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Isogeometric analysis of Mindlin plate with local gap and overlapping feature
ZHAO Gang, DU Xiaoxiao, WANG Wei
School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2016-03-21; Accepted: 2016-05-06; Published online: 2016-05-12 18:12
Foundation item: National Natural Science Foundation of China (51305016, 61572056); Young Talent Project for Central Universities in Beijing 2013 (29201437)
Corresponding author. WANG Wei, E-mail: jrrt@buaa.edu.cn
Abstract: In the frame of isogeometric analysis (IGA), non-uniform rational B-spline (NURBS) serves as both the description of geometries in computer aided design (CAD) and the shape function in finite element analysis (FEA). The common problems of gap and overlapping in NURBS models increase the difficulty of analysis. Based on Mindlin plate theory, the NURBS plate models with gap and overlapping were analyzed by IGA method. The Nitsche method was employed to solve non-conforming problem along models' interfaces. The simulation results of benchmark problem obtained by IGA method were compared with analytical solutions to verify the validity of the presented method. It is demonstrated that Nitsche based IGA method can be used to analyze non-conforming Mindlin plate models with local gap and overlapping feature. Higher NURBS polynomial degrees will produce more accurate IGA results and faster convergence.
Key words: isogeometric analysis (IGA)     Mindlin plate     non-conforming     Nitsche method     gap and overlapping

Hesch和Betsch[2]成功地将Mortar法用来求解等几何分析中非协调问题，Mortar法实质是拉格朗日方法的一种。祝雪峰[3]也对Mortar法求解非协调问题做了相关研究。Nguyen等[4]基于Nitsche方法实现了三维实体的非协调网格等几何分析计算。Apostolatos等[5]将拉格朗日方法、罚函数法和Nitsche方法用在了二维平面非协调网格单元等几何分析中，并对比发现Nitsche方法在计算时要比前2种方法更加优越。Ruess等[6]基于有限胞元法对非协调以及裁剪模型进行了等几何分析。Guo和Ruess[7]研究了Kirchhoff-Love薄壳结构中的非协调等几何问题。Du等[8]对非协调Reissner-Mindlin板的静态弯曲问题进行了研究。但是，目前大部分的研究还是针对多张面片边界完全一致的情况，并没有考虑实际工程模型中面片边界存在缝隙和重叠的情况。

1 多域问题描述与控制方程 1.1 多域问题描述

 图 1 2-域问题示意图 Fig. 1 Schematic diagram of problem domain with two parts
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1.2 Nitsche弱形式

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2 等几何离散 2.1 NURBS及其基函数

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B样条基函数具有许多优良的特性可以引用在等几何分析中，如规范性、非负性和局部支撑性等。此外，B样条基函数可以在内节点处保持Cp-k连续，k表示节点重复度。高连续性将使等几何分析的计算结果更加精确。

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NURBS相关方法大部分都可以使用其对应的高一维的齐次空间的B样条来进行计算，但是NURBS的导数不能采用这一方法。本文使用RXW分别表示NURBS基函数、基函数的分子和分母。基函数Rξ的一阶偏导数可以表示为

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NURBS基函数的高阶导数可以通过相同的方式获得。节点插入算法和升阶算法是NURBS方法中的2个非常重要的算法，其在等几何分析中的作用相当于传统有限元中的h-型细化和p-型细化。等几何分析中的k-型细化是节点插入算法和升阶算法的有效结合。此外，徐岗等[12]还提出了基于最小化后验误差估计的r-型细化方法。

2.2 Mindlin板理论

 图 2 Mindlin板模型及其坐标系统 Fig. 2 Model and coordinate system of a Mindlin plate
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Mindlin板中广义应变εp可以表示为

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2.3 离散方程推导

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2.4 交接面积分

 图 3 交界线Γ*1到Γ*2上的高斯点映射 Fig. 3 Mapping of Gauss points from interface Γ*1 to Γ*2
 图 4 求解整体刚度矩阵K程序流程图 Fig. 4 Program flowchart of solving global stiffness matrix K
3 数值算例

 图 5 含缝隙与重叠部分的NURBS方板模型 Fig. 5 NURBS based square plate with gap and overlapping
3.1 固支方板静态弯曲

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 图 6 固支方板挠度 Fig. 6 Deflections of clamped square plate
 图 7 固支方板挠度绝对误差 Fig. 7 Absolute errors of deflection of clamped square plate
3.2 固支方板自由振动

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 模态阶数 h/l=0.1 h/l=0.2 等几何分析 Liew等[16] 等几何分析 Liew等[16] 1 3.297 0 3.295 4 2.688 4 2.687 5 2 6.270 5 6.285 8 4.680 9 4.690 7 3 6.304 7 6.285 8 4.702 1 4.690 7 4 8.810 6 8.809 8 6.298 6 6.298 5 5 10.377 1 10.378 8 7.174 3 7.176 7 6 10.484 3 10.477 8 7.278 3 7.275 9 7 12.539 4 12.552 9 8.507 0 8.515 5 8 12.570 0 12.552 9 8.522 7 8.515 5 9 15.296 0 15.291 8 10.008 9 10.012 6 10 15.305 7 15.291 8 10.017 7 10.012 6

 图 8 不同次数下的非协调方板前10阶频率相对误差 Fig. 8 Relative errors of first ten mode frequencies of non-conforming square plate with different polynomial degrees
 图 9 不同次数下的第1阶和第2阶非协调方板频率参数收敛 Fig. 9 Convergence of first two mode frequency parameters of non-conforming square plate with different polynomial degrees

 图 10 非协调固支方板的模态振型 Fig. 10 Mode shapes of non-conforming clamped square plate
4 结论

1) Nitsche方法可以实现对含缝隙与重叠部分几何模型的等几何分析。

2) 推导的Mindlin板静态弯曲和自由振动刚度方程可以实现对平板挠度和转角以及自然频率进行预测。

3) NURBS次数越高，等几何分析的计算结果越精确。由于模型存在缝隙和重叠，计算结果与精确解会有一定误差。

4) 收敛性可以达到最优收敛，并且次数越高，收敛性速度越快。

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#### 文章信息

ZHAO Gang, DU Xiaoxiao, WANG Wei

Isogeometric analysis of Mindlin plate with local gap and overlapping feature

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(3): 432-440
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0221