Analysis of moderately-thick and deep shells subjected to thermal and mechanical loadings

An analytical solution to a boundary value problem of moderately-thick and deep doubly curved shells of rectangular planform subjected to thermal and mechanical loadings is presented. Sanders' kinematic relations that incorporate transverse shear deformations in moderately deep shell theory are considered in the shell formulations, yielding five coupled second order partial differential equations in five unknowns displacements. These are then solved in conjunction with admissible boundary conditions. The numerical results presented herein should serve as bench-mark solutions for future comparisons.     

Quintic splines in vibrations of nonuniform annular plates

Free axisymmetric vibrations of annular plates of non-uniform thickness have been studied on the basis of classical theory of plates. The fourth order differential equation governing the transverse motion is solved using the quintic spline interpolation technique. Characteristic equations for plates of linearly varying thickness have been obtained for three combinations of boundary conditions at the inner and outer boundaries. Frequencies, mode shapes and moments have been computed for different values of taper constant and radii ratio, for the first two modes of vibration.     

Stress,buckling and vibration of hybrid bodies of revolution

The analysis is applicable to bodies of revolution composed of thin shell segments, thick segments and discrete rings. The thin shell segments are discretized by the finite difference energy method and the thick or solid segments are treated as assemblages of 8-node isoparametric quadrilateral finite elements of revolution. Suitable compatibility conditions are formulated through which these dissimilar segments are joined without introduction of large spurious discontinuity stresses. Plasticity and primary or secondary creep are included. Axisymmetric prebuckling displacements may be moderately large. The nonlinear axisymmetric problem is solved in two nested iteration loops at each load level or time step. In the inner loop the simultaneous nonlinear equations corresponding to a given tangent stiffness are solved by the Newton-Raphson method. In the outer loop the plastic and creep strains and tangent stiffness are calculated by a subincremental procedure. The linear response to nonaxisymmetric loading is obtained by superposition of Fourier harmonics. Many examples are given to demonstrate the scope of the computer program, BOSOR6, derived from the analysis and to illustrate certain stress concentration effects in shell-type structures which cannot adequately be treated with use of thin shell theory.     

A coupled boundary element/finite difference method for fluid–structure interaction with application to dynamic analysis of outer hair cells

Outer hair cells (OHC) in the inner ear, which resemble fluid-filled and fluid-surrounded cylinders, are known to exhibit motility and play a critical role in our keen sense of hearing. In this study, we investigate the OHC frequency response using a mathematical model of the OHC, which consists of a two-layered anisotropic cylindrical lateral wall, and both the intracellular and extracellular fluids. We use the boundary integral equations to model the intracellular and extracellular fluids, and these are coupled to the anisotropic cylindrical shell equations (discretized using the finite difference method). Since the geometry is axisymmetric, the dynamic analysis is performed by decomposing the motion into Fourier modes in the circumferential direction. For the boundary element method, this leads to two sequences of line integrals along the generator of the domain, and the singular kernels need to be evaluated with elliptic integrals. The coupled fluid–structure equations are solved for several modes of deformation (axisymmetric, cylindrical beam-bending, and pinched modes), and the frequency responses are obtained. The frequency response of the model with viscous fluid is found to be significantly different from that using inviscid fluid. For the small length scale of the OHC (which is of micron size), the viscosity of the fluid is found to have significant damping effects on the OHC frequency response.     

双层圆柱壳振动传递及声辐射试验研究

对有限长的敷设阻尼材料的加筋双层圆柱壳进行振动和声辐射试验,测量了壳体的结构响应和水中的辐射声压。对全部敷设隔声去耦材料、内壳全部敷设隔声去耦材料、外壳全部敷设以及外壳部分敷设4种工况下的近场声压进行了对比,分析了单频激振时内外壳间的振动传递特性。结果表明:托板在内外壳的振动传递中起着较大的作用;且内外壳全部敷设隔声去耦材料对抑制振动和声辐射很有效。     

Identification of tumor region parameters using evolutionary algorithm and multiple reciprocity boundary element method

In the paper the inverse problems consisting in the simultaneous estimation of unknown thermophysical and/or geometrical parameters (thermal conductivity, perfusion coefficient, metabolic heat source, location, size) of the tumor region are solved. The additional information concerning the knowledge of local skin surface temperature at the selected set of points is assumed to be known. The problem of thermal processes proceeding in the domain considered is described by the system of the Pennes equations and boundary conditions given on the outer and contact surfaces. On the stage of numerical solution the evolutionary algorithm coupled with the multiple reciprocity boundary element method has been applied.     

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators: Bending and free vibrations

This paper addresses the bending and free vibrations of multilayered cylindrical shells with piezoelectric properties using a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

Rapid heating vibrations of FGM annular sector plates

In this paper, thermally induced vibration of annular sector plate made of functionally graded materials is analyzed. All of the thermomechanical properties of the FGM media are considered to be temperature dependent. Based on the uncoupled linear thermoelasticity theory, the one-dimensional transient Fourier type of heat conduction equation is established. The top and bottom surfaces of the plate are under various types of rapid heating boundary conditions. Due to the temperature dependency of the material properties, heat conduction equation becomes nonlinear. Therefore, a numerical method should be adopted. First, the generalized differential quadrature method (GDQM) is implemented to discretize the heat conduction equation across the plate thickness. Next, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson time marching technique. The obtained thermal force and thermal moment resultants at each time step from temperature profile are applied to the equations of motion. The equations of motion, based on the first-order shear deformation theory (FSDT), are derived with the aid of the Hamilton principle. Using the GDQM, two-dimensional domain of the sector plate and suitable boundary conditions are divided into a number of nodal points and differential equations are turned into a system of ordinary differential equations. To obtain the unknown displacement vector at any time, a direct integration method based on the Newmark time marching scheme is utilized. Comparison investigations are performed to validate the formulation and solution method of the present research. Various examples are demonstrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the plate, temperature dependency, sector opening angle, values of the radius, in-plane boundary conditions, and type of rapid heating boundary conditions on thermally induced response of the FGM plate under thermal shock.

     

Numerical studies of “side-by-side” and other modes for the Taylor problem in a finite annulus

A numerical scheme for solving the Navier-Stokes equations for axisymmetric flow in a Couette apparatus is presented. It is time-dependent, and uses primitive variables with finite-differencing on a stretched, staggered grid. The model gives good agreement with the experiments. Anomalous modes have been found and are initially established by a technique in which Schaeffer's [1] end-wall boundary conditions are used to set up a steady state, and then the boundary conditions are gradually changed to those of the finite-length, Taylor apparatus. When both the inner and outer walls of the annulus rotate, a new class of “side-by-side” modes is predicted.     

Electric potential in the earth’s ionosphere: a numerical model

Modeling the global distribution of the electric potential in the Earth’s ionosphere is based on the solution of a 2D continuity equation in the ionospheric-magnetospheric current circuit. The potential distribution is described by the boundary value problem for an elliptic system of partial differential equations on the spherical shell approximating the ionosphere, which is divided into three subregions with nonlocal boundary conditions. Implementation of the boundary conditions, which reflect the continuity of the common current circuit and potential equalization at the boundaries of the polar caps, is leading to the mutual dependence of the potential distribution within the northern and southern caps and their influence on the potential distribution in the midlatitude region. The problem is solved by an iterative method with a regularizing operator which is inverted using the separation of the variables and the fast Fourier transform with respect to the azimuthal variable and the sweep method with respect to the latitudinal one.     

Thermal stress analysis of reentry vehicle nosetips at angle of attack

The ASAAS (Asymmetric Stress Analysis of Axisymmetric Solids) computer program is applied to the prediction of thermal stresses in a reentry vehicle nosetip subjected to asymmetric temperature distributions and angle-of-attack loadings during reentry. This three-dimensional stress analysis computer program has the unique capability of properly accounting for the circumferential variation of temperature dependent material properties. It is based on a meridional finite element discretization of an axisymmetric solid combined with a Fourier series representation of circumferentially dependent variables. In contrast with similar methods where material properties cannot be expressed as a function of temperature, this method requires that the large system of equations be solved simultaneously. This is handled efficiently in ASAAS by several newly developed approaches to stiffness matrix generation and simultaneous equation solution.

The nosetip analyzed is fabricated from an orthotropic, temperature dependent graphite material that is susceptible to thermal shock. Analyses are performed at several points in a trajectory and the effects of both aerodynamic heating and pressure loading are considered. The complete states of temperature, stress, strain and displacement throughout the graphite material are obtained as a product of ASAAS. Results are presented in the form of meridional contour plots at selected circumferential stations.

The usefulness of ASAAS in performing angle-of-attack analysis of nosetips is evaluated by comparison to an axisymmetric analysis based on single ray heating. In addition, the convergence of solutions with increasing numbers of harmonics is demonstrated and computer run times are discussed.     

Thick shells of revolution: Derivation of the dynamic stiffness matrix of continuous elements and application to a tested cylinder

This paper describes a procedure for calculating the dynamic stiffness matrix of tubular shells with free edge boundary conditions. Such an analysis forms the basis for the Continuous Element Method. The method is used to formulate a thick axisymmetric shell element which takes into account rotatory inertia, transverse shear deformation and non-axisymmetric loadings.     

A new energy harvester using a cross-ply cylindrical membrane shell integrated with PVDF layers

In this paper, we proposed a smart cylindrical membrane shell panel (SCMSP) model for vibration-based energy harvester. The SCMSP is made of an orthotropic elastic core covered by outer PVDF layers with transverse polarization vector. Electrodynamics governing equations of motion are derived by applying extended Hamilton’s principle. The governing equations are based on Donnell’s linear thin shell theory. The SCMSP displacement fields are expanded by means of double Fourier series satisfying immovable edges with free rotation boundary conditions and coupled system of linear partial differential equations are obtained. The discretized linear ordinary differential equations of motion are obtained using Galerkin method. The output power is taken as an indicating criterion for the generator. A parametric study for MEMS applications is conducted to predict the power generated due to radial harmonic ambient vibration. Optimal resistance value is also obtained for the particular electrode distribution that gives maximum output power. A low vibration amplitude (5?Pa), and a low-frequency (471.79?Hz) vibration source is targeted for the resonance operation, in which the output power of 0.4111?μW and peak-to-peak voltage of 0.2952?V are predicted.     

Comparison of shell theories in the analysis of cylindrical shells with discontinuity in the thickness

Cylindrical shells with discontinuity in the thickness and that are subjected to axisymmetric loading have been analysed. Two types of finite elements are used: the first is based on thin shell theory and the second on thick shell theory. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. The effect of these loads for various end conditions and various step-ratios in the thickness have been analysed. Numerical results are presented and compared for both the theories. It has been shown that the transverse normal stress acting along the thickness direction is not negligible compared to other stresses at places of discontinuity either in the thickness or in the loading. The weight of the shell is kept constant for various step-ratios.     

Non-linear dynamics and stability of circular cylindrical shells conveying flowing fluid

The non-linear dynamics and stability of simply supported, circular cylindrical shells containing inviscid, incompressible fluid flow is analyzed. Geometric non-linearities of the shell are considered by using the Donnell's non-linear shallow shell theory. A viscous damping mechanism is considered in order to take into account structural and fluid dissipation. Linear potential flow theory is applied to describe the fluid–structure interaction. The system is discretized by Galerkin's method and is investigated by using two models: (i) a simpler model obtained by using a base of seven modes for the shell deflection, and (ii) a relatively high-dimensional dynamic model with 18 modes. Both models allow travelling-wave response of the shell and shell axisymmetric contraction. Boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied. Stability, bifurcation and periodic responses are analyzed by means of the computer code AUTO for the continuation of the solution of ordinary differential equations. Non-stationary motions are analyzed with direct integration techniques. An accurate analysis of the shell response is performed by means of phase space representation, Fourier spectra, Poincaré sections and their bifurcation diagrams. A complex dynamical behaviour has been found. The shell bifurcates statically (divergence) in absence of external dynamic loads by using the flow velocity as bifurcation parameter. Under harmonic load a shell conveying flow can give rise to periodic, quasi-periodic and chaotic responses, depending on flow velocity, amplitude and frequency of harmonic excitation.     

Domain decomposition for near-wall turbulent flows

Modeling near-wall high-Reynolds-number turbulent flows is a time-consuming problem. A domain decomposition approach is developed to overcome the problem. The original computational domain is split into a near-wall (inner) subdomain and an outer subdomain. The developed approach is applied to a model 2D equation simulating major peculiarities of near-wall high-Reynolds-number flows. On the base of the Calderon–Ryaben’kii potential theory it is possible to consider the near-wall (inner) problem independently on the outer problem. The influence of the inner problem can exactly be represented by a pseudo-differential equation formulated on the intermediate boundary. In a 1D case, it leads to the wall functions represented by Robin boundary conditions, which can be determined either analytically or numerically. It is important that the wall functions (or boundary conditions) are mesh independent and can be realized in a separate routine. Thus, the original problem can only be solved in the outer domain with some specific nonlocal boundary conditions called nonlocal wall functions. The technique can be extended to 3D problems straightforward.     

A boundary collocation method for cracked plates

A boundary collocation method is developed for analyzing cracked thin plates. Complex stress functions which satisfy the equilibrium equations of an infinite domain having a single crack are first derived. As the functions have also satisfied the stress singularity at the crack tips, it is only necessary to enforce the stress functions to satisfy the boundary conditions along the edges of the plates and the surfaces of the cracks, if there is more than one crack. This is achieved by the collocation least square approach. The unknown coefficients of the stress functions having been determined, the stress intensity factors can then be computed according to the related formulae. Examples of rectangular and circular plates with a different number of cracks and under different loadings are used to demonstrate the accuracy, versatility and advantages of the method.     

The method of boundary condition transfer in application to modeling near-wall turbulent flows

Generalized wall functions in application to high-Reynolds-number turbulence models are derived. The wall functions are based on transfer of a boundary condition from a wall to some intermediate boundary near the wall (usually the first nearest to the wall mesh point but that is not obligatory). The boundary conditions on the intermediate boundary are of Robin-type and represented in a differential form. The wall functions are obtained in an analytical easy-to-implement form, can take into account source terms such as pressure gradient and buoyancy forces, and do not include free parameters. The log-profile assumption is not used in this approach. Both Dirichlet and Newman boundary-value problems are considered. A method for complementing solution near the wall is suggested. Although the generalized wall functions are obtained for the k– model, generalization to other turbulence models is straightforward. The general approach suggested is applicable to studying high-temperature regimes with variable laminar viscosity and density. A robust numerical algorithm is proposed for implementation of Robin-type wall functions. Test results made for a channel flow and axisymmetric impinging jet have showed reasonably good accuracy, reached without any case-dependent turning, and a weak dependence of the solution on the location of the intermediate boundary where the boundary conditions are set. It is demonstrated that the method of boundary condition transfer applied to low-Reynolds-number turbulence models can be used as a decomposition method.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

On the use of cubic splines to solve certain circular plate problems

It is well known that the evaluation of the deformation due to a uniform load on a circular annulus that is of varying thickness and clamped at both inner and outer radii reduces to the solution of a two-point boundary value problem. A cubic spline solution to the governing equations is obtained and an accurate comparison is made with a shooting method of solution. Computing times for both methods are also compared. The evidence is that cubic spline methods are viable for the solution of problems of the type considered here.