Analytical Solutions of Peridynamic Equations. Part I: Transient Heat Diffusion
Analytical Solutions of Peridynamic Equations. Part I: Transient Heat Diffusion
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DOI：
10.1007/s42102-022-00080-7
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发表时间：
2022-03
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影响因子：
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通讯作者：
Ziguang Chen;Xuhao Peng;S. Jafarzadeh;F. Bobaru
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文献类型：
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作者：
Ziguang Chen;Xuhao Peng;S. Jafarzadeh;F. Bobaru
本文利用分离变量技术，构造了瞬态扩散环动力模型的形式化解析解。我们证明了通过在解的时间指数部分插入“周动力（非局部）因子”可以直接从相应的经典解中得到无穷级数的非局部解。我们找到了非局部因素的解析表达式。在二维矩形域中，这些可以写成贝塞尔函数的形式。非局部因子取决于视界大小，并在视界大小趋于零时收敛为1，恢复相应偏微分方程解的经典形式。我们还证明，当时间趋于无穷时，对于固定视界，非局部解收敛于经典解。我们考虑了具有Dirichlet和Neumann边界条件的瞬态扩散问题的例子。并将其解析解与相应的经典解进行了比较。虽然我们在这里给出的大多数解析解都是形式化的，但对于一些情况，我们能够证明级数解的一致收敛。这是第一个通过分离变量，在任意边界条件下，在一维或二维有限域中给出周期动力学瞬态扩散问题的解析（正式）解，并显示它们与经典/局部问题的相应解的联系的贡献。
In this paper, we construct formal analytical solutions for peridynamic models of transient diffusion using the separation of variables technique. We show that the infinite series nonlocal solutions can be obtained directly from corresponding classical solutions by inserting “peridynamic (nonlocal) factors” in the time-exponential part of the solution. We find analytical expressions for the nonlocal factor. In 2D rectangular domains, these can be written in terms of Bessel functions. The nonlocal factor depends on the horizon size and converges to value one as the horizon size goes to zero, recovering the classical form of the solution for the corresponding partial-differential equations. We also show that, as time goes to infinity, the nonlocal solution converges to the classical one, for a fixed horizon. We consider examples of transient diffusion problems with Dirichlet and Neumann boundary conditions. Their analytical solutions are compared with the corresponding classical solutions. While most of the analytical solutions we present here are formal, for a number of cases, we are able to prove uniform convergence of the series solutions. This is the first contribution that presents analytical (formal) solutions to peridynamic transient diffusion problems in 1D or 2D finite domains by separation of variables, with arbitrary boundary conditions, and shows their connections to the corresponding solutions to the classical/local problem.