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NASA NTRS1996

Non-Parabolic Hydrodynamic Formulations for the Simulation of Inhomogeneous Semiconductor Devices

Hydrodynamic models are becoming prevalent design tools for small scale devices and other devices in which high energy effects can dominate transport. Most current hydrodynamic models use a parabolic band approximation to obtain fairly simple conservation equations. Interest in accounting for band structure effects in hydrodynamic device simulation has begun to grow since parabolic models cannot fully describe the transport in state of the art devices due to the distribution populating non-parabolic states within the band. This paper presents two different non-parabolic formulations or the hydrodynamic model suitable for the simulation of inhomogeneous semiconductor devices. The first formulation uses the Kane dispersion relationship ((hk)(exp 2)/2m = W(1 + alphaW). The second formulation makes use of a power law ((hk)(exp 2)/2m = xW(exp y)) for the dispersion relation. Hydrodynamic models which use the first formulation rely on the binomial expansion to obtain moment equations with closed form coefficients. This limits the energy range over which the model is valid. The power law formulation readily produces closed form coefficients similar to those obtained using the parabolic band approximation. However, the fitting parameters (x,y) are only valid over a limited energy range. The physical significance of the band non-parabolicity is discussed as well as the advantages/disadvantages and approximations of the two non-parabolic models. A companion paper describes device simulations based on the three dispersion relationships; parabolic, Kane dispersion and power law dispersion.

Smith, A. W.↗

NASA NTRS1999

Pulsating Hydrodynamic Instability and Thermal Coupling in an Extended Landau/Levich Model of Liquid-Propellant Combustion

Hydrodynamic (Landau) instability in combustion is typically associated with the onset of wrinkling of a flame surface, corresponding to the formation of steady cellular structures as the stability threshold is crossed. In the context of liquid-propellant combustion, such instability has recently been shown to occur for critical values of the pressure sensitivity of the burning rate and the disturbance wavenumber, significantly generalizing previous classical results for this problem that assumed a constant normal burning rate. Additionally, however, a pulsating form of hydrodynamic instability has been shown to occur as well, corresponding to the onset of temporal oscillations in the location of the liquid/gas interface. In the present work, we consider the realistic influence of a nonzero temperature sensitivity in the local burning rate on both types of stability thresholds. It is found that for sufficiently small values of this parameter, there exists a stable range of pressure sensitivities for steady, planar burning such that the classical cellular form of hydrodynamic instability and the more recent pulsating form of hydrodynamic instability can each occur as the corresponding stability threshold is crossed. For larger thermal sensitivities, however, the pulsating stability boundary evolves into a C-shaped curve in the disturbance-wavenumber/ pressure-sensitivity plane, indicating loss of stability to pulsating perturbations for all sufficiently large disturbance wavelengths. It is thus concluded, based on characteristic parameter values, that an equally likely form of hydrodynamic instability in liquid-propellant combustion is of a nonsteady, long-wave nature, distinct from the steady, cellular form originally predicted by Landau.

Margolis, S. B.↗

NASA NTRS1999

Pulsating Hydrodynamic Instability in a Dynamic Model of Liquid-Propellant Combustion

Margolis, Stephen B.↗

NASA NTRS1999

Pulsating Hydrodynamic Instability and Thermal Coupling in an Extended Landau/Levich Model of Liquid-Propellant Combustion: Inviscid Analysis - 1

Hydrodynamic (Landau) instability in combustion is typically associated with the onset of wrinkling of a flame surface, corresponding to the formation of steady cellular structures as the stability threshold is crossed. In the context of liquid-propellant combustion, such instability has recently been shown to occur for critical values of the pressure sensitivity of the burning rate and the disturbance wavenumber, significantly generalizing previous classical results for this problem that assumed a constant normal burning rate. Additionally, however, a pulsating form of hydrodynamic instability has been shown to occur as well, corresponding to the onset of temporal oscillations in the location of the liquid/gas interface. In the present work, we consider the realistic influence of a non-zero temperature sensitivity in the local burning rate on both types of stability thresholds. It is found that for sufficiently small values of this parameter, there exists a stable range of pressure sensitivities for steady, planar burning such that the classical cellular form of hydrodynamic instability and the more recent pulsating form of hydrodynamic instability can each occur as the corresponding stability threshold is crossed. For larger thermal sensitivities, however, the pulsating stability boundary evolves into a C-shaped curve in the (disturbance-wavenumber, pressure-sensitivity) plane, indicating loss of stability to pulsating perturbations for all sufficiently large disturbance wavelengths. It is thus concluded, based on characteristic parameter values, that an equally likely form of hydrodynamic instability in liquid-propellant combustion is of a non-steady, long-wave nature, distinct from the steady, cellular form originally predicted by Landau.

Margolis, Stephen B.↗

DOE OSTI2025

Stemflow Hydrodynamics

Stemflow hydrodynamics is the study of water movement along the exterior surface area of plants. Its primary goal is to describe water velocity and water depth along the stem surface area. Its significance in enriching the rhizosphere with water and nutrients is not in dispute. Yet, the hydrodynamics of stemflow have been entirely overlooked. This review seeks to fill this knowledge gap by drawing from thin film theories to seek outcomes at the tree scale. The depth‐averaged conservation equations of water and solute mass are derived at a point. These equations are then supplemented with the conservation of momentum that is required to describe water velocities or relations between water velocities and water depth. Relevant forces pertinent to momentum conservation are covered and include body forces (gravitational effects), surface forces (wall friction), line forces (surface tension), and inertial effects. The inclusion of surface tension opens new vistas into the richness and complexity of stemflow hydrodynamics. Flow instabilities such as fingering, pinching of water columns into droplets, accumulation of water within fissures due to surface tension and their sudden release are prime examples that link observed spatial patterns of stemflow fronts and morphological characteristics of the bark. Aggregating these effects at the tree‐ and storm‐ scales are featured using published experiments. The review discusses outstanding challenges pertaining to stemflow hydrodynamics, the use of dynamic similarity and 3D printing to enable the interplay between field studies and controlled laboratory experiments.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2026

Schwinger-Keldysh effective action for hydrodynamics with approximate symmetries

We study hydrodynamic theories with approximate symmetries in the recently developed effective action approach on the Schwinger-Keldysh contour. We employ the method of spurious symmetry transformation for small explicit symmetry-breaking parameters to systematically constrain symmetry-breaking effects in the nonequilibrium effective action for hydrodynamics. We apply our method to the hydrodynamic theory of chiral symmetry in quantum chromodynamics at finite temperature and density and its explicit breaking by quark masses. We show that the spurious symmetry and the Kubo-Martin-Schwinger relation dictate that the Ward-Takahashi identity for the axial symmetry, i.e., the partial conservation of axial vector current (PCAC) relation, contains a relaxational term proportional to the axial chemical potential, whose kinetic coefficient is at least of the second order in the quark mass. In the phase where the chiral symmetry is spontaneously broken, and the pseudo-Nambu-Goldstone pions appear as hydrodynamic variables, this relaxation effect is subleading compared to the conventional pion mass term in the PCAC relation, which is of the first order in the quark mass. On the other hand, in the chiral symmetry restored phase, we show that our relaxation term, which is of the second order in the quark mass, becomes the leading contribution to the axial charge relaxation. Therefore, the leading axial charge relaxation mechanism is parametrically different in the quark mass across a chiral phase transition.

Goldstone bosons↗

DOE OSTI2026

Emergent viscous hydrodynamics from a single quantum particle

We investigate an explicit example of how spatial decoherence can lead to hydrodynamic behavior in the late-time, long-wavelength regime of open quantum systems. We focus on the case of a single nonrelativistic quantum particle linearly coupled to a thermal bath of noninteracting harmonic oscillators at temperature T , a la Caldeira and Leggett. Taking advantage of decoherence in the position representation, we expand the reduced density matrix in powers of the off-diagonal spatial components, so that high-order terms are suppressed at late times. Truncating the resulting power series at second order leads to a set of dissipative transient hydrodynamic equations similar to the nonrelativistic limit of equations widely used in simulations of the quark-gluon plasma formed in ultrarelativistic heavy-ion collisions. Transport coefficients are directly determined by the damping constant γ , which quantifies the influence of the environment. The asymptotic limit of our hydrodynamic equations reduces to the celebrated Navier-Stokes equations for a compressible fluid in the presence of a drag force. Furthermore, our results shed new light on the onset of hydrodynamic behavior in open quantum systems where a system with few degrees of freedom is coupled to a large thermal environment.

Hydrodynamics↗

DOE OSTI2024

Relativistic Guiding-Center Motion: Action Principle, Kinetic Theory and Hydrodynamics

We treat the guiding-center dynamics in a varying external Maxwell field using a relativistically covariant action principle which reproduces the known Vandervoort expression for the drift velocity and extends it to curved spacetime. We derive the corresponding kinetic theory and ideal hydrodynamic theory. In contrast to conventional five-equation hydrodynamics, the guiding-center hydrodynamics needs only three equations due to a constraint on the motion across magnetic field. Furthermore, we argue that such a hydrodynamics is applicable to strongly coupled plasmas where kinetic theory fails.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Generalized Hydrodynamics: A Perspective

Conventional hydrodynamics describes systems with few long-lived excitations. In one dimension, however, many experimentally relevant systems feature a large number of long-lived excitations and conserved quantities even at high temperature, because they are proximate to integrable limits. Such models cannot be treated using conventional hydrodynamics. The framework of generalized hydrodynamics (GHD) was recently developed to treat the dynamics of one-dimensional models: It combines ideas from integrability, hydrodynamics, and kinetic theory to come up with a quantitative theory of transport. GHD has successfully settled several long-standing questions about one-dimensional transport; it has also been leveraged to study dynamical questions beyond the transport of conserved quantities and to systems that are not integrable. In this article, we introduce the main ideas and predictions of GHD, survey some of the most recent theoretical extensions and experimental tests of the GHD framework, and discuss some open questions in transport that the GHD perspective might elucidate. Published by the American Physical Society 2025

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Assessing the Role of Hydrodynamics in Enhancing Height-Above-the-Nearest-Drainage Derived Synthetic Rating Curves: A Comparative Study in the Wu River Basin, Taiwan

The conventional approach to generating synthetic rating curves (SRC) using the Height-Above-the-Nearest-Drainage (HAND) method typically relies on the assumption of uniform flow, such as Manning's equation, to establish stage-discharge ratings. The zero-physics application of the uniform flow equation is insufficient for capturing detailed hydraulic features (e.g., backwater effect) and neglects the hydraulic effects from adjacent channels. This lack of hydrodynamic computation can impact the accuracy and effectiveness of riverine flood risk estimation and management. To reduce this foreseeable error, we introduce the HAND-hd workflow, which integrates sophisticated hydrodynamic computations in the production of HAND-based SRC with hydrodynamic features (SRC hd ). The results indicate that SRC hd demonstrates consistent agreement with both gauge observations and benchmark solutions. Additionally, the comparative analysis suggests that SRC hd provides notable improvements in stage-discharge ratings over conventional HAND-based SRCs, particularly in channels with mild bed gradients, where it reduces water stage prediction errors and percent biases. In steeper channel segments, SRC hd maintains comparable accuracy to conventional methods. The comprehensive evaluation in this study emphasizes the potential discrepancies and inaccuracies associated with the adoption of the uniform flow assumption in the conventional HAND-SRCs and addresses the necessity of including hydrodynamic physics in the application of HAND-based SRC (e.g., inundation map) in channels with mild gradients.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Stochastic relativistic viscous hydrodynamics from the Metropolis algorithm

We propose an algorithm for simulating stochastic relativistic fluid dynamics based on Metropolis updates. Each step of the algorithm begins with an update based on ideal hydrodynamics. This is followed by proposing random (spatial) momentum transfers between fluid cells, keeping the total energy fixed. These proposals are then accepted or rejected using the change in entropy as a statistical weight. The algorithm reproduces relativistic viscous hydrodynamics in the “density frame,” which is a formulation of viscous hydrodynamics we review and clarify here. This formulation is first order in time and requires no auxiliary dynamical fields such as Π 𝜇⁢𝜈 . The only parameters are the shear and bulk viscosities and the equation of state. Here, by adopting the 3+1 split of general relativity, we extend the Metropolis algorithm to general space-time coordinates, such as Bjorken coordinates, which are commonly used to simulate heavy-ion collisions.

Hydrodynamic noise↗

DOE OSTI2026

Shock wave formation in the thermosphere by an earthgrazing fireball: Empirical evidence for volatile-enhanced hydrodynamic shielding

Hydrodynamic shielding is a theoretically well-established but observationally elusive and experimentally difficult-to-replicate phenomenon with implications that extend far beyond meteor physics. Rare earthgrazing meteoroids with infrasound signatures that penetrate to the ground can be used to probe hydrodynamic shielding that leads to strong shock formation at high altitude. Here, we report the first coordinated optical and multi-station infrasound observations of a centimeter-scale earthgrazing fireball that generated sustained cylindrical line shock at thermospheric altitudes near 92 km. The event was recorded by numerous optical stations and three infrasound arrays, allowing trajectory reconstruction, ablation behavior, acoustic source localization, and shock characteristics. Optical observations indicate early mechanical erosion and ablation/evaporation at exceptionally low dynamic pressure, consistent with a cometary or a porous, volatile-bearing CM chondritic object. Independent infrasound detections localize shock generation to multiple points along a 164 km trajectory segment near perigee. Weak-shock modeling yields a consistent blast radius of ∼30 m, implying an acoustic-equivalent source size far exceeding the physical dimensions of the ∼45 g nucleus. We demonstrate that classical gas dynamics and ablation-driven hydrodynamic shielding alone cannot account for these observations under ambient thermospheric conditions. We show that volatile release provides the additional flow-field density enhancement required to amplify hydrodynamic shielding, reduce the effective local Knudsen number, and sustain a shock envelope capable of radiating detectable infrasound. Furthermore, these results demonstrate that small, volatile-rich meteoroids can transiently establish continuum-like flow in rarefied environments.

Astrometry↗

DOE OSTI2025

Coupled Aerodynamic and Hydrodynamic Hybrid Simulation of Floating Offshore Wind Turbines

The development and innovation of floating offshore wind energy in the U.S. requires detailed high-fidelity observations and measurements of turbine and platform loading due to wind, waves, and currents. However, full-scale and quasi-full-scale experiments require significant financial and temporal investments for construction, experimental testing, and long-term field campaigns. To support the commercial advancement of the offshore wind energy industry, specialized wind tunnel and wave basin experimental facilities are critical to be able to test FOWT designs at small scale under controlled conditions prior to full-scale deployment. Oregon State University (OSU) is internationally known as a leader in water and energy research, development, and testing. The O.H. Hinsdale Wave Research Laboratory (HWRL) and the Wallace Energy Systems and Renewables Facility (WESRF) at OSU have extensive experience building, modeling, monitoring, controlling, and actuating scaled systems. Experiments on wave-structure interaction have been performed at the HWRL since its establishment in 1972. Studies have included the interaction of waves with coastal structures (breakwaters, seawalls, buildings, cylinders, bridges, fixed foundations of offshore wind turbines, etc.) and with floating structures (e.g., wave energy converters, maneuvering of vessels, etc.). Hinsdale is actively used by marine energy technology developers, both for private testing and OSU-collaborative research projects. However, despite the availability of several large-scale facilities for hydrodynamic testing (at OSU and elsewhere in the U.S.), existing experimental laboratories are generally limited in their ability to accurately generate combined wind and wave conditions. The simulation of both wind and waves in experimental testing is complicated due to a number of constraints, including: [i] incompatible similitude laws governing the wind and waves for scaled experiments, [ii] producing accurate wind over a large enough control volume via fans, and [iii] generating wind that reasonably represents the atmospheric boundary layer in existing wave basins/flumes. Hence, physical test data providing insight into the simultaneous wave- and wind-structure response of floating offshore wind components can be difficult to generate. Given the aforementioned challenges in classic hydrodynamic experiments, the motivation of this project is to establish a real-time hybrid simulation (RTHS) approach that can apply aero- and hydro-dynamic loading by augmenting wave-only experimental facilities with virtual aerodynamic forces through numerical models representing the remaining dynamic forces. RTHS is a physical-numerical approach that partitions a prototype system into physical and numerical sub-assemblies that interact with each other through actuators and sensors in real time. In coupling physical and numerical models, the hybrid simulation approach applied herein is ideal for problems with: (1) structures subjected to different scaling laws, such as floating offshore wind turbines subjected to combined aero/hydro-dynamic loading, (2) structures that are too large or complex to be tested entirely in a laboratory setting, such as deep-water mooring applications, and (3) component testing, where the behavior of a portion of the assembly is uncertain but still interacts with other portions of the structure, such as testing the fatigue life of turbine blades. Few U.S. experimental facilities are able to test simultaneous aero- and hydro-dynamic loading and none can accurately produce aero/hydro-dynamic response on scaled FOWT models due to conflicting similitude laws between the wind (commonly Reynolds) and the waves (commonly Froude). To aid in accelerating the development of the U.S. floating offshore industry, there is a significant need to develop a flexible, modular framework that can expand the capacities of existing wave-only laboratories. The project goal is to demonstrate a hydrodynamic real-time hybrid simulation (hydro-RTHS) framework that couples numerical wind and physical waves acting on a FOWT, thus representing simultaneous aero/hydro-dynamic loading. The FOWT is partitioned into a full-scale numerical sub-assembly associated with the aerodynamics and a model-scale physical sub-assembly associated with the hydrodynamics. The numerical-physical partition associated with hydro-RTHS mitigates scaling constraints by supplying different scaling laws to the physical and numerical sub-assemblies. Herein, length, force, and time are scaled and exchanged between the sub-assemblies using Froude scaling to represent the open-channel flow in the physical sub-assembly. Other similitude laws could also be utilized depending on the problem definition. It is envisioned that the ability to model FOWTs under waves and wind, with mitigation of similitude distortions, would result in reduced development costs (currently, FOWT concept development is performed with full-size pro- totypes at enormous expense and risk) and increase the reliability of the FOWT industry (since extreme wave and wind conditions and contingency events can be tested safely in a controlled environment).

16 TIDAL AND WAVE POWER↗

DOE OSTI2025

A Gigaparsec-scale Hydrodynamic Volume Reconstructed with Deep Learning

The next generation of spectroscopic surveys will map the large-scale structure of the Universe at high redshifts (2 ≤ z ≤ 5) using millions of quasar spectra, enabling major advances in constraining both the standard cosmological model and its extensions. Robust cosmological analyses of these data sets require numerical simulations that both cover gigaparsec volumes and resolve features on ∼10 kpc scales and smaller. However, running such large-volume, high-resolution hydrodynamic simulations is computationally prohibitive. We present a generative deep learning model that enhances a low-resolution, gigaparsec-scale (960 h −1 Mpc) hydrodynamic simulation using a smaller (80 h −1 Mpc) high-resolution input hydrodynamic simulation as training data. The resulting enhanced simulation reproduces the line-of-sight power spectrum to within ∼10% and the three-dimensional power spectrum at the ∼20% level at intermediate to small scales (k ≲ 2 h Mpc −1 ). Our method shows strong promise for producing realistic simulations for cosmological analyses with current surveys such as the Dark Energy Spectroscopic Instrument and upcoming next-generation experiments, but further improvements are needed to accurately recover the large-scale modes. We publicly release the enhanced hydrodynamic simulation, along with a halo catalog from a companion N-body dark matter simulation to support the calibration of data analysis pipelines for these large-scale surveys.

Convolutional neural networks↗

DOE OSTI2025

An efficient hybrid downscaling framework to estimate high-resolution river hydrodynamics

Flow depth and velocity are the most important hydrodynamic variables that govern various river functions, including water resources, navigation, sediment transport, and biogeochemical cycling. Existing high-resolution flow depth simulations rely on either computationally expensive river hydrodynamic models (RHMs) or data-driven models with formidable training costs, whereas data-driven modeling of flow velocity has rarely been explored. Here, using the hybrid Low-fidelity, Spatial analysis, and Gaussian process learning (LSG) model, we developed a downscaling approach to construct high-resolution flow depth and velocity from a two-dimensional (2-D) RHM simulation at coarse resolution. The LSG models were trained and tested in an urban watershed in Houston using two different hurricane-driven flood events. The high-resolution (as fine as 30 m resolution) and low-resolution (mostly 1000 m resolution) meshes include 664 724 and 14 536 grid cells, respectively. The results showed that through downscaling, the simulation errors were reduced to less than one-fourth and one-third of the errors of the low-resolution 2-D RHM for flow depth and velocity, respectively. Our analysis further revealed that the dominant uncertainty sources of the downscaled hydrodynamics are different, with flow velocity dominated by the dimensionality reduction error, which we reduced by using a regionalized training procedure. The downscaling approach achieves an 84-fold acceleration in computational time compared to the high-resolution 2-D RHM, making high-fidelity ensemble flood modeling feasible. More importantly, the developed method provides an opportunity to couple large-scale hydrodynamical processes with local physical, chemical, and biological processes in river models.

Tan, Zeli [Pacific Northwest National Laboratory (↗

NASA NTRS1971

Collisionless solar wind protons: A comparison of kinetic and hydrodynamic descriptions

Kinetic and hydrodynamic descriptions of a collisionless solar wind proton gas are compared. Heat conduction and viscosity are neglected in the hydrodynamic formulation but automatically included in the kinetic formulation. The results of the two models are very nearly the same, indicating that heat conduction and viscosity are not important in the solar wind proton gas beyond about 0.1 AU. It is concluded that the hydrodynamic equations provide a valid description of the collisionless solar wind protons, and hence that future models of the quiet solar wind should be based on a hydrodynamic formulation.

Leer, E.↗

NASA NTRS1989

Experimental investigation of the hydrodynamic forces on the shroud of a centrifugal pump impeller

Fluid-induced forces acting on a rotating impeller are known to cause rotor-dynamic problems in turbomachines. The forces generated by leakage flow along the front shroud surface of a centrifugal turbomachine impeller play an important role among these fluid-induced forces. The present research was aimed to gain a better understanding of these shroud forces. An experimental apparatus was designed and constructed to simulate the impeller shroud leakage flow. Hydrodynamic forces and steady and unsteady pressure distributions on the rotating shroud were measured as functions of eccentricity, width of shroud clearance, face seal clearance and shaft rotating speed. The forces measured from the dynamometer and manometers agreed well. The hydrodynamic force matrices were found skew-symmetric and statically unstable. This is qualitatively similar to the result of previous hydrodynamic volute force measurements. Nondimensionalized normal and tangential forces decrease slightly as Reynolds number increases. As the width of the shroud clearance decreases and/or the eccentricity increases, the hydrodynamic forces increase nonlinearly. There was some evidence found that increased front seal clearance could reduce the radial shroud forces and the relative magnitude of the destabilizing tangential force. Subharmonic pressure fluctuations were also observed which may adversely affect the behavior of the rotor system.

Zhuang, Fei↗

NASA NTRS1993

Explicit 3D continuum fracture modeling with smooth particle hydrodynamics

Impact phenomena shaped our solar system. As usual for most solar system processes, the scales are far different than we can address directly in the laboratory. Impact velocities are often much higher than we can achieve, sizes are often vastly larger, and most impacts take place in an environment where the only gravitational force is the mutual pull of the impactors. The Smooth Particle Hydrodynamics (SPH) technique has been applied in the past to the simulations of giant impacts. In these simulations, the colliding objects were so massive (at least a sizeable fraction of the Earth's mass) that material strength was negligible compared to gravity. This assumption can no longer be made when the bodies are much smaller. To this end, we have developed a 3D SPH code that includes a strength model to which we have added a von Mises yielding relation for stresses beyond the Hugoniot Elastic Limit. At the lower stresses associated with brittle failure, we use a rate-dependent strength based on the nucleation of incipient flaws whose number density is given by a Weibull distribution. Following Grady and Kipp and Melosh et al., we introduce a state variable D ('damage'), 0 less than D less than 1, which expresses the local reduction in strength due to crack growth under tensile loading. Unfortunately for the hydrodynamics, Grady and Kipp's model predicts which fragments are the most probable ones and not the ones that are really formed. This means, for example, that if a given laboratory experiment is modeled, the fragment distribution obtained from the Grady-Kipp theory would be equivalent to a ensemble average over many realizations of the experiment. On the other hand, the hydrodynamics itself is explicit and evolves not an ensemble average but very specific fragments. Hence, there is a clear incompatibility with the deterministic nature of the hydrodynamics equations and the statistical approach of the Grady-Kipp dynamical fracture model. We remedy these shortcomings by making the incipient flaw distribution explicit, i.e., particles carry activation strains which are distributed at random with a probability of occurrence given by the Weibull distribution. If the local principal axis strain exceeds this limit, damage starts to grow. By growing explicit cracks together with statistical cracks (damage) at the sub-particle scale, we ensure that material strength and fragmentation is independent of model resolution. We tested our scheme by simulating laboratory impact experiments on basalt spheres.

Benz, W.↗