A nonlinear convolution scheme to simulate bridge aerodynamics Teng Wu ⇑ , Ahsan Kareem Nathaz Modeling Laboratory, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, IN, USA article info Article history: Received 17 January 2013 Accepted 11 June 2013 Available online 13 September 2013 Keywords: Bridge aerodynamics Nonlinearity Convolution Volterra abstract A linear convolution scheme involving first-order (linear) kernels for linear bridge aerodynamics is first reviewed and the significance of the selection of proper input parameters is emphasized. Following the concept of nonlinear indicial response function, the linear convolution scheme is extended to the nonlin- ear convolution scheme involving higher-order (nonlinear) kernels for the treatment of nonlinear bridge aerodynamics using a ‘‘peeling-an-onion’’ type procedure. Utilizing an impulse function as input, a com- prehensive kernel identification scheme is developed. A numerical example of a long-span suspension bridge is investigated to verify the fidelity of the proposed nonlinear convolution scheme. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction A main source of nonlinearity in bridge aerodynamics results from flow separation around the deck. For streamlined sections like an airfoil, flow separation occurs only in the case of large an- gles of attack (dynamic stall) or the shock motions in the transonic region (the shock motion itself also induces nonlinearity). For bluff sections like a bridge deck, flow separation is prevalent as the fluid motion around the deck cannot negotiate sudden changes in the deck profile. The resulting nonlinearity can be viewed from four viewpoints: (i) non-proportional relationship between amplitudes of input and output; (ii) single-frequency input exciting multiple frequencies; (iii) amplitude dependence of aerodynamic and aero- elastic forces and (iv) hysteretic behavior of aerodynamic forces versus angles of attack [1]. Nonlinear effects are usually exploited to offer a possible explanation for any differences observed be- tween the linear analysis results and experiments [2] although it is difficult to delineate their relative contributions. In order to take into account the increasing nonlinear behavior of bridge aerodynamics observed in wind-tunnel tests, several numerical schemes such as the ‘‘band superposition’’ [3], ‘‘hybrid’’ [4], ‘‘rheological’’ [5] and ‘‘artificial neural network’’ [6] have been proposed over the last decade to advance conventional linear anal- ysis framework [7,8]. Generally, these numerical schemes have been unable to represent completely nonlinear bridge aerodynam- ics [6,9], which limits their utility and calls for a comprehensive nonlinear analysis framework. The consideration of nonlinearity is usually carried out in the time domain benefitting from its ability to take into account the nonlinear effects readily. In the time domain, the convolution of a linear kernel, e.g., the unit-step response function, is well known as the Duhamel’s integral. In this study, the linear convolution scheme concerning first-order kernels for linear analysis of bridge aerodynamics is reviewed together with a selection of proper input variables. Then, it is extended to the nonlinear convolution scheme involving higher-order kernels for nonlinear analysis of bridges un- der winds based on the concept of nonlinear indicial response function. A nonlinear convolution scheme is represented utilizing a Volterra-type formalism, which ensures convergence of its trun- cated form. To facilitate this formalism, a comprehensive kernel identification scheme is developed utilizing the impulse function as input. Finally, a numerical example of a long-span suspension bridge with vertical and torsional degrees of freedom is investi- gated to verify the fidelity of the simulation based on the proposed nonlinear convolution scheme, where the amplitude dependence of kernels is also discussed. 2. Linear convolution scheme This study focuses on the simulation based on a two-dimen- sional (2-D) representation of the deck and the strip theory. 2.1. Input information of bridge aerodynamics The selection of proper input variables for bridge aerodynamics based on convolution integrals is a critical issue. In the case of gust-induced effects, the input information is straightforward, i.e., the gust fluctuations in each degree of freedom. However, in the case of motion-induced effects, the input information is often misunderstood in bridge aerodynamics. 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.06.004 ⇑ Corresponding author. Tel.: +1 574 904 4290. E-mail addresses: twu@nd.edu (T. Wu), kareem@nd.edu (A. Kareem). Computers and Structures 128 (2013) 259–271 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc