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At least 505 records · Page 28

Many-body expansion based machine learning models for octahedral transition metal complexes

Abstract Graph-based machine learning (ML) models for material properties show great potential to accelerate virtual high-throughput screening of large chemical spaces. However, in their simplest forms, graph-based models do not include any 3D information and are unable to distinguish stereoisomers such as those arising from different orderings of ligands around a metal center in coordination complexes. In this work we present a modification to revised autocorrelation descriptors, a molecular graph featurization method, for predicting spin state dependent properties of octahedral transition metal complexes (TMCs). Inspired by analytical semi-empirical models for TMCs, the new modeling strategy is based on the many-body expansion (MBE) and allows one to tune the captured stereoisomer information by changing the truncation order of the MBE. We present the necessary modifications to include this approach in two commonly used ML methods, kernel ridge regression and feed-forward neural networks. On a test set composed of all possible isomers of binary TMCs, the best MBE models achieve mean absolute errors (MAEs) of 2.75 kcal mol −1 on spin-splitting energies and 0.26 eV on frontier orbital energy gaps, a 30%–40% reduction in error compared to models based on our previous approach. We also observe improved generalization to previously unseen ligands where the best-performing models exhibit MAEs of 4.00 kcal mol −1 (i.e. a 0.73 kcal mol −1 reduction) on the spin-splitting energies and 0.53 eV (i.e. a 0.10 eV reduction) on the frontier orbital energy gaps. Because the new approach incorporates insights from electronic structure theory, such as ligand additivity relationships, these models exhibit systematic generalization from homoleptic to heteroleptic complexes, allowing for efficient screening of TMC search spaces.

Meyer, Ralf (ORCID:0000000322360261)↗

The seventh blind test of crystal structure prediction: structure ranking methods

A seventh blind test of crystal structure prediction has been organized by the Cambridge Crystallographic Data Centre. The results are presented in two parts, with this second part focusing on methods for ranking crystal structures in order of stability. The exercise involved standardized sets of structures seeded from a range of structure generation methods. Participants from 22 groups applied several periodic DFT-D methods, machine learned potentials, force fields derived from empirical data or quantum chemical calculations, and various combinations of the above. In addition, one non-energy-based scoring function was used. Results showed that periodic DFT-D methods overall agreed with experimental data within expected error margins, while one machine learned model, applying system-specific AIMnet potentials, agreed with experiment in many cases demonstrating promise as an efficient alternative to DFT-based methods. For target XXXII, a consensus was reached across periodic DFT methods, with consistently high predicted energies of experimental forms relative to the global minimum (above 4 kJ mol −1 at both low and ambient temperatures) suggesting a more stable polymorph is likely not yet observed. The calculation of free energies at ambient temperatures offered improvement of predictions only in some cases (for targets XXVII and XXXI). Several avenues for future research have been suggested, highlighting the need for greater efficiency considering the vast amounts of resources utilized in many cases.

Chemistry↗

Hex Element Aspect Ratio and Formulation

When modeling thin structures, some analysts use shell elements. Others choose hex elements. Both element types have their advantages and disadvantages. This memo describes a weakness of using hex elements to model bending. Specifically, the accuracy degrades as the element aspect ratio increases. This study illustrates this effect for the mean quadrature and the selective deviatoric formulations. For the test case considered, the selective deviatoric formulation produces larger errors.

42 ENGINEERING↗

Heterogeneous mantle effects on the behavior of SmKS waves and outermost core imaging

Seismic traveltime anomalies of waves that traverse the uppermost 100–200 km of the outer core have been interpreted as evidence of reduced seismic velocities (relative to radial reference models) just below the core–mantle boundary (CMB). These studies typically investigate differential traveltimes of SmKS waves, which propagate as P waves through the shallowest outer core and reflect from the underside of the CMB m times. The use of SmKS and S(m-1)KS differential traveltimes for core imaging are often assumed to suppress contributions from earthquake location errors and unknown and unmodelled seismic velocity heterogeneity in the mantle. The goal of this study is to understand the extent to which differential SmKS traveltimes are, in fact, affected by anomalous mantle structure, potentially including both velocity heterogeneity and anisotropy. Velocity variations affect not only a wave's traveltime, but also the path of a wave, which can be observed in deviations of the wave's incoming direction. Since radial velocity variations in the outer core will only minimally affect the wave path, in contrast to other potential effects, measuring the incoming direction of SmKS waves provides an additional diagnostic as to the origin of traveltime anomalies. Here we use arrays of seismometers to measure traveltime and direction anomalies of SmKS waves that sample the uppermost outer core. We form subarrays of EarthScope's regional Transportable Array stations, thus measuring local variations in traveltime and direction. We observe systematic lateral variations in both traveltime and incoming wave direction, which cannot be explained by changes to the radial seismic velocity profile of the outer core. Moreover, we find a correlation between incoming wave direction and traveltime anomaly, suggesting that observed traveltime anomalies may be caused, at least in part, by changes to the wave path and not solely by perturbations in outer core velocity. Modelling of 1-D ray and 3-D wave propagation in global 3-D tomographic models of mantle velocity anomalies match the trend of the observed traveltime anomalies. Overall, we demonstrate that observed SmKS traveltime anomalies may have a significant contribution from 3-D mantle structure, and not solely from outer core structure.

58 GEOSCIENCES↗

Heterogeneous mantle effects on the behavior of SmKS waves and outermost core imaging

Seismic traveltime anomalies of waves that traverse the uppermost 100–200 km of the outer core have been interpreted as evidence of reduced seismic velocities (relative to radial reference models) just below the core–mantle boundary (CMB). These studies typically investigate differential traveltimes of SmKS waves, which propagate as P waves through the shallowest outer core and reflect from the underside of the CMB m times. The use of SmKS and S(m- 1)KS differential traveltimes for core imaging are often assumed to suppress contributions from earthquake location errors and unknown and unmodelled seismic velocity heterogeneity in the mantle. The goal of this study is to understand the extent to which differential SmKS traveltimes are, in fact, affected by anomalous mantle structure, potentially including both velocity heterogeneity and anisotropy. Velocity variations affect not only a wave’s traveltime, but also the path of a wave, which can be observed in deviations of the wave’s incoming direction. Since radial velocity variations in the outer core will only minimally affect the wave path, in contrast to other potential effects, measuring the incoming direction of SmKS waves provides an additional diagnostic as to the origin of traveltime anomalies. Here we use arrays of seismometers to measure traveltime and direction anomalies of SmKS waves that sample the uppermost outer core. We form subarrays of EarthScope’s regional Transportable Array stations, thus measuring local variations in traveltime and direction. We observe systematic lateral variations in both traveltime and incoming wave direction, which cannot be explained by changes to the radial seismic velocity profile of the outer core. Moreover, we find a correlation between incoming wave direction and traveltime anomaly, suggesting that observed traveltime anomalies may be caused, at least in part, by changes to the wave path and not solely by perturbations in outer core velocity. Modelling of 1-D ray and 3-D wave propagation in global 3-D tomographic models of mantle velocity anomalies match the trend of the observed traveltime anomalies. Overall, we demonstrate that observed SmKS traveltime anomalies may have a significant contribution from 3-D mantle structure, and not solely from outer core structure.

58 GEOSCIENCES↗

Twisted bilayer graphene as topological heavy fermion: II. Analytical approximations of the model parameters

The recently-introduced topological heavy fermion (THF) model [1] of twisted bilayer graphene (TBG) aims to reconcile the quantum-dot-like electronic structure of the latter observed by scanning tunneling microscopy, with its electron delocalization seen in transport measurements. The THF model achieves this by coupling localized (heavy) fermions with anomalous conduction electrons. Originally, the parameters of the THF model were obtained numerically from the Bistritzer–Macdonald (BM) model of TBG [1]. In this work, we derive analytical expressions for the THF model parameters as a function of the twist angle, the ratio between the tunneling amplitudes at the AA and AB regions (w 0 /w 1 ), and the screening length of the interaction potential. By numerically computing the THF model parameters across an extensive experimentally-relevant parameter space, we show that the resulting approximations are remarkably good, i.e., within the 30% relative error for almost the entire parameter space. At the single-particle level, the THF model accurately captures the energy spectrum of the BM model over a large phase space of angles and tunneling amplitude ratios. When interactions are included, we also show that the THF description of TBG is good around the magic angle for realistic values of the tunneling amplitude ratios (0.6 ≤ w 0 /w 1 ≤ 1.0), for which the hybridization between the localized and conduction fermions γ is smaller than the onsite repulsion of the heavy fermions U 1 (i.e., |γ| < U 1 ).

Approximation methods↗

Testing general relativity on cosmological scales at redshift z ∼ 1.5 with quasar and CMB lensing

ABSTRACT We test general relativity (GR) at the effective redshift $\bar{z} \sim 1.5$ by estimating the statistic EG, a probe of gravity, on cosmological scales $19 - 190\, h^{-1}{\rm Mpc}$. This is the highest redshift and largest scale estimation of EG so far. We use the quasar sample with redshifts 0.8 < z < 2.2 from Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey Data Release 16 as the large-scale structure (LSS) tracer, for which the angular power spectrum $C_\ell ^{qq}$ and the redshift-space distortion parameter β are estimated. By cross-correlating with the Planck 2018 cosmic microwave background (CMB) lensing map, we detect the angular cross-power spectrum $C_\ell ^{\kappa q}$ signal at $12\, \sigma$ significance. Both jackknife resampling and simulations are used to estimate the covariance matrix (CM) of EG at five bins covering different scales, with the later preferred for its better constraints on the covariances. We find EG estimates agree with the GR prediction at $1\, \sigma$ level over all these scales. With the CM estimated with 300 simulations, we report a best-fitting scale-averaged estimate of $E_G(\bar{z})=0.30\pm 0.05$, which is in line with the GR prediction $E_G^{\rm GR}(\bar{z})=0.33$ with Planck 2018 CMB + BAO matter density fraction Ωm = 0.31. The statistical errors of EG with future LSS surveys at similar redshifts will be reduced by an order of magnitude, which makes it possible to constrain modified gravity models.

79 ASTRONOMY AND ASTROPHYSICS↗

Regional evapotranspiration from an image-based implementation of the Surface Temperature Initiated Closure (STIC1.2) model and its validation across an aridity gradient in the conterminous US

Recent studies have highlighted the need for improved characterizations of aerodynamic conductance and temperature (g A and T 0 ) in thermal remote-sensing-based surface energy balance (SEB) models to reduce uncertainties in regional-scale evapotranspiration (ET) mapping. By integrating radiometric surface temperature (TR) into the Penman–Monteith (PM) equation and finding analytical solutions of gA and T0, this need was recently addressed by the Surface Temperature Initiated Closure (STIC) model. However, previous implementations of STIC were confined to the ecosystem-scale using flux tower observations of infrared temperature. This study demonstrates the first regional-scale implementation of the most recent version of the STIC model (STIC1.2) that integrates the Moderate Resolution Imaging Spectroradiometer (MODIS) derived TR and ancillary land surface variables in conjunction with NLDAS (North American Land Data Assimilation System) atmospheric variables into a combined structure of the PM and Shuttleworth–Wallace (SW) framework for estimating ET at 1 km × 1 km spatial resolution. Evaluation of STIC1.2 at 13 core AmeriFlux sites covering a broad spectrum of climates and biomes across an aridity gradient in the conterminous US suggests that STIC1.2 can provide spatially explicit ET maps with reliable accuracies from dry to wet extremes. When observed ET from one wet, one dry, and one normal precipitation year from all sites were combined, STIC1.2 explained 66 % of the variability in observed 8-day cumulative ET with a root mean square error (RMSE) of 7.4 mm/8-day, mean absolute error (MAE) of 5 mm/8-day, and percent bias (PBIAS) of -4 %. These error statistics showed relatively better accuracies than a widely used but previous version of the SEB-based Surface Energy Balance System (SEBS) model, which utilized a simple NDVI-based parameterization of surface roughness (z OM ), and the PM-based MOD16 ET. SEBS was found to overestimate (PBIAS = 28 %) and MOD16 was found to underestimate ET (PBIAS =-26 %). The performance of STIC1.2 was better in forest and grassland ecosystems as compared to cropland (20 % underestimation) and woody savanna (40 % overestimation). Model inter-comparison suggested that ET differences between the models are robustly correlated with g A and associated roughness length estimation uncertainties which are intrinsically connected to T R uncertainties, vapor pressure deficit (D A ), and vegetation cover. A consistent performance of STIC1.2 in a broad range of hydrological and biome categories, as well as the capacity to capture spatio-temporal ET signatures across an aridity gradient, points to the potential for this simplified analytical model for near-real-time ET mapping from regional to continental scales.

Bhattarai, Nishan (ORCID:0000000327493549)↗

Modelling Charged Defects in Non-Cubic Semiconductors for Radiation Effects Studies in Next Generation Materials

This final report summarizes the results of the Laboratory Directed Research and Development (LDRD) Project Number 212587 entitled "Modeling Charged Defects in Non-Cubic Semiconductors for Radiation Effects Studies in Next Generation Electronic Materials" . The goal of this project was to extend a predictive capability for modeling defect level energies using first principle density functional theory methods (e.g., for radiation effects assessments) to semiconductors with non-cubic crystal structures. Computational methods that proved accurate for predicting defect levels in standard cubic semiconductors, were found to have shortcomings when applied to the lowered symmetry structures prevalent in next generation electronic materials such as SiC, GaN, and Ga 2 0 3 , stemming from an error in the treatment of the electrostatic boundary conditions. I describe methods to generalized the local moment countercharge (LMCC) scheme to position a charge in bulk supercell calculations of charged defects, circumventing the problem of measuring a dipole in a periodically replicated bulk calculation.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Methodology for Path Planning with Dynamic Data-Driven Flight Capability Estimation

This paper presents methodology to enable path planning for an unmanned aerial vehicle that uses dynamic data-driven flight capability estimation. The main contribution of the work is a general mathematical approach that leverages offline high-fidelity physics-based modeling together with onboard sensor measurements to achieve dynamic path planning. The mathematical framework, expressed as a constrained partially observable Markov decision process, accounts for vehicle capability constraints and is robust to modeling error and disturbances in both the vehicle process and measurement models. Vehicle capability constraints are incorporated using probabilistic support vector machine surrogates of high-fidelity physics-based models that adequately capture the richness of the vehicle dynamics. Sensor measurements are treated in a general manner and can include combinations of multiple modalities such as that through global positioning system signals, inertial mass unit outputs, and structural strain data of the airframe. Results are presented for a simulated three-dimensional environment and point-mass airplane model. The vehicle can dynamically adjust its trajectory according to the observations it receives about its current state of health, thereby retaining a high probability of survival and mission success.

42 ENGINEERING↗

Quantifying Environmental and Line-of-sight Effects in Models of Strong Gravitational Lens Systems

Matter near a gravitational lens galaxy or projected along the line of sight (LOS) can affect strong lensing observables by more than contemporary measurement errors. We simulate lens fields with realistic three-dimensional mass configurations (self-consistently including voids), and then fit mock lensing observables with increasingly complex lens models to quantify biases and uncertainties associated with different ways of treating the lens environment (ENV) and LOS. We identify the combination of mass, projected offset, and redshift that determines the importance of a perturbing galaxy for lensing. Foreground structures have a stronger effect on the lens potential than background structures, due to nonlinear effects in the foreground and downweighting in the background. There is dramatic variation in the net strength of ENV/LOS effects across different lens fields; modeling fields individually yields stronger priors for H {sub 0} than ray tracing through N-body simulations. Models that ignore mass outside the lens yield poor fits and biased results. Adding external shear can account for tidal stretching from galaxies at redshifts z⩾z{sub lens}, but it requires corrections for external convergence and cannot reproduce nonlinear effects from foreground galaxies. Using the tidal approximation is reasonable for most perturbers as long as nonlinear redshift effects are included. Even then, the scatter in H {sub 0} is limited by the lens profile degeneracy. Asymmetric image configurations produced by highly elliptical lens galaxies are less sensitive to the lens profile degeneracy, so they offer appealing targets for precision lensing analyses in future surveys like LSST and Euclid.

79 ASTRONOMY AND ASTROPHYSICS↗

The AWAKEN wind farm benchmark, Part 2: Modeling results

Accurately modeling wind farm performance in complex atmospheric flows remains a challenge. This paper presents the modeling results of the American WAKE experimeNt (AWAKEN) wind farm benchmark, a collaborative effort involving 16 research groups from academia and industry within the International Energy Agency Wind Technology Collaboration Programme Task 57. The study evaluates a diverse suite of simulation tools, ranging from fast-running engineering wake models to high-fidelity large-eddy simulations, against a diurnal case study observed during the AWAKEN campaign. The benchmark utilized a three-phase structure to progressively assess model performance as observational data availability increased. Initial blind predictions showed that higher-fidelity models did not uniformly outperform simpler simulation tools. A distinct spatial bias was observed where models struggled to resolve the interplay between a low-level jet, wakes, and terrain-induced flow acceleration. In subsequent phases, leveraging additional measurements for model improvement led to a reduction in mean absolute error across the model ensemble; however, this effect was most pronounced in engineering wake models, where targeted calibration reduced error by up to 40~\%. Overall, the study demonstrates that inflow characterization remains a primary prerequisite for accuracy, particularly for models relying on coarse forcing datasets. While the limited ability to resolve local terrain-flow interactions under single-day conditions represent a recognized constraint, the overall findings on wake modeling and real-world validation still provide valuable guidance for model application and for mitigating this limitation.

Bodini, Nicola↗

dynamics of organic-mineral interactions at the metal oxide-solution interface as studied via binding energetics (Final report)

This project focused on addressing longstanding fundamental and experimental uncertainties on how dissolved organic substances (DOS) interact with metal oxide surface under environmentally relevant conditions. By leveraging a custom-built real-time, in-tandem flow adsorption microcalorimetry-UV-Vis/fluorescence spectroscopy platform, we characterized the binding energetics, kinetics and mechanistic pathways driving DOS-metal oxide interactions at temporal resolution on the order of 1-5 seconds. We studied a diverse suite of model organic compounds/substances – including monocarboxylates (e.g. acetate and benzoate), di-carboxylates (oxalate and succinate), amino acids, amino-based nanparticles and natural organic matter – interacting at the mineral-water interface of structurally- and/or chemically distinct metal oxides (including SiO2, boehmite, ferrihydrite, and γ-Al2O3). Our results indicated that DOS-metal oxide interactions are governed by multi-step reaction pathways, often switching between distinct, resolvable enthalpy- and entropy-driven non-electrostatic or electrostatic configurations. To quantify these interactions, we developed and implemented an analytical workflow that integrates peak deconvolution and Monte-Carlo based error propagation to determine site-specific thermodynamic and kinetic parameters for individual binding/debinding events. In addition to resolving apparent first-order rate constants of each event, we were able to quantify associated apparent equilibrium constants as well as free energy, enthalpy and entropy contribution to the activation and subsequent progression of the binding/debinding process across compounds, compound class and metal oxide surfaces. The kinetic-thermodynamic data produced in this study captured how the interplay between oxide surface reactivity and DOS molecular structure jointly drives binding-debinding dynamics. Notably, that at pH below PZC of the oxide surface, neutral species were heavily involved in monocarboxylate binding, while anionic species drove dicarboxylate binding. Also, that among amino acids 1) positional isomers show distinctive binding characteristics to each other while enantiomers show no significant differences in binding characteristics, 2) molecules that bind via outer-sphere complexation show a larger entropic shift between binding and debinding with no impact on oxide surface while 3) inner-sphere interactions increased anion exchange capacity of the oxide surface. The new insights and data from this work has great potential for improving predictive modeling of carbon dynamics and specifically organic-mineral interactions in environmental and industrial systems.

54 ENVIRONMENTAL SCIENCES↗

A robust hierarchical solver for ill-conditioned systems with applications to ice sheet modeling

A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strong-coupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using a thousand processors.

97 MATHEMATICS AND COMPUTING↗

Experimental and modeling analysis of detonation in circular arcs of the conventional high explosive PBX 9501

We examine the diffraction dynamics of a two-dimensional (2D) detonation in a circular arc of the conventional HMX-based, high performance, solid explosive PBX 9501, for which the detonation reaction zone length scale is estimated to be of the order of 100–150 µm. In this configuration, a steady propagating detonation will develop, sweeping around the arc with constant angular speed. We report on results from three PBX 9501 arc experiments, exploring the variation in linear speed on the inner and outer arc surfaces for the steady wave along with the structure of the curved detonation front, as a function of varying inner surface radius and arc thickness. Comparisons of the properties of the motion of the steady wave for each arc configuration are then made with a spatially-distributed PBX 9501 reactive burn model, calibrated to detonation performance properties in a 2D planar slab geometry. We show that geometry-induced curvature of the detonation near the inner arc surface has a significant effect on the detonation motion even for conventional high explosives. We also examine the detonation driving zone structure for each arc case, and thus the subsonic regions of the flow that determine the influence of the arc geometry on the detonation propagation. In addition, streamline paths and reaction progress isolines are calculated. Overall, we conclude that a common approximation for modeling conventional high explosive detonation, wherein the shock-normal detonation speed is assumed equal to the Chapman–Jouguet speed, can lead to significant errors in describing the speed at which the detonation propagates.

42 ENGINEERING↗

Revealing the Formation Energy–Exfoliation Energy–Structure Correlation of MAB Phases Using Machine Learning and DFT

MAB phases are popular as high-temperature materials with high damage tolerance and excellent electrical conductivity that are used to exfoliate two-dimensional (2D) transition metal borides (MBenes); a promising material for developing next- generation nanodevices. In this report, we explore the correlation between the formation energy, exfoliation energy and structural factors of MAB phases using DFT and machine learning. In order to do so, we developed three different machine learning models based on the support vector machine, deep neural network, and random forest regressor to study the stability of the MAB phases by calculating their formation energies. Furthermore, our support vector machine and deep neural network models are capable of predicting the formation energies with mean absolute errors less than 0.1 eV/atom. We demonstrated that the stability of a MAB phase for a given transition metal decreases when the A element changes from Al to Tl. The density functional theory (DFT) re- vealed that M-A and M-B bond strength strongly correlates with the stability of MAB phases. In addition, the exfoliation possibility of 2D MBenes becomes higher when the A element changes from Al to Tl due to weakening of M-A and M-B bonds.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Automatic Time Step Control to Resolve Hydromechanically Driven Fault Reactivation, Spontaneous Nucleation, and Seismic Arrest

Abstract A physical understanding of the progression from flow‐driven (quasi‐static) poromechanical deformation to dynamic fault rupture is critical to the resilient operations of several engineering systems. These processes are bridged by a progression from fault reactivation to the spontaneous nucleation of unstable sliding. Toward addressing this challenge, novel automatic time step size control methods are developed to enable accurate and efficient simulation of these dynamics and transitions from the first principles. The controllers combine local models for discretization error and Coulomb failure conditions to automatically adjust the time step size across several orders of magnitude. The methods do not require additional empirical or theoretical input and can resolve the pre‐rupture, interseismic, and seismic periods to the allowed accuracy. The computational results reveal that the proposed methods automatically capture the onset of reactivation and nucleation for homogeneous and heterogeneous fields. Hydrodynamic and structural heterogeneity lead to disparate critical nucleation sizes compared to those predicted by theory. The results highlight its potential in predicting induced seismicity in realistic subsurface engineering systems and at practical scales.

Environmental Sciences & Ecology↗

An Adaptive Mesh Algorithm: Mesh Structure and Generation

The purpose of Adaptive Mesh Refinement is to minimize spatial errors over the computational space not to minimize the number of computational elements. The additional result of the technique is that it may reduce the number of computational elements needed to retain a given level of spatial accuracy. Adaptive mesh refinement is a computational technique used to dynamically select, over a region of space, a set of computational elements designed to minimize spatial error in the computational model of a physical process. The fundamental idea is to increase the mesh resolution in regions where the physical variables are represented by a broad spectrum of modes in k-space, hence increasing the effective global spectral coverage of those physical variables. In addition, the selection of the spatially distributed elements is done dynamically by cyclically adjusting the mesh to follow the spectral evolution of the system. Over the years three types of AMR schemes have evolved; block, patch and locally refined AMR. In block and patch AMR logical blocks of various grid sizes are overlaid to span the physical space of interest, whereas in locally refined AMR no logical blocks are employed but locally nested mesh levels are used to span the physical space. The distinction between block and patch AMR is that in block AMR the original blocks refine and coarsen entirely in time, whereas in patch AMR the patches change location and zone size with time. The type of AMR described herein is a locally refi ned AMR. In the algorithm described, at any point in physical space only one zone exists at whatever level of mesh that is appropriate for that physical location. The dynamic creation of a locally refi ned computational mesh is made practical by a judicious selection of mesh rules. With these rules the mesh is evolved via a mesh potential designed to concentrate the nest mesh in regions where the physics is modally dense, and coarsen zones in regions where the physics is modally sparse.

97 MATHEMATICS AND COMPUTING↗