等几何分析方法求解静电场非齐次边值问题

Isogeometric analysis for electrostatic problem with inhomogeneous boundary conditions

等几何分析方法中采用非均匀有理B样条基函数离散求解域,而非均匀有理B样条基函数不具有插值性,使得原有的等几何分析方法求解非齐次边值问题较为困难。针对静电场非齐次边值问题,在等几何分析方法的虚功方程中引入拉格朗日乘子,使静电场控制方程弱化为关于控制点变量和拉格朗日乘子的代数方程。修正后的等几何方法虽然增加了计算自由度,但将其扩展到非齐次边值问题的求解。数值算例表明:修正后的等几何方法能够很好地求解静电场非齐次边值问题,且将计算结果与解析解和经典的有限元方法比较表明,此方法具有自由度少、精度高、收敛速度快的优点。

Preliminary isogeometric analysis （IGA） is not suitable for electrostatic problem with inhomogeneous boundary conditions,due to the non-interpolatory feature of non-uniform rational B-splines （NURBS） employed in IGA. In this paper, virtual work equation with Lagrange multiplier is established for inhomogeneous boundary conditions and weakened to augmented algebraic equations containing control variables involved with NURBS elements and discretized Lagrange multipliers corresponding to inhomogeneous boundary. The modified method is suitable for electrostatic problem with inhomogeneous boundary conditions even though the degree of freedom is little increased. The numerical results,compared with analytical solution and conventional finite element method, show that the proposed method is effective to inhomogeneous boundary and yields much better results, consumes less amount of DOFs, and obtains quicker convergence than traditional method.