Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

This paper deals with the randomized heat equation defined on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.     

Hadamard’s formula and couplings of SLEs with free field

The relation between level lines of Gaussian free fields (GFF) and SLE₄-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE₄ is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering

This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.     

Time–accurate Modular CFD‐CSD Coupling for Aeroelastic Rotor Simulations

This paper addresses the timewise accuracy of different coupling approaches applied to instationary aeroelastic simulations of rotors in forward flight. Two different approaches which are widely discussed in literature are examined: the tight or strong coupling, and the fully integrated or monolithic coupling. Strong coupling means an exchange of fluid loads and structural deformations at each time step which is effectuated in a fully modular manner. We will address aspects of conservativity and time‐accuracy, and will present results for a helicopter forward flight scenario. However, objections concerning the correct solution of the global non‐linear three field problem – structure, grid deformation, aerodynamics – remain. These objections are normally rejected by the monolithic approach. Here, a common set of partial differential equations is derived and solved in a single code. However, a truly monolithic system of equations is only needed for stability analysis, and it can be decomposed in a three field problem respecting appropriate boundary conditions for each domain. Thus, modularity can be maintained, conceiving a quasi‐monolithic procedure, when both domains are simultaneously solved in a common non‐linear iteration loop on a per time‐step basis. First results will be shown for a 2D flutter testcase.     

On the stability of the Kalman–Bucy filter with stationary time varying parameters

Linear filtering systems with stationary parameters are considered in continuous time. We generalize the conditionally Gaussian result of Lipster – Shiryaev and Haussmann – Pardoux. Moreover, we show that the Kalman – Bucy filter is exponentially stable under weak conditions of stabile ability and detect ability. This filter will have a well defined asymptotic behaviour     

Isotropic models for a heat transfer in periodic composites

A macroscopic model for the non – stationary heat transfer processes in a periodic honeycomb – type anisotropic rigid conductor is formulated. The main aim of this contribution is to show that the macroscopic properties of this conductor are isotropic. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimality conditions for wave speed in anisotropic smectic C∗ liquid crystals

We discuss minimality conditions for the speed of monotone travelling waves in a sample of smectic C^∗ liquid crystal subject to a constant electric field, dealing with both isotropic and anisotropic cases. Such conditions are important in understanding the properties of domain wall switching across a smectic layer, and our focus here is on examining how the presence of anisotropy can affect the speed of this switching. We obtain an estimate of the influence of anisotropy on the minimal speed, sufficient conditions for linear and non‐linear minimal speed selection mechanisms to hold in different parameter regimes, and a characterisation of the boundary separating the linear and non‐linear regimes in parameter space.     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

Local and non‐local boundary quenching

The quenching problem is examined for a one‐dimensional heat equation with a non‐linear boundary condition that is of either local or non‐local type. Sufficient conditions are derived that establish both quenching and non‐quenching behaviour. The growth rate of the solution near quenching is also given for a power‐law non‐linearity. The analysis is conducted in the context of a nonlinear Volterra integral equation that is equivalent to the initial–boundary value problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Eigenvalues of a λ – Rational Sturm– Liouville Problem

The Sturm – Liouville problem (1.1), (1.2) is considered, which depends rationally on the eigenvalue parameter. Estimates for the eigenvalues and especially for the embedded eigenvalues are proved. Moreover, conditions for the non – existence of embedded eigenvalues are given.     

Non‐homogeneous boundary value problem for the Chan–Bodner–Linholm model

This paper proves global in time existence and uniqueness of large solutions for a problem in non‐linear inelasticity with non‐homogeneous boundary conditions. The proof is based on the non‐linear non‐autonomous semigroup method. Copyright © 2000 John Wiley & Sons, Ltd.     

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

Novel closed – form solutions in buckling of inhomogeneous columns under distributed variable loading

In this study we consider buckling of columns with variable stiffness, under axially distributed loading varying polynomially. The objective is to obtain closed – form solutions for the buckling load. The problem is posed in inverse setting: determine the column’s stiffness, so that it has the given, polynomial, buckling mode. Four sets of boundary conditions are investigated. Some perplexing results are obtained, namely, that irrespective of boundary conditions, the critical load of the column is the same; this occurs in conjunction with the fact that the obtained distribution for stiffness is different for each set of boundary conditions.     

Stochastic distributed systems with point observations and boundary control: an abstract theory

There already exists a fairly complete theory for the problems of estimation and stochastic optimal control for linear distributed parameter systems, with Gaussian or non Gaussian noise disturbance. In [8] and [12] generalizations of the familiar finite dimensional results of the Kalman-Bucy filter and the separation principle are obtained using an abstract input-output Hilbert space representation for the system. However, in [8] and [12] all the input operators are assumed to be bounded and so it does not cover the important practical cases of control and noise on submanifolds of the spatial domain or point observations. Here we introduce unbounded system operators in the abstract input-output Hilbert space representation and thus extend all the results of [8] and [12] to allow for point observations and noise and control on submanifolds including the boundary. The theory is illustrated by several examples.     

面内功能梯度三角形板等几何面内振动分析

基于平面应变理论,利用等几何有限元方法分析了弹性边界条件下面内功能梯度三角形板的面内振动特性.板的材料属性沿厚度方向呈均匀分布,而在面内方向呈任意指数梯度变化.采用非均匀有理B样条(NURBS)基函数对三角形结构进行等几何建模和位移描述,实现了三角形板几何设计和振动分析的无缝衔接.在三角形板边界上引入虚拟弹簧约束并通过调节虚拟弹簧刚度,实现任意边界条件的施加.通过不同的单元细化方案和对比算例,验证了等几何方法的灵活性、准确性和快速收敛性.系统研究了边界条件、材料属性和几何参数对三角形板振动特性的影响.同时给出了弹性边界条件下面内功能梯度三角形板的振动特性解,具有重要参考价值.     

Automatic control of the temperature in thermoviscoelasticity

A viscoelastic body which undergoes a thermomechanical process is considered. The evolution of the state variables, i.e., the displacement and temperature fields, is described by a coupled hyperbolic–parabolic integrodifferential system. The heat flux through a portion of the boundary is controlled automatically by a thermostat device acting on the external temperature according to suitable measurements of the boundary strain and/or the boundary temperature. Two types of control are analysed: relay switch and Preisach hysteresis operator. In both cases, the resulting model is an initial and boundary value problem for the evolution system with non‐linear and non‐local boundary conditions for the temperature. Existence and uniqueness results are proved. Copyright © 1999 John Wiley & Sons, Ltd.     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generation of Gevrey class semigroup by non‐selfadjoint Euler–Bernoulli beam model

Asymptotic and spectral properties of a non‐selfadjoint operator that is a dynamics generator for the Euler–Bernoulli beam model of a finite length are studied in this paper. The hyperbolic equation, which governs the vibrations of the Euler–Bernoulli beam model, is supplied with a one‐parameter family of physically meaningful boundary conditions containing damping terms. The initial boundary‐value problem is equivalent to the evolution equation that generates a strongly continuous semigroup in the state space of the system. It is found that the semigroup, being non‐analytic, belongs to Gevrey class semigroups. This means that the differentiability of such semigroup is slightly weaker than that of an analytic semigroup. In the forthcoming works, the results of the present paper will be applied (a) to the solution of the exact controllability problem for Euler–Bernoulli beam and (b) to spectral analysis of a planar network of serially connected Euler–Bernoulli beams modelling ‘flying wing configurations’ in aeronautic engineering. Copyright © 2006 John Wiley & Sons, Ltd.