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 … Show more
“…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.…”
Section: Case 2: Elastic-plastic Two Bar Truss Under Random Seismic Ementioning
confidence: 90%
“…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.…”
Section: Case 2: Elastic-plastic Two Bar Truss Under Random Seismic Ementioning
“…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.…”
Section: Case 2: Elastic-plastic Two Bar Truss Under Random Seismic Ementioning
confidence: 90%
“…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.…”
Section: Case 2: Elastic-plastic Two Bar Truss Under Random Seismic Ementioning
“…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…”
Section: Numerical Implementationmentioning
confidence: 99%
“…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.…”
Section: Case 4 (Probability Density Function With Double Peaks)mentioning
confidence: 99%
“…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.…”
The traditional probability density evolution equations of stochastic systems are usually in high dimensions. It is very hard to obtain the solutions. Recently the development of a family of generalized density evolution equation (GDEE) provides a new possibility of tackling nonlinear stochastic systems. In the present paper, a numerical method different from the finite difference method is developed for the solution of the GDEE. In the proposed method, the formal solution is firstly obtained through the method of characteristics. Then the solution is approximated by introducing the asymptotic sequences of the Dirac δ function combined with the smart selection of representative point sets in the random parameters space. The implementation procedure of the proposed method is elaborated. Some details of the computation including the selection of the parameters are discussed. The rationality and effectiveness of the proposed method is verified by some examples. Some features of the numerical results are observed.
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