Numerical solutions for spherical elastic-plastic wave propagation

Zusammenfassung Die Fortpflanzung elastisch-plastischer Belastungs- und Entlastungswellen in einem idealen elastisch-plastischer Medium, welches einen kugelformigen Hohlraum enthält und dem Trescaschen Fliesskriterium folgt, wird betrachtet, wobei ein zeitabhängiger Druck auf der Hohlraumfläche angenommen wird. Die numerische Methode, basiert auf endliche Differenzen, behandelt die verschiedenen Wellenfronten exakt. Es wird gezeigt, dass die Geschwindigkeiten der Entlastungswellen von der Abnahmegeschwindigkeit des angewandten Druckes abhängen. Die Resultate für die Hohlraumerweiterung werden mit denjenigen verglichen, welche mit einer bestehenden Methode erhalten werden.

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Notation _r radial stress - _t tangential stress - _r non-dimensional radial stress (= _r /K) - _t non-dimensional tangential stress (= _t K) - K yield stress in tension - , Lamé's constants - bulk modulus (=(3 +2 )/3) - Poisson's ratio - material density - C elastic wave velocity - C _p plastic wave velocity - r radial co-ordinate - r ₀ cavity radius - r non-dimensional radial co-ordinate (=r/r ₀) - t time - t non-dimensional time (=C t/r ₀) - u radial displacement - u non-dimensional radial displacement (=u/r ₀) - v particle velocity - v non-dimensional particle velocity (v/C) - p pressure - p non-dimensional pressure (=p/K) Presented at the 12th International Congress of Applied Mechanics, Stanford, Aug. 26–31, 1968.     

Numerical solutions of some parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we introduce a variational form by defining a new function. Both continuous and discrete Galerkin procedures are illustrated in this paper. The error estimates are also derived.     

Numerical solutions to continuous linear programming problems

Summary A method for finding approximate solutions for continuous linear programming problems is suggested. The required conditions to be met are: a) the matrix associated with the integrals in the constraints is constant; b) all functions involved are of bounded variation; c) the matrices involved satisfy certain boundedness conditions, and d) there exist feasible solutions.The approximations converge almost everywhere to an optimal solution. The optimal solution is shown to be a function of bounded variation.The method is illustrated by means of a numerical example. Here, the approximate solution reveals also the structure of the exact (analytical) solution and makes its construction possible.     

Differential stability of solutions to parametric optimization problems

A method for the differential stability of solutions to a class of solutions to a class of parametric optimization problem is prposed. Any solution of the parametric optimization problem is given as a fixed point of the metric projection onto the set of admissible coefficients. A new result on the differential stability of the metric projection in Sobolev space H²(Ω)onto a set of admissible parameters is obtained. The stability results with respect to perturbations of observations for the solutions to a coefficient estimation problem for a second-order elliptic equation are derived.     

Differential stability of solutions to constrained optimization problems

In this paper the differential stability of solutions to constrained optimization problems is investigated. The form of right-derivatives of optimal solutions to such problems, with respect to a real parameter, is derived. The right-derivative of the optimal control with respect to parameter for an optimal control problem for parabolic equation is obtained in the form of the optimal solution to an auxiliary optimal control problem. A method for determination of the second right-derivative of the optimal solutions to constrained optimization problems is proposed. Several examples are provided.This research has been supported by INRIA, in the framework of a collaboration between the project Théorie des Systèmes and the Systems Research Institute of the Polish Academy of Sciences.     

Numerical validation of solutions of linear complementarity problems

Summary. This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists of two sufficient conditions that can be tested on a digital computer. If the first condition is satisfied then a given multidimensional interval centered at an approximate solution of the problem is guaranteed to contain an exact solution. If the second condition is satisfied then the multidimensional interval is guaranteed to contain no exact solution. This study is based on the mean value theorem for absolutely continuous functions and the reformulation of linear complementarity problems as nonsmooth nonlinear systems of equations. Received August 21, 1997 / Revised version July 2, 1998     

Numerical verification of solutions for some unilateral problems

We proposed some numerical methods for the automatic proof of existence of solutions for some unilateral problems. In this paper, our goal is to establish a new procedure for numerical verification of some unilateral problems.     

Asymptotic properties of solutions to some three-dimensional wave problems

Green's function for the Helmholtz equation of a bounded domain D with Neumann boundary conditions is considered. The boundary of D is smooth and bounded, and its Gaussian curvature is positive. An estimate of Green's function is obtained in the nonphysical region 0 >.Jm K >–|Re K |^1/3.This estimate shows that the nonspectral singularities of Green's function lie below the parabola Jm K0 ; |Jm K|=|Re K|^1/3.On the basis of these results, the authors investigate the behavior as t of Green's function for the wave equation in the domain Dx(0相似文献   

Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $\left\{\begin{array}{l}i\varepsilon_{t}+\varepsilon_{xx}=n\varepsilon+\alpha|\varepsilon|^{2}\varepsilon,\\n_{t}=(|\varepsilon|^{2})_{x}, x\in R.\end{array} \right.$ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $L$ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $L$. As the modulus of the Jacobian elliptic function $k\rightarrow 1$, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $\alpha=0$, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.     

Well-posedness and stability of solutions for set optimization problems

In this paper, some characterizations for the generalized l-B-well-posedness and the generalized u-B-well-posedness of set optimization problems are given. Moreover, the Hausdorff upper semi-continuity of l-minimal solution mapping and u-minimal solution mapping are established by assuming that the set optimization problem is l-H-well-posed and u-H-well-posed, respectively. Finally, the upper semi-continuity and the lower semi-continuity of solution mappings to parametric set optimization problems are investigated under some suitable conditions.     

Numerical methods for oscillatory solutions to hyperbolic problems

Difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied. Analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths. © 1993 John Wiley & Sons, Inc.     

On the stability of solutions to quadratic programming problems

We consider the parametric programming problem (Q _p ) of minimizing the quadratic function f(x,p):=x ^T Ax+b ^T x subject to the constraint Cx≤d, where x∈ℝ ⁿ , A∈ℝ ^n×n , b∈ℝ ⁿ , C∈ℝ ^m×n , d∈ℝ ^m , and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q _p ) are denoted by M(p) and M ^loc (p), respectively. It is proved that if the point-to-set mapping M ^loc (·) is lower semicontinuous at p then M ^loc (p) is a nonempty set which consists of at most ? _m,n points, where ? _m,n = is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones. Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000     

Numerical linear stability on controlled solutions of Boussinesq Navier-Stokes

A numerical linear stability analysis on optimally controlled solutions in a thermoconvective problem is presented. The thermoconvective problem is a stationary Boussinesq Navier-Stokes with a gaussian heating at the bottom with a large horizontal temperature gradient. The numerical method is Chebyshev collocation. The optimal control permits to find a boundary condition such that the vorticity is minimized. For some values of the penalizing parameter the controlled solution is more stable than the corresponding uncontrolled one. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces.     

Regular solutions of transmission and interaction problems for wave equations

Consider n bounded domains Ω ? ? and elliptic formally symmetric differential operators A₁ of second order on Ω_i Choose any closed subspace V in $ \prod\limits_{i = 1}^n {L^2 \left({\Omega _i } \right)} $, and extend (A_i)_i=1,…,n by Friedrich's theorem to a self-adjoint operator A with D(A^1/2) = V (interaction operator). We give asymptotic estimates for the eigenvalues of A and consider wave equations with interaction. With this concept, we solve a large class of problems including interface problems and transmission problems on ramified spaces.^25,32 We also treat non-linear interaction, using a theorem of Minty²⁹.     

Numerical solutions to dynamic portfolio problems with upper bounds

In this paper, we apply value function iteration to solve a multi-period portfolio choice problem. Our problem uses power utility preferences and a vector autoregressive process for the return of a single risky asset. In contrast to the observation in van Binsbergen and Brandt (Comput Econ 29:355–368, 2007) that value function iteration produces inaccurate results, we achieve highly accurate solutions through refining the conventional value function iteration by two innovative ingredients: (1) approximating certainty equivalents of value functions by regression, and (2) taking certainty equivalent transformation on expected value functions in optimization. We illustrate that the new approach offers more accurate results than those exclusively designed for improvement through a Taylor series expansion in Garlappi and Skoulakis (Comput Econ 33:193–207, 2009). In particular, both van Binsbergen and Brandt (Comput Econ 29:355–368, 2007) and Garlappi and Skoulakis (Comput Econ 33:193–207, 2009) comparing their lower bounds with other lower bounds, we more objectively assess our lower bounds by comparing with upper bounds. Negligible gaps between our lower and upper bounds across various parameter sets indicate our proposed lower bound strategy is close to optimal.     

Asymptotic stability of solitary wave solutions to the regularized long-wave equation

We study the asymptotic stability of solitary wave solutions to the regularized long-wave equation (RLW) in . RLW is an equation which describes the long waves in water. To prove the result, we make use of the monotonicity of the local H¹-norm and apply the Liouville property of (RLW) as in Merle and Martel (J. Math. Pures Appl. 79 (2000) 339; Arch. Rational Mech. Anal. 157 (2001) 219).