On the treatment of time-dependent boundary conditions in splitting methods for parabolic differential equations

Splitting methods for time-dependent partial differential equations usually exhibit a drop in accuracy if boundary conditions become time-dependent. This phenomenon is investigated for a class of splitting methods for two-space dimensional parabolic partial differential equations. A boundary-value correction discussed in a paper by Fairweather and Mitchell for the Laplace equation with Dirichlet conditions, is generalized for a wide class of initial boundary-value problems. A numerical comparison is made for the ADI method of Peaceman-Rachford and the LOD method of Yanenko applied to problems with Dirichlet boundary conditions and non-Dirichlet boundary conditions.     

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

A direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.     

On the well-posedness of the initial value problem in elasto-plastodynamics for a linear comparison solid

Summary This work deals with the well-posedness of the linearized initial value problem in solid elastoplastodynamics. By well-posedness we understand existence, uniqueness and continuous dependence of the solution with respect to initial and boundary data. The initial value problem is studied in its most general form using a technique from Kreiss [1]. The cases of elastic and elastic-plastic materials with associated and non-associated plastic flow rules are considered. The analysis is carried under the assumption that the material response remains within the same constitutive cone everywhere (i.e.; exclude loading/unloading subregions). A complete answer to the well-posedness question is given without the requirement of major symmetry of the constitutive tensor. First it is shown that in 2-D the necessary and sufficient condition for well-posedness is the strong ellipticity condition. Up to now this condition had been known only to be a sufficient condition.     

On a mixed initial boundary-value problem of linear elastodynamics

We prove that the solutions of a nonstandard mixed problem of elastodynamics in an unbounded domain satisfy the basic theorems of elastodynamics provided the elasticity tensor is strongly elliptic and the acoustic tensor satisfies the hyperbolicity condition. We also give a counterexample showing that our results are sharp.     

Computational algorithms for solving the coefficient inverse problem for parabolic equations

Among inverse problems for partial differential equations, we distinguish coefficient inverse problems, which are associated with the identification of coefficients and/or the right-hand side of an equation using some additional information. When considering time-dependent problems, the identification of the coefficient dependences on space and on time is usually separated into individual problems. In some cases, we have linear inverse problems (e.g. identification problems for the right-hand side of an equation); this situation essentially simplify their study. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a non-linear inverse problem, linearized approximations in time are constructed using standard finite difference approximations in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems.     

New formulation of the transparent boundary conditions for the parabolic equation

New approximate transparent boundary conditions for the nonstationary parabolic (Schrödinger) equation are derived using the method of multiple scales.     

A stochastic approach to the problem of stability of a spherical shell with initial imperfections

A probabilistic method is applied to the problem of stability of a spherical shell with initial imperfections, subjected to uniform external pressure. The distribution law of the normal deflections of the shell is obtained following the Smoluchovsky equation, at a given density of the initial deflections. On this basis the probability for the deflections to be in a given interval is found. A relation for the probability for a jump transition to a new stability form is obtained for a spherical shell with a given probability distribution of the initial imperfections. The relations obtained can be used for solving different kinds of reliability problems, as well as problems for prediction of the shell stability forms together with the probability for their realization.     

Analysis of the stability and conditions for bulging of gas main line sections laid with an initial curvature

A method is proposed for calculating for the stability of gas main line sections installed with an initial, upwardly convex, curvature. Conditions for the start of stability loss are formulated, and calculated dependences are obtained for determining the deflections and bending moments at the bulged sections.Sverdlovsk. Translated from Problemy Prochnosti, No. 9, pp. 118–121, September, 1989.     

On the nonlinear stability of the MHD anisotropic Bénard problem

In this paper, by the Lyapunov direct method, we study the nonlinear Lyapunov stability of the conduction–diffusion solution of the anisotropic magnetic Bénard problem, for a fully ionized fluid. We show that, if the conduction–diffusion solution is linearly stable, it is asymptotically nonlinearly stable.     

Determination of the initial condition in parabolic equations from integral observations

We study the problem of determining the initial condition in parabolic equations with time-dependent coefficients from integral observations which can be regarded as generalizations of point-wise interior observations. Our approach is new in the sense that for determining the initial condition we do not assume that the data available in the whole space domain at the final moment or in a subset of the space domain during a certain time interval, but some integral observations during a time interval. We propose a variational method in combination with Tikhonov regularization for solving the problem and then discretize it by finite difference splitting methods. The discretized minimization problem is solved by the conjugate gradient method and tested on computer to show its efficiency. Also as a by-product of the variational method, we propose a numerical scheme for estimating the degree of ill-posedness of the problem.     

A finite element collocation method for the exact control of a parabolic problem

A finite element method is given for the problem of exact control of a linear parabolic equation. The basis functions consist of piecewise bicubic polynomials and the differential equation is satisfied at Gaussian collocation points within each element. The overdetermined system of equations obtained is solved by the method of least squares, and a convergence argument is given for the complete procedure. Numerical results are given for two problems of boundary control.     

On the numerical solution of the Dirichlet initial boundary-value problem for the heat equation in the case of a torus

The initial-boundary-value problem for the heat equation in the case of a toroidal surface with Dirichlet boundary conditions is considered. This problem is reduced to a sequence of elleptic boundary-value problems by a Laguerre transformation. The special integral representation leads to boundary-integral equations of the first kind and the toroidal surface gives one-dimensional integral equations with a logarithmic singularity. The numerical solution is realized by a trigonometric quadrature method in cases of open or closed smooth boundaries. The results of some numerical experiments are presented.     

Finite elements for an optimal control problem governed by a parabolic equation

A stationary variational formulation of the necessary conditions for optimality is derived for an optimal control problem governed by a parabolic equation and mixed boundary conditions. Then a mixed finite element model with elements in space and time is utilized to solve a simple numerical example whose analytical and finite difference solutions are given elsewhere. Numerical results show that the proposed method with C^° continuity elements constitutes a powerful numerical technique for solution of optimal control problems of distributed parameter systems.     

The initial boundary-value problem for geometrically nonlinear elastic-plastic shells

Starting from three-dimensional theory, a global minimum principle for stresses in elastic-plastic shells subjected to arbitrary conservative loading histories is presented. Finite displacements but small strains are considered. For numerical illustration, the stress state in an elastic-plastic cylindrical shell under nonproportional loading is calculated.     

A mixed boundary-value problem of elasticity with parabolic boundary

Summary The mixed boundary-values problem of elasticity with parabolic boundary when the prescribed displacement is parabolic has been reduced to the solution of the non-homogeneous Hilbert problem following the method of complex variable. The result has been compared with that obtained byParia [5] for parabolic boundary with prescribed constant displacement as a particular case.

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Zusammenfassung Das gemischte Randwertproblem der Elastizitätstheorie wird für parabolische Berandung auf die Lösung des inhomogenen Hilbert-Problems unter Benützung der komplexen Methode zurückgeführt. Die Randverschiebung wird durch ein Polynom zweiten Grades dargestellt. Das Ergebnis wird mit dem vonParia [5] verglichen, das als Sonderfall konstante Verschiebungen am parabolischen Rand vorschreibt.

     

On stability and convergence of finite element approximations of Biot's consolidation problem

Stability and convergence analysis of finite element approximations of Biot's equations governing quasistatic consolidation of saturated porous media are, discussed. A family of decay functions, parametrized by the number of time steps, is derived for the fully discrete backward Euler–Galerkin formulation, showing that the pore-pressure oscillations, arising from an unstable approximation of the incompressibility constraint on the initial condition, decay in time. Error estimates holding over the unbounded time domain for both semidiscrete and fully discrete formulations are presented, and a post-processing technique is employed to improve the pore-pressure accuracy.     

Neutral stability curves for a thermal convection problem

Summary An exact solution of an eigenvalue problem arising in a thermal convection problem of Bénard type is given. Its numerical realisation gives the neutral curves at various values of an electrical dimensionless numberEl. Our results validate those obtained by Turnbull [1] by means of approximative Galerkin method.

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Neutrale Stabilitätskurven für ein Konvektionsproblem

Zusammenfassung Es wird eine exakte Eigenwertlösung des in der thermischen Konvektion vom Bénard-Typ auftretenden Problems gefunden. Das numerische Ergebnis stellt die von einer elektrischen KennzahlEl abhängige neutrale Kurvenschar dar. Diese mit einem exakten Verfahren gefundenen Ergebnisse entsprechen den von Turnbull [2] mit einem Galerkin-Verfahren näherungsweise berechneten Resultaten.

     

On a problem of the vibration of functionally graded conical shells with mixed boundary conditions

This paper presents a theoretical approach to solve vibration problems of functionally graded (FG) truncated conical shells under mixed boundary conditions. The material properties of FG shell are assumed to vary continuously through the thickness of the conical shell. The fundamental relations, motion and strain compatibility equations of FG truncated conical shells are derived by means of the Airy stress function method. Two cases of mixed boundary conditions are investigated. The basic equations are solved by using Galerkin method and fundamental cyclic frequencies of FG truncated conical shells are obtained. The results are compared and validated with the results available in the literature. The detailed parametric studies are carried out to investigate the influences of radius-to-thickness ratio, lengths-to-radius ratio, material composition and mixed boundary conditions on the fundamental cyclic frequencies of truncated conical shells.