The method of harmonic balance (HB) is a spectrally accurate method used to obtain periodic steady state solutions to dynamical systems subjected to periodic perturbations. We adapt HB to solve for the stress response of the Giesekus model under large amplitude oscillatory shear (LAOS) deformation. HB transforms the system of differential equations to a set of nonlinear algebraic equations in the Fourier coefficients. Convergence studies find that the difference between the HB and true solutions decays exponentially with the number of harmonics ($H$) included in the ansatz as $e^{-m H}$. The decay coefficient $m$ decreases with increasing strain amplitude, and exhibits a "U" shaped dependence on applied frequency. The computational cost of HB increases slightly faster than linearly with $H$. The net result of rapid convergence and modest increase in computational cost with increasing $H$ implies that HB outperforms the conventional method of using numerical integration to solve differential constitutive equations under oscillatory shear. Numerical experiments find that HB is simultaneously about three orders of magnitude cheaper, and several orders of magnitude more accurate than numerical integration. Thus, it offers a compelling value proposition for parameter estimation or model selection.

翻译：谐波平衡法（HB）是一种谱精度方法，用于求解受周期性扰动的动力系统的周期稳态解。我们将HB方法应用于求解Giesekus模型在大振幅振荡剪切（LAOS）变形下的应力响应。HB将微分方程组转化为关于傅里叶系数的非线性代数方程组。收敛性研究表明，HB解与真实解之间的差异随假设中包含的谐波数（$H$）呈指数衰减，衰减形式为$e^{-m H}$。衰减系数$m$随应变振幅增大而减小，且对施加频率呈现"U"形依赖关系。HB方法的计算成本随$H$增加的速率略高于线性。快速收敛与计算成本随$H$适度增长的共同结果表明，在振荡剪切条件下，HB方法优于传统使用数值积分求解微分本构方程的方法。数值实验发现，HB方法同时具有约三个数量级的计算成本优势以及数个数量级的精度优势。因此，该方法为参数估计或模型选择提供了极具吸引力的价值主张。