Modified BFGS Update (H-Version) Based on the Determinant Property of Inverse of Hessian Matrix for Unconstrained Optimization

The study presents the modification of the Broyden-Flecher-Goldfarb-Shanno (BFGS) update (H-Version) based on the determinant property of inverse of Hessian matrix (second derivative of the objective function), via updating of the vector s ( the difference between the next solution and the current solution), such that the determinant of the next inverse of Hessian matrix is equal to the determinant of the current inverse of Hessian matrix at every iteration. Moreover, the sequence of inverse of Hessian matrix generated by the method would never approach a near-singular matrix, such that the program would never break before the minimum value of the objective function is obtained. Moreover, the new modification of BFGS update (H-version) preserves the symmetric property and the positive definite property without any condition.

Finite Element Analysis with Iterated Multiscale Analysis for Mechanical Parameters of Composite Materials with Multiscale Random Grains

Based on the iterated statistically multiscale analysis (SMSA), we present the convergence of the equivalent mechanical parameters (effective moduli), obtain the error result, and prove the symmetric, positive and definite property of the equivalent mechanical parameters tensor computed by the finite element method. The numerical results show the proved results and illustrate that the SMSA-FE algorithm is a rational method for predicting the equivalent mechanical parameters of the composite material with multiscale random grains. In conclusion, we discuss the future work for the inhomogeneous composite material with multiscale random grains.

The Corrosion and Protection of Magnesium and its Light Alloys

The chemist regards magnesium as a highly reactive metal for such reasons as the inflammability of its powder or foil in air, its active displacement of hydrogen gas from many aqueous chloride solutions and its position near the reactive end of the electrode-potential series. All these suggest that the metal would be unsuitable for constructional engineering. Yet engineers use alloys, rich in magnesium, up to 98 per cent. of the metal, for an increasing number of services, although the alloying elements do not, as a rule, greatly cut down, and may even increase, the corrosion rate. Their industrial use is possible because the liability to corrode, reckoned over a reasonably prolonged period, is not a definite property of a metal such as conductivity which is subject only to relatively small changes with alteration of environment, but is highly specific to metal-liquid and metal-gas systems. Moreover, these systems may undergo important changes with time owing to the intervention of films of corrosion products, and the rate of attack may be governed by the physical characteristics of these films which will vary with the adjacent liquid and gases. Thus in stagnant caustic alkalies magnesium may be reckoned as almost incorrodible because of the intervention of a film of hydroxide of the self-healing type which, in these conditions, is highly impervious to magnesium ions; but in the presence of alkali chlorides the corrosion product is physically different and rapid corrosion occurs. Many dilute acids attack magnesium rapidly but hydrofluoric acid scarcely at all, no doubt owing to the formation of a protective film of fluoride.