【作者】 史作强；

【导师】 谢定裕； 候一钊；

【作者基本信息】 清华大学 ， 数学， 2008， 博士

【摘要】 在研究流体和弹性体结构相互作用的问题时,浸没边界法(Immersed BoundaryMethod)是一种非常重要而且常用的计算方法。在很多领域尤其是生物力学领域有着广泛的应用。但是浸没边界法本身也有一些局限,影响了其应用的范围,其中一个就是当使用显式格式时浸没边界法的数值刚性特别大,这就使得在计算时,时间步长必须取得非常小,那么如果想研究系统长时间的演化,计算量就会变得不可忍受,大大限制了浸没边界法的应用范围。为了克服这个困难,很多的半隐式和近似隐式格式被发展了出来,但是迄今为止还没有一种格式有令人满意的表现。在本文中,我们针对二维的定常Stokes流动,非定常Stokes流动以及Navier-Stokes流动,分别提出了几种高效的半隐式格式。这些半隐式格式具有非常好的数值稳定性,可以采用比显式格式大的多的时间步长,同时每一步的计算量与显式格式是相当的,这样在进行长时间的计算时,就可以大大的降低计算量。我们的半隐式格式的构造分为两步:第一步是找到一种无条件稳定的半隐式格式,这种半隐式格式稳定性非常好,但是单步计算量仍嫌太大,在实际计算中相对于显式格式并没有明显的优势。为了降低每一步的计算量,在第二步中我们对这种无条件稳定的半隐式格式进行小尺度分解(Small Scale Decomposition)。然后只对小尺度分解之后得到的首阶项使用隐式格式处理,经过一系列的简化,首阶项可以表示成卷积的形式,这样我们的半隐式格式就可以在傅立叶空间中显式求解,从而获得与显式格式相当的单步计算量。数值结果也证明了我们构造出的高效半隐式格式在大大改善数值稳定性的同时单步的计算量并没有很大的增加。在实际计算中,我们给出的高效半隐式格式是显式格式的一种非常好的替代方法。更多还原

【Abstract】 The Immersed Boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to have some limitations to limit its application. One of them is that it require small time steps to maintain stability when solved with an explicit method. If we want to study the long time evolution, the computational cost will be too expensive to afford. In order to overcome this difficulty, many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose some efficient semi-implicit schemes to remove this stiffness from the Immersed Boundary method for the steady Stokes, unsteady Stokes and Navier-Stokes equations respectively. These semi-implicit schemes have much better stability than the explicit schemes and their computational costs of each step are comparable to those of the explicit schemes. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semi-implicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the Small Scale Decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semi-implicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the Immersed Boundary method.更多还原