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DOE OSTI2022

NH-PINN: Neural homogenization-based physics-informed neural network for multiscale problems

Physics-informed neural network (PINN) is a data-driven approach to solving equations. It is successful in many applications; however, the accuracy of the PINN is not satisfactory when it is used to solve multiscale equations. Homogenization approximates a multiscale equation by a homogenized equation without multiscale property; it includes solving cell problems and the homogenized equation. The cell problems are periodic, and we propose an oversampling strategy that significantly improves the PINN accuracy on periodic problems. The homogenized equation has a constant or slow dependency coefficient and can also be solved by PINN accurately. We hence proposed a 3-step method, neural homogenization based PINN (NH-PINN), to improve the PINN accuracy for solving multiscale problems with the help of homogenization.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Sensitivity analysis of homogenized cross section in AP1000 lattices

The two-step method for light water reactor simulation consists of performing lattice-level calculations to determine homogenized properties of the lattice for a variety of configurations and burnups then performing a core-level calculation which relies on those pre-calculated lattice properties. Calculating the homogenized lattice properties for the specific conditions needed in the core-level calculation relies on interpolating from the pre-calculated lattice properties. Although many studies have explored different models that can be used for this interpolation, there is a need to explore how to appropriately parameterize the lattice for those interpolation models. The present study uses linear sensitivity methods to determine the sensitivity of homogenized cross sections to both instantaneous and historical values of state variables. The results from this study can be used to inform parameterizations which more appropriately capture history effects in homogenized cross section interpolation. Two different lattices are used for this analysis from the AP1000 reactor, one includes burnable absorbers. It was found that early in the cycle, the historical values of state variables have little effect on the instantaneous value of the homogenized cross sections explored here - especially in the case of the lattice containing burnable absorbers. However, it was found that later in the cycle, the thermal fission homogenized cross section was sensitive to the historic states of the lattice. In fact, using the average of a state variable as a predictor for interpolation models may be insufficient for capturing historic effects due to more recent states having larger sensitivity measures than less recent states. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2016

Application of SPH Factors to PHWR Lattice Homogenization

Pressurized Heavy-Water Reactors (PHWRs) are heavy-water moderated and cooled. They consist of a horizontal, cylindrical, non-pressurized calandria vessel which contains the heavy-water moderator and is penetrated axially by fuel channels. Fuel channels consist of two concentric tubes separated by a gas gap: an inner tube called the pressure tube, and an outer tube called the calandria tube. The pressure tubes hold the fuel bundles, which are cooled by the flow of coolant at high temperature and pressure. The coolant flows in opposite directions in adjacent channels. The gap between the pressure and calandria tubes ensures that the temperature of the moderator remains relatively low, below 80 deg. C. For a CANDU 6 reactor, there are 380 fuel channels, each holding 12 37-element fuel bundles, approximately 10-cm in diameter and 50-cm long each. The fuel pins are arranged in three concentric rings (consisting of 6, 12 and 18 pins respectively), plus a central pin. The distance between channels (lattice pitch) is approximately 28.6 cm. Production core-level neutronics calculations are usually performed in a Cartesian geometry using few group diffusion and a simplified core model, whereby neutronics properties are homogenized over large regions called nodes. For Light Water reactors (LWRs), the two-dimensional axial (x-y) projection of a node corresponds to one fuel assembly. For PHWRs, a node usually consists of one fuel bundle and its associated moderator. For LWRs, advanced core-level calculations based on advanced node-level homogenization methods, such as Generalized Equivalence Theory (GET) or Super-homogenisation (SPH), improve the accuracy of full-core calculations compared to - standard homogenization. For rectangular fuel-pin arrangements, SPH can be used to perform pin-cell level homogenization which allows subsequent core-level diffusion calculations to be performed using fine Cartesian meshes. For PHWRs, the distribution of fuel pins in concentric rings makes the use of Cartesian mesh refinement impractical. However, SPH can still be applied, albeit using a coarser node subdivision, by dividing the heterogeneous node into 3x3 sub-regions in such a way that the central sub-region includes the entire fuel channel. This work investigates the use of SPH homogenization using a 3x3 subdivision of the PHWR node. Preliminary results show that the use of SPH factors does not yield more accurate results than standard homogenization when applied to PHWR nodes using a 3x3 subdivision. The explanation appears to lie in the fact that SPH factors are dependent on node-boundary leakage. Future investigations to confirm the leakage-dependence of SPH factors are recommended. Accuracy improvements may also be achievable by using smaller SPH regions at the periphery of the PHWR node, in an attempt to reduce the SPH factors' dependence on node-boundary leakage. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2025

Homogenization of Dendritic Structures and the High-Temperature Strength of the Refractory High-Entropy Alloy MoNbTaVW

MoNbTaVW, a pioneering refractory high-entropy alloy (RHEA), is renowned for its exceptional strength at elevated temperatures. Like most RHEAs, it solidifies into a dendritic microstructure with steep concentration gradients, necessitating heat treatment for equilibration. Steep concentration gradients typically complicate mechanical property analysis and modeling, key challenges for high-throughput alloy discovery. This study examines the effects of high-temperature homogenization (1800 °C for 8 hours) on the microstructure and mechanical properties of MoNbTaVW across temperatures up to 1200 °C. Postmortem microstructural analysis, coupled with chemical, crystallographic, and mechanical assessments, revealed that yield strength was surprisingly insensitive to homogenization, with less than a 3 pct difference between as-cast and heat-treated materials. Nanoindentation confirmed minimal nanohardness changes across the dendritic structure at room temperature. However, homogenization significantly enhanced high-temperature work hardening, producing higher peak compressive strength. Both as-cast and homogenized MoNbTaVW exhibited room-temperature strengths of 1400 MPa, exceeding previously reported values. These findings demonstrate that high-temperature treatment enables microstructural homogenization without compromising strength, a unique behavior among RHEAs. In conclusion, while not universally applicable, this insight highlights the importance of understanding microstructural development in experimental alloys and offers a pathway to design alloys that achieve homogeneous-like properties without heat treatment.

Rietema, C. J. [Lawrence Livermore National Labora↗

DOE OSTI2017

Mechanisms and Rates of U(VI) Reduction by Fe(II) in Homogeneous Aqueous Solution and the Role of U(V) Disproportionation

Molecular-level pathways in the aqueous redox transformation of uranium by iron remain unclear, despite the importance of this knowledge for predicting uranium transport and distribution in natural and engineered environments. As the relative importance of homogeneous versus heterogeneous pathways is difficult to probe experimentally, here we apply computational molecular simulation to isolate rates of key one electron transfer reactions in the homogeneous pathway. By comparison to experimental observations the role of the heterogeneous pathway also becomes clear. Density functional theory (DFT) and Marcus theory calculations for all primary monomeric species at pH values ≤7 show for UO22+ and its hydrolysis species UO2OH+ and UO2(OH)20 that reduction by Fe2+ is thermodynamically favorable, though kinetically limited for UO22+. An inner-sphere encounter complex between UO2OH+ and Fe2+ was the most stable for the first hydrolysis species and displayed an electron transfer rate constant ket = 4.3 × 103 s-1. Three stable inner- and outer-sphere encounter complexes between UO2(OH)20 and Fe2+ were found, with electron transfer rate constants ranging from ket = 7.6 × 102 to 7.2 × 104 s-1. Homogeneous reduction of these U(VI) hydrolysis species to U(V) is, therefore, predicted to be facile. In contrast, homogeneous reduction of UO2+ by Fe2+ was found to be thermodynamically unfavorable, suggesting the possible importance of U(V)-U(V) disproportionation as a route to U(IV). Calculations on homogeneous disproportionation, however, while yielding a stable outer-sphere U(V)-U(V) encounter complex, indicate that this electron transfer reaction is not feasible at circumneutral pH. Protonation of both axial O atoms of acceptor U(V) (i.e., by H3O+) was found to be a prerequisite to stabilize U(IV), consistent with the experimental observation that the rate of this reaction is inversely correlated with pH. Thus, despite prevailing notions that U(V) is rapidly eliminated by homogeneous disproportionation, this pathway is irrelevant at environmental conditions.

Environmental Molecular Sciences Laboratory↗

DOE OSTI2011

Testing homogeneity with the fossil record of galaxies

The standard Friedmann model of cosmology is based on the Copernican Principle, i.e. the assumption of a homogeneous background on which structure forms via perturbations. Homogeneity underpins both general relativistic and modified gravity models and is central to the way in which we interpret observations of the CMB and the galaxy distribution. It is therefore important to probe homogeneity via observations. We describe a test based on the fossil record of distant galaxies: if we can reconstruct key intrinsic properties of galaxies as functions of proper time along their worldlines, we can compare such properties at the same proper time for our galaxy and others. We achieve this by computing the lookback time using radial Baryon Acoustic Oscillations, and the time along galaxy world line using stellar physics, allowing us to probe homogeneity, in principle anywhere inside the past light cone. Agreement in the results would be an important consistency test — although it would not in itself prove homogeneity. Any significant deviation in the results however would signal a breakdown of homogeneity.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

A 14 h {sup −3} Gpc{sup 3} study of cosmic homogeneity using BOSS DR12 quasar sample

The BOSS quasar sample is used to study cosmic homogeneity with a 3D survey in the redshift range 2.2 < z < 2.8. We measure the count-in-sphere, N (< r ), i.e. the average number of objects around a given object, and its logarithmic derivative, the fractal correlation dimension, D {sub 2}( r ). For a homogeneous distribution N (< r ) ∝ r {sup 3} and D {sub 2}( r ) = 3. Due to the uncertainty on tracer density evolution, 3D surveys can only probe homogeneity up to a redshift dependence, i.e. they probe so-called ''spatial isotropy'. Our data demonstrate spatial isotropy of the quasar distribution in the redshift range 2.2 < z < 2.8 in a model-independent way, independent of any FLRW fiducial cosmology, resulting in 3 − ( D {sub 2}) < 1.7 × 10{sup −3} (2 σ) over the range 250 < r < 1200 h {sup −1} Mpc for the quasar distribution. If we assume that quasars do not have a bias much less than unity, this implies spatial isotropy of the matter distribution on large scales. Then, combining with the Copernican principle, we finally get homogeneity of the matter distribution on large scales. Alternatively, using a flat ΛCDM fiducial cosmology with CMB-derived parameters, and measuring the quasar bias relative to this ΛCDM model, our data provide a consistency check of the model, in terms of how homogeneous the Universe is on different scales. D {sub 2}( r ) is found to be compatible with our ΛCDM model on the whole 10 < r < 1200 h {sup −1} Mpc range. For the matter distribution we obtain 3 − ( D {sub 2}) < 5 × 10{sup −5} (2 σ) over the range 250 < r < 1200 h {sup −1} Mpc, consistent with homogeneity on large scales.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2013

Automated treatment planning for a dedicated multi-source intra-cranial radiosurgery treatment unit accounting for overlapping structures and dose homogeneity

Purpose: The purpose of this work is to advance the two-step approach for Gamma Knife{sup ®} Perfexion™ (PFX) optimization to account for dose homogeneity and overlap between the planning target volume (PTV) and organs-at-risk (OARs).Methods: In the first step, a geometry-based algorithm is used to quickly select isocentre locations while explicitly accounting for PTV-OARs overlaps. In this approach, the PTV is divided into subvolumes based on the PTV-OARs overlaps and the distance of voxels to the overlaps. Only a few isocentres are selected in the overlap volume, and a higher number of isocentres are carefully selected among voxels that are immediately close to the overlap volume. In the second step, a convex optimization is solved to find the optimal combination of collimator sizes and their radiation duration for each isocentre location.Results: This two-step approach is tested on seven clinical cases (comprising 11 targets) for which the authors assess coverage, OARs dose, and homogeneity index and relate these parameters to the overlap fraction for each case. In terms of coverage, the mean V{sub 99} for the gross target volume (GTV) was 99.8% while the V{sub 95} for the PTV averaged at 94.6%, thus satisfying the clinical objectives of 99% for GTV and 95% for PTV, respectively. The mean relative dose to the brainstem was 87.7% of the prescription dose (with maximum 108%), while on average, 11.3% of the PTV overlapped with the brainstem. The mean beam-on time per fraction per dose was 8.6 min with calibration dose rate of 3.5 Gy/min, and the computational time averaged at 205 min. Compared with previous work involving single-fraction radiosurgery, the resulting plans were more homogeneous with average homogeneity index of 1.18 compared to 1.47.Conclusions: PFX treatment plans with homogeneous dose distribution can be achieved by inverse planning using geometric isocentre selection and mathematical modeling and optimization techniques. The quality of the obtained treatment plans are clinically satisfactory while the homogeneity index is improved compared to conventional PFX plans.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2016

SU-F-T-190: Homogeneity Study of Proton and Heavy Ion Scanning Beams Based On Different Compositions of Spot Sizes (FWHM) and Grid Sizes

Purpose: Different particle scanning beam delivery systems have different delivery accuracies. This study was performed to determine, for our particle treatment system, an appropriate composition (n=FWHM/GS) of spot size(FWHM) and grid size (GS), which can provide homogenous delivered dose distributions for both proton and heavy ion scanning beam radiotherapy. Methods: We analyzed the delivery errors of our beam delivery system using log files from the treatment of 28 patients. We used a homemade program to simulate square fields for different n values with and without considering the delivery errors and analyzed the homogeneity. All spots were located on a rectilinear grid with equal spacing in the × and y directions. After that, we selected 7 energy levels for both proton and carbon ions. For each energy level, we made 6 square field plans with different n values (1, 1.5, 2, 2.5, 3, 3.5). Then we delivered those plans and used films to measure the homogeneity of each field. Results: For program simulation without delivery errors, when n≥1.1 the homogeneity can be within ±3%. For both proton and carbon program simulations with delivery errors and film measurements, the homogeneity can be within ±3% when n≥2.5. Conclusion: For our facility with system errors, the n≥2.5 is appropriate for maintaining homogeneity within ±3%.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2021

Methodology for Sensitivity Analysis of Homogenized Cross-Sections to Instantaneous and Historical Lattice Conditions with Application to AP1000® PWR Lattice

In the two-step method for nuclear reactor simulation, lattice physics calculations are performed to compute homogenized cross-sections for a variety of burnups and lattice configurations. A nodal code is then used to perform full-core analysis using the pre-calculated homogenized cross-sections. One source of uncertainty introduced in this method is that the lattice configuration or depletion conditions typically do not match a pre-calculated one from the lattice physics simulations. Therefore, some interpolation model must be used to estimate the homogenized cross-sections in the nodal code. This current study provides a methodology for sensitivity analysis to quantify the impact of state variables on the homogenized cross-sections. This methodology also allows for analyses of the historical effect that the state variables have on homogenized cross-sections. An application of this methodology on a lattice for the Westinghouse AP1000® reactor is presented where coolant density, fuel temperature, soluble boron concentration, and control rod insertion are the state variables of interest. The effects of considering the instantaneous values of the state variables, historical values of the state variables, and burnup-averaged values of the state variables are analyzed. Using these methods, it was found that a linear model that only considers the instantaneous and burnup-averaged values of state variables can fail to capture some variations in the homogenized cross-sections.

Price, Dean (ORCID:0000000309990111)↗

NASA NTRS1983

Homogeneity of lava flows - Chemical data for historic Mauna Loan eruptions

Chemical analyses of basalts collected from the major historic eruptions of Mauna Loa volcano show that many of the flow fields are remarkably homogeneous in composition. Despite their large size (lengths 9-85 km), large areal extents (13-114 sq km), and various durations of eruption (1-450 days), many of the flow fields have compositional variability that is within, or close to, the analytical error for most elements. The flow fields that are not homogeneous vary mainly in olivine content in an otherwise homogeneous melt. Some are composite flow fields made up of several, apparently homogeneous subunits erupted at different elevations along the active volcanic rifts. Not all volcanoes produce lavas that are homogeneous like those of Mauna Loa. If studies such as this are to be used to evaluate compositional diversity in lavas where there is a lack of sampling control, such as on other planets, it is necessary to understand why some flow units and flow fields are compositionally homogeneous and others are not, and to develop criteria for distinguishing between them.

Rhodes, J. M.↗

NASA NTRS1986

Chemical zoning and homogenization of olivines in ordinary chondrites and implications for thermal histories of chondrules

The extent and degree of homogenization of chemical zoning of olivines in type 3 ordinary chondrites is studied in order to obtain some constraints on cooling histories of chondrites. Based on Mg-Fe and CaO zoning, olivines in type 3 chondrites are classified into four types. A single chondrule usually contains olivines with the same type of zoning. Microporphyritic olivines show all four zoning types. Barred olivines usually show almost homogenized chemical zoning. The cooling rates or burial depths needed to homogenize the chemical zoning are calculated by solving the diffusion equation, using the zoning profiles as an initial condition. Mg-Fe zoning of olivine may be altered during initial cooling, whereas CaO zoning is hardly changed. Barred olivines may be homogenized during initial cooling because their size is relatively small. To simulated microporphyritic olivine chondrules, cooling from just below the liquidus at moderately high rates is preferable to cooling from above the liquidus at low rates. For postaccumulation metamorphism of type 3 chondrites to keep Mg-Fe zoning unaltered, the maximum metamorphic temperature must be less than about 400 C if cooling rates based on Fe-Ni data are assumed. Calculated cooling rates for both Fa and CaO homogenization are consistent with those by Fe-Ni data for type 4 chondrites. A hot ejecta blanket several tens of meters thick on the surface of a parent body is sufficient to homogenize Mg-Fe zoning if the temperature of the blanket is 600-700 C. Burial depths for petrologic types of ordinary chondrites in a parent body heated by Al-26 are broadly consistent with those previously proposed.

Miyamoto, Masamichi↗

NASA NTRS2023

Homogeneity Study of ZrC, NbC, and TaC Binary Carbide Fuels for the Application of Nuclear Thermal Propulsion

In alignment with NASA and DARPA goals, efforts towards Nuclear Thermal Propulsion (NTP) have increased in recent years. As progress is made, a main challenge still exists; the fuel in an NTP system must survive the extreme environment experienced during operating conditions. Three fuel forms have been studied including a ceramic-metallic matrix, a ceramic-ceramic matrix, and solid solution carbides. Solid solution carbides were studied previously as a fuel for NTP and was shown to be a promising technology; however, the programs were canceled before demonstration in an NTP engine test. Of the refractory metal carbides of interest, zirconium carbide (ZrC), niobium carbide (NbC) and tantalum carbide (TaC) were studied. ZrC, NbC, and TaC powders were consolidated in monocarbide (ZrC, NbC, TaC) and bi-carbide (ZrC-NbC, ZrC-TaC, NbC-TaC) forms using spark plasma sintering (SPS), with 50-50 mol% compositions of the bi-carbides of interest. The time and temperature conditions to achieve a completely homogeneous solid solution for the mixed carbides is of interest for NTP fuel applications, since this homogeneity is considered to be essential for this fuel form. Homogeneity is typically not achieved during sintering, as surface diffusion (predominant for sintering to near ideal density) tends to occur at lower temperatures than bulk volumetric diffusion (required for solute interdiffusion to achieve a homogeneous solid solution). High temperature isochronal annealing was performed to identify the conditions to achieve a chemically homogeneous sample. The solute interpenetration vs annealing temperature was quantified using energy dispersion spectroscopy (EDS) in a scanning electron microscope (SEM). X-ray diffraction (XRD) was used to accurately measure the phase stability and lattice parameter as a function of annealing conditions. The diffusion data from each sample was then used to determine self-diffusion coefficients for interdiffusion of the Nb, Zr, and Ta solutes and compared to existing literature values. The minimum annealing conditions to achieve complete chemical homogeneity for each carbide was determined.

nuclear thermal propulsion↗

DOE OSTI2017

On domain symmetry and its use in homogenization

The present study focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D 3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D 3 symmetry to reduce the computational expense by performing the “unit strain” periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian D 2 , i.e. orthotropic, square unit cell and nonabelian D 3 , i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D 3 symmetric hexagonal unit cell.

36 MATERIALS SCIENCE↗

DOE OSTI2015

A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory

The development of reliable methods for upscaling fine scale models of elastic media has long been an important topic for rock physics and applied seismology. Several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures. In such cases, the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation. However, such theories cannot be applied directly to homogenize elastic media with more complex, arbitrary spatial heterogeneity. We therefore propose a numerical homogenization algorithm based on multiscale finite element methods for simulating elastic wave propagation in heterogeneous, anisotropic elastic media. Specifically, our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions. Homogenized, effective medium parameters were then computed using these basis functions, and the approach applied a numerical discretization that is similar to the rotated staggered-grid finite difference scheme. Comparisons of the results from our method and from conventional, analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry. Additional tests examined anisotropic models with arbitrary spatial heterogeneity where the average size of the heterogeneities ranged from several centimeters to several meters, and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100. Comparisons to finite-difference simulations proved that the numerical homogenization was equally accurate for these complex cases.

58 GEOSCIENCES↗

DOE OSTI2017

SuPer-Homogenization (SPH) Corrected Cross Section Generation for High Temperature Reactor

The deterministic full core simulators require homogenized group constants covering the operating and transient conditions over the entire lifetime. Traditionally, the homogenized group constants are generated using lattice physics code over an assembly or block in the case of prismatic high temperature reactors (HTR). For the case of strong absorbers that causes strong local depressions on the flux profile require special techniques during homogenization over a large volume. Fuel blocks with burnable poisons or control rod blocks are example of such cases. Over past several decades, there have been a tremendous number of studies performed for improving the accuracy of full-core calculations through the homogenization procedure. However, those studies were mostly performed for light water reactor (LWR) analyses, thus, may not be directly applicable to advanced thermal reactors such as HTRs. This report presents the application of SuPer-Homogenization correction method to a hypothetical HTR core.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2026

Reproducibility, Replicability, and Research Quality in Homogeneous Catalysis

Researchers from all sectors of homogeneous catalysis convened in response to concerns regarding reproducibility in science to analyze the issue and provide recommendations. In addition to an in- person workshop, the group engaged the broader homogeneous catalysis community through a webinar series and virtually during the workshop. Results of the project affirm that homogeneous catalysis is not in a reproducibility crisis, as evidenced by the field’s current and past contributions to society that have led to economic growth and advances in a range of industries from agriculture to consumer goods to human health. However, it is not uncommon for researchers to encounter obstacles related to reproducibility. Ensuring reproducibility remains the responsibility of the community, both in current work and in training future researchers. This report is intended to engage key stakeholders, including disciplinary societies, publishers, employers, research leaders, and researchers, in practices that maximize reproducible homogeneous catalysis and ensure continued innovation and translatable discoveries. Recommendations made herein are also framed to be applicable beyond homogeneous catalysis, empowering the broader chemical if not scientific community.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2014

A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach

We present a new stochastic analysis for steady and transient one-dimensional heat conduction problem based on the homogenization approach. Thermal conductivity is assumed to be a random field K consisting of random variables of a total number N. Both steady and transient solutions T are expressed in terms of the homogenized solution (symbol) and its spatial derivatives (equation), where homogenized solution (symbol) is obtained by solving the homogenized equation with effective thermal conductivity. Both mean and variance of stochastic solutions can be obtained analytically for K field consisting of independent identically distributed (i.i.d) random variables. The mean and variance of T are shown to be dependent only on the mean and variance of these i.i.d variables, not the particular form of probability distribution function of i.i.d variables. Variance of temperature field T can be separated into two contributions: the ensemble contribution (through the homogenized temperature (symbol)); and the configurational contribution (through the random variable Ln(x)Ln(x)). The configurational contribution is shown to be proportional to the local gradient of (symbol). Large uncertainty of T field was found at locations with large gradient of (symbol) due to the significant configurational contributions at these locations. Numerical simulations were implemented based on a direct Monte Carlo method and good agreement is obtained between numerical Monte Carlo results and the proposed stochastic analysis.

99 GENERAL AND MISCELLANEOUS↗