Regularity of the solution and well-posedness of a mixed problem for an elliptic system with quadratic nonlinearity in gradients

A boundary value problem for an elliptic system of equations is studied that arises in the analysis of a new hydrodynamic model describing charge transport in a planar semiconductor MESFET (metal semiconductor field effect transistor). The problem has a number of features, specifically, the equations of the system involve squared components of the gradients of the unknown functions; the boundary conditions are of a mixed character, i.e., Dirichlet and Neumann conditions are set on different portions of the boundary; and the boundary of the domain is a nonsmooth curve, namely, a rectangle. Under a certain optimal condition, the C ^1,α-regularity of a weakened solution of the problem is justified and its existence is proved, while its uniqueness is shown under additional constraints. The results are used to justify the stabilization method as applied to finding approximate stationary solutions of the hydrodynamic model.     

A decomposition scheme for acoustic obstacle scattering in a multilayered medium

We consider the acoustic wave scattering by an impenetrable obstacle embedded in a multilayered background medium, which is modelled by a linear system constituted by the Helmholtz equations with different wave numbers and the transmission conditions across the interfaces. The aim of this article is to construct an efficient computing scheme for the scattered waves for this complex scattering process, with a rigorous mathematical analysis. First, we construct a set of functions by a series of coupled transmission problems, which are proven to be well-defined. Then, the solution to our complex scattering in each layer is decomposed as the summation in terms of these functions, which are essentially the contributions from two interfaces enclosing this layer. These contributions physically correspond to the scattered fields for simple scattering problems, which do not involve the multiple scattering and are coupled via the boundary conditions. Finally, we propose an iteration scheme to compute the wave field in each layer decoupling the multiple scattering effects, with the advantage that only the solvers for the well-known transmission problems and an obstacle scattering problem in a homogeneous background medium are applied. The convergence property of this iteration scheme is proven.     

On barotropic trapped wave solutions with no-slip boundary conditions

Barotropic trapped wave solutions of a linearized system of the ocean dynamics equations are described for a semi-infinite, f-plane model basin of constant depth bordering a straight, vertical coast, for some “typical” values of the model parameters. No-slip boundary conditions are considered. When the wave length is shorter than the Rossby deformation radius, the main features of the wave solutions are as follows: the Kelvin wave exponential offshore decay scale essentially decreases as the wave length decreases, and an additional wave solution propagating in the opposite direction appears.     

On the evolution of two-dimensional patterns in an inviscid liquid sheet

We perform an analysis of the pattern formation for a moving sheet of inviscid fluid. The sheet, which is assumed to have an infinite horizontal extent, moves at some prescribed velocity into a passive surrounding gas. The sheet’s thickness is assumed much smaller than the horizontal scale of the fluid motion. By considering a system that is symmetric with respect to the horizontal planes, long scale asymptotics are used to reduce the full governing equations in three dimensions to a set of three coupled nonlinear partial differential equations for the horizontal components of the velocity field and the height of the interface profile. The interfacial conditions consisting of the kinematic and normal stress balance are incorporated into these evolution equations. Investigations are carried out as function of the sole dimensionless parameter, namely the Weber number. A small amplitude stability analysis around the planar gas–liquid interface reveals that wave patterns in the form of traveling plane waves occur subcritically, and are therefore unstable. The reduced evolution equations are solved numerically for fixed values of the Weber number. Since the reduced system of equations is homogeneous, the wave motion is generated by initial conditions. Five initial conditions have been imposed: one-dimensional rolls, two-dimensional squares, two-dimensional hexagons, two-dimensional ridges, and smooth peaks. The ensuing evolution of the liquid sheet’s shape and corresponding flow fields are described by illustrations of the changes in the sheet’s morphology with time.     

The Diffraction of an Acoustic Wave by an Embedded Quarter-Space

The subject of this paper is the diffraction of a plane harmonicwave when it falls upon a quarter-space with different transmissionproperties from the rest of unbounded space. A matching procedureallows asymptotic expressions for the field on the two planeinterfaces to be calculated in a fairly simple way. In orderto obtain these expressions, certain assumptions are made aboutthe asymptotic form of the field on the interface. These assumptionsare plausible and lead to consistent results. We begin with the problem of wave propagation in two weldedquarter-spaces due to excitation on the plane boundary. Thisproblem has an exact solution and provides an illustration ofthe method of matching asymptotic fields (not the method ofmatched asymptotic expansions). We then move on to the problemof a plane wave normally incident on our embedded quarter-spaceand derive exact expressions for the asymptotic field on theinterfaces. Finally, we include an analysis of oblique incidence.     

Optimization of component reliability in the design phase of capital goods

We introduce a quantitative model to support the decision on the reliability level of a critical component during its design. We consider an OEM who is responsible for the availability of its systems in the field through service contracts. Upon a failure of a critical part in a system during the exploitation phase, the failed part is replaced by a ready-for-use part from a spare parts inventory. In an out-of-stock situation, a costly emergency procedure is applied. The reliability levels and spare parts inventory levels of the critical components are the two main factors that determine the downtime and corresponding costs of the systems. These two levels are decision variables in our model. We formulate the portions of Life Cycle Costs (LCC) which are affected by a component’s reliability and its spare parts inventory level. These costs consist of design costs, production costs, and maintenance and downtime costs in the exploitation phase. We conduct exact analysis and provide an efficient optimization algorithm. We provide managerial insights through a numerical experiment which is based on real-life data.     

Periodic switching strategies for an isoperimetric control problem with application to nonlinear chemical reactions

This paper deals with an isoperimetric optimal control problem for nonlinear control-affine systems with periodic boundary conditions. As it was shown previously, the candidates for optimal controls for this problem can be obtained within the class of bang-bang input functions. We consider a parametrization of these inputs in terms of switching times. The control-affine system under consideration is transformed into a driftless system by assuming that the controls possess properties of a partition of unity. Then the problem of constructing periodic trajectories is studied analytically by applying the Fliess series expansion over a small time horizon. We propose analytical results concerning the relation between the boundary conditions and switching parameters for an arbitrary number of switchings. These analytical results are applied to a mathematical model of non-isothermal chemical reactions. It is shown that the proposed control strategies can be exploited to improve the reaction performance in comparison to the steady-state operation mode.     

The scattering of an harmonic elastic wave by a volume inclusion with a thin interlayer

The boundary element method is used to investigate the propagation of harmonic elastic waves in an infinite matrix with a volume inclusion with a thin interlayer between the inclusion and the matrix. A boundary-integral formulation of the problem is based on a consideration of a two-phase medium, consisting of the matrix and the inclusion, on the contact surface of which conditions of proportional dependence between the forces and jumps in the displacements, which model the interlayer, are satisfied. These conditions are taken into account implicitly in the boundary integral equations obtained, which are subsequently regularized and discretized on the grid of boundary elements introduced. The numerical results obtained demonstrate the effect of the interlayer on the dynamic contact stresses for a spherical inclusion in the field of a plane longitudinal wave.     

基于EM算法的混合模型中子总体个数的研究

混合模型已成为数据分析中最流行的技术之一,由于拥有数学模型,它通常比聚类分析中的传统的方法产生的结果更精确,而关键因素是混合模型中子总体个数,它决定了数据分析的最终结果。期望最大化(EM)算法常用在混合模型的参数估计,以及机器学习和聚类领域中的参数估计中,是一种从不完全数据或者是有缺失值的数据中求解参数极大似然估计的迭代算法。学者们往往采用AIC和BIC的方法来确定子总体的个数,而这两种方法在实际的应用中的效果并不稳定,甚至可能会产生错误的结果。针对此问题,本文提出了一种利用似然函数的碎石图来确定混合模型中子总体的个数的新方法。实验结果表明,本文方法确定的子总体的个数在大部分理想的情况下可以得到与AIC、BIC方法确定的聚类个数相同的结果,而在一般的实际数据中或条件不理想的状态下,碎石图方法也可以得到更可靠的结果。随后,本文将新方法在选取的黄石公园喷泉数据的参数估计中进行了实际的应用。     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

Boundary stabilization of a 3-dimensional structural acoustic model

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).     

Numerical grids used in a coastal ocean model with breaking wave effects

In coastal ocean modeling, traditional single-block rectangular (Cartesian) grids have been most commonly used for their simplicity. In many cases, these grids may be not well suited (even at very high resolutions) for regions with complicated physical fields, open boundaries, coastlines, and bottom bathymetry. The numerical curvilinear nearly orthogonal/orthogonal, single/multi-block coastline-following grids for the Mediterranean Sea, Monterey Bay and the South China Sea (SCS) are presented. These grids can be used in coastal ocean modeling to enhance model numerical solutions and save computer resources by giving better treatment of regions with high gradients such as areas of complicated coastlines and steep slopes of shelf breaks, complicated bottom topography, open boundaries, and multi-scale physical phenomenon. Grid generation techniques are used to designed these grids. This kind of grids can also easily increase horizontal resolutions in the subregion of the model domain, without increasing the computational expense, with a higher resolution over the entire domain.A three dimensional coastal ocean model with breaking wave effects is also presented and applied. The ocean system is a primitive equation modeling system with grid generation routines and a turbulent closure which is capable of taking surface breaking wave effects into account. The system also includes a grid package which allows model numerical grids to be coupled with the ocean model. The model code is written for multi-block grids, but a single-block grid is used for the South China Sea (SCS). The model with breaking wave effects and a grid of 121 × 121 grid points are used to simulate the winter circulation of the SCS as an example. The model output of the 60-day run shows the observed upwelling locations in the sea surface salinity field.     

Investigation of the wave field in an effective model of a layered elastic-fluid medium

For a medium that consists of alternating elastic and fluid layers, an effective model is constructed and investigated. This model is a special case of the Biot medium. The wave field is represented as Fourier and Mellin integrals. In the Mellin integral, the contour of integration is replaced by a stationary contour. In the expressions obtained, the order of integration is changed and the inner integral is calculated. The outer integral is equal to two residues. The corresponding poles are roots of two equations of fourth order. These roots lie on the right half-plane and may be complex or real. The representation obtained for the wave field is in agreement with the expressions derived by the Smirnov-Sobolev method. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 175–192.     

Analysis of critical motions of floating structures

Validation of numerical methods for describing the motion of a ship in sea conditions by adequate experiments is a major research field in ocean engineering. For the development of a method for the systematic determination of critical and safe operational conditions and for the classification of capsize scenarios bifurcation analyses are performed. The computational effort for these analyses is enormous using a full model describing the nonlinear dynamics of a floating body. Therefore, a method for model reduction is currently being developed at the Institute of Mechanics and Ocean Engineering at TUHH. Bases for the validation of this new method are experiments conducted in the institute's wave tank. The determination of position and attitude of the body is performed with an integrated measurement system: An inertial measurement unit and a video system are combined using an extended Kalman Filter. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic solutions for an asymptotically linear wave equation with resonance

In this paper, we study the existence and multiplicity of nontrivial periodic solutions for an asymptotically linear wave equation with resonance, both at infinity and at zero. The main features are using Morse theory for the strongly indefinite functional and the precise computation of critical groups under conditions which are more general.     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

The passage of a bending wave through an inhomogeneity in an absolutely rigid rib backing an elastic plate

The wave properties of a system consisting of an elastic plate and an absolutely rigid infinite rib with a defect on a segment are examined. An elastic inclusion and a gap are two kinds of defects under study. The Green's function method is applied to the diffraction problem and transforms it to singular integro-differential equations on an interval. For the case of short defects, the nonresonance and resonance asymptotics of the scattering pattern are obtained. These results show that the coefficient of penetration for a gap is much larger than that for an elastic inclusion if the frequency is nonresonant. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210. 1994, pp. 22–29. Translated by I. V. Andronov.