Presence Of Stiffness Research Articles

Applicability of the H/V spectral ratio of microtremors in assessing site effects on seismic motion

AbstractThe authors examine the reliability of site response estimations obtained by the horizontal to vertical (H/V) spectral ratios of microtremors by means of cross‐validation with the ratio of the horizontal spectra of earthquake motion with respect to reference sites. The data comprise microtremor and ground motion records recorded at 150 sites of Yokohama strong motion array. The use of non‐supervised pattern recognition techniques aims to group the sites with more objectivity. Attributes defining the overall shape of the amplification spectra serve as input in the computation of Euclidean distance similarity coefficients amongst sites. The implementation of the Ward clustering scheme leads to the attainment of a meaningful tree diagram. Its analysis shows the possibility of summarizing the results into six general patterns. A good coincidence of site effects estimates at 80 per cent of the sites becomes apparent. However, this coincidence appears poor for sites characterized by H/V amplification ratios around 2 or smaller and predominant periods longer than 0.5 s. In such cases, the presence of stiff, sandy sediments in the soil profile proves common. To proscribe H/V estimations, relying solely on the small spectral ratios criterion seems inadequate. Copyright © 2001 John Wiley & Sons, Ltd.

Development of structures under the influence of heterogeneous flow field around rigid inclusions: insights from theoretical and numerical models

Rocks that are mechanically heterogeneous due to the presence of stiff or rigid inclusions floating in a ductile matrix, commonly show a variety of micro- to macro-scale structures developing under the influence of heterogeneous flow field in the neighbourhood of the inclusions. It is of fundamental importance to apprehend the nature of strain heterogeneity around inclusions to understand progressive development of structures associated with rigid inclusions such as strain shadow, foliation drag, porphyroclast mantle, porphyroblast inclusion trails, intragranular fractures, etc. The development of these diverse types of structures can be analyzed with the help of a suitable hydrodynamic theory. In this paper, we review different continuum models that have been proposed to characterize the heterogeneous flow field around rigid inclusions, focusing on recent developments. Recent studies reveal that Jeffery's [Proc. R. Soc. Lond. A 120 (1922) 161.] theory dealing with the motion of ellipsoidal rigid bodies in an infinitely extended viscous medium is more general in nature, and applicable for modeling the heterogeneous flow around both equant and inequant shapes of inclusions and ideal or non-ideal shear deformation of the matrix. The application of this theory, therefore, has advantages over other models, based on Lamb's [Lamb, H., 1932. Hydrodynamics. Cambridge University Press, Cambridge.] theory dealing with spherical inclusions. The review finally illustrates numerical simulations based on hydrodynamic theories, highlighting the controls of physical and kinematic factors on the progressive development of the structures mentioned above.

The error committed by stopping the Newton iteration in the numerical solution of stiff initial value problems

This paper concerns the numerical solution of initial value problems, for non-linear ordinary differential equations, by step-by-step methods that are implicit. For non-stiff problems, fixed-point iteration, also called Picard iteration, is a classical process for solving the system of (algebraic) equations occurring in each time step of such step-by-step methods. The order of the error committed by stopping this process, after a fixed number of iteration steps, is well understood. For stiff problems, Picard iteration is not a useful process, so that one almost invariably employs Newton's method or a variant thereof. This paper analyzes the stopping error for such Newton-type iterations. We aim at an understanding of this error that is comparable to the insight one has for the case of Picard iteration. The error committed by stopping a Newton-type iteration, after a specified number of iteration steps, is estimated and related to the pure discretization error of the underlying step-by-step method. Also the question of existence and uniqueness of solutions to the system of (algebraic) equations is studied. All conclusions in the paper are relevant to non-linear problems that may be arbitrarily stiff. The estimates obtained for these problems reflect order reduction phenomena, in the presence of stiffness, confirmed by numerical experiments. The focus is on linear multistep methods and Runge-Kutta methods, but part of the conclusions are also relevant to other situations.

A New Order Selection Strategy for Ordinary Differential Equation Solvers

The normal order selection strategy used by a variable order, variable stepsize ordinary differential equation solver is based on selecting the formula that can use the largest stepsize while meeting the accuracy constraint. This strategy does not take into account the stiffness or nonstiffness of the differential system. By considering how the stiffness of the differential system influences order selection, a new selection strategy is proposed that is biased in the presence of stiffness toward the lower-order formulas. This new order selection strategy has been incorporated into a version of a widely used ordinary differential equation solver LSODE. The modified solver is shown to be greatly more efficient for problems that the original solver cannot solve efficiently, namely, problems where the Jacobian of the differential system has some eigenvalues of large modulus close to the imaginary axis. On other problems that the unmodified solver solves efficiently, the modified solver still performs with comparable efficiency.

Software for a method of finite approximations for the numerical solution of differential equations

solution of the given nonlinear differential equation, is obtained by means of software of a fairly simple structure. Problems with complex initial, boundary or mixed conditions, involving combinations of function and derivative values, can be dealt with by means of our program. Encouraging results have been obtained in the solution of equations with regions of rapid variation, oscilla- tory behaviour and in the presence of stiffness. As well as in the treatment of nonlinear boundary value problems where the solution is not unique. Examples are given in the last section of the paper. Applications to eigenvalue problems and to partial differential equations are reported elsewhere. Finite approximations method based on tim Tau method: different approaches in the design of numerical algorithms

Effect of Stiffness and Nonuniformity on Vibroscopic Determination of Filament Cross-Sectional Area

The presence of stiffness in a stretched string vibrating between fixed supports raises all the natural frequencies over those for a perfectly flexible string. The presence of nonuniformity in cross-sectional area along the string changes some or all of the frequencies from those for a perfectly uniform string, the amount and direction of change depending on the form of the nonuniformity. These changes in frequency may introduce error into the calculation of the average cross-sectional area of a fiber by means of the vibroscopic technique, wherein the mechanical resonance of a stretched fiber is observed under oscillating mechanical or electrostatic forces. In this paper the natural frequencies are calculated with both the stiffness and the nonuniformity taken into account to first-order terms. The relation between the average cross-sectional area and the natural frequencies is analyzed. Consideration is given to the manner in which the changes in frequency and shifts of the nodes, due to nonuniformity, determine the detailed form of the area variation. The characterization of nonuniformities in terms of certain arbitrary parameters is shown to lead to the possibility of calculating the average area by measuring the natural frequencies in only the first and second modes and of calculating the maximum or minimum area by measuring also the nodal shift in the second mode.