中文摘要
本项目首先分别就二维热传导方程和N-S方程浸入边界虚拟区域法的近似解存在性问题进行了研究，给出了对应的数学证明；数值模拟时，提出了引入虚拟边界条件控制虚拟流动以处理出现的奇异问题的方法，算例表明：虚拟边界条件的引入有助于控制Lagrange乘子的变化，从而改善数值模拟的精度；其次，把研究的基于边界Lagrange乘子的虚拟区域法，嵌入到用于求解电磁散射场的精确控制算法中，构造出虚拟区域与精确控制耦合算法；在此基础上，通过构造Lagrange乘子空间，使Lagrange乘子相关的物面线积分问题变为基于Collocation的近似问题，避免了边界积分的求交运算，成功地进行了多物体相互干扰情形的组合调试，计算结果与现有文献结果吻合；接着，通过介质边界的近似处理，把发展的算法推广用于散射体加表面涂层多介质问题，得到了初步的调试结果，展示出算法的可行性。最后，对与涂层局部加密相关的无网格自适应和点云运动处理技术以及代数虚拟区域法等问题也进行了研究，算例展示出新的进展。由于本项目发展的算法引入了虚拟区域，具有前置处理简单，适合处理复杂的实际问题，因此，具有广阔的应用前景。
英文摘要
The mathematical existence theorems of the approximate solutions of the two-dimensional heat equations and the two-dimensional N-S equations with immersed boundary method are first obtained in this project. Fictitious boundary conditions for controlling fictitious flows inside the body are introduced in order to deal with singularities near the corner of bodies. Numerical experiments have shown that the use of fictitious boundary conditions on a decomposed domain has good effect on controlling the magnitude of Lagrange multipliers with the improvement of accuracy. Secondly, coupling exact controllability and fictitious domain methods with Lagrangian multipliers is investigated for solutions of the electromagnetic scattering. In order to avoid the difficulties of computing integral on the irregular mesh,collocation based approximation of problem with distributed multipliers is further implemented.With this new method,the numerical results presented have a good agreement with the ones appeared in the open references. Then, the method is further developed for numerical solutions of the electromagnetic scattering by coated shapes. Finally, studies of refinement and adaptation of the gridless clouds and algebraic fictitious method are also presented with the numerical cases. The developed methods with the ingredient of fictitious domain have a wide of applications occurred in engineering due to their simpler treatment and well accommodation for complicated bodies.
结题摘要
本项目首先分别就二维热传导方程和N-S方程浸入边界虚拟区域法的近似解存在性问题进行了研究，给出了对应的数学证明；数值模拟时，提出了引入虚拟边界条件控制虚拟流动以处理出现的奇异问题的方法，算例表明：虚拟边界条件的引入有助于控制Lagrange乘子的变化，从而改善数值模拟的精度；其次，把研究的基于边界Lagrange乘子的虚拟区域法，嵌入到用于求解电磁散射场的精确控制算法中，构造出虚拟区域与精确控制耦合算法；在此基础上，通过构造Lagrange乘子空间，使Lagrange乘子相关的物面线积分问题变为基于Collocation的近似问题，避免了边界积分的求交运算，成功地进行了多物体相互干扰情形的组合调试，计算结果与现有文献结果吻合；接着，通过介质边界的近似处理，把发展的算法推广用于散射体加表面涂层多介质问题，得到了初步的调试结果，展示出算法的可行性。最后，对与涂层局部加密相关的无网格自适应和点云运动处理技术以及代数虚拟区域法等问题也进行了研究，算例展示出新的进展。由于本项目发展的算法引入了虚拟区域，具有前置处理简单，适合处理复杂的实际问题，因此，具有广阔的应用前景。