Particle trajectory and mass transport of finite-amplitude waves in water of uniform depth

A set of governing equations in Lagrangian form is derived for propagating gravity waves in water of uniform depth. The Lindstedt–Poincaré perturbation method is used to obtain approximations up to fifth order. Recognizing the Lagrangian frequency to be a position function for all particles is a key to find these higher-order approximations. The present solution has zero pressure at the free surface and satisfies exactly the dynamic boundary condition. Under the present approximations, the Lagrangian frequency is composed of two parts. The first part is constant for all particles and equivalent to the term in the fifth-order Stokes' wave theory [J.D. Fenton, A fifth-order Stokes theory for steady waves, J. Waterway, Port, Coastal Ocean Eng. 111 (1985) 216–234]. The second part is a function of the depth. All the particles move as open (nonclosed) loops and have mean drift displacements that decrease exponentially with the water depth. Thus, a new fourth-order mass transport velocity is found.     

Projection-algebraic approximation of linear and nonlinear operator differential equations in Banach spaces

We develop a Calogero-type projection-algebraic method of discrete approximations for linear differential equations in Banach spaces and analyze the convergence of finite-dimensional approximations based on the functional-analytic approach to discrete approximations and methods of operator theory in Banach spaces. Applications of the obtained results to the functional-interpolation scheme of the projection-algebraic method of discrete approximations are considered. Based on a generalized Leray–Schauder-type theorem, we consider the projection-algebraic scheme of discrete approximations and analyze its solvability and convergence for a special class of nonlinear operator equations.     

Error estimates for averaging method in multifrequency systems with constant delay

We prove new error estimates for the averaging method for multifrequency systems of higher approximations with delay in slow and fast variables. The main assumption is imposed on the vector of frequencies and its derivatives along the solution of the averaged system of zero approximation. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 233–243, April–June, 2006.     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems

By introducing a new parametric transformation and a suitable nonlinear frequency expansion, the modified Lindstedt–Poincaré method is extended to derive analytical approximations for limit cycles in three-dimensional nonlinear autonomous dynamical systems. By considering two typical examples, it can be seen that the results of the present method are in good agreement with those obtained numerically even if the control parameter is moderately large. Moreover, the present prediction is considerably more accurate than some published results obtained by the multiple time scales method and the normal form method.     

Transitions from Strongly to Weakly Nonlinear Motions of Damped Nonlinear Oscillators

We construct analytical approximations for the transition from strongly nonlinear, early-time oscillations to weakly nonlinear, late-time motions of single degree of freedom, damped, nonlinear oscillators. Two methods are developed. The first relies on (a) derivation of an analytic solution for the initial value problem of an exactly integrable damped system, (b) development of separate early- and late-time approximations to the damped motion using the integrable solution, and (c) patching of the two approximations in the time domain by imposing continuity conditions on the composite solution at the point of matching. The second approach relies on a multiple-scales application of the method of nonsmooth transformations first developed by Pilipchuck, but complemented with a corrected frequency-amplitude relation. This improved relation is obtained by developing two separate frequency-amplitude asymptotic expansions in the frequency-amplitude plane, that are valid for large and small amplitudes, respectively, and then matching them using two-point diagonal Padé approximants. Comparisons between analytical approximations and numerical results validate the two approaches developed     

Application of Uniform Asymptotics to the Second Painlevé Transcendent

In this work we propose a new method for investigating connection problems for the class of nonlinear second‐order differential equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort. Our method relies on finding uniform approximations of differ ential equations of the generic form as the complex‐valued parameter . The details of the treatment rely heavily on the locations of the zeros of the function F in this limit. If they are isolated, then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of F coalesce as , then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the “classical” connection problem associated with the second Painlevé transcendent. Future papers will show how the technique can be applied with very little change to the other Painlevé equations, and to the wider problem of the asymptotic behavio ur of the general solution to any of these equations. (Accepted May 15, 1997)     

A method to study the nonaxisymmetric plastic deformation of solids of revolution with allowance for the stress mode

The paper proposes a method to allow for the stress mode in analyzing the thermoelastoplastic stress-strain state of compound bodies of revolution under asymmetric loading and heating. Use is made of a semianalytic finite-element method and the method of successive approximations. Some numerical results are presented Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 26–35, September 2008.     

Error Bounds for Semi-Galerkin Approximations of Nonhomogeneous Incompressible Fluids

We consider spectral semi-Galerkin approximations for the strong solutions of the nonhomogeneous Navier–Stokes equations. We derive an optimal uniform in time error bound in the H¹ norm for approximations of the velocity. We also derive an error estimate for approximations of the density in some spaces L^r. P. Braz e Silva was supported for this work by FAPESP/Brazil, #02/13270-1 and is currently supported in part by CAPES/MECD-DGU Brazil/Spain, #117/06. M. Rojas-Medar is partially supported by CAPES/MECD-DGU Brazil/Spain, #117/06 and project BFM2003-06446-CO-01, Spain.     

Drag of a strongly heated sphere at small Reynolds numbers

Using an asymptotic small-perturbation method, the flow around a strongly heated sphere at small Reynolds numbers Re ≪ 1 is considered with account for thermal stresses in the gas in the higher-order approximations, beyond the Stokes one. It is assumed that the value of the Prandtl number Pr is arbitrary and the temperature dependence of the viscosity is described by a power law with an arbitrary exponent. In the O(Re²) and O(Re³ ln(Re)) approximations, the drag force of a heated sphere is found over a wide range of the ratios of sphere’s temperature to the gas free-stream temperature T _W /T _∞. The limits of applicability of the first (in Re) approximation are investigated, including the negative-drag effect, attributable to the action of the thermal stresses. The results are compared with numerical calculations of the flow around a hot sphere. The limits of applicability of the approximations found are examined. Similar results are obtained for the standard Navier-Stokes equations in which the thermal stresses are neglected.     

The back-scattering from a circular loop

Summary The variational method is used to determine the echo area of a perfectlyconducting, thin, circular loop. The equivalent source of the scattered field is approximated by the current which flows parallel with the axis of the wire. An interesting feature of this problem is that for the broadside aspect the current distribution can be determined exactly within the limits of the approximations made for thin wires; consequently. the accuracy of the approximations in this case can be readily checked. Comparisons are made between the calculated and measured values of the echo area at the broadside and edge aspects of the loop. Good agreement between the two values is obtained. Hence, it appears that approximations which have been successfully used for straight wire radiators may be extended to cases where the wire is curved. Echo area values of the loop are compared with those of the circular plate, the sphere, and the straight wire. The equivalent dipole sources associated with small loop scatterers are treated.     

Solution of the Stochastic Boundary-Value Problem of Steady-State Creep for a Thick-Walled Tube Using the Small-Parameter Method

The physically and statistically nonlinear problem of steady-state creep for a thick-walled tube loaded by internal pressure is solved in the third approximation using the small-parameter method. The variances of random creep strain rates and displacements are calculated. The results obtained are compared with the solution of the same problem in the first and second approximations. A reliability assessment method for the tube using the strain failure criteria is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 161–171, January–February, 2006.     

Second-order effects and Saint Venant’s principle in the torsion problem of a nonlinear elastic rod

The torsion problem of a circular nonlinear elastic rod loaded by end moments is considered. The solution constructed by the method of successive approximations taking into account second-order effects is compared with the solution obtained by a semi-inverse method. It is shown that the dead-loading assumption breaks the symmetry of the Cauchy stress tensor in a certain region. A refined formulation of Saint Venant’s principle is proposed for the problem of determining integral strain characteristics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 129–136, November–December, 2006.     

NEW TREATMENT OF ESSENTIAL BOUNDARY CONDITIONS IN EFG METHOD BY COUPLING WITH RPIM

One of major difficulties in the implementation of meshfree methods using the moving least square (MLS) approximation, such as element-free Galerkin method (EFG), is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. Another class of meshfree methods based on the radial basis point interpolation can satisfy the essential boundary conditions exactly since its approximation function passes through each node in an influence domain and thus its shape functions possess the properties of delta function. In this paper, a coupled element-free Galerkin(EFG)-radial point interpolation method (RPIM) is proposed to enhance their advantages and avoid their disadvantages. Discretized equations of equilibrium are obtained in the RPIM region and the EFG region, respectively. Then a collocation approach is introduced to couple the RPIM and the EFG method. This method satisfies the linear consistency exactly and can maintain the stiffness matrix symmetric. Numerical tests show that this method gives reasonably accurate results consistent with the theory.     

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.     

Domain decomposition method with hybrid approximations applied to solve problems of elasticity

To solve two-dimensional boundary-value problems of elasticity, two iteration algorithms of the domain decomposition method are proposed: parallel Neumann–Neumann and sequential Dirichlet–Neumann. They are based on the hybrid boundary–finite-element approximations. The algorithms are proved to converge. The optimal parameters are selected using the minimum-residual and steepest-descent methods. Some plane problems of elasticity are solved as examples, and stationary and nonstationary iteration algorithms in these examples are analyzed for efficiency Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 18–29, November 2008.     

Stress state of compound solids of revolution made of damaged orthotropic materials with different tensile and compressive moduli

A technique is proposed to allow for damages and different tensile and compressive moduli of orthotropic materials in stress-strain analysis of compound bodies of revolution under nonaxisymmetric loading and heating. The technique combines the semi-analytic finite-element method and the method of successive approximations __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 57–68, November 2006.     

A model-reduction method for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty

The present research work proposes a new systematic approach to the problem of model reduction for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty. The problem of interest is addressed within the context of functional equation theory, and in particular, through a system of invariance functional equations for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system’s slow invariant manifold attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original discrete-time system dynamics on the slow manifold. The analyticity property of the solution to the invariance functional equations enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the system’s transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, the proposed method is evaluated through an illustrative biological reactor example.