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At least 397 records · Page 22

The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach. II

This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, S{sub {kappa}}{sup 3} ({kappa} > 0) and H{sub k}{sup 3} ({kappa} < 0), to the standard spherical waves in E{sup 3}. The curvature {kappa} is considered as a parameter and for any {kappa} we show how the radial Schroedinger equation can be transformed into a {kappa}-dependent Gauss hypergeometric equation that can be considered as a {kappa}-deformation of the (spherical) Bessel equation. The specific properties of the spherical waves in the spherical case are studied with great detail. These have a discrete spectrum and their wave functions, which are related with families of orthogonal polynomials (both {kappa}-dependent and {kappa}-independent), and are explicitly obtained.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Efficient Development of High Fidelity Structured Volume Grids for Hypersonic Flow Simulations

A new technique for the control of grid line spacing and intersection angles of a structured volume grid, using elliptic partial differential equations (PDEs) is presented. Existing structured grid generation algorithms make use of source term hybridization to provide control of grid lines, imposing orthogonality implicitly at the boundary and explicitly on the interior of the domain. A bridging function between the two types of grid line control is typically used to blend the different orthogonality formulations. It is shown that utilizing such a bridging function with source term hybridization can result in the excessive use of computational resources and diminishes robustness. A new approach, Anisotropic Lagrange Based Trans-Finite Interpolation (ALBTFI), is offered as a replacement to source term hybridization. The ALBTFI technique captures the essence of the desired grid controls while improving the convergence rate of the elliptic PDEs when compared with source term hybridization. Grid generation on a blunt cone and a Shuttle Orbiter is used to demonstrate and assess the ALBTFI technique, which is shown to be as much as 50% faster, more robust, and produces higher quality grids than source term hybridization.

Alter, Stephen J.↗

A Christoffel function weighted least squares algorithm for collocation approximations

Here, we propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation framework. Our investigation is motivated by applications in the collocation approximation of parametric functions, which frequently entails construction of surrogates via orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density defining the orthogonal polynomial family. Our proposed algorithm instead samples with respect to the (weighted) pluripotential equilibrium measure of the domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

97 MATHEMATICS AND COMPUTING↗

Natural geometric representation for electron local observables

An existence of the quartic identities for the electron local observables that define orthogonality relations for the 3D quantities quadratic in the electron observables is found. It is shown that the joint solution of the quartic and bilinear identities for the electron observables defines a unique natural representation of the observables. In the natural representation the vector type electron local observables have well-defined fixed positions with respect to a local 3D orthogonal reference frame. It is shown that the natural representation of the electron local observables can be defined in six different forms depending on a choice of the orthogonal unit vectors. The natural representation is used to determine the functional dependence of the electron wave functions on the local observables valid for any shape of the electron wave packet. -- Highlights: •Quartic identities that define the orthogonality relations for the electron local observables are found. •Joint solution of quartic and bilinear identities defines a unique natural representation of the electron local observables. •Functional dependence of the electron wave functions on the electron local observables is determined.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Coherent orthogonal polynomials

We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis (|x〉), for an alternative countable basis (|n〉). The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L{sup 2} functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L{sup 2} and, in particular, generalized coherent polynomials are thus obtained. -- Highlights: •Fundamental characteristic of orthogonal polynomials (OP): existence of a Lie algebra. •Differential recurrence relations of OP determine a unitary representation of a non-compact Lie group. •2nd order Casimir originates a 2nd order differential equation that defines the corresponding OP family. •Generalized coherent polynomials are obtained from OP.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Association Mapping of Cell Wall Synthesis Regulatory Genes and Cell Wall Quality in Switchgrass

Inefficient conversion of biomass to biofuels is one of the main barriers for biofuel production from such materials. Approximately half of polysaccharides in biomass remain unused by typical biochemical conversion methods. Conversion efficiency is influenced by the composition and structure of cell walls of biomass. Grasses such as wheat, maize, and rice, as well as dedicated perennial bioenergy crops, like switchgrass, make up ~55% of biomass that can be produced in the United States. Grass cell walls have a different composition and patterning compared with dicotyledonous plants, including the well-studied model plant, Arabidopsis. This project identified genetic determinants of cell wall composition in grasses using both naturally occurring genetic variation of switchgrass and gene network reconstruction and functional assays in rice. In addition, the project linked functional data in rice and other species to switchgrass improvement efforts through curation of the most abundant class of regulators in the switchgrass genome. Characterizing natural diversity of switchgrass for variation in cell wall composition and properties, also known as quality, provides an unbiased avenue for identifying biologically viable diversity in switchgrass cell walls. To characterizing natural diversity, this project generated cell wall composition and enzymatic deconstruction data for ~450 genotypes of the Switchgrass Southern Association Collection (SSAC), a diverse collection composed of 36 switchgrass accessions from the southern U.S. distribution of switchgrass. Comparing these data with other measures of cell wall quality for the same samples demonstrated the complementary nature of the diverse characterization platforms now being used for biomass characterization. Association of the composition data with ~3.2K single nucleotide variant markers identified six significant single nucleotide variant markers co-associated with digestibility and another compositional trait. These markers might be used to select switchgrass genotypes with improved composition in breeding programs for biofuel and forage production. Because the SSAC continues to be characterized by collaborators in the bioenergy community, the data generated will be used to identify additional markers in higher resolution genotyping data to approach identifying the genes and alleles that cause natural variation in switchgrass cell wall quality. For example, these markers can be surveyed in the 2100-member Oklahoma Southern and Northern Lowland switchgrass collections that this project also characterized. An orthogonal approach to biodiversity studies, using comparative functional genomics permits systematic querying of how much regulatory information is likely to be transferable from dicots to grasses and use of accumulated functional genomics resources for better-characterized grass species, such as rice, itself a biomass source in global agriculture and in certain regions. The project generated and tested a number of specific hypotheses regarding cell wall transcription factors and enzymes of grasses. To aid identification of cell wall regulators, the project assembled a novel, highdepth and -quality gene association network using a general linearized model scoring system to combine rice gene network data. Using known or putative orthologs of Arabidopsis cell wall biosynthesis genes and regulators, the project pulled from this network a cell wall sub-network that includes 96 transcription factors. Reverse genetics of a co-ortholog of the Arabidopsis MYB61 transcription factor in rice revealed that this regulatory node has evolved the ability to regulate grass-specific cell wall synthesis enzymes. A transcription factor with such activity has not been previously characterized to our knowledge, representing a major conclusion of this work. Changes in gene expression in a protoplast-based assay demonstrated positive or negative roles in cell wall regulation for eleven other transcription factors from the rice gene network. Eight of fifteen (53%) of these have not previously been examined for this function. Some of these may represent novel grass-diverged cell wall regulators, while others are likely to have this function across angiosperms. A parallel effort of this project to expand knowledge of enzymes that have evolved to function in grass cell wall synthesis, revealed that a grass-diverged enzyme in rice, OsAT 5 , ferulates monolignols that are naturally incorporated into grass cell walls. This finding opens potential natural selection avenues for improving biomass composition for downstream processing by weak base pretreatment. Thus, this project has significantly expanded knowledge of cell wall synthesis and regulation in rice, information that can be used in reverse genetics and synthetic biology approaches to re-engineer cell walls for improved production of biofuel and high-value products. To lay the foundation for translating these results directly for switchgrass improvement, the project employed a comparative phylogenetic analysis of the major group of cell wall transcription factors that have been found to function in cell wall regulation, the R 2 R 3 MYBs. This analysis concluded that known cell wall regulators are largely conserved across switchgrass, rice, maize, poplar, and Arabidopsis. This interpretation is also largely consistent with the gene network analysis described above, though both approaches provide evidence that some co-orthologs of Arabidopsis regulators have diminished or increased in importance based on gene expression patterns. Also, several clades containing dicot cell wall regulators have expanded, consistent with the evolution of new cell wall regulators. This latter result is supported by functional analysis of the R 2 R 3 MYB protein SWAM 1 in a collaboration between this project and the DOE-funded group of Dr. S. Hazen at the University of Massachusettes. The curation of the switchgrass genome through this project provides specific targets for future engineering of switchgrass cell wall regulation and may also facilitate identification of regulators that underlie the molecular markers that are genetically linked to differences in cell wall quality. With the goal of spurring further research and technological developments in lignocellulosic biofuel production, this work has been communicated to the bioenergy and cell wall communities though various presentations and publications. To date, three manuscripts have been published, two others are near to publication, three others are in an advanced state, and two to four more are likely to be written based on analyses still in progress. In addition, project participants have presented thirteen posters and talks at regional, national, and international meetings about aspects of this project. In sum, the work supported by this funding has made and communicated significant progress in identifying the genes that grasses use for cell wall synthesis and regulation, information that will be used by project participants and others to improve the efficiency of conversion of lignocellulosic biomass to biofuels.

09 BIOMASS FUELS↗

On the description of conical intersections—A continuous representation of the local topography of seams of conical intersection of three or more electronic states: A generalization of the two state result

For conical intersections of two states (I,J = I + 1) the vectors defining the branching or g-h plane, the energy difference gradient vector g{sup I,J}, and the interstate coupling vector h{sup I,J}, can be made orthogonal by a one parameter rotation of the degenerate electronic eigenstates. The representation obtained from this rotation is used to construct the parameters that describe the vicinity of the conical intersection seam, the conical parameters, s{sup I,J}{sub x} (R), s{sup I,J}{sub y} (R), g{sup I,J}(R), and h{sup I,J}(R). As a result of the orthogonalization these parameters can be made continuous functions of R, the internuclear coordinates. In this work we generalize this notion to construct continuous parametrizations of conical intersection seams of three or more states. The generalization derives from a recently introduced procedure for using non-degenerate electronic states to construct coupled diabatic states that represent adiabatic states coupled by conical intersections. The procedure is illustrated using the seam of conical intersections of three states in parazolyl as an example.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A recursive algorithm for Zernike polynomials

The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.

Davenport, J. W.↗

The Orbital Acceleration Research Experiment

The hardware and software of NASA's proposed Orbital Acceleration Research Experiment (OARE) are described. The OARE is to provide aerodynamic acceleration measurements along the Orbiter's principal axis in the free-molecular flow-flight regime at orbital attitude and in the transition regime during reentry. Models considering the effects of electromagnetic effects, solar radiation pressure, orbiter mass attraction, gravity gradient, orbital centripetal acceleration, out-of-orbital-plane effects, orbiter angular velocity, structural noise, mass expulsion signal sources, crew motion, and bias on acceleration are examined. The experiment contains an electrostatically balanced cylindrical proofmass accelerometer sensor with three orthogonal sensing axis outputs. The components and functions of the experimental calibration system and signal processor and control subsystem are analyzed. The development of the OARE software is discussed. The experimental equipment will be enclosed in a cover assembly that will be mounted in the Orbiter close to the center of gravity.

Blanchard, R. C.↗

Plates and shells containing a surface crack under general loading conditions

The severity of the underlying assumptions of the line-spring model (LSM) are such that verification with three-dimensional solutions is necessary. Such comparisons show that the model is quite accurate, and therefore, its use in extensive parameter studies is justified. Investigations into the endpoint behavior of the line-spring model have led to important conclusions about the ability of the model to predict stresses in front of the crack tip. An important application of the LSM was to solve the contact plate bending problem. Here the flexibility of the model to allow for any crack shape is exploited. The use of displacement quantities as unknowns in the formulation of the problem leads to strongly singular integral equations, rather than singular integral equations which result from using displacement derivatives. The collocation method of solving the integral equations was found to be better and more convenient than the quadrature technique. Orthogonal polynomials should be used as fitting functions when using the LSM as opposed to simpler functions such as power series.

Joseph, Paul F.↗

Improved Blending-Function Algebraic Generation Of Grids

Method of algebraic generation of grids used in computing flows incorporates blending-function techniques to obtain high degree of orthogonality of each grid on its bounding surfaces and to blend grid smoothly, making it conform to all boundaries. Demonstrated in two dimensions, and applied in continuing development of software for real-time interactive generation of three-dimensional grids with color graphical displays. Software used routinely in generating grids for computing flows of practical interest.

Ching-Chung Luh, Raymond↗

VHF Wide-Band, Dual-Polarization Microstrip-Patch Antenna

The figure depicts selected aspects of a very-high-frequency (VHF) microstrip patch antenna designed and built to satisfy requirements specific to an airborne synthetic-aperture radar system for measuring the thickness of sea ice. One of the requirements is that the antenna be capable of functioning over the relatively wide frequency band of 127 to 172 MHz corresponding to a fractional bandwidth of about 30 percent relative to a nominal mid-band frequency of 149.5 MHz. Another requirement is that the antenna be capable of functioning in either or both of two orthogonal linear polarizations. In addition, the antenna is required to be as compact and lightweight as possible. In a basic design according to generally accepted microstrip-patch-antenna engineering practice, one would ordinarily use a relatively thick dielectric substrate and multiple feed probes to obtain the desired combination of wide-band and dual-polarization capabilities. However, the combination of a thick substrate and multiple feeds would give rise to higher-order electromagnetic nodes, thereby undesirably contributing to cross polarization and to reduction of the isolation between feed probes. To counter these adverse effects while satisfying the requirements stated above, the design of this antenna incorporates several improvements over the basic design.

Huang, John↗

Stability Analysis of the Flow Over a Swept Forward-Facing Step Using PIV Base Flows

Step excrescences are a type of surface imperfection encountered on swept wings of commercial aircraft, commonly because accessibility requirements prevent creating the wing’s surface from a single panel. If the step height is too large, super-critical, the flow can undergo an early transition to turbulence, which can be highly detrimental for the performance of the wing. In the case of a forward-facing step on a swept wing in a low-disturbance environment, stationary crossflow vortices can develop a significant amplitude upstream of the step and hence dominate the structure of the boundary-layer flow over the step. The main goal of the present investigation is to illuminate the path to transition supported by the fascinatingly complex flow field in the direct downstream vicinity of a step with a super critical height. The high-resolution, stereographic Particle Image Velocimetry (PIV) measurement dataset presently available for this flow field provides a complete description of the laminar flow for the execution of BiGlobal stability analysis in a plane parallel to the step. Although the notorious sensitivity of stability results to the description of the base flow demands a very careful uncertainty analysis of those results, it is argued that this very fact can be leveraged to produce new insight into the supported perturbation dynamics. In performing the analysis, several unsteady mode families are discovered that display the explosive perturbation expected for early transition to be induced. In considering domain widths equal to an integer-multiple of the incident crossflow-vortex wavelength and analyzing an extent of 5 crossflow-vortex wavelengths parallel to the step, it is found that the stability results converge while increasing the domain width. It is demonstrated, moreover, that the results for the wider domains can be approximated by appropriately averaging the results on neighboring single-crossflow-vortex-wavelength domains covering the same region. Besides being useful for computational purposes, this observed property suggests interpreting the instability mechanism as a distorted primary mechanism rather than a “proper” secondary mechanism. This follows in the context of the secondary in-stability analysis of three-dimensional boundary layers, because the secondary mechanism is usually characterized by being localized in a pocket of strong shear, while the distorted primary mechanism typically has an infinite support in the direction parallel to the step. Even though the growth rates are found to be sensitive to the interrogation-window size inherent to the PIV post-processing procedure, the spatial structure of the eigenfunctions is found to be relatively insensitive. Lastly, the spatial structure of the eigenfunctions corresponding to all velocity components are matched with the shape functions determined by computing the Spectral Proper Orthogonal Decomposition (SPOD) of a time-resolved measurement of the perturbation content.

forward-facing step↗

Transition probabilities and oscillator strengths of E1 transitions in Fe XII and Fe XIV

Non-orthogonal orbitals in the multiconfiguration Hartree-Fock approach are used to calculate line strengths, oscillator strengths and transition probabilities for E1 transitions among the fine-structure levels of the 3s{sup 2}3p{sup 3}, 3s3p{sup 4}, 3s{sup 2}3p{sup 2}3d, 3s3p{sup 3}3d, 3p{sup 5} and 3s{sup 2}3p3d{sup 2} configurations in Fe XII and 3s{sup 2}3p, 3s3p{sup 2}, 3s{sup 2}3d, 3p{sup 3}, 3s3p3d, 3p{sup 2}3d, 3s3d{sup 2}, 3p3d{sup 2}, 3s{sup 2}4s, 3s{sup 2}4p, 3s3p4s and 3s{sup 2}4d configurations in Fe XIV. The lifetimes of excited levels belonging to these configurations of Fe XII and Fe XIV are also presented. An accurate representation of the levels has been obtained using spectroscopic and correlation radial functions. The wavefunctions exhibit large correlations and significant dependence of one-electron valence orbitals due to both the total and intermediate terms. The relativistic corrections are included through the one-body and two-body operators in the Breit-Pauli Hamiltonian. Progressively larger calculations are performed to check for important electron correlation contributions and for convergence of results. The atomic wavefunctions give excitation energies which are in close agreement with experiment. The present oscillator strengths and transition probabilities compare very well with previous large scale calculations. - Highlights: {yields} Fe XII and Fe XIV oscillator strengths and transition probabilities for E1 transitions. {yields} Energies and lifetimes of excited levels of Fe XII and Fe XIV. {yields} Multiconfiguration Hartree-Fock calculations with non-orthogonal radial functions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Algebraic grid generation with boundary orthogonality control

This paper presents a new approach in applying the blending function method to the generation of structured finite-difference meshes. It is shown that the surface equation developed by Coons (1967) can be easily adapted to generate two-dimensional meshes, and its extension to hyperspace allows the construction of three-dimensional meshes. The robust method in its basic form blends among the boundary curves/surfaces conforming to their intrinsic slopes. When parameter-controlled boundary-slope corrections are introduced, it is possible to establish varying degrees of orthogonality along the boundaries. Additional controls are made possible by moving points in the parameter space, which affects the movement of points in the physical space. The combination of these controls provides a simple and yet powerful means for exploiting the blending-function method to satisfy the needs of practical CFD applications. Results from a validation program for two-dimensions are shown for a variety of geometries.

Luh, Raymond Ching-Chung↗

Rainbow Fourier Transform

We present a novel technique for remote sensing of cloud droplet size distributions. Polarized reflectances in the scattering angle range between 135deg and 165deg exhibit a sharply defined rainbow structure, the shape of which is determined mostly by single scattering properties of cloud particles, and therefore, can be modeled using the Mie theory. Fitting the observed rainbow with such a model (computed for a parameterized family of particle size distributions) has been used for cloud droplet size retrievals. We discovered that the relationship between the rainbow structures and the corresponding particle size distributions is deeper than it had been commonly understood. In fact, the Mie theory-derived polarized reflectance as a function of reduced scattering angle (in the rainbow angular range) and the (monodisperse) particle radius appears to be a proxy to a kernel of an integral transform (similar to the sine Fourier transform on the positive semi-axis). This approach, called the rainbow Fourier transform (RFT), allows us to accurately retrieve the shape of the droplet size distribution by the application of the corresponding inverse transform to the observed polarized rainbow. While the basis functions of the proxy-transform are not exactly orthogonal in the finite angular range, this procedure needs to be complemented by a simple regression technique, which removes the retrieval artifacts. This non-parametric approach does not require any a priori knowledge of the droplet size distribution functional shape and is computationally fast (no look-up tables, no fitting, computations are the same as for the forward modeling).

Alexandrov, Mikhail D.↗

Symmetry-Breaking Dendrimer Synthons in Colloidal Crystal Engineering with DNA

Breaking symmetry in colloidal crystals is challenging due to the inherent chemical and structural isotropy of many nanoscale building blocks. If a non-particle component could be used to anisotropically encode such building blocks with orthogonal recognition properties, one could expand the scope of structural and compositional possibilities of colloidal crystals beyond what is thus far possible with purely particle-based systems. In this work, we report the synthesis and characterization of novel DNA dendrimers that function as symmetry-breaking synthons, capable of programming anisotropic and orthogonal interactions within colloidal crystals. When the DNA dendrimers have identical sticky ends, they hybridize with DNA-functionalized nanoparticles to yield three distinct colloidal crystals, dictated by dendrimer size, including a structure not previously reported in the field of colloidal crystal engineering, Si 2 Sr. When used as symmetry-breaking synthons (when the sticky ends deliberately consist of orthogonal sequences), the synthesis of binary and ternary colloidal alloys with structures that can only be realized through directional interactions is possible. Furthermore, by modulating the extent of shape anisotropy within the DNA dendrimers, the local distribution of the nanoparticles within the crystals can be directed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

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↗