution we show that not all AEM design problems are ill-posed, and in such cases a conventional deep neural network can perform better than DIMs. We recommend that a deep neural network is always employed as a simple baseline approach when addressing AEM design problems. Furthermore, we publish python code for our AEM simulators and our DIMs to enable easy replication of our results, and benchmarking of new DIMs by the AEM community.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗