Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “dependability modeling”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

A time-dependent chloride diffusion model for predicting initial corrosion time of reinforced concrete with slag addition

The effect of granulated blast furnace slag (GBFS) addition on the threshold chloride concentration (TCC) for rebar corrosion was investigated. A modified diffusion model, coupled with a time-dependent effective diffusion coefficient and surface chloride concentration, was proposed to predict the chloride profile. The corrosion initiation time was estimated based on the model predictions and the measured TCC. The results indicate that adding GBFS decreases the TCC by lowering the pH value of the pore solution. The evolution of corrosion potential and current density is found to obey a 3-parameter Weibull distribution. MIP tests show that adding GBFS contributes to refinement of pore structure by decreasing the fraction of large capillary pores. The time-dependent model exhibits good predictive strength and helps understand how GBFS addition delays the corrosion initiation by retarding the chloride diffusion, though a lower TCC is obtained.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Bayesian inference of nuclear symmetry energy from measured and imagined neutron skin thickness in Sn 116 , 118 , 120 , 122 , 124 , 130 , 132 , Pb 208 , and Ca 48

The neutron skin thickness Δr np in heavy nuclei has been known as one of the most sensitive terrestrial probes of the nuclear symmetry energy E sym (ρ) around $\frac{2}{3}$ of the saturation density ρ 0 of nuclear matter. Existing neutron skin data mostly from hadronic observables suffer from large uncertainties and their extraction from experiments are often strongly model dependent. While waiting eagerly for the promised model-independent and high-precision neutron skin data for 208 Pb and 48 Ca from the parity-violating electron scattering experiments (PREX-II and CREX at JLab as well as MREX at MESA), within the Bayesian statistical framework using the Skyrme-Hartree-Fock model we infer the posterior probability distribution functions (PDFs) of the slope parameter L of the nuclear symmetry energy at ρ 0 from imagined Δr np ( 208 Pb)=0.15, 0.20, and 0.30 fm with a 1σ error bar of 0.02, 0.04, and 0.06 fm, respectively, as well as Δr np ( 48 Ca)=0.12, 0.15, and 0.25 fm, with different 1σ error bar of 0.01 and 0.02 fm, respectively. The results are compared with the PDFs of L inferred using the same approach from the available Δr np data for 116, 118, 120, 122, 124, 130, 132 Sn from hadronic probes. They are also compared with results from a recent Bayesian analysis of the radius and tidal deformability data of canonical neutron stars from GW170817 and NICER. The neutron skin data for Sn isotopes gives L = 45.5 $^{+ 26.5}_{-21.6}$ MeV surrounding its mean value or L = 53 . 4 $^{+ 18.6}_{ -29.5}$ MeV surrounding its maximum a posteriori value, respectively, with the latter smaller than but consistent with the L = 66 $^{+ 12}_{-20}$ MeV from the neutron star data within their 68% confidence intervals. We found that Δr np = 0.17 –0.18 fm in 208 Pb with an error bar of about 0.02 fm leads to a PDF of L compatible with that from analyzing the Sn data. To provide additionally useful information on L extracted from the Δr np of Sn isotopes, the experimental error bar of Δr np in 208 Pb should be at least smaller than 0.06 fm aimed by some current experiments. In addition, the Δr np ( 48 Ca) needs to be larger than 0.15 fm but smaller than 0.25 fm to be compatible with the Sn and/or neutron star results. To further improve our current knowledge about L and distinguish its PDFs in the examples considered, even higher precisions leading to significantly less than ±20 MeV error bars for L at 68% confidence level are necessary.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Platform Of Optimal Experiment Management

The platform of optimal experiment management, POEM, powered with automated machine learning to accelerate the discovery of optimal solutions, and automatically guide the design of experiments to be evaluated. POEM currently supports 1) random model explorations for experiment design, 2) sparse grid model explorations with Gaussian Polynomial Chaos surrogate model to accelerate experiment design ,3) time-dependent model sensitivity and uncertainty analysis to identify the importance features for experiment design, 4) model calibrations via Bayesian inference to integrate experiments to improve model performance, and 5) Bayesian optimization for optimal experimental design. In addition, POEM aims to simplify the process of experimental design for users, enabling them to analyze the data with minimal human intervention, and improving the technological output from research activities.

Wang, Congjian [Idaho National Laboratory (INL), I↗
DOE OSTI2022

Effects of local order parameter dependent transport coefficient in diblock copolymers under applied electric fields

In this work, we present an approach for constructing thermodynamically consistent time-dependent models relevant to thin films of diblock copolymers in applied electric fields. The approach is based on the principles of linear irreversible thermodynamics, and, in this work, it is applied to study the effects of electric fields on thin films of incompressible diblock copolymers. Enforcement of local incompressibility constraint at all times leads to a local order parameter dependent transport coefficient in the model for the diblock copolymers. The dependence of the transport coefficient on the local order parameter is used to relate it with the diffusion constant of Rouse chains and leads to sensitivity of the model to initial conditions. In addition, transient behavior is found to be affected when compared with an ad hoc model assuming a constant transport coefficient. Numerical results such as electric field induced alignment of lamellae domains due to the field are found to be in qualitative agreement with experiments. This approach opens up a systematic way of developing kinetic models for simulating effects of electrolytes added to thin films containing diblock copolymers in the presence of applied electric fields.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Fully implicit crystal plasticity models representing orientations with modified Rodrigues parameters

Here, this work describes a crystal plasticity formulation combining several mathematical, numerical, and implementation choices to produce a highly efficient model. Specifically, the key choices in the implementation are (1) representing orientations with modified Rodrigues parameters, (2) implementing a fully coupled implicit time integration for the elastic stretch, the crystal orientations, and the model internal variables, (3) implementing the model in the NEML2 constitutive modeling framework, based on PyTorch, to vectorize the calculations and port the computation to GPUs and other hardware accelerators, and (4) an exact implementation of the consistent tangent matrix, even for arbitrary coupling to other field variables beyond the displacements, like temperature, neutron fluence, etc. The first two features of the model are, to our knowledge, novel. The paper considers each of these choices individually as well as the final model as a whole. This includes a full description of modified Rodrigues parameters, their advantages over other representations of orientations, the mathematical formulae and tools required to implement a model with modified Rodrigues parameters, and a detailed description of the geometry of the space of modified Rodrigues parameters (in an appendix). It also includes a description of a fully implicit time integration scheme for the orientations and the advantages in representing orientations with modified Rodrigues parameters in implementing such a model. The work then assess, via numerical examples, the advantages of fully coupled implicit time integration versus more common decoupled and explicit time integration schemes. These studies demonstrate the computational advantages of fully coupled integration versus other time integration algorithms, though the performance of the competing models depends on the complexity of the underlying single crystal model. The study concludes by demonstrating that the choice of time integration method affects the sharpness of the predicted texture, with explicit methods for integrating the orientations overestimating texture sharpness and implicit methods underestimating texture sharpness.

Crystal plasticity↗
DOE OSTI2023

Bayesian-Motivated Probabilistic Model of Hurricane-Induced Multimechanism Flood Hazards

Multimechanism floods (MMFs) are caused by the simultaneous occurrence of more than one flood mechanism such as storm surge, precipitation, tides, and waves. MMFs can lead to more severe or differing impacts than single-mechanism floods. As a result, comprehensive risk assessments require the ability to assess the multivariate probabilistic behaviors of hazards from MMFs. Here this study introduces a novel Bayesian-motivated approach for the probabilistic assessment of hurricane-induced hazards from the combination of the surge, precipitation, tides, and river antecedent flow. A Bayesian network (BN) is developed to capture the physical (conditional) relationship between variables and facilitate the generation of a hazard curve for river discharge that captures the contributions from multiple flood drivers. A case study located along the Delaware River is used to illustrate the proposed approach. Five computationally efficient representative predictive models are developed to estimate the conditional distributions required for the BN as a means of demonstrating the overall framework. The predictive models used in this study act as placeholders and can be replaced with more sophisticated and high-fidelity models depending on the desired accuracy level. While the predictive models are intended to be representative and illustrative, the model performance is evaluated using three historical storms that affected the area. Overall, the proposed framework is shown to be transparent, effective, and adaptable.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Investigating effects of relativistic kinematics, dimensionality, interactions, and short-range correlations on the ratio of quartic over quadratic nuclear symmetry energies

While ample evidence for the so-called empirical parabolic law of the Equation of State (EOS) of isospin asymmetric nuclear matter (ANM) has been obtained in many studies within both non-relativistic and relativistic nuclear many-body theories using various interactions, it has been unclear if there is any fundamental physics reason for the small quartic symmetry energy compared to the quadratic one even as the ANM approaches pure neutron matter. Within both relativistic and non-relativistic Free Fermi Gas (FFG) models in coordinate spaces of arbitrary dimension d with and without considering Short-Range Correlations (SRC) as well as non-linear Relativistic Mean Field (RMF) models, we study effects of relativistic kinematics, dimensionality, interactions and SRC on the ratio Ψ(ρ) of quartic over quadratic symmetry energies in ANM EOSs. We found that the ratio Ψ(ρ) in the FFG model depends strongly on the dimension d. While it is very small already in the normal 3D space, it could be even smaller in spaces with reduced dimensions for sub-systems of particles in heavy-ion reactions and/or whole neutron stars due to constraints, collectivities and/or symmetries. Here, we also found that the ratio Ψ(ρ) could theoretically become very large only at the ultra-relativistic limit far above the density reachable in neutron stars. On the other hand, nuclear interaction directly and/or indirectly through SRC-induced high-momentum nucleons affect significantly the density dependence of Ψ(ρ) compared to the relativistic FFG model prediction. The SRC affects significantly not only the kinetic energy of symmetric nuclear matter but also the ratio Ψ(ρ) while the relativistic corrections are found negligible. The results may help better understand the EOS of dense neutron-rich matter.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Microstructure-Dependent Rate Theory Model of Radiation-Induced Segregation in Binary Alloys

Conventional rate theory often uses the mean field concept to describe the effect of inhomogeneous microstructures on the evolution of radiation induced defect and solute/fission product segregation. However, the spatial and temporal evolution of defects and solutes determines the formation and spatial distribution of radiation-induced second phase such as precipitates and gas bubbles/voids, especially in materials with complicated microstructures and subject to high dose radiation. In this work, a microstructure-dependent model of radiation-induced segregation (RIS) has been developed to investigate the effect of inhomogeneous thermodynamic and kinetics properties of defects on diffusion and accumulations of solute A in AB binary alloys. Four independent concentrations: atom A, interstitial A, interstitial B, and vacancy on [A, B] sublattice are used as field variables to describe temporal and spatial distribution and evolution of defects and solute A. The independent concentrations of interstitial A and interstitial B allow to describe their different generation rates, thermodynamic and kinetic properties, and release the assumptions of interstitial generation and sink strength used in the conventional rate theory. Microstructure and concentration dependent chemical potentials of defects are used to calculate the driving forces of defect diffusions. With the model, the effects of defect chemical potentials and mobilities on the RIS in polycrystalline AB model alloys have been simulated. The results demonstrated the model capability in predicting defect evolution in materials with inhomogeneous thermodynamic and kinetic properties of defects. The model can be extended to materials with complicated microstructures such as a wide range of grain size distribution, coating structure and multiphases as well as radiation-induced precipitation subject to severe radiation damage.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Microstructure-dependent rate theory model of defect segregation and phase stability in irradiated polycrystalline LiAlO 2

We report gamma lithium aluminate (LiAlO 2 ) is a breeder material for tritium and is one of key components in a tritium-producing burnable absorber rod (TPBAR). Dissolution and precipitation of second phases such as LiAl 5 O 8 and voids are observed in irradiated LiAlO 2 . Such microstructure changes cause the degradation of thermomechanical properties of LiAlO 2 and affect tritium retention and release kinetics, and hence, the TPBAR performance. In this work, a microstructure-dependent model of radiation-induced segregation (RIS) has been developed for investigating the accumulation of species and phase stability in polycrystalline LiAlO 2 structures under irradiation. Three sublattices (i.e. [Li, Al, V] I [O, V o ] II [Li i , Al i , O i , V i ] III ), and concentrations of six diffusive species (i.e. Li; vacancy of Li or Al at [Li, Al, V] I sublattice, O vacancy at [O, V o ] II sublattice, and Li, Al and O interstitials at [Li i , Al i , O i , V i ] III interstitial sublattices; are used to describe spatial and temporal distributions of defects and chemistry. Microstructure-dependent thermodynamic and kinetic properties including the generation, reaction, and chemical potentials of defects and defect mobility are taken into account in the model. The parametric studies demonstrated the capability of the developed RIS model to assess the effect of thermodynamic and kinetic properties of defects on the segregation and depletion of species in polycrystalline structures and to explain the phase stability observed in irradiated LiAlO 2 samples. The developed RIS model will be extended to study the precipitation of LiAl 5 O 8 and voids and tritium retention by integrating the phase-field method.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Direct sensitivity analysis on the parameterization of crystal plasticity models

Various methods for calibrating crystal plasticity finite element (CPFE) models lead to non-unique input parameter values, which subsequently introduce uncertainty in the predicted mechanical response. Sensitivity analysis (SA) conducted on crystal plasticity models is used to identify how variability in these parameters contribute to output uncertainty. Traditional SA on CPFE parameters uses simplified surrogate models to save computational time. However, the accuracy of the surrogate models depends on the quantity of training data used, and any modeling error can propagate into the SA results, potentially affecting their reliability. In this work, the elementary effects test (EET) method, a global SA technique using direct CPFE simulations was employed, and the results obtained were compared with the First Order Second Moment (FOSM) method. ExaConstit, an open-source GPU-enabled CPFE code, was used to perform the simulations and direct SA. The EET method was accurately able to capture the non-linear effects of all the input parameters on the output and is a valuable approach for reliably attributing parameter sensitivities in CPFE models. Based on the results, efficient strategies to perform future parameter calibration and SA are discussed. Additionally, the SA trends observed in different single crystal orientations closely mirrored the activity of the slip systems.

Elementary Effects Test↗
DOE OSTI2021

Low-energy probes of no-scale SU(5) super-GUTs

We explore the possible values of the μ → eγ branching ratio, BR(μ → eγ), and the electron dipole moment (eEDM), d e , in no-scale SU(5) super-GUT models with the boundary conditions that soft supersymmetry-breaking matter scalar masses vanish at some high input scale, M in , above the GUT scale, M GUT . We take into account the constraints from the cosmological cold dark matter density, Ω CDM h 2 , the Higgs mass, M h , and the experimental lower limit on the lifetime for p → K + ν¯, the dominant proton decay mode in these super-GUT models. Reconciling this limit with Ω CDM h 2 and M h requires the Higgs field responsible for the charge-2/3 quark masses to be twisted, and possibly also that responsible for the charge-1/3 and charged-lepton masses, with model-dependent soft supersymmetry-breaking masses. We consider six possible models for the super-GUT initial conditions, and two possible choices for quark flavor mixing, contrasting their predictions for proton decay with versions of the models in which mixing effects are neglected. We find that τ(p → K + ν¯) may be accessible to the upcoming Hyper-Kamiokande experiment, whereas all the models predict BR(μ → eγ) and d e below the current and prospective future experimental sensitivities or both flavor choices, when the dark matter density, Higgs mass and current proton decay constraints are taken into account. However, there are limited regions with one of the flavor choices in two of the models where μ → e conversion on a heavy nucleus may be observable in the future. Our results indicate that there is no supersymmetric flavor problem in the class of no-scale models we consider.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Bayesian chain graph models to characterize microbe-environment dynamics

Microbiome data require statistical models that can simultaneously decode microbes' reaction to the environment and interactions among microbes. While a multiresponse linear regression model seems like a straight-forward solution, we argue that treating it as a graphical model is problematic given that the regression coefficient matrix does not encode the conditional dependence structure between response and predictor nodes. This observation is especially important in biological settings when we have prior knowledge on the edges from specific experimental interventions that can only be properly encoded under a conditional dependence model. Here, we propose a chain graph model with two sets of nodes (predictors and responses) whose solution yields a graph with edges that indeed represent conditional dependence, thus agreeing with the experimenter's intuition on the average behavior of nodes under treatment. The solution to our model is sparse via the Bayesian linear regression (LASSO). In addition, we propose an adaptive extension so that different shrinkages can be applied to different edges to incorporate edge-specific prior knowledge. Our model is computationally inexpensive through an efficient Gibbs sampling algorithm and can account for binary, counting, and compositional responses via an appropriate hierarchical structure. We test the performance of our model in a variety of simulated datasets, thereby showing superior performance to state-of-the-art approaches. We further apply our model to human gut and soil microbial compositional datasets, and we highlight that CG-LASSO can estimate biologically meaningful network structures in the data.

compositional data↗
DOE OSTI2025

An attention-based neural ordinary differential equation framework for modeling inelastic processes

To preserve strictly conservative behavior as well as model the variety of dissipative behavior displayed by solid materials, we propose a significant enhancement to the internal state variable-neural ordinary differential equation (ISV-NODE) framework. In this data-driven, physics-constrained modeling framework internal states are inferred rather than prescribed. The ISV-NODE consists of: (a) a stress model dependent on observable deformation and inferred internal state, and (b) a model of the evolution of the internal states. The enhancements to ISV-NODE proposed in this work are multifold: (a) a partially input convex neural network stress potential provides polyconvexity in terms of observed strain while leaving the inferred state unconstrained, and (b) an internal state flow model uses common latent features to inform novel attention-based gating and drives the flow of internal state only in dissipative regimes. We demonstrated that this architecture can accurately model dissipative and conservative behavior across an isotropic, isothermal elastic-viscoelastic-elastoplastic spectrum with three exemplars, while maintaining fundamental principles by design.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Building Initial Dynamic System Models for Digital Twins of the Cryogenic Moderator System at the ORNL Spallation Neutron Source

This work describes the initial development of dynamic system models of the cryogenic moderator system (CMS) of the Spallation Neutron Source (SNS) at ORNL as a part of the ORNL LDRD funded project Building TRANSFORM to Accelerate Digital Twin Applications for Nuclear Systems, LOIS 10563. The goal of the work is to start the dynamic system modeling effort with the end goal of using them for real-time applications as digital twins. The CMS is a cryogenic liquid hydrogen flow loop that provides moderation of the neutrons that are generated by the SNS. For optimal neutron production, the CMS needs to maintain a steady and controlled density of cryogenic hydrogen in the moderator section thus requiring precise temperature and pressure control. Due to the varied time scales and system characteristics, control of the system is complex, and diagnostics are also difficult. Difficulty in accessing the flow loop during operations, limited instrumentation and unknown design details of the equipment combine to make the case for having sophisticated digital twin models of the system. Operationally the CMS also provides a strong use case for digital twins due to the constant need of optimization and for troubleshooting/diagnostics. The large amount of data collected which are freely available for using in building the model and verifying and validating the model also makes it a great candidate for a proof-of-concept for digital twins. The project extends ORNL's capacity of development and implementation of the open-source dynamic system modeling tool TRANSFORM for engineering design and digital twin/real-time applications. Specific system configuration data for the CMS have been gathered and an initial dynamic model was created in the TRANSFORM library using Dymola as the solution platform. Models of increasing complexity are created to demonstrate the need for a multi-layered approach in digital twin modeling depending on the scale and phenomena being focused on. The dynamic modeling is shown to bring the dynamic operational aspects to the design process for systems as well as serve as a digital twin to the hardware and allow for models to be tuned and compared against real time operational data. These aims should help to push forward strategic goals of application of digital twins and increase the impact of ORNL systems modeling capabilities with TRANSFORM/Modelica for various advanced energy systems.

42 ENGINEERING↗
DOE OSTI2023

Drivers of Dry Day Sensitivity to Increased CO 2

Persistent precipitation deficits are among the most impactful consequences of global warming. Here we focus on changes in the annual number of dry days (NDD) and in the annual maximum length of dry spells due to a quadrupling of atmospheric CO 2 . We use atmosphere-only simulations to decompose the projected changes into additive contributions. A fast adjustment leads to a global increase in NDD despite notable regional exceptions (e.g., South Asia and Sahel). The effect of the uniform component of the surface ocean warming is model-dependent but shapes the regional distribution of the NDD response in each model. Finally, the ocean warming pattern also contributes to large uncertainties, likely through contrasting changes in large-scale circulation. Our results thus highlight the complexity of the NDD response, with policy-relevant practical implications for mitigation and adaptation strategies.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Austenitic parent grain reconstruction in martensitic steel using deep learning

In this work we develop a deep convolutional architecture to estimate the prior austenite structure from observed martensite electron backscatter diffraction micrographs. A novel data augmentation strategy randomizes the global reference coordinate system which makes it possible to train our model from only four micrographs. The model is much faster than algorithmic approaches and generalizes well when applied to micrographs of a different material. Empirical evidence suggests the efficacy of the model depends on the scale of the microstructure and receptive field of the vision model. Furthermore, this work demonstrates that modern computer vision approaches are well suited for capturing complex spatial-orientation patterns present in orientation imaging micrographs.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Estimation of biquadratic and bicubic Heisenberg effective couplings from multiorbital Hubbard models

Abstract We studied a multi-orbital Hubbard model at half-filling for two and three orbitals per site on a two-site cluster via full exact diagonalization, in a wide range for the onsite repulsion U , from weak to strong coupling, and multiple ratios of the Hund coupling J H to U . The hopping matrix elements among the orbitals were also varied extensively. At intermediate and large U , we mapped the results into a Heisenberg model. For two orbitals per site, the mapping is into a S = 1 Heisenberg model where by symmetry both nearest-neighbor ( S i ⋅ S j ) and ( S i ⋅ S j ) 2 are allowed, with respective couplings J 1 and J 2 . For the case of three orbitals per site, the mapping is into a S = 3/2 Heisenberg model with ( S i ⋅ S j ), ( S i ⋅ S j ) 2 , and ( S i ⋅ S j ) 3 terms, and respective couplings J 1 , J 2 , and J 3 . The strength of these coupling constants in the Heisenberg models depend on the U , J H , and hopping amplitudes of the underlying Hubbard model. Our study provides a first crude estimate to establish bounds on how large the ratios J 2 / J 1 and J 3 / J 1 can be. We show that those ratios appear rather limited and, as a qualitative guidance, we conclude that J 2 / J 1 is less than 0.4 and J 3 / J 1 is less than 0.2, establishing bounds on effective models for strongly correlated Hubbard systems. Moreover, the intermediate Hubbard U regime was found to be the most promising to enhance J 2 / J 1 and J 3 / J 1 .

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Modeling of the high column density systems in the Lyman-Alpha forest

The Lyman-α forests observed in the spectra of high-redshift quasars can be used as a tracer of the cosmological matter density to study baryon acoustic oscillations (BAO) and the Alcock-Paczynski effect. Extraction of cosmological information from these studies requires modeling of the forest correlations. While the models depend most importantly on the bias parameters of the intergalactic medium (IGM), they also depend on the numbers and characteristics of high-column-density systems (HCDs) ranging from Lyman-limit systems with column densities log N HI /1cm -2 > 17 to damped Lyman-α systems (DLAs) with log N HI /1cm -2 > 20.2. These HCDs introduce broad damped absorption characteristic of a Voigt profile. Consequently they imprint a component on the power spectrum whose modes in the radial direction are suppressed, leading to a scale-dependent bias. Using mock data sets of known HCD content, we test a model that describes this effect in terms of the distribution of column densities of HCDs, the Fourier transforms of their Voigt profiles and the bias of the halos containing the HCDs. Our results show that this physically well-motivated model describes the effects of HCDs with an accuracy comparable to that of the ad-hoc models used in published forest analyses. We also discuss the problems of applying the model to real data, where the HCD content and their bias is uncertain.

Lyman alpha forest↗