Number Theoretical Method For Numerical Multiple Integrals And The Application In Structural Reliability Analysis

Monte Carlo method (MCM) is an efficient method in Numerical Multiple Integrals and estimation of failure probabilities for structural reliability. The method and the program structure are simple, the convergence probability and convergence rate are independent with the dimension S and it has good adaptability in different situations. However, convergence rate of MCM is low and MCM requires long calculation. Especially is that MCM got the error in the sense of probability, not the usual sense of error. For MCM in computation of multidimensional integrals, the result is dependent on the uniformity of the quasi-random sequences of numbers for a fixed N. Therefore, the replacement of quasi-random sequences of numbers by uniformly distributed sets with low discrepancy (Number Theoretical method (NTM)) can improve the efficiency and accuracy. And the result with uniformly distributed sets got the usual sense of error.Result of three numerical multiple integrals with GLP (good lattice point) sets, Halton sets, Halton Leaped sets, Hammersley sets, Hua-Wang sets and quasi-random sequences show that GLP sets got less error and needs less calculations. Quasi-random sequences got the largest error. Four examples for estimation of failure probabilities show results with GLP sets, Halton Leaped sets and Hua-Wang sets got the less error, but Halton Leaped sets needs longer calculations. Hua-Wang sets leads more error in numerical multiple integrals. So GLP sets is the most efficient for estimation of failure probabilities overall. However, result with quasi-random sequences leads to biggest error and is uncertainty.Importance sampling is widely used in MCM, examples show that importance sampling can also improve the efficiency of Number Theoretical method for Numerical Multiple Integrals and estimation of failure probabilities for structural reliability...