A CONSISTENT AND EFFICIENT METHOD FOR IMPOSING MESHFREE ESSENTIAL BOUNDARY CONDITIONS VIA HELLINGER-REISSNER VARIATIONAL PRINCIPLE
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摘要: 无网格法具有高阶连续光滑的形函数, 在结构分析中呈现出显著的精度优势. 但无网格形函数在节点处一般没有插值性, 导致伽辽金无网格法难以直接施加本质边界条件. 采用变分一致尼兹法施加边界条件的数值解具有良好的收敛性和稳定性, 因而得到了非常广泛的应用, 然而该方法仍然需要引入人工参数来保证算法的稳定性. 本文以赫林格−赖斯纳变分原理为基础, 建立了一种变分一致的本质边界条件施加方法. 该方法采用混合离散近似赫林格−赖斯纳变分原理弱形式中的位移和应力, 其中位移采用传统无网格形函数进行离散, 而应力则在背景积分单元中近似为相应阶次的多项式. 此时的无网格离散方程可视为一种新型的尼兹法施加本质边界条件, 其中修正变分项采用再生光滑梯度和无网格形函数进行混合离散, 稳定项则内嵌于赫林格−赖斯纳变分原理弱形式中, 无需额外增加稳定项, 消除了对人工参数的依赖性. 该方法无需计算复杂耗时的形函数导数, 并满足积分约束条件, 保证了数值求解的精度. 数值结果表明, 所提方法能够保证伽辽金无网格法的计算精度最优误差收敛率, 与传统的尼兹法相比明显提高了计算效率.
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关键词:
- 无网格法 /
- 赫林格−赖斯纳变分原理 /
- 本质边界条件 /
- 再生光滑梯度 /
- 变分一致性
Abstract: Galerkin meshfree methods with arbitrary order smooth shape functions exhibit superior accuracy advantages in structural analysis. However, the smooth meshfree shape functions generally do not have the interpolatory property and thus the enforcement of essential boundary conditions in Galerkin meshfree methods is not trivial. The variationally consistent Nitsche’s method shows very good performance regarding convergence and stability and is widely used to impose essential boundary conditions. In this work, a consistent and efficient method is proposed to impose meshfree essential boundary conditions. The proposed method is based upon the Hellinger-Reissner (HR) principle, where the displacements are represented by the conventional meshfree shape functions and the stresses are approximated by reproducing kernel smoothed gradients defined in each background integration cells. The resulting meshfree discrete equations share almost identical forms with those derived from Nitsche’s method. It is shown that the stabilized term in Nitsche’s method is a natural outcome from the HR variational principle, but there is absolutely no need to use any artificial parameter to maintain the coercivity of stiffness matrix. Moreover, under the reproducing kernel gradient smoothing framework, the costly derivatives of conventional meshfree shape functions are completely avoided and the integration constraint is automatically fulfilled. Numerical results demonstrate that the proposed approach and Nitsche’s method yield comparable solution accuracy, nonetheless, much higher efficiency is observed for the proposed methodology that imposes the essential boundary conditions for Galerkin meshfree formulation via the HR variational principle. -
图 2 背景积分单元示意图和优化的数值积分方案
Figure 2. Illustration of background integration cells and optimized quadrature rules
图 9 带孔无限大平板问题无网格离散
Figure 9. Meshfree discretizations of the plate with a hole problem
图 12 带孔无限大平板问题
$ {\sigma _{xx}} $ 应力云图Figure 12. Contour plot of stress
$ {\sigma _{xx}} $ for the plate with a hole problem表 1 二次基函数无网格法分片试验结果
Table 1. The results of patch test with quadratic basis functions
Linear patch test Quadratic patch test L2-Err He-Err L2-Err He-Err GI-penalty $ 7.7 \times {10^{ - 6}} $ $ 2.7 \times {10^{ - 4}} $ $ 1.2 \times {10^{ - 5}} $ $ 2.6 \times {10^{ - 4}} $ GI-LM $ 1.0 \times {10^{ - 4}} $ $ 4.7 \times {10^{ - 3}} $ $ 1.5 \times {10^{ - 4}} $ $ 4.3 \times {10^{ - 3}} $ GI-Nitsche $ 8.1 \times {10^{ - 6}} $ $ 2.8 \times {10^{ - 4}} $ $ 1.3 \times {10^{ - 5}} $ $ 2.8 \times {10^{ - 4}} $ RKGSI-penalty $ 7.9 \times {10^{ - 8}} $ $ 2.0 \times {10^{ - 6}} $ $ 1.4 \times {10^{ - 7}} $ $ 2.1 \times {10^{ - 6}} $ RKGSI-LM $ 8.6 \times {10^{ - 5}} $ $ 4.0 \times {10^{ - 3}} $ $ 1.4 \times {10^{ - 4}} $ $ 3.7 \times {10^{ - 3}} $ RKGSI-Nitsche $ 2.1 \times {10^{ - 15}} $ $ 4.0 \times {10^{ - 14}} $ $ 2.2 \times {10^{ - 15}} $ $ 2.7 \times {10^{ - 14}} $ RKGSI-HR $ 2.0 \times {10^{ - 15}} $ $ 3.2 \times {10^{ - 14}} $ $ 2.2 \times {10^{ - 15}} $ $ 2.1 \times {10^{ - 14}} $ 
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表 2 三次基函数无网格法分片试验结果
Table 2. The results of patch test with cubic basis functions
Quadratic patch test Cubic patch test L2-Err He-Err L2-Err He-Err GI-penalty $ 9.1 \times {10^{ - 6}} $ $ 2.1 \times {10^{ - 4}} $ $ 1.2 \times {10^{ - 5}} $ $ 2.0 \times {10^{ - 4}} $ GI-LM $ 2.9 \times {10^{ - 4}} $ $ 9.3 \times {10^{ - 3}} $ $ 4.0 \times {10^{ - 4}} $ $ 9.3 \times {10^{ - 3}} $ GI-Nitsche $ 1.1 \times {10^{ - 5}} $ $ 2.8 \times {10^{ - 4}} $ $ 1.4 \times {10^{ - 5}} $ $ 2.7 \times {10^{ - 4}} $ RKGSI-penalty $ 1.4 \times {10^{ - 7}} $ $ 2.1 \times {10^{ - 6}} $ $ 2.0 \times {10^{ - 7}} $ $ 2.7 \times {10^{ - 6}} $ RKGSI-LM $ 3.0 \times {10^{ - 4}} $ $ 9.8 \times {10^{ - 3}} $ $ 4.2 \times {10^{ - 4}} $ $ 9.8 \times {10^{ - 3}} $ RKGSI-Nitsche $ 3.6 \times {10^{ - 15}} $ $ 1.0 \times {10^{ - 13}} $ $ 4.6 \times {10^{ - 15}} $ $ 9.5 \times {10^{ - 14}} $ RKGSI-HR $ 3.1 \times {10^{ - 15}} $ $ 1.0 \times {10^{ - 13}} $ $ 3.5 \times {10^{ - 15}} $ $ 7.4 \times {10^{ - 14}} $
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B1 二次基函数无网格法优化的数值积分方案
B1. The optimized quadrature rules for quadratic basis function
$ \xi $ $ \eta $ $ \gamma $ $ w $ $ {w_B} $ 
$\dfrac{ {\text{2} } }{ {\text{3} } }$ $ \dfrac{{\text{1}}}{{\text{6}}} $ $ \dfrac{{\text{1}}}{{\text{6}}} $ $ \dfrac{{\text{1}}}{{\text{3}}} $ − 
$ \dfrac{{\text{1}}}{{\text{2}}} $ $ \dfrac{{\text{1}}}{{\text{2}}} $ 0 $ \dfrac{{\text{1}}}{{\text{6}}} $ − 1 0 0 $ \dfrac{{\text{2}}}{{\text{3}}} $ $ \dfrac{{\text{1}}}{{\text{3}}} $
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B2 三次基函数无网格法优化的数值积分方案
B2. The optimized quadrature rules for cubic basis function
$ \xi $ $ \eta $ $ \gamma $ $ w $ $ {w_B} $ 
$\eta_a $ $ \dfrac{1-\xi_{a}}{2} $ $ \dfrac{1-\xi_{a}}{2} $ ${w}_{a}$ − $ \eta_b $ $ \eta_b $ $ \dfrac{1-\xi_{b}}{2} $ ${w}_{b}$ − $\begin{aligned}{\xi_{a}=0.108\;103\;018\;168\;070}, \quad {w_{a}=0.223\;381\;589\;678\;011} \\{\xi_{b}=0.816\;847\;572\;980\;459}, \quad w_{b}=0.109\;951\;743\;655\;322\end{aligned}$ 
$\dfrac{{\text{1}}}{{\text{3}}} $ $ \dfrac{{\text{1}}}{{\text{3}}} $ $ \dfrac{{\text{1}}}{{\text{3}}} $ $ \dfrac{{\text{9}}}{{\text{20}}} $ − 1 0 0 $ -\dfrac{{\text{1}}}{{\text{30}}} $ $ \dfrac{{\text{1}}}{{\text{20}}} $ $ \dfrac{{\text{1}}}{{\text{2}}} $ $ \dfrac{{\text{1}}}{{\text{2}}} $ 0 $ \dfrac{{\text{4}}}{{\text{135}}} $ $ \dfrac{{\text{16}}}{{\text{46}}} $ $ \dfrac{7+\sqrt{21}}{14} $ $ \dfrac{7+\sqrt{21}}{14} $ 0 $ \dfrac{{\text{49}}}{{\text{540}}} $ $ \dfrac{{\text{49}}}{{\text{180}}} $
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