A linear system solver based on a modified Krylov subspace method for breakdown recovery

Despite its usefulness in solving eigenvalue problems and linear systems of equations, the nonsymmetric Lanczos method is known to suffer from a potential breakdown problem. Previous and recent approaches for handling the Lanczos exact and near-breakdowns include, for example, the look-ahead schemes by Parlett-Taylor-Liu [23], Freund-Gutknecht-Nachtigal [9], and Brezinski-Redivo Zaglia-Sadok [4]; the combined look-ahead and restart scheme by Joubert [18]; and the low-rank modified Lanczos scheme by Huckle [17]. In this paper, we present yet another scheme based on a modified Krylov subspace approach for the solution of nonsymmetric linear systems. When a breakdown occurs, our approach seeks a modified dual Krylov subspace, which is the sum of the original subspace and a new Krylov subspaceK _m (w _j ,A ^T ) wherew _j is a newstart vector (this approach has been studied by Ye [26] for eigenvalue computations). Based on this strategy, we have developed a practical algorithm for linear systems called the MLAN/QM algorithm, which also incorporates the residual quasi-minimization as proposed in [12]. We present a few convergence bounds for the method as well as numerical results to show its effectiveness.Research supported by Natural Sciences and Engineering Research Council of Canada.     

Near- and mid-infrared spectroscopy of sol–gel derived ormosils: vinyl and phenyl silicates

The suitability of ormosils as photonic materials was investigated. Vinyl and phenyl silicates were synthesised below 100°C. A detailed assignment of mid-infrared vibrational absorption bands is given. This allowed assignment of overtone and combination bands in the near-infrared region and an assessment of residual water contamination, which is low and can be expelled by evacuation. These ormosils have low intrinsic and extrinsic optical absorption in the visible spectral region and at useful wavelengths in the near-infrared.     

A residual method of finite differencing for the elliptic transport problem and its application to cavity flow

A residual method of finite differencing the governing differential equation for the elliptic transport problem is presented. The new finite differencing technique is applied to (1) the one-dimensional transport problem and (2) the cavity flow problem for numerical illustrations. The results indicate the validity of the residual method of finite differencing. The usual method of term-by-term finite differencing, and considerations such as central differencing, hybrid differencing and upwind differencing are not needed in the present residual method.     

Evaluation of tidal residual currents in a wide estuary,using a finite element method

From the linearized, time-independent, constant depth, shallow water tidal equations in an f-plane for a two-layer estuary, two independent modal Helmholtz equations are derived. These modal equations are solved using a fifth-degree finite element technique. The first and second space derivatives of the complex modal tidal elevations, and thus the modal currents and their first derivatives, are evaluated directly from the solution at each node of the finite element mesh. The Stokes drift, which is the major part of the residual tidal flow, is evaluated from these nodal values of the currents and their derivatives. Good agreement is obtained with the exact analytical solution for a wedge-shaped estuary with a wedge angle of π/3, using a mesh of 64 equilateral triangles with sides approximately 1/10 of the wavelength 2πC₂/σ of a Kelvin wave solution for the short-wavelength mode.     

C－半群的谱与其生成元谱之间的关系

本文首次给出了Ｃ－半群的预解集和谱的概念，得到了Ｃ－半群的谱与其生成元谱的一系列结果．     

Effect of laser energy on the deformation behavior in microscale laser bulge forming

Microscale laser bulge forming is a high strain rate microforming method using high-amplitude shock wave pressure induced by pulsed laser irradiation. The process can serve as a rapidly established and high precision technique to impress microfeatures on thin sheet metals and holds promise of manufacturing complex miniaturized devices. The present paper investigated the forming process using both numerical and experimental methods. The effect of laser energy on microformability of pure copper was discussed in detail. A 3D measuring laser microscope was adopted to measure deformed regions under different laser energy levels. The deformation measurements showed that the experimental and numerical results were in good agreement. With the verified simulation model, the residual stress distribution at different laser energy was predicted and analyzed. The springback was found as a key factor to determine the distribution and magnitude of the compressive residual stress. In addition, the absorbent coating and the surface morphology of the formed samples were observed through the scanning electron microscope. The observation confirmed that the shock forming process was non-thermal attributed to the protection of the absorbent coating.     

Elastic-plastic analyses on the residual stresses and curvature of the film/substrate bilayer system

During thermal cycling, the residual stresses are often generated in the film/substrate bilayer due to the material mismatch between the substrate and the film. If the thickness of the film is relatively high, the thermal residual stresses in it may be of different signs. When the film is subjected to elastic-plastic deformation, two plastic zones with different thicknesses may be generated in the film at a significantly high temperature difference. In this paper, a theoretical model which reflects the complete history of thermal residual stresses and curvatures in the elastoplastic film/substrate bilayer system is developed. Solutions are derived to estimate the residual stresses and curvature in the film as functions of temperature difference. The case of Al/Si system is used to illustrate the implementation of this model. Results show that the critical temperature difference at which the second plastic zone near the film surface is generated near the Al film surface is dependent on the film thickness. The strain hardening of the film has an obvious influence on the magnitude of residual stresses within the film at high temperature difference.     

Laser gas-assisted processing of carbon coated and TiC embedded Ti-6Al-4V alloy surface

Laser gas-assisted treatment of Ti-6Al-4V alloy surface is carried out. The alloy surface is initially coated by a carbon layer, in which the TiC particles are embedded prior to laser processing of the surface. The carbon coating with the presence of TiC particles on the workpiece surface is expected to result in carbonitride compound in the surface vicinity after the laser treatment process. Optical and scanning electron microscopes are used to examine the morphological and the metallurgical changes in the laser treated layer. The residual stress formed in the surface region after the laser treatment process is critical for the practical applications of the resulting surface. Therefore, the residual stress formed in the laser treated region is predicted from the analytically equation. The X-ray diffraction technique is incorporated to obtain the residual stress formed in the surface region. It is found that the residual stress predicted agrees with the X-ray diffraction data. The dense structures consisting of TiC_xN_1−x, TiN_x, Ti₂N, and TiC compounds are formed in the surface region of the treated layer. This, in turn, significantly increases the microhardness at the surface.     

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.