Modeling elastic shells immersed in fluid

We describe a numerical method to simulate an elastic shell immersed in a viscous incompressible fluid. The method is developed as an extension of the immersed boundary method using shell equations based on the Kirchhoff‐Love and the planar stress hypotheses. A detailed derivation of the shell equations used in the numerical method is presented. This derivation, as well as the numerical method, uses techniques of differential geometry. Our main motivation for developing this method is its use in constructing a comprehensive, three‐dimensional computational model of the cochlea (the inner ear). The central object of study within the cochlea is the basilar membrane, which is immersed in fluid and whose elastic properties rather resemble those of a shell. We apply the method to a specific example, which is a prototype of a piece of the basilar membrane, and study the convergence of the method in this case. Some typical features of cochlear mechanics are already captured in this simple model. In particular, numerical experiments have shown a traveling wave propagating from the base to the apex of the model shell in response to external excitation in the fluid. © 2004 Wiley Periodicals, Inc.     

Solving the 2D Maxwell equations by a Laguerre spectral method

A spectral method for solving the 2D Maxwell equations with relaxation of electromagnetic parameters is presented. The method is based on an expansion of the solution in terms of Laguerre functions in time. The operation of convolution of functions, which is part of the formulas describing the relaxation processes, is reduced to a sum of products of the harmonics. The Maxwell equations transform to a system of linear algebraic equations for the solution harmonics. In the algorithm, an inner parameter of the Laguerre transformis used. With large values of this parameter, the solution is shifted to high harmonics. This is done to simplify the numerical algorithm and to increase the efficiency of the problem solution. Results of a comparison of the Laguerre method and a finite-difference method in accuracy both for a 2D medium structure and a layered medium are given. Results of a comparison of the spectral and finite-difference methods in efficiency for axial and plane geometries of the problem are presented.     

Viscous Flows in a Half Space Caused by Tangential Vibrations on Its Boundary

The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a half‐space, its boundary is a nondeformable plane that exhibits purely tangential vibrations. Such a simple geometrical setting allows us to study general boundary velocity fields and to obtain general results. From a practical viewpoint, such boundary conditions may be seen as the tangential vibrations of the material points of a stretchable plane membrane. In contrast to the classical boundary layer theory, we aim to build a global solution. To achieve this goal we employ the Vishik–Lyusternik approach, combined with two‐timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions. This solution corresponds to general boundary conditions and to three different settings of the main small parameter. Our solution always include the inner part and outer part that both contain oscillating and non‐oscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full Navier–Stokes equations or the Stokes equations (both with the unit viscosity) and can be interpreted as a steady or unsteady streaming. In contrast to the existing theories of a steady streaming, our solutions do not contain any secular (infinitely growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular torsional vibrations of an infinite rigid disc are considered. These examples are still brief and illustrative, while the core of the paper is devoted to the adaptation of the Vishik–Lyusternik method to the development of the general theory of vibrational boundary layers.     

Transversally inhomogeneous sharply tuned linear cochlear model

A linear mechanism of sharp frequency selectivity in the inner ear is developed. The cochlea is assumed to be a slightly varying wave guide with inhomogeneous cross section. The tectorial membrane is considered as an additional mass loading the narrow strip of the basilar membrane that underlies the rows of outer hair cells. A high concentration of mass along the middle line of the cochlea partition provides the sharpening of the tuning curve without significantly altering the phase. The dissipation of energy is assumed to hold in fluid boundary layers near the cochlea partition. The responses of the model with one and the same set of input parameters are compared with different experimental data obtained during the last decade, in the basal and apical parts of the cochlea. This paper demonstrates that a system such as cochlea is capable of performing sharply tuned linear frequency analysis without adding any outside energy to the input waveform to be analyzed. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 213–240. Translated by S. M. Novoselova.     

Investigation of frequencies of natural vibrations of a transversally isotropic cylindrical panel with a circular hole

We consider the problem of natural vibrations of a hingedly supported transversally isotropic cylindrical panel with a circular hole. The deformation of the shell is described by modified equations of the Timoshenko theory of shells. The numerical solution of the problem is constructed by the indirect method of boundary integral equations based on the sequential representation of Green functions.     

Solving fractional integral equations by the Haar wavelet method

Haar wavelets for the solution of fractional integral equations are applied. Fractional Volterra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples.     

Mathematical simulation of nonlinear oscillations of viscoelastic pipelines conveying fluid

A mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid is developed in the paper. The Boltzmann–Volterra integral model with weakly singular kernels of heredity is used to describe the processes of pipeline strain. Using the Bubnov–Galerkin method, the mathematical model of the problem is reduced to the study of a system of ordinary integro-differential equations, where time is an independent variable. The solution of integro-differential equations is determined by a numerical method based on the elimination of the singularity in the relaxation kernel of the integral operator. Using the numerical method for unknowns, a system of algebraic equations is obtained. To solve a system of algebraic equations, the Gauss method is used. A computational algorithm is developed to solve the problems of the dynamics of viscoelastic pipelines with a flowing fluid. The algorithm of the proposed method makes it possible to investigate in detail the effect of rheological parameters on the character of vibrational strength of viscoelastic pipelines with a fluid flow, in particular, in the study of free oscillations of pipelines based on the theory of ideally elastic shells. On the basis of the computational algorithm developed, a set of applied computer programs has been created, which makes it possible to carry out numerical studies of pipeline oscillations. The influence of singularity in the heredity kernels and the geometric parameters of the pipeline on the vibrations of structures possessing viscoelastic properties is numerically investigated. It is shown that an account of viscoelastic properties of pipeline material leads decrease in the amplitude and frequency of oscillation. It is established that to reveal the influence of viscoelastic properties of structure material on the pipeline oscillations, it is necessary to use the Abel-type weakly singular kernels of heredity. The obtained results of numerical simulation can be used in the enterprises of oil and gas industries, as well as in design organizations.     

The modeling and numerical simulation of gas flow networks

Summary. A network formulation is introduced for the modeling and numerical simulation of complex gas transmission systems like a multi-cylinder internal combustion engine. Several simulation levels are discussed which result in different network representations of a specific system. Basic elements of a network are chambers of finite volume, straight pipes and connections like valves or nozzles. The pipe flow is modeled by the unsteady, one-dimensional Euler equations of gas dynamics. Semi-empirical approaches for the chambers and the connections yield differential-algebraic equations (DAEs) in time. The numerical solution is based on a TVD scheme for the pipe equations and a predictor-corrector method for the DAE-system. Simulation results for an internal combustion engine demonstrate the practical interest of the new approach. Received May 12, 1994 / Revised version received August 26, 1994     

Stability study and control of helicopter blade flapping vibrations

The response of a two-degree-of-freedom, controlled, autoparametric system to harmonic excitations is studied and solved. The objective of this research is to investigate the effect of linear absorber on the vibrating system and the saturation control of a linear absorber to reduce vibrations due to rotor blade flapping motion. The method of multiple scale perturbation technique is applied to obtain the periodic response equation near the primary resonance in the presence of internal resonance of the system. The stability of the obtained numerical solution is investigated using both phase plane methods and frequency response equations. Variation of some parameters leads to the bending of the frequency response curves and hence to the jump phenomenon occurrence. The reported results are compared to the available published work.     

Space-time discontinuous Galerkin method for the solution of fluid-structure interaction

The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.     

Frequency dependent iteration method for forced nonlinear oscillators

A new iteration method for nonlinear vibrations has been developed by decomposing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.     

The Stability of a Flexible Vertical Rod on a Vibrating Support

The classical Kapitsa problem of the inverted flexible pendulum is generalized. We consider a thin homogeneous vertical rod with a free top end and pivoted or rigid attached lower end under the weight of the pendulum’s action and vertical harmonic vibrations of the support. In both cases of attachment, we have stability conditions for the vertical rod position. We take the influence of axial and bending rod vibrations and describe the bending vibrations using the Bernoulli–Euler beam model. The solution is built as a Fourier expansion by eigenfunctions of auxiliary boundary-value problems. As a result, the problem is reduced to the set of ordinary differential equations with periodic coefficients and a small parameter. The asymptotic method of two-scale expansions is used for its solution and to determine the critical level of vibration. The influence of longitudinal waves in the rod essentially decreases the critical load. The single-mode approximation has an acceptable accuracy. With pivoting support at the lower end of the rod, we find the explicit approximate solution. For the rigid attachment, we conduct numerical analysis of the critical level of vibrations depending on the problem parameters.     

Numerical analysis of the coupling of free fluid with a poroelastic material

This paper presents a numerical solution of the coupled system of the time-dependent Stokes and fully dynamic Biot equations. The numerical scheme is based on standard inf-sup stable finite elements in space and the Backward Euler scheme in time. We establish stability of the scheme and derive error estimates for the fully discrete coupled scheme. To handle realistic parameters which may cause nonphysical oscillations in the pore fluid pressure, a heuristic stabilization technique is considered. Numerical errors and convergence rates for smooth problems as well as tests on realistic material parameters are presented.     

Antiplane Problem of Electroelasticity for a Hollow Piezoceramic Cylinder Excited by a System of Surface Electrodes

An antiplane mixed boundary-value problem of electroelasticity is considered for a hollow piezoceramic cylinder with an arbitrary system of active surface electrodes generating its harmonic vibrations. The problem is solved using a method elaborated earlier for investigating vibrations of a solid piezoceramic cylinder with a system of active surface electrodes. The scheme of numerical solution of the obtained singular integro-differential equations of the boundary-value problem is based on the quadrature method. Calculation results are presented that describe the amplitude-frequency characteristics of a hollow cylinder and the behavior of some mechanical and electric quantities both within the cylinder and on its boundary.     

Numerical simulation of airfoil vibrations induced by turbulent flow

The subject of this paper is the numerical simulation of the interaction between two-dimensional incompressible viscous flow and a vibrating airfoil. A solid elastically supported airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the stabilized finite element solution of the Reynolds averaged Navier–Stokes equations with algebraic models of turbulence, coupled with the system of ordinary differential equations describing the airfoil motion. Since the computational domain is time dependent and the grid is moving, the Arbitrary Lagrangian–Eulerian (ALE) method is used. The developed method was applied to the simulation of flow-induced airfoil vibrations.     

A numerical study of a spectral problem in solid-fluid type structures

This article presents a numerical study of a spectral problem that models the vibrations of a solid–fluid structure. It is a quadratic eigenvalue problem involving incompressible Stokes equations. In its numerical approximation we use Lagrange finite elements. To approximate the velocity, degree 2 polynomials on triangles are used, and for the pressure, degree 1 polynomials. The numerical results obtained confirm the theory, as they show in particular that the known theoretical bound for the maximum number of nonreal eigenvalues admitted by such a system is optimal. The results also take account of the dependence of vibration frequencies with respect to determined physical parameters, which have a bearing on the model. © 1995 John Wiley & Sons, Inc.     

Modelling two-dimensional flow past arbitrary cylindrical bodies using boundary element formulations

This paper presents an efficient method of solving Queen's linearized equations for steady plane flow of an incompressible, viscous Newtonian fluid past a cylindrical body of arbitrary cross-section. The numerical solution technique is the well known direct boundary element method. Use of a fundamental solution of Oseen's equations, the ‘Oseenlet’, allows the problem to be reduced to boundary integrals and numerical solution then only requires boundary discretization. The formulation and solution method are validated by computing the net forces acting on a single circular cylinder, two equal but separated circular cylinders and a single elliptic cylinder, and comparing these with other published results. A boundary element representation of the full Navier-Stokes equations is also used to evaluate the drag acting on a single circular cylinder by matching with the numerical Oseen solution in the far field.     

Vibrations of a fractal elastic string

We construct the solution of the fractional space-time equations that describe the vibrations of a quasi-one-dimensional fractal elastic string. We give the solution of the Cauchy problem for fractional differential equations with initial conditions. We carry out a numerical analysis and construct the graphic variation of the displacement function of a fractal elastic string. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 142–147     

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.     

Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders

With the use of the 3D theory of elasticity, we investigate the problem of free torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at its end faces. We have proposed a numerical-analytic approach for the solution of this problem. The original partial differential equations of the theory of elasticity with the use of spline approximation and collocation are reduced to an eigenvalue problem for a system of ordinary differential equations of high order in the radial coordinate. This system is solved by the stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present numerical results for the case of orthotropic and inhomogeneous material of the cylinder for some kinds of boundary conditions.