【摘要】 该文通过提高积分精度改进了非均匀格点三角形求积元的收敛性。以球面映射点为初值,以能够精确积分的多项式最高阶数为目标,采用数值优化方法,得到了一系列新的积分点位置和积分权系数。这样的积分法则所能够精确积分的多项式最高阶数为插值多项式阶数的1.6倍左右。初步的数值实验表明,对于平面位势问题和平面弹性问题,改进的单元具有更好的收敛性。此外还讨论了进一步改进单元性质的方法以及这些方法可能存在的上限。更多还原

【Abstract】 An improvement of the convergence of triangular quadrature elements is achieved via the increase of quadrature precision. Taking a planar triangular grid points mapped from equi-areal-point-set on a spherical triangle as initial values and the highest order of polynomials that can be accurately integrated as the objective, a new set of quadrature points and weights in a triangle is obtained using numerical optimization methods. The new quadrature rule raises the quadrature precision from the order of the interpolation by approximately 1.6 times. Preliminary numerical examples show that the new triangular quadrature element exhibits a higher convergence rate. Further improvements and possible limitations of this strategy are also discussed.更多还原