多参数传热反问题的RBF-MLPG方法研究

MLPG(Meshless Local Petrov-Galerkin) method have good stability and high computational precise, and can be very good application at the the irregular domain. The present meshless method has been the study focus to compute heat transfer and fluid flow problem and inverse problem, but it exits some shortcomings,such as low precise for computational method of convection term and more computational time for inverse problem,etc. In present project, the radial basis function(RBF) is applied as the trail function of MLPG method, the purpose is to construct the computational method of the second boundary condition of RBF-MLPG method based on the Hermite method, the stability and accuracy of present method is investigated. Based on the computational method of convection term in the Finite volume method(FVM) or Finite element method(FEM), the upwind scheme of convection term of RBF-MLPG method is constructed to overcome the false diffusion, and the robustness, stability and accuracy of present method is studied. The RBF-MLPG method for inverse problem of irregular domain with multi-variables is constructed. the stability, accuracy and optimized method are investigated. The reduce order model (ROM) of the inverse problem with multi-variables is consturcted to improve the computational efficiency. In-depth study of this project will provide a new method for computational heat transfer problems and inverse problems, and the investigated results have important significance to further improve the theory of meshless method and its engineering application.

MLPG(meshless local Petrov-Galerkin)方法具有稳定性好、计算精度高、对不规则区域适应性好等优点。近年来采用该方法计算传热与流动问题及其反问题已成为研究热点，但仍存在对流项处理方法造成的计算精度低和用于反问题时计算量大等缺点。本项目研究以径向基函数（RBF）作为MLPG方法的插值函数，应用Hermite方法构建RBF-MLPG方法的第二类边界条件处理方法，研究其稳定性及精度；借鉴有限容积法或有限元方法处理对流项的思想，构建RBF-MLPG方法的对流项迎风格式，克服假扩散现象，研究格式的健壮性、稳定性与精度；构建多参数不规则区域传热反问题的RBF-MLPG计算方法，研究其稳定性、计算精度及优化方法；研究建立多参数传热反问题的降阶模型，提高计算效率。本项目的深入研究将为传热问题及反问题的计算提供一种新方法，研究结果对丰富无网格方法理论及其工程应用具有重要意义。

无网格局部Petrov-Galerkin(MLPG)方法具有较高稳定性和精度，对不规则区域适应性好等优点，但计算量较大，在对流项占优问题及反问题计算研究较少。本项目利用MLPG方法在非规则区域求解方便与FVM在规则区域计算准确高效的优点，提出了MLPG-FVM的耦合方法用于非规则区域导热问题求解，该方法不仅能保证计算精度高，而且还能减少计算耗时。针对传统流线型迎风格式（SUPG）中影响计算结果稳定性和精确性的稳定参数进行研究。本项目提出了一个新的稳定参数，结果表明：新的稳定参数使SUPG方法能更有效地求解对流占优问题，相比于其他稳定参数，该参数对应的结果具有稳定性好，计算精度高及越界现象不明显的特点。根据变分多尺度（VMS）思想，将原影算子和投影算子之差作为稳定项添入无网格方法中，提出变分多尺度无网格方法（VMS-MLPG方法），该方法在不引入额外变量和不增加额外存储空间的基础上保持了经典变分多尺度方法的优点，结果说明：此方法是对提高计算结果的稳定性和精确性有显著的效果。通过把变分多尺度（VMS）思想结合混合公式克服压力产生的振荡，提出了基于两局部高斯积分的变分多尺度方法，将速度投影逼近用于简化经典VMS中尺度空间的构造，并对非线性的Navier-Stocks方程进行研究，结果说明：该方法的数值解不仅保证了稳定性还在精确性上有明显提高。将MLPG方法与一种经典的迭代法CGM结合，提出MLPG-CGM的耦合方法并应用于非规则区域导热反问题的研究，CGM在反问题中的计算结果依赖初值的选择，而MLPG方法只在计算区域内布置一些离散的节点，不需节点与节点之间的相互联系，故其相对于传统方法而言对不规则区域有更好的适应性，并能获得精度较高的计算结果，因此，可将MLPG方法在非规则区域中的计算结果作为初值，用CGM进行迭代求解。反演结果显示：采用两种方法结合处理非规则区域导热反问题，能获得准确的数值解。