# 埋め込み境界法

## 定式化

{\displaystyle {\begin{aligned}&\rho \left({\frac {\partial {u}({x},t)}{\partial {t}}}+{u}\cdot \nabla {u}\right)=\mu \Delta u(x,t)-\nabla p+f(x,t)\\&\nabla \cdot u=0\end{aligned}}}

${\displaystyle f(x,t)=\sum _{j=1}^{N}\delta _{a}(x-Z_{j})F_{j}}$

ここで、δaディラックのデルタ関数を長さa のスケールで平滑化した関数である。一方、構造体の変形は、次式に基づいて行われる。

${\displaystyle {\frac {dZ_{j}}{dt}}=\int \delta _{a}(x-Z_{j})u(x,t)dx}$

## 参考資料

1. C. S. Peskin, The immersed boundary method, Acta Numerica, 11, pp. 1– 39, 2002.
2. R. Mittal and G. Iaccarino, Immersed Boundary Methods, Annual Review of Fluid Mechanics, vol. 37, pp. 239-261, 2005.
3. Y. Mori and C. S. Peskin, Implicit Second Order Immersed Boundary Methods with Boundary Mass Computational Methods in Applied Mechanics and Engineering, 2007.
4. L. Zhua and C. S. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, Journal of Computational Physics, vol. 179, Issue 2, pp.452-468, 2002.
5. P. J. Atzberger, P. R. Kramer, and C. S. Peskin, A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Journal of Computational Physics, vol. 224, Issue 2, 2007.
6. A. M. Roma, C. S. Peskin, and M. J. Berger, An adaptive version of the immersed boundary method, Journal of Computational Physics, vol. 153 n.2, pp.509-534, 1999.