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TURBOMAT-FR: Turbomachinery Aeroelastic Forced Response Analysis Automation Using MATLAB®: PART 1

The primary aim of this effort is to develop a software tool named “Turbomachinery Aeroelastic Analysis Tool for Forced Response (TURBOMAT-FR)” using MATLAB® (The MathWorks, Inc.) scripts. The tool will aid in performing routine forced response analysis of turbomachinery fan blades with the aid of the modal summation method (MSUM). The MSUM approach requires a modal solution consisting of modal displacements, modal frequencies, and modal stresses. The MATLAB® scripts facilitate easy extraction of the required quantities from the computational fluid dynamics (CFD) and MSC Nastran (Hexagon AB) solution files. These quantities are subsequently utilized by the MSUM approach to perform a complete forced response analysis.The MATLAB® scripts in TURBOMAT-FR are utilized in postprocessing of the MSC Nastran finite element analysis outputs in three types of solutions or steps, namely (1) stresses and displacements from static and steady response solution, (2) unsteady stresses and displacements from direct forced response solution, and (3) modal solution containing modal displacements and modal stresses. In step (4), separate MATLAB® scripts are also developed to read the processed outputs in step (3) and perform forced response analysis using MSUM. The TURBOMAT-FR MATLAB® scripts mentioned previously are presented in two reports designated “Part 1” and “Part 2”. The current report, Part 1, presents the scripts needed to accomplish steps (1) and (2), noted previously. This will be exclusively useful for performing only the static and steady response analysis and the forced response analysis using the direct forced response solution approach. Part 2 contains the TURBOMAT-FR MATLAB® scripts needed to accomplish steps (3) and (4). The MATLAB® scripts given in Part 2 facilitate the calculation of unsteady stresses using MSUM and provide an automated procedure associated with aeroelastic forced response analysis with different pressure files for the same structural model. The unsteady stresses calculated using MSUM in Part 2 can be compared with those presented in Part 1. A cantilevered blade-alone model subjected to a realistic pressure loading to a boundary was used for static and unsteady direct forced response analysis to demonstrate the effectiveness of the TURBOMATFR MATLAB® scripts.

MATLAB-Scripts↗

Stress, and pathogen response gene expression in modeled microgravity

Purpose: Immune suppression in microgravity has been well documented. With the advent of human exploration and long-term space travel, the immune system of the astronaut must be optimally maintained. It is important to investigate the expression patterns of cytokine genes, because they are directly related to immune response. Heat shock proteins (HSPs), also called stress proteins, are a group of proteins that are present in the cells of every life form. These proteins are induced when a cell responds to stressors such as heat, cold and oxygen deprivation. Microgravity is another stressor that may regulate HSPs. Heat shock proteins trigger immune response through activities that occur both inside the cell (intracellular) and outside the cell (extracellular). Knowledge about these two gene groups could lead to establishment of a blueprint of the immune response and adaptation-related genes in the microgravity environment. Methods: Human peripheral blood cells were cultured in 1g (T flask) and modeled microgravity (MMG, rotating-wall vessel) for 24 and 72 hours. Cell samples were collected and subjected to gene array analysis using the Affymetrix HG_U95 array. Data was collected and subjected to a two-way analysis of variance. The genes related to immune and stress responses were analyzed. Results and Conclusions: HSP70 was up-regulated by more than two fold in microgravity culture, while HSP90 was significantly down-regulated. HSP70 is not typically expressed in all kinds of cells, but it is expressed at high levels in stress conditions. HSP70 participates in translation, protein translocation, proteolysis and protein folding, suppressing aggregation and reactivating denatured proteins. Increased serum HSP70 levels correlate with a better outcome for heat-stroke or severe trauma patients. At the same time, elevated serum levels of HSP70 have been detected in patients with peripheral or renal vascular disease. HSP90 has been identified in the cytosol, nucleus and endoplasmic reticulum, and exists in many tissue types. HSP90 associates with actin filaments in certain conditions and aids cell motility. The down-regulation of HSP90 could lead to deleterious effects in the lymphocytes, thereby contributing to suppressed immune function in microgravity. Interleukins such as IL 1 alpha, IL11 receptor chain alpha, IL7R, and IL4R were significantly down regulated in modeled microgravity. Further analysis of the genes involved in immune response at the protein level may provide a basis for prophylactic and countermeasure strategies to augment the human immune system for space exploration.

Sundaresan, Alamelu↗

Solving Two-Equation Turbulence Models With a Perspective on Solving Transport Equations

There are three principal objectives of this report. The first objective is to investigate frequently used two-equation models when solving the RANS equations. In this study, we consider the 2006Wilcox and the 2003 Menter Shear Stress Transport (SST) models. Also, a simple change of the Wilcox model is introduced to improve computational predictions for transonic flows. Computations for flows over two different airfoils are examined to compare these turbulence models. Effects on modeling the physics of each flow due to variations in the models, such as using either strainrate or vorticity in the turbulence production term, are considered and discussed. The second objective of this report is to not only explore but also evaluate the performance of the solution algorithm for the mean flow and the transport equations. The RANS and turbulence modeling equations aresolved in a weakly coupled manner with a diagonal implicit Runge-Kutta (DIRK) solution algorithm. Throughout this report, emphasis is given to reducing the residuals to machine zero in all flow calculations, so as to eliminate the error due to numerical integration of the discrete governing equations. The final objective is to provide a perspective on solving transport equations for turbulence modeling. Aspects of solving such stiff systems of equations, as well as various numerical difficulties and possible techniques to overcome them, are considered. Discussion is also provided concerning what is called ’numerical compatibility’, which is an essential requirement when designing a solution algorithm for solving transport equations.

turbulence modeling↗

Large-Eddy Simulation of Flow Over Boeing Gaussian Bump Using Multiagent Reinforcement Learning Wall Model: Preprint

We develop a wall model for large-eddy simulation (LES) that takes into account various pressure-gradient effects using multi-agent reinforcement learning. The model is trained using low-Reynolds-number flow over periodic hills with agents distributed on the wall at various computational grid points. It utilizes a wall eddy-viscosity formulation as the boundary condition to apply the modeled wall shear stress. Each agent receives states based on local instantaneous flow quantities at an off-wall location, computes a reward based on the estimated wall-shear stress, and provides an action to update the wall eddy viscosity at each time step. The trained wall model is validated in wall-modeled LES of flow over periodic hills at higher Reynolds numbers, and the results show the effectiveness of the model on flow with pressure gradients. The analysis of the trained model indicates that the model is capable of distinguishing between the various pressure gradient regimes present in the flow. To further assess the robustness of the developed wall model, simulations of flow over the Boeing Gaussian bump are conducted at a Reynolds number of 2 x 10^6, based on the free-stream velocity and the bump width. The results of mean skin friction and pressure on the bump surface, as well as the velocity statistics of the flow field, are compared to those obtained from equilibrium wall model (EQWM) simulations and published experimental data sets. The developed wall model is found to successfully capture the acceleration and deceleration of the turbulent boundary layer on the bump surface, providing better predictions of skin friction near the bump peak and exhibiting comparable performance to the EQWM with respect to the wall pressure and velocity field. We also conclude that the subgrid-scale model is crucial to the accurate prediction of the flow field, in particular the prediction of separation.

boundary layer↗

Entrance and exit region friction factor models for annular seal analysis

The Mach number definition and boundary conditions in Nelson's nominally-centered, annular gas seal analysis are revised. A method is described for determining the wall shear stress characteristics of an annular gas seal experimentally. Two friction factor models are developed for annular seal analysis; one model is based on flat-plate flow theory; the other uses empirical entrance and exit region friction factors. The friction factor predictions of the models are compared to experimental results. Each friction model is used in an annular gas seal analysis. The seal characteristics predicted by the two seal analyses are compared to experimental results and to the predictions of Nelson's analysis. The comparisons are for smooth-rotor seals with smooth and honeycomb stators. The comparisons show that the analysis which uses empirical entrance and exit region shear stress models predicts the static and stability characteristics of annular gas seals better than the other analyses. The analyses predict direct stiffness poorly.

Elrod, David Alan↗

Modeling Step-Strain Relaxation and Cyclic Deformations of Elastomers

Data for step-strain relaxation and cyclic compressive deformations of highly viscous short elastomer cylinders are modeled using a large strain rubber viscoelastic constitutive theory with a rate-independent friction stress term added. In the tests, both small and large amplitude cyclic compressive strains, in the range of 1% to 10%, were superimposed on steady state compressed strains, in the range of 5% to 20%, for frequencies of 1 and 10 Hz. The elastomer cylinders were conditioned prior to each test to soften them. The constants in the viscoclastic-friction constitutive theory are determined by employing a nonlinear least-squares method to fit the analytical stresses for a Maxwell model, which includes friction, to measured relaxation stresses obtained from a 20% step-strain compression test. The simulation of the relaxation data with the nonlinear model is successful at compressive strains of 5%, 10%, 15%, and 20%. Simulations of hysteresis stresses for enforced cyclic compressive strains of 20% +/- 5% are made with the model calibrated by the relaxation data. The predicted hysteresis stresses are lower than the measured stresses.

Johnson, A.R.↗

Data-driven closure modeling for hypersonic turbulent flows

The Reynolds-averaged Navier–Stokes (RANS) equations remain a workhorse technology for simulating compressible fluid flows of practical interest. Due to model-form errors, however, RANS models can yield erroneous predictions that preclude their use on mission-critical problems. This report summarizes work performed from FY22-FY24 focused on improving RANS models for hypersonic flows using data-driven modeling and scientific machine learning. In this work we: 1. Investigate the current capabilities of RANS models in Sandia’s parallel aerodynamics and re-entry code (SPARC) for hypersonic flows with a focus on shock boundary layer interactions (SBLIs), 2. Assess several established corrections that exist in the literature aimed at improving predictions for SBLIs, 3. Develop improved models for the Reynolds stress tensor using tensor-basis neural networks, 4. Develop a neural-network-based variable turbulent Prandtl number model to reduce errors in wall heating in SBLIs. 5. Begin future investigations including employing the LIFE framework to improve wall heating predictions in SBLIs as well as the ensemble Kalman filter. We find that current RANS models in SPARC are deficient for complex SBLI flows. In particular, no current model jointly predicts wall heat flux, wall shear stress, and wall pressure with reasonable accuracy. Existing corrections help, but do not alleviate this issue altogether. The development of improved models for the Reynolds stress tensor via tensor-basis neural networks results in more predictive RANS models across a suite of low-speed and high-speed cases. For hypersonic boundary layers, the inclusion of the wall-normal Reynolds stress via TBNNs has an appreciable impact on the wall-normal momentum balance and wall quantities. However, we find that improvements to the Reynolds stress tensor do not address the over-prediction in wall heat flux in SBLIs. We find that a neural-network-based variable turbulent Prandtl number model systematically and substantially improves wall heating predictions for a range of SBLI cases.

97 MATHEMATICS AND COMPUTING↗

Dynamic fragmentation in impacts - Hydrocode simulation of laboratory impacts

The dynamic fragmentation in impacts into solids is examined using a physical model for the formation and growth of cracks in rocks. The physical model is then inserted into a numerical model (hydrocode) of stress wave propagation and interaction, from which the outcome of a given impact event can be computed. The hydrocode model predicts fragment sizes due to impact in terms of shock waves propagating in a homogeneous elastic medium containing a distribution of crack nucleation centers known as Weibull flaws.

Melosh, H. J.↗

Out of plane gradients in laminated composites

The research under this grant focused on three different, but related, research projects. All three projects were concerned with predicting the interlaminar stresses that develop in laminated composites in the presence of free edges. The first project was dedicated to the development of a new semi-analytical approximate solution for predicting interlaminar stresses. The second project was devoted to a study of the influence of boundary conditions, geometric parameters, and material modelling on interlaminar stresses in stiffened composite panels. The third project was dedicated to the development of a new finite element for efficient analysis of interlaminar stresses in curved, stiffened panels. A summary of each of the projects is given.

Herakovich, Carl T.↗

Viscoelastic analysis of mussel threads reveals energy dissipative mechanisms

Mussels use byssal threads to secure themselves to rocks and as shock absorbers during cyclic loading from wave motion. Byssal threads combine high strength and toughness with extensibility of nearly 200%. Researchers attribute tensile properties of byssal threads to their elaborate multi-domain collagenous protein cores. Because the elastic properties have been previously scrutinized, we instead examined byssal thread viscoelastic behaviour, which is crucial for withstanding cyclic loading. By targeting protein domains in the collagenous core via chemical treatments, stress relaxation experiments provided insights on domain contributions and were coupled with in situ small-angle X-ray scattering to investigate relaxation-specific molecular reorganizations. Results show that when silk-like domains in the core were disrupted, the stress relaxation of the threads decreased by nearly 50% and lateral molecular spacing also decreased, suggesting that these domains are essential for energy dissipation and assume a compressed molecular rearrangement when disrupted. A generalized Maxwell model was developed to describe the stress relaxation response. The model predicts that maximal damping (energy dissipation) occurs at around 0.1 Hz which closely resembles the wave frequency along the California coast and implies that these materials may be well adapted to the cyclic loading of the ambient conditions.

36 MATERIALS SCIENCE↗

Utah FORGE 2439: A Multi-Component Approach to Characterizing In-Situ Stress

Core-based in-situ stress estimation, Triaxial Ultrasonic Velocity (labTUV) data, and Deformation Rate Analysis (DRA) data for Utah FORGE well 16A(78)-32 using triaxial ultrasonic velocity and deformation rate analysis. Report documenting a multi-component approach to characterizing in-situ stress at the U.S. DOE FORGE EGS site: laboratory, modeling and field measurement. Core-based methods for in-situ stress estimation were applied using samples from 5 intervals within the Utah FORGE 16A(78)-32 well. At three of these locations, Triaxial Ultrasonic Velocity (labTUV) tests were performed, resulting in experimentally-determined relationships between wave velocities and stresses. Non-monotonic increase in the velocity-stress relationships are inferred provide evidence of stress history and are therefore used to estimate in-situ stress magnitudes. Additionally, Deformation Rate Analysis (DRA) tests were run on core plugs from various orientations at each of the 5 sampling locations. These, too, provide evidence of stress history based on stress-strain behavior. A novel Weight of Evidence (WoE) method was developed as a means of synthesizing in-situ stress evidence from these two types of tests. Results indicate the minimum horizontal stress gradient ranges from 0.58 psi/ft to 0.69 psi/ft, with 4 of the 5 values between 0.66 psi/ft and 0.69 psi/ft. The vertical stress gradient ranges from 1.05 psi/ft to 1.12 psi/ft, with 4 of the 5 zones given results between 1.09 psi/ft and 1.12 psi/ft. The maximum horizontal stress gradient ranges from 0.98 psi/ft to 1.34 psi/ft, with 4 of the 5 zones falling between 0.98 psi/ft and 1.24 psi/ft. The stress regime thus appears to be on the edge between normal faulting and strike-slip faulting, potentially flipping back and forth between the two regimes due to variability of rock properties, structures such as faults, and/or thermal anomalies.

15 GEOTHERMAL ENERGY↗

On the correlation between the stress exponent for creep determined by nanoindentation and the mechanism of action enabling stress relief in indium

Instrumented indentation performed at room temperature with a Berkovich and 10 μm radius sphere has been used to measure the stress exponent for creep before and after the strain burst observed in well-annealed, high-purity indium. Before the strain burst, the measured values are successfully rationalized using a new model based on stress directed diffusional flow along the interface between the indenter tip and test specimen. After the strain burst, the measured stress exponents are found to be representative of dislocation glide and climb assisted glide. Here these results are compared and contrasted to the previous experimental investigations and modeling efforts of Feng et al., Lucas et al., and Li et al. Collectively, the experimental observations and rationalization presented here provide significant new insight into the mechanisms of action that control the competition for stress relief in small, constrained volumes of crystalline metals subjected to high homologous temperatures.

36 MATERIALS SCIENCE↗

Viscometric flow of dense granular materials under controlled pressure and shear stress

This study examines the flow of dense granular materials under external shear stress and pressure using discrete element method simulations. In this method, the material is allowed to strain along all periodic directions and adapt its solid volume fraction in response to an imbalance between the internal state of stress and the external applied stress. By systematically varying the external shear stress and pressure, the steady rheological response is simulated for: (1) rate-independent quasi-static flow; and (2) rate-dependent inertial flow. The simulated flow is viscometric with non-negligible first and second normal stress differences. While both normal stress differences are negative in inertial flows, the first normal stress difference switches from negative to slightly positive, and second normal stress difference tends to zero in quasi-static flows. The first normal stress difference emerges from a lack of coaxiality between a second-rank contact fabric tensor and strain rate tensor in the flow plane, while the second normal stress difference is linked to an excess of contacts in the shear plane compared with the vorticity direction. A general rheological model of second order (in terms of strain rate tensor) is proposed to describe the two types of flow, and the model is calibrated for various values of interparticle friction from simulations on nearly monodisperse spheres. The model incorporates normal stress differences in both regimes of flow and provides a complete viscometric description of steady dense granular flows.

42 ENGINEERING↗

Shock-wave-induced turbulent boundary-layer separation at hypersonic speeds

An experiment is described that tests and guides computations of the interaction of a shock wave with a turbulent boundary layer. Numerical solutions of the time-averaged Navier-Stokes equations for the entire flow field employing algebraic eddy viscosity and turbulent Prandtl number models for shear stress and heat flux are presented and used to illustrate the dependence of the computations on the particulars of the turbulence models. To guide modifications in the models, the mean flow profiles and surface measurements of pressure, shear, and heat flux are analyzed critically. The results show that the models of eddy viscosity require substantial modifications in the interaction region. Improved solutions employing the experimentally modified models are presented.

Horstman, C. C.↗

Radio science investigations with Mars Observer

Mars Observer radio science investigations focus on two major areas of study: the gravity field and the atmosphere of Mars. Measurement accuracies expressed as an equivalent spacecraft velocity are expected to be of the order of 100 microns/s (for both types of investigations) from use of an improved radio transponder for two-way spacecraft tracking and a highly stable on-board oscillator for atmospheric occultation measurements. Planned gravity investigations include a combination of classical and modern elements. A spherical harmonic (or equivalent) field model of degree and order in the range 30-50 will be obtained, while interpretation will be in terms of internal stress and density models for the planet, using the topography to be obtained from the Mars Observer laser altimeter. Atmospheric investigations will emphasize precision measurement of the thermal structure and dynamics in the polar regions, which are regularly accessible as a result of the highly inclined orbit. Studies based on the measurements will include polar processes, cycling of the atmosphere between the poles, traveling baroclinic disturbances, small-scale waves and turbulence, the planetary boundary layer, and (possibly) the variability and altitude of the ionosphere.

Tyler, G. L.↗

Stress-strain and work hardening relationships of 304L AM alloy

A new approach to analytically derive constitutive stress-strain relationships from modeling the work hardening behavior of alloys was developed for assessing the strength and ductility of the Ti-6Al-4V alloy. This new approach is now successfully applied for assessing the quasi-static stress-strain behavior of an additively manufactured 304L sample. A predictive capability of this modelling approach may then be extended to model material stress-strain behavior at higher strain rates of loading.

36 MATERIALS SCIENCE↗

Hawai‘i Play Fairway (Final Report)

Most of Hawai'i's geothermal resources are blind—their manifestations, such as hot springs and steam vents, do not appear on the ground surface because the heated water flows far below. With the exception of $K\bar{i}lauea East Rift Zone$, in most areas of Hawai'i, high lateral permeability in the first kilometer below ground surface prevents surface thermal features from developing. As a methodology for discovering these blind resources, Play Fairway Analysis (PFA) involves finding potential locations of blind hydrothermal systems and describing potential geothermal sources in rift-zone settings. Using the PFA to find Hawai'i's geothermal resources, the University of Hawai'i (UH) conducted the Hawai‘i Play Fairway Project, Hawai'i's first statewide geothermal resource assessment since 1985. Sponsored by the U.S. Department of Energy, the Hawai'i Play Fairway Project provided an updated resource assessment, a roadmap for additional exploration activities, and the identification of areas for further exploration. Benefitting from UH's core competency in earth sciences and experienced geothermal researchers, the project comprised three phases. During the first phase, the team identified, compiled, and ranked existing geologic, groundwater, and geophysical datasets relevant to subsurface heat, fluid and permeability. Using a Bayesian statistical approach, the team developed a statistical methodology to integrate these data into a resource probability map. The team evaluated the confidence in the probability value and considered development viability of areas with geothermal resources. With these analyses, the team identified 10 locations in the Hawaiian Islands for exploration activities. For the second phase, the team collected new groundwater data in 10 locations across the state and new geophysical data on $L\bar{a}na‘i, Maui$, and central Hawai'i Island and modeled topographically induced stress to better characterize subsurface permeability. Analyzing the subsurface stresses, the team evaluated the potential for fracture-induced permeability. The team inverted the MT and gravity data to produce 3D models of resistivity and density, respectively, on $L\bar{a}na‘i$, across $Haleakal\bar{a}'s$ SW rift (Maui), and surrounding Mauna Kea (Hawai‘i Island). The team developed and applied a new method for incorporating depth information about resistivity, density, and potential for fracture-induced permeability into the statistical method for computing resource probability in these three focus areas. The team incorporated the new groundwater results with the new geophysical results and the calculations of potential for fracture-induced permeability to produce updated maps of resource probability and confidence. Through combining data from the first and second phases, the team determined locations for further exploration during the third phase. For MT and gravity surveys, the team recommended $Kaua'i's$ $L\bar{i}hu'e$ $Basin$, the east rift of $Maui's$ $Haleakal\bar{a}$ volcano, and the southwest rift of Hawai'i Island's Mauna Loa volcano. The MT and gravity surveys aimed to enable improved confidence in the resource potential in these locations. For drilling deep groundwater well(s), the team recommended Southeast Mauna Kea and $L\bar{a}na's$ $P\bar{a}l\bar{a}wai$ $Basin$. During the third phase, further exploration involved drilling a groundwater well in $L\bar{a}na's$ $P\bar{a}l\bar{a}wai$ $Basin$ and performing more geophysical surveys. We deepened an existing water well proximal to our target area on $L\bar{a}na'i$ due to funding constraints that precluded us from spudding a new well that would exceed 1km depth. Drilling was preceded by a number of substantial elements including: writing an Environmental Assessment and the subsequent legal process, performance of deviation logging, lowering a camera down the well, coordinating site preparation with $P\bar{u}lama$ $L\bar{a}na'i$, shipping the UH-owned rig interisland, procuring supplies, and leading 3 community meetings on $L\bar{a}na'i$. Drilling occurred 24/7 the entire month of June 2019 over which time $L\bar{a}na'i$ $Well$ 10 was deepened from 427 m to 1057 m, with continuous core collected. We measured a roughly linear temperature gradient averaging 42°C/km and a maximum bottom hole temperature of 66°C. This gradient is more than twice the background for Hawai'i and within a range of gradients measured in this depth range for some exploration wells within KERZ. We consider these results encouraging for $L\bar{a}na'i's$ resource potential and recommend following with a slim hole within $L\bar{a}na'i's$ caldera (our target zone) to ~ 2 km. Further, the positive implications such results have for the island of O‘ahu are substantial - the shield stage of O'ahu's volcanoes ended 1-2 My earlier. However, O'ahu uses more electricity than the rest of the islands combined, and the utility recently called for 500-700MW of firm, dispatchable renewable electricity on O'ahu by 2033. In Phase 3, we also collected limited new encouraging groundwater data, and updated our thoughts on the probabilities of fluid and permeability at resource depths (Pr F = 1; Pr P = mostly unconstrained). Ultimately, we advocate for using our final probability of heat, and confidence in this probability, to drive the next phase of exploration. We contend further development of geothermal in Hawai‘i will enable the state to achieve its 100% renewable policy objective and Hawai'i to transition off of fossil fuels through geothermal discovery and development. The project not only produced a large amount of data and expanded the existing knowledge of Hawai'i's geothermal resources, but also produced publications, theses, presentations, core photos, datasets, media reports, television interviews, community events, and a blog. Students and new professionals benefitted from the project's hands-on research experiences and educational opportunities and earned awards and recognition.

15 GEOTHERMAL ENERGY↗

Modeling of Ice Flow and Internal Layers Along a Flow Line Through Swiss Camp in West Greenland

An anisotropic ice flow line model is applied to a flow line through Swiss Camp (69.57 N, 49.28 W) in West Greenland to estimate the dates of internal layers detected by Radio-Echo Sounding measurements. The effect of an anisotropic ice fabric on ice flow is incorporated into the steady state flow line model. The stress-strain rate relationship for anisotropic ice is characterized by an enhancement factor based on the laboratory observations of ice deformation under combined compression and shear stresses. By using present-day data of accumulation rate, surface temperature, surface elevation and ice thickness along the flow line as model inputs, a very close agreement is found between the isochrones generated from the model and the observed internal layers with confirmed dates. The results indicate that this part of Greenland ice sheet is primarily in steady state.

Wang, W. L.↗