Rigorous Closed-form Solution Research Articles

Deep Unrolled Low-Rank Tensor Completion for High Dynamic Range Imaging.

The major challenge in high dynamic range (HDR) imaging for dynamic scenes is suppressing ghosting artifacts caused by large object motions or poor exposures. Whereas recent deep learning-based approaches have shown significant synthesis performance, interpretation and analysis of their behaviors are difficult and their performance is affected by the diversity of training data. In contrast, traditional model-based approaches yield inferior synthesis performance to learning-based algorithms despite their theoretical thoroughness. In this paper, we propose an algorithm unrolling approach to ghost-free HDR image synthesis algorithm that unrolls an iterative low-rank tensor completion algorithm into deep neural networks to take advantage of the merits of both learning- and model-based approaches while overcoming their weaknesses. First, we formulate ghost-free HDR image synthesis as a low-rank tensor completion problem by assuming the low-rank structure of the tensor constructed from low dynamic range (LDR) images and linear dependency among LDR images. We also define two regularization functions to compensate for modeling inaccuracy by extracting hidden model information. Then, we solve the problem efficiently using an iterative optimization algorithm by reformulating it into a series of subproblems. Finally, we unroll the iterative algorithm into a series of blocks corresponding to each iteration, in which the optimization variables are updated by rigorous closed-form solutions and the regularizers are updated by learned deep neural networks. Experimental results on different datasets show that the proposed algorithm provides better HDR image synthesis performance with superior robustness compared with state-of-the-art algorithms, while using significantly fewer training samples.

Analytical study of stiffened multibay planar coupled shear walls

Placing stiffening coupling beams at some levels of a coupled shear wall (CSW) structure is a proper way to enhance the coupling effects and reduce the interstory drifts of the structure. The resulting structural systems are called stiffened CSWs. To date, rigorous closed-form solutions for stiffened multibay planar CSWs subjected to lateral loading have not yet been derived. To solve this problem, a new approach based on the displacement method was proposed to establish the governing equations and boundary conditions, in which the rotation of the cross sections of walls was decomposed into two components: one associated with the axial deformations of wall piers and the other associated with the flexural and shear deformations of coupling beams. The closed-form solutions for the lateral displacement and rotation components were derived for standard load cases, which were verified to have high accuracy. The analytical solutions indicate that the distributions of the lateral displacements and rotation components along the structural height only depend on four dimensionless parameters, which are related to the composition ratio of the moment of inertia of the wall piers about their common centroid, the ratios of the span stiffness of the ordinary coupling beams to that of the wall piers and that of the stiffening beams, and the relative stiffness level, respectively. Parametric studies were further conducted to investigate the influence of these governing parameters on the degree of coupling, interstory drift ratios, and rotation components. It was found that the optimal stiffening levels for achieving the maximum degree of coupling commonly range from 0.06 to 0.50, and those for achieving the minimum of the maximum interstory drift ratio commonly range from 0.59 to 0.80. These findings provide useful implications for the optimization design of stiffened multibay CSWs.

An Analytical Solution of the Electric Field Excited by a Vertical Electric Dipole Above a Lossy Half-Space: From Radio to Microwave Frequencies

In this article, an analytical solution of the electromagnetic radiation from a vertical electric dipole (VED) above a lossy half-space is proposed. The expression for the traditional solution is associated with the Sommerfeld-type integral, which is difficult to evaluate due to its highly oscillating and slowly decaying integrand. The solution for the intermediate Hertz potential is decomposed into two integrals and a rigorous closed-form solution in the near and far-field regions are presented for each term. Then, the scattered electric-field components are calculated from the intermediate Hertz potential. A numerical evaluation of the solution for different lossy half-spaces, such as seawater, wet earth, dry earth, and lake water, validates the accuracy of the proposed solution at various frequencies, from radio to microwave frequencies, as well as different distances from the antenna in the near and far-field regions.

Solitary waves in two-dimensional nonlinear lattices

Our understanding of nonlinear dynamics critically hinges on rigorous closed-form solutions to nonlinear wave equations; few closed-form solutions have been achieved for physics-based interaction models beyond one-dimensional space. In this paper, we investigate the dynamics of a two-dimensional lattice with harmonic, weakly nonlinear, and strongly nonlinear interactions. Assuming nearest neighbor interaction, we derive the continuum approximation of the discrete system in the long-wavelength regime while keeping the Hamiltonian structure of the system. For a hexagonal lattice with nontrivial shear resistance, we surprisingly find that solitary wave solutions exist in certain directions related to the underlying symmetries of the lattice. The properties of the solitary waves are also studied by numerical simulations of the original discrete system. Besides being of fundamental scientific interest, the solitary wave solutions in nonlinear hexagonal lattices are anticipated to have applications in the design of shock absorbers, acoustic lens, or nondestructive structural testing devices, among many others.

Performance Bounds of Quaternion Estimators.

The quaternion widely linear (WL) estimator has been recently introduced for optimal second-order modeling of the generality of quaternion data, both second-order circular (proper) and second-order noncircular (improper). Experimental evidence exists of its performance advantage over the conventional strictly linear (SL) as well as the semi-WL (SWL) estimators for improper data. However, rigorous theoretical and practical performance bounds are still missing in the literature, yet this is crucial for the development of quaternion valued learning systems for 3-D and 4-D data. To this end, based on the orthogonality principle, we introduce a rigorous closed-form solution to quantify the degree of performance benefits, in terms of the mean square error, obtained when using the WL models. The cases when the optimal WL estimation can simplify into the SWL or the SL estimation are also discussed.

Semianalytical Calculation of the Collapse Load of Eccentrically Compressed, No-Tension Beams with an Annular Cross Section

AbstractA second-order equilibrium analysis is developed for the calculation of the collapse load of an eccentrically compressed beam with an annular cross section in the presence of a no-tension material. Unlike the case of a rectangular cross section, extensively treated in the literature, no rigorous closed-form solution is possible, in this case, but semianalytical results can be obtained, which may be useful for engineering applications.

Rigorous solution for optical diffraction of a sub-wavelength real-metal slit

We present a rigorous closed-form solution of the Sommerfeld integral for the optical scattering of a metal sub-wavelength slit. The two-dimensional (2D) field solution consists of the Surface Plasmon Polariton (SPP) mode at the metal surface and the 2D scattered field, which is the cylindrical harmonic of first order emitted by the electrical dipole and convolved with the 1D transient SPP along the interface. The creeping wave or quasi-cylindrical wave detected in the previous experiment is not an extra evanescent surface wave, but is the asymptotic behavior of the 2D scattered field at the proximity of the slit. Furthermore, our solution predicts a strong resonant enhancement of the scattered field at the proximity of the slit, depending on the materials and wavelength.

Impulse Representation of Sound Field due to a Rigid Wedge

An impulse representation for calculating a diffraction wave due to a rigid wedge is described. The method is an approximation of the Biot-Tolstoy rigorous closed-form solution for the diffraction of point source radiation by an infinite rigid wedge. The band-limited time-domain function can be reconstructed to the original waveform if it satisfies the sampling theorem, which assumes that sampling takes place at the lowest permissible sampling rate. Therefore, if the energy is concentrated between the first sampling intervals immediately after the rise time of the time-domain function, the rigorous solution can be approximated as a delta function. This paper shows the description methods of the diffraction field near the ridge in three-dimensional space. Using the proposed impulse representation, numerical simulation was performed and the calculation accuracy was examined.

Vertically Loaded Single Piles in Gibson Soil

The available closed-form solutions for vertically loaded piles have been, strictly speaking, limited to homogeneous soil, or nonhomogeneous soil with the shear modulus as a power of depth. The latter solutions —based on a zero shear modulus at ground surface—are generally sufficiently accurate for normally consolidated soil. For overconsolidated soil, however, there is generally a nonzero shear modulus at the surface, which may affect pile response. In this note, rigorous closed-form solutions are established to account for the nonhomo- geneity of soil profile with nonzero shear modulus at ground surface. The solutions are developed using a load transfer approach, and are shown to give satisfactory results in comparison with a more rigorous continuum- based numerical approach, when the proposed load transfer factors are adopted.

Equivalent systems for variable thickness circular plates

The research work in this paper is made up of a rigorous closed form solution speaking of the deflection of thin circular plates with variable thickness in one dimension. The solution is given in the form of an equivalent system of flat plates that replaces the original variable thickness plate. Due to the linearity of the governing equation, the sum of the deflections, bending moments, or bending stresses of the flat plates by the equivalent system is equal to that of the corresponding elements of the original variable thickness plate. A simplified equivalent system of constant flexural rigidity is also obtained which is used in reducing the mathematical complexity of the variable thickness problem. It should also be noted that any known method of analysis, closed form or numerical, may be used to solve the equivalent system of flat plates. Various cases of loading and thickness variation of circular plates are solved using both exact and simplified equivalent systems and the results are compared.

Proof of the Closed-Form Solution to the Mori Formula for Transport Coefficients

Three decades ago, the quantum Langevin-like equation (Mori formula) was used to reformulate the linear response theory (Kubo formula) in the continued fraction formalism. Since then, the Mori formula has often been used to evaluate transport coefficients, but only within the Born approximation. Very recently, a rigorous closed-form solution to the Mori formula for transport coefficients has been obtained. In the present study, we prove the equivalence of the closed-form solution to the Mori formula and the Kubo formula using the general relations between correlation functions and then comment on the Born approximation.

Study of pore pressures induced in laboratory dynamic consolidation

Knowledge of the pore pressure behavior during dynamic impacts on moderately low permeability soils is essential in averting possible liquefaction and effective field implementation of dynamic consolidation. Although field observations of dynamically induced pore pressures are abundant in the literature, analytical or numerical approaches for pore pressure prediction are scarce. Herein, the authors advance a simple technique to analytically model the laboratory dynamic consolidation by modifying the classical Terzaghi's static consolidation theory. Since the analytical prediction of dynamic surface stress and experimental verification are performed in a companion paper, the surface stress due to a dynamic impact is assumed to be known in this work. Then, the time dependent stress pulse is de-synthesized into a number of constant load steps to predict the subsequent pore pressure behavior. Since this methodology accounts for even the dynamic stress attenuation within the soil sample, a rigorous closed form solution has to be replaced by a numerical solution. It is shown how this solution rapidly converges when the load steps are made sufficiently small. Finally, the analytical predictions of dynamic pore pressure are verified by well controlled laboratory experiments performed on a special apparatus set up at the University of South Florida.

Equivalent Systems for Variable Thickness Plates

The research work in this paper consists of a rigorous closed form solution regarding the deflection of thin rectangular plates with variable thickness in one dimension. The rigorous closed form solution is given in the form of an equivalent system of flat plates that replaces the original variable thickness plate. The sum of the deflections of the flat plates is equal to the deflection of the original variable thickness plate. A simplified equivalent system is also obtained in order to reduce the mathematical complexity of the problem. Various variable thickness rectangular plates have been solved by using the method stated above, and the results are compared with those obtained by using the finite difference method.

Stress intensity factors for transversely loaded elastic plates and their application to predictions of crack arrest

Running cracks in elastic plates have been observed to be arrested as they pass under a zone of externally applied transverse compression. The compression is applied over circular zones on each face of the plate, directly opposite one another. This phenomenon has been observed in a variety of materials, and the purpose of this paper is to examine the mechanics of this crack arrest from a theoretical standpoint. The analysis was conducted for two cases: (1) a very thin plate, in which rigorous closed form solutions are obtained for the stress intensity factors. K, and (2) a thick plate, for which an approximate technique was devised to obtain an expression for the stress intensity factor. This expression was in terms of triple integrals, which required excessive computer time for their evaluation. Attention was therefore focused on the average stress intensity through the plate thickness, which can be evaluated economically. A comparison of the average K for the thick plate with the corresponding thin plate results showed a good agreement, which indicated that the approximations and numerical calculations provided reasonable results. The average K results were then applied in conjunction with fracture initiation and arrest criteria to the prediction of the conditions under which crack arrest could be achieved in an edge-cracked half-plane in tension, which is felt to be representative of an actual structure. The results predicted that a compression zone diameter of about one plate thickness was optimum for crack arrest, and that this technique is most suitable for high-strength, low-toughness materials with a high Poisson's ratio.