Finite Expansion Research Articles

An Analytical Solution for Fully Developed Forced Convection in Triangular Ducts

In this paper, an exact analytical solution for fully developed convective heat transfer in equilateral triangular ducts under constant heat flux at the walls is presented. The previous studies have been performed using numerical methods and to the knowledge of authors, this study is the first EXACT analytical solution about the heat convection in triangular ducts. Here, the finite series expansion method is used to derive the closed form of dimensionless temperature distribution. The Nusselt number and dimensionless temperature at the center of the cross section were calculated equal to 28/9 and 5/9, respectively. The present analytical solution could be useful in analysis of the heat convection in microfluidics and designing compact heat exchangers.

Three-dimensional micro-diffraction modeling

Squared elementary cells with correlated radiant point sources are presented as basic structures for characterizing the propagation of the field emitted by two-dimensional planar sources of any shape and in arbitrary state of spatial coherence. The field is transported on a finite expansion of nonparaxial modes, whose propagation in the micro-diffraction domain is discussed under both the diffraction and the interference conditions.

Thermodynamics and thermal decomposition for shape memory effects with crystallization based on dissipation and logarithmic strain

The present effort provides a 3-D thermodynamic framework generalizing the 1-D modeling of 2-way shape memory materials described by Westbrook et al. (J. Eng. Mater. Technol. 312:041010, 2010) and Chung et al. (Macromolecules 41:184–192, 2008), while extending the strain-induced crystallization and shape memory approaches of Rao and Rajagopal (Interfaces Free Bound. 2:73–94, 2000; Int. J. Solids Struct. 38:1149–1167, 2001), Barot and Rao (Z. Angew. Math. Phys. 57:652–681, 2006), and Barot et al. (Int. J. Eng. Sci. 46:325–351, 2008) to include finite thermal expansion within a logarithmic strain basis. The free energy of newly-formed orthotropic crystallites is assumed additive, with no strains in their respective configurations of formation. A multiplicative decomposition is assumed for the assumed thermoelastic orthotropic expansional strains of the respective crystallites. The properties of the crystallites are allowed to depend both on current temperature and their respective temperatures of formation. The entropy production rate relation is written in the frame rotating with the logarithmic spin and produces stress and entropy relations incorporating the integrated configurational free energies, and a driving term for the crystallization analogous to that obtained by the previous studies of Rao et al. The salient attributes of the 1-D modeling of Westbrook et al. are recovered, and applications are discussed.

Non-diffusive ignition of a gaseous reactive mixture following time-resolved, spatially distributed energy deposition

Systematic asymptotic methods are applied to the compressible conservation and state equations for a reactive gas, including transport terms, to develop a rational thermomechanical formulation for the ignition of a chemical reaction following time-resolved, spatially distributed thermal energy addition from an external source into a finite volume of gas. A multi-parameter asymptotic analysis is developed for a wide range of energy deposition levels relative to the initial internal energy in the volume when the heating timescale is short compared to the characteristic acoustic timescale of the volume. Below a quantitatively defined threshold for energy addition, a nearly constant volume heating process occurs, with a small but finite internal gas expansion Mach number. Very little added thermal energy is converted to kinetic energy. The gas expelled from the boundary of the hot, high-pressure spot is the source of mechanical disturbances (acoustic and shock waves) that propagate away into the neighbouring unheated gas. When the energy addition reaches the threshold value, the heating process is fully compressible with a substantial internal gas expansion Mach number, the source of blast waves propagating into the unheated environmental gas. This case corresponds to an extremely large non-dimensional hot-spot temperature and pressure. If the former is sufficiently large, a high activation energy chemical reaction is initiated on the short heating timescale. This phenomenon is in contrast to that for more modest levels of energy addition, where a thermal explosion occurs only after the familiar extended ignition delay period for a classical high activation reaction. Transport effects, modulated by an asymptotically small Knudsen number, are shown to be negligible unless a local gradient in temperature, concentration or velocity is exceptionally large.

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, multiple time scales, or chaotic dynamics. In particular, we prove that the differential equations for the coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of truncation order. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincare section of the system and that, as the scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

New Berry-Esseen bounds for non-linear functionals of Poisson random measures

This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations on the Poisson space, we derive a bound, which is strictly smaller than what is available in the literature. This is applied to sequences of multiple integrals and sequences of Poisson functionals having a finite chaotic expansion. This leads to new Berry-Esseen bounds in a Poissonized version of de Jong's theorem for degenerate U-statistics. Moreover, geometric functionals of intersection processes of Poisson $k$-flats, random graph statistics of the Boolean model and non-linear functionals of Ornstein-Uhlenbeck-Lévy processes are considered.

Discussion: “Vibration Analysis of Composite Beams With End Effects Via the Formal Asymptotic Method” (Kim, J.-S., and Wang, K. W., 2010, ASME J. Vib. Acoust., 132(4), p. 041003)

In their paper [1], Kim and Wang presented the vibration analysis of composite beams by using a finite element-based formal asymptotic expansion method. After a detailed mathematical treatment of the formal asymptotic method-based beam analysis (FAMBA), they considered various examples of composite beams with solid and closed thin-walled cross sections for verification purposes and the numerical results were compared with those from 3D FEM and other methods, such as the variational-asymptotic beam section analysis (VABS) [2].One particular case, the 15 deg CUS thin-walled composite box beam (whose ply material properties are listed in Table 3 of [1]), was analyzed with VABS by taking the stiffness model given in [2] and applying it to a typical Rankine–Timoshenko beam model. The normalized first and second eigenvalues computed by 3D FEM, FAMBA, VABS, etc. were plotted against the beam length-to-height ratio in Figs. 6 and 7. While it is clear that FAMBA-second predictions are reasonably close to those of 3D FEM, the authors (Kim and Wang) also show, and hence claimed, that results of VABS deviate significantly from those of 3D FEM, even qualitatively so for the second eigenvalue.The normalized first, second, and eighth mode shapes in Figs. 8 and 9 and 10 in [1] also show VABS as being unable to match 3D FEM predictions, with the deviation most obvious in Figs. 8(b), 9(a), and 10(b) in [1]. This is solely attributed by the authors to VABS inaccurately capturing the bending-shear coupling of the box beam. These claims call for the following comments:1. The stiffness model given in [2] (as calculated by versions of VABS till v.3.1), when used in tandem with a 1D beam analysis code based on a geometrically exact intrinsic beam theory (GEBT), provide results that are both qualitatively and quantitatively in close agreement to 3D FEM unlike the results reported in [1]. This can be seen in Figs. 1–5 which have now been recalculated by the authors (Kim and Wang). We can safely assume that the fault lies neither with the Rankine–Timoshenko model that the authors used nor the geometrically exact nonlinear beam theory (since the former seemed to have worked with other beam models and the latter is designed to capture all the geometrical nonlinearities obtainable by a beam model). Therefore, it seems logical to conclude that the source of discrepancy was an error in applying the given stiffness model correctly to the macroscopic analysis.2. The discrepancy has been examined by the authors (Kim and Wang). It is found that the sign convention used in VABS [2] was not consistently applied to the calculation of VABS results in [1]. The vibration analysis results from VABS and others have been corrected and replotted, as shown in Figs. 1–5.In conclusion, we find that although there are minor differences in the prediction of the bending-shear coupling for the CUS box beam between VABS [2] and FAMBA [1], both stiffness models yield very acceptable results in terms of vibration analysis. Specifically, VABS does not significantly deviate, qualitatively or quantitatively, from 3D FEM predictions. Additionally, the authors (Kovvali, Hodges, and Yu) would like to note that since the publication of [1], VABS has been updated and now provides results that are still further improved for some problems; see [3].

Coupled Model and Grid Adaptivity in Hierarchical Reduction of Elliptic Problems

In this paper we propose a surrogate model for advection---diffusion---reaction problems characterized by a dominant direction in their dynamics. We resort to a hierarchical model reduction where we couple a modal representation of the transverse dynamics with a finite element approximation along the mainstream. This different treatment of the dynamics entails a surrogate model enhancing a purely 1D description related to the leading direction. The coefficients of the finite element expansion along this direction introduce a generally non-constant description of the transversal dynamics. Aim of this paper is to provide an automatic adaptive approach to locally select the dimension of the modal expansion as well as the finite element step in order to satisfy a prescribed tolerance on a goal functional of interest.

A meshless procedure for transient heat conduction in functionally graded materials

AbstractA meshless numerical procedure is developed for analyzing the transient heat conduction problem in non‐homogeneous functionally graded materials. In the proposed method the time derivative of temperature is approximate by the finite difference. At each time step the original nonlinear boundary value problem is transform into a hierarchy of non‐homogeneous linear problem by used the homotopy analysis method. In this method a sought solution is presented by using a finite expansion in Taylor series, which consecutive elements are solutions of series linear value problems defining differential deformations. Each of linear boundary value problems with the corresponding boundary conditions is solved by using the method of fundamental solutions and radial basis functions which are used for interpolation of the inhomogeneous term. The accuracy of the obtained approximate solution is controlled by the number of components of the Taylor series, while the convergence of the process is monitored by an additional parameter of the method. Numerical experiments demonstrate the efficiency and accuracy of the present scheme in the solution of the heat conduction problem in nonlinear functionally graded materials. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Edge contact effect on thermal elastohydrodynamic lubrication of finite contact lines

Minimum lubricant film thickness and maximum pressure every so often appear close to roller ends. This study combines the Boussinesq–Cerruti half-space equations with a free boundary correction procedure for precise modeling of edge contact conditions. The thermal EHL model developed associates this representation to a standard finite difference of the energy equation, and to a modified finite difference expansion of the Couette term of the Reynolds equation. To complete the model, the Carreau expression describes the shear-thinning response of the lubricant. The investigation includes different roller profile corrections. The results show that a large radius crowning modification combined with a rounding of the corners constitutes the most effective profile adjustment.

First and second sound in a highly elongated Fermi gas at unitarity

We consider a Fermi gas at unitarity trapped by a highly elongated harmonic potential and solve the equations of two fluid hydrodynamics at finite temperature. The propagation of sound waves as well as the discretized solutions in the presence of weak axial trapping are considered. The relevant thermodynamic functions entering the hydrodynamic equations are discussed in the superfluid and normal regimes in terms of universal scaling functions. Both first sound and second sound solutions are calculated as a function of temperature and the role of the superfluid density is explicitly pointed out. The density fluctuations in the second sound wave are found to be large enough to be measured as a consequence of the finite thermal expansion coefficient of the gas. Emphasis is given to the comparison with recent experimental data.

A finite element method perturbation expansion for a coupled structural-acoustic system: two dimensional case

The structural acoustic coupled vibration problem is very important in many engineering applications such as quality control of vehicles. Formulating the problem using the finite element method leads to a nonsymmetric generalized eigenvalue problem. We show that the problem can be reformulated into uncoupled structural and acoustic problems by introducing a coupling strength parameter \({\varepsilon}\) as a multiplier applied to the off-diagonal coupling terms. The discretized uncoupled problems then lead to a pair of symmetric generalized eigenvalue problems which can be efficiently and independently solved. The solutions of the uncoupled problems are then used to compute the coupled solution using the perturbation method and the introduced coupling strength parameter. We confirm the adequacy of the method by investigating numerical examples for a two dimensional uniform mesh, whose exact solution is known, as well as arbitrary meshes for a car-like example.

Accurate 2d finite element calculations for hydrogen in magnetic fields of arbitrary strength

Recent observations of hundreds of hydrogen-rich magnetic white dwarf stars with magnetic fields up to 105 T (103 MG) have called for more comprehensive and accurate databases for wavelengths and oscillator strengths of the H atom in strong magnetic fields for all states evolving from the field-free levels with principal quantum numbers n≤10. We present a code to calculate the energy eigenvalues and wave functions of such states which is capable of covering the entire regime of field strengths B=0T to B∼109T. We achieve this high flexibility by using a two-dimensional finite element expansion of the wave functions in terms of B-splines in the directions parallel and perpendicular to the magnetic field, instead of using asymptotically valid basis expansions in terms of spherical harmonics or Landau orbitals. We have paid special attention to the automation of the program such that the data points for the magnetic field strengths at which the energy of a given state are calculated can be selected automatically. Furthermore, an elaborate method for varying the basis parameters is applied to ensure that the results reach a pre-selected precision, which also can be adjusted freely. Energies and wave functions are stored in a convenient format for further analysis, e.g. for the calculation of transition energies and oscillator strengths. The code has been tested to work for 300 states with an accuracy of better than 10−6 Rydberg across several symmetry subspaces over the entire regime of magnetic field strengths. Program summaryProgram title: H2dbCatalogue identifier: AERB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AERB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 158867No. of bytes in distributed program, including test data, etc.: 322671Distribution format: tar.gzProgramming language: Fortran95.Computer: 1 HP Compaq dc5750.Operating system: Linux.RAM: At least 2 GBClassification: 2.1.External routines: Lapack (http://www.netlib.org/lapack/)Nature of problem:The hydrogen problem in the presence of a magnetic field of arbitrary strength shall be solved for all states up to a principal quantum number of n=10. We obtain the full energy vs. field strength function within a certain precision.Solution method:We expand the wave functions in a 2d B-spline basis, vary the corresponding energy functional for the B-spline coefficients and solve the resulting generalised eigenvalue problem. The B-spline basis parameters are adapted iteratively to ensure the overall precision of our results.Restrictions:Non-relativistic Hamiltonian, non-moving atom.Unusual features:Automated analysis of the states at magnetic field strengths from B=0T to B=109T.Running time:Seconds to minutes per single result; hours to days for a full analysis.

Numerical Simulation on Stress Distribution and Failure of Composites Holes with Cold Expansion

A numerical investigation was conducted to apply cold expansion methods on composites structures under the guidance of cold expansion techniques of metal materials and interference-fit techniques of composites structures. Finite element expansion models with sleeves or without sleeves are established to study the stress distributions and failure conditions around cold expanded holes with different expansion values (0.5%, 1% and 1.5%). Results show that no damage occurs with expansion value 0.5% and using a sleeve in the expansion process can cause uniform stress distributions.

A fast wavelet expansion technique for evaluation of portfolio credit risk under the Vasicek multi-factor model

This paper presents a new methodology to compute value at risk (VaR) and the marginal VaR contribution (VaRC) in the Vasicek multi-factor model of portfolio credit loss. The wavelet approximation method can be useful to compute non-smooth distributions, often arising in small or concentrated portfolios. This paper contributes to this technique by extending the wavelet approximation method for the Vasicek one-factor model to the multi-factor model. Key features of the new algorithm presented in this paper are (i) a finite series expansion of the wavelet scaling coefficients, (ii) fast calculation methods to accelerate convergence of those series and (iii) an efficient spline interpolation method to calculate the Laplace transforms. This paper also illustrates the effectiveness of the algorithm through numerical examples.

Photonic Crystal Slab Add-Drop Filter

A new type of optical add drop filter (ADF) based on slab photonic crystals resonant cavities is proposed. ADF operation is based on coupling between the photonic crystal waveguides. Using the finite difference time domain (FDTD) method and plane wave expansion (PWE) method, the ADF characteristics and band structure of the filter, respectively are obtained. The proposed structure is optimized to work as an ADF. Dropping efficiency at 1560 nm and quality factor ( Q ) of our proposed structure are 90% and 195, respectively. The quantities of quality factor and transmission efficiency are suitable for optical applications. This structure is highly attractive for photonic integrated circuits (PICs).

Domain decomposition Fourier finite element method for the simulation of 3D marine CSEM measurements

We present a novel numerical method based on domain decomposition for the simulation of 3D geophysical marine controlled source electromagnetic (CSEM) measurements. Parts of the computational domain where it is reasonable to represent geoelectric properties in 2D, are discretized combining 2D mixed finite elements (FE) and Fourier expansion. The remaining part is discretized utilizing standard 3D FE methods. The method delivers high-accuracy simulations of marine CSEM problems with arbitrary 3D geometries while it considerably reduces the computational complexity of full 3D FE simulations for typical marine CSEM problems. For the particular scenarios considered in this work, the total CPU time required by the novel method is reduced approximately by a factor of five with respect to that needed by full 3D FE formulations.

Near-Optimal Encoding for Sigma-Delta Quantization of Finite Frame Expansions

In this paper we investigate encoding the bit-stream resulting from coarse Sigma-Delta quantization of finite frame expansions (i.e., overdetermined representations) of vectors. We show that for a wide range of finite-frames, including random frames and piecewise smooth frames, there exists a simple encoding algorithm—acting only on the Sigma-Delta bit stream—and an associated decoding algorithm that together yield an approximation error which decays exponentially in the number of bits used. The encoding strategy consists of applying a discrete random operator to the Sigma-Delta bit stream and assigning a binary codeword to the result. The reconstruction procedure is essentially linear and equivalent to solving a least squares minimization problem.

Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high‐index asymptotics

Weber integrals and Beltrami integrals are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functions with Gaussian or exponentially cut power‐law densities. Finite series representations of the integrals are derived for integer power‐law index μ, which admit high‐precision evaluation at low and moderate Bessel index n. At high n, numerically tractable uniform asymptotic approximations are obtained on the basis of the Debye expansion of modified spherical Bessel functions in the case of Weber integrals. The high‐n approximation of Beltrami integrals can be reduced to Legendre asymptotics. The Airy approximation of Weber and Beltrami integrals is derived as well, and numerical tests are performed over a wide range of Bessel indices by comparing the exact finite series expansions of the integrals with their high‐index asymptotics. Copyright © 2013 John Wiley & Sons, Ltd.

A Dynamic Cylindrical Cavity Expansion Model for the Penetration of Confined Concrete Targets

The anti-projectile performance of the confined concrete is better than that of the normal concrete. When the radius of the confining tube is relatively small, the confined concrete would be in comminuted stage during the penetrating process of projectiles. Based on the assumption that comminuted concrete materials obey Griffith strength theory, a finite cylindrical cavity expansion model for confined concrete targets is proposed. Numerical results show that the lateral confinement affects little on the cavity stress for relatively small confining rigidity, and the cavity stress increases significantly with the increase of the radius ratio of cavity to confining tube for relatively large lateral confining rigidity.