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:调心平衡法(HB)是一种光谱精确的方法,用于获得定期扰动的动态系统定期稳定状态的解决方案。我们调整HB,以解决大型振幅振幅振动振动剪切(LAOS)下吉塞库斯模型的应激反应。HB将差异方程系统转换成Fourier系数中的一组非线性代数方程。一致研究发现,HB与真实解决方案之间的差别随着以$-mH$(H$)为Ansatz 中包含的调和器的数量而急剧消减。以增压振动振动振动振动剪切变形(LAOS)下Giesekus模型的减缩系数为$-mH}。HB的计算成本以美元略微高于线性增。计算成本增加的净结果意味着HB在使用数字集解以解的常规方法(HC$),以美元计数分立方阵阵阵阵阵阵阵阵阵阵阵阵阵阵阵阵阵阵,以更精确度的数值推算方式进行若干次级的数值实验,以“高等等等压”。