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At least 163 records · Page 9

On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system

We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier–Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.

Weber, Franziska↗

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. Here this work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Asymptotic-preserving semi-implicit finite volume scheme for extended magnetohydrodynamics

A Finite Volume (FV) scheme is developed for solving the extended magnetohydrodynamic (XMHD) equations, yielding accurate results in the ideal, resistive, and Hall MHD limits. This is accomplished by first re-writing the XMHD equations such that it allows the algorithm to retain the use of ideal MHD Riemann solvers and the constrained transport method to preserve divergence-free magnetic fields. Incorporation of electron inertia and displacement current introduces additional numerical stiffness which motivates a semi-implicit FV scheme that re-formulates the XMHD model as a relaxation system. The equations are then advanced in time using an explicit 2nd-order Runge–Kutta scheme with operator splitting applied to the implicit source term updates at each sub-stage. For additional numerical stability, a density-dependent slope limiter is implemented to increase flux diffusivity at low density regions where non-ideal effects become significant. The algorithm is subsequently implemented in a scalable adaptive mesh refinement (AMR) framework. As the new algorithm retains many aspects of the ideal MHD formulations, it asymptotes naturally to the ideal MHD limit. Moreover, it shows promising results at the resistive and Hall MHD limits. This is verified against reference test problems for ideal, resistive and Hall MHD.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Diagonally implicit Runge–Kutta schemes: Discrete energy-balance laws and compactness properties

Abstract We study diagonally implicit Runge–Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to illustrate cases that satisfy the required structural stability properties and cases that do not. In addition, under some mild structural conditions on the problem we can guarantee compactness of families of discrete solutions with respect to time discretization.

Mathematics↗

Simulation of Multiphase Flow and Poromechanical Effects Around Injection Wells in CO 2 Storage Sites

In geological CO 2 storage operations, wellbore deformations and leakage pathways formations can occur around injection and abandoned wells subjected to high rates and long-term CO 2 injection. To guide engineering design and prevent CO 2 leakage risks, a full understanding of the underlying physics and robust numerical models is necessary to evaluate the response of underground formations in the near wellbore region and in the reservoir. In this study, a multi-scale and multi-physics open-source simulator (GEOS) is used to simulate multiphase flow and poromechanical deformations over time in three dimensions. The governing equations for mechanical deformations of the rock body and multiphase compositional fluid flow within the rock matrix are solved with a fully coupled finite element and finite volume approach. The Drucker–Prager model with friction hardening is applied to simulate elastoplastic deformation and a multiphase fluid model with power-law correlations for relative permeability is used to model the migration of CO 2 plume, which are coupled with numerical implicit scheme. Simulation results are verified against multiple analytical solutions for multiphase flow and wellbore problems, thus demonstrating the accuracy of this advanced simulator. In two engineering applications, here we highlight the impact of elastoplastic deformation and coupled modeling for assessing induced displacements and stress perturbations, which are more pronounced in the near wellbore regions. This work focuses on short-term processes in the vicinity of injection wells where stress evolutions, rock deformations and multiphase compositional flow and transport are simulated jointly to ensure wellbore stability and prevent damage. This fully coupled geomechanical model can simulate multiphase flow and any associated poromechanical effects within the CO 2 storage site and in the surrounding formations. Such a large-scale, long-term, multi-physics simulation model is useful in many ways: it can guide operational decisions for CO 2 injection, assess the containment potential and risks of a site, and analyze the wellbore stability and integrity during and after CO 2 injection.

58 GEOSCIENCES↗

Construction of explicit and implicit symmetric TVD schemes and their applications

A one-parameter family of explicit and implicit total variation diminishing (TVD) schemes is developed which permits incorporation of an expanded group of slope and flux limiters. The numerical technique is intended for use in calculations which include a time-differencing scheme and an optional Lax-Wendroff scheme. Methods of extending the TVD models to nonlinear scalar equations and systems of hyperbolic conservation equations are described. Sample results are presented from calculations of shocked flows around NACA 0012 and NACA 0018 airfoils.

Yee, H. C.↗

Simulation of unsteady blood flows in a patient-specific compliant pulmonary artery with a highly parallel monolithically coupled fluid-structure interaction algorithm

Computational fluid dynamics (CFD) is increasingly used to study blood flows in patient-specific arteries for understanding certain cardiovascular diseases. The techniques work quite well for relatively simple problems, but need improvements when the problems become harder in the case when (1) the geometry becomes complex (from a few branches to a full pulmonary artery), (2) the model becomes more complex (from fluid-only calculation to coupled fluid-structure interaction calculation), (3) both the fluid and wall models become highly nonlinear, and (4) the computer on which we run the simulation is a supercomputer with tens of thousands of processor cores. To push the limit of CFD in all four fronts, in this paper, we develop and study a highly parallel algorithm for solving a monolithically coupled fluid-structure system for the modeling of the interaction of the blood flow and the arterial wall. As a case study, we consider a patient-specific, full size pulmonary artery obtained from CT (Computed Tomography) images, with an artificially added layer of wall with a fixed thickness. The fluid is modeled with a system of incompressible Navier-Stokes equations and the wall is modeled by a geometrically nonlinear elasticity equation. As far as we know this is the first time the unsteady blood flow in a full pulmonary artery is simulated without assuming a rigid wall. We report the proposed numerical algorithm and software scale well beyond 10,000 processor cores on a supercomputer for solving the fluid-structure interaction problem discretized with a stabilized finite element method in space and an implicit scheme in time involving hundreds of millions of unknowns.

97 MATHEMATICS AND COMPUTING↗

A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations

The neutron multigroup diffusion equations is an important approximation of the neutron transport equation, and has been widely used for studying the motion of neutrons and their interactions with the stationary background materials. Solving the multigroup diffusion equations faces challenging because many variables are coupled together and a lot of computing resources are required. In this paper, we focus on the development of a scalable parallel algorithm framework for the system of equations arising from the discretization of the multigroup diffusion equations using a finite element method in space and using an implicit scheme in time. The parallel agorithm framweork consists of an inexact Jacobian-free Newton for the nonlinear equations, a Krylov subspace method for the Jacobian system and a parallel Schwarz preconditioner for the acceleration of the algorithm convergence. A coarse space based on smoothed aggregation is introduced to construct a two-level method to improve the parallel performance of the Schwarz preconditioner. We numerically shown that compared with the one-level method, the proposed two-level method works significantly better in terms of the number of linear iterations and the compute time for a system of equations with millions of unknowns and a supercomputer with thousands of processors.

97 MATHEMATICS AND COMPUTING↗

GPU-accelerated kinetic Hall thruster simulations in WarpX

Abstract Two-dimensional (axial-azimuthal) simulations of a Hall thruster are performed using the open-source particle-in-cell code WarpX. The simulation conditions are chosen to match those of the axial-azimuthal benchmark first reported by Charoy et al. in 2019. A range of numerical and solver parameters is investigated in order to find those which yield the best performance. It is found that WarpX completes the benchmark case in 3.8 days on an Nvidia V100 GPU, and in as low as 1.5 days on a more recent Nvidia H100 GPU. Of the numerical parameters investigated, it is determined that the field-solve tolerance and particle resampling thresholds have the largest effect on the simulation wall time and that particle resampling may artificially widen electron velocity distribution functions, leading to unphysical heating. A semi-implicit scheme for the electrostatic field solve is tested and is found to produce results consistent to within 10% of the benchmark in less than twelve hours. The scaling properties of the electrostatic solver to multiple GPUs are also assessed on a uniform plasma test problem. The results of this work are discussed in the context of advancements in GPU hardware and the suitability of kinetic Hall thruster simulations for engineering applications.

Marks, Thomas A.↗

Verification of iterative matrix solutions for multipoint kinetics equations

Multipoint kinetic equation systems have been solved numerically using matrix algebra software and 1st to 4th order implicit schemes based on single-step matrix propagators. These matrix schemes have been validated successfully on demanding point kinetics benchmarks with various prescribed reactivity insertions. Verification tests have also been performed on a simple 3-region fast reactor core model with asymmetric step or ramp reactivity insertions and cross-checked with the multipoint kinetics SACRE code developed at INL and based on existing and validated stiff ODE solver packages. Accurate results can be obtained at a limited time expense. The reason for intriguing results obtained in some transients has been elucidated and linked to the multipoint matrix coefficient interpolation method during the transient.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

An enriched finite element model with q-refinement for radiative boundary layers in glass cooling

Radiative cooling in glass manufacturing is simulated using the partition of unity finite element method. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary simplified P{sub 1} approximation for the radiation in non-grey semitransparent media. To integrate the coupled equations in time we consider a linearly implicit scheme in the finite element framework. A class of hyperbolic enrichment functions is proposed to resolve boundary layers near the enclosure walls. Using an industrial electromagnetic spectrum, the proposed method shows an immense reduction in the number of degrees of freedom required to achieve a certain accuracy compared to the conventional h-version finite element method. Furthermore the method shows a stable behaviour in treating the boundary layers which is shown by studying the solution close to the domain boundaries. The time integration choice is essential to implement a q-refinement procedure introduced in the current study. The enrichment is refined with respect to the steepness of the solution gradient near the domain boundary in the first few time steps and is shown to lead to a further significant reduction on top of what is already achieved with the enrichment. The performance of the proposed method is analysed for glass annealing in two enclosures where the simplified P{sub 1} approximation solution with the partition of unity method, the conventional finite element method and the finite difference method are compared to each other and to the full radiative heat transfer as well as the canonical Rosseland model.

97 MATHEMATICS AND COMPUTING↗

Development of the FHR advanced natural circulation analysis code and application to FHR safety analysis

The University of California, Berkeley (UCB) is performing thermal hydraulics safety analysis to develop the technical basis for design and licensing of fluoride-salt-cooled, high-temperature reactors (FHRs). FHR designs investigated by UCB use natural circulation for emergency, passive decay heat removal when normal decay heat removal systems fail. The FHR advanced natural circulation analysis (FANCY) code has been developed for assessment of passive decay heat removal capability and safety analysis of these innovative system designs. The FANCY code uses a one-dimensional, semi-implicit scheme to solve for pressure-linked mass, momentum and energy conservation equations. Graph theory is used to automatically generate a staggered mesh for complicated pipe network systems. Heat structure models have been implemented for three types of boundary conditions (Dirichlet, Neumann and Robin boundary conditions). Heat structures can be composed of several layers of different materials, and are used for simulation of heat structure temperature distribution and heat transfer rate. Control models are used to simulate sequences of events or trips of safety systems. A proportional-integral controller is also used to automatically make thermal hydraulic systems reach desired steady state conditions. A point kinetics model is used to model reactor kinetics behavior with temperature reactivity feedback. The underlying large sparse linear systems in these models are efficiently solved by using direct and iterative solvers provided by the SuperLU code on high performance machines. Input interfaces are designed to increase the flexibility of simulation for complicated thermal hydraulic systems. In conclusion, this paper mainly focuses on the methodology used to develop the FANCY code, and safety analysis of the Mark 1 pebble-bed FHR under development at UCB is performed.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Complex Dependence of Calcite Crack Kinetics on Salinity: The Role of DLVO and Hydration Forces

Abstract Subcritical crack growth (SCG) plays an important role in many geological processes such as delayed earth rupture and rock weathering. The complex dependency of SCG on the in‐crack fluid chemistry, however, is still poorly understood. In this study, we utilize the newly developed surface force‐based fracture theory (SFFT) to elucidate the relative contributions of surface forces and solute transport to the crack growth kinetics of calcite in NaCl solutions. Expanding on Barenblatt's cohesive crack model, SFFT introduces an effective stress intensity at the crack tip that encompasses all the relevant intermolecular forces across the crack in addition to the external far‐field stresses. The nonlinear system of equations portraying the crack opening profile, the solute distribution in a propagating crack, and the crack growth velocity are numerically solved via an implicit scheme. After carefully calibrating the model for calcite‐water systems, the SFFT is used to predict the SCG response of calcite at different NaCl concentrations, based on various hypotheses. These predictions are then compared to existing SCG data from the literature. We demonstrate that the experimentally observed variation of SCG rate with NaCl concentration cannot be explained solely by DLVO forces (electrostatic and Van der Waals interactions). This can be remediated by introducing an exponentially decaying hydration force with a nonlinear, nonmonotonic dependence on NaCl concentration. Furthermore, we demonstrate that accounting for both diffusive and advective transport of ions is important in explaining the absence of a stage‐II SCG response for calcite in electrolyte solutions. Plain Language Summary Subcritical crack growth (SCG) refers to the slow propagation of cracks in materials under a stress below the threshold for catastrophic failure. SCG is a key process in many geological events, for example, delayed earth ruptures and rock weathering. New initiatives such as underground CO 2 and H 2 storage in carbonate reservoirs further call for better understanding of SCG in carbonate minerals subjected to varying fluid chemistry. This study examines the SCG of calcite, a key mineral found in carbonate rocks, intergranular cement in sandstones, and filling material in mineral veins and faults, determining their deformation and strength. A mathematical model is developed to describe how the crack opens and propagates, how solutes (like salts) distribute within the crack, and how the crack surfaces interact with each other. We used the model to predict calcite SCG in water at different salt concentrations and compared it with experimental data. Our results revealed that the hydration force is the dominating factor in determining the complex, non‐linear dependency of SCG on salinity. We also found that both the movement of ions by diffusion and by bulk water flow are crucial for explaining the SCG rates, especially when the cracks grow quickly. Key Points Surface Force‐Based Fracture Theory predicts the complex subcritical crack growth patterns of calcite crystals immersed in NaCl solutions Results highlight the dominant role of hydration forces in altering the fracture behavior of calcite compared to VdW and electric double‐layer forces Advective solute transport explains the absence of stages‐II and ‐III subcritical crack growth responses in solid‐liquid systems

DLVO↗

Ion temperature and gas pressure effects on the magnetized sheath dynamics during plasma immersion ion implantation

Here, a collisional magnetized plasma with finite ion temperature is considered to examine the effects of the ion temperature and gas pressure on the plasma-sheath dynamics. We use the two-fluid model of plasma-sheath where the nonlinear equations of a dynamic sheath are solved using a full implicit scheme of finite difference method along with some convenient initial and boundary conditions at the plasma center and target. It is found that the ion temperature only has a significant effect on the characteristics of low voltage sheath, while the gas pressure (collision rate) seriously affects the dynamic characteristics of the low and high voltage plasma-sheath. One can see, increasing the ion temperature in low voltage plasma-sheath causes to increase the temporal curve of the ion dose and the ion impact energy on the target, reduces the temporal curve of the sheath width, and has no any effect on the temporal curve of the ion incident angle on the target. However, rising the gas pressure in low and high voltage plasma-sheath reduces all of these temporal curves.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

The effects of electron thermal radiation on laser ablative shock waves from aluminum plasma into ambient air

The effect of electron thermal radiation on 7 ns laser ablative shock waves from aluminum (Al) plasma into an ambient atmospheric air has been numerically investigated using a one-dimensional, three-temperature (electron, ion, and radiation) radiation hydrodynamic code MULTI. The governing equations in Lagrangian form are solved using an implicit scheme for planar, cylindrical, and spherical geometries. The shockwave velocities (V{sub sw}) obtained numerically are compared with our experimental values obtained over the intensity range of 2.0 × 10{sup 10} to 1.4 × 10{sup 11 }W/cm{sup 2}. It is observed that the numerically obtained V{sub sw} is significantly influenced by the thermal radiation effects which are found to be dominant in the initial stage up to 2 μs depending on the input laser energy. Also, the results are found to be sensitive to the co-ordinate geometry used in the simulation (planar, cylindrical, and spherical). Moreover, it is revealed that shock wave undergoes geometrical transitions from planar to cylindrical nature and from cylindrical to spherical nature with time during its propagation into an ambient atmospheric air. It is also observed that the spatio-temporal evolution of plasma electron and ion parameters such as temperature, specific energy, pressure, electron number density, and mass density were found to be modified significantly due to the effects of electron thermal radiation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Comment on ‘Numerical modeling of tokamak breakdown phase driven by pure Ohmic heating under ideal conditions’

In this comment, we point out possible critical numerical flaws of recent particle simulation studies (Jiang et al 2016 Nucl. Fusion 56 126017, Peng et al 2018 Nucl. Fusion 58 026007) on the electrical gas breakdown in a simple one-dimensional periodic slab geometry. We show that their observations on the effects of the ambipolar electric fields during the breakdown, such as the sudden reversal of the ion flow direction, could not be real physical phenomena but resulting from numerical artifacts violating the momentum conservation law. We show that an incomplete implementation of the direct-implicit scheme can cause the artificial electric fields and plasma transports resulting in fallacies in simulation results. We also discuss that their simple plasma model without considering poloidal magnetic fields seriously mislead the physical mechanism of the electrical gas breakdown because it cannot reflect important dominant plasma dynamics in the poloidal plane (Yoo et al 2018 Nat. Commun. 9 3523).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Comment on 'Numerical modeling of tokamak breakdown phase driven by pure Ohmic heating under ideal conditions'

In this comment, we point out possible critical numerical flaws of recent particle simulation studies (Jiang et al 2016 Nucl. Fusion 56 126017, Peng et al 2018 Nucl. Fusion 58 026007) on the electrical gas breakdown in a simple one-dimensional periodic slab geometry. We show that their observations on the effects of the ambipolar electric fields during the breakdown, such as the sudden reversal of the ion flow direction, could not be real physical phenomena but resulting from numerical artifacts violating the momentum conservation law. We show that an incomplete implementation of the direct-implicit scheme can cause the artificial electric fields and plasma transports resulting in fallacies in simulation results. We also discuss that their simple plasma model without considering poloidal magnetic fields seriously mislead the physical mechanism of the electrical gas breakdown because it cannot reflect important dominant plasma dynamics in the poloidal plane (Yoo et al 2018 Nat. Commun. 9 3523).

tokamak↗