State-space based discretize-then-differentiate adjoint sensitivity method for transient responses of non-viscously damped systems

Highlights

• The design sensitivity analysis of transient responses is considered.

• The adjoint variable method (AVM) is derived in a state-space formulation.

• The discretize-then-differentiate AVM is adopted for non-viscously damped systems.

• The time-history analysis relies on a modified precise integration method.

• Computational considerations of proposed method are investigated and compared.

Abstract

In this paper, a new design sensitivity analysis (DSA) method for transient responses of non-viscously damped systems is developed. The damping force of the non-viscously damped systems depends on the past history of motion and is represented by convolution integrals over suitable kernel functions. The equations of motion are first transformed into a state-space formulation and the time-history analysis of the responses relies on a modified precise integration method (MPIM). The residual equation in each discrete time step is achieved by a recurrence formula of the MPIM. The sensitivities of the transient responses are calculated with an adjoint variable method (AVM). In particular, a discretize-then-differentiate approach is adopted for the DSA. The accuracy, consistency, implementation effort and computational complexity are discussed. Two numerical examples are illustrated to show the performances of the proposed method compared with other two methods. The results indicate that, when calculating the sensitivities of the transient responses for non-viscously damped systems based on the MPIM, the order of differentiation and discretization has no obvious effect on the consistency. The proposed state-space based discretize-then-differentiate AVM is more efficient than other two methods.

Introduction

Design sensitivity analysis (DSA) is used to quantify the influence of measurable outputs with respect to a set of design variables [1]. It plays a vital role in a wide variety of fields, such as model updating [2], gradient-based optimization [3], damage detection [4] and structural reliability [5]. In practice, engineering structures are often subjected to dynamic loadings, such as impact, shock and seismic excitations, which may cause severe undesired noises and vibrations. Therefore, efficient and accurate DSA methods must be developed for transient response problems. In principle, methods for the DSA can be divided into three broad categories [6]: finite difference method (FDM), direct differentiate method (DDM) and adjoint variable method (AVM). The FDM is easy to implement, but it suffers from severe computational burden and numerical error associated with the step size. Therefore, it is typically used for comparison and verification of the sensitivity results obtained from new methodologies. The AVM is more efficient than the DDM when the number of design variables is large and the number of active constraints is small [7], e.g., in topology optimization that typically involves thousands or millions of design variables, and vice versa.

In the AVM, the sensitivity is computed by first adding some residual equations to the response function to built an augmented function and subsequently by differentiating the resulting augmented function. In general, there are two ways to perform the AVM according to the order of discretization and the differentiation, namely differentiate-then-discretize method and discretize-then-differentiate method, respectively. The differentiate-then-discretize method first differentiates the augmented response function and then transforms the adjoint variable problem into a terminal value problem which is similar to the prime problem. And finally the prime and terminal problems are calculated at each discretized time point. On the contrary, the discretize-then-differentiate method consists in differentiating directly the discretized version of the augmented response functions for each time point [8]. The literatures of the gradient-based optimization methods that rely on the differentiate-then-discretize method are well-developed. Some dealt with the design of frequency response problems [9], [10], [11] which aim to improve the performances of the structural system under harmonic excitations. Others are related to the transient response problems [12], [13], [14], [15], which are used to minimizing the dynamic responses, such as displacement, velocity, acceleration and stress, generated by the structure under external loads.

While the focus on differentiate-then-discretize AVM continues to dominate the literature [16], there have been some notable exceptions, particularly in the areas of sensitivity analyses of transient responses. One of the earliest papers to explore the use of discretize-then-differentiate AVM in combination with topology optimization was a 2012 paper by Le et al. [17]. Here the authors used this algorithm to avoid numerical inconsistencies and optimized material microstructural to achieve desired energy propagation in a two-phase composite plate. Since then, a number of authors have also investigated the performances of the discretize-then-differentiate approach on sensitivity analysis of transient responses. Jensen et al. [18] pointed out that, for viscously damped systems, the traditional differentiate-then-discretize AVM may lead to inconsistent sensitivities. It was found that, besides the appropriate choice of time step and summation rule for the discrete approximation, the more cumbersome discretize-then-differentiate AVM is also an effective way to resolve the inconsistent sensitivities. Later on, Nakshatrala and Tortorelli [19] presented a transient dynamic topology optimization framework for rate independent small deformation elastoplastic materials systems and derived the adjoint sensitivity expression based on the discretize-then-differentiate approach. A year later, they further extended the discretize-then-differentiate method to iteratively updated the micro-structural design parameters and developed a multiscale topology optimization framework for nonlinear structural systems. Other DSA methods for the nonlinear transient problems based on the discretize-then-differentiate AVM were also studied [8], [20].

Mathematically, the damping force can not be ignored in the DSA of transient responses [21]. However, most of the above mentioned studies are developed in assumption of undamped or viscously damped models. In practice, viscoelastic materials have been extensively applied in controlling vibration and noise of engineering structures. The viscoelastic effects are of great importance to get a feasible design in structural optimization algorithms [22]. The non-viscous damping model, which assumes that the dissipative forces depend on the past history of motion via convolution integrals over suitable kernel functions [23], provides an effective way to accurately describe the dissipative forces of viscoelastic materials. It has been considered to be the most general damping model within the scope of linear systems [24]. Currently, most of the studies on the DSA for non-viscously damped systems are focused on the eigensolution problems [25], [26], [27] and frequency response problems [28], [29], [30]. As to the problems of transient responses, Li et al. [31] extended the discrete Fourier transform and inverse discrete Fourier transform algorithms to non-viscously damped systems and presented the AVM and DDM to calculate the sensitivity of transient response. James and Waisman [32] derived a novel discretize-then-differentiate time-dependent AVM and incorporated into a topology optimization framework accounting for viscoelastic creep responses. Yun and Youn [33] developed a Newmark-$\beta$ based discretize-then-differentiate AVM on calculating the transient response sensitivity for non-viscously damped systems and then utilized it to optimize the microstructural of viscoelastically damped structures subjected to dynamic loads. Recently, Ding et al. [34] proposed a DSA method of transient responses for non-viscously damped systems based on the DDM to obtain both first- and second-order derivatives.

For viscously damped systems, the differentiate-then-discretize AVM may lead to inconsistency errors [35] and the alternative discretize-then-differentiate AVM can resolve this issue [18]. The effect of the order of the discretization and the differentiation on the accuracy of DSA is still uncertain for non-viscously damped systems. Since it is difficult to directly differentiate the equations of motion of non-viscously damped systems with some convolution integral terms, most of the published studies are based on the discretize-then-differentiate method, which firstly discretize the convolution integral terms and then approximate them using certain summation scheme. However, this approximation may introduce large numerical error. Ding et al. [36] developed a state-space based differentiate-then-discretize AVM on DSA of transient responses for non-viscously damped systems. The equations of motion of non-viscously damped systems are equivalently transformed into a state-space form without approximation and a modified precise integration method (MPIM) [37] is adopted to derive the adjoint variables. It is proved that, compared with the Newmark-$\beta$ based discretize-then-differentiate AVM [33], the state-space based differentiate-then-discretize AVM shows obvious advantages on the computational accuracy, efficiency and the implementation effort [36]. But the comparison is somewhat unfair, because the time integration methods are different and the MPIM is much more accurate and efficient than the Newmark-$\beta$ method on calculating the transient responses of non-viscously damped systems.

The aim of this work is to develop a novel state-space based discretize-then-differentiate AVM on DSA of transient responses for non-viscously damped systems. Firstly, the equations of motion of non-viscously damped systems are transformed into a state-space formulation to eliminate the convolution integral terms. Secondly, the transient responses are obtained by the MPIM and the residual equations are discretized at each time point. Then, the augmented response functions are differentiated and the adjoint variables are calculated to deduce the transient response sensitivities for the non-viscously damped systems. Finally, theoretical and numerical comparisons are made to investigate the performances of the proposed state-space based discretize-then-differentiate AVM with the existing methods. It is hoped that this study will clarify the effects of the order of the differentiation and the discretization on the computational considerations of the state-space based AVMs for transient response sensitivities of non-viscously damped systems. The paper is organized as follows: in Section 2, the problem of DSA is formulated by introducing some theoretical backgrounds on the non-viscously damped systems and inconsistent sensitivities. In Section 3, the state-space based differentiate-then-discretize AVM is reviewed covering the introduction of the MPIM. A novel state-space based discretize-then-differentiate AVM for transient responses of non-viscously damped systems is presented in Section 4, which is also the main contribution of this paper. In Section 5, the flowchart and some computational considerations including the numerical error, implementation effort and computational complexity are studied and compared. The proposed approach together with two other methods are applied on two numerical examples to compare the performances in Section 6. Finally, important concluding remarks are drawn in Section 7.