动态Preisach迟滞模型的数学分析及数值解
Mathematical analysis and numerical solution of models with dynamic Preisach hysteresis
摘要：
滞后现象是一种在各种物理系统中观察到的现象，它导致非线性和多值行为，使得它们的建模和控制变得困难。即使经典或速率无关的滞后模型的分析和数学性质是已知的，这不是动态模型的情况下，目前的方法缺乏一个适当的功能分析框架，这是必不可少的制定最优化问题和发展稳定的数值，都是至关重要的，在实践中。本文论述了动态Preisach迟滞模型的描述和数学分析。为此，我们完成了一个广泛接受的定义的动力学模型通常用于描述磁场H和磁感应强度B之间的本构关系，其中，B的值不仅取决于H的当前值，但也在过去的历史和它的速度。我们首先从数学上分析了模型的一些重要性质，并将它们与静态Preisach模型的已知结果进行了比较。然后，我们考虑了一个由电磁场方程驱动的具有动态迟滞的抛物问题。在适当的假设下，我们证明了问题的弱形式的适定性，并数值求解。最后，我们报告了一个数值试验，以评估收敛的顺序，并说明数值解的行为，为不同的配置的动态Preisach模型
英文摘要：
Hysteresis is a phenomenon that is observed in a great variety of physical systems, which leads to a nonlinear and multivalued behavior, making their modeling and control difficult. Even though the analysis and mathematical properties of classical or rate-independent hysteresis models are known, this is not the case for dynamic models where current approaches lack a proper functional analytic framework which is essential to formulate optimization problems and develop stable numerics, both being crucial in practice. This paper deals with the description and mathematical analysis of the dynamic Preisach hysteresis model. Toward that end, we complete a widely accepted definition of the dynamic model commonly used to describe the constitutive relation between the magnetic field H and the magnetic induction B, in which, the values of B not only depends on the present values of H but also on the past history and its velocity. We first analyze mathematically some important properties of the model and compare them with known results for the static Preisach model. Then, we consider a parabolic problem with dynamic hysteresis motivated by electromagnetic field equations. Under suitable assumptions, we show the well posedness of a weak formulation of the problem and solve it numerically. Finally, we report a numerical test in order to assess the order of convergence and to illustrate the behavior of the numerical solution for different configurations of the dynamic Preisach model
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