Quadratic Boundary Research Articles

Wave trapping by porous breakwater near a rigid wall under the influence of ocean current

The present work investigates the water wave interaction with bottom-standing thick porous trapezoidal-shaped structures and shore-side vertical rigid wall in the presence of uniform ocean currents. This study has been done to understand the impact of different physical parameters like friction, porosity, and ocean currents along with different structural parameters (width and height) on different phenomena like wave energy reflection, wave forces, wave energy dissipation, etc. The quadratic boundary element method-based numerical technique has been used to solve the boundary value problem. The structural porosity is modeled using Sollitt and Cross’s model of water wave interaction with thick porous structures. Several results associated with the wave energy reflection and energy dissipation have been analyzed. Also, the wave force exerted by the incoming waves has been investigated to check the stability and sustainability of the right vertical rigid wall and porous structure. The Doppler-shift effect is observed in wave transformation characteristics due to the presence of ocean currents. The impact of following and opposing ocean currents can be seen in the graph of wave energy reflection, dissipation, and wave forces. The periodic patterns can be observed clearly in wave characteristics like wave energy reflection, dissipation, and wave forces when plotted against the non-dimensional separation gap between the porous breakwater and shore-side rigid seawall.

Study on the film thickness and surface wave velocity of the thin liquid film formed by a round jet obliquely impinging on a horizontal plate

For a thin liquid film (in a supercritical flow) prior to the formation of a non-circular hydraulic jump formed by a round jet obliquely impinging on a horizontal plate, the time-averaged film thickness and the surface wave velocity are extracted based on the measured transient film thickness. On the one hand, the effect of many factors, including the jet velocity, impingement angle, azimuthal angle, liquid viscosity, and surface tension, on the time-averaged film thickness and surface wave velocity are discussed. When the jet Reynolds number increases to about 1.4×104, the film thickness profile suddenly increases, and the transition of liquid flow from laminar to turbulent occurs. Meanwhile, a rapid increase is observed downstream of the turbulent film thickness profile. The influence of surface tension on the time-averaged film thickness and surface wave velocity is negligible for thin liquid films before non-circular hydraulic jumps. Nonetheless, the surface tension has a significant influence on the interface profile of non-circular hydraulic jumps. Furthermore, a “crescent” kink region upstream of the jump can be identified when the surface tension is lower than 40.6 mN/m. On the other hand, experimental results are used to verify the prediction accuracy of existing approximate solutions. The laminar approximate solution with a quadratic boundary layer velocity profile can accurately predict the film thickness distribution of most laminar thin liquid films, except downstream of the thin liquid films with a dynamic viscosity higher than 9.71 mPa s. The surface wave velocities are found to be close to the predicted surface velocities of the approximate solutions.

An isoparametric quadratic boundary element for coupled stretching-bending analysis of thick laminated composite plates with transverse shear deformation

Recently, we derived the Green's function for an infinite thick laminated composite plate subjected to concentrated forces and moments. By serving this Green's function as the fundamental solutions, in this paper for the first time we develop the associated boundary element method (BEM), which is applicable to general problems of thick laminated plates with various types of deformations. For example, the in-plane stretching and out-of-plane bending deformations may be coupled together, which occurs for the unsymmetric laminated composite plates. Besides, the transverse shear deformations are also incorporated based upon the first order plate theory. To complete the entire development of the proposed BEM, all the involved issues, such as the boundary integral equations, the fundamental solutions and their derivatives, the mathematical formulation for the selected isoparametric quadratic boundary element, and the treatments of singular integrals are elaborated in this paper. The calculations of complete solutions at boundary or internal points are provided as well. The accuracy and efficiency of the proposed BEM are demonstrated through numerical examples, which show that it outperforms the conventional finite element method and is applicable for general thick laminated plates with unrestricted coupling effects.

Large deviations for quadratic functionals of stable Gauss–Markov chains and entropy production

In this paper, we establish a large deviation principle for the entropy production rate of possible non-stationary, centered stable Gauss–Markov chains, verifying the Gallavotti–Cohen symmetry. We reach this goal by developing a large deviation theory for quasi-Toeplitz quadratic functionals of multivariate centered stable Gauss–Markov chains, which differ from a perfect Toeplitz form by the addition of quadratic boundary terms.

Global C2,α estimates for the Monge-Ampère equation on polygonal domains in the plane

We classify global solutions of the Monge-Amp\\`ere equation $\\det D^2 u=1$ on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global $C^\{2,\\alpha\}$ estimates for the non-degenerate Monge-Amp\\`ere equation in convex polygonal domains in $\\Bbb\{R\}^2$ provided a globally $C^2$, convex strict subsolution exists.

Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

<p style='text-indent:20px;'>In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.</p>

Optimal control of fractional Sturm–Liouville wave equations on a star graph

ABSTRACT In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

Trainable segmentation for transmission electron microscope images of inorganic nanoparticles.

We present a trainable segmentation method implemented within the python package ParticleSpy. The method takes user labelled pixels, which are used to train a classifier and segment images of inorganic nanoparticles from transmission electron microscope images. This implementation is based on the trainable Waikato Environment for Knowledge Analysis (WEKA) segmentation, but is written in python, allowing a large degree of flexibility and meaning it can be easily expanded using other python packages. We find that trainable segmentation offers better accuracy than global or local thresholding methods and requires as few as 100 user-labelled pixels to produce an accurate segmentation. Trainable segmentation presents a balance of accuracy and training time between global/local thresholding and neural networks, when used on transmission electron microscope images of nanoparticles. We also quantitatively investigate the effectiveness of the components of trainable segmentation, its filter kernels and classifiers, in order to demonstrate the use cases for the different filter kernels in ParticleSpy and the most accurate classifiers for different data types. A set of filter kernels is identified that are effective in distinguishing particles from background but that retain dissimilar features. In terms of classifiers, we find that different classifiers perform optimally for different image contrast; specifically, a random forest classifier performs best for high-contrast ADF images, but that QDA and Gaussian Naïve Bayes classifiers perform better for low-contrast TEM images.

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Analytical method for the second-order homogenization of two-phase composites within Mindlin–Toupin strain gradient elasticity theory is proposed. Direct approach and self-consistent approximation are used to reduce the homogenization problem to the problem of determination of averaged Cauchy stresses, double stresses and static moments of Cauchy stresses inside the inclusions under prescribed quadratic boundary conditions. The ellipsoidal shape of inclusions and orthotropic properties of phases are assumed. Extended equivalent inclusion method with linear eigenstrain is proposed to derive the explicit relations between the Eshelby-like tensors and corresponding concentration tensors that are used to define the averaged field variables inside the inclusions. Obtained analytical solutions allow to evaluate the effective classical and gradient elastic moduli of composite materials accounting for the phases properties, volume fraction, shape and size of inclusions. Presented solution for the effective gradient moduli covers the full range of volume fraction and correctly predicts the absence of gradient effects for the homogeneous classical Cauchy medium. Examples of calculations for the composites with spherical inclusions are given. Micro-scale definition of the well-known simplified strain gradient elasticity theory is provided. Namely, it is shown that this theory is the phenomenological continuum model for the composites with isotropic matrix and with the small volume fraction of stiff isotropic spherical inclusions, which have the same Poisson’s ratio to those one of matrix phase.

Optimal control in poroelasticity

In this paper we address optimal control problems subject to fluid flows through deformable, porous media. In particular, we focus on linear quadratic elliptic-parabolic control problems, with both distributed and boundary controls, and prove existence and uniqueness of optimal control. Furthermore, we derive the first order necessary optimality conditions. These problems have important biological and biomechanical applications. For example, optimizing the pressure of the flow, and investigating the influence and control of pertinent biological parameters are relevant in the case of the lamina cribrosa – a porous tissue at the base of the optic nerve head inside the eye – which is modeled by poroelasticity – where these factors are believed to be related to the development of ocular neurodegenerative diseases such as glaucoma. Moreover, the study and results will be applicable to other situations such as the poroelastic modeling of cartilages, bones, and engineered tissues.

A new insight into the analysis of plane elasticity with coupling of numerical manifold and boundary element methods

In this research, a combination of the numerical manifold method (NMM) and boundary element method (BEM) is presented for solving plane elasticity problems to benefit from the advantages of both methods. For this goal, the domain of the problems is divided into two regions, each region is analyzed separately, and finally, equilibrium and compatibility conditions are controlled in the interface by the iteration method. 1D quadratic boundary elements and 2D quadratic, cubic and quartic ones in the form of triangular shapes are utilized in the boundary element and manifold methods, respectively. The proposed method is verified by calculating the displacements and stresses resulting from the application of static loads in five different numerical problems. Comparison of the numerical results shows that the suggested method is very efficient and accurate.

Bilinear boundary optimal control with final observation for the heat equation

In this paper, we consider a bilinear optimal control with final observation for the heat equation. We first give the existence and uniqueness results of weak solutions under an integral constraints on the control and the state. We prove the existence and uniqueness of the solution to a quadratic boundary optimal control problem by means of compactness Sobolev space embedding. Then we provide a characterization of the optimal control via the Euler-Lagrange first-order optimality conditions. The unique optimal control is characterized in terms of the solution of the optimality system, which consists of the state system coupled with an adjoint system.

Optimal control of a fractional Sturm–Liouville problem on a star graph

ABSTRACT This paper is devoted to elliptic fractional boundary value problems of Sturm–Liouville type on an interval and on a general star graph. We first give some existence and uniqueness results on an open bounded real interval, prove the existence of solutions to a quadratic boundary optimal control problem and provide a characterization via optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem on a star graph with mixed Dirichlet and Neumann boundary controls.

Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods

In this paper we deal with the determination of the strain gradient elasticity coefficients of composite material in the framework of the homogenization methods. Particularly we aim to eliminate the persistence of the strain gradient effects when the method based on quadratic boundary conditions is considered. Such type of boundary conditions is often used to determine the macroscopic strain gradient elastic coefficients but leads to contradictory results, particularly when a RVE is made up of a homogeneous material. The resulting macroscopic equivalent material exhibits strain gradient effects while it should be expected of Cauchy type. The present contribution is to provides new relationship to correct the approach based on the quadratic boundary condition. To this purpose, we start from the asymptotic homogenization approach, we establish a connection with the method based on quadratic boundary conditions and we highlight the correction required to eliminate the persistence of the strain gradient effects. An application to a composite with fibers is provided to illustrate the method.

Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior

A computational homogenization method to determine the effective parameters of Mindlin’s Strain Gradient Elasticity (SGE) model from a local heterogeneous Cauchy linear material is developed. The devised method, which is an extension of the classical one based on the use of Quadratic Boundary Conditions, intents to correct the well-known non-physical problem of persistent gradient effects when the Representative Volume Element (RVE) is homogeneous. Those spurious effects are eliminated by introducing a microstructure-dependent body force field in the homogenization scheme together with alternative definitions of the localization tensors. With these modifications, and by a simple application of the superposition principle, the higher-order stiffness tensors of SGE are computed from elementary numerical calculations on RVE. Within this new framework, the convergence of SGE effective properties is investigated with respect to the size of the RVE. Finally, a C1-FEM procedure for simulating the behavior of the effective material at the macro scale is developed. We show that the proposed model is consistent with the solutions arising from asymptotic analysis and that the computed effective tensors verify the expected invariance properties for several classes of anisotropy. We also point out an issue that the present model shares with asymptotic-based solutions in the case of soft inclusions. Applications to anisotropic effective strain-gradient materials are provided, as well as comparisons between fully meshed structures and equivalent homogeneous models.

Isotropic Damage Analysis of Thermoelastic Problems using BIE

This work presents a quadratic boundary integral equation formulation for isotropic damage analysis of components subjected to mechanical, thermal and centrifugal loads. To evaluate domain-related integrals due to the damage effects, the radial integration method (RIM) based on the use of the approximating the normalized displacements in the domain integrals by a series of prescribed radial basis functions (RBF) is adopted. The density of micro-defects is assumed to be small in the material. The scalar damage parameter expressed in an exponential evolution equation is utilized. Numerical examples including a rectangular plate, thick-walled cylinder and rotating disk problems under the thermal loads are given.

A Thin Phase Screen Model for Surface Plasmon Polaritons

We develop a thin phase screen model for the transmission of a surface plasmon polariton through a curved boundary between two different co-planar metal surfaces. The result is illustrated by applying it to the determination of the intensity distribution of the electric field of a surface plasmon polariton that has been transmitted through a quadratic (parabolic) boundary, through a cubic boundary, and through a periodic boundary.

Reconstruction of potential and boundary conditions for second order difference equations

Assume the eigenvalues and the weights are given for a difference boundary value problem and that the form of the boundary conditions at the endpoints is known. In particular, it is known whether the endpoints are fixed (i.e. Dirichlet or non-Dirichlet boundary conditions) or whether the endpoints are free to move (i.e. boundary conditions with affine dependence on the eigenparameter). This work illustrates how the potential as well as the exact boundary conditions can be uniquely reconstructed. The procedure is inductive on the number of unit intervals. This paper follows along the lines of S. Currie and A. Love, Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions, \emph{Quaestiones Mathematicae}, 40 (2017), no. 7, 861−877. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in the above reference, an additional spectrum is required more often than was the case in the unique reconstruction of the potential alone.

Nonlinear Deformation of Three-Dimensional Piecewise Homogeneous Media in Stress Waves

The integral representation of the relevant boundary problems together with the difference approximation schemes of the time solution is used to simulate the nonlinear wave dynamics of threedimensional heterogeneous deformable media. The time solution of the integral equations is based on the Kelvin—Somigliana fundamental solution, the Newmark -scheme, the collocation approximation, and the predictor-corrector of the method of the theory of elastoplastic media flow with anisotropic hardening. The parametrization of the domain surface occupied by the medium, i.e., its internal boundaries, is made using the quadratic boundary elements. Similar volume elements are used to calculate the integrals with the bulk (inertial and viscous) forces and determine the plastic deformation zones. The complex histories of the combined shock loading of the composite piecewise homogeneous media slowly changing in time in the presence of the local zones of singular perturbation of the solution are considered. The discrete domain method was developed, based on which the practice-relevant nonlinear problems of stress wave propagation in heterogeneous media with the concentrators were solved.

Stability analysis and observer design for one-sided Lipschitz descriptor systems with time-varying delay

This paper investigates the problem of stability analysis and observer design for nonlinear descriptor systems with time-varying delay. In the systems, the nonlinear function satisfies the one-sided Lipschitz condition and the quadratic internal boundary condition. The disturbance is considered in both the state and the output equation. Using one-sided Lipschitz condition, the quadratic internal boundary condition, and the generalized Lyapunov method, we establish the non-strict bilinear matrix inequality (BMI)-based condition. We change the condition into strict bilinear matrix inequality (BMI) condition. Furthermore, we give the linear matrix inequality-based condition to ensure the gradual convergence of state estimation error and to accomplish robustness against L2 norm bounded disturbances by utilizing change of variables for straight forward computation of the observer gain matrix. Finally, a numerical example is given to verify the effectiveness of the observer design scheme.