Topology optimization of non-linear viscous dampers for energy-dissipating structures subjected to non-stationary random seismic excitation

Abstract

Non-linear fluid viscous dampers have found widespread applications in engineering practice for seismic mitigation of civil structures. Topology optimization has emerged as an appealing means to achieve the optimal design of non-linear viscous dampers in terms of both layouts and parameters. However, the conventional methodologies are mainly restricted to deterministic dynamic excitations. This research is devoted to the topology optimization of non-linear viscous dampers for energy-dissipating structures with consideration of non-stationary random seismic excitation. On the basis of the equivalent linearization—explicit time-domain method (EL-ETDM), which has been recently proposed for non-stationary stochastic response analysis of non-linear systems, an adjoint variable method-based (AVM-based) EL-ETDM is further proposed for non-stationary stochastic sensitivity analysis of energy-dissipating structures with non-linear viscous dampers. The stochastic response and sensitivity results obtained by EL-ETDM with high efficiency are utilized for topology optimization of non-linear viscous dampers with the gradient-based method of moving asymptotes. The optimization problem is formulated as the minimization of the maximum standard deviation of a critical response subjected to a specified maximum number of viscous dampers, and the p-norm function is employed for approximation of the non-smooth objective function. The existence information of each potential viscous damper as well as the damper parameters are characterized by continuous design variables, and the solid isotropic material with penalization technique is utilized to achieve clear existences of viscous dampers. Two numerical examples are presented to illustrate the feasibility of the proposed topology optimization framework.

Appendix A

Differentiating Eq. (11) with respect to $\theta$, one has

$$\frac{{\partial {\mathbf{C}}_{{\text{e}}} (\tau )}}{\partial \theta } = \sum\nolimits_{k = 1}^{N} {{\mathbf{E}}_{k} \frac{{\partial c_{k} (\tau )}}{\partial \theta }{\mathbf{E}}_{k}^{{\text{T}}} } \,{ (}\theta = \eta ,\alpha ,\tilde{\rho }_{1} ,\tilde{\rho }_{2} , \ldots ,\tilde{\rho }_{N} {)}$$

(A1)

where the sensitivity of the equivalent dam** parameter with respect to $\theta$, ${{\partial c_{k} (\tau )} \mathord{\left/ {\vphantom {{\partial c_{k} (\tau )} {\partial \theta }}} \right. \kern-\nulldelimiterspace} {\partial \theta }}$, can be derived from Eq. (8) as follows:

$$\frac{{\partial c_{k} (\tau )}}{\partial \eta } = \tilde{\rho }_{k}^{q} \overline{C}_{{\text{d}}} \frac{{2^{{{{(\alpha + 1)} \mathord{\left/ {\vphantom {{(\alpha + 1)} 2}} \right. \kern-\nulldelimiterspace} 2}}} }}{\sqrt \pi }\Gamma \left( {\frac{\alpha }{2} + 1} \right)\left( {\sigma_{{v_{k} (\tau )}}^{\alpha - 1} + \eta (\alpha - 1)\sigma_{{v_{k} (\tau )}}^{\alpha - 2} \frac{{\partial \sigma_{{v_{k} (\tau )}} }}{\partial \eta }} \right)\,{ (}k = 1,2, \ldots ,N{)}$$

(A2)

$$\frac{{\partial c_{k} (\tau )}}{\partial \alpha } = \tilde{\rho }_{k}^{q} \eta \overline{C}_{{\text{d}}} \frac{{2^{{{{(\alpha + 1)} \mathord{\left/ {\vphantom {{(\alpha + 1)} 2}} \right. \kern-\nulldelimiterspace} 2}}} }}{\sqrt \pi }\Gamma \left( {\frac{\alpha }{2} + 1} \right)\left[ {\sigma_{{v_{k} (\tau )}}^{\alpha - 1} \ln \left( {\sigma_{{v_{k} (\tau )}} } \right) + (\alpha - 1)\sigma_{{v_{k} (\tau )}}^{\alpha - 2} \frac{{\partial \sigma_{{v_{k} (\tau )}} }}{\partial \alpha } + \frac{{\sigma_{{v_{k} (\tau )}}^{\alpha - 1} }}{2}\left( {ln2 + \psi (\frac{\alpha }{2} + 1)} \right)} \right]\,{ (}k = 1,2, \ldots ,N{)}$$

(A3)

$$\frac{{\partial c_{k} (\tau )}}{{\partial \tilde{\rho }_{j} }} = \left\{ {\begin{array}{*{20}c} {\tilde{\rho }_{k}^{q} \eta \overline{C}_{{\text{d}}} \frac{{2^{{{{(\alpha + 1)} \mathord{\left/ {\vphantom {{(\alpha + 1)} 2}} \right. \kern-\nulldelimiterspace} 2}}} }}{\sqrt \pi }\Gamma \left( {\frac{\alpha }{2} + 1} \right)(\alpha - 1)\sigma_{{v_{k} (\tau )}}^{\alpha - 2} \frac{{\partial \sigma_{{v_{k} (\tau )}} }}{{\partial \tilde{\rho }_{j} }} \, (k \ne j)} \\ {\eta \overline{C}_{{\text{d}}} \frac{{2^{{{{(\alpha + 1)} \mathord{\left/ {\vphantom {{(\alpha + 1)} 2}} \right. \kern-\nulldelimiterspace} 2}}} }}{\sqrt \pi }\Gamma \left( {\frac{\alpha }{2} + 1} \right)\left( {\tilde{\rho }_{k}^{q} (\alpha - 1)\sigma_{{v_{k} (\tau )}}^{\alpha - 2} \frac{{\partial \sigma_{{v_{k} (\tau )}} }}{{\partial \tilde{\rho }_{j} }} + q\tilde{\rho }_{k}^{q - 1} \sigma_{{v_{k} (\tau )}}^{\alpha - 1} } \right) \, (k = j)} \\ \end{array} } \right.\,{ (}k = 1,2, \ldots ,N; \, j = 1,2, \ldots ,N{)}$$

(A4)

in which $\psi ( \bullet )$ is the digamma function or psi function defined as $\psi ( \bullet ) = {{\Gamma^{\prime}( \bullet )} \mathord{\left/ {\vphantom {{\Gamma^{\prime}( \bullet )} {\Gamma ( \bullet )}}} \right. \kern-\nulldelimiterspace} {\Gamma ( \bullet )}}$ (Batir 2007).