Strategy for selecting representative points via tangent spheres in the probability density evolution method

Abstract

SUMMARYA strategy of selecting efficient integration points via tangent spheres in the probability density evolution method (PDEM) for response analysis of non-linear stochastic structures is studied. The PDEM is capable of capturing instantaneous probability density function of the stochastic dynamic responses. The strategy of selecting representative points is of importance to the accuracy and efficiency of the PDEM. In the present paper, the centers of equivalent non-overlapping tangent spheres are used as the basis to construct a representative point set. An affine transformation is then conducted and a hypersphere sieving is imposed for spherically symmetric distributions. Construction procedures of centers of the tangent spheres are elaborated. The features of the point sets via tangent spheres, including the discrepancy and projection ratio, are observed and compared with some other typical point sets. The investigations show that the discrepancies of the point sets via tangent spheres are in the same order of magnitude as the point sets by the number theoretical method. In addition, it is observed that rotation transformation could greatly improve the projection ratios. Numerical examples show that the proposed method is accurate and efficient for situations involving up to four random variables.

A PDEM based new methodology for stochastic dynamic stability control of nonlinear structures with fractional-type viscoelastic dampers

“…The probability of stability can be obtained easily according to Eq. (33). Again, it is seen that the stability performance of the truss is greatly enhanced when fractional-type VE dampers are applied.…”

“…Thus a total of 151 representative points are generated by the Tangent Sphere Method [33] in the two dimensional random-variate space. As is observed, the nonlinearities in both material and geometry are encountered simultaneously in such a simple example.…”

Solution of generalized density evolution equation via a family of δ sequences

“…Introducing the strategy of selecting points uniformly scattered in the multi-dimensional space, such as the points via tangent sphere [25] and those via number theoretical method [26] and so on, the approximation of Eq. 25a becomes…”

“…5. Secondly, 1261 points are selected over the square [−4, 4] 2 via tangent spheres [25] with P 0 = 6.20e−06, then the approximation of the joint PDF and comparisons between the approximation and the exact solution are pictured in Fig. 6.…”

“…Investigations show that for many practical problems the finite difference method employing the Lax-Wendroff scheme or the modified scheme with TVD nature combined with the strategies of selecting 'uniformly scattered' point sets exhibits good performance [24][25][26]. But this does not exclude the attempt of developing alternative numerical methods.…”