Discrete Interpolation Research Articles

Extension of the Discrete Ordinates Interpolation Method to Combined Convection-Radiation Problems with Fluid Flow in Three-Dimensional Geometries Filled with Gray Participating Media

Combined convection-radiation problems in gray absorbing/emitting/scattering media are investigated numerically. The discrete ordinates interpolation method (DOIM) is incorporated into commercial software (FLUENT) using userdefined functions (UDFs). As preliminary tests, combined conduction/natural convection-radiation problems in a rectangular geometry are solved to validate the accuracy. The original DOIM is modified to solve the radiative heat transfer in an irregular geometry with internal obstacles. Then the combined turbulent forced convection-radiation problem is solved, and the radiation effects are discussed through the profiles of velocity and temperature. Fine agreement with other reference solutions is found. The DOIM promises reliable accuracy and excellent versatility for incorporation into fluid flow codes. Further computational issues are also discussed.

SU‐E‐J‐176: Further Investigation of Improved Reconstruction Algorithm for Tomosynthesis Plus Few Discrete Projections: Noisy Projections and TV Filtration

Purpose: Investigations over the past few years have demonstrated some of the benefits and shortfalls of tomosynthesis in radiation therapy for patient positioning or dose tracking. Tomosynthesis generates excellent image quality in one reconstruction plane; however, there is a loss of edge and frequency information in the additional planes due to the spatial incompleteness of the projection data. In our previous study, we demonstrated improved results from a new discrete frequency interpolation technique (DFIT), which is used to incorporate tomosynthesis limited arc projection data and three additional “filling” projections. Here, we continue investigation into DFIT by studying noisy projections for the data and incorporation of total variation (TV) noise reduction into the reconstruction processing. Methods: For this investigation, we simulated a Shepp‐Logan phantom to generate 93 parallel beam projections with ±5% uniform random noise. 90 of these projections were over a 90‐degree tomosynthetic arc. The remaining three projections were evenly spaced (i.e. 22.5, 45, and 67.5 degrees from the end of the arc) to help “fill in” the incomplete data. DFIT+TV was used to reconstruct images by interpolating frequency values into the gaps of the Fourier spectrum, inverse Fourier transforming from frequency to image space, adding the resultant image to a filtered backprojection (FBP) image, and finally applying a TV noise reduction step. Results: The images by traditional FBP, DFIT, FBP+TV and DFIT+TV are compared visually and using horizontal and vertical line profile plots. Additionally, contrast results are examined. The DFIT+TV images show improved edge delineation in the usual tomosynthesis “blurred” direction and improved reconstructed contrast. Conclusions: The addition of limited “filling” discrete projections and DFIT+TV can be used to improve the reconstruction accuracy and quality of short arc imaging. Future work will study DFIT+TV in the cone beam geometry.Research supported by Nucletron, B.V.

SU‐E‐J‐177: Improved Reconstruction Algorithm for Tomosynthesis Plus Few Discrete Projections

Purpose: Investigations over the past few years have demonstrated some of the benefits and shortfalls of tomosynthesis in radiation therapy for patient positioning or dose tracking. Tomosynthesis generates excellent image quality in one reconstruction plane; however, there is a loss of edge and frequency information in the additional planes due to the spatial incompleteness of the projection data. In this study, we investigate the results of a new discrete frequency interpolation technique (DFIT), which is used to incorporate tomosynthesis limited arc projection data and three additional “filling” projections. Methods: For the initial investigations, we simulated a Shepp‐Logan phantom with parallel beam geometry to generate 93 projections. 90 of these projections were over a tomosynthetic arc at one projection per degree. The remaining three projections were generated to fill in the next 90 degrees evenly (i.e. 22.5, 45, and 67.5 degrees from the end of the arc) to help “fill in” the incomplete data. Simple filtered backprojection (FBP) results show no improvement as FBP is designed for uniform distribution of projections. Per the projection slice theorem, the tomosynthesis data fills in a portion of the frequency plane. The remaining sparsely sampled area of the frequency plane is filled using linear interpolation from the end arc projections and the 3 discrete projections. The DFIT image is the sum of the FBP image and the inverse Fourier transform of the interpolated frequencies Results: The images by traditional FBP and DFIT are compared visually and using line profile plots. Additionally, contrast results are examined. The DFIT images show improved edge delineation in the usual tomosynthesis “blurred” direction and improves reconstructed contrast. Conclusions: The addition of limited “filling” discrete projections and DFIT can be used to improve the reconstruction accuracy and quality of short arc imaging.Research supported by Nucletron, B.V.

Research on Variable Time-Domain Discrete Interpolation Control in Flexible Roll Forming

Due to the complex motion control in flexible roll forming when processing the sheet metal with variable cross-section, it is required for high real-time and high precision control during the processing. Controlling servo motor by adopting the configuration programming is a common approach to achieve roller motion, but it will not be able or difficult to realize by the configuration programming for complex contour sheet metal without the rules such as hyperbolic, or the molding product in non-symmetric cross-section shape, etc. Based on the process characteristic of flexible roll forming and its motion control technology, this paper adopts the variable time-domain discrete interpolation control method to process the variable cross-section sheet metal. Experiments indicate that this control method provides a good operating method for the sheet metal forming with complex variable cross-section shape in flexible roll forming.

Discrete ordinates interpolation method incorporated into a flow and energy solver for solution of combined heat transfer problems

Combined conduction–radiation and natural convection–radiation in two-dimensional enclosures containing gray absorbing/emitting medium are numerically investigated. The discrete ordinates interpolation method (DOIM) is used to solve the radiative transfer equation (RTE). It is incorporated into a commercial software (FLUENT ®) by using user-defined function (UDF) to be used in a finite volume-based code for fluid flow computation. Two issues are critically examined: accuracy and versatility. Cases of combined conduction–radiation are considered first and the results are compared with other benchmark solutions to validate the accuracy. Additional problems are also tested to verify the capability of handling unstructured grid system and irregular geometry. Combined natural convection–radiation problem is then examined varying the optical thickness. The radiation effect is investigated through the profiles of velocity, temperature distributions and streamlines. The results are compared with discrete ordinates (DO) solutions, Rosseland solutions and P1 solutions which are offered by FLUENT ® package. The accuracy and other numerical characteristics of DOIM are scrutinized. The DOIM shows very successful results from the viewpoint of accuracy and grid compatibility. It is proved to be a reliable future numerical tool for combined heat transfer problems in engineering applications.

Error inequalities for quintic and biquintic discrete Hermite interpolation

In this paper we shall develop a class of discrete Hermite interpolates in one and two independent variables. Further, we offer explicit error bounds in ℓ∞ norm for the quintic and biquintic discrete Hermite interpolates. Some numerical examples are included to illustrate the results obtained.

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [ − 1 , 1 ] . It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [ − 1 , 1 ] . Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O ( h α | ln h | ) and O ( h | ln h | ) in the classes of functions H α ( [ − 1 , 1 ] ) and C 1 ( [ − 1 , 1 ] ) , respectively.

Discrete Interpolation Norms with Applications

We describe norm representations for interpolation spaces generated by finite-dimensional subspaces of Hilbert spaces. These norms are products of integer and noninteger powers of the Grammian matrices associated with the generating pair of spaces for the interpolation space. We include a brief description of some of the algorithms which allow the efficient computation of matrix powers. We consider in some detail the case of fractional Sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques. Numerical experiments are included.

Discrete ordinates interpolation method for radiative heat transfer problems in three-dimensional enclosures filled with non-gray or scattering medium

The discrete ordinates interpolation method (DOIM) is applied to three groups of problems of radiative heat transfer in three-dimensional rectangular enclosures containing non-gray or scattering medium. The original DOIM is first extended to a gray gas model using a new geometric interpolation scheme. It is applied to participating media for different scattering phase functions and optical thicknesses. For the non-gray gas model, the DOIM coupled with the narrow band-based weighted-sum-of-gray-gases (WSGG) model is developed. A few test problems with real gases such as pure H 2O and a mixture of CO 2, H 2O and N 2 are taken. The wall heat flux is calculated and compared with the exact solutions or reference values. All results of test problems are found to be reliable in this study. The DOIM closely reproduces the Monte Carlo reference solutions for different scattering phase functions and optical thicknesses. The non-gray gas results are compared with reference calculations based on the statistical narrow band model and they also show good agreements. The DOIM shows a remarkable merit in the computation time and the grid compatibility, to prove its usefulness for engineering applications.

Finite element formulation of the first- and second-order discrete ordinates equations for radiative heat transfer calculation in three-dimensional participating media

Two finite element methods (FEMs), FEDOM1 and FEDOM2 (standing for the first and the second finite element discrete ordinates methods, respectively), are formulated and numerically tested. The reference second-order discrete equation is modified in its scattering terms and is applied to the problems of absorbing/emitting and anisotropically scattering media by using the FEM. Numerical features of the developed FEMs are compared with one of the discrete ordinates interpolation method (DOIM), which uses a finite difference scheme. Prediction results of radiative heat transfer by these two FEMs are compared with reference solutions and verified in three-dimensional enclosures containing participating media. The results of FEDOM1 and FEDOM2 agree well with exact solutions for the problem of absorbing/emitting medium with various range of optical thickness. Generally, the two FEMs show more accurate results than DOIM. And FEDOM1 shows more accurate results than FEDOM2 in most of the test problems. Both of the developed FEMs show reasonable results compared with published Monte Carlo solutions for the tested absorbing/emitting and anisotropically scattering media. Although the FEDOM2 is not as accurate as the FEDOM1, it shows its own advantages that it reduces CPU time and memory space of dependent variable to half.

Discrete Fourier Analysis, Cubature, and Interpolation on a Hexagon and a Triangle

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planar region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.

Construction of blending surfaces by parametric discrete interpolation PDE splines

In this paper the problem of blending parametric surfaces is discussed. We present a constructing method of blending surfaces by parametric discrete interpolation PDE splines. These functions are obtained from some boundary conditions and a given interpolation data point set, by minimizing a functional associated with a partial differential equation (PDE) in a parametric finite element space. We formulate the corresponding problem and we establish a variational characterization of the solution and some convergence results. Finally, we show a graphical example.

Nonaxisymmetric radiative transfer in inhomogeneous cylindrical media with anisotropic scattering

In this paper, the control volume finite element method (CVFEM) is applied for the first time to solve nonaxisymmetric radiative transfer in inhomogeneous, emitting, absorbing and anisotropic scattering cylindrical media. Mathematical formulations as well as numerical implementation are given and the final discretized equations are based on similar meshes used for convective and conductive heat transfer in computational fluid dynamic analysis. In order to test the efficiency of the developed method, four nonaxisymmetric problems have been examined. Also, the grid dependence and the false scattering of the CVFEM are investigated and compared with the finite volume method and the discrete ordinates interpolation method.

A finite-volume variational multiscale method coupled with a discrete interpolation filter for large-eddy simulation of isotropic turbulence and fully developed channel flow

The variational multiscale (VMS) approach proposed by Hughes et al. is applied to the Smagorinsky model and developed within the context of a physical-space unstructured finite-volume solver. A specific explicit filtering procedure is also developed, based on the idea of the discrete interpolation filters, originally proposed by Marsden et al. Tests on decaying isotropic turbulence at infinite Reynolds number and fully developed channel flow at friction Reynolds number Reτ=395 are performed, using the dynamic Smagorinsky model for comparison. For the VMS version of the Smagorinsky model, the conservative choice of not adapting the model’s constant is made, giving good indications on the required adjustment directions in both isotropic flows and nonisotropic flows. Promising results are obtained in general by the VMS approach with the two test cases. An overdissipative behavior is observed in the isotropic turbulence test case, which does not have a strong spectral connotation and could probably be cured adjusting the model’s constant. A very good prediction of the levels of unsteadiness is produced by VMS in the channel flow test case, showing the potential of this approach for nonisotropic flows. In both of the test cases the VMS approach gives performances of the same order of accuracy of the dynamic Smagorinsky model used for comparison, being at the same time less computationally expensive.

DISCRETE INTERPOLATION BETWEEN MONOTONE PROBABILITY AND FREE PROBABILITY

We construct a sequence of states called m-monotone product states which give a discrete interpolation between the monotone product of states of Muraki in monotone probability and the free product of states of Avitzour and Voiculescu in free probability. We derive the associated basic limit theorems and develop the combinatorics based on non-crossing ordered partitions with monotone order starting from depth m. We deduce an explicit formula for the Cauchy transforms of the m-monotone central limit measures and for the associated Jacobi coefficients. A new type of combinatorics of inner blocks in non-crossing partitions leads to explicit formulas for the mixed moments of m-monotone Gaussian operators, which are new even in the case of monotone independent Gaussian operators with arcsine distributions.

Optical Joint Transform Encryption Using Binary Phase Difference Key Mask

This study describes optical encryption based on joint transform correlator architectures using a binary phase difference key mask, designed by discrete sinc function interpolation. The key mask has the required properties of space-limited and phase-mostly spectrum for accurate image recovery. Gray-level image encryption and decryption performance measures are discussed as well.

Optimal key mask design for optical encryption based on joint transform correlator architecture

We examine perfect recovery in the optical encryption system based on joint transform correlator architecture, which requires the key mask to be space-limited and phase-only in the frequency domain. Accordingly, a discrete sinc function interpolation is used to generate a binary phase difference mask for image encryption and decryption. Furthermore, the optimal binary phase difference mask is derived from the interpolation process best approximating the ideal sinc function interpolation. The simulation results confirm better recovery of the decrypted image for applying the proposed key masks to the optical encryption system. Especially, the optimal binary phase difference mask significantly enhances the recovery performance.

Deficient Discrete Quartic Spline Interpolation

In the present paper, the existence, uniqueness and convergence properties of discrete quartic spline interpolation over non-uniform mesh have been studied which match the given functional values at mesh points, interior points and second difference at boundary points.

A dynamic finite volume scheme for large-eddy simulation on unstructured grids

In recent years there has been considerable progress in the application of large-eddy simulation (LES) to increasingly complex flow configurations. Nevertheless a lot of fundamental problems still need to be solved in order to apply LES in a reliable way to real engineering problems, where typically finite-volume codes on unstructured meshes are used. A self-adaptive discretisation scheme, in the context of an unstructured finite-volume flow solver, is investigated in the case of isotropic turbulence at infinite Reynolds number. The Smagorinsky and dynamic Smagorinsky sub-grid scale models are considered. A discrete interpolation filter is used for the dynamic model. It is one of the first applications of a filter based on the approach presented by Marsden et al. In this work, an original procedure to impose the filter shape through a specific selection process of the basic filters is also proposed. Satisfactory results are obtained using the self-adaptive scheme for implicit LES. When the scheme is coupled with the sub-grid scale models, the numerical dissipation is shown to be dominant over the sub-grid scale component. Nevertheless the effect of the sub-grid scale models appears to be important and beneficial, improving in particular the energy spectra. A test on fully developed channel flow at Re τ = 395 is also performed, comparing the non-limited scheme with the self-adaptive scheme for implicit LES. Once again the introduction of the limiter proves to be beneficial.

Recovery of image blocks using the method of alternating projections

A technique for block-loss restoration in block-based image and video coding, dubbed recovery of image blocks using the method of alternating projections (RIBMAP), is developed. The algorithm is based on orthogonal projections onto constraint sets in a Hilbert space. For the recovery of a linear dimension N size block, a total of 8N vectors are extracted from the surrounding area of an N x N missing block. These vectors form a library from which the best matching spatial information for the missing block is extracted. Recovery vectors, including both undamaged and restored damaged pixels, are introduced. The vectors are used to find highly correlated information relating to the lost pixels. To assure continuity with the surrounding undamaged area, three additional convex constraints are formulated. Adherance to these sets is imposed using alternating projections. Simulation results using orthogonal projections demonstrate that RIBMAP recovers spatial structure faithfully. Simulation comparisons with other procedures are presented: Ancis and Giusto's hybrid edge-based average-median interpolation technique, Sun and Kwok's projections onto convex sets-based method, Hemami and Meng's interblock correlation interpolation approach, Shirani et al.'s modified interblock correlation interpolation scheme, and Alkachouh and Bellanger's fast discrete cosine transformation-based spatial domain interpolation algorithm. Characteristic of the results are those of the "Lena" JPEG image when one fourth of periodically spaced blocks in the image have errors. The peak signal-to-noise ratio of the restored image is 28.68, 29.99, 31.86, 31.69, 31.57, and 34.65 dB using that of Ancis and Giusto, Sun and Kwok, Hemami and Meng, Shirani et al., Alkachouh and Bellanger, and RIPMAP, respectively.