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At least 1,063 records · Page 59
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

Methodological Aspects of Shielding and Activation Analyses for ITER Diagnostics Systems - Paper 125

It is expected that the diagnostics systems of the ITER tokamak will provide valuable measurement data about the first in its kind D-T fusion plasma with emission of 14 MeV neutrons. These systems must withstand the neutron and secondary photon radiation along the ITER operation. Under the neutron irradiation the ITER materials are getting radioactive and therefore an adequate shielding must be provided along all the time of ITER operation and shut-down for the reliable machine operation and for the safe personnel access to its components, such as the diagnostics ports. These are the prerequisites for the shielding and activation analysis of the ITER diagnostics systems discussed in this paper. This paper provides insight into the current tasks of neutronics support for the development of ITER diagnostics systems. The tasks themselves range from relatively straightforward ones such as Monte Carlo radiation transport calculations for shielding analyses of the diagnostics systems, to more advanced tasks of Shut-Down Dose Rate (SDDR) analyses for which the dedicated Rigorous 2-Step mesh-based (R2Smesh) and Direct 1-Step (D1S) methodologies were applied. The accuracy and precision of the ITER neutronics analysis is dependent on the methodological approaches and the implying modeling assumptions. In addition to the SDDR challenges, another distinguishing feature of fusion neutronics is a wide use of CAD-based neutronics modeling to design components of fusion devices such as ITER and DEMO. Basically, there are two possibilities of CAD geometry settings in Monte Carlo (MC) radiation transport models, either by the geometry format conversion of CAD models into the format of MC models by means of the interface programs such as McCad and MCAM or by the direct use of CAD geometry engine in MC calculations such as it was realized in the DAGMC code approach. Several conversion interfaces and the DAGMC software library have been tested on the 3D ITER model developed for the CAD-MCNP benchmark. It was concluded that the considered approaches have reached the maturity level to allow their application to real ITER design analyses.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2021

Photoelectrochemically Self Improving Si/GaN Photocathode: Figure 4a Raw Data

Calculated formation energy of (0001 ̅) c-plane GaN surfaces. The density-functional theory (DFT) calculations were performed with the generalized gradient approximation, using the plane-wave pseudopotential quantum-ESPRESSO package. Ultrasoft pseudopotentials were used to describe the interaction between valence electrons and ionic cores, where Ga 3d electrons were explicitly treated as valence electrons. The wave functions and electronic density were expanded in a plane-wave basis set truncated at a cutoff energy of 30 Ry and 240 Ry, respectively. The surface was modeled using a repeated slab geometry using a (2×2) unit cell with a thickness of seven GaN bilayers, where the Ga-terminated bottom side was passivated by pseudo-hydrogen with charge of 1.25e. In addition, the (1010) surface was modeled using 8 GaN atomic layers with a lateral dimension of 9.57 Å x 10.38 Å. A vacuum width of at least 15.0 Å was introduced between consecutive slabs, and a k-point mesh of 5x5x1 and 2x2x1 was used to sample the surface Brillouin zone for the polar and non-polar surfaces, respectively. Calculated formation energy of (0001 ̅) c-plane GaN surfaces with different oxygen configuration within the anion and cation-rich limits: ideal GaN surfaces of c-plane and m-plane; configurations with 100% replacement of nitrogen by oxygen on the surface c-plane and m-plane; 100% replacement of nitrogen by oxygen in the subsurface bilayer c-plane and m-plane; and configurations with 50% replacement of nitrogen of the surface and subsurface bilayers by oxygens c-plane and m-plane. In both cases, the ideal surface is chosen as the reference.

photocathode↗
DOE OSTI2021

Photoelectrochemically Self Improving Si/GaN Photocathode: Figure 4b Raw Data

Calculated formation energy of (1010 ̅) m-plane GaN surfaces. The density-functional theory (DFT) calculations were performed with the generalized gradient approximation, using the plane-wave pseudopotential quantum-ESPRESSO package. Ultrasoft pseudopotentials were used to describe the interaction between valence electrons and ionic cores, where Ga 3d electrons were explicitly treated as valence electrons. The wave functions and electronic density were expanded in a plane-wave basis set truncated at a cutoff energy of 30 Ry and 240 Ry, respectively. The surface was modeled using a repeated slab geometry using a (2×2) unit cell with a thickness of seven GaN bilayers, where the Ga-terminated bottom side was passivated by pseudo-hydrogen with charge of 1.25e. In addition, the (1010) surface was modeled using 8 GaN atomic layers with a lateral dimension of 9.57 Å x 10.38 Å. A vacuum width of at least 15.0 Å was introduced between consecutive slabs, and a k-point mesh of 5x5x1 and 2x2x1 was used to sample the surface Brillouin zone for the polar and non-polar surfaces, respectively. Calculated formation energy of (1010 ̅) m-plane GaN surfaces with different oxygen configuration within the anion and cation-rich limits: ideal GaN surfaces of c-plane and m-plane; configurations with 100% replacement of nitrogen by oxygen on the surface c-plane and m-plane; 100% replacement of nitrogen by oxygen in the subsurface bilayer c-plane and m-plane; and configurations with 50% replacement of nitrogen of the surface and subsurface bilayers by oxygens c-plane and m-plane. In both cases, the ideal surface is chosen as the reference.

photocathode↗
NASA NTRS1996

Multigrid Methods for EHL Problems

In many bearings and contacts, forces are transmitted through thin continuous fluid films which separate two contacting elements. Objects in contact are normally subjected to friction and wear which can be reduced effectively by using lubricants. If the lubricant film is sufficiently thin to prevent the opposing solids from coming into contact and carries the entire load, then we have hydrodynamic lubrication, where the lubricant film is determined by the motion and geometry of the solids. However, for loaded contacts of low geometrical conformity, such as gears, rolling contact bearings and cams, this is not the case due to high pressures and this is referred to as Elasto-Hydrodynamic Lubrication (EHL) In EHL, elastic deformation of the contacting elements and the increase in fluid viscosity with pressure are very significant and cannot be ignored. Since the deformation results in changing the geometry of the lubricating film, which in turn determines the pressure distribution, an EHL mathematical model must simultaneously satisfy the complex elasticity (integral) and the Reynolds lubrication (differential) equations. The nonlinear and coupled nature of the two equations makes numerical calculations computationally intensive. This is especially true for highly loaded problems found in practice. One novel feature of these problems is that the solution may exhibit sharp pressure spikes in the outlet region. To this date both finite element and finite difference methods have been used to solve EHL problems with perhaps greater emphasis on the use of the finite difference approach. In both cases, a major computational difficulty is ensuring convergence of the nonlinear equations solver to a steady state solution. Two successful methods for achieving this are direct iteration and multigrid methods. Direct iteration methods (e.g Gauss Seidel) have long been used in conjunction with finite difference discretizations on regular meshes. Perhaps one of the best examples of the application of such methods is the recent Effective Influence Method of Dowson and Wang. Multigrid methods have also been used with great success by Venner and Venner and Lubrecht with a good summary being given by Venner. As both these finite difference discretization based approaches appear to provide an efficient way of solving EHL problems, it is important to understand their relative merits. This paper is a first attempt at providing such an understanding in the context of EHL point contact problem, (contact of two spheres), in which the contact zone is a point and an ellipse or circle for unloaded and loaded dry contacts respectively. Since the film thickness and the contact width are generally small compared to the local radius of curvature of the two surfaces, the reduced geometry of the surfaces in the contact area can be accurately approximated to the contact between a paraboloid and a flat surface. The layout of the remainder of this paper is as follows. In section 2 we introduce the form of the equations to be solved. The Effective Influence Newton Method is described in Section 3 while Section 4 describes the Multigrid method to be used. Sections 5 and 6 describe the test problems to be used in the comparison between the two methods and compare the performance of the two methods. Section 7 concludes the paper with an argument of the two methods and suggests some future research directions.

Nurgat, Elyas↗
NASA NTRS1993

Space marching calculations about hypersonic configurations using a solution-adaptive mesh algorithm

A solution-adaptive marching algorithm is developed and applied to a three-dimensional parabolized Navier-Stokes equation solver. The resulting algorithm obtains accurate solutions by using a spatial-marching/adaptive grid procedure. The adaptation step redistributes grid points line by line in both crossflow directions, with grid point motion controlled by forces analogous to tensional and torsional spring forces with the tensional force proportional to the error measure or weighting functions. The solution-adaptive marching procedure is applied to the hypersonic flow about two generic aircraft configurations. The first of these is an all-body-type geometry with elliptical cross sections and is studied at angles of attack of 0.5, and 15 deg. The second geometry is a generic blended-wing-body design. Results are presented that demonstrate the improvements in flowfield resolution obtainable with the solution-adaptive marching procedure over conventional fixed grid techniques. Computed pitot pressure profiles obtained using the solution-adaptive algorithm show improved agreement with experimental data compared to predictions obtained using a fixed grid.

Harvey, Albert D.↗
NASA NTRS2012

Optimal Inlet Shape Design of N2B Hybrid Wing Body Configuration

The N2B hybrid wing body aircraft was conceptually designed to meet environmental and performance goals for the N+2 generation transport set by the Subsonic Fixed Wing project of NASA Fundamental Aeronautics Program. In the present study, flow simulations are conducted around the N2B configuration by a Reynolds-averaged Navier-Stokes flow solver using unstructured meshes. Boundary conditions at engine fan face and nozzle exhaust planes are provided by the NPSS thermodynamic engine cycle model. The flow simulations reveal challenging design issues arising from boundary layer ingestion offset inlet and airframe-propulsion integration. Adjoint-based optimal designs are then conducted for the inlet shape to minimize the airframe drag force and flow distortion at fan faces. Design surfaces are parameterized by NURBS, and the cowl lip geometry is modified by a spring analogy approach. By the drag minimization design, flow separation on the cowl surfaces are almost removed, and shock wave strength got remarkably reduced. For the distortion minimization design, a circumferential distortion indicator DPCP(sub avg) is adopted as the design objective and diffuser bottom and side wall surfaces are perturbed for the design. The distortion minimization results in a 12.5 % reduction in the objective function.

Kim, Hyoungjin↗
DOE OSTI2025

Comparison Study of Conventional and Adaptive Mesh Refinement in Organic Material Decomposition Models

This study compares conventional mesh refinement techniques, specifically Uniform Mesh Refinement (UMR), with a new Adaptive Mesh Refinement (AMR) method, applied to Organic Material Decomposition (OMD) models. The proposed benefit of AMR is that only areas that require refinement, based on minimizing a specific field gradient, are refined thus decreasing model wall time compared to conventional UMR methods. This work specifically focuses on comparing UMR and AMR methods on decomposing (both No-Flow and Porous-Flow material models) Polymeric Methylene Diisocyanate (PMDI) polyurethane foam. Throughout the work, the geometry increased in complexity to assess the refinement methods performance at varying levels geometric intricacy. While AMR has been shown to work well in a variety of applications, the UMR approach proved to be computationally faster, for many of the geometries and foam decomposition models, than AMR. However, it was observed that at higher levels of refinement, greater than 3 UMR, AMR begins to be computationally better. Additionally, the settings used to perform AMR greatly impact its performance, and lessons learned, in terms of OMD models, are shared. Due to physics involved in material decomposition, specifically the evolution of state variables, these problems don’t fully benefit from the advantages of AMR.

36 MATERIALS SCIENCE↗
NASA NTRS1992

A finite element formulation for scattering from electrically large 2-dimensional structures

A finite element formulation is given using the scattered field approach with a fictitious material absorber to truncate the mesh. The formulation includes the use of arbitrary approximation functions so that more accurate results can be achieved without any modification to the software. Additionally, non-polynomial approximation functions can be used, including complex approximation functions. The banded system that results is solved with an efficient sparse/banded iterative scheme and as a consequence, large structures can be analyzed. Results are given for simple cases to verify the formulation and also for large, complex geometries.

Ross, Daniel C.↗
NASA NTRS2017

Simulating Matrix Crack and Delamination Interaction in a Clamped Tapered Beam

Blind predictions were conducted to validate a discrete crack methodology based on the Floating Node Method to simulate matrix-crack/delamination interaction. The main novel aspects of the approach are: (1) the implementation of the floating node method via an 'extended interface element' to represent delaminations, matrix-cracks and their interaction, (2) application of directional cohesive elements to infer overall delamination direction, and (3) use of delamination direction and stress state at the delamination front to determine migration onset. Overall, good agreement was obtained between simulations and experiments. However, the validation exercise revealed the strong dependence of the simulation of matrix-crack/delamination interaction on the strength data (in this case transverse interlaminar strength, YT) used within the cohesive zone approach applied in this work. This strength value, YT, is itself dependent on the test geometry from which the strength measurement is taken. Thus, choosing an appropriate strength value becomes an ad-hoc step. As a consequence, further work is needed to adequately characterize and assess the accuracy and adequacy of cohesive zone approaches to model small crack growth and crack onset. Additionally, often when simulating damage progression with cohesive zone elements, the strength is lowered while keeping the fracture toughness constant to enable the use of coarser meshes. Results from the present study suggest that this approach is not recommended for any problem involving crack initiation, small crack growth or multiple crack interaction.

De Carvalho, N. V.↗
DOE OSTI2013

Pore-Scale Simulation of Intragranular Diffusion: Effects of Incomplete Mixing on Macroscopic Manifestations

Diffusive mass transfer into and out of intragranular micropores ("intragranular diffusion") plays an important role in the large-scale transport of some groundwater contaminants. We are interested in understanding the combined effect of pore-scale advection and intragranular diffusion on solute transport at the effective porous medium scale. To study this problem, we have developed a three-dimensional pore-scale model of fluid flow and solute transport that incorporates diffusion into and out of intragranular pore spaces. Our model is based on the Smoothed Particle Hydrodynamics (SPH) simulation method, which represents fluid and solid phases by a mesh-free particle discretization. In the pore spaces, fluid flow is simulated by discretizing the Navier-Stokes equations using the SPH approach. Solute transport is represented by advection, diffusion within the fluid phase, and diffusion between the fluid and solid phases. Our model is implemented on large-scale parallel computing hardware, allowing us to simulate millions of computational particles and represent fully three-dimensional systems of pores and grains with arbitrarily complex physical geometry. We have used this model system to perform numerical experiments using various model porous media systems, which allows us to draw comparisons between macroscopic measures computed from the pore-scale simulations (such as breakthrough curves) and those predicted by macroscopic formulations that assume complete mixing over the representative volume. In this paper we present results of 3D simulations of pore-scale flow and transport, including cases with and without intragranular diffusion, in two model porous media, one with randomly-packed uniform spherical grains and a second with randomly-packed spheres drawn from a binary grain size distribution. Breakthrough curves were computed from the 3D simulations at various transport distances. Comparable breakthrough curves were computed using 1D macroscopic models with parameters determined independently to the degree possible. Based on comparisons of the pore-scale and macroscopic model results, we draw two primary conclusions. First, non-Fickian behavior is persistent and ubiquitous at the scales considered, and most cases are better represented by a multi-rate mass transfer model even when there is no distinct secondary porosity (i.e., no intragranular diffusion). This suggests that diffusive mass transfer processes between preferential flow paths and relatively immobile zones within the primary porosity may have significant impact on transport, particular in low-concentration tails. Second, the application of mass transfer rate parameters based on an assumption of well-mixed concentrations at the pore-scale tend to overestimate the amount of mass transfer that occurs in heterogeneous pore geometries in which preferential flow leads to incomplete pore-scale lateral mixing.

Environmental Molecular Sciences Laboratory↗
DOE OSTI2026

Spatially Accelerated Winding Numbers for Curved Geometry

The generalized winding number (GWN) is a scalar field that supports robust containment queries on curved geometry, including non-watertight, overlapping, and nested boundary representations. While queries can be easily parallelized over samples, direct evaluation on parametric curves and surfaces remains costly for large and complex models. Fast, state-of-the-art GWN approaches leverage a spatial index to approximate the GWN, typically coupled with a Taylor expansion which approximates the GWN contribution for far clusters of geometric primitives. However, such methods operate only on discrete inputs such as triangle meshes and point clouds, and would introduce containment errors near boundaries if applied to curved input. We extend support for fast GWN evaluation over arbitrary collections of NURBS curves in 2D and trimmed NURBS patches in 3D via a Bounding Volume Hierarchy that stores efficiently precomputed moment data in the hierarchy nodes. When querying the hierarchy, approximations for far clusters are used alongside direct evaluation for nearby NURBS primitives, achieving sub-linear complexity while preserving the geometric features in the vicinity of the query point. Central to our performance improvements is an adaptive subdivision strategy for NURBS primitives during a preprocessing phase, creating better spatial partitions while retaining the same accuracy for containment decisions as a direct evaluation. We demonstrate the performance and accuracy of our approach across a large collection of 2D and 3D datasets.

Computer science↗
DOE OSTI2014

Mechanical Analysis of W78/88-1 Life Extension Program Warhead Design Options

Life Extension Program (LEP) is a program to repair/replace components of nuclear weapons to ensure the ability to meet military requirements. The W78/88-1 LEP encompasses the modernization of two major nuclear weapon reentry systems into an interoperable warhead. Several design concepts exist to provide different options for robust safety and security themes, maximum non-nuclear commonality, and cost. Simulation is one capability used to evaluate the mechanical performance of the designs in various operational environments, plan for system and component qualification efforts, and provide insight into the survivability of the warhead in environments that are not currently testable. The simulation efforts use several Sandia-developed tools through the Advanced Simulation and Computing program, including Cubit for mesh generation, the DART Model Manager, SIERRA codes running on the HPC TLCC2 platforms, DAKOTA, and ParaView. Several programmatic objectives were met using the simulation capability including: (1) providing early environmental specification estimates that may be used by component designers to understand the severity of the loads their components will need to survive, (2) providing guidance for load levels and configurations for subassembly tests intended to represent operational environments, and (3) recommending design options including modified geometry and material properties. These objectives were accomplished through regular interactions with component, system, and test engineers while using the laboratory's computational infrastructure to effectively perform ensembles of simulations. Because NNSA has decided to defer the LEP program, simulation results are being documented and models are being archived for future reference. However, some advanced and exploratory efforts will continue to mature key technologies, using the results from these and ongoing simulations for design insights, test planning, and model validation.

42 ENGINEERING↗
DOE OSTI2021

Modeling fission product diffusion in TRISO fuel particles with BISON

Diffusion of fission products in intact TRISO particles depends on particle geometry, fission product source rates, time, temperature, and temperature-dependent diffusion coefficients. Simulating this diffusion process requires models for source rates and diffusion coefficients, plus computation of the temperature field if not prescribed. In addition, simulation quality depends on discretization of the geometry, appropriate time stepping, and the accuracy of the solution method. In this paper, we explore the simulation of fission product diffusion in TRISO fuel particles using the finite element method via the fuel performance code Bison. Recent material model development has occurred in Bison for each material present in tri-structural isotropic (TRISO) fuel particles: the buffer, inner pyrolytic carbon, silicon carbide, and outer pyrolytic carbon layers, as well as the fuel kernel. Also, new mesh generation and fission product release fraction capabilities have been added. Diffusion capabilities are shown to converge to the correct solution via formal verification tests. A large number of code benchmarking problems are also given, with good results, showing that Bison’s computed release fractions closely match those of other software tools. Finally, a significant validation effort is detailed in which fission product release, measured as part of the AGR-1 capsule experiments, is compared to Bison outputs. Bison outputs compare very well to the experimental data and to PARFUME results.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2019

An Approach to Upscaling SPPARKS Generated Synthetic Microstructures of Additively Manufactured Metals

Additive manufacturing (AM) of metal parts can save time, energy, and produce parts that cannot otherwise be made with traditional machining methods. Near final part geometry is the goal for AM, but material microstructures are inherently different from those of wrought materials as they arise from a complex temperature history associated with the additive process. It is well known that strength and other properties of interest in engineering design follow from microstructure and temperature history. Because of complex microstructure morphologies and spatial heterogeneities, properties are heterogeneous and reflect underlying microstructure. This report describes a method for distributing properties across a finite element mesh so that effects of complex heterogeneous microstructures arising from additive manufacturing can be systematically incorporated into engineering scale calculations without the need for conducting a nearly impossible and time consuming effort of meshing material details. Furthermore, the method reflects the inherent variability in AM materials by making use of kinetic Monte Carlo calculations to model the AM process associated with a build.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Reverse-Mode Algorithmic Differentiation of an OpenMP-Parallel Compressible Flow Solver

y Reverse-mode algorithmic differentiation (AD) is an established method for obtaining adjoint derivatives of computer simulation applications. In computational fluid dynamics (CFD), adjoint derivatives of a cost function output such as drag or lift with respect to design parameters such as surface coordinates or geometry control points are a key ingredient for shape optimization, uncertainty quantification and flow control. The computational cost of CFD applications and their derivatives makes it essential to use high-performance computing hardware efficiently, including multi- and many-core architectures. Nevertheless, OpenMP is not supported in most AD tools, and previously shown methods achieve poor scalability of the derivative code. We present the AD of an OpenMP-parallelized finite volume compressible flow solver for unstructured meshes. Our approach enables us to reuse the parallelization of the original code in the computation of adjoint derivatives. The method works by identifying code segments that can be differentiated in reverse-mode without changing their memory access pattern. The OpenMP parallelization is integrated into the derivative code during the build process in a way that is robust to modifications of the original code and independent of the OpenMP support of the differentiation tool. We show the scalability of our adjoint CFD solver on test cases ranging from thousands to millions of finite volume mesh cells on CPUs with up to 16 threads as well as on an Intel XeonPhi card with 236 threads. We demonstrate that our approach is more practical to implement for production-sized CFD codes and produces more efficient adjoint derivative code than previously shown AD methods.

Huckelheim, Jan↗
DOE OSTI2022

304L Can Crush Validation Studies

Accurate prediction of ductile behavior of structural alloys up to and including failure is essential in component or system failure assessment, which is necessary for nuclear weapons alteration and life extensions programs of Sandia National Laboratories. Modeling such behavior requires computational capabilities to robustly capture strong nonlinearities (geometric and material), rate- dependent and temperature-dependent properties, and ductile failure mechanisms. This study's objective is to validate numerical simulations of a high-deformation crush of a stainless steel can. The process consists of identifying a suitable can geometry and loading conditions, conducting the laboratory testing, developing a high-quality Sierra/SM simulation, and then drawing comparisons between model and measurement to assess the fitness of the simulation in regards to material model (plasticity), finite element model construction, and failure model. Following previous material model calibration, a J 2 plasticity model with a microstructural BCJ failure model is employed to model the test specimen made of 304L stainless steel. Simulated results are verified and validated through mesh and mass-scaling convergence studies, parameter sensitivity studies, and a comparison to experimental data. The converged mesh and degree of mass-scaling are the mesh discretization with 140,372 elements, and a mass scaling with a target time increment of 1.0e-6 seconds and time step scale factor of 0.5, respectively. Results from the coupled thermal-mechanical explicit dynamic analysis are comparable to the experimental data. Simulated global force vs displacement (F/D) response predicts key points such as yield, ultimate, and kinks of the experimental F/D response. Furthermore, the final deformed shape of the can and field data predicted from the analysis are similar to that of the deformed can, as measured by 3D optical CMM scans and DIC data from the experiment.

36 MATERIALS SCIENCE↗
NASA NTRS1986

Statistical characterization of sea surface geometry for a wave slope field discontinuous in the mean square

Statistics of two-dimensional wave groups, of steep wave events, and of a cascade pattern manifested in the surface geometry in a developed sea state are derived. A systematic view is presented of the spectral moment problem, highlighting its physical meaning and showing its relationship with the theory of random fields. A solution of this problem is suggested that is appropriate to the case of wind-generated surface waves. The solution method treats the surface elevation field as specified on a spatial (temporal) running grid, developing an averaging procedure which employs the Taylor microscale as the mesh size. The technique is illustrated by first exposing errors in direct calculations of the effective surface impedance for a coherently reflected L-band radio wave. The technique is then used to study wave groups and steep waves for a Gaussian, two-dimensional, time-varying surface. Finally, the theory is applied to estimate breaking wave statistics,

Glazman, R. E.↗
NASA NTRS1993

Structured adaptive grid generation using algebraic methods

The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration of points to the grid redistribution scheme. The evaluation of the weighting mesh is accomplished by utilizing the weight function representing the solution variation and the equidistribution law. The selection of the weight function plays a key role in grid adaptation. A new weight function utilizing a properly weighted boolean sum of various flowfield characteristics is defined. The redistribution scheme is developed utilizing Non-Uniform Rational B-Splines (NURBS) representation. The application of NURBS representation results in a well distributed smooth grid by maintaining the fidelity of the geometry associated with boundary curves. Several algebraic methods are applied to smooth and/or nearly orthogonalize the grid lines. An elliptic solver is utilized to smooth the grid lines if there are grid crossings. Various computational examples of practical interest are presented to demonstrate the success of these methods.

Yang, Jiann-Cherng↗