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DOE OSTI2023

Graph-based quantum response theory and shadow Born–Oppenheimer molecular dynamics

Graph-based linear scaling electronic structure theory for quantum-mechanical molecular dynamics simulations [A. M. N. Niklasson et al., J. Chem. Phys. 144, 234101 (2016)] is adapted to the most recent shadow potential formulations of extended Lagrangian Born–Oppenheimer molecular dynamics, including fractional molecular-orbital occupation numbers [A. M. N. Niklasson, J. Chem. Phys. 152, 104103 (2020) and A. M. N. Niklasson, Eur. Phys. J. B 94, 164 (2021)], which enables stable simulations of sensitive complex chemical systems with unsteady charge solutions. The proposed formulation includes a preconditioned Krylov subspace approximation for the integration of the extended electronic degrees of freedom, which requires quantum response calculations for electronic states with fractional occupation numbers. For the response calculations, we introduce a graph-based canonical quantum perturbation theory that can be performed with the same natural parallelism and linear scaling complexity as the graph-based electronic structure calculations for the unperturbed ground state. Further, the proposed techniques are particularly well-suited for semi-empirical electronic structure theory, and the methods are demonstrated using self-consistent charge density-functional tight-binding theory both for the acceleration of self-consistent field calculations and for quantum-mechanical molecular dynamics simulations. Graph-based techniques combined with the semi-empirical theory enable stable simulations of large, complex chemical systems, including tens-of-thousands of atoms.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2026

Finite deformation implementation of a mixed-mode single-integral type cohesive zone with reorienting surfaces of separation

To model material ductile failure and crack propagation, cohesive zone elements can be embedded along potential fracture paths in a finite element simulation. When damage criteria are met, elements in the mesh decohere, simulating the formation and propagation of a crack. In this paper, we present a novel computational algorithm based on finite deformation theory, essential to modeling crack initiation and growth in solids undergoing large deformations. This new algorithm was formulated within a Lagrangian frame of reference to extend previous cohesive zone algorithms to include modeling crack growth in finite deformation contexts. The local coordinate system, necessary for defining an embedded cohesive zone, is constructed based upon the current configuration and is updated within the nonlinear iteration process, thereby resulting in the convergence of the solution for a growing crack in a large deformation quasi-static setting. The model’s accuracy was demonstrated by comparing finite element model simulation results with the analytic case of a constant surface separation, as shown in the verification examples. The power and efficacy of the algorithm to capture large deformations during crack growth were then demonstrated with a double cantilever beam example case. It indicates that the model can be applied to a variety of physical circumstances for predicting crack initiation and growth with delamination and fracture.

42 ENGINEERING↗

DOE OSTI2023

Shadow energy functionals and potentials in Born–Oppenheimer molecular dynamics

In Born–Oppenheimer molecular dynamics (BOMD) simulations based on the density functional theory (DFT), the potential energy and the interatomic forces are calculated from an electronic ground state density that is determined by an iterative self-consistent field optimization procedure, which, in practice, never is fully converged. The calculated energies and forces are, therefore, only approximate, which may lead to an unphysical energy drift and instabilities. Here, we discuss an alternative shadow BOMD approach that is based on backward error analysis. Instead of calculating approximate solutions for an underlying exact regular Born–Oppenheimer potential, we do the opposite. Instead, we calculate the exact electron density, energies, and forces, but for an underlying approximate shadow Born–Oppenheimer potential energy surface. In this way, the calculated forces are conservative with respect to the approximate shadow potential and generate accurate molecular trajectories with long-term energy stabilities. We show how such shadow Born–Oppenheimer potentials can be constructed at different levels of accuracy as a function of the integration time step, δt, from the constrained minimization of a sequence of systematically improvable, but approximate, shadow energy density functionals. For each energy functional, there is a corresponding ground state Born–Oppenheimer potential. These pairs of shadow energy functionals and potentials are higher-level generalizations of the original “zeroth-level” shadow energy functionals and potentials used in extended Lagrangian BOMD. The proposed shadow energy functionals and potentials are useful only within this extended dynamical framework, where also the electronic degrees of freedom are propagated as dynamical field variables together with the atomic positions and velocities. The theory is quite general and can be applied to MD simulations using approximate DFT, Hartree–Fock, or semi-empirical methods, as well as to coarse-grained flexible charge models.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

An Improved Convection Parameterization with Detailed Aerosol–Cloud Microphysics for a Global Model

Abstract A new microphysical treatment that includes aerosol–cloud interactions and secondary ice production (SIP) mechanisms is implemented in the convection scheme of the Community Atmosphere Model, version 6 (CAM6). The approach is to embed a 1D Lagrangian parcel model in the bulk convective plume of the existing deep convection parameterization. Aerosol activation, growth processes including collision/coalescence, and three processes of SIP mechanisms, two of which are normally overlooked in atmospheric models, are represented in this embedded parcel model. These microphysical processes are treated with a hybrid bin/bulk scheme and a high spatial and temporal resolution for the integration of the embedded parcel in 1D, allowing vertical velocity to determine the microphysical evolution following the in-cloud motion during ascent. Simulations of an observed case (Midlatitude Continental Convective Clouds Experiment) of a mesoscale convective system in Oklahoma, United States, with a single-column model (SCAM) version of CAM, are compared with aircraft in situ and ground-based observations of microphysical properties from the convection and precipitation. Results from the validation show the new microphysical scheme has a good representation of the ice initiation in the bulk convective plume, including the known and empirically quantified pathways of primary and secondary initiation, with benefits for the accuracy of properties of its supercooled cloud liquid. The sensitivity simulations and use of tagging tracers for the validated simulation confirm that the newly included SIP mechanisms are of paramount importance for convective microphysics and can be successfully treated in the global model.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Background-field Method and QCD Factorization

One method for deriving a factorization for QCD processes is to use successive integration over fields in the functional integral. In this approach, we separate the fields into two categories: dynamical fields with momenta above a relevant cutoff, and background fields with momenta below the cutoff. The dynamical fields are then integrated out in the background of the low-momentum background fields. This strategy works well at tree level, allowing us to quickly derive QCD factorization formulas at leading order. However, to extend the approach to higher loops, it is necessary to rigorously define the functional integral over dynamical fields in an arbitrary background field. This framework was carefully developed for the calculation of the effective action in a background field at the two-loop level in the classic paper by Abbott «The Background Field Method Beyond One Loop», Nucl. Phys. B 185 , 189 (1981). Building on this work, I specify the renormalized background-field Lagrangian and define the notion of the quantum average of an operator in a background field, consistent with the “separation of scales” scheme mentioned earlier. As examples, I discuss the evolution of the twist-2 gluon light-ray operator and the one-loop gluon propagator in a background field near the light cone.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Potassium carbonate decomposition modeling within rotating detonation engines for direct power extraction applications

We report that recently, there has been a significant interest in detonation-based combustion systems, such as rotating detonation engines (RDE), due to potential performance advantages in propulsion and energy applications. This includes improvements in thermodynamic efficiency, inherently high gas velocities, and the possibility of achieving an increase in total pressure (i.e. pressure gain) through the combustion process. Due to approximating a constant volume combustion process, these devices also tend to exhibit extremely high local gas temperatures relative to a comparable constant pressure combustor. These advantages overlap well with desirable performance characteristics of direct power extraction (DPE) technologies, such as a magnetohydrodynamic (MHD) generator. Typically, in DPE systems hot combustion products are seeded with an easily ionizable material such as potassium carbonate (K 2 CO 3 ) in order to boost the electrical conductivity. However, due to the short gas residence times within an RDE, it was unclear whether forming an electrically conductive combustion plasma would be feasible for integration with a downstream MHD generator. A model is presented which describes the heating, decomposition, and ionization of solid K 2 CO 3 particles and aqueous solutions of K 2 CO 3 in water, for a given initial particle diameter. This model was combined with available computational fluid dynamics (CFD) data for an oxygen-methane RDE in a one-way coupled Eulerian-Lagrangian framework to predict particle trajectories and the corresponding heating, decomposition, and ionization histories. Electrical conductivities were computed using a previously developed model, and a method was proposed to determine an equivalent average electrical conductivity. Results show that particle sizes below ~30 µm are able to fully decompose before reaching the exit of the RDE. While the one-way coupled nature of the simulations precluded rigorous evaluation of the effects of seed material on detonability, a substantial temperature reduction is expected at the detonation wave front due to heating and decomposition. A preliminary comparison is presented between the RDE and an equivalent constant pressure adiabatic combustor, showing a potential performance advantage for the RDE.

42 ENGINEERING↗

DOE OSTI2024

Froggatt-Nielsen meets the SMEFT

We study the matching of Froggatt-Nielsen theories of flavour onto the Standard Model Effective Field Theory (SMEFT), upon integrating out a heavy Beyond-the-Standard-Model (BSM) scalar ‘flavon’ whose vacuum expectation value breaks an Abelian flavour symmetry at energies Λ FN well above the electroweak scale, Λ FN > Λ SM . We include matching contributions to the infrared d SM = 6 (Warsaw basis) SMEFT sourced from ultraviolet contact terms suppressed up to order 1/${\Lambda}_{\textrm{UV}}^2$ in the Froggatt-Nielsen Lagrangian, where Λ UV > Λ FN is an arbitrary ultraviolet scale where further unspecified BSM particles are dynamical. This includes tree-level (one-loop) ultraviolet diagrams with d FN = 6 (5) effective vertices. We first do so with a toy model, but then generalize our findings to arbitrary Frogatt-Nielsen charges. Our results indicate a rich and non-trivial signature of Froggatt-Nielsen theories on the (otherwise) model-independent operators of the SMEFT, and we briefly speculate on extending our analysis to broader classes of BSM flavour models, e.g. non-Abelian and/or gauged theories. We thus take an important step towards determining how to use rapidly developing theoretical and experimental SMEFT technologies to gain unambiguous insight into the SM’s longstanding fermion flavour puzzle.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Stability of trapped solutions of a nonlinear Schrödinger equation with a nonlocal nonlinear self-interaction potential

This work focuses on the study of the stability of trapped soliton-like solutions of a (1 + 1)-dimensional nonlinear Schrödinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [|Ψ(x,t)| 2 +|Ψ(-x,t)| 2 ] κ where κ is an arbitrary nonlinearity parameter. Although the system with κ = 1 (i.e. fully integrable case) was first reported by Yang (2018 Phys. Rev. E 98 042202), here in the present work, we extend this model to the one in which κ is arbitrary. This allows us to compare the stability properties of the now trapped solutions to previously found solutions of the more usual NLSE with κ ≠ 1 which are moving soliton solutions. We show that there is a simple, one-component, nonlocal Lagrangian and corresponding action governing the dynamics of the system. Using a collective coordinate method derived from the action as well as assuming the validity of Derrick's theorem, we find that these trapped solutions are stable for 0 < κ < 2 and unstable when κ > 2. At the critical value of κ, i.e. κ = 2, the solution can either collapse or blowup linearly in time when q 0 = 0, where q 0 is the center of the initial density ρ(x, t = 0) = ψ*ψ of the solution. For q 0 ≠ 0 the displaced solution collapses. When κ > 2 initial small displacements from the origin also lead to collapse of the wave function. This phenomenon is not seen in the usual NLSE.

collective coordinates↗

DOE OSTI2021

Approaching the self-dual point of the sinh-Gordon model

One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 1/b self-duality of its S-matrix, of which there is no trace in its Lagrangian formulation. Here b is the coupling appearing in the model’s eponymous hyperbolic cosine present in its Lagrangian, cosh(bΦ). In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for b << 1 and intermediate values of b, but as the self-dual point b = 1 is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff Ec dependence, which disappears according only to a very slow power law in Ec. Standard renormalization group strategies — whether they be numerical or analytic — also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of b = 1. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how ‘quantum mechanical’ vs ‘quantum field theoretic’ the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of b as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase b > 1 of the Lagrangian formulation of model may be different from what is naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Joint ptycho-tomography with deep generative priors

Abstract Joint ptycho-tomography is a powerful computational imaging framework to recover the refractive properties of a 3D object while relaxing the requirements for probe overlap that is common in conventional phase retrieval. We use an augmented Lagrangian scheme for formulating the constrained optimization problem and employ an alternating direction method of multipliers (ADMM) for the joint solution. ADMM allows the problem to be split into smaller and computationally more efficient subproblems: ptychographic phase retrieval, tomographic reconstruction, and regularization of the solution. We extend our ADMM framework with plug-and-play (PnP) denoisers by replacing the regularization subproblem with a general denoising operator based on machine learning. While the PnP framework enables integrating such learned priors as denoising operators, tuning of the denoiser prior remains challenging. To overcome this challenge, we propose a denoiser parameter to control the effect of the denoiser and to accelerate the solution. In our simulations, we demonstrate that our proposed framework with parameter tuning and learned priors generates high-quality reconstructions under limited and noisy measurement data.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

A physics-driven Σ-Y atomization model for heavy-duty engine simulations

The atomization of a liquid jet is a multi-scale and multi-physics problem of interest for many engineering applications. Particularly, it drives the fuel-oxidizer mixing and dictates the efficiency of combustion engines. High-fidelity multi-phase simulations remain challenging due to the excessive computational cost required to capture all the atomization scales. Thus, atomization models are necessary to represent sub-grid liquid structures. Here, in this work, a modified Σ-Y model in the context of Eulerian-Lagrangian Spray Atomization (ELSA) is used to transport the surface area density of the spray. The model's predictive performance is assessed under various operating conditions relevant to heavy-duty engines using the Engine Combustion Network (ECN) Spray C and Spray D research-grade injectors. The classic droplet collision formulation of the Σ-Y model alone does not replicate the response of the measured spray surface area to changes in injector, ambient pressure, and injection pressure, requiring individual tuning of the model's parameters. Instead, a transition between dense and dilute spray breakup mechanisms is proposed in terms of the average droplet spacing. The collision breakup mechanism represents the dilute spray, whereas the droplet size in the dense spray is driven by a competition between the integral scale of turbulence and the balance between the turbulent kinetic energy and the surface energy of the droplets. Such an approach minimizes the requirements for model tuning. Moreover, the role of the constants of a compressible standard k-ε RANS model is assessed in the injector's internal flow and external spray simulation framework, and an updated set is proposed. The results are validated against x-ray radiography and Ultra-Small Angle X-ray Scattering (USAXS) data and highlight the predictive capabilities of the proposed physics-driven Σ-Y model, which is compatible with engine simulation turnaround times.

42 ENGINEERING↗

DOE OSTI2021

Simulation of dynamic crystal plasticity with a Lagrangian discontinuous Galerkin hydrodynamic method

Here we present a new Lagrangian modal discontinuous Galerkin (DG) hydrodynamic method that supports a dynamic dislocation based crystal plasticity model for simulating the mechanical behavior of crystallographic materials, both single crystal and polycrystalline, under dynamic conditions. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated by Taylor series polynomials of varying degree. These polynomials describe macro-scale hydrodynamic behavior while their evolution is determined by evaluating the dynamic crystal plasticity model at material points within the element. The dynamic crystal plasticity model is sensitive to the time increment size, with too large of time increments leading to instability in the model. To mitigate this, the temporal evolution of the dynamic crystal plasticity model is achieved with the combination of a sub-incrementing scheme with Heun’s third-order time integration scheme, which is also used to temporally evolve the governing equations. The implementation of the dynamic crystal plasticity model within the DG framework is tested using a 2D approximation of the Taylor impact test with a single crystal material, using quadratic elements that have faces that can bend. In addition to the standard continuous material modeling, we propose a new simulation method that would represent the heterogeneous behavior of polycrystalline microstructures within an element by varying the position and material properties of the material points within the element. This method is demonstrated using random orientation distributions on materials points that are arranged in both structured and random configurations.

42 ENGINEERING↗

DOE OSTI2026

A robust framework for frictional fault contact in geological formations using a stabilized augmented Lagrangian approach

Numerical simulations are essential to evaluate the performance and safety of engineered subsurface systems such as geological carbon storage sites, enhanced geothermal fields, and oil and gas reservoirs. A key challenge lies in accurately modeling the frictional contact behavior along fault surfaces. This problem involves inequality constraints that arise from the physics of frictional slip, requiring specialized numerical methods to handle the resulting highly nonlinear and path-dependent behavior. Here, in this work, we address this challenge using an Augmented Lagrangian Method (ALM) implemented via the Uzawa algorithm. The formulation employs mixed finite element spaces, combining low-order piecewise linear displacements within the 3D domain cells with piecewise constant tractions defined on the fault surfaces. Furthermore, to ensure stability and satisfy the inf-sup condition, the discrete displacement space is enriched with face bubble functions on both sides of the contact interfaces. This approach offers several advantages over other stabilization techniques that rely on additional terms, and it integrates naturally in the Uzawa framework.

58 GEOSCIENCES↗

DOE OSTI2022

Computation of conventional and alternative jet fuel sensitivity to lean blowout

Large Eddy Simulations (LES) are performed to compute the sensitivity of a conventional (A-2) and an alternate bio-jet (C-1) fuel to Lean Blowout (LBO). A realistic aviation gas turbine engine combustor configuration is considered. Reliable experimental LBO data and OH* chemiluminescence data for the conventional and alternate jet fuel in the combustor configuration have recently become available. The present work utilizes a highly automated, on-the-fly meshing strategy, along with adaptive mesh refinement, to demonstrate the feasibility of capturing the realistic combustion processes. A Lagrangian framework, with initial conditions specified using measurements of spray statistics, is used to model the fuel spray. Newly developed compact reaction mechanisms based on fuel surrogates are validated for the A-2 and the C-1 fuels. The compact reaction mechanisms are implemented using a detailed finite rate chemistry solver. Spray statistics computed by the present LES simulations compare well with available measurements at stable flame conditions near the lean blowout limit. The computed shape of the stable flame as represented by line integrated OH concentrations compares well with the experimental OH* chemiluminescence data. Lean blowout is reached by gradually decreasing the fuel flow rate in the computations, similar to that in the experiments. The results of the LES simulations effectively capture the fuel composition effects and estimate the sensitivity of the LBO limits to the fuel type. The computed trends in LBO limits agree within engineering accuracy with the experimental results for conventional and alternative aviation fuels. The methodology for predicting the fuel composition effects on the lean blowout limits in a fully resolved realistic, complex combustor is established for the first time.

Adaptive mesh refinement↗

DOE OSTI2021

Contributions of biofilm-induced flow heterogeneities to solute retention and anomalous transport features in porous media

We report that microbial biofilms are ubiquitous within porous media and the dynamics of their growth influence surface and subsurface flow patterns which impacts the physical properties of porous media and large-scale transport of solutes. Here, a two-dimensional pore-scale numerical model was used to evaluate the impact of biofilm-induced flow heterogeneities on conservative transport. Our study integrates experimental biofilm images of Paenibacillus 300A strain in a microfluidic device packed with cylindrical grains in a hexagonal distribution, with mathematical modeling. Biofilm is represented as a synthetic porous structure with locally varying physical properties that honors the impact of biofilm on the porous medium. We find that biofilm plays a major role in shaping the observed conservative transport dynamics by enhancing anomalous characteristics. More specifically, when biofilm is present, the pore structure in our geometry becomes more spatially correlated. We observe intermittent behavior in the Lagrangian velocities that switches between fast transport periods and long trapping events. Our results suggest that intermittency enhances solute spreading in breakthrough curves which exhibit extreme anomalous slope at intermediate times and very marked late solute arrival due to solute retention. The efficiency of solute retention by the biofilm is controlled by a transport regime which can extend the tailing in the concentration breakthrough curves. These results indicate that solute retention by the biofilm exerts a strong control on conservative solute transport at pore-scale, a role that to date has not received enough attention.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

The geometric theory of charge conservation in particle-in-cell simulations

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Finally, our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Super‐Droplet Method to Simulate Lagrangian Microphysics of Nuclear Fallout in a Homogeneous Cloud

Abstract Nuclear detonations produce hazardous local and global particles or fallout. Predicting fallout size, chemical components, and location is necessary to inform officials and determine immediate guidance for the public. However, existing nuclear detonation fallout models prescribe the particle size distributions based on limited observations. In this work, we apply the super‐droplet method, which is a numerical modeling technique developed for cloud microphysics, to simulate size distributions of particles in a mushroom cloud formed post‐detonation of a nuclear device. We model fallout formation and evolution with homogeneous nucleation and condensation of a single species and a Monte Carlo coagulation algorithm. We verify the numerical methods representing coagulation and condensation processes against analytical test problems. Additionally, we explore several scenarios for the integral system mass and yield in equivalent kilotons (kt) of TNT (trinitrotoluene). The fallout size distribution median diameter d pg follows a scaling law based on the integral system mass m v 0 kg and yield Y kt: nm. We test the effect of cloud turbulence, enhanced nucleation and growth, and vapor volatility with a sensitivity study. The range in median diameter predictions for simulations of historical tests performed over the Pacific encompass the measurements of particles sampled from the cloud caps. Predicted median particle size ranges up to 217, 123, 86, and 35 nm for historical tests with yields of 0.2, 0.7, 2, and 10 Mt, respectively. This work can be expanded in many different directions to build a more predictive model for fallout formation.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗

DOE OSTI2020

Semi-implicit method to solve compressible multiphase fluid flows without acoustic time step restrictions

A new 3D multiphase numerical capability is presented here for simulating multiphase flow regimes at all Mach numbers (M). The new method is a semi-implicit extension of the finite volume discrete equation method (DEM) of Chinnaya et al. (2004), which originally used explicit time-stepping. The capability is also developed to work with another extension of the DEM to moving grids for arbitrary Lagrangian-Eularian (ALE) methods detailed in Dunn (2011). Rather than solving all phase equations simultaneously, the DEM reduces the equations to a system of single-phase Riemann solves, where each phase has its own velocity and thermodynamic state. Exchanges between the phases are modeled through source terms accounting for the phase interactions. Furthermore, since the original multiphase scheme uses an explicit time-advancement scheme, it has time step restrictions dictated by the speed of sound, which limits the model's ability to simulate weakly compressible flows which typically need to be integrated for longer time periods. Here, we extend the current multiphase formulation by implementing a pressure-correcting step to enable implicit calculations and remove acoustic time step limitations. The new semi-implicit algorithm allows use of relatively large time steps compared to an explicit method. Validation and benefits of the new implicit time-step method are illustrated using several examples including weakly compressible flows and strong shock waves. The scheme presented here is general and may be used for a variety of applications which require capabilities for handling multiphase flow at a wide range of Mach numbers. However, the goal of this research is to simulate all stages of high energy explosions, including the shock formation (high Mach numbers) and evolution of the buoyant cloud (low Mach numbers).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗