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At least 379 records · Page 21

Zonal multigrid solution of compressible flow problems on unstructured and adaptive meshes

The simultaneous use of adaptive meshing techniques with a multigrid strategy for solving the 2-D Euler equations in the context of unstructured meshes is studied. To obtain optimal efficiency, methods capable of computing locally improved solutions without recourse to global recalculations are pursued. A method for locally refining an existing unstructured mesh, without regenerating a new global mesh is employed, and the domain is automatically partitioned into refined and unrefined regions. Two multigrid strategies are developed. In the first, time-stepping is performed on a global fine mesh covering the entire domain, and convergence acceleration is achieved through the use of zonal coarse grid accelerator meshes, which lie under the adaptively refined regions of the global fine mesh. Both schemes are shown to produce similar convergence rates to each other, and also with respect to a previously developed global multigrid algorithm, which performs time-stepping throughout the entire domain, on each mesh level. However, the present schemes exhibit higher computational efficiency due to the smaller number of operations on each level.

Mavriplis, Dimitri J.↗

Multi-Objective design of interlocking metasurfaces using conditional diffusion models

Unit cell design remains a major challenge for interlocking metasurfaces, a promising joining technology for dissimilar materials, due to the complex, competing, multivariate design space and the need for rapid adaptation to varying performance requirements. This study explores Conditional Diffusion Models as a design optimization tool for interlocking metasurfaces. Given the complex, competing, multivariate design space for interlocking metasurfaces, unit cell design remains a major challenge for this joining technology. We trained a conditional diffusion model on 25,000 finite element analysis-simulated interlocking metasurface unit cells to generate designs with tailored thermo-mechanical properties (tensile strength, shear strength, and thermal conductivity) based on specified performance criteria. The model demonstrated a success rate of approximately 72 % in producing designs that met specified property bounds. The conditional diffusion model generated both thermally resistive and conductive designs, revealing clear trends in design characteristics: taller, dendritic structures were advantageous for tensile loads, while shorter, robust designs excelled in shear applications. Our findings indicate that the model's performance is more influenced by the breadth of the design space than by the quantity of training data, highlighting the importance of expansive design domains for generating innovative solutions. This work establishes conditional diffusion models as a highly efficient and adaptable tool for rapid interlocking metasurface unit cell design, paving the way for advancements in multi-material joining technologies, as well as highlighting the justification to leverage conditional diffusion models as design tools across complex design domains.

Conditional diffusion models↗

Using adaptive grid in modeling rocket nozzle flow

The mechanical behavior of a rocket motor internal flow field results in a system of nonlinear partial differential equations which cannot be solved analytically. However, this system of equations called the Navier-Stokes equations can be solved numerically. The accuracy and the convergence of the solution of the system of equations will depend largely on how precisely the sharp gradients in the domain of interest can be resolved. With the advances in computer technology, more sophisticated algorithms are available to improve the accuracy and convergence of the solutions. An adaptive grid generation is one of the schemes which can be incorporated into the algorithm to enhance the capability of numerical modeling. It is equivalent to putting intelligence into the algorithm to optimize the use of computer memory. With this scheme, the finite difference domain of the flow field called the grid does neither have to be very fine nor strategically placed at the location of sharp gradients. The grid is self adapting as the solution evolves. This scheme significantly improves the methodology of solving flow problems in rocket nozzles by taking the refinement part of grid generation out of the hands of computational fluid dynamics (CFD) specialists and place it into the computer algorithm itself.

Chow, Alan S.↗

An adaptive finite element strategy for complex flow problems

Adaptive finite element methods for steady and unsteady flow problems in two-dimensional domains are described. Details of a data management scheme are given that provide for the rapid implementation of various CFD algorithms on changing unstructured meshes. The results of several numerical experiments on subsonic and supersonic flow problems are discussed.

Oden, J. T.↗

Finite element analysis in fluids; Proceedings of the Seventh International Conference on Finite Element Methods in Flow Problems, University of Alabama, Huntsville, Apr. 3-7, 1989

Recent advances in computational fluid dynamics are examined in reviews and reports, with an emphasis on finite-element methods. Sections are devoted to adaptive meshes, atmospheric dynamics, combustion, compressible flows, control-volume finite elements, crystal growth, domain decomposition, EM-field problems, FDM/FEM, and fluid-structure interactions. Consideration is given to free-boundary problems with heat transfer, free surface flow, geophysical flow problems, heat and mass transfer, high-speed flow, incompressible flow, inverse design methods, MHD problems, the mathematics of finite elements, and mesh generation. Also discussed are mixed finite elements, multigrid methods, non-Newtonian fluids, numerical dissipation, parallel vector processing, reservoir simulation, seepage, shallow-water problems, spectral methods, supercomputer architectures, three-dimensional problems, and turbulent flows.

Chung, T. J.↗

PalymSys (TM): An extended version of CLIPS for construction and reasoning using blackboards

This paper describes PalymSys(TM) -- an extended version of the CLIPS language that is designed to facilitate the implementation of blackboard systems. The paper first describes the general characteristics of blackboards and shows how a control blackboard architecture can be used by AI systems to examine their own behavior and adapt to real-time problem-solving situations by striking a balance between domain and control reasoning. The paper then describes the use of PalymSys in the development of a situation assessment subsystem for use aboard Army helicopters. This system performs real-time inferencing about the current battlefield situation using multiple domain blackboards as well as a control blackboard. A description of the control and domain blackboards and their implementation is presented. The paper also describes modifications made to the standard CLIPS 6.02 language in PalymSys(TM) 2.0. These include: (1) a dynamic Dempster-Shafer belief network whose structure is completely specifiable at run-time in the consequent of a PalymSys(TM) rule, (2) extension of the run command including a continuous run feature that enables the system to run even when the agenda is empty, and (3) a built-in communications link that uses shared memory to communicate with other independent processes.

Bryson, Travis↗

Elliptic Volume Grid Generation for Viscous CFD Parametric Design Studies

This paper presents a robust method for the generation of zonal volume grids of design parametrics for aerodynamic configurations. The process utilizes simple algebraic techniques with parametric splines coupled with elliptic volume grid generation to generate isolated zonal grids for changes in body configuration needed to perform parametric design studies. Speed of the algorithm is maximized through the algebraic methods and reduced number of grid points to be regenerated for each design parametric without sacrificing grid quality and continuity within the volume domain. The method is directly applicable to grid reusability, because it modifies existing ow adapted volume grids and enables the user to restart the CFD solution process with an established flow field. Use of this zonal approach reduces computer usage time to create new volume grids for design parametric studies by an order of magnitude, as compared to current methods which require the regeneration of an entire volume grid. A sample configuration of a proposed Single Stage-to-Orbit Vehicle is used to illustrate an application of this method.

Alter, Stephen J.↗

GPR Imaging for Deeply Buried Objects: A Comparative Study Based on FDTD Models and Field Experiments

Conventional use of Ground Penetrating Radar (GPR) is hampered by variations in background environmental conditions, such as water content in soil, resulting in poor repeatability of results over long periods of time when the radar pulse characteristics are kept the same. Target objects types might include voids, tunnels, unexploded ordinance, etc. The long-term objective of this work is to develop methods that would extend the use of GPR under various environmental and soil conditions provided an optimal set of radar parameters (such as frequency, bandwidth, and sensor configuration) are adaptively employed based on the ground conditions. Towards that objective, developing Finite Difference Time Domain (FDTD) GPR models, verified by experimental results, would allow us to develop analytical and experimental techniques to control radar parameters to obtain consistent GPR images with changing ground conditions. Reported here is an attempt at developing 20 and 3D FDTD models of buried targets verified by two different radar systems capable of operating over different soil conditions. Experimental radar data employed were from a custom designed high-frequency (200 MHz) multi-static sensor platform capable of producing 3-D images, and longer wavelength (25 MHz) COTS radar (Pulse EKKO 100) capable of producing 2-D images. Our results indicate different types of radar can produce consistent images.

Tilley, roger↗

Extended Magnetohydrodynamics with Embedded Particle-in-Cell Simulation of Ganymede's Magnetosphere

We have recently developed a new modeling capability to embed the implicit particle-in-cell (PIC) model iPIC3D into the Block-Adaptive-Tree-Solarwind-Roe-Upwind-Scheme magnetohydrodynamic (MHD) model. The MHD with embedded PIC domains (MHO-EPIC) algorithm Is a two-way coupled kinetic-fluid model. As one of the very first applications of the MHD-EPIC algorithm, we simulate the Interaction between Jupiter's magnetospherlc plasma and Ganymede's magnetosphere. We compare the MHO-EPIC simulations with pure Hall MHD simulations and compare both model results with Galileo observations to assess the Importance of kinetic effects In controlling the configuration and dynamics of Ganymede's magnetosphere. We find that the Hall MHD and MHO-EPIC solutions are qualitatively similar, but there are significant quantitative differences. In particular. the density and pressure inside the magnetosphere show different distributions. For our baseline grid resolution the PIC solution is more dynamic than the Hall MHD simulation and it compares significantly better with the Galileo magnetic measurements than the Hall MHD solution. The power spectra of the observed and simulated magnetic field fluctuations agree extremely well for the MHD-EPIC model. The MHO-EPIC simulation also produced a few flux transfer events (FTEs) that have magnetic signatures very similar to an observed event. The simulation shows that the FTEs often exhibit complex 3-0 structures with their orientations changing substantially between the equatorial plane and the Galileo trajectory, which explains the magnetic signatures observed during the magnetopause crossings. The computational cost of the MHO-EPIC simulation was only about 4 times more than that of the Hall MHD simulation.

Toth, Gabor↗

Flow Characterization of the NASA Langley Unitary Plan WindTunnel, Test Section 2: Computational Results

Flow in the empty Unitary Planform Wind Tunnel at the NASA Langley Research Center issimulated with computational fluid dynamics methods. The objectives are to assess CFD’s truepredictive capability and to generate flow-field maps upstream of the tunnel’s test section foruse in associated vehicle tests. Multiple CFD solvers, grid adaption methods, and turbulencemodels are used by five teams doing the simulations. The simulation domain is as large aspractical: from the start of the settling chamber, just downstream of the last set of turningvanes, to well downstream of the test section. Simulations are done at three Mach numbers andseveral Reynolds numbers that cover most of the tunnel’s operating range. The sensitivity ofthe CFD solutions to simulation process details, including turbulence model, spatial resolutionand settling-chamber inflow velocity profile, is characterized. Significant sensitivities in somedetails of the predicted test section flow are described. CFD predicts that details of flow inthe settling chamber can affect the flow in the test section. Turning of the flow through thetunnel contraction and past nozzle-block hardware generates streamwise vortices, some ofwhich survive into the core of the test section, principally at low Mach number. Preliminarycomparisons to experimental measurements are given. The major flow characteristics andthe dynamical vortices predicted by CFD exist in the experiment. Several details in CFD andexperiment are observed to differ, including the effects of the vortices and features that appearto be Mach waves in the test section. Sensitivity to grid resolution and turbulence modelingis noted in the vortices; no potential cause for the differences in Mach waves has yet beenidentified in the CFD sensitivity studies.

wind tunnel↗

Understanding Structures of Cyber Competition in an Era of Major Power Rivalry

Over the past two decades, the cyber domain has emerged and evolved into a key strategic domain for nations across the globe. Security strategies are espoused by heads of state that focus on how to manage the increasingly enormous, crosscutting impact that the cyber domain has on national security across economic, military, intelligence, intellectual property, and countless other facets. These strategies frequently evolve, and even change entirely, as leaders adapt to new technologies and administrations change. Even if these strategies did not change at all, they would take inordinate amounts of time to effectively implement within the organizational structures of government. The time required to go from setting department and agency-level goals, to the time small teams have well-oiled processes and expertise to accomplish tactical objectives that meet the strategic vision is lengthy. With near certainty, by the time objectives and vision are implemented, the landscape, strategy, or both has changed entirely. While this churn will likely never cease due to the rapidly changing nature of the cyber domain, this problem raises an important question: can governmental structures be organized to rapidly adapt to changing cyber strategies? As offices responsible for particular missions in cyberspace shuffle about within the bureaucracy, are technical capabilities enabled or enhanced? No matter how advanced a particular technical capability is or how adept the staff is at solving problems, they will be ineffective if placed haphazardly within the organization: the right authorities may not exist for their office, the correct lines of interpersonal communication may not be established properly, or insufficient resources have not been allocated to effectively deploy a brilliant technical solution. This concept of organizational agility in the context of national cyber capabilities is important when taking into account the National Defense Strategy’s emphasis on cyber capability and the ability of the United States to compete and rapidly adapt to new challenges posed by rivals. As a nation, we are at a point where technology evolves rapidly enough to warrant thoughtful and nimble changes to the bureaucratic structures that support how cyber operations are carried out. Taking these questions and cross-comparing them to the organizational structures across China, Russia, and the United States provides for an interesting thought experiment. As non-democratic regimes, China and Russia have differing priorities and internal power dynamics than the United States and thus organize their governments differently. By combining known and broadcasted strategies of these nations with the observed technical capabilities demonstrated in the public domain, we can begin to see how organizational structures map to strategic goals and directly enable technical capabilities. Insight can be gained by introducing organizational structures into traditional analysis focusing solely around strategies and capabilities; additionally, otherwise unknown capabilities or intents might be discovered or inferred by analyzing organizational structures alone. Analyzing cyber operations from this oft-overlooked perspective could potentially provide useful insight and more concrete actions that can be undertaken to realize the National Defense Strategy’s goals.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗

In-Flight Suppression of an Unstable F/A-18 Structural Mode Using the Space Launch System Adaptive Augmenting Control System

NASA's Space Launch System (SLS) Flight Control System (FCS) includes an Adaptive Augmenting Control (AAC) component which employs a multiplicative gain update law to enhance the performance and robustness of the baseline control system for extreme off-nominal scenarios. The SLS FCS algorithm including AAC has been flight tested utilizing a specially outfitted F/A-18 fighter jet in which the pitch axis control of the aircraft was performed by a Non-linear Dynamic Inversion (NDI) controller, SLS reference models, and the SLS flight software prototype. This paper describes test cases from the research flight campaign in which the fundamental F/A-18 airframe structural mode was identified using post-flight frequency-domain reconstruction, amplified to result in closed loop instability, and suppressed in-flight by the SLS adaptive control system.

VanZwieten, Tannen S.↗

Adaptive Augmenting Control Flight Characterization Experiment on an F/A-18

This paper summarizes the Adaptive Augmenting Control (AAC) flight characterization experiments performed using an F/A-18 (TN 853). AAC was designed and developed specifically for launch vehicles, and is currently part of the baseline autopilot design for NASA's Space Launch System (SLS). The scope covered here includes a brief overview of the algorithm (covered in more detail elsewhere), motivation and benefits of flight testing, top-level SLS flight test objectives, applicability of the F/A-18 as a platform for testing a launch vehicle control design, test cases designed to fully vet the AAC algorithm, flight test results, and conclusions regarding the functionality of AAC. The AAC algorithm developed at Marshall Space Flight Center is a forward loop gain multiplicative adaptive algorithm that modifies the total attitude control system gain in response to sensed model errors or undesirable parasitic mode resonances. The AAC algorithm provides the capability to improve or decrease performance by balancing attitude tracking with the mitigation of parasitic dynamics, such as control-structure interaction or servo-actuator limit cycles. In the case of the latter, if unmodeled or mismodeled parasitic dynamics are present that would otherwise result in a closed-loop instability or near instability, the adaptive controller decreases the total loop gain to reduce the interaction between these dynamics and the controller. This is in contrast to traditional adaptive control logic, which focuses on improving performance by increasing gain. The computationally simple AAC attitude control algorithm has stability properties that are reconcilable in the context of classical frequency-domain criteria (i.e., gain and phase margin). The algorithm assumes that the baseline attitude control design is well-tuned for a nominal trajectory and is designed to adapt only when necessary. Furthermore, the adaptation is attracted to the nominal design and adapts only on an as-needed basis (see Figure 1). The MSFC algorithm design was formulated during the Constellation Program and reached a high maturity level during SLS through simulation-based development and internal and external analytical review. The AAC algorithm design has three summary-level objectives: (1) "Do no harm;" return to baseline control design when not needed, (2) Increase performance; respond to error in ability of vehicle to track command, and (3) Regain stability; respond to undesirable control-structure interaction or other parasitic dynamics. AAC has been successfully implemented as part of the Space Launch System baseline design, including extensive testing in high-fidelity 6-DOF simulations the details of which are described in [1]. The Dryden Flight Research Center's F/A-18 Full-Scale Advanced Systems Testbed (FAST) platform is used to conduct an algorithm flight characterization experiment intended to fully vet the aforementioned design objectives. FAST was specifically designed with this type of test program in mind. The onboard flight control system has full-authority experiment control of ten aerodynamic effectors and two throttles. It has production and research sensor inputs and pilot engage/disengage and real-time configuration of up to eight different experiments on a single flight. It has failure detection and automatic reversion to fail-safe mode. The F/A-18 aircraft has an experiment envelope cleared for full-authority control and maneuvering and exhibits characteristics for robust recovery from unusual attitudes and configurations aided by the presence of a qualified test pilot. The F/A-18 aircraft has relatively high mass and inertia with exceptional performance; the F/A-18 also has a large thrust-to-weight ratio, owing to its military heritage. This enables the simulation of a portion of the ascent trajectory with a high degree of dynamic similarity to a launch vehicle, and the research flight control system can simulate unstable longitudinal dynamics. Parasitic dynamics such as slosh and bending modes, as well as atmospheric disturbances, are being produced by the airframe via modification of bending filters and the use of secondary control surfaces, including leading and trailing edge flaps, symmetric ailerons, and symmetric rudders. The platform also has the ability to inject signals in flight to simulate structural mode resonances or other challenging dynamics. This platform also offers more test maneuvers and longer maneuver times than a single rocket or missile test, which provides ample opportunity to fully and repeatedly exercise all aspects of the algorithm. Prior to testing on an F/A-18, AAC was the only component of the SLS autopilot design that had not been flight tested. The testing described in this paper raises the Technology Readiness Level (TRL) early in the SLS Program and is able to demonstrate its capabilities and robustness in a flight environment.

VanZwieten, Tannen S.↗

Ultrafast Laser Welding and Terahertz Time-domain Spectroscopy of Soda Lime Silicate Glass

Joining and welding techniques of similar and dissimilar materials have reached their limits due to the inflexibility of adapting dielectric materials, e.g., glasses and ceramics, reproducibility for transparent materials, and cycle process time. The ability to join pieces of glasses using laser irradiation allows rapid processing of vacuum insulating glass (VIG) for a widening range of commercial energy-efficient applications. We have used a 1064-nm laser wavelength with 15 picoseconds pulse width and a 155-kHz repetition rate to weld commercial plate glass pieces. The welded samples were characterized via x-ray fluorescence (XRF), time-of-flight secondary ion mass spectrometry (TOF–SIMS), polarimetry, and terahertz time-domain spectroscopy (THz-TDS) to determine the integrity of the weld and chemical changes in the welded areas. Our data shows no significant changes in glass chemistry and structure on average over nano- to micro-scales occurred from the welding process within the limits of the characterization tools. Our results demonstrate successful welding of glass plates without significant changes in the glass chemistry, structure, or properties in the laser-modified region or strains in the bulk glass.

Matthies, Kathleen [ORNL] (ORCID:0009000213135441)↗

DNS of Flame Stabilization Dynamics of a Swirl-Stabilized Spray Burner Using Sustainable Aviation Fuels

Concern for emission reduction has motivated the development of new cost-effective alternative sustainable aviation fuels (SAFs). Drop-in SAFs with blends of the certified and the alternative fuels are beneficial because they do not require engine modifications for use in current aviation engines. Combustion characteristics are of particular concern when comparing a new fuel with Jet-A. Lean blow-off and flame stabilization have been identified to be governed by the fuel propensity to autoignition, i.e. derived cetane number. However, fundamental investigation is required to identify the impact of derived cetane number on the combustion mode of flame stabilization in a realistic combustor. The combustion mode has also a direct impact on turbulent combustion modeling closures. DNS simulations are performed in the low-Mach solver of the Pele Suite called PeleLMeX. Lagrangian multi-phase modeling is used to capture the liquid spray injection of Jet-A (reference fuel) and C1 as a representative of a low cetane number SAF. Adaptive Mesh Refinement (AMR) is used to enable a more efficient simulation of a more realistic domain size and embedded boundary treatment is used to model a bluff-body geometry. Local extinction and edge flame propagation were observed for both fuels. The edge flame propagation mode was quantified in terms of a Damkohler number defined as the ratio between progress variable reaction rate and its diffusive flux. Initial analysis suggests that a mixed mode combustion occurs for the edge flame propagation, with flame propagation assisted by ignition and autoignition co-existing for both Jet-A and C1 flames. The analysis of the extinction region shows a larger progress variable for Jet-A which can further increase the local displacement speed and present a faster reignition of the stoichiometric mixture fraction.

adaptive mesh refinement↗

A formally certified end-to-end implementation of Shor’s factorization algorithm

Quantum computing technology may soon deliver revolutionary improvements in algorithmic performance, but it is useful only if computed answers are correct. While hardware-level decoherence errors have garnered significant attention, a less recognized obstacle to correctness is that of human programming errors—“bugs.” Techniques familiar to most programmers from the classical domain for avoiding, discovering, and diagnosing bugs do not easily transfer, at scale, to the quantum domain because of its unique characteristics. To address this problem, we have been working to adapt formal methods to quantum programming. With such methods, a programmer writes a mathematical specification alongside the program and semiautomatically proves the program correct with respect to it. The proof’s validity is automatically confirmed—certified—by a “proof assistant.” Formal methods have successfully yielded high-assurance classical software artifacts, and the underlying technology has produced certified proofs of major mathematical theorems. As a demonstration of the feasibility of applying formal methods to quantum programming, we present a formally certified end-to-end implementation of Shor’s prime factorization algorithm, developed as part of a framework for applying the certified approach to general applications. By leveraging our framework, one can significantly reduce the effects of human errors and obtain a high-assurance implementation of large-scale quantum applications in a principled way.

Science & Technology - Other Topics↗

Numerical Investigation of Fluid Flow and Space Charge in Liquid Argon Time Projection Chamber (LArTPC) Detectors

Overview This project focused on developing a high-fidelity numerical framework to simulate the multiphysics environment within Liquid Argon Time Projection Chamber (LArTPC) detectors. The primary objective was to characterize the complex interplay between ion transport, background fluid dynamics, and electric field distortions—a critical factor for the calibration and sensitivity of next-generation High Energy Physics experiments, such as DUNE. Technical Achievements The research successfully yielded a hybrid numerical space-charge solver utilizing a Cell-Centered Finite Volume Method (FVM) for ion transport coupled with a Finite Element Method (FEM) for electric potential. Key accomplishments include: • Verification & Validation: The 3-D solver was rigorously verified against 1-D analytical solutions, demonstrating high numerical accuracy in predicting space-charge-induced field deviations. • Field Distortion Analysis: 3D simulations revealed that space charge effects introduce significant non-uniformities in the electric field. Critically, the research identified that background LAr flow velocities, when comparable to ion drift velocities, markedly exacerbate these distortions. • Technology Transfer: The resulting source code and comprehensive user manuals were successfully transferred to collaborators at Fermilab, providing a portable computational tool for the broader scientific community. Challenges and Future Directions While the space-charge solver achieved all performance metrics, the integrated fluid dynamics modeling encountered convergence challenges stemming from the extreme 200-fold disparity in length scales between the detector's 37 mm inlet pipes and the 8-meter global domain. To address this, the project has identified a clear technical pivot toward Hierarchical Geometric Adaptive Mesh Refinement (HG-AMR). By implementing an h-type refinement strategy with hanging nodes, future iterations of this solver will be capable of resolving localized high-gradient inlet flows without the prohibitive computational costs of regular grids. This advancement, combined with data-driven uncertainty quantification based on MicroBooNE-style calibration, will enable the precise modeling of detector responses in large-scale cryogenic environments where direct measurement remains difficult. Impact The computational tools developed under this award provide a foundation for enhancing the energy resolution and spatial reconstruction of noble liquid detectors. By bridging the gap between theoretical fluid dynamics and experimental field calibration, this work supports the DOE’s mission to advance the frontiers of neutrino physics and dark matter detection.

42 ENGINEERING↗

A numerical investigation of the finite element method in compressible primitive variable Navier-Stokes flow

The results of a comprehensive numerical investigation of the basic capabilities of the finite element method (FEM) for numerical solution of compressible flow problems governed by the two-dimensional and axis-symmetric Navier-Stokes equations in primitive variables are presented. The strong and weak points of the method as a tool for computational fluid dynamics are considered. The relation of the linear element finite element method to finite difference methods (FDM) is explored. The calculation of free shear layer and separated flows over aircraft boattail afterbodies with plume simulators indicate the strongest assets of the method are its capabilities for reliable and accurate calculation employing variable grids which readily approximate complex geometry and capably adapt to the presence of diverse regions of large solution gradients without the necessity of domain transformation.

Cook, C. H.↗