Dargush G F
Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York Buffalo, New York 14260, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Dec;86(6 Pt 2):066606. doi: 10.1103/PhysRevE.86.066606. Epub 2012 Dec 28.
The principle of mixed convolved action provides a new rigorous weak variational formalism for a broad range of initial value problems in mathematical physics and mechanics. Here, the focus is initially on classical single-degree-of-freedom oscillators incorporating either Kelvin-Voigt or Maxwell dissipative elements and then, subsequently, on systems that utilize fractional-derivative constitutive models. In each case, an appropriate mixed convolved action is formulated, and a corresponding weak form is discretized in time using temporal shape functions to produce an algorithm suitable for numerical solution. Several examples are considered to validate the mixed convolved action principles and to investigate the performance of the numerical algorithms. For undamped systems, the algorithm is found to be symplectic and unconditionally stable with respect to the time step. In the case of dissipative systems, the approach is shown to be robust and to be accurate with good convergence characteristics for both classical and fractional-derivative based models. As part of the derivations, some interesting results in the calculus of Caputo fractional derivatives also are presented.
混合卷积作用原理为数学物理和力学中的广泛初值问题提供了一种新的严格弱变分形式。这里，最初关注的是包含开尔文 - 伏伊特或麦克斯韦耗散元件的经典单自由度振荡器，随后关注使用分数阶导数本构模型的系统。在每种情况下，都制定了适当的混合卷积作用，并使用时间形状函数对相应的弱形式进行时间离散化，以产生适用于数值求解的算法。考虑了几个例子来验证混合卷积作用原理并研究数值算法的性能。对于无阻尼系统，发现该算法关于时间步长是辛的且无条件稳定。在耗散系统的情况下，该方法被证明是稳健的，并且对于基于经典模型和分数阶导数模型都具有良好的收敛特性且准确。作为推导的一部分，还给出了卡普托分数阶导数微积分中的一些有趣结果。
