翼型跨音速流动计算和外形设计的变域变分有限元方法

在气体动力学问题研究中经常会碰到诸如激波、翼型设计等未知界面问题。未知界面的存在为该类问题的理论分析和数值求解带来了很大困难。刘高联针对未知界面问题发展了一种变域变分有限元方法，该方法将未知界面看作是一个变化区域的边界，采用变域变分将未知界面结合在变分泛函中，使其与求解流场的控制方程结合起来，从而将未知界面的求解和流场的求解完全耦合进行，因而是一种处理未知界面的独特工具，极适合于气动外形的设计求解。本文运用变域变分有限元方法对翼型跨音速流动正、反命题进行了数值研究。由于在跨音速翼型绕流中存在激波，所以为了得到压缩激波解，采用了“人工密度”办法。几个算例均得到了满意的计算结果和设计结果，证明了本文方法的有效性和优越性。

Variable Domain Variational Finite Element Method for Simulation and Design of Transonic Airfoil Flows

Unknown interface problems, such as shock wave and shape designing etc., are usually occurring in aerodynamics researches. The existence of unknown interfaces causes it difficult to theoretically and numerically solve these problems. In order to overcome this difficulty, Liu Gao-Lian developed a variable domain variation finite element method, in which the unknown interfaces were regarded as the boundaries of one variable domain and considered as unknown parameters in the variable domain variation, so the solving of the unknown interfaces equations were coupled with the solving of the governing equations in fluid flow. This method is a specific tool of handling unknown interfaces and is implemented to simulate the transonic fluid flows over airfoil and design airfoil shape. Moreover, in order to capture the shocks, the artificial density was used. The reasonable and satisfied results show that the current method is effective and robust.