Explicit Algebraic Expression Research Articles

Data-driven turbulence modeling for fluid flow and heat transfer in peripheral subchannels of a rod bundle

This study presents a comparison of the performance of machine learning (ML) techniques, specifically multi-dimensional gene expression programming (MGEP), tensor basis neural network (TBNN), and also proposes a novel universally interpretable machine learning architecture to model the turbulent scalar flux (UIML-s) to enhance turbulence models for fluid flows at different Prandtl numbers in channels with complex shapes of walls in the channel cross section. In particular, peripheral subchannels of rod bundles are of primary interest. However, the accuracy of mean velocity and scalar distributions predicted by commonly used turbulence models still poses a challenge compared to data extracted from high-fidelity eddy-resolving numerical simulations, particularly for engineering applications involving complex geometry flows. In the present study, by utilizing an explicit algebraic expression for the nonlinear Reynolds-stress term obtained through both the evolutionary MGEP optimization and TBNN, the secondary flow structure has been adequately predicted in the cross-wise mean velocity distributions in the square duct and the rectangular channel with three longitudinal rods. This structure is also observed in the data from the concurrent runs performed by direct numerical simulation (DNS) but is completely absent in the results produced by a baseline Reynolds-averaged Navier–Stokes (RANS) closure, which employs the linear eddy viscosity model for the Reynolds stress tensor. Comparison of MGEP and TBNN has shown their nearly equal performance in a square duct flow; however, MGEP works better for the more complex geometry channel with three rods. Furthermore, based on the velocity field produced by the RANS-MGEP model, the ML modification of the gradient diffusion hypothesis, integrated into the aforementioned novel RANS-ML model called as UIML-s, significantly improves the mean scalar distributions in a flow with three bumps serving as a prototype for the peripheral subchannel of rod bundle. The normalized root mean squared error decreases from 13.5% to 7.6%, bringing the predicted distributions closer to the DNS data, particularly in the near-wall region. Another approach, MGEP-s, also yields the acceptable results, which are nearly identical to those from UIML-s. These findings highlight the potential of using data-driven calibration of turbulence models with nonlinear closures to enhance the predictability for RANS simulations of fluid flows, heat, and mass transfer in channels with complex geometry.

New Theoretical Model for Mass Sensitivity of Love Wave Sensors

In this work we analyse basic characteristics of Love wave sensors implemented in waveguide structures composed of a lossy viscoelastic surface layer deposited on a lossless elastic substrate. It has to be noted that Love wave sensors working at ultrasonic frequencies have the highest mass density sensitivity $S_σ^(v_p )$ among all known ultrasonic sensors, such as QCM, Lamb wave or Rayleigh wave sensors. In this paper we have established an exact analytical formula for the mass density sensitivity $S_σ^(v_p )$ of the Love wave sensors in the form of an explicit algebraic expression. Subsequently, using this developed analytical formula, we compared theoretically the mass density sensitivity $S_σ^(v_p )$ for various Love wave waveguide structures, such as: (1) lossy PMMA surface layer on lossless Quartz substrate and (2) lossy PMMA on lossless Diamond substrate. The performed analysis shows that the mass density sensitivity $S_σ^(v_p )$ (real and imaginary part) for a sensor with a structure PMMA on Diamond is five times higher than that of a PMMA on Quartz structure. It was found that the mass density sensitivity $S_σ^(v_p )$ for Love wave sensors increases with the increase of the ratio: bulk shear wave velocity in the substrate to bulk shear wave velocity in the surface layer.

Local index formulae on noncommutative orbifolds and equivariant zeta functions for the affine metaplectic group

We consider the algebra A of bounded operators on L2(Rn) generated by quantizations of isometric affine canonical transformations. The algebra A includes as subalgebras noncommutative tori of all dimensions and toric orbifolds. We define the spectral triple (A,H,D) with H=L2(Rn,Λ(Rn)) and the Euler operator D, a first order differential operator of index 1. We show that this spectral triple has simple dimension spectrum: For every operator B in the algebra Ψ(A,H,D) generated by the Shubin type pseudodifferential operators and the elements of A, the zeta function ζB(z)=Tr(B|D|−2z) has a meromorphic extension to C with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.

Assessment of advanced RANS turbulence models for the stability analysis of low specific speed pump-turbines

Reversible pump-turbines, which can switch between pump and turbine operation, provide important support for the stabilization of the electrical grid. In order to fulfill their task, fast transition between the operation modes is required. This includes stable synchronization with the electrical grid during the process of turbine start at small guide vane openings (GVOs). However, the special geometry of pump-turbine impellers tends to produce unstable flow behavior, which can affect the synchronization process in a negative way.The prediction of the stability behavior with the help of CFD during the design process is not yet fully developed. The simulation of small specific speed pump-turbines operating at small GVOs sets high requirements on the underlying numerical model. To capture all relevant flow structures accordingly, 3-dimensional unsteady computation of the entire pump-turbine domain is required. Further, it has been shown that turbulence modeling is a key factor in order to reproduce the 4-quadrant characteristic in the unstable S-shape region. Vortex structures in the vaneless space and runner channels are considered the root-cause for the occurring instabilities. Standard two-equation models are not able to predict the stability behavior under these difficult circumstances (highly separated flows, small specific speed, small GVO angle). Models, which consider the anisotropic behavior of Reynolds-stresses have shown to produce better results.Considering the accurate prediction of mixing processes at off-design conditions, the focus was oriented to the anisotropy present in the turbulent structures. Standard models are often not sufficient to accurately predict vortices, which can have a huge impact on the performance, since based on the assumption of isotropic turbulence. Accordingly, they tend to dissipate and diffuse vortices too quickly. Improved models, which take the anisotropic nature of the Reynolds-stresses into account, can help in this context. The models can introduce the anisotropy via an explicit algebraic expression, or model directly the transport equations for the Reynolds-stresses. For this work the performance of a full Reynolds-stress model with improved coupling was investigated.In order to obtain the predictions using advanced turbulence models a particularly robust framework based on a pressure-based fully coupled approach was used. The goal of this work is the development and testing of an improved full Reynolds-stress model for the application in the stability analysis of pump-turbines. The focus lies thereby on separation behavior and mixing/vortex dissipation.

Shear stress distribution in tapered I-beams: Analytical expression and finite element validation

Shear stress distribution in tapered I-beams is remarkably different from that of prismatic beams as it is largely influenced by the vertical components of the inclined flange forces. Tapered I-beams can have a linear (or nonlinear) variation of web depths or flange widths or a combination of both, with monosymmetric or bisymmetric cross-sections. Evaluation of shear stress distribution at a section in tapered I-beams is a challenging task, and also closed form expressions have not been reported in literature. Hence, this paper presents a mechanics-based approach to arrive at a closed-form analytical expression for evaluating the shear stress distribution in generic 3D tapered I-beams. The major highlight of the proposed approach is that it accurately predicts the shear stress distribution in I-beams with simultaneously tapering web and flanges. The proposed approach was validated with 3D finite element models for a range of problems, and it was observed that the errors are always less than 4% at sections with L/D ratio greater than 3 and with flange inclination less than 10°. In particular, for the case of bisymmetric web-tapered beams and flange-tapered beams, the errors were less than 1%. Finally, the proposed approach lends itself to an explicit algebraic expression that will be useful for design engineers.

Hopfions in a lattice dimer model

In this paper, we study topological properties of 3D lattice dimer model. We demonstrate, that the dimer model on a bipartite lattice possesses topological defects, which are exactly characterized by Hopf invariant. We derive its explicit algebraic expression in terms of effective magnetic field of a dimer configuration. Thus, we solve the problem of topological classification of possible states in 3D lattice dimer model. Furthermore, since the lattice dimer model is known to be dual to spin ice, our work can be viewed as a proposal to search for hopfions in classical, as well as, artificial spin ice and related materials.

Nonlinear stability of moderately thick functionally graded sandwich shells with double curvature in thermal environment

Theoretical closed-form solutions and numerical results for nonlinear stability of the moderately thick functionally graded sandwich shells subjected to thermomechanical loadings are presented in this study. Two proposed material distribution models supported by elastic foundations are examined. The nonlinear strain field is deduced from the first-order shear deformation theory taking the stretching, bending and shear effects into consideration. The Bubnov–Galerkin procedure and harmonic balance principle are utilized to bring about the explicit algebraic expression for the shell static behaviors from governing equations derived from Hamilton's principle. Mechanical buckling loads and critical thermal rises for the shells in spherical, cylindrical, and hyperbolic paraboloid forms are obtained. The effect of geometry, elastic foundations, volume fraction index, material distribution models, buckling modes, and imperfections on the shell stability behaviors are considered in parametric studies. The yielding plateau in the thermal analysis of the spherical shells in case of temperature dependent characteristics is recognized for the first time. Verification studies are also conducted.

Approximate expressions for solutions to two kinds of transcendental equations with applications

In a broad spectrum of physics and engineering applications, transcendental equations have to be solved in order to determine their roots. Exact and explicit algebraic expression of solutions to such equations is, in general, impossible. Analytical approximate solutions to two kinds of transcendental equations with wide applications are presented. These approximate root formulas are systematically established by using the Padé approximant and show high accuracy. As an application of the proposed approximations, a highly accurate expression of the effective mass of the spring for a spring-mass system is obtained. The method described in this paper is also applied to other transcendental equations in physics and engineering applications.

Algebraic solution of differential geometric guidance command and time delay control

According to the three-dimensional geometry of the engagement, the explicit algebraic expression of differential geometric guidance command (DGGC) is proposed. Compared with the existing solutions, the algebraic solution is much simpler and better for the further research of the characteristics of DGGC. Time delay control (TDC) is a useful method to tackle the uncertainty problem of a control system. Based on TDC, taking the target maneuvering acceleration as a disturbance, the estimation algorithm of the target maneuvering acceleration is presented, which can be introduced in DGGC to improve its performance. Then, the augmented DGGC (ADGGC) is obtained. The numerical simulation of intercepting a high maneuvering target is conducted to demonstrate the effectiveness of ADGGC.

Torus knots and the rational DAHA

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n−1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov–Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q,t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.

Global Hopf bifurcation of differential equations with threshold type state-dependent delay

We develop global Hopf bifurcation theory of differential equations with state-dependent delay using the S1-equivariant degree and investigate a two-degree-of-freedom mechanical model of turning processes. For the model of turning processes we show that the extreme points of each vibration component of the non-constant periodic solutions can be embedded into a manifold with explicit algebraic expression. This observation enables us to establish analytical upper and lower bounds of the amplitudes of the periodic solutions in terms of the system parameters and to exclude certain periods. Using the achieved global bifurcation theory we reveal that if the relative frequency between the natural frequency and the turning frequency varies in a certain interval, then generically every bifurcated continuum of periodic solutions must terminate at a bifurcation point. This termination means that the underlying system with parameters in the stability region near the vertical asymptotes of the stability lobes is less subject to chatter. In the process, several sufficient conditions for the non-existence of non-constant periodic solutions are also obtained.

New explicit algebraic stress and flux model for active scalar and simulation of shear stratified cylinder wake flow

On the numerical simulation of active scalar, a new explicit algebraic expression on active scalar flux was derived based on Wikstrom, Wallin and Johansson model (aWWJ model). Reynolds stress algebraic expressions were added by a term to account for the buoyancy effect. The new explicit Reynolds stress and active scalar flux model was then established. Governing equations of this model were solved by finite volume method with unstructured grids. The thermal shear stratified cylinder wake flow was computed by this new model. The computational results are in good agreement with laboratorial measurements. This work is the development on modeling of explicit algebraic Reynolds stress and scalar flux, and is also a further modification of the aWWJ model for complex situations such as a shear stratified flow.

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.

Explicit algebraic influence coefficients: one-dimensional aquifer model

Explicit algebraic expressions of influence coefficients of an aquifer system are addressed. The methodology is straightforwardly exemplified using a one-dimensional (1D) semi-confined homogeneous aquifer model. It consists of the integration of the governing equation from two different viewpoints. On the one hand, the analytical solution to the governing equation is presented under an integral form involving the appropriate Green function. On the other hand, equations resulting from a finite difference discretization are processed to deduce a fully explicit algebraic expression of the nodal hydraulic head. This latter, arranged after the previously developed continuum solution, helps with deriving an explicit algebraic expression of influence coefficients corresponding to the studied aquifer model. Finally, the study is extended to an inhomogeneous aquifer model.

Selection of a fixed-income portfolio

A nonlinear programming problem in the space of two scalar parameters is applied to approximate the problem of optimization of a fixed-income portfolio by the quantile quality criterion. Since such securities have limited maturity time, portfolio is formed through reinvestment of the avails obtained upon maturity of securities. A quantile criterion is applied to determine the investment risk due to the uncertainty in future reinvestment. The geometry of the set of admissible values of parameters is studied and an explicit algebraic expression is derived for the optimum of the aim function in the nonlinear programming problem.

Antarctic ice volume and deep-sea temperature during the last 50 Myr: a model study

AbstractA simple quasi-analytical model is used to study the sensitivity of the Antarctic ice sheet to climate change. The model is axisymmetrical and has a profile that only depends on the ice-sheet radius. The climatic conditions are represented by three parameters: the altitude of the runoff line, the accumulation rate above the runoff line, and the balance gradient below the runoff line. The ice sheet may extend into the sea. At the grounding line the ice velocity is assumed to be proportional to the water depth. For this set-up, an explicit algebraic expression for the total mass budget of the ice sheet can be derived. After calibration of the model with respect to the present-day ice sheet, equilibrium states are studied for a wide range of temperatures. The model predicts a maximum ice volume (+3.4%) for a temperature that is 2.5 K above the present value. For a temperature increase of 7 K, mass loss by runoff and calving are about the same. In this case the ice volume is about 82% of the current value. The ice-sheet model is used to correct the Cenozoic deep-sea temperature record (δ18O record from benthic foraminifera in ocean sediments) for Antarctic ice volume. The model is forced with the oxygen isotope record, which is then corrected for the calculated ice volume. Therefore, the resulting deep-sea temperature and Antarctic ice-volume curves are mutually consistent. It is concluded that for the last 35×106 years the δ18O record truly is a mixed temperature/ice-volume record, in which the contributions from these parameters have the same order of magnitude.

Explicit analytical expressions for frequency equation and mode shapes of composite beams

A systematic procedure for the derivation of exact expressions for the frequency equation and mode shapes of composite beams undergoing free vibration is presented by using the symbolic computing package reduce. The effect of material coupling between the bending and torsional modes of deformation, which usually exists in composite beams due to ply orientation, is taken into account while developing the theory. The governing differential equations of motion of the bending–torsion coupled composite beam are solved analytically for bending displacements and torsional rotations in free vibration. For subsequent developments, the important case of the cantilever beam is chosen because of its application to aircraft wings. The boundary conditions for displacements and forces for the cantilever are imposed and the frequency equation is obtained, but first in the form of a determinant, and then, in the form of an explicit algebraic expression. The expressions for the mode shapes are also derived in explicit analytical form. The method is demonstrated by an illustrative example of a composite beam for which some comparative results are available in the literature. The future potential of this method, particularly in the context of aeroelastic optimisation of composite wings, is considerable because it is very accurate, computationally efficient and importantly, free from ill-conditioning problems usually associated with numerical matrix manipulation.

Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers

In this article we continue our investigations of one particle quantum scattering theory for Schrödinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In “Kirchhoff's rule for quantum wires” [J. Phys. A: Math. Gen. 32, 595–630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary n × n matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian. We also suggest an approach by which the S-matrix in our context may be obtained from “scattering experiments”, another aspect of the inverse problem. Finally we extend our previous discussion, how our approach is related to von Neumann's theory of selfadjoint extensions.

Algebraic Estimates, Stability of Local Zeta Functions, and Uniform Estimates for Distribution Functions

A method of ``algebraic estimates'' is developed, and used to study the stability properties of integrals of the form \int_B|f(z)|^{-\d}dV, under small deformations of the function f. The estimates are described in terms of a stratification of the space of functions \{R(z)=|P(z)|^{\e}/|Q(z)|^{\d}\} by algebraic varieties, on each of which the size of the integral of R(z) is given by an explicit algebraic expression. The method gives an independent proof of a result on stability of Tian in 2 dimensions, as well as a partial extension of this result to 3 dimensions. In arbitrary dimensions, combined with a key lemma of Siu, it establishes the continuity of the mapping c\ra \int_B|f(z,c)|^{-\d}dV_1\cdots dV_n when f(z,c) is a holomorphic function of (z,c). In particular the leading pole is semicontinuous in f, strengthening also an earlier result of Lichtin.

Cure kinetics of light‐activated polymers

The purpose of this article is to develop a mathematical model that describes photocure and photopolymerization kinetics. This model is neither wholly phenomenological nor mechanistic, but contains elements of both. We draw an analogy between the classical Avrami approach for first-order phase transformations and the kinetic phenomena that occur during photocuring, and take into account cure inhibition due to a decrease in mobility of the constituents. The result is an explicit algebraic two-parameter expression for the extent of cure versus time. More importantly, the two parameters have physical significance, and to some extent can be predicted a priori. The model is compared to three sets of unrelated data, and excellent agreement is obtained, except for part of the data at the onset of the reactions. Physical insight obtained by comparing our theory with experiments supports the existence of preferred sites for initiating the reaction, and indicates that irradiation is the rate-limiting step in the overall cure process in the limit of small irradiation intensities. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 2887–2894, 1998