无网格局部强弱法求解不规则域问题

无网格局部彼得洛夫-伽辽金(meshless local Petrov-Galerkin,MLPG)法是一种具有代表性的无网格方法,在计算力学领域得到广泛应用.然而,这种方法在边界上需执行积分运算,通常很难处理不规则求解域问题.为了克服MLPG法的这种局限性,提出了无网格局部强弱(meshless local strong-weak,MLSW)法.MLSW法采用MLPG法离散内部求解域,采用无网格介点(meshless intervention-point,MIP)法施加自然边界条件,并采用配点法施加本质边界条件,避免执行边界积分运算,可适用于求解各类复杂的不规则域问题.从理论上讲,这种结合式方法,既保持了MLPG法稳定而精确计算的优势,同时兼备配点型方法在处理复杂结构问题时简洁而灵活的优势,实现了弱式法和强式法的优势互补.此外,MLSW法采用移动最小二乘核(moving least squares core,MLSc)近似法来构造形函数,是对传统移动最小二乘(moving least squares,MLS)近似法的一种改进.MLSc使用核基函数代替通常的基函数,有利于数值求解的精确性和稳定性,而且其导数近似计算变得更为简单.数值算例结果初步表明:这种新方法实施简单,求解稳定、精确,表现出适合工程运用的潜力.

MESHLESS LOCAL STRONG-WEAK (MLSW) METHOD FOR IRREGULAR DOMAIN PROBLEMS

The meshless local Petrov-Galerkin (MLPG) method is a representative meshless method, and is widely applied in computational mechanics. But, this method is necessary to execute the boundary integral operation, and it is always difficult to solve irregular domain problems. In order to remove this kind of limitation of the MLPG method, a meshless local strong-weak (MLSW) method is presented. The proposed method uses the MLPG method for domain discretization, adopts the meshless intervention-point (MIP) method for imposing the natural boundary conditions, and employs a collocation method for imposing the essential boundary conditions. Thus, the boundary integral is completely eliminated, and it favours to solve all kinds of irregular domain problem. Theoretically, the MLSW method deduced by coupling algorithm, not only has inherited the advantage of the MLPG method, which is always stable and accurate fornumerical solution, but also has attained the superiority of the collocation-type method, which is naturally simple and flexible to cope with the domain of complex structure. Thereby, the method realizes advantageous complementarities of the weak-form method and the strong-form method. In addition, the MLSW method uses a moving least squares core (MLSc) approximation for constructing meshless shape function, which is an improvement for the traditional moving least squares (MLS) approximation. By replacing common basis function with core basis function, MLSc approximation is more stable and accurate, and also realizes a simple calculation for derivatives approximation. Early results with numerical tests have showed that the proposed new method is easy for numerical implementation, is accurate and stable for numerical solution, and is promising for engineering application.