用有限元法计算二维重力场垂直分量及重力位二阶导数

有限元法可以计算密度分布、形态复杂物体的重力场垂直分量 g 及重力位二阶导数 W_xy、W_yy。取一个包围密度体的足够大的区域，求解 g 的边值问题可表为 与上述边值问题相应的变分问题是泛函取极值。用有限元解上述变分问题时，将区域Ω剖分为三角单元，在单元 e 内进行二次函数插值。首先计算各单元的 F_ε（g），然后相加组成总体的 F（g），它是各节点待求的 g 的函数。对F（g）求极值，得一线性代数方程组。解方程组可得各节点的g。对F进行微商，即可得重力位二阶导数。

Calculation of vertical component of 2D gravitational field and second derivative of gravitational potential by finite element method

Density distribution, vertical component g of gravitational field of a complex form object and second derivative w_xv, w_yy of gravitational potential can be calculated by finite element method.Taking a sufficient Large area that is surrounding the density body, the boundary value of g can be expressed as followsThe calculus of variations corresponding to the above boundary value will be to get the extreme value by functional analysisWhen the calculus of various described above is solved by finite ele- ment method, the area S2 is seperated into triangular elements and second functional interpolation is performed in element e.F_e(g) of each element is calculated first, then added together to get total F(g) which is the function of g to be calculated for each node.Taking the extreme value of F(g)，a linear algebraic equation is derived; g of each node can be derived by solving the equation; after taking numerical derivative for g,second derivative of gravitational potential can be obtained.