Self-similar solution of the plane problem of the evolution of a hydraulic fracture crack in an elastic medium

The plane problem of the evolution of a hydraulic fracture crack in an elastic medium is considered. It is established that a self-similar solution is only possible at a constant rate of fluid injection. The solution for the value of the crack opening is presented in the form of a series expansion in Chebyshev polynomials of the second kind, and expansion coefficients are obtained as a solution of the algebraic set of equations which arise when projecting the balance equation for injected fluid mass on Chebyshev polynomials. When there is no part of the region unfilled with fluid (a fluid lag), the gradient of the crack opening at the crack tip turns out to be singular when the finiteness of the medium stress intensity factor is taken into account. According to the estimate made, the rate of convergence of the series expansion for the solution in Chebyshev polynomials is fairly rapid for a small injection intensity.     

The coupled thermoelastic problem for a plane with a rectilinear crack

We solve the thermoelastic problem for a plane with a rectilinear heat-conducting crack whose conductivity depends on its opening. By modeling the crack as a thin inclusion of variable thickness we reduce the problem to a system of singular integrodifferential equations for the potential densities of the temperature field. We study the behavior of the unknown functions at the ends of the contour of integration and, using a numerical-iteration method, we also determine the solution of the problem. We find an approximate asymptotic solution in the case of a weakly conducting crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 54–58.     

Case study: problem formulation and solution for a real-world sports scheduling problem

This case study paper describes a problem faced every year by the Devon Cricket League in England. Every league match requires two officiating umpires. There are various preferences relating to the deployment of umpires that fall short of being definite constraints. The requirement was to produce a computer system that would allocate umpires in a satisfactory manner without the need for human interaction, since the users of the system will be unqualified amateurs. This necessitated the construction of a very complex objective function, using imprecise and ill-defined information. The biggest challenge was thus one of problem formulation; the solution technique was relatively straightforward. The system has been put into successful use and it is expected that its use will continue indefinitely. The work raises ethical issues regarding work done for customers who are inexperienced amateurs.     

Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators

A system of two non-linear differential equations is considered that simulates the dynamics of two completely identical weakly coupled oscillators both in the case of dissipative and active coupling. The system of normal modes is investigated. All the self-similar periodic solutions, including the asymmetric solutions describing the natural ascillations of oscillators with dissimilar amplitude's, are found analytically. The stability is investigated as well as the local bifurcations of the self-similar cycles when there is a change in stability. In particular, the possibility of the creation of two-dimensional invariant tori is pointed out. In the case of active coupling, it is shown that the basic version of the natural oscillations is a stable antiphase cycle that was observed in Huygens experiments.     

Numerical solution of a singular integral equation arising in a cruciform crack problem

A singular integral equation arising in a cruciform crack problem is investigated in the present paper. Based on the convex technique, the piecewise Taylor-series expansion method is extended by introducing a weight parameter. An approximate solution of the singular integral equation is constructed and its convergence and error estimate are made. The variations of the approximate solutions associating with stress intensity factors are analyzed by considering internal pressures of power and sine functions, respectively. By comparing with the known methods, the observations reveal that a good approximation can be achieved using less derivative times, less discretization points, and a suitable weight parameter. The obtained results show that the crack growth is dependent on applied mechanical loadings.     

A simple solution of the bridged crack problem

A simple, yet effective solution is derived to the problem of a penny-shaped crack, whose faces are bridged by fibers, and subjected to normal loading. The mathematical apparatus developed by the author earlier (Fabrikant, 1989, 1991) is used to reduce the governing integral equation to a sequence of integrals of elementary functions, which can be easily solved by the method of collocations. The case of uniform loading is considered as an example. The accuracy of the obtained solution is rigorously verified by different criteria.     

一个耦合系统自由边界问题的整体古典解

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒｗｅｄｉｓｃｕｓｓｔｈｅｇｌｏｂａｌｃｌａｓｓｉｃａｌｓｏｌｕｔｉｏｎｏｆａｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｑｕａｓｉｓｔａｔｉｏｎａｒｙｐｒｏｂｌｅｍ .Ｔｈｅｐｒｏｂｌｅｍｃｏｍｅｓｆｒｏｍｔｈｅｄｉｓｃｕｓｓｉｏｎｏｆａｇｒｏｗｔｈｍｏｄｅｌｏｆｓｅｌｆｍａｉｎｔａｉｎｉｎｇｐｒｏｔｏｃｅｌｌ(ｓｅｅ [1— 3])ｉｎｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｃａｓｅ .Ｔｈｅｐｒｏｔｏｃｅｌｌｃａｎｂｅｖｉｓｕａｌｉｚｅｄａｓｈａｖｉｎｇａｐｏｒｏｕｓｓｔ…     

New formulation and solution of the plane elastostatic traction problem

The traction problem of elastostatics in two dimensions is formulated in terms of integral equations via the Airy stress function. The integral equations are solved numerically using piecewise constant approximations to the unknown functions. The validity of the formulation is demonstrated by its application to a simple problem with a known solution.     

On the solution of a coupled thermo-mechanical problem for non-homogeneous Timoshenko-type shells

In this work a coupled thermo-mechanical problem of non-homogeneous shells with variable thickness and variable Young modulus (a so-called Timoshenko type model) is considered. The problem is reduced to uniformly correct problem in the form of a first order difference operator equation. In addition, a similar approach can easily be applied to the Kirchhoff-Love model.     

Iterative methods for the solution of a singular control formulation of a GMWB pricing problem

Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a guaranteed minimum withdrawal benefit (GMWB), where the underlying asset is assumed to follow a jump diffusion process. We use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges. Our methods take advantage of the special structure of the GMWB problem in order to obtain a rapidly convergent iteration. The direct control method has a scaling parameter which must be set by the user. We give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round-off error. Example computations verify that these estimates are of the correct order. Finally, we compare the scaled direct control formulation to a formulation based on a block version of the penalty method (Huang and Forsyth in IMA J Numer Anal 32:320?C351, 2012). We show that the scaled direct control method has some advantages over the penalty method.     

A finite element solution of an added mass formulation for coupled fluid-solid vibrations

Summary. A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed. Received August 31, 1998 / Published online July 12, 2000     

An analytical solution for the mechanically coupled diffusion problem in thin-film electrodes

This paper presents a quasi-static solution for the mechanically coupled diffusion problem in a Lithium-ion battery thin film electrode during charge. The concentration field and the stresses are calculated under the assumption of a constant flux. The influence of the electrode's stiffness, its thickness, and of the current density on the results are discussed. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm

The Vehicle Routing Problem (VRP) is one of the most well studied problems in operations research, both in real life problems and for scientific research purposes. During the last 50 years a number of different formulations have been proposed, together with an even greater number of algorithms for the solution of the problem. In this paper, the VRP is formulated as a problem of two decision levels. In the first level, the decision maker assigns customers to the vehicles checking the feasibility of the constructed routes (vehicle capacity constraints) and without taking into account the sequence by which the vehicles will visit the customers. In the second level, the decision maker finds the optimal routes of these assignments. The decision maker of the first level, once the cost of each routing has been calculated in the second level, estimates which assignment is the better one to choose. Based on this formulation, a bilevel genetic algorithm is proposed. In the first level of the proposed algorithm, a genetic algorithm is used for calculating the population of the most promising assignments of customers to vehicles. In the second level of the proposed algorithm, a Traveling Salesman Problem (TSP) is solved, independently for each member of the population and for each assignment to vehicles. The algorithm was tested on two sets of benchmark instances and gave very satisfactory results. In both sets of instances the average quality is less than 1%. More specifically in the set with the 14 classic instances proposed by Christofides, the quality is 0.479% and in the second set with the 20 large scale vehicle routing problems, the quality is 0.826%. The algorithm is ranked in the tenth place among the 36 most known and effective algorithms in the literature for the first set of instances and in the sixth place among the 16 algorithms for the second set of instances. The computational time of the algorithm is decreased significantly compared to other heuristic and metaheuristic algorithms due to the fact that the Expanding Neighborhood Search Strategy is used.     

On the uniqueness of the solution of a unified formulation for the boundary conditions in the Dam problem

We study a free boundary problem corresponding to a model for the Dam problem with generalized boundary conditions. Concentrating ourselves on a two dimensional example, we establish various properties of the solution, study the regularity of the free boundary and prove the uniqueness of the solution.

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Sunto Studiamo un problema di frontiera libero corrispondente al modello relativo al problema della diga con condizioni al contorno generalizzate. Concentrandoci su di un esempio in due dimensioni, stabiliamo varie proprietà della soluzione, studiamo la regolarità della frontiera libera e proviamo l’unicità della soluzione.

     

Classical solution to a 1-periodic initial value problem of parabolic type coupled with operators

We consider a class of initial value problem of parabolic type coupled with operators, which arises naturally from image recovery. We obtain the existence and uniqueness of 1-periodic classical solution to the problem. The result is a prerequisite for establishing the existence of viscosity solutions to some PDE based image denoising and enhancement models.     

Self-similar solutions of the Laplacian growth problem in the half-plane

We investigate a version of the Laplacian growth problem with zero surface tension in the upper half-plane. Using the method of time-dependent conformal maps, we find families of self-similar exact solutions that are expressible in terms of the hypergeometric function.     

A stream function-vorticity formulation coupled with boundary integrals for the two-dimensional exterior Stokes problem

In this work, we derive a stream function-vorticity variational formulation coupled with boundary integrals for the exterior Stokes problem in two dimensions, when the right-hand side has a bounded support. The stream function-vorticity formulation is expressed in a bounded region containing the support of the right-hand side, and the boundary conditions on the artificial boundary are obtained by an integral representation. We prove that this coupled formulation is equivalent to the original Stokes problem.