Harmonic-Modal Hybrid Reduced Order Model for the Efficient Integration of Non-Linear Soil Dynamics

Claudia Germoso,Jean Louis Duval,Francisco Chinesta

doi:10.3390/app10196778

Abstract

Nonlinear behavior of soils during a seismic event has a predominant role in current site response analysis. Soil response analysis, and more concretely laboratory data, indicate that the stress-strain relationship of soils is nonlinear and exhibits hysteresis. An equivalent linearization method, in which non-linear characteristics of shear modulus and damping factor of soils are modeled as equivalent linear relations of the shear strain is usually applied, but this assumption, however, may lead to a conservative approach of the seismic design. In this paper, we propose an alternative analysis formulation, able to address forced response simulation of soils exhibiting their characteristic nonlinear behavior. The proposed approach combines ingredients of modal and harmonic analyses enabling efficient time-integration of nonlinear soil behaviors based on the offline construction of a dynamic response parametric solution by using Proper Generalized Decomposition (PGD)-based model order reduction technique.

Highlights

Structural solid dynamics is usually formulated either in the time or in the frequency domains [1]
In our former works, we considered a PGD (Proper Generalized Decomposition) formulation for constructing a parametric transfer function [9] that allowed efficient solutions of transient dynamics operating in the time-domain
The PGD method was used to calculate the parametric solution for the displacement field ξ (ω, ζ 1, ζ 2, ζ 3, ζ 4, ζ 5 ), as detailed in

Summary

Introduction

Structural solid dynamics is usually formulated either in the time or in the frequency domains [1]. Where M, C, and K are, respectively, the mass, damping, and stiffness matrices; U the vector that contains the nodal displacements; and F the nodal excitations (forces). We assume the mechanical system described by N nodal degrees of freedom, which gives the size of the different matrices and vectors involved in Equation (1). The direct integration of the previous discrete form, which consists of N second order coupled ordinary differential equations, can be performed using either, explicit or implicit time integration schemes. Addressing fast transient dynamics can be usually accomplished using explicit integrations that require satisfying stability conditions affecting the largest integration time-step ∆tmax , closely related to the size of the elements involved in the mesh M covering the domain Ω.

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Conclusion