Relation Preserving Research Articles

A note on dynamic relational systems

It is known that weak and strong definabilities are defined for multiple-source approximation systems by Khan and Banerjee. This paper presents, in a more general setting a discussion on definabilities of sets in dynamic relational systems. We prove that the inverse-image of a weak or strong definable set with respect to relation preserving function is also weak or strong definable set, respectively. On the way, we show that the inverse-image of a reduct of attribute set is also a reduct under the object function of an information system homomorphism. Further, we give the connections between definable sets and the topology determined by the intersections of the topologies of reflexive relations. A quasi-uniformity is a filter on the cartesian product of a given universe satisfying certain conditions. In fact, every quasi-uniformity is a dynamic relational system where the relations are reflexive. In this respect, we discuss on the connections between approximation systems and (quasi) uniformities.

A dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations

In this work, we have derived an optimized upwind compact difference scheme for achieving excellent spatial resolution. The derived numerical scheme adds numerical diffusion which is strictly restricted to a high wavenumber region in order to control numerical instabilities, as well as to achieve de-aliasing ability. More importantly, the derived numerical scheme has excellent dispersion relation preserving (DRP) property. The applicability of the proposed scheme in simulating real flow problems has been demonstrated by solving flow inside a lid driven cavity, a transitional flow past AG24 aerofoil and a two-dimensional decaying turbulence flow.

Machine learning based decision support for many-objective optimization problems

Multiple Criteria Decision-Making (MCDM) based Multi-objective Evolutionary Algorithms (MOEAs) are increasingly becoming popular for dealing with optimization problems with more than three objectives, commonly termed as many-objective optimization problems (MaOPs). These algorithms elicit preferences from a single or multiple Decision Makers (DMs), a priori or interactively, to guide the search towards the solutions most preferred by the DM(s), as against the whole Pareto-optimal Front (POF). Despite its promise for dealing with MaOPs, the utility of this approach is impaired by the lack of-objectivity; repeatability; consistency; and coherence in DM?s preferences. This paper proposes a machine learning based framework to counter the above limitations. Towards it, the preference-structure of the different objectives embedded in the problem model is learnt in terms of: a smallest set of conflicting objectives which can generate the same POF as the original problem; the smallest objective sets corresponding to pre-specified errors; and the objective sets of pre-specified sizes that correspond to minimum error. While the focus is on demonstrating how the proposed framework could serve as a decision support for the DM, its performance is also studied vis-i?-vis an alternative approach (based on dominance relation preservation), for a wide range of test problems and a real-world problem. The results mark a new direction for MCDM based MOEAs for MaOPs.

A Stability Criterion for Semi-Discrete Difference Schemes of Hyperbolic Conservation Laws on Uniform Grids

In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.

Wave properties of fourth-order fully implicit Runge–Kutta time integration schemes

A global spectral analysis of fully implicit, two-stage, fourth-order Runge–Kutta method (IRK2) is presented here. The essential numerical properties of the IRK2 time-integration scheme, viz., nodal amplification factor, phase speed and group velocity have been quantified and plotted at all the nodal points of the computational domain as functions of the CFL (Courant–Friedrichs–Lewy number) and wavenumber for various explicit and implicit spatial discretization schemes. Results show that IRK2 scheme produces neutral stability at all interior nodes for all CFL numbers. This has been shown that while using IRK2 scheme one has to use refined grid at high CFL numbers for dispersion relation preservation property.

Direct numerical simulation of two-dimensional wall-bounded turbulent flows from receptivity stage

Deterministic route to turbulence creation in 2D wall boundary layer is shown here by solving full Navier-Stokes equation by dispersion relation preserving (DRP) numerical methods for flow over a flat plate excited by wall and free stream excitations. Present results show the transition caused by wall excitation is predominantly due to nonlinear growth of the spatiotemporal wave front, even in the presence of Tollmien-Schlichting (TS) waves. The existence and linear mechanism of creating the spatiotemporal wave front was established in Sengupta, Rao and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] via the solution of Orr-Sommerfeld equation. Effects of spatiotemporal front(s) in the nonlinear phase of disturbance evolution have been documented by Sengupta and Bhaumik [Phys. Rev. Lett. 107, 154501 (2011)], where a flow is taken from the receptivity stage to the fully developed 2D turbulent state exhibiting a k(-3) energy spectrum by solving the Navier-Stokes equation without any artifice. The details of this mechanism are presented here for the first time, along with another problem of forced excitation of the boundary layer by convecting free stream vortices. Thus, the excitations considered here are for a zero pressure gradient (ZPG) boundary layer by (i) monochromatic time-harmonic wall excitation and (ii) free stream excitation by convecting train of vortices at a constant height. The latter case demonstrates neither monochromatic TS wave, nor the spatiotemporal wave front, yet both the cases eventually show the presence of k(-3) energy spectrum, which has been shown experimentally for atmospheric dynamics in Nastrom, Gage and Jasperson [Nature 310, 36 (1984)]. Transition by a nonlinear mechanism of the Navier-Stokes equation leading to k(-3) energy spectrum in the inertial subrange is the typical characteristic feature of all 2D turbulent flows. Reproduction of the spectrum noted in atmospheric data (showing dominance of the k(-3) spectrum over the k(-5/3) spectrum in Nastrom et al.) in laboratory scale indicates universality of this spectrum for all 2D turbulent flows. Creation of universal features of 2D turbulence by a deterministic route has been established here for the first time by solving the Navier-Stokes equation without any modeling, as has been reported earlier in the literature by other researchers.

Space-time discretizing optimal DRP schemes for flow and wave propagation problems

The present paper reports constrained optimization of explicit Runge–Kutta (RK) schemes, coupled with optimal upwind compact scheme to achieve dispersion relation preservation (DRP) property for high performance computing. Essential ideas of optimization employed in arriving at the proposed time integration scheme are extension of the earlier work reported in Rajpoot et al. (J Comput Phys 2010;229:3623–51). This is in turn an application of the correct error evolution equation in Sengupta et al. (J Comput Phys 2007;226:1211–8). Resultant DRP scheme demonstrated the idea for explicit spatial central difference schemes. Present work is similar, extending it for near-spectral accuracy compact schemes. Practical utility of the developed method is demonstrated by solution of model problems and for flow problems by solving Navier–Stokes equation, some of which cannot be solved by conventional schemes, as the problem of rotary oscillation of cylinder. Developed method is calibrated with: (i) flow past a circular cylinder performing rotary oscillation at Re = 150 and (ii) flow inside a 2D lid-driven cavity (LDC) at Reynolds numbers of Re = 1000 and Re = 10,000. Quantitative and qualitative comparisons show excellent match for rotary oscillation cylinder cases with the experimental results of Thiria et al. (J Fluid Mech 2006;560:123–47). Results for LDC for Re = 1000 are compared with that in Botella & Peyret (Comp Fluids 1998;27:421–33) and results for Re = 10,000 are compared with recent published ones showing triangular vortex in the core.

Distinguishing variance embedding

Nonlinear dimensionality reduction is a challenging problem encountered in a variety of high dimensional data analysis. Based on the different geometric intuitions of manifolds, maximum variance unfolding (MVU) and Laplacian eigenmaps are designed for detecting the different aspects of data set. In this paper, combining the ideas of MVU and Laplacian eigenmaps, we propose a new nonlinear dimensionality reduction method called distinguishing variance embedding (DVE), which unfolds the data manifold by maximizing the global variance subject to the proximity relation preservation constraint originated in Laplacian eigenmaps. We illustrate the algorithm on easily visualized examples of curves and surfaces, as well as on actual images of faces, handwritten digits, and rotating objects.

Numerical simulation of sound generation in a mixing layer by the finite difference lattice Boltzmann method

We simulated aerodynamic sound in a two-dimensional mixing layer using the finite difference lattice Boltzmann method (FDLBM). We introduced a finite difference scheme, called the dispersion relation preserving (DRP) scheme, into the FDLBM to carry out an accurate simulation of aerodynamic problems. The scheme is designed such that the dispersion relation of the finite difference scheme is the same as that of the original partial differential equations and is useful for acoustic simulations. A turbulent flow field was simulated by large-eddy simulation (LES), using the Smagorinsky model, and the results were compared with those from a direct simulation based on the Navier–Stokes equations to confirm the usefulness of this method. The combination of the FDLBM and the DRP scheme was shown to be very effective for direct simulations of aerodynamic sound.

Combining local and global information for nonlinear dimensionality reduction

Nonlinear dimensionality reduction is a challenging problem encountered in a variety of high dimensional data analysis, including machine learning, pattern recognition, scientific visualization, and neural computation. Based on the different geometric intuitions of manifolds, maximum variance unfolding (MVU) and Laplacian eigenmaps are designed for detecting the different aspects of dataset. In this paper, combining the ideas of MVU and Laplacian eigenmaps, we propose a new nonlinear dimensionality reduction method called distinguishing variance embedding (DVE). DVE unfolds the dataset by maximizing the global variance subject to the proximity relation preservation constraint originated in Laplacian eigenmaps. We illustrate the algorithm on easily visualized examples of curves and surfaces, as well as on the actual images of rotating objects, faces, and handwritten digits.

A new combined stable and dispersion relation preserving compact scheme for non-periodic problems

A new compact scheme is presented for computing wave propagation problems and Navier–Stokes equation. A combined compact difference scheme is developed for non-periodic problems (called NCCD henceforth) that simultaneously evaluates first and second derivatives, improving an existing combined compact difference (CCD) scheme. Following the methodologies in Sengupta et al. [T.K. Sengupta, S.K. Sircar, A. Dipankar, High accuracy schemes for DNS and acoustics, J. Sci. Comput. 26 (2) (2006) 151–193], stability and dispersion relation preservation (DRP) property analysis is performed here for general CCD schemes for the first time, emphasizing their utility in uni- and bi-directional wave propagation problems – that is relevant to acoustic wave propagation problems. We highlight: (a) specific points in parameter space those give rise to least phase and dispersion errors for non-periodic wave problems; (b) the solution error of CCD/NCCD schemes in solving Stommel Ocean model (an elliptic p.d.e.) and (c) the effectiveness of the NCCD scheme in solving Navier–Stokes equation for the benchmark lid-driven cavity problem at high Reynolds numbers, showing that the present method is capable of providing very accurate solution using far fewer points as compared to existing solutions in the literature.

Design and analysis of a new filter for LES and DES

Spatial filters have been rigorously analyzed here for stabilization and dispersion relation preservation properties using spectral-matrix theory. Traditional LES faces problems of instability, aliasing and additional complexity of SGS modeling. Here, an alternative is proposed where unfiltered governing equation is solved, followed by spatial upwind filtering, without requiring any SGS models. This filter retains the resolution of high accuracy methods and removes energy, like the action of hyper-viscosity used in spectral methods. As examples, solutions for (a) accelerated flow past a NACA-0015 airfoil held normal to the flow and (b) transitional flow past a NLF airfoil, have been reported.

A Comprehensive New Methodology for Formulating FDTD Schemes With Controlled Order of Accuracy and Dispersion

Numerical dispersion errors in the wave-equation-finite-difference-time-domain (WE-FDTD) method have been treated by higher order schemes, coefficient modification schemes, dispersion relation preserving and non-standard schemes. In this work, a unified methodology is formulated for the systematic generation of WE-FDTD schemes tailored to the spectrum of the excitation. The methodology enables the scheme designer to gradually trade order of accuracy (OoA) for lower dispersion errors in a controlled manner at the cost of sacrificing low frequency behavior, that is not deemed critical for this type of excitation. The methodology is shown to encompass both existing and new schemes. Stability analysis is carried out concurrently with the generation of each scheme. Using a stencil size of 3 and 5 temporal and spatial samples, respectively, long term errors of a scheme designed for a specific pulse are compared with the standard (4,4) scheme that has the same computational complexity, via simulation of a modulated pulse that propagates over a million time steps.

Schema and constraints-based matching and merging of Topic Maps

In this paper, we propose a multi-strategic matching and merging approach to find correspondences between ontologies based on the syntactic or semantic characteristics and constraints of the Topic Maps. Our multi-strategic matching approach consists of a linguistic module and a Topic Map constraints-based module. A linguistic module computes similarities between concepts using morphological analysis, string normalization and tokenization and language-dependent heuristics. A Topic Map constraints-based module takes advantage of several Topic Maps-dependent techniques such as a topic property-based matching, a hierarchy-based matching, and an association-based matching. This is a composite matching procedure and need not generate a cross-pair of all topics from the ontologies because unmatched pairs of topics can be removed by characteristics and constraints of the Topic Maps. Merging between Topic Maps follows the matching operations. We set up the MERGE function to integrate two Topic Maps into a new Topic Map, which satisfies such merge requirements as entity preservation, property preservation, relation preservation, and conflict resolution. For our experiments, we used oriental philosophy ontologies, western philosophy ontologies, Yahoo western philosophy dictionary, and Wikipedia philosophy ontology as input ontologies. Our experiments show that the automatically generated matching results conform to the outputs generated manually by domain experts and can be of great benefit to the following merging operations.

Analysis of a new high resolution upwind compact scheme

A new high-resolution upwind compact scheme is presented and analyzed, along with several previously proposed schemes, for numerical dispersion-dissipation, anisotropy, phase damping, dispersion relation preservation property and numerical stability. The schemes are tested on problems of the propagation of a initially discontinuous wave and of the transport of a sharp scalar cone. A benchmark problem of aeroacoustics is solved with the present scheme, giving satisfactory resolution. The scheme compares well with the existing schemes for the range of properties considered desirable in high resolution schemes.

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.

One-dimensional model of valveless pumping in a closed loop and a numerical solution

The development of a theoretical model of valveless pumping and its numerical solution is presented in this work, applied for the case of a closed hydraulic loop, consisting of a soft and a rigid tube. A periodic compression and decompression of the soft tube causes a unidirectional flow, under certain conditions. The integration of the governing flow equations (continuity and momentum), over the tube cross-sectional area results in a quasi-one-dimensional unsteady model. A system of nonlinear partial differential equations of the hyperbolic type is solved numerically, employing three finite difference schemes: Lax-Wendroff, MacCormack, and Dispersion Relation Preserving, the last being the most accurate one. When the excitation takes place far from the midlength of the soft tube, a phase difference between the pressures at the two edges of each tube is developed, being in advance the one that is closer to the excitation area. Increasing the tube occlusion or the length of the excited part of the loop the mean flow rate increases and maximizes at the natural frequency of the loop. The direction of the maximum mean flow rate, for a given tube occlusion, is from the excitation area toward the edge of the stiff tube, which is located closer to the excitation area. Varying the excitation frequency both above and below the resonance frequency, local flow rate extremes appear, manifesting the complex character of the valveless pumping phenomenon.

A new flux–vector splitting compact finite volume scheme

A new high resolution finite volume method is proposed here that uses flux–vector splitting as the building block to represent the physical processes. The high accuracy projection in space is obtained here, inter alia, by using some compact schemes of Sengupta et al. [1] and Lele [2] – that have extremely high accuracy in spectral plane. This produces a new class of finite volume (FV) scheme for spatial discretization, that has been analyzed for spectral accuracy, numerical stability and dispersion relation preservation (DRP) property following the full-domain analysis method of [1] using two different time integration strategies. These combination of spatial and temporal methods have been tested for two linear wave problems, reported in [1,3] and the solution of Burgers’ equation is compared with the analytical results presented in Adams and Shariff [4]. The shock capturing method is based on the proposed technique in [5], that does not require the usage of any nonlinear differencing techniques or limiters. To show that the proposed method can also handle irregular grids, the Burgers’ equation is solved using non-uniform grids. Also to demonstrate the utility of the proposed scheme to solve practical problems, the benchmark problem of nonlinear wave propagation in a one-dimensional shock-tube has been solved with the initial data as given in [23] by solving the Euler equation. The analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method, at the same time providing a neutrally stable scheme when used with four stage Runge–Kutta scheme. Also the same scheme displays extended ranges of wave numbers and circular frequencies over which DRP property is valid. The computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems.

Analysis of central and upwind compact schemes

Central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space, numerical stability and dispersion relation preservation. A von Neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously. This allows one to evaluate various boundary closures and their effects on the domain interior. The same method can be used for stability analysis performed for the semi-discrete initial boundary value problems (IBVP). This analysis tells one about the stability for every resolved length scale. Some well-known compact schemes that were found to be G-K-S and time stable are shown here to be unstable for selective length scales by this analysis. This is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming. To demonstrate the asymptotic stability of the resultant schemes, numerical solution of the wave equation is compared with analytical solution. Furthermore, some of these schemes are used to solve two-dimensional Navier–Stokes equation and a computational acoustic problem to check their ability to solve problems for long time. It is found that those schemes, that were found unstable for the wave equation, are unsuitable for solving incompressible Navier–Stokes equation. In contrast, the proposed compact schemes with improved boundary closure and an explicit higher-order upwind scheme produced correct results. The numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite DRP property.

Quench simulation in superconducting cables using optimized DRP scheme

A high order finite difference scheme is introduced to deal with thermo-hydraulic quench simulation in superconducting magnets, specifically in the case of cable-in-conduit conductors (CICC). Liquid helium flow in CICC is governed by the unsteady convection-diffusion conservation equations, in the low Mach number and high Reynolds number flow region. In the past several years many methods have been considered for quench simulation in superconducting magnets, for example, finite element, finite volume, and finite difference methods. One of the main challenges is the accurate simulation of flow during a superconductor quench, given the large gradients in pressure and temperature that arise during conductor normalization. These simulations are key to proper magnet protection design. The Dispersion Relation Preserving (DRP) scheme is applied to the quench simulation problem. The computational results are compared to those obtained using a well-proven commercial quench simulation code (Gandalf), as well as experimental data. Preliminary results are promising.