Identification of Viscoelastic Functions for Hot-Mix Asphalt Mixtures Using a Modified Harmony Search Algorithm Sungho Mun 1 and Sangyum Lee 2 Abstract: This study proposes a modified harmony search (MHS) algorithm for determining the time-domain viscoelastic function of hot- mix asphalt (HMA) concrete materials. This MHS technique, employing a global optimization technique as well as a Wiechert model for the relaxation function, substantially enhances accuracy and consistency in the determination of viscoelastic functions of several HMA mixtures. In addition, this study shows how to determine a time-domain Prony series representation from the complex modulus in the frequency domain using the MHS algorithm. This can be efficiently used for numerical analysis with techniques such as the finite-element method. The results from lab frequency sweep tests of unmodified and lime-modified HMA at various asphalt contents were consistent with the functions obtained from the MHS algorithm. DOI: 10.1061/(ASCE)CP.1943-5487.0000078. © 2011 American Society of Civil Engineers. CE Database subject headings: Viscoelasticity; Asphalts; Mixtures; Algorithms. Author keywords: Harmony search; Frequency sweep test; Wiechert model; Prony series representation. Introduction Many techniques exist for predicting the viscoelastic material func- tion in the time domain from a linear viscoelastic (LVE) response function in the frequency domain (Emri and Tschoegl 1995; Park and Schapery 1999; Park and Kim 2001; Mun et al. 2007). The methods for the conversion from frequency to time domain and for the interconversion between LVE functions have been investi- gated based on the theory of linear differential and integral equations. Prony series representation is widely used for time-domain LVE functions because of its ability to describe a wide range of visco- elastic responses and the computational efficiency and stability associated with its exponential basis functions (Taylor et al. 1970; Kaliske and Rothert 1997). Several techniques have been developed to fit the Prony series representation to the LVE response function. For example, Schapery (1961) used a collocation method, and Emri and Tschoegl (1995) developed a recursive computer algorithm that generates line spectra from experimental response functions. Ramkumar et al. (1997) used a regularization method based on quadratic programming to minimize the oscillations of the experimental data, while Park and Schapery (1999) and Schapery and Park (1999) developed novel methods of interconversion based on numerical and approximate analytical approaches. Park and Kim (2001) solved the problem of negative Prony series coefficients or oscillations by fitting the Prony series with a power-law presmoothing function, and Mun et al. (2007) proposed a two-step approach to overcome the problems of oscillations in the fitted curve and negative Prony series coefficients. It is widely accepted that the number of coefficients in the Prony series repre- sentation has to correspond to the number of decades in analysis (Park and Schapery 1999; Schapery and Park 1999; Park and Kim 2001). The other techniques of expressing relaxation spectra in time domain are available (e.g., Lee and Knauss 2000; Sorvari and Malinen 2006; Sorvari and Malinen 2007; Mun and Zi 2010). This study proposes a modified harmony search (MHS) algo- rithm to determine a time-domain Prony series representation of hot-mix asphalt (HMA) concrete, resulting in nonnegative Prony coefficients, which are spring constants in a series representation model. This algorithm is based on a time-varying coefficient (TVC) (Naka et al. 2003; Ratnaweera et al. 2004; Tripathi et al. 2007) and the conceptualization of a behavioral phenomenon in the improvisation process of searching for a better state of musical harmony (Geem et al. 2001; Mun and Geem 2009a, b). This method reduces the number of Prony coefficients as compared with other methods (Park and Kim 2001; Chehab 2002; Mun et al. 2007), and thus reduces the numerical computation costs of the convolution integral that represents the LVE constitutive relation- ship between stress and strain. The remainder of the paper is organized into the following sec- tions: the details of the linear viscoelastic model and material parameters; the MHS algorithm along with some constraints; the determination of Prony series coefficients; and concluding remarks that summarize the paper. Linear Viscoelastic Model A frequency sweep test was performed to obtain the LVE material properties of various HMA mixtures. Viscoelastic material proper- ties, such as complex and dynamic modulus, can be determined from this test. Frequency Sweep Test The frequency sweep test consists of a haversine loading applied to a cylindrical specimen of an HMA mixture. The load amplitude is 1 Assistant Professor, School of Civil Engineering, Seoul National Univ. of Science and Technology, 138 Gongneung-gil, Nowon-gu, Seoul, 139- 743, South Korea; formerly, Senior Researcher, Expressway and Transpor- tation Research Institute, Korea Expressway Corporation, 50-5, Sancheok- ri, Dongtan-myeon, Hwaseong-si, Gyeonggi-do, 445-812, South Korea (corresponding author). E-mail: smundyna@gmail.com 2 Director, Road Management Division, Seoul Metropolitan Govern- ment, Deoksugung-gil 15, Jung-gu, Seoul, 100-110, South Korea. E-mail: slee11@seoul.go.kr Note. This manuscript was submitted on November 27, 2009; approved on May 24, 2010; published online on June 7, 2010. Discussion period open until August 1, 2011; separate discussions must be submitted for individual papers. This paper is part of the Journal of Computing in Civil Engineering, Vol. 25, No. 2, March 1, 2011. ©ASCE, ISSN 0887-3801/ 2011/2-139–148/$25.00. JOURNAL OF COMPUTING IN CIVIL ENGINEERING © ASCE / MARCH/APRIL 2011 / 139