Calculation Method for the Relaxation Time Spectra of Viscoelastic Fluids Using Dynamic Moduli

Abstract

The influence of different error functions (standard deviation, normalized standard deviation, and chi-square) as objective functions in the calculation of the relaxation time spectrum is studied, as well as the influence of different minimization procedures available in MATLAB (fmincon, fminimax, and patternsearch). Two simulated spectra were extracted from the literature to validate the proposed methodology, each with its corresponding dynamic modulus. These dynamic moduli were then used to calculate the spectrum, which was subsequently compared with the initial simulated spectra. This calculus implies a well-known ill-posed problem solved by simultaneously minimizing the regularized error function and a constraint function. The proposed regularization is based on Tikhonov regularization, with a second spectrum differential as the operator and an optimal regularization parameter generated using a modified L Curve. A new way of comparing the different procedures used is proposed, which involves the calculation of the mean between the resulting spectra and its subsequent use as an initial point in the different minimizations studied until convergence is achieved. Finally, the results show that the best objective function studied is the normalized standard deviation and that the best minimization procedure is fmincon.

Keywords

• dynamic moduli
• relaxation time spectra
• ill-posed problem
• matlab