A variable boundary method for modelling two dimensional free surface flows with moving boundaries

A coordinate transformation method is proposed for modelling unsteady, depth-averaged shallow water equations for a open channel flow with moving lateral boundaries. The transformation technique which maps the changing domain onto a fixed domain and solves the governing equations in the mapped domain, facilitates the numerical treatment of an irregular boundary configuration. The transformed equations are solved on a staggered grid with a conditionally stable, explicit finite difference scheme. Several numerical experiments are carried out corresponding to different situations, viz., flow with constant discharge, flow with constant discharge and a closed boundary at the downstream, flow in a converging channel with constant discharge and finally flow with varying discharge. The experiments are used to verify the model ability to predict free surface elevation, circulatory pattern and displacement of the boundaries. The simulated results such as displaced area, depth, displacement–time and flow-field are used to evaluate the effects of excess discharge at the upstream on the movement of lateral boundaries.     

Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains

This paper presents a signal and image recovery scheme by the method of alternating projections onto convex sets in optimum fractional Fourier domains. It is shown that the fractional Fourier domain order with minimum bandwidth is the optimum fractional Fourier domain for the method employing alternating projections in signal recovery problems. Following the estimation of optimum fractional Fourier transform orders, incomplete signal is projected onto different convex sets consecutively to restore the missing part. Using a priori information in optimum fractional Fourier domains, superior results are obtained compared to the conventional Fourier domain restoration. The algorithm is tested on 1-D linear frequency modulated signals, real biological data and 2-D signals presenting chirp-type characteristics. Better results are obtained in the matched fractional Fourier domain, compared to not only the conventional Fourier domain restoration, but also other fractional Fourier domains.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Modelling of Lamb wave propagation in composite plate excited by surface bonded piezoelectrical actuators

Transient propagation of Lamb waves in a multilayered infinite composite plate with the arbitrary elastic anisotropy of each layer is considered in this paper. Integral representations of wave fields, an algorithm of computing the Green's matrix of the multilayered medium in Fourier domain and some methods of the Fourier transform inversion are used as basic analytical research instruments. The piezoelectrical transducers frequently used for Lamb wave excitation are modelled in this paper as surface sources with stresses concentrated on the boundaries of actuators. The contribution of each wave mode to a whole amplitude, focusing of Lamb waves and influence of the geometry of the surface source are discussed in this paper by numerical examples. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Image deblurring by sparsity constraint on the Fourier coefficients

This paper is concerned with the image deconvolution problem. For the basic model, where the convolution matrix can be diagonalized by discrete Fourier transform, the Tikhonov regularization method is computationally attractive since the associated linear system can be easily solved by fast Fourier transforms. On the other hand, the provided solutions are usually oversmoothed and other regularization terms are often employed to improve the quality of the restoration. Of course, this weighs down on the computational cost of the regularization method. Starting from the fact that images have sparse representations in the Fourier and wavelet domains, many deconvolution methods have been recently proposed with the aim of minimizing the ?₁-norm of these transformed coefficients. This paper uses the iteratively reweighted least squares strategy to introduce a diagonal weighting matrix in the Fourier domain. The resulting linear system is diagonal and hence the regularization parameter can be easily estimated, for instance by the generalized cross validation. The method benefits from a proper initial approximation that can be the observed image or the Tikhonov approximation. Therefore, embedding this method in an outer iteration may yield further improvement of the solution. Finally, since some properties of the observed image, like continuity or sparsity, are obviously changed when working in the Fourier domain, we introduce a filtering factor which keeps unchanged the large singular values and preserves the jumps in the Fourier coefficients related to the low frequencies. Numerical examples are given in order to show the effectiveness of the proposed method.     

Meshless method for shallow water equations with free surface flow

Solving problems with free surface often encounters numerical difficulties related to excessive mesh distortion as is the case of dambreak or breaking waves. In this paper the Natural element method (NEM) is used to simulate a 2D shallow water flows in the presence of theses strong gradients. This particle-based method used a fully Lagrangian formulation based on the notion of natural neighbors. In the present study we consider the full non-linear set of Shallow Water Equations, with a transient flow under the Coriolis effect. For the numerical treatment of the nonlinear terms we used a Lagrangian technique based on the method of characteristics. This will allow avoiding divergence of Newton-Raphson scheme, when dealing with the convective terms. We also define a thin area close to the boundaries and a computational domain dedicated for nodal enrichment at each time step. Two numerical test cases were performed to verify the well-founded hopes for the future of this method in real applications.     

The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface

We consider the Dirichlet problem for the Laplace equation in a starlike domain, i.e. a domain which is normal with respect to a suitable polar co-ordinates system. Such a domain can be interpreted as a non-isotropically stretched unit circle. We write down the explicit solution in terms of a Fourier series whose coefficients are determined by solving an infinite system of linear equations depending on the boundary data. Numerical experiments show that the same method works even if the considered starlike domain belongs to a two-fold Riemann surface.     

The point source method for inverse scattering in the time domain

Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non‐time‐harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time‐dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi‐frequency data. Copyright © 2006 John Wiley & Sons, Ltd.     

Low regularity well-posedness for the viscous surface wave equation

In this paper, we prove the local well-posedness of the viscous surface wave equation in low regularity Sobolev spaces. The key points are to establish several new Stokes estimates depending only on the optimal boundary regularity and to construct a new iteration scheme on a known moving domain. Our method could be applied to some other fluid models with free boundaries.     

A new regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation in a one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under the a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

A mapping technique for numerical computations of bed evolutions

Free surface flow is one of the most difficult problems in engineering to be solved, since velocity and pressure fields depend on the free surface. On the other hand, the position of the free surface is unknown previously. Furthermore, the boundary condition on the free surface is expressed by a complicated equation. In an alluvial stream, where the boundaries of the domain are not fixed, addition of free surface at the bed will increase this difficulty. A domain mapping technique is developed in this paper to study the bed evolutions. The flow is considered 2D, choosing two coordinates in streamwise and upward directions. With a proper transformation, the hydrodynamics and sediment transport governing equations in irregular domain will be mapped into a simple rectangular one. The new domain can be discretize by finite elements. The transformed governing equations are solved to obtain desired variables in the mapped domain. With a proper transformation, there is no need of inverse mapping to obtain the free water surface profile and bedform evolution and migration in the actual domain. The model has been applied to streams with movable bed and the results show a good agreement with the experimental experiences.     

Spectral regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation with one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

A robust desirability function method for multi-response surface optimization considering model uncertainty

A robust desirability function approach to simultaneously optimizing multiple responses is proposed. The approach considers the uncertainty associated with the fitted response surface model. The uniqueness of the proposed method is that it takes account of all values in the confidence interval rather than a single predicted value for each response and then defines the robustness measure for the traditional desirability function using the worst case strategy. A hybrid genetic algorithm is developed to find the robust optima. The presented method is compared with its conventional counterpart through an illustrated example from the literature.     

Two-Phase Image Segmentation by Nonconvex Nonsmooth Models with Convergent Alternating Minimization Algorithms

Two-phase image segmentation is a fundamental task to partition an image into foreground and background. In this paper, two types of nonconvex and nonsmooth regularization models are proposed for basic two-phase segmentation. They extend the convex regularization on the characteristic function on the image domain to the nonconvex case, which are able to better obtain piecewise constant regions with neat boundaries. By analyzing the proposed non-Lipschitz model, we combine the proximal alternating minimization framework with support shrinkage and linearization strategies to design our algorithm. This leads to two alternating strongly convex subproblems which can be easily solved. Similarly, we present an algorithm without support shrinkage operation for the nonconvex Lipschitz case. Using the Kurdyka-Łojasiewicz property of the objective function, we prove that the limit point of the generated sequence is a critical point of the original nonconvex nonsmooth problem. Numerical experiments and comparisons illustrate the effectiveness of our method in two-phase image segmentation.     

A new modified panoramic UAV image stitching model based on the GA-SIFT and adaptive threshold method

This paper presents panoramic unmanned aerial vehicle (UAV) image stitching techniques based on an optimal Scale Invariant Feature Transform (SIFT) method. The image stitching representation associates a transformation matrix with each input image. In this study, we formulate stitching as a multi-image matching problem, and use invariant local features to find matches between the images. An improved Geometric Algebra (GA-SIFT) algorithm is proposed to realize fast feature extraction and feature matching work for the scanned images. The proposed GA-SIFT method can locate more feature points with greater accurately than the traditional SIFT method. The adaptive threshold value method proposed solves the limitation problem of high computation load and high cost of stitching time by greater feature points extraction and stitching work. The modified random sample consensus method is proposed to estimate the image transformation parameters and to determine the solution with the best consensus for the data. The experimental results demonstrate that the proposed image stitching method greatly increases the speed of the image alignment process and produces a satisfactory image stitching result. The proposed image stitching model for aerial images has good distinctiveness and robustness, and can save considerable time for large UAV image stitching.     

Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method

An improved meshless radial point interpolation method, for the analysis of nonlinear transient heat conduction problems is proposed. This method is implemented for the heat conduction analysis of functionally graded materials (FGMs) with non-homogenous and/or temperature dependent heat sources. The conventional meshless RPIM is an appropriate numerical technique for the analysis of engineering problems. One advantage of this method is that it is based on the global weak formulation, and also the associated shape functions possess the Kronecker delta function property. However, in the original form, the evaluation of the global domain integrals requires the use of a background mesh. The proposed method benefits from a meshless integration technique, which has the capability of evaluating domain integrals with a better accuracy and speed in comparison with the conventional integration methods, and therefore a truly meshless technique is attained. This integration technique is especially designed for the fast and accurate evaluation of several domain integrals, with different integrands, over a single domain. Some 2D and 3D examples are provided to assess the efficiency of the proposed method.     

Numerical studies to propose a ghost particle removed SPH (GR-SPH) method

This paper aims to propose a new modified SPH method with novel treatments at the boundaries. Although SPH methods decrease contradictions due to grid distortions compared to traditional mesh-based methods, the penetration of particles through boundaries and the consistency problem make the simulation of problems with definite boundaries a concern. The use of ghost boundary particles and the insertion of artificial forces at the boundaries are the most popular boundary consistency treatments proposed thus far. The use of artificial forces causes the mixing of molecular and finite theories, which can violate the conservation of momentum. This paper shows how the use of ghost boundary particles can violate the continuity equation in problems with non-zero velocity divergence. This study proposes a novel ghost particle removed SPH (GR-SPH) method that discards all ghost particles and artificial forces at the boundaries. Liner layers and liner particles have been defined inside the domain instead of ghost boundary particles in such a way that the so-called violations can partially be remedied. Based on the continuity equation and kernel function unity specification, a novel truncation correction factor has been defined for density renormalization to override the consistency problem at the boundaries. In addition, a new method is proposed to detect the particles near complex wall boundaries and evaluate the normal distance from boundaries. Finally, some benchmark problems have been solved to show the capabilities of the new modified SPH method for the prediction of both particle location and pressure distribution with acceptable accuracy. The GR-SPH method facilitates programming, with fewer particles contributing to the computations. Comparison of its outcomes with published results shows that the new treatments executed at the boundaries are effective.     

The method of lines for Poisson's equation with nonlinear or free boundary conditions

Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.This work was supported by the U.S. Army Research Office under Grant DA-AG29-76-G-0261     

Regularization of backward heat conduction problem

We study the backward heat conduction problem in an unbounded region. The problem is ill-posed, in the sense that the solution if it exists, does not depend continuously on the data. Continuous dependence of the data is restored by cutting-off high frequencies in Fourier domain. The cut-off parameter acts as a regularization parameter. The discrepancy principle, for choosing the regularization parameter and double exponential transformation methods for numerical implementation of regularization method have been used. An example is presented to illustrate applicability and accuracy of the proposed method.