【作者】 俞红；

【导师】 谭永基；

【作者基本信息】 复旦大学 ， 应用数学， 2009， 硕士

【摘要】 本文中我们提出了用摄动和特征函数展开的方法处理高维侵蚀反问题并将其降维的新方法。一个轴对称体由于内壁受到侵蚀,丧失其轴对称性,几何形状变得十分复杂。这样原来的一维或二维的轴对称问题就可能变成一般的二维甚至三维的问题。由于几何复杂性的增加导致了计算复杂性的增加,在数值求解这类反问题时,通常需要计算大量的边界变动的正问题。特别是在高维的情况下计算复杂性和计算量的恶性增长为问题的求解带来极大的困难。本文通过摄动方法将问题化为边界固定,形状轴对称但边界条件非轴对称的问题,进而用特征函数系展开的方法将这类问题化为若干低维问题进行求解。用这样的方法可以将二维不定边界问题降维为一维轴对称问题;将三维不定边界的问题化为二维轴对称固定边界问题,有时甚至可以化为一维轴对称问题,克服了几何复杂性和高维计算复杂性的困难。大量计算实例表明此方法是精确有效的。更多还原

【Abstract】 In this paper,we advance a new method which is a combination of perturbation method and expansion with eigenfunction to deal with a high-dimension inverse problem of corrosion and reduce its dimension.The corrosion will destroy the original approximate axis-symmetry and complicate the shape,of the container.Thus the original one-dimension or two-dimension axis-symmetric problem probably become a two-dimension or three-dimension problem.Since the increase of complexity of geometry leads to the increase of complexity of computation,when we solve this kind of inverse problem numerically we often need to solve a lot of direct problems with flexible boundary.Especially,when we deal with a high-dimension problem the sick increase of complexity and amount of computation leads to great difficulty in solving the problem.In this paper,we use perturbation method to transfer the problem into a problem with fixed boundary and axis-symmetric domain. Then the boundary condition is not axis-symmetric and we use expansion with eigenfunction to transfer it into a group of lower-dimension problems. Using this method,we can transfer a two-dimension problem with unknown boundary into a group of one-dimension axis-symmetric problems and we also can transfer a three-dimension problem with unknown boundary into a group of two-dimension axis-symmetric problems with fixed boundary,sometimes even into a group of one-dimension problems.This method overcome the difficulty brought by the complexity of geometry and high-dimension.Large numbers of computational results show that the methods are effective and accurate.更多还原