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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Numerical modeling of physical vapor transport under microgravity conditions: Effect of thermal creep and stress

One of the most promising applications of microgravity (micro-g) environments is the manufacture of exotic and high-quality crystals in closed cylindrical ampoules using physical vapor transport (PVT) processes. The quality enhancements are believed to be due to the absence of buoyant convection in the weightless environment - resulting in diffusion-limited transport of the vapor. In a typical experiment, solid-phase sample material is initially contained at one end of the ampoule. The sample is made to sublime into the vapor phase and deposit onto the opposite end by maintaining the source at an elevated temperature with respect to the deposit. Identification of the physical factors governing both the rates and uniformity of crystal growth, and the optimization of the micro-g technology, will require an accurate modeling of the vapor transport within the ampoule. Previous micro-g modeling efforts have approached the problem from a 'classical' convective/diffusion formulation, in which convection is driven by the action of buoyancy on thermal and solutal density differences. The general conclusion of these works have been that in low gravity environments the effect of buoyancy on vapor transport is negligible, and vapor transport occurs in a diffusion-limited mode. However, it has been recently recognized than in the non-isothermal (and often low total pressure) conditions encountered in ampoules, the commonly-assumed no-slip boundary condition to the differential equations governing fluid motion can be grossly unrepresentative of the actual situation. Specifically, the temperature gradients can give rise to thermal creep flows at the ampoule side walls. In addition, temperature gradients in the vapor itself can, through the action of thermal stress, lead to bulk fluid convection.

Mackowski, Daniel W.↗
NASA NTRS1987

A creep-rupture model of filament-wound spherical pressure vessels

The creep-rupture model is that of a quasi-isotropic filament-wound spherical pressure vessel, subjected to internal pressurization. The matrix material of the composite system is assumed to be linearly viscoelastic. Internal damage resulting from the relaxation of the matrix and the corresponding increase in microcracks is represented by a functional relationship between circumferential strain and transverse modulus. The numerical solution to this nonlinear problem is an iterative technique, whereby the elastic-viscoelastic correspondence principle is employed. In the Laplace domain, the associated elastic solution is obtained and this solution is inverted by the multidata method to yield the time-dependent solution.

Dozier, Jan D.↗
DOE OSTI2024

Modeling In-Reactor Temperature Gradients in Lithium Aluminate Ceramic Pellets

In reactor thermal performance of lithium aluminate ceramic pellets are calculated to determine operating temperatures if internal components shift during irradiation. A 2D conduction model was setup to model this scenario with additional physics, including thermal expansion of components to reactor temperatures, creep down of the cladding due to differential pressure between the inside of the rod and reactor coolant system, and net radiation exchange between components. The non-linear coupling of the governing equations presented convergence challenges due to the very different effects from a small temperature change on the radiation vs radial and circumferential conduction equations. Initial temperature guesses for the multidimensional solution were provided from conservative 1D models simulating bounding scenarios. Convergence was greatly improved by taking small steps towards the root solutions and averaging with initial guesses. The final method robustly converged all 132 points at an axial plane after a few dozen iterations. The temperature results show that the additional physics have a secondary effect as the dominant heat transfer mechanism is radial conduction across the gas gap followed by circumferential conduction around components. Circumferential conduction in the lithium aluminate pellets is enhanced by the relatively high thermal conductivity of this ceramic.

Carstens, Nathan A.↗
NASA NTRS1983

Engine cyclic durability by analysis and material testing

The problem of calculating turbine engine component durability is addressed. Nonlinear, finite-element structural analyses, cyclic constitutive behavior models, and an advanced creep-fatigue life prediction method called strainrange partitioning were assessed for their applicability to the solution of durability problems in hot-section components of gas turbine engines. Three different component or subcomponent geometries are examined: a stress concentration in a turbine disk; a louver lip of a half-scale combustor liner; and a squealer tip of a first-stage high-pressure turbine blade. Cyclic structural analyses were performed for all three problems. The computed strain-temperature histories at the critical locations of the combustor linear and turbine blade components were imposed on smooth specimens in uniaxial, strain-controlled, thermomechanical fatigue tests of evaluate the structural and life analysis methods.

Kaufman, A.↗
NASA NTRS1984

Engine cyclic durability by analysis and material testing

The problem of calculating turbine engine component durability is addressed. Nonlinear, finite-element structural analyses, cyclic constitutive behavior models, and an advanced creep-fatigue life prediction method called strainrange partitioning were assessed for their applicability to the solution of durability problems in hot-section components of gas turbine engines. Three different component or subcomponent geometries are examined: a stress concentration in a turbine disk; a louver lip of a half-scale combustor linear; and a squealer tip of a first-stage high-pressure turbine blade. Cyclic structural analyses were performed for all three problems. The computed strain-temperature histories at the critical locations of the combustor linear and turbine blade components were imposed on smooth specimens in uniaxial, strain-controlled, thermomechanical fatigue tests of evaluate the structural and life analysis methods.

Kaufman, A.↗
DOE OSTI2025

Using X-ray radiography to study oxygen flow in a proton exchange membrane electrolyzer operating under balanced pressure conditions

Of the various water electrolyzer technologies, the proton exchange membrane electrolyzer (PEMWE) is one of the best solutions for producing clean hydrogen without releasing CO 2 . In order to allow for widespread use of clean hydrogen, it is necessary to decrease its cost, which is intrinsically related to system operation. Current PEMWE plants operate in differential mode, directly pressurizing hydrogen and benefiting from thermodynamic compression, which increases overall system efficiency. However, high differential pressure above 30 bar can cause membrane stress, resulting in membrane creeping and failure. Pressurizing the water and operating at balanced pressure allows hydrogen to be produced at higher pressures while preserving the integrity of the membrane and porous layers. Nevertheless, the impact of pressurizing water on PEMWE performance must be better understood to maximize performance under balanced pressure conditions. Here, this study examined the impact of water pressure on two-phase flow. A high-pressure electrolyzer setup was developed to perform operando X-ray radiography and examine oxygen transport with high temporal resolution. The imaging segmentation process, developed to capture bubble properties in the channel, was applied to a specific experiment. The results clearly showed that as pressure increased up to 30 bars, the initial bubbly flow transitioned to slug flow, which led to channel saturation with oxygen. This work demonstrates that two-phase flow in an electrolyzer can be studied using X-ray radiography, which has the advantages of fast measurements and the ability to probe dense materials, such as those required for pressurized electrolyzers.

Balanced high pressure operation↗
DOE OSTI2020

An Integrated Multiscale Experimental-Numerical Analysis on Reconsolidation of Salt-Clay Mixture for Disposal of Heat-Generating Waste (Final NEUP Technical Report)

The overall purpose of this research is to improve understanding of THMC coupling effect on the reconsolidation of granular (or crushed) salt-clay mixture used for seal systems of shafts and drifts in salt repositories. This proposed work is partially motivated by the recent work on the Waste Isolation Pilot Plant (WIPP) that shows the promising sealing capability of clay-salt mixture compared to crushed salt. In particular, primary emphasis is to develop a fully integrated multiscale experiment-numerical study to determine and explain what leads to the superior sealing ability of the clay-salt mixture. These research activities are designed to seek further understanding of (1) why clay additives may enhance the fluid trapping and (2) whether this flow barrier effect may prevail under different combinations of temperature, confining pressure, deviatoric stress and other foreseeable environmental factors. If successful, this enhanced flow trapping ability of the seal provides significant improvement to the seal and repository performance and therefore make the repository safer in the long-term. The experiment component includes microstructural investigation and macroscopic tests on a reconsolidated salt-clay mixture. In the microstructural study, the goal is to (1) characterize microscopic distributions of distinct phases (e.g., clay, salt crystal boundaries, trapped brine, and pore) to examine the connectivity of the pore network inside the salt-clay mixture with different amounts of clay additive and moisture content and (2) analyze multiscale imaging data to reconstruct the polycrystalline microstructures for numerical simulations. Meanwhile, macroscopic tests are performed to analyze how clay alters the failure/creep mechanisms in the salt-clay mixture. Microscopic and macroscopic experimental observations will both be used to calibrate and validate a multiscale model that explicitly simulates the capillary and multiphase flow in the connected pores and the deformation due to the presence of intra-crystalline brine at the pore scale via a new polyhedral discrete element–lattice Boltzmann method (DEM-LBM) coupling model. The pore-scale simulations are homogenized via an upscaling procedure that converts pore-scale information (e.g. force exerted on grain boundary, sliding, pressure-solution) to continuum measures (e.g. Cauchy stress, Darcy’s flow) at each integration point in the macroscopic multiphase TMHC model. This multiscale scheme will allow coupling be- tween high-fidelity simulations of brine-salt-clay interaction and the macroscopic TMHC model. The multiscale model helps the understanding of how the trapped brine inclusion affects the pressure-solution mechanism with the presence of clay and moisture. This work brings new insight into the sealing capacity of salt-clay mixture under elevated temperature over a long period of time - a key to evaluating the potential of salt-clay mixture usage for salt repositories.

42 ENGINEERING↗
DOE OSTI2021

An Approach to Bayesian Optimization for Design Feasibility Check on Discontinuous Black-Box Functions

The paper presents a novel approach to applying Bayesian Optimization (BO) in predicting an unknown constraint boundary, also representing the discontinuity of an unknown function, for a feasibility check on the design space, thereby representing a classification tool to discern between a feasible and infeasible region. Bayesian optimization is a low-cost black-box global optimization tool in the Sequential Design Methods where one learns and updates knowledge from prior evaluated designs, and proceeds to the selection of new designs for future evaluation. However, BO is best suited to problems with the assumption of a continuous objective function and does not guarantee true convergence when having a discontinuous design space. This is because of the insufficient knowledge of the BO about the nature of the discontinuity of the unknown true function. In this paper, we have proposed to predict the location of the discontinuity using a BO algorithm on an artificially projected continuous design space from the original discontinuous design space. The proposed approach has been implemented in a thin tube design with the risk of creep-fatigue failure under constant loading of temperature and pressure. The stated risk depends on the location of the designs in terms of safe and unsafe regions, where the discontinuities lie at the transition between those regions; therefore, the discontinuity has also been treated as an unknown creep-fatigue failure constraint. The proposed BO algorithm has been trained to maximize sampling toward the unknown transition region, to act as a high accuracy classifier between safe and unsafe designs with minimal training cost. The converged solution has been validated for different design parameters with classification error rate and function evaluations at an average of <1% and ~150, respectively. Finally, the performance of our proposed approach in terms of training cost and classification accuracy of thin tube design is shown to be better than the existing machine learning (ML) algorithms such as Support Vector Machine (SVM), Random Forest (RF), and Boosting.

Engineering↗
DOE OSTI2020

Computational thermomechanics for crystalline rock. Part II: Chemo-damage-plasticity and healing in strongly anisotropic polycrystals

We present a thermal–mechanical–chemical-phase field model that captures the multi-physical coupling effects of precipitation creeping, crystal plasticity, anisotropic fracture, and crack healing in polycrystalline rock at various temperature and strain-rate regimes. This model is solved via a fast Fourier transfer solver with an operator-split algorithm to update displacement, temperature and phase field, and chemical concentration incrementally. In nuclear waste disposal in salt formation, brine inside the crystal salt may migrate along the grain boundary and cracks due to the gradient of interfacial energy and pressure. This migration has a significant implication on the permeability evolution, creep deformation, and crack healing within rock salt but is difficult to incorporate implicitly via effective medium theories compared with computational homogenization. As such, we introduce a thermodynamic framework and a corresponding computational implementation that explicitly captures the brine diffusion along the grain boundary and crack at the grain scale. Meanwhile, the anisotropic fracture and healing are captured via a high-order phase field that represents the regularized crack region in which a newly derived non-monotonic driving force is used to capture the fracture and healing due to the solution–precipitation. Numerical examples are presented to demonstrate the capacity of the thermodynamic framework to capture the multiphysics material behaviors of rock salt.

42 ENGINEERING↗
NASA NTRS1996

Cross-Effects in Microgravity Flows

Film growth by chemical/physical vapor deposition is a process of considerable interest in microgravity experiments. The absence of natural convection should allow better control of film growth processes but, in highly non-isothermal ampoules, thermal slip (creep) can become a matter of significant concern. The reported research is a theoretical and experimental investigation of the flow of gas/vapor mixtures under non-continuum conditions. The Boltzmann equation has been solved for a monatomic gas under non-condensing conditions and the various phenomenological coefficients have been computed. Computations for realistic potentials as well as for velocity and creep slip have been completed and the creep slip has been found to be dependent on the type of gas confirming the accuracy of previous variational results. The variational technique has been extended and planar flows calculated via the Burnett solutions. Velocity, diffusion and creep slips have been computed for gas mixtures and previously unknown dependencies of the creep slip on the mixture properties have been observed. Also for gas mixtures, an integral representation of the linearized Boltzmann operator has been developed for use in numerical and variational calculations for all intermolecular force laws. Two, two-bulb capillary systems have been designed, built and tested for the measurements of cross-flows; one of glass for isothermal measurements and one of stainless steel for non-isothermal measurements. Extensive data have been collected for Ar-He and N2-He mixtures at a variety of pressures and mole ratios. Viscosity, velocity slip coefficients and tangential momentum accommodation coefficients have been obtained from measurements with a spinning rotor gauge via a new theory that has been formulated for the spinning rotor gauge in the slip regime. The FIDAP fluid dynamics code has been applied to condensing flows in ampoules in the continuum regime and agreement obtained with the earlier work of Duval.

Loyalka, Sudarshan K.↗
NASA NTRS2003

The Effect of Stabilization Treatments on Disk Alloy CH98

Gas turbine engines for future subsonic transports will probably have higher pressure ratios which will require nickelbase superalloy disks with 1300 to 1400 F temperature capability. Several advanced disk alloys are being developed to fill this need. One of these, CH98, is a promising candidate for gas turbine engines and is being studied in NASA s Advanced Subsonic Technology (AST) program. For large disks, residual stresses generated during quenching from solution heat treatments are often reduced by a stabilization heat treatment, in which the disk is heated to 1500 or 1600 F for several hours followed by a static air cool. The reduction in residual stress levels lessens distortion during machining of disks. However, previous work on CH98 has indicated that stabilization treatments can also decrease creep capability. In this study, a systematic variation of stabilization temperature and time was investigated to determine its effect on 1300 F tensile and, more importantly, creep behavior. Dwell crack growth rates were also measured for selected stabilization conditions. As these advanced disk alloys may be given a supersolvus solution or a subsolvus solution heat treatment for a given application, it was decided that both options would be studied.

Gayda, John↗
NASA NTRS2003

The Effect of Tungsten and Niobium on the Stress Relaxation Rates of Disk Alloy CH98

Gas turbine engines for future subsonic transports will probably have higher pressure ratios which will require nickel-base superalloy disks with 1300 to 1400 F temperature capability. Several advanced disk alloys are being developed to fill this need. One of these, CH98, is a promising candidate for gas turbine engines and is being studied in NASA s Advanced Subsonic Technology (AST) program. For large disks, residual stresses generated during quenching from solution heat treatment are often reduced by a stabilization heat treatment, in which the disk is heated to 1500 to 1600 F for several hours followed by a static air cool. The reduction in residual stress levels lessens distortion during machining of disks. However, previous work on CH98 has indicated that stabilization treatments decrease creep capability. Additions of the refractory elements tungsten and niobium improve tensile and creep properties after stabilization, while maintaining good crack growth resistance at elevated temperatures. As the additions of refractory elements increase creep capability, they might also effect stress relaxation rates and therefore the reduction in residual stress levels obtained for a given stabilization treatment. To answer this question, the stress relaxation rates of CH98 with and without tungsten and niobium additions are compared in this paper for temperatures and times generally employed in stabilization treatments on modern disk alloys.

Gayda, John↗
DOE OSTI2023

In Situ Study on Cu-to-Cu Thermal Compression Bonding

Cu-to-Cu thermal compression bonding (TCB) has emerged as a promising solution for ultrafine pitch packaging in 3D integrated circuit technologies. Despite the progress made by conventional Cu-to-Cu TCB methods in achieving good mechanical strength of the Cu bonds, the bonding processes generally require high temperature and high pressure, which may degrade the performance and reliability of the device. Therefore, it is imperative to investigate the processing parameters to understand the bonding mechanism and achieve effective TCB at a low temperature and low pressure. Here, we developed an in situ TCB technique inside a scanning electron microscope. The in situ TCB method enables a real-time observation of bonding development, which provides critical insights into how the texture and microstructure of Cu bumps may influence the creep and surface diffusion during the bonding process. This work features a strategy to advance our understanding of the bonding mechanisms and provides insight into tailoring the microstructure of Cu for bonding at a low temperature and low pressure.

36 MATERIALS SCIENCE↗
NASA NTRS2003

The Effect of Tungsten Additions on Disk Alloy CH98

Gas turbine engines for future subsonic transports will probably have higher pressure ratios which will require nickelbase superalloy disks with 1300 to 1400 F temperature capability. Several advanced disk alloys are being developed to fill this need. One of these, CH98, is a promising candidate for gas turbine engines and is being studied in NASA s Advanced Subsonic Technology (AST) program. For large disks, residual stresses generated during quenching from solution heat treatments are often reduced by a stabilization heat treatment, in which the disk is heated in the range of 1500 to 1600 F for several hours followed by a static air cool and age. The reduction in residual stress levels lessens distortion during machining of disks. Previous work on CH98 has indicated that stabilization treatments will decrease creep capability, however, tungsten additions appear to improve the creep capability of stabilized and aged CH98. In this study, a systematic variation of tungsten additions to CH98 was investigated. Specifically, the 1300 F tensile, creep, and fatigue crack growth properties of stabilized CH 98 were assessed with varying levels of tungsten additions.

Gayda, John↗
DOE OSTI2025

The Role of Dislocations in the Anelasticity of the Upper Mantle

Dislocation‐based dissipation mechanisms potentially control the viscoelastic response of Earth's upper mantle across a variety of geodynamic contexts, including glacial isostatic adjustment, postseismic creep, and seismic‐wave attenuation. However, there is no consensus on which dislocation‐based, microphysical process controls the viscoelastic behavior of the upper mantle. Although both intergranular (plastic anisotropy) and intragranular (backstress) mechanisms have been proposed, there is currently insufficient laboratory data to discriminate between those mechanisms. Here, we present the results of forced‐oscillation experiments in a deformation‐DIA apparatus at confining pressures of 3–7 GPa and temperatures of 298–1370 K. Our experiments tested the viscoelastic response of polycrystalline olivine—the main constituent of the upper mantle—at stress amplitudes from 70 to 2,800 MPa. Mechanical data are complemented by microstructural analyses of grain size, crystallographic preferred orientation, and dislocation density. We observe amplitude‐ and frequency‐dependent attenuation and modulus relaxation and find that numerical solutions of the backstress model match our results well. Therefore, we argue that interactions among dislocations, rather than intergranular processes (e.g., plastic anisotropy or grain boundary sliding), control the viscoelastic behavior of polycrystalline olivine in our experiments. In addition, we present a linearized version of the constitutive equations of the backstress model and extrapolate it to conditions typical of seismic‐wave propagation in the upper mantle. Our extrapolation demonstrates that the backstress model can explain the magnitude of seismic‐wave attenuation in the upper mantle, although some modification is required to explain the weak frequency dependence of attenuation observed in nature and in previous experimental work.

Hein, Diede [Univ. of Minnesota, Minneapolis, MN (↗
DOE OSTI2023

Machining of Alloy 709 Creep-fatigue Specimens from G. O. Carlson Heat

Alloy 709 has been selected as the next candidate material for Section III, Division 5 qualification in the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code (BPVC) for elevated-temperature nuclear construction. The qualification data package requires an assortment of information and material data including tensile, creep, and creep-fatigue performance at elevated temperatures from different material heats. The goal of the project is to generate a dataset to support qualification of Alloy 709 material for ASME BPVC design code. Working towards the project goal, Idaho National Laboratory needs to conduct a series of tests on three different heats to support the A709 code case development. At present, there is a gap in the data package for one of the heats: Heat number 58776 manufactured by G.O. Carlson. To address this data gap, Argonne National Laboratory transmitted five plates of A709 heat 58776 manufactured by G.O. Carlson to Idaho National Laboratory. These plates were solution annealed at 1150°C and heat treated at 775°C for 10 hours. The objective of this specification is to procure a series of creep-fatigue specimens to support the qualification data package. The creep-fatigue specimen design captures the cyclic material performance at elevated temperature. The material performance data generated from specimens machined herein will support the ASME BPVC code case development and establish design limits, and design life curves for Alloy 709 material.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2023

A Summary of the Microstructural Characterization Results for the A709 Commercial Heats

To improve high temperature strength for advanced nuclear reactors, it is necessary to qualify additional structural materials for introduction into the American Society of Mechanical Engineers (AMSE) Boiler and Pressure Vessel Code (BPVC). Specifically, Alloy 709 was identified as a material for Sodium-cooled Fast Reactors to enhance time-dependent mechanical properties compared to 316H stainless steel, a code qualified high temperature alloy. Idaho National Laboratory (INL) and Oak Ridge National Laboratory (ORNL) performed initial microstructure and mechanical property testing on a third, commercial heat of Alloy 709 stainless steel. The Alloy 709 plate was produced by Allegheny Technology Inc. (ATI) and in the solution annealed condition. The chemical composition was within the limits for an American Society of Testing and Materials (ASTM) A213 specification for seamless tube for material UNS S31025. Grain size was measured as an average ASTM size of 6.5, which was larger than the ASTM 7 required according to ASTM A213 for seamless tube. The chemical composition was within the limits for an ASTM A213, and the room temperature tensile tests and hardness data met ASTM A213 specifications. Creep-fatigue results showed similar behavior to previous heats of Alloy 709.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2021

Linear and Nonlinear Solvers for Simulating Multiphase Flow within Large-Scale Engineered Subsurface Systems

Simulation of multiphase flow in the subsurface is well-known to be computationally challenging. While there have been many studies that have explored approaches to overcoming these challenges, they often utilize relatively simple case studies. In this paper, we focus on the unique numerical challenges posed by modeling large-scale engineered subsurface systems, characterized by discrete features embedded in a heterogeneous natural subsurface setting. The man-made features such as shafts, tunnels, and barriers often cause multiple challenges in modeling the domain for multiphase porous media flow. This flow scenario can have a wide range of applications such as nuclear waste repositories, enhanced recovery of a petroleum reservoir, geothermal engineering, and carbon sequestration. An example of these severe numerical challenges is the case of performance assessment (PA) for Waste Isolation Pilot Plant (WIPP), the only operating deep geological repository in the US, which simulates extreme material properties of bedded salt rock formation and extreme contrast due to open excavation next to the formation. The models have extremes not only of permeability and porosity but also of the constitutive models needed for multiphase flow; additionally, they have process models like salt creep closure reducing porosity over time, fracturing in clay and anhydrite interbeds of the bedded salt, gas generation from the waste materials, and unintentional human borehole intrusions in some scenarios. Numerical simulations require the solution of coupled systems of nonlinear PDEs; in our work, we use the open-source simulator PFLOTRAN which is based on Finite Volume discretization. The solution of the nonlinear equations requires use of the Newton-Raphson iteration at each time step, which entails the solution of the linearized Jacobian system at each iteration. The effects of all the processes (i.e., large number of unknowns, highly nonlinear constitutive relations, large contrasts in material properties in short distances) lead to an ill-conditioned Jacobian matrix that severely challenges traditional linear solver, i.e., stabilized biconjugate gradient with block Jacobi incomplete LU preconditioner (BCGS-ILU) leading to non-convergence for traditional Newton-Raphson nonlinear solver causing unacceptably long computation time for each model. This paper presents linear solvers such as constrained pressure residual (CPR) two-stage preconditioner with alternate-block-factorization (ABF) and quasi- implicit pressure and explicit saturation (QIMPES) decouplers and flexible generalized residual solver (FGMRES). The new general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), is also introduced to resolve extreme nonlinearities in the models. We demonstrate the effectiveness of each method relative to the default BCGS-Newton solver. The two best cases had nearly 50 times speed-up and achieved completion of a simulation in 14 hours that never completed due to non-convergence with the default solver. We also investigate the strong scalability of each method and discuss some of the deficiencies found for Block Jacobi preconditioner using parallel domain decomposition, and node packing effects of modern processor architecture.

Preconditioner, Nonlinear, Porous media, Multiphas↗