Academic literature on the topic '010201 Approximation Theory and Asymptotic Methods'
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Journal articles on the topic "010201 Approximation Theory and Asymptotic Methods"
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HEWETT, D. P. "Shadow boundary effects in hybrid numerical-asymptotic methods for high-frequency scattering." European Journal of Applied Mathematics 26, no. 5 (June 30, 2015): 773–93. http://dx.doi.org/10.1017/s0956792515000315.
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The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost of conventional numerical methods for high-frequency wave scattering problems by enriching the numerical approximation space with oscillatory basis functions, chosen based on partial knowledge of the high-frequency solution asymptotics. In this paper, we propose a new methodology for the treatment of shadow boundary effects in HNA boundary element methods, using the classical geometrical theory of diffraction phase functions combined with mesh refinement. We develop our methodology in the context of scattering by a class of sound-soft non-convex polygons, presenting a rigorous numerical analysis (supported by numerical results) which proves the effectiveness of our HNA approximation space at high frequencies. Our analysis is based on a study of certain approximation properties of the Fresnel integral and related functions, which govern the shadow boundary behaviour.
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Lambaré, Gilles, Jean Virieux, Raul Madariaga, and Side Jin. "Iterative asymptotic inversion in the acoustic approximation." GEOPHYSICS 57, no. 9 (September 1992): 1138–54. http://dx.doi.org/10.1190/1.1443328.
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We propose an iterative method for the linearized prestack inversion of seismic profiles based on the asymptotic theory of wave propagation. For this purpose, we designed a very efficient technique for the downward continuation of an acoustic wavefield by ray methods. The different ray quantities required for the computation of the asymptotic inverse operator are estimated at each diffracting point where we want to recover the earth image. In the linearized inversion, we use the background velocity model obtained by velocity analysis. We determine the short wavelength components of the impedance distribution by linearized inversion of the seismograms observed at the surface of the model. Because the inverse operator is not exact, and because the source and station distribution is limited, the first iteration of our asymptotic inversion technique is not exact. We improve the images by an iterative procedure. Since the background velocity does not change between iterations. There is no need to retrace rays, and the same ray quantities are used in the iterations. For this reason our method is very fast and efficient. The results of the inversion demonstrate that iterations improve the spatial resolution of the model images since they mainly contribute to the increase in the short wavelength contents of the final image. A synthetic example with one‐dimensional (1-D) velocity background illustrates the main features of the inversion method. An example with two‐dimensional (2-D) heterogeneous background demonstrates our ability to handle multiple arrivals and a nearly perfect reconstruction of a flat horizon once the perturbations above it are known. Finally, we consider a seismic section taken from the Oseberg oil field in the North Sea off Norway. We show that the iterative asymptotic inversion is a reasonable and accurate alternative to methods based on finite differences. We also demonstrate that we are able to handle an important amount of data with presently available computers.
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Chiappinelli, Raffaele. "Variational Methods for NLEV Approximation Near a Bifurcation Point." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–32. http://dx.doi.org/10.1155/2012/102489.
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We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains ofℝN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.
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Grebenikov, E. A. "Concerning New Perturbation Methods in Solar System Dynamics." International Astronomical Union Colloquium 165 (1997): 399–404. http://dx.doi.org/10.1017/s0252921100046868.
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In this paper a new method of construction of the perturbation motion theory of celestial bodies, based on the averaging principle in view of frequency resonances, is stated. The first approximation of the asymptotic theory is the exact solution of the dynamics averaging equations, in which are included “secular” and “long-periodic” terms. The high-degree approximations are the exact solution of a known Krylov-Bogoliubov generalized equation. It is shown that these iterations are expressed in the analytical form by multiple Fourier series.
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Kal'chuk, I. V., Yu I. Kharkevych, and K. V. Pozharska. "Asymptotics of approximation of functions by conjugate Poisson integrals." Carpathian Mathematical Publications 12, no. 1 (June 12, 2020): 138–47. http://dx.doi.org/10.15330/cmp.12.1.138-147.
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Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.
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Hrabova, U. Z., and I. V. Kal'chuk. "Approximation of the classes $W^{r}_{\beta,\infty}$ by three-harmonic Poisson integrals." Carpathian Mathematical Publications 11, no. 2 (December 31, 2019): 321–34. http://dx.doi.org/10.15330/cmp.11.2.321-334.
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In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.
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Razavy, M., and B. Lenoach. "Reciprocity principle and the approximate solution of the wave equation." Bulletin of the Seismological Society of America 76, no. 6 (December 1, 1986): 1776–89. http://dx.doi.org/10.1785/bssa0760061776.
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Abstract In this paper, we examine the validity of reciprocity when certain approximations are applied to the wave equation. We focus on some of the approximation techniques which are frequently used in investigating seismic waves, particularly in the context of synthetic seismograms, namely, asymptotic ray theory and finite difference methods. The main result is that the reciprocity property is not necessarily preserved when an approximation is used. This conclusion is shown to be valid for both ray theory and finite difference techniques. We give some concrete examples as well as numerical results to illustrate the practical effect of nonreciprocity. We also show that reciprocity breaks down when analytical approximation is required to solve the problem in question.
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Raudenbush, Stephen W., and Anthony S. Bryk. "Examining Correlates of Diversity." Journal of Educational Statistics 12, no. 3 (September 1987): 241–69. http://dx.doi.org/10.3102/10769986012003241.
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Statistical methods are presented for studying “correlates of diversity”: characteristics of educational organizations which predict dispersion on the dependent variable. The conceptual framework for these methods distinguishes between variance heterogeneity that arises from educational program effects and heterogeneity that merely reflects heterogeneity of variance of inputs. The estimation theory is empirical Bayes, requiring probabilistic models both for the data and for the random dispersion parameters from each of many groups. Two strategies are considered, one based on exact distribution theory and the second based on an asymptotic normal approximation. The accuracy of the approximation is evaluated analytically and its use illustrated by an analysis of mathematics achievement data from a random sample of U.S. high schools.
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FEDOROV, A. V., N. D. MALMUTH, and V. G. SOUDAKOV. "Supersonic scattering of a wing-induced incident shock by a slender body of revolution." Journal of Fluid Mechanics 585 (August 7, 2007): 305–22. http://dx.doi.org/10.1017/s0022112007006714.
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The lift force acting on a slender body of revolution that separates from a thin wing in supersonic flow is analysed using Prandtl–Glauert linearized theory, scattering theory and asymptotic methods. It is shown that this lift is associated with multi-scattering of the wing-induced shock wave by the body surface. The local and global lift coefficients are obtained in simple analytical forms. It is shown that the total lift is mainly induced by the first scattering. Contributions from second, third and higher scatterings are zero in the leading-order approximation. This greatly simplifies calculations of the lift force. The theoretical solution for the flow field is compared with numerical solutions of three-dimensional Euler equations and experimental data at free-stream Mach number 2. There is agreement between the theory and the computations for a wide range of shock-wave strength, demonstrating high elasticity of the leading-order asymptotic approximation. Theoretical and experimental distributions of the cross-sectional normal force coefficient agree satisfactorily, showing robustness of the analytical solution. This solution can be applied to the moderate supersonic (Mach numbers from 1.2 to 3) multi-body interaction problem for crosschecking with other computational or engineering methods.
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Prat, V., S. Mathis, K. Augustson, F. Lignières, J. Ballot, L. Alvan, and A. S. Brun. "Asymptotic theory of gravity modes in rotating stars." Astronomy & Astrophysics 615 (July 2018): A106. http://dx.doi.org/10.1051/0004-6361/201832576.
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Context. Differential rotation has a strong influence on stellar internal dynamics and evolution, notably by triggering hydrodynamical instabilities, by interacting with the magnetic field, and more generally by inducing transport of angular momentum and chemical elements. Moreover, it modifies the way waves propagate in stellar interiors and thus the frequency spectrum of these waves, the regions they probe, and the transport they generate. Aims. We investigate the impact of a general differential rotation (both in radius and latitude) on the propagation of axisymmetric gravito-inertial waves. Methods. We use a small-wavelength approximation to obtain a local dispersion relation for these waves. We then describe the propagation of waves thanks to a ray model that follows a Hamiltonian formalism. Finally, we numerically probe the properties of these gravito-inertial rays for different regimes of radial and latitudinal differential rotation. Results. We derive a local dispersion relation that includes the effect of a general differential rotation. Subsequently, considering a polytropic stellar model, we observe that differential rotation allows for a large variety of resonant cavities that can be probed by gravito-inertial waves. We identify that for some regimes of frequency and differential rotation, the properties of gravito-inertial rays are similar to those found in the uniformly rotating case. Furthermore, we also find new regimes specific to differential rotation, where the dynamics of rays is chaotic. Conclusions. As a consequence, we expect modes to follow the same trend. Some parts of oscillation spectra corresponding to regimes similar to those of the uniformly rotating case would exhibit regular patterns, while parts corresponding to the new regimes would be mostly constituted of chaotic modes with a spectrum rather characterised by a generic statistical distribution.
Dissertations / Theses on the topic "010201 Approximation Theory and Asymptotic Methods"
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(6858680), Lida Ahmadi. "Asymptotic Analysis of the kth Subword Complexity." Thesis, 2019.
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The Subword Complexity of a character string refers to the number of distinct substrings of any length that occur as contiguous patterns in the string. The kth Subword Complexity in particular, refers to the number of distinct substrings of length k in a string of length n. In this work, we evaluate the expected value and the second factorial moment of the kth Subword Complexity for the binary strings over memory-less sources. We first take a combinatorial approach to derive a probability generating function for the number of occurrences of patterns in strings of finite length. This enables us to have an exact expression for the two moments in terms of patterns' auto-correlation and correlation polynomials. We then investigate the asymptotic behavior for values of k=a log n. In the proof, we compare the distribution of the kth Subword Complexity of binary strings to the distribution of distinct prefixes of independent strings stored in a trie.
The methodology that we use involves complex analysis, analytical poissonization and depoissonization, the Mellin transform, and saddle point analysis.
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(9757565), Yo-Sing Yeh. "Efficient Knot Optimization for Accurate B-spline-based Data Approximation." Thesis, 2020.
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Many practical applications benefit from the reconstruction of a smooth multivariate function from discrete data for purposes such as reducing file size or improving analytic and visualization performance. Among the different reconstruction methods, tensor product B-spline has a number of advantageous properties over alternative data representation. However, the problem of constructing a best-fit B-spline approximation effectively contains many roadblocks. Within the many free parameters in the B-spline model, the choice of the knot vectors, which defines the separation of each piecewise polynomial patch in a B-spline construction, has a major influence on the resulting reconstruction quality. Yet existing knot placement methods are still ineffective, computationally expensive, or impose limitations on the dataset format or the B-spline order. Moving beyond the 1D cases (curves) and onto higher dimensional datasets (surfaces, volumes, hypervolumes) introduces additional computational challenges as well. Further complications also arise in the case of undersampled data points where the approximation problem can become ill-posed and existing regularization proves unsatisfactory.
This dissertation is concerned with improving the efficiency and accuracy of the construction of a B-spline approximation on discrete data. Specifically, we present a novel B-splines knot placement approach for accurate reconstruction of discretely sampled data, first in 1D, then extended to higher dimensions for both structured and unstructured formats. Our knot placement methods take into account the feature or complexity of the input data by estimating its high-order derivatives such that the resulting approximation is highly accurate with a low number of control points. We demonstrate our method on various 1D to 3D structured and unstructured datasets, including synthetic, simulation, and captured data. We compare our method with state-of-the-art knot placement methods and show that our approach achieves higher accuracy while requiring fewer B-spline control points. We discuss a regression approach to the selection of the number of knots for multivariate data given a target error threshold. In the case of the reconstruction of irregularly sampled data, where the linear system often becomes ill-posed, we propose a locally varying regularization scheme to address cases for which a straightforward regularization fails to produce a satisfactory reconstruction.
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(8964155), Ahmad Bassam Barhoumi. "ORTHOGONAL POLYNOMIALS ON S-CURVES ASSOCIATED WITH GENUS ONE SURFACES." Thesis, 2020.
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We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.
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(6943460), Roozbeh Gharakhloo. "Asymptotic Analysis of Structured Determinants via the Riemann-Hilbert Approach." Thesis, 2020.
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In this work we use and develop Riemann-Hilbert techniques to study the asymptotic behavior of structured determinants. In chapter one we will review the main underlying
definitions and ideas which will be extensively used throughout the thesis. Chapter two is devoted to the asymptotic analysis of Hankel determinants with Laguerre-type and Jacobi-type potentials with Fisher-Hartwig singularities. In chapter three we will propose a Riemann-Hilbert problem for Toeplitz+Hankel determinants. We will then analyze this Riemann-Hilbert problem for a certain family of Toeplitz and Hankel symbols. In Chapter four we will study the asymptotics of a certain bordered-Toeplitz determinant which is related to the next-to-diagonal correlations of the anisotropic Ising model. The analysis is based upon relating the bordered-Toeplitz determinant to the solution of the Riemann-Hilbert problem associated to pure Toeplitz determinants. Finally in chapter ve we will study the emptiness formation probability in the XXZ-spin 1/2 Heisenberg chain, or equivalently, the asymptotic analysis of the associated Fredholm determinant.
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(8726829), Vaseem A. Shaik. "The Motion of Drops and Swimming Microorganisms: Mysterious Influences of Surfactants, Hydrodynamic Interactions, and Background Stratification." Thesis, 2020.
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Microorganisms and drops are ubiquitous in nature: while drops can be found in sneezes, ink-jet printers, oceans etc, microorganisms are present in our stomach, intestine, soil, oceans etc. In most situations they are present in complex conditions: drop spreading on a rigid or soft substrate, drop covered with impurities that act as surfactants, marine microbe approaching a surfactant laden drop in density stratified oceanic waters in the event of an oil spill etc. In this thesis, we extract the physics underlying the influence of two such complicated effects (surfactant redistribution and density-stratification) on the motion of drops and swimming microorganisms when they are in isolation or in the vicinity of each other. This thesis is relevant in understanding the bioremediation of oil spill by marine microbes.We divide this thesis into two themes. In the first theme, we analyze the motion of motile microorganisms near a surfactant-laden interface in homogeneous fluids. We begin by calculating the translational and angular velocities of a swimming microorganism outside a surfactant-laden drop by assuming the surfactant is insoluble, incompressible, and non-diffusing, as such system is relevant in the context of bioremediation of oil spill. We then study the motion of swimming microorganism lying inside a surfactant-laden drop by assuming the surfactant is insoluble, compressible, and has large surface diffusivity. This system is ideal for exploring the nonlinearities associated with the surfactant transport phenomena and is relevant in the context of targeted drug delivery systems wherein one uses synthetic swimmers to transport the drops containing drug. We then analyze the motion of a swimming organism in a liquid film covered with surfactant without making any assumptions about the surfactant and this system is relevant in the case of free-standing films containing swimming organisms as well as in the initial stages of the biofilm formation. In the second theme, we consider a density-stratified background fluid without any surfactants. In this theme, we examine separately a towed drop and a swimming microorganism, and find the drag acting on the drop, drop deformation, and the drift volume induced by the drop as well as the motility of the swimming microorganism.
Books on the topic "010201 Approximation Theory and Asymptotic Methods"
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Kolassa, John Edward. Series approximation methods in statistics. 2nd ed. New York: Springer, 1997.
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Series approximation methods in statistics. New York: Springer-Verlag, 1994.
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Gil, Amparo. Numerical methods for special functions. Philadelphia, Pa: Society for Industrial and Applied Mathematics, 2007.
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Bruijn, N. G. Asymptotic Methods in Analysis. Dover Publications, Incorporated, 2014.
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Kolassa, John E. Series Approximation Methods in Statistics. Springer London, Limited, 2013.
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Kolassa, John E. Series Approximation Methods in Statistics. Springer London, Limited, 2013.
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Series Approximation Methods in Statistics. Springer, 2006.
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Bruijn, N. G. de. Asymptotic Methods in Analysis. Dover Publications, Incorporated, 2014.
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Analytical and approximate methods: International conference at the Kygyz-Russian-Slavic University Bishkek, Kyrgyzstan, September 23-24, 2002. Aachen: Shaker, 2003.
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Numerical Methods for Special Functions. Society for Industrial Mathematics, 2007.
Find full textBook chapters on the topic "010201 Approximation Theory and Asymptotic Methods"
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Cam, Lucien Le. "Approximation by Exponential Families." In Asymptotic Methods in Statistical Decision Theory, 370–98. New York, NY: Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4612-4946-7_14.
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Cam, Lucien Le. "An Approximation Theorem for Certain Sequential Experiments." In Asymptotic Methods in Statistical Decision Theory, 346–69. New York, NY: Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4612-4946-7_13.
Full textConference papers on the topic "010201 Approximation Theory and Asymptotic Methods"
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Johansen, Per, Daniel B. Roemer, Henrik C. Pedersen, and Torben O. Andersen. "Analytical Thermal Field Theory Applicable to Oil Hydraulic Fluid Film Lubrication." In ASME/BATH 2014 Symposium on Fluid Power and Motion Control. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/fpmc2014-7844.
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An analytical thermal field theory is derived by a perturbation series expansion solution to the energy conservation equation. The theory is valid for small values of the Brinkman number and the modified Peclet number. This condition is sufficiently satisfied for hydraulic oils, whereby the analytical approach provides an alternative to existing computationally expensive numerical methods. The paper presents the dimensional analysis, which provides the foundation for the derivation of the analytical approximation. Subsequently, the perturbation method is applied in order to find an asymptotic expansion of the thermal field. The series solution is truncated at first order in order to obtain a closed form approximation. Finally a numerical thermohydrodynamic simulation of a piston-cylinder interface is presented, and the results are used for a comparison with the analytical theory in order to validate the modelling approach.
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Gommerstadt, B. Y. "The J and M Integrals for a Cylindrical Cavity in a Time-Harmonic Wave Field." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-65353.
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The invariant integrals are being widely used in the study of defects and fracture mechanics, mostly in elastostatics. However, the properties and the interpretation of these integrals in elastodynamics, especially in the case of time-harmonic excitation, have remained unexplored. Their study has a variety of engineering and geophysical implications, in particular, for the further development of non-destructive evaluation techniques. This contribution is focused on the derivation of the time average J integral for a cylindrical inhomogeneity and M integral for a cylindrical cavity placed in a monochromatic plane elastic wave of arbitrary wavelength. It is shown in the context of antiplane linear elasticity, that the J integral or the material force acting on the inhomogeneity resembles the radiation pressure force exerted on a dielectric cylinder by the normally incident electromagnetic wave. Based on the existing solution of this electrodynamic problem and the corresponding acoustic problem, the J integral is expressed as a function of the nondimensional wave number in the form of the partial wave expansion of the scattering theory. Employing the same classical method as for the J integral, the closed-form solution for the time average M integral for a traction-free cavity is also obtained as a function of the nondimensional wave number. The M integral, i.e., the expansion moment per unit length on an infinitely long circular cavity, is represented in terms of the scattering phase shifts as in the case of the J integral. Rather different expressions for the cavity are also derived for both integrals, which can be used more conveniently for numerical calculations, and these calculations are carried out for J and M integrals in a wide spectrum of frequencies. Asymptotic approximations of both integrals for low and high frequencies are presented. The long wavelength approximation, including the monopole and dipole contributions, has been provided for the J integral in the form of simple analytical expression. The value of M integral in the vanishing frequency limit is also presented. In the opposite short wavelength limit, the corresponding asymptotic values are derived for both integrals. These solutions which are valid for the empty cavity are extended to the case of inviscid fluid-filled cavity. The obtained results can be used in the area of non-destructive evaluation for the flaw characterization by ultrasonic scattering methods. The derived frequency dependence of the J and M integral can be related to the measurable far-field scattering amplitudes. This relationship is relevant to the inverse-scattering approach, which can be applied to the characterization of materials in an attempt to infer geometrical characteristics of flow structures.