Can numerical methods compete with analytical solutions of linear constitutive models for large amplitude oscillatory shear flow?

Abstract

We consider the corotational Maxwell model which is perhaps the simplest constitutive model with a nontrivial oscillatory shear response that can be solved analytically. The exact solution takes the form of an infinite series. Due to exponential convergence, accurate analytical approximations to the exact solution can be obtained by truncating the series after a modest number (\(\varvec{\approx }\) 10–20) of terms. We compare the speed and accuracy of this truncated analytical solution (AS) with a fast numerical method called harmonic balance (HB). HB represents the periodic steady-state solution using a Fourier series ansatz. Due to the linearity of the constitutive model, HB leads to a tridiagonal linear system of equations in the Fourier coefficients that can be solved very efficiently. Surprisingly, we find that the convergence properties of HB are superior to AS. In terms of computational cost, HB is about 200 times cheaper than AS. Thus, the answer to the question posed in the title is affirmative.