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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 OSTI2016

Planck intermediate results: XLIV. Structure of the Galactic magnetic field from dust polarization maps of the southern Galactic cap

Using data from the Planck satellite, we study in this paper the statistical properties of interstellar dust polarization at high Galactic latitudes around the south pole (b < -60°). Our aim is to advance the understanding of the magnetized interstellar medium (ISM), and to provide a modelling framework of the polarized dust foreground for use in cosmic microwave background (CMB) component-separation procedures. We examine the Stokes I, Q, and U maps at 353 GHz, and particularly the statistical distribution of the polarization fraction (p) and angle (ψ), in order to characterize the ordered and turbulent components of the Galactic magnetic field (GMF) in the solar neighbourhood. The Q and U maps show patterns at large angular scales, which we relate to the mean orientation of the GMF towards Galactic coordinates (l 0 ,b 0 ) = (70° ± 5°,24° ± 5°). The histogram of the observed p values shows a wide dispersion up to 25%. The histogram of ψ has a standard deviation of 12° about the regular pattern expected from the ordered GMF. We build a phenomenological model that connects the distributions of p and ψ to a statistical description of the turbulent component of the GMF, assuming a uniform effective polarization fraction (p 0 ) of dust emission. To compute the Stokes parameters, we approximate the integration along the line of sight (LOS) as a sum over a set of N independent polarization layers, in each of which the turbulent component of the GMF is obtained from Gaussian realizations of a power-law power spectrum. We are able to reproduce the observed p and ψ distributions using a p 0 value of 26%, a ratio of 0.9 between the strengths of the turbulent and mean components of the GMF, and a small value of N. The mean value of p (inferred from the fit of the large-scale patterns in the Stokes maps) is 12 ± 1%. We relate the polarization layers to the density structure and to the correlation length of the GMF along the LOS. We emphasize the simplicity of our model (involving only a few parameters), which can be easily computed on the celestial sphere to produce simulated maps of dust polarization. Finally, our work is an important step towards a model that can be used to assess the accuracy of component-separation methods in present and future CMB experiments designed to search the B mode CMB polarization from primordial gravity waves.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Vibronic-structure tracking: A shortcut for vibrationally resolved UV/Vis-spectra calculations

The vibrational coarse structure and the band shapes of electronic absorption spectra are often dominated by just a few molecular vibrations. By contrast, the simulation of the vibronic structure even in the simplest theoretical models usually requires the calculation of the entire set of normal modes of vibration. Here, we exploit the idea of the mode-tracking protocol [M. Reiher and J. Neugebauer, J. Chem. Phys. 118, 1634 (2003)] in order to directly target and selectively calculate those normal modes which have the largest effect on the vibronic band shape for a certain electronic excitation. This is achieved by defining a criterion for the importance of a normal mode to the vibrational progressions in the absorption band within the so-called “independent mode, displaced harmonic oscillator” (IMDHO) model. We use this approach for a vibronic-structure investigation for several small test molecules as well as for a comparison of the vibronic absorption spectra of a truncated chlorophyll a model and the full chlorophyll a molecule. We show that the method allows to go beyond the often-used strategy to simulate absorption spectra based on broadened vertical excitation peaks with just a minimum of computational effort, which in case of chlorophyll a corresponds to about 10% of the cost for a full simulation within the IMDHO approach.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1989

A new adaptive control approach for aerospace vehicles with parameter uncertainties

A new stochastic adaptive control structure is developed for the problem of combined parameter estimation and control of aerospace vehicles with changing parameters. Parameter uncertainties are modeled as first-order Gauss-Markov processes, and are introduced to the system dynamics through a small parameter. It is assumed that an accurate inertial measurement unit gives perfect measurements of the state variables. Since the stochastic system is assumed to be Gauss-Markov, the density function of the parameters given these measurements is conditionally Gaussian. Based on this conditionally Gaussian density, the problem of minimizing a quadratic cost over an infinite time horizon can be set up within the framework of stochastic optimal control theory. The optimal feedback control law is derived from a straightforward expansion of the Hamilton-Jacobi-Bellman equation, based on the LQG solution. The resulting nonlinear controller is applied to the pitch axis control of a space platform with uncertain moments of inertia and is shown to produce marked improvement over a fixed controller.

Hahn, Yungsun↗
NASA NTRS1972

Numerical study of strong plasma shock waves produced in an electromagnetic shock tube.

A set of two-fluid Navier-Stokes equations with classical physical transport coefficients is used to compute the evolution and structure of collisional plasma shock waves in an electromagnetic shock tube. Shock speeds up to 200 cm/microsec and shocked-plasma temperatures of the order of kilovolts are studied. A strong transverse bias magnetic field is employed, which significantly alters the size and shape of the shock profiles, when compared with the zero bias field case. The wave structure is different from that of a two-fluid gasdynamic shock without transverse magnetic field. Especially significant is the effect of the small ion Larmor radius in reducing the gasdynamic shock thickness by at least an order of magnitude in the transverse shock case, which permits collisional shocks to have thicknesses much smaller than the post-shock mean free path. These collisional shock waves produce very hot ions, the ion temperature increasing with shock speed, although the formation time and distances also increase substantially with shock speed.

Schneider, S. H.↗
DOE OSTI2021

Bayesian forecasts for dark matter substructure searches with mock pulsar timing data

Dark matter substructure, such as primordial black holes (PBHs) and axion miniclusters, can induce phase shifts in pulsar timing arrays (PTAs) measurements due to gravitational effects. In order to gain a more realistic forecast for the detectability of such models of dark matter with PTAs, we propose a Bayesian inference framework to search for phase shifts generated by PBHs and perform the analysis on mock PTA data. Furthermore, for most PBH masses the constraints on the dark matter abundance agree with previous (frequentist) analyses (without mock data) to $\mathcal{O}$(1) factors. This further motivates a dedicated search for PBHs (and dense small scale structures) in the mass range from 10 -8 M ⊙ to well above 10 2 M ⊙ with the Square Kilometer Array. Moreover, with a more optimistic set of timing parameters, future PTAs are predicted to constrain PBHs down to 10 -11 M ⊙ . Lastly, we discuss the impact of backgrounds, such as Supermassive Black Hole Mergers, on detection prospects, suggesting a future program to separate a dark matter signal from other astrophysical sources.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Aberrations and Emittance Growth in the DARHT 2nd Axis Downstream Transport

The emittance of the DARHT 2 nd Axis has been inferred from solenoid scans performed in the downstream transport (DST) region using a short kicked pulse. The beam spot size is measured by viewing optical transition radiation (OTR) in the near field as a function of the field (current) of a solenoid magnet (S4). The imaging station containing the OTR target is located about 100 cm downstream of the solenoid magnet. The emittance is then inferred using a beam optics code such as LAMDA or XTR by fitting the data to initial conditions upstream of the S4 solenoid magnet. The initial conditions are the beam size, beam convergence and emittance. The beam energy and current are measured. In preparation for a solenoid scan, the magnets upstream of the solenoid are adjusted to produce a round beam with no beam losses due to scraping in the beam tube. This is different from the standard tune in which the beam tune is adjusted to suppress the effects of ions and rf in the septum dump. In this standard tune, approximately 10% of the beam is lost due to scraping as the beam enters the small 3.75” ID beam tube after the septum. The normalized emittance inferred from recent solenoid scans typically ranges from 600 to 800 π(mm-mrad). This larger beam size increases the sensitivity to any non-linear fields in the Collins quadrupoles that are mounted along the small diameter beam tube. The primary magnet used to adjust the beam size in this region is the S3 solenoid magnet. Measurements made of the beam shape as the beam size was decreased showed significant structure consistent with non-linear fields. Using the measured magnetic fields in the Collins quadrupoles including higher order multipoles, the beam transport through the Collins quadrupoles is simulated and compared to the observed OTR images. The simulations are performed using the beam optics codes TRANSPORT [1] and TURTLE [2]. Estimates of the emittance growth and beam losses are made as a function of the S3 magnet setting. The increase in the spot size on the x-ray production target resulting from this emittance growth is examined for different DST tunes.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2019

Multilayer Divide-Expand-Consolidate Coupled-Cluster Method: Demonstrative Calculations of the Adsorption Energy of Carbon Dioxide in the Mg-MOF-74 Metal–Organic Framework

The implementation and evaluation of a multilayer extension of the divide-expand-consolidate (DEC) scheme within the LSDalton program is presented. The DEC scheme is a linear-scaling, fragmentation-based local coupled-cluster (CC) method that provides a means of overcoming the scaling wall associated with canonical CC electronic structure calculations on large molecular systems. Taking advantage of the local nature of correlation effects, the correlation energy for the full molecule is calculated from a set of independent fragments using localized molecular orbitals. However, when only a small subsystem of a larger system is of interest, for example, adsorption sites or catalytically active sites, the majority of the computational time may be spent evaluating the correlation energy of fragments which have little effect on the properties in the area of interest (AOI). The multilayer DEC (ML-DEC) scheme addresses this by taking advantage of the independent nature of the fragments in order to evaluate the correlation energy of various regions of the system at different levels of theory. Regions far from the AOI are evaluated at lower (cheaper) levels of theory such as Hartree–Fock (HF) or Møller–Plesset second-order perturbation theory (MP2), while the area immediately surrounding the AOI is treated with a higher level CC model. Through the ML-DEC scheme, the computational cost of CC calculations on these types of systems can be significantly reduced while maintaining the accuracy of higher-level calculations. Results from HF/RI-MP2 and RI-MP2/CCSD ML-DEC calculations of the binding energy of a fatty acid dimer are presented. We find that the ML-DEC scheme is capable of reproducing DEC energy differences at a target level of theory, provided that the region treated at the target level of theory is chosen to be sufficiently large. Time-to-solution is found to be significantly reduced, particularly in the RI-MP2/CCSD calculations. Finally, the ML-DEC scheme is applied to the calculation of CO 2 adsorption in a Mg-MOF-74 channel.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Turbulence patterns and neutrino flavor transitions in high-resolution supernova models

During the shock-wave propagation in a core-collapse supernova (SN), matter turbulence may affect neutrino flavor conversion probabilities. Such effects have been usually studied by adding parametrized small-scale random fluctuations (with arbitrary amplitude) on top of coarse, spherically symmetric matter density profiles. Recently, however, two-dimensional (2D) SN models have reached a space resolution high enough to directly trace anisotropic density profiles, down to scales smaller than the typical neutrino oscillation length. In this context, we analyze the statistical properties of a large set of SN matter density profiles obtained in a high-resolution 2D simulation, focusing on a post-bounce time (2 s) suited to study shock-wave effects on neutrino propagation on scales as small as O(100) km and possibly below. We clearly find the imprint of a broken (Kolmogorov-Kraichnan) power-law structure, as generically expected in 2D turbulence spectra. We then compute the flavor evolution of SN neutrinos along representative realizations of the turbulent matter density profiles, and observe no or modest damping of the neutrino crossing probabilities on their way through the shock wave. In order to check the effect of possibly unresolved fluctuations at scales below O(100) km, we also apply a randomization procedure anchored to the power spectrum calculated from the simulation, and find consistent results within ± 1σ fluctuations. These results show the importance of anchoring turbulence effects on SN neutrinos to realistic, fine-grained SN models.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

BC8 Silicon (Si-III) is a Narrow-Gap Semiconductor

Large-volume, phase-pure synthesis of BC8 silicon ($Ia\bar{3}$, cI16) has allowed bulk measurements of optical, electronic, and thermal properties. Unlike previous reports that conclude BC8-Si is semimetallic, we demonstrate that this phase is a direct band gap semiconductor with a very small energy gap and moderate carrier concentration and mobility at room temperature, based on far- and midinfrared optical spectroscopy, temperature-dependent electrical conductivity, Seebeck and heat capacity measurements. Samples exhibit a plasma wavelength near 11 μm, indicating potential for infrared plasmonic applications. Thermal conductivity is reduced by 1–2 orders of magnitude depending on temperature as compared with the diamond cubic (DC-Si) phase. The electronic structure and dielectric properties can be reproduced by first-principles calculations with hybrid functionals after adjusting the level of exact Hartree–Fock (HF) exchange mixing. These results clarify existing limited and controversial experimental data sets and ab initio calculations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2010

INTERMITTENCY AND MULTIFRACTALITY SPECTRA OF THE MAGNETIC FIELD IN SOLAR ACTIVE REGIONS

We present the results of a study of intermittency and multifractality of magnetic structures in solar active regions (ARs). Line-of-sight magnetograms for 214 ARs of different flare productivity observed at the center of the solar disk from 1997 January until 2006 December are utilized. Data from the Michelson Doppler Imager (MDI) instrument on board the Solar and Heliospheric Observatory operating in the high resolution mode, the Big Bear Solar Observatory digital magnetograph, and the Hinode SOT/SP instrument were used. Intermittency spectra were derived from high-order structure functions and flatness functions. The flatness function exponent is a measure of the degree of intermittency. We found that the flatness function exponent at scales below approximately 10 Mm is correlated with flare productivity (the correlation coefficient is -0.63). The Hinode data show that the intermittency regime is extended toward small scales (below 2 Mm) as compared to the MDI data. The spectra of multifractality, derived from the structure functions and flatness functions, are found to be broader for ARs of higher flare productivity as compared to those of low flare productivity. The magnetic structure of high-flaring ARs consists of a voluminous set of monofractals, and this set is much richer than that for low-flaring ARs. The results indicate the relevance of the multifractal organization of the photospheric magnetic fields to the flaring activity. The strong intermittency observed in complex and high-flaring ARs is a hint that we observe a photospheric imprint of enhanced sub-photospheric dynamics.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Quantifying Quantum Chaos through Microcanonical Distributions of Entanglement

A characteristic feature of “quantum chaotic” systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this feature, we introduce a framework that allows us to compare the properties of eigenstates in local systems with those of pure random states. In particular, our framework defines a notion of distance between quantum state ensembles that utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). This notion gives rise to a quantitative metric for quantum chaos that not only accounts for averages of the distributions but also higher moments. The differences in moments are compared on a highly resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. As a result, the metric can distinguish between chaotic and integrable behaviors and, in addition, quantify and compare the of chaos (in terms of proximity to RMT behavior) between two systems that are assumed to be chaotic. We implement our framework in local, minimally structured, Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed-field Ising model (MFIM). Importantly, for Hamiltonian systems, we find that the reference random distribution must be appropriately constrained to incorporate the effect of energy conservation in order to describe the ensemble properties of midspectrum eigenstates. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that “maximally chaotic” Hamiltonians may exist in fine-tuned pockets of parameter space. Published by the American Physical Society 2024

Physics↗
DOE OSTI2016

A novel approach to background subtraction in contrast-enhanced dual-energy digital mammography with commercially available mammography devices: Noise minimization

Purpose: Dual-energy image subtraction represents a useful tool to improve the detectability of small lesions, especially in dense breasts. A feature it shares with all x-ray imaging techniques is the appearance of fluctuations in the texture of the background, which can obscure the visibility of interesting details. The aim of the work is to investigate the main noise sources, in order to create a better performing subtraction mechanism. In particular, the structural noise cancellation was achieved by means of a suitable extension of the dual-energy algorithm. Methods: The effect of the cancellation procedure was tested on an analytical simulation of a target with varying structural composition. Subsequently, the subtraction algorithm was also applied to a set of actual radiographs of a breast phantom exhibiting a nonuniform background pattern. The background power spectra of the outcomes were computed and compared to the ones obtained from a standard subtraction algorithm. Results: The comparison between the standard and the proposed cancellations showed an overall suppression of the magnitudes of the spectra, as well as a flattening of the frequency dependence of the structural component of the noise. Conclusions: The proposed subtraction procedure provides an effective cancellation of the residual background fluctuations. When combined with the polychromatic correction already described in a companion publication, it results in a high performing dual-energy subtraction scheme for commercial mammography units.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2022

MgADP Promotes Myosin Head Movement toward Actin at Low [Ca 2+ ] to Increase Force Production and Ca 2+ -Sensitivity of Contraction in Permeabilized Porcine Myocardial Strips

Myosin cross-bridges dissociate from actin following Mg 2+ -adenosine triphosphate (MgATP) binding. Myosin hydrolyses MgATP into inorganic phosphate (P i ) and Mg 2+ -adenosine diphosphate (ADP), and release of these hydrolysis products drives chemo-mechanical energy transitions within the cross-bridge cycle to power muscle contraction. Some forms of heart disease are associated with metabolic or enzymatic dysregulation of the MgATP-MgADP nucleotide pool, resulting in elevated cytosolic [MgADP] and impaired muscle relaxation. We investigated the mechanical and structural effects of increasing [MgADP] in permeabilized myocardial strips from porcine left ventricle samples. Sarcomere length was set to 2.0 µm at 28 °C, and all solutions contained 3% dextran T-500 to compress myofilament lattice spacing to near-physiological values. Under relaxing low [Ca 2+ ] conditions (pCa 8.0, where pCa = -log 10 [Ca 2+ ]), tension increased as [MgADP] increased from 0-5 mM. Complementary small-angle X-ray diffraction measurements show that the equatorial intensity ratio, I 1,1 /I 1,0 , also increased as [MgADP] increased from 0 to 5 mM, indicating myosin head movement away from the thick-filament backbone towards the thin-filament. Ca 2+ -activated force-pCa measurements show that Ca 2+ -sensitivity of contraction increased with 5 mM MgADP, compared to 0 mM MgADP. These data show that MgADP augments tension at low [Ca 2+ ] and Ca 2+ -sensitivity of contraction, suggesting that MgADP destabilizes the quasi-helically ordered myosin OFF state, thereby shifting the cross-bridge population towards the disordered myosin ON state. Together, these results indicate that MgADP enhances the probability of cross-bridge binding to actin due to enhancement of both thick and thin filament-based activation mechanisms.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2020

Chespa: Streamlining Expansive Chemical Space Evaluation of Molecular Sets

Thousands of chemical properties can be calculated for small molecules, which can be used to place the molecules within the context of a broader “chemical space.” These definitions vary based on compounds of interest and the goals for the given chemical space definition. Here, we introduce a customizable (i.e., modular) Python module, chespa, built to easily assess different chemical space definitions through cluster-ing of compounds in these spaces and visualize trends of these clusters. To demonstrate this, chespa currently streamlines prediction of vari-ous molecule descriptors (predicted chemical properties, molecular substructures, AI-based chemical space, and chemical class ontology) in order to test 6 different chemical space definitions. Furthermore, we investigated how these varying definitions trend with mass spectrometry (MS)-based observability, i.e., the ability of a molecule to be observed with MS (e.g., as a function of the molecule ionizability), using an example data set from the U.S. EPA's Non-Targeted Analysis Collaborative Trial (ENTACT), where blinded samples had been analyzed previously, providing 1,398 data points. Improved understanding of observability would offer many advantages in small molecule identifica-tion, such as (i) a priori selection of experimental conditions based on suspected sample composition, (ii) the ability to reduce the number of candidate structures during compound identification by removing those less likely to ionize, and, in turn, (iii) a reduced false discovery rate and increased confidence in identifications. Factors controlling observability are not fully understood, making prediction of this property non-trivial and a prime candidate for chemical space analysis. Chespa is available at github.com/pnnl/chespa.

Nunez, Jamie↗
DOE OSTI2025

Enhancing the accuracy of XPS calculations: Exploring hybrid basis set schemes for CVS-EOMIP-CCSD calculations

Reliable computational methodologies and basis sets for modeling x-ray spectra are essential for extracting and interpreting electronic and structural information from experimental x-ray spectra. In particular, the trade-off between numerical accuracy and computational cost due to the size of the basis set is a major challenge, since molecular orbitals undergo extreme relaxation in the core-hole state. To gain clarity on the changes in electronic structure induced by the formation of a core-hole, the use of sufficiently flexible basis for expanding the orbitals, particularly for the core region, has been shown to be essential. This work focuses on the refinement of core-hole ionized state calculations using the equation-of-motion coupled cluster family of methods through an extensive analysis on the effectiveness of “hybrid” and mixed basis sets. In this investigation, we utilize the CVS-EOMIP-CCSD method in combination and construct hybrid basis sets piecewise from readily available Dunning’s correlation consistent basis sets in order to calculate x-ray ionization energies (IEs) for a set of small gas phase molecules. Our results provide insights into the impact of basis sets on the CVS-EOMIP-CCSD calculations of K-edge IEs of first-row p-block elements. Furthermore, these insights enable us to understand more about the basis set dependence of the core IEs computed and allow us to establish a protocol for deriving reliable and cost-effective theoretical estimates for computing IEs of small molecules containing such elements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Deep learning simulations of the microwave sky

Here we present 500 high-resolution, full-sky millimeter-wave deep learning (DL) simulations that include lensed CMB maps and correlated foreground components. We find that these MillimeterDL simulations can reproduce a wide range of non-Gaussian summary statistics matching the input training simulations, while only being optimized to match the power spectra. The procedure we develop in this work enables the capability to mass produce independent full-sky realizations from a single expensive full-sky simulation, when ordinarily the latter would not provide enough training data. We also circumvent a common limitation of high-resolution DL simulations that they be confined to small sky areas, often due to memory or GPU issues; we do this by developing a “stitching” procedure that can recover the large-scale, high-order statistics and avoid discontinuities or repeated features in the maps. In addition, since our network takes as input a full-sky lensing convergence map, it can in principle take a full-sky lensing convergence map from any large-scale structure (LSS) simulation and generate the corresponding lensed CMB and correlated foreground components at millimeter wavelengths; this is especially useful in the current era of combining results from both CMB and LSS surveys, which require a common set of simulations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1999

Optimization of Aerospace Structure Subject to Damage Tolerance Criteria

The objective of this cooperative agreement was to seek computationally efficient ways to optimize aerospace structures subject to damage tolerance criteria. Optimization was to involve sizing as well as topology optimization. The work was done in collaboration with Steve Scotti, Chauncey Wu and Joanne Walsh at the NASA Langley Research Center. Computation of constraint sensitivity is normally the most time-consuming step of an optimization procedure. The cooperative work first focused on this issue and implemented the adjoint method of sensitivity computation in an optimization code (runstream) written in Engineering Analysis Language (EAL). The method was implemented both for bar and plate elements including buckling sensitivity for the latter. Lumping of constraints was investigated as a means to reduce the computational cost. Adjoint sensitivity computation was developed and implemented for lumped stress and buckling constraints. Cost of the direct method and the adjoint method was compared for various structures with and without lumping. The results were reported in two papers. It is desirable to optimize topology of an aerospace structure subject to a large number of damage scenarios so that a damage tolerant structure is obtained. Including damage scenarios in the design procedure is critical in order to avoid large mass penalties at later stages. A common method for topology optimization is that of compliance minimization which has not been used for damage tolerant design. In the present work, topology optimization is treated as a conventional problem aiming to minimize the weight subject to stress constraints. Multiple damage configurations (scenarios) are considered. Each configuration has its own structural stiffness matrix and, normally, requires factoring of the matrix and solution of the system of equations. Damage that is expected to be tolerated is local and represents a small change in the stiffness matrix compared to the baseline (undamaged) structure. The exact solution to a slightly modified set of equations can be obtained from the baseline solution economically without actually solving the modified system. Sherrnan-Morrison-Woodbury (SMW) formulas are matrix update formulas that allow this. SMW formulas were therefore used here to compute adjoint displacements for sensitivity computation and structural displacements in damaged configurations.

Akgun, Mehmet A.↗