Imperfect Boundary Conditions Research Articles

Effects of boundary conditions and partial drainage on cyclic simple shear test results—a numerical study

AbstractOwing to imperfect boundary conditions in laboratory soil tests and the possibility of water diffusion inside the soil specimen in undrained tests, the assumption of uniform stress/strain over the sample is not valid. This study presents a qualitative assessment of the effects of non‐uniformities in stresses and strains, as well as effects of water diffusion within the soil sample on the global results of undrained cyclic simple shear tests. The possible implications of those phenomena on the results of liquefaction strength assessment are also discussed. A state‐of‐the‐art finite element code for transient analysis of multi‐phase systems is used to compare results of the so‐called ‘element tests’ (numerical constitutive experiments assuming uniform stress/strain/pore pressure distribution throughout the sample) with results of actual simulations of undrained cyclic simple shear tests using a finite element mesh and realistic boundary conditions. The finite element simulations are performed under various conditions, covering the entire range of practical situations: (1) perfectly drained soil specimen with constant volume, (2) perfectly undrained specimen, and (3) undrained test with possibility of water diffusion within the sample. The results presented here are restricted to strain‐driven tests performed for a loose uniform fine sand with relative density Dr=40%. Effects of system compliance in undrained laboratory simple shear tests are not investigated here. Copyright © 2004 John Wiley & Sons, Ltd.

Impact of a Digital Filter as a Weak Constraint in MM5 4DVAR: An Observing System Simulation Experiment

In this study, a digital filter is introduced into the fifth-generation Pennsylvania State University‐National Center for Atmospheric Research Mesoscale Model (MM5) four-dimensional variational data assimilation (4DVAR) system as a weak constraint to control high-frequency oscillations, which negatively affect assimilation performance. To assess the impact of the digital filter and to understand how the digital-filter 4DVAR functions, a series of observing system simulation experiments are conducted with the assimilation of global positioning system (GPS) refractivity soundings for a cyclogenesis case over the Antarctic region. It is shown that the use of a digital filter, centered at the midpoint of the assimilation period, is effective in suppressing the highfrequency waves. The imbalance during the early period of assimilation is further reduced by utilizing an additional short-span filter, starting at the beginning of the assimilation period. The filtering of the wind field is found to be the most effective in suppressing high-frequency oscillations. It is also revealed that the imposed weak constraint significantly reduces the wave-reflection problem caused by imperfect upper boundary conditions. It is concluded that the weakly constrained 4DVAR with digital filters not only reduces dynamic imbalance, but also significantly improves the qualities of analysis and forecast. Without projecting its solution onto the highfrequency waves, which diminish rapidly with forecast time, the constrained 4DVAR is able to yield additional improvement in the model initial condition in the larger-scale range and hence utilizes the available observations more effectively when compared with the unconstrained 4DVAR.

The effect of imperfect boundary conditions on the collapse behavior of anisotropic shells

A rigorous solution is presented for the case of stiffened anisotropic shells with general imperfections under combined axial compression, internal or external pressure and torsion. Donnell type equations are used to describe the behavior of the layered composite shell. The stiffeners are represented by smeared theory and for the discrete end rings, Cohen’s ring equations are used. The circumferential dependence is eliminated by a truncated Fourier series. The resulting 2-point boundary value problem is solved numerically via the “parallel shooting method”. The nonlinear collapse behavior is studied using different combinations of axisymmetric and asymmetric imperfections. Comparison with Koiter type imperfection predictions display the range of validity of the asymptotic results. It is shown that besides initial geometric imperfections also nonuniform harmonically varying boundary conditions can have severe degrading effect on the load carrying capacity of anisotropic shells.

Imperfections and buckling in cylindrical shells with consistent residual stresses

Metal silo walls are often constructed from isotropic plates, and the controlling design condition is buckling under axial compression. It has long been recognised that this buckling strength is highly sensitive to imperfections in the cylindrical shell, but most attention has been paid to geometric imperfections and imperfect boundary conditions. Imperfections in the form of residual stresses have only rarely been investigated, and the challenges facing a rigorous treatment of them have often not been faced. This paper adopts a rigorous treatment technique to investigate residual stresses and their effect on the axial compression buckling strength under elastic conditions. It achieves this by considering consistent stress and displacement fields arising from local geometrical incompatibilities, and adopting their consequent geometric imperfections. The calculations of the strength of imperfect shells with residual stresses are compared with corresponding calculations for the same imperfections but with the residual stresses ‘annealed’ out of the analysis. The results show that consistent residual stresses generally appear to strengthen a thin shell relative to the corresponding strength with only geometric imperfections present.

Heat Transfer Contributions of Pins and Endwall in Pin-Fin Arrays: Effects of Thermal Boundary Condition Modeling

Short pin-fin arrays are often used for cooling turbine airfoils, particularly near the trailing edge. An accurate heat transfer estimation from a pin-fin array should account for the total heat transfer over the entire wetted surface, which includes the pin surfaces and uncovered endwalls. One design question frequently raised is the actual magnitudes of heat transfer coefficients on both pins and endwalls. Results from earlier studies have led to different and often contradicting conclusions. This variation, in part, is caused by imperfect or unrealistic thermal boundary conditions prescribed in the individual test models. Either pins or endwalls, but generally not both, were heated in those previous studies. Using a mass transfer analogy based on the naphthalene sublimation technique, the present experiment is capable of revealing the individual heat transfer contributions from pins and endwalls with the entire wetted surface thermally active. The particular pin-fin geometry investigated, S/D = X/D = 2.5 and H/D = 1.0, is considered to be one of the optimal array arrangement for turbine airfoil cooling. Both inline and staggered arrays with the identical geometric parameters are studied for 5000 ≤ Re ≤ 25,000. The present results reveal that the general trends of the row-resolved heat transfer coefficients on either pins or endwalls are somewhat insensitive to the nature of thermal boundary conditions prescribed on the test surface. However, the actual magnitudes of heat transfer coefficients can be substantially different, due to variations in the flow bulk temperature. The present study also concludes that the pins have consistently 10 to 20 percent higher heat transfer coefficient than the endwalls. However, such a difference in heat transfer coefficient imposes very insignificant influence on the overall array-averaged heat transfer, since the wetted area of the uncovered endwalls is nearly four times greater than that of the pins.

Transient Response in a One-Dimensional Model of Thermoelastic Contact

We consider two layers of different materials with the initial gap between them in the field of temperature with imperfect boundary conditions in Barber’s sense. The model we discuss is that of two contacting rods (Barber and Zhang, 1988) which in the case of a single rod was devised and discussed by Dundurs and Comninou (1976, 1979). In this paper we try to make a step further in the investigation of the essentially nonlinear problem. Though we consider a system of the linear equations of thermoelasticity the nonlinearity is induced by the boundary conditions dependent on the solution. We present an algorithm for solving the system of equations based on Laplace’s transform technique. The method of solution can be used also in the dynamical problems with inertial terms taken into account. The numerical results have been obtained by a kind of computational simulation.

Numerical study on mode-transition of natural convection in differentially heated inclined enclosures

The present paper reports a numerical investigation of natural convection and the associated mode-transition and hysteresis phenomena in a two-dimensional differentially heated inclined enclosure. Unsteady Navier-Stokes/Boussinesq equations governing the fluid flow and heat transfer are solved numerically. Rayleigh numbers in steady flow regime, 1 × 10 3 ⩽ Ra ⩽ 2 × 10 4, and the angles of inclination of 0° ⩽ γ ⩽ 90° are considered in the computations. Major concerns are the effects of inclination on flow-mode transition. Enclosures of length-to-height aspect ratio As = 4, 3 and 1 are studied. Air of Pr = 0.71 in an enclosure of As = 4 is chosen as the flow model to examine the influences of the inclination at various Rayleigh numbers. Hysteresis phenomena for Ra ⩾ 2000 are demonstrated. Parameter maps of Ra vs γ for As = 4 with γ increasing and decreasing are proposed, in which flow regimes characterized by various modes are designated. Effects of initial condition on the flow pattern formation are examined for As = 3. In a model of square enclosure ( As = 1), effect of the imperfect thermal boundary conditions is examined to investigate the possible causes for inconsistency of the predictions with the experiments. The present study provides more physical insight into the natural convection in enclosures.

On Nonuniqueness and Stability in Barber’s Model of Thermoelastic Contact

In the recent our paper we have derived an algorithm which lets us find numerical solutions of special cases of a nonstationary problem of thermoelasticity with imperfect boundary conditions of Barber’s model. In this paper we show the results of computational simulation for a case of the thermal resistance function. We have obtained a number of solutions for different situations and have discussed the unique, nonunique, stable, and unstable solutions. We have found cases when the unique and nonunique solutions alternate. The results have been presented in the form of diagrams.

De-embedding correction for imperfect absorbing boundary conditions in FDTD

The finite difference time domain (FDTD) technique is a very powerful method for field evaluation in complex structures. In many cases knowledge of the field is unnecessary, because that of the scattering parameter suffices. We introduce a very simple, fast, and easy-to-implement method to correct the absorbing boundary condition (ABC) errors in the scattering parameter evaluation arising from any ABC, however imperfect: this method allows universal use of any simple ABC for any transmission media, therefore facilitating the development of general purpose codes. The technique is based on the evaluation of the reflection coefficient due to the absorbing boundary condition and a subsequent correction in the frequency domain.

Discussion of: Modal Response of a Beam with Imperfect Boundary Conditions

Discussion of: Modal Response of a Beam with Imperfect Boundary Conditions

MODAL RESPONSE OF A BEAM WITH IMPERFECT BOUNDARY CONDITIONS

The introduction of non-contacting dynamic measurement techniques such as holographic interferometry and laser vibrometry has led to increased usage of model testing. Without having to add mass, stiffness, or damping due to sensor attachment, small scale models can be employed, thus permitting the use of less expensive specimens, fixtures, and excitation sources. An additional driver towards model testing is the dynamic characterization of exotic materials for which only small samples are available. One of the disadvantages of small scale testing is the necessity to provide accurate placement of the specimen in a clamping fixture (the most often used boundary condition for beams and plates). This work is an effort to characterize the effect of imperfect boundary conditions on the resulting data.

Ultrasonic inspection of very thin adhesive layers

It is shown here that ultrasonic measurements can provide accurate information on an adhesive layer whose thickness is negligible compared with the wavelength. In this case, the adhesive layer can be considered as a simple interface between two elastic solids. This kind of contact enables small discontinuities of stresses or displacements between both substrates. A simple formulation is introduced to take into account these discontinuities expressed in terms of imperfect boundary conditions. It is shown that this formulation is consistent with exact calculation in layered media. The range of applications of this model and its limits is also investigated. This model is used to infer quantitative data on the changes in mechanical properties of a thin bonding layer (about 5 μm) from reflection coefficient measurements during its hardening process. The results obtained are in excellent agreement with ultrasonic measurement in bulk.

Modes of a bar

An aluminum rod (≈ 30 cm) of circular cross section (≈ 1.2‐cm diameter) can be excited electrodynamically in its longitudinal, torsional, and flexural resonant modes using a pair of coils glued to the ends and placed in a magnetic field. Several harmonics of all three “free‐free” modes can be easily observed using a signal generator and oscilloscope. The nondispersive modes (longitudinal and torsional) are harmonic to better than ± 0.5%, and the anharmonicity of the flexural modes is apparent. The Young's modulus of the bar determined by the flexural modes and the longitudinal modes also agree to better than + 0.05%, so this laboratory exercise demonstrates excellent quantitative agreement with theory. The exercise can be extended to provide an introduction to Rayleigh's method [J. W. Strutt (Lord Rayleigh), The Theory of Sound (Dover, New York, 1945), 2nd ed., Vol. I, Secs. 155 and 185] for calculation of resonance frequency shift due to the imperfect boundary conditions created by the mass of the coils and their adhesive. The agreement between theory and experiment increases to ± 0.1% when the mass loading is included. [Work supported by N.P.S. Direct Funded Research Program.]

The angular and frequency characteristics of reflectivity from a solid layer embedded between two solids with imperfect boundary conditions

In this paper, a comprehensive model of an isotropic, homogeneous, solid layer embedded in between two half-space solids with imperfect interfacial conditions is presented. The Thomson’s matrix technique for transfer of boundary conditions across a thin layer, based on recurrence formulas, coupled with separate normal and tangential rigidity representation for each interface was used. The reflection and transmission coefficients for oblique incidence of both longitudinal as well as transverse waves as a function of angle of incidence, incident frequency, interface condition, and material properties can be numerically computed through this model. Several studies of the reflection factors for a three-medium case is used to provide insight on the adhesively bonded structures. The ultrasonic quality evaluation of the commonly used aluminum–epoxy-resin–aluminum bond situation is dealt with here in detail.

Some Aspects of Reliability and Safety of Space Frames

This article discusses the structural reliability and safety of steel space frames. Since steel space frames are composed of a large number of members of comparatively small size and are covering large areas without any internal supporting columns, the influence of some imperfect boundary conditions as well as differences introduced through lack of fit during the erection process may lead to occasional local overstressing in some members. This overstressing in compressive members may exceed their critical buckling capacity leading to collapse and redistribution of the stresses in the remaining structure. If the members cannot resist the increased stresses then the local collapse may spread to adjacent members and eventually cause overall collapse. The authors wish to put forward some recommendations for prevention of this dangerous phenomenon.

An observed r.f. hysteresis characteristic of particular gunn diodes

Certain Gunn diodes, when mounted in half-wave resonant cavities, exhibited marked hysteresis in r.f. power when the bias voltage was increased above threshold and then reduced. Observations and measurements are described of the dependence on circuit loading of this phenomenon, and the associated I-V curve of the diode. Explanations of the phenomenon are proposed, based on recent investigations into the effect of imperfect cathode boundary conditions on the behaviour of Gunn devices. Some possible practical applications of the hysteresis are suggested, assuming that controlled manufacture of such devices might eventually be possible.

The Gunn effect under imperfect cathode boundary conditions

The usual assumption in the theory of the Gunn effect of a well-behaved cathode is replaced by a more general set of boundary conditions adapted to the treatment of cathode contacts with some interface resistance. The treatment is in terms of a characteristic: that is, the current-field characteristic of the cathode interface itself. For contacts with an interface resistance, the control characteristic lies at least partially below the bulk current-field characteristic. This leads to the primary nucleation of depletion rather than accumulation layers. Depending on the detailed shape of the control characteristic and the location of its crossover point (if any) with the bulk characteristic, a large variety of device behavior can be explained. Examples are Gunn's preoscillation cathode fall, Kennedy's prethreshold amplification behavior, the occurrence of excessive threshold fields for the onset of oscillations, and oscillations with lower amplitude and higher frequency than for well-behaved transit time oscillations, despite good doping homogeneity.

On the wave motion and stability of dynamic systems with imperfect boundaries

General normal-mode methods and variational expressions are developed which are useful for studying linear wave motion, diffusion and stability of a wide variety of dynamic systems having non-uniform internal energy dissipation and imperfect boundaries. The motion of these systems is here restricted to that representable by one-dimensional 2 N-th-order equations of motion and a set of boundary impedance equations. Orthogonality relations are developed among eigenmodes to facilitate eigenfunction solutions of transient motion, and of motion excited by either deterministic or random forces. Variational expressions are established which facilitate approximate solutions of the eigenvalues and eigenfunctions when exact calculations are not feasible. The normal-mode solutions obtained are general in the sense that for the same system under different boundary conditions (including those perfect ones) one need only change values of boundary impedances in the general solutions. The variational expressions obtained are stationary with no restrictions on the boundary conditions of the trial fields. The methods are directly applicable to a wide variety of electromagnetic, acoustic, elastic, thermal and chemical systems. Three specific examples are considered, namely the influence of imperfect boundaries on a vibrating string, on plane magnetohydro-dynamic waves and on the ‘flute’ instability at a confined plasma boundary. General remarks are made about the significance of imperfect boundaries on plasma waves and instability, the range of validity of the developed methods and on the normal-mode solutions and variational principles for the vector wave motion and stability of general anisotropic media (with emphasis on magnetoplasmas) having tensor imperfect boundary conditions.