A complete closed form vectorial solution to the Kepler problem

The paper gives an exact vectorial solution to the Kepler problem. A vectorial regularization that linearizes the Kepler problem is given using a Sundman transformation. Closed form expressions describing the Keplerian motion are deduced. A unified approach to the classic Kepler problem is offered, by studying both rectilinear and non-rectilinear Keplerian motions with the same instrument. The approach is an elementary one and only simple vectorial computations are involved.     

A problem for diffusion with relaxation in polymers

A nonlinear problem for penetrant diffusion with relaxation in polymers is considered. A numerical approach to solving this type of problems is developed. The proposed numerical scheme based on a finite element domain approximation and a time difference method can be used for numerical simulation of the considered penetrant diffusion in 2-D and 3-D domains. A numerical procedure and a corresponding computer code are created and tested for some examples in 1-D and 2-D domains.     

The langevin approach to the dumbbell kinetic problem in rheology

An alternative formalism to the dumbbell kinetic problem is proposed which is believed to be more fundamental than the classical Liouville one. The new formulation provides a logical approach to non-conservative systems and systems with varying frictional coefficients. A non-linear dumbbell with internal viscosity and varying frictional factor for the beads is examined. It is proved that the centre of gravity of the dumbbell must move affinely with the solvent continuum. A useful class of approximation is suggested to reduce the constitutive equation to an explicit form. The response of the model is computed for a number of flow fields. For shear flows, the introduction of the internal viscosity results in a shear-thinning phenomenon. The onset of non-Newtonian behaviour occurs at the correct order of magnitude of the dimensionless shear rate. Also, a negative second normal stress difference is found which varies with shear rate. In an oscillatory shear flow, the internal viscosity leads to a finite limiting value of the dynamic viscosity at high frequencies. In elongational flow the effects of the varying frictional coefficient dominate that of the internal viscosity. Interesting phenomena include the presence of a hysteresis loop and the ability of the dumbbell to maintain an extended configuration at moderate elongational rates. Clearly, these are relevant in turbulent drag reduction applications. The model has sufficient merits to deserve more investigation.     

A semi-inverse shape optimization problem in linear anti-plane shear

The design of a semi-infinite fillet for efficient stress transmission is considered. The problem is treated within the context of anti-plane shear deformations of a homogeneous, isotropic, linearly elastic solid. Under a remote state of simple shear, it is desired to determine the shape of the traction-free lateral boundaries of a symmetric plane domain so that the shear stress distribution on the finite end is as uniform as possible. A semi-inverse approach for a particular class of semi-infinite profiles is used to examine this issue.     

Developing flow in inclined laminar-laminar stratified systems: Investigation of the multiple holdup problem

In this study, the multiple holdup solutions problem for stratified laminar-laminar flow in a channel is investigated. A stationary but developing monodimensional flow model is adopted here to follow the evolution of the holdup value from a non-equilibrium inlet condition to the final downstream and fully developed solution. A first order Ordinary Differential Equation (ODE) solver is used to perform this analysis under the assumption that the flow remains supercritical all along the pipe. The possibility of having a hydraulic shock during the longitudinal evolution of the system is investigated too. A second order ODE model is then proposed to handle situations with shocks, by including the effects of the longitudinal viscous stress diffusion when large interface level gradients occur. The results are also discussed regarding the approach of minimization of a potential function, showing a good consistency between the two methods.     

A numerical study on the solution of the Cauchy problem in elasticity

This work deals with the Cauchy problem in two-dimensional linear elasticity. The equations of the problem are discretized through a standard FEM approach and the resulting ill-conditioned discrete problem is solved within the frame of the Tikhonov approach, the choice of the required regularization parameter is accomplished through the Generalized Cross Validation criterion. On this basis a numerical experimentation has been performed and the calculated solutions have been used to highlight the sensitivity to the amount of known data, the noise always present in the data, the regularity of boundary conditions and the choice of the regularization parameter. The aim of the numerical study is to implicitly device some guidelines to be used in the solution of this kind of problems.     

A substructure‐based iterative inner solver coupled with Uzawa's algorithm for the Stokes problem

A domain decomposition method with Lagrange multipliers for the Stokes problem is developed and analysed. A common approach to solve the Stokes problem, termed the Uzawa algorithm, is to decouple the velocity and the pressure. This approach yields the Schur complement system for the pressure Lagrange multiplier which is solved with an iterative solver. Each outer iteration of the Uzawa procedure involves the inversion of a Laplacian in each spatial direction. The objective of this paper is to effectively solve this inner system (the vector Laplacian system) by applying the finite‐element tearing and interconnecting (FETI) method. Previously calculated search directions for the FETI solver are reused in subsequent outer Uzawa iterations. The advantage of the approach proposed in this paper is that pressure is continuous across the entire computational domain. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer Uzawa iterations and inner FETI iterations is reported. Results show that the total number of inner iterations is almost numerically scalable since it grows asymptotically with the mesh size and the number of subdomains. Copyright © 2003 John Wiley & Sons, Ltd.     

Inverse problem for the viscoelastic medium with discontinuous wave impedance

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A model problem for ultra-wide band ionospheric sounding

A 1D model problem of wide-band pulse reflection from a plasma layer with linearly growing electron density is considered. Exact, analytical expression is found, describing the plasma response to the Dirac delta excitation. Both short and long times in the reflected pulse are separately considered. This approach is extended to arbitrary electron density profiles and its relation to traditional vertical ionospheric sounding techniques is discussed.     

First-order optimality condition of basis pursuit denoise problem

A new first-order optimality condition for the basis pursuit denoise （BPDN） problem is derived. This condition provides a new approach to choose the penalty param- eters adaptively for a fixed point iteration algorithm. Meanwhile, the result is extended to matrix completion which is a new field on the heel of the compressed sensing. The numerical experiments of sparse vector recovery and low-rank matrix completion show validity of the theoretic results.     

On the nonlocal bending problem with fractional hereditariness

Meccanica - Nonlocal hereditariness in Bernoulli–Euler beam is investigated in this paper. An approach to solve that problem is proposed and some analytical solutions are provided. To this...     

Non-local criteria for the borehole problem: Gradient Elasticity versus Finite Fracture Mechanics

Two nonlocal approaches are applied to the borehole geometry, herein simply modelled as a circular hole in an infinite elastic medium, subjected to remote biaxial loading and/or internal pressure. The former approach lies within the framework of Gradient Elasticity (GE). Its characteristic is nonlocal in the elastic material behaviour and local in the failure criterion, hence simply related to the stress concentration factor. The latter approach is the Finite Fracture Mechanics (FFM), a well-consolidated model within the framework of brittle fracture. Its characteristic is local in the elastic material behaviour and non-local in the fracture criterion, since crack onset occurs when two (stress and energy) conditions in front of the stress concentration point are simultaneously met. Although the two approaches have a completely different origin, they present some similarities, both involving a characteristic length. Notably, they lead to almost identical critical load predictions as far as the two internal lengths are properly related. A comparison with experimental data available in the literature is also provided.

     

A note on the reservoir routing problem

The reservoir routing problem has been recognized and managed for quite some time, but this paper presents a consistent mathematical approach by applying only the mass conservation principle (instead of various ‘ad hoc’ approaches). Application of the mass conservation principle facilitates the presentation of integral and differential formulations based on either the water volume or the water level of an arbitrarily-shaped reservoir. For the solution, new functions with a clear physical meaning are introduced. An error analysis is performed using a new approach based on spectral operators (traditionally, Richardson extrapolation has been used). It becomes evident that the integral formulations are more advantageous in terms of stability; however, they take more time to program and to calculate. The calculations in the examples are based on actual data from an artificial water reservoir (lake) Botonega in Croatia using the authors’ computer program that was created in the Mathcad environment.     

Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates

A novel asymptotic approach to the theory of non-homogeneous anisotropic plates is suggested. For the problem of linear static deformations we consider solutions, which are slowly varying in the plane of the plate in comparison to the thickness direction. A small parameter is introduced in the general equations of the theory of elasticity. According to the procedure of asymptotic splitting, the principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. Three-dimensional conditions of compatibility make the analysis more efficient and straightforward. We obtain the system of equations of classical Kirchhoff's plate theory, including the balance equations, compatibility conditions, elastic relations and kinematic relations between the displacements and strain measures. Subsequent analysis of the edge layer near the contour of the plate is required in order to satisfy the remaining boundary conditions of the three-dimensional problem. Matching of the asymptotic expansions of the solution in the edge layer and inside the domain provides four classical plate boundary conditions. Additional effects, like electromechanical coupling for piezoelectric plates, can easily be incorporated into the model due to the modular structure of the analysis. The results of the paper constitute a sound basis to the equations of the theory of classical plates with piezoelectric effects, and provide a trustworthy algorithm for computation of the stressed state in the three-dimensional problem. Numerical and analytical studies of a sample electromechanical problem demonstrate the asymptotic nature of the present theory.     

A fictitious domain approach for the Stokes problem based on the extended finite element method

In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM is used for the fluid flow. The interface between the fluid and the structure is localized by a level‐set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf‐sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given, and several numerical tests are performed to illustrate the capabilities of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Whole time domain solution of inverse heat conduction problem in multi-layer media

The unknown surface conditions in composite media is estimated by minimizing the nonlinear least squares error between the computed and measured temperatures over the whole time domain. This approach shown to be stable, efficient and accurate. The unknown surface conditions are assumed to have an abrupt change at unknown time. A sensitivity analysis is conducted to learn more insight into the nature of difficulties that can be encountered in the estimation of the parameters associated with the inverse problem. The stability and accuracy of the method is demonstrated by several numerical examples which provide very strict test conditions. Received on 7 March 1997     

Quadratic minimization for equilibrium problem variational inclusion and fixed point problem

The purpose of this paper is to find the solutions to the quadratic mini- mization problem by using the resolvent approach. Under suitable conditions, some new strong convergence theorems are proved for approximating a solution of the above min- imization problem. The results presented in the paper extend and improve some recent results.     

On the kinematic minimum principle for the rate problem in classical plasticity

An alternative form is given of the kinematic minimum principle for the rate problem in classical, small displacement plasticity. This study was motivated by the theorem given by G. Maier (1969) based on a quadratic programming approach, but the result is derived from the classical theorem and has a simpler form than Maier's theorem.     

有限变形的弧长算法

近些年来,人们提出了很多方法来解决结构静力非线性跟踪分析问题,其中,弧长算法应用最为广泛,但是,其中仍存在很多问题,本文针对梁板壳结构计算中的有限变形弧长算法,首先引入了将位移自由度与转动自由度分离提法,在此前提下对前人已有的处法加以改造,建立一个N＋1维的增量弧长方程组进行跟踪求解,本文同时引入了无量纲化,增量板长自动调节系数,奇点的判定准则,最终提出一个实用的弧长算法,本文在结尾将给出几个算例以显示该算法良好的跟踪性能。