马小乐, 曹伟. 一种无数值积分的间断Galerkin有限元方法[J]. 空气动力学学报, 2018, 36(4): 596-604. DOI: 10.7638/kqdlxxb-2016.0065
引用本文: 马小乐, 曹伟. 一种无数值积分的间断Galerkin有限元方法[J]. 空气动力学学报, 2018, 36(4): 596-604. DOI: 10.7638/kqdlxxb-2016.0065
MA Xiaole, CAO Wei. Discontinuous Galerkin method with quadrature-free formulation[J]. ACTA AERODYNAMICA SINICA, 2018, 36(4): 596-604. DOI: 10.7638/kqdlxxb-2016.0065
Citation: MA Xiaole, CAO Wei. Discontinuous Galerkin method with quadrature-free formulation[J]. ACTA AERODYNAMICA SINICA, 2018, 36(4): 596-604. DOI: 10.7638/kqdlxxb-2016.0065

一种无数值积分的间断Galerkin有限元方法

Discontinuous Galerkin method with quadrature-free formulation

  • 摘要: 在使用间断Galerkin有限元方法的计算过程中,需要构造相应的积分表达式作为数值求解的出发点,继而会引入体积分和面积分。对于这些积分项的值,一般需要通过数值积分的方法获得。当需要使用高阶间断Galerkin有限元方法时,数值积分计算精度的要求会相应地增加,其所需的计算量将变得很大,而数值积分的计算量又在很大程度上决定了间断Galerkin有限元方法的计算效率。针对这一问题,通过建立Lagrange插值多项式基函数和Jacobi正交多项式基函数的一定关系,构造了一种无数值积分的间断Galerkin有限元方法显式半离散格式,并对不同条件下的线性和非线性一维、二维守恒律进行了直接数值模拟,得到了理想的数值结果。该方法不再需要通过数值积分来计算每个单元的积分项,而且有效地达到了间断Galerkin有限元方法的高阶精度,其对于构造更为高效的高阶间断Galerkin有限元计算方法具有非常显著的意义。

     

    Abstract: In the calculation process of using discontinuous Galerkin finite element method, the corresponding integral expression needs to be constructed as the starting point of numerical solving method, then the volume integral and surface integral are introduced. Under normal circumstances, the value of these integral items is acquired by using numerical integration method. When high-order discontinuous Galerkin finite element method needs to be used, the demand for numerical integration calculation accuracy increases accordingly. This demand results in large calculation amount, and the numerical integration calculation amount largely determines the computational efficiency of discontinuous Galerkin finite element method. In order to solve this problem, an explicit semi-discretization of quadrature-free discontinuous Galerkin finite element method was structured by establishing the relationship between Lagrange interpolation polynomial basis function and Jacobi orthogonal polynomial basis function. An explicit semi-discretization was applied to the direct numerical simulation of linear and nonlinear one-dimensional, two-dimensional conservation laws with different condition, and ideal numerical results were obtained. Using this method, there's no need to calculate the integral items of each element by numerical integration, and the high precision of discontinuous Galerkin finite element method can be achieved effectively. The method has very significant meaning for structuring high efficient high-order discontinuous Galerkin finite element method.

     

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