多穿孔板的多尺度公式化
On a multiscale formulation for multiperforated plates
AI总结：
针对多穿孔板高梯度与边界层正则性损失导致的数值模拟昂贵问题，提出基于异构多尺度方法的多尺度公式化，分离宏观远场与介观近场，并证明变分公式化的适定性与截断误差估计。
摘要：
多穿孔板表现出高梯度以及集中在边界层中的正则性损失，对此直接数值模拟变得非常昂贵。对于椭圆方程，在边界一定距离处的解仅以有效方式受到影响，宏观和介观行为可以分离。以泊松方程为例，引入了一种符合异构多尺度方法精神的多尺度公式化。基于匹配渐近展开方法，将解分离为定义在仅具有缓慢变化边界的域中的宏观远场，以及定义在可能变化无限周期单元上的缩放坐标中的介观近场。近场具有多项式行为，该行为与多穿孔板中线上宏观变量的迹耦合。引入了一种在条带中使用Beppo-Levi空间的变分公式化，并证明了其适定性。讨论了截断无限条带时的变分框架，并估计了截断误差。
英文摘要：
Multiperforated plates exhibit high gradients and a loss of regularity concentrated in a boundary layer for which a direct numerical simulation becomes very expensive. For elliptic equations the solution at some distance of the boundary is only affected in an effective way and the macroscopic and mesoscopic behaviour can be separated. A multiscale formulation in the spirit of the heterogeneous multiscale method is introduced on the example of the Poisson equation. Based on the method of matched asymptotic expansion the solution is separated into a macroscopic far field defined in a domain with only slowly varying boundary and a mesoscopic near field defined in scaled coordinates on possibly varying infinite periodicity cells. The near field has a polynomial behaviour that is coupled to the traces of the macroscopic variable on the mid-line of the multiperforated plate. A variational formulation using a Beppo-Levi space in the strip is introduced and its well-posedness is shown. The variational framework when truncating the infinite strip is discussed and the truncation error is estimated.
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