A new model order reduction strategy adapted to nonlinear problems in earthquake engineering

Abstract

SUMMARYEarthquake dynamic response analysis of large complex structures, especially in the presence of nonlinearities, usually turns out to be computationally expensive. In this paper, the methodical developments of a new model order reduction strategy (MOR) based on the proper orthogonal decomposition (POD) method as well as its practical applicability to a realistic building structure are presented. The seismic performance of the building structure, a medical complex, is to be improved by means of base isolation realized by frictional pendulum bearings. According to the new introduced MOR strategy, a set of deterministic POD modes (transformation matrix) is assembled, which is derived based on the information of parts of the response history, so-called snapshots, of the structure under a representative earthquake excitation. Subsequently, this transformation matrix is utilized to create reduced-order models of the structure subjected to different earthquake excitations. These sets of nonlinear low-order representations are now solved in a fractional amount of time in comparison with the computations of the full (non-reduced) systems. The results demonstrate accurate approximations of the physical (full) responses by means of this new MOR strategy if the probable behavior of the structure has already been captured in the POD snapshots.

A Structural Pounding Formulation Using Systematic Modal Truncation

“…, Φ ] (cf. [12]). The transformation into the reduced space is physically motivated by the fact that, dependent on the frequency range of the excitation force, the whole deformation energy of the vibrating system is associated with a low number of vibration modes.…”

“…[12]). Considering second-order Taylor forward and backwards expansion, the approximation of the velocity is obtained by the sum of the two expansions, and the approximation of the acceleration is obtained by the difference of the two expansions.…”

“…The initial accelerationü 0 is obtained by inserting the initial conditions u 0 andu 0 into the global equation of motion (12). The coefficient matrix ((1/Δ 2 )M + (1/2Δ )C) is positive definite and can be LU factorized a priori, which accelerates the solution procedure.…”

“…Implicit time integration schemes, as, for example, the Newmark method (cf. [12]), require, on the one hand, an iterative procedure within one time integration step, and, therefore, more computational effort within this integration step. On the other hand, in case of the right choice of certain parameters, they are unconditionally stable.…”

“…However, they applied those strategies only to dynamical single body systems without consideration of multibody pounding; see [11][12][13][14][15][16][17]. Only limited research studies have been conducted considering model reduction techniques for structural pounding problems.…”