【作者】 陈思；

【导师】 董石麟； 周岱；

【作者基本信息】 上海交通大学 ， 固体力学， 2007， 博士

【摘要】 ?无网格方法是新一代的计算方法,这种方法计算空间导数时不需要借助于事先划定的网格,从而避免了高维拉普拉斯网格法中网格缠结和扭曲等问题。光滑质点流体动力学的方法(SPH)是一种应用广泛的无网格方法,但这种方法存在计算精度低、求解二阶空间导数不稳定、边界条件难以处理等缺点。当质点分布均匀、对称时,本文证明:线性函数的真实值与SPH近似值的比值为常数;多项式函数的阶导数的真实值与SPH近似值的比值为常数;多项式函数的二阶导数和与SPH近似值的比值为常数。采用上述结论可以得到一种高精度的SPH离散格式。本文采用移动最小二乘法保证质点分布均匀、对称。如果采用完备性的方法,则可使计算更加简洁。采用完备性方法时,需引进背景函数,但不再需要质点分布均匀、对称。如果采用线性函数作为背景函数,在计算一阶导数时,可得到与Belytschko类似的结论。借助弹性力学变分公式推导出一种新的SPH离散方法,并提出扩张核函数的概念。利用扩张核函数求空间函数的二阶导数,可降低核函数的可导性要求。采用高精度的SPH方法推导了计算流体力学和计算固体力学相关公式,并通过算例检验这种方法的精度。其中固体力学的算例精度令人满意,流体力学的算例与经典SPH方法比较,精度提高不明显。主要原因在于流体力学模拟受边界影响较大,而文中所提高精度SPH格式尚无配套的高精度边界条件处理方法。本文将SPH方法和动力松弛法(D.R.)相结合,利用SPH的特点,提出加速D.R.方法收敛的新技巧。同时,利用扩张核函数的概念处理固体力学边界条件获得成功。更多还原

【Abstract】 Meshless methods are regarded as next generation of computationalmethods. The key idea of meshless methods is to provide numericalsolutions for integral equations or PDEs with a set of arbitrarilydistributed nodes without using any mesh that provides the connectivityof these nodes. The smoothed particle hydrodynamics (SPH) is a widelyused messless method. However, there are fewer efficient SPH methodsto treat solid‐liquid boundary and the error of SPH approximation couldlead to computational failure in the case that the computation domainincreases or in the case that the derivative order is high.The dissertation proves three lemmas: When particles in the supportdomain of smoothing kernel function are distributed evenly andsymmetrically, the ratio of the real value of linear function to its SPHapproximation is constant, the ratio of the real value of the nthderivative of the polynomial function P?????? to its SPH approximation isconstant and the ratio of the real value of ????P?? to its SPHapproximation is constant. A new kind of high precision SPH methodbased on the lemmas mentioned above is achieved in this dissertation. The dissertation introduces the moving least squares method (MLS)which can make particles in the support domain distribute evenly andsymmetrically. However, the computation can be simplified byintroducing background function instead of MLS. If the backgroundfunction is linear function, the formula deduced in this dissertation isequal to that of Belytschko when calculating 1st derivative.Through the application of SPH approximation to the elasticmechanics equation of variation, it is discovered that the secondderivative of SPH approximation can be obtained by integral action. Bythis method, a new concept of widened‐kernel function is introduced.In this dissertation, the high accuracy SPH approximation is appliedto the elastic mechanics equations and Navier‐Stokes equations. Thecomparisons between the results computed by ANSYS and the methoddeduced in the present study prove that the present method is accurate.However, the precision improvement is limited in CFD because there isno high precision method to deal with boundary conditions.Moreover, the combination of SPH and Dynamic Relaxation (D.R.) isinvestigated. Meanwhile, the boundary conditions are dealt with by thewidened‐kernel function.更多还原