Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.

Grouping and the pitch of a mistuned fundamental component:effects of applying simultaneous multiple mistunings to the other harmonics

Mistuning a harmonic produces an exaggerated change in its pitch. This occurs because the component becomes inconsistent with the regular pattern that causes the other harmonics (constituting the spectral frame) to integrate perceptually. These pitch shifts were measured when the fundamental (F0) component of a complex tone (nominal F0 frequency = 200 Hz) was mistuned by +8% and -8%. The pitch-shift gradient was defined as the difference between these values and its magnitude was used as a measure of frame integration. An independent and random perturbation (spectral jitter) was applied simultaneously to most or all of the frame components. The gradient magnitude declined gradually as the degree of jitter increased from 0% to ±40% of F0. The component adjacent to the mistuned target made the largest contribution to the gradient, but more distant components also contributed. The stimuli were passed through an auditory model, and the exponential height of the F0-period peak in the averaged summary autocorrelation function correlated well with the gradient magnitude. The fit improved when the weighting on more distant channels was attenuated by a factor of three per octave. The results are consistent with a grouping mechanism that computes a weighted average of periodicity strength across several components. © 2006 Elsevier B.V. All rights reserved.

Crystal orientation effects in scratch testing with a spherical indenter

Spherical scratch tests were conducted in individual grains of a randomly oriented polycrystalline body-centered-cubic (bcc) Ti-Nb alloy. For each grain, scratch tests were conducted at four different levels of normal load, which resulted in varying amounts of plastic strain during indentation. The results show a dependence of the horizontal load component on the crystallographic orientation and on the amount of plastic strain. The component of the horizontal force that resulted from plastic deformation was found to correlate with the active slip systems for the particular grain orientation. © 2010 Materials Research Society.

A study of the effect of harmonics in the system on the performance of residual current-operated circuit-breakers and electromechanical overcurrent relays

Residual current-operated circuit-breakers (RCCBs) have proved useful devices for the protection of both human beings against ventricular fibrillation and installations against fire. Although they work well with sinusoidal waveforms, there is little published information on their characteristics. Due to shunt connected non-linear devices, not the least of which is the use of power electronic equipment, the supply is distorted. Consequently, RCCBs as well as other protection relays are subject to non-sinusoidal current waveforms. Recent studies showed that RCCBs are greatly affected by harmonics, however the reasons for this are not clear. A literature search has also shown that there are inconsistencies in the analysis of the effect of harmonics on protection relays. In this work, the way RCCBs operate is examined, then a model is built with the aim of assessing the effect of non-sinusoidal current on RCCBs. Tests are then carried out on a number of RCCBs and these, when compared with the results from the model showed good correlation. In addition, the model also enables us to explain the RCCBs characteristics for pure sinusoidal current. In the model developed, various parameters are evaluated but special attention is paid to the instantaneous value of the current and the tripping mechanism movement. A similar assessment method is then used to assess the effect of harmonics on two types of protection relay, the electromechanical instantaneous relay and time overcurrent relay. A model is built for each of them which is then simulated on the computer. Tests results compare well with the simulation results, and thus the model developed can be used to explain the relays behaviour in a harmonics environment. The author's models, analysis and tests show that RCCBs and protection relays are affected by harmonics in a way determined by the waveform and the relay constants. The method developed provides a useful tool and the basic methodology to analyse the behaviour of RCCBs and protection relays in a harmonics environment. These results have many implications, especially the way RCCBs and relays should be tested if harmonics are taken into account.

Excitation of higher spatial harmonics by a moving light pattern in sillenites

We investigate numerically the dependence of higher harmonics of the space-charge field on the detuning frequency between the pump waves, which form a running interference pattern. Bistability and hysteresis of harmonics are predicted for a contrast of the interference pattern m =(0.25-0.3). For contrasts m˜1 and small detuning frequencies we show the existence of a narrow resonance, connected with the nonlinear excitation of a slowly decreasing sequence of spatial harmonics. For experiments we use a BSO crystal in the optical configuration which avoids nonlinear optical distortions. The experimental data show good qualitative agreement with theory.

Load losses in synchronous machines:a study of the pole face loss caused by armature reaction M.M.F. harmonics

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Pitch shifts on mistuned harmonics in the presence and absence of corresponding in-tune components

Mistuning a harmonic produces an exaggerated change in its pitch, a component-pitch shift. The origin of these pitch shifts was explored by manipulations intended to alter the grouping status of a mistuned target component in a periodic complex tone. In experiment 1, which used diotic presentation, reinstating the corresponding harmonic (in-tune counterpart) caused the pitch shifts on the mistuned target largely to disappear for components 3 and 4, although they remained for component 2. A computational model of component-pitch shifts, based on harmonic cancellation, was unable to explain the near-complete loss of pitch shifts when the counterpart was present; only small changes occurred. In experiment 2, the complex tone and mistuned component 4 were presented in the left ear and the in-tune counterpart was presented in the right. The in-tune counterpart again reduced component-pitch shifts, but they were restored when a captor complex into which the counterpart fitted as harmonic 3 was added in the right ear; presumably by providing an alternative grouping possibility for the counterpart. It is proposed that component-pitch shifts occur only if the mistuned component is selected to contribute to the complex-tone percept; these shifts are eliminated if it is displaced by a better candidate.

Comparative numerical study of energy deposition in femtosecond laser microfabrication with fundamental and second harmonics of Yb-doped laser

We present the results of comparative numerical study of femtosecond laser inscription for fundamental and second harmonic of Yb-doped laser. We have found that second harmonic is more efficient in terms of amount of absorbed energy which leads to lower inscription threshold. Hence this regime is more attractive for applications in femtosecond laser microfabrication. We observed the different size of modified domain on initial pulse energy and different spectrum dynamics during the pulse propagation for fundamental and second harmonics.

Efficiency of energy deposition by fundamental and second harmonics in femtosecond laser inscription

We present the results of numerical modelling of energy deposition in single-shot femtosecond laser inscription for fundamental and second harmonics, which shows that second harmonic is more efficient considering the amount of absorbed energy

Comparative numerical study of efficiency of energy deposition in femtosecond microfabrication with fundamental and second harmonics of Yb-doped fiber laser

We present the results of comparative numerical study of energy deposition in single shot femtosecond laser inscription for fundamental and second harmonic of Yb-doped fiber laser. We have found that second harmonic is more efficient in absorbing energy which leads to lower inscription threshold. Hence this regime is more attractive for applications in femtosecond laser microfabrication.

Mean zonal flows generated by librations of a rotating spherical cavity

Longitudinal librations represent oscillations about the axis of a rotating axisymmetric fluid-filled cavity. An analytical theory is developed for the case of a spherical cavity in the limit when the libration frequency is small in comparison with the rotation rate, but large in comparison with the inverse of the spin-up time. It is shown that longitudinal librations create a steady zonal flow through the nonlinear advection in the Ekman layers. The theory can be applied to laboratory experiments as well as to solid planets and satellites with a liquid core. Copyright © Cambridge University Press 2010.

Bistable attractors in a model of convection-driven spherical dynamos

The range of existence and the properties of two essentially different chaotic attractors found in a model of nonlinear convection-driven dynamos in rotating spherical shells are investigated. A hysteretic transition between these attractors is established as a function of the rotation parameter t. The width of the basins of attraction is also estimated. © 2012 The Royal Swedish Academy of Sciences.

Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells

Bistability and hysteresis of magnetohydrodynamic dipolar dynamos generated by turbulent convection in rotating spherical fluid shells is demonstrated. Hysteresis appears as a transition between two distinct regimes of dipolar dynamos with rather different properties including a pronounced difference in the amplitude of the axisymmetric poloidal field component and in the form of the differential rotation. The bistability occurs from the onset of dynamo action up to about 9 times the critical value of the Rayleigh number for onset of convection and over a wide range of values of the ordinary and the magnetic Prandtl numbers including the value unity. Copyright © EPLA, 2009.