论文标题
具有内部变量配方的粘弹性标量波传播的有限元近似和分析
Finite Element Approximation and Analysis of Viscoelastic Scalar Wave Propagation with Internal Variable Formulations
论文作者
论文摘要
我们考虑使用用于建模粘弹性固体的遗传积分项的线性标量波方程。该Volterra积分中的内核是衰减指数的总和（所谓的Maxwell或Zener模型），这允许引入两种类型的内部变量家族之一，每个变量都根据普通的微分方程（ODE）演化。每个衰减指数都有一种这样的颂歌，这些ODES的引入意味着可以从管理方程式中删除Volterra积分。内部变量的两种类型的区分是通过未知数出现在Volterra积分中，还是它的时间导数出现。我们将最终问题称为位移和速度形式。我们通过在空间中使用连续的Galerkin有限元近似和隐式的“曲柄 - 尼科尔森”类型的有限差异方法来定义这些形式的每种形式的完全离散的公式。我们证明了稳定性和先验界限，并且（使用Fenics环境，https://fenicsproject.org/）给出了一些数值结果。这些界限不需要Grönwall的不平等，因此可以被认为是高质量的，可以在没有错误的指数构成错误的情况下长期整合。据我们所知，这是第一次将这两种配方与伴随的高质量稳定性和误差界的证据一起描述。结果将结果扩展到矢量值粘弹性问题很简单，最后总结了。通过从https://github.com/yongseok7717获取Python来源，或通过运行自定义构建的Docker容器（提供指令），可以将数值结果重现。
We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according to an ordinary differential equation (ODE). There is one such ODE for each decaying exponential, and the introduction of these ODEs means that the Volterra integral can be removed from the governing equation. The two types of internal variable are distinguished by whether the unknown appears in the Volterra integral, or whether its time derivative appears; we call the resulting problems the displacement and velocity forms. We define fully discrete formulations for each of these forms by using continuous Galerkin finite element approximations in space and an implicit `Crank-Nicolson' type of finite difference method in time. We prove stability and a priori bounds, and (using the FEniCS environment, https://fenicsproject.org/) give some numerical results. These bounds do not require Grönwall's inequality and so can be regarded to be of high quality, allowing confidence in long time integration without an a priori exponential build up of error. As far as we are aware this is the first time that these two formulations have been described together with accompanying proofs of such high quality stability and error bounds. The extension of the results to vector-valued viscoelasticity problems is straightforward and summarised at the end. The numerical results are reproducible by acquiring the python sources from https://github.com/Yongseok7717, or by running a custom built docker container (instructions are given).
