Efficient evaluation of performance-based earthquake engineering equations

In this paper attention is given to the efficient numerical evaluation of the Pacific Earthquake Engineering Research (PEER) performance-based earthquake engineering framework equations. In particular, potential problems in determining an adequate yet efficient region of integration are discussed. An algorithm called “Magnitude-oriented Adaptive Quadrature” (MAQ) is developed, which is an integration algorithm with both locally and globally adaptive capabilities. MAQ allows efficient integration over the entire integration domain and requires only an error tolerance and maximum number of function evaluations to be specified. The advantages of utilizing the MAQ algorithm over other conventional integration methods such as Romberg integration and conventional adaptive quadrature are illustrated for the numerical computation of (1) expected annual loss; and (2) annual rate of collapse. It is shown that for determination of the expected annual loss a 4.5- to 8.8-fold reduction in the computational demand is obtained using MAQ compared to conventional integration methods. For annual rate of collapse the computational demand reductions range from 30% to two-fold. The computational reductions are a function of the error tolerance prescribed, with greater computational reductions as stricter tolerances are enforced.