MODEL ORDER REDUCTION FOR NONLINEAR DYNAMIC ANALYSIS OF PARAMETERIZED CURVED BEAM STRUCTURES BASED ON ISOGEOMETRIC ANALYSIS
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摘要: 模型降阶方法通过构造全阶模型的低阶近似模型有效提升了求解效率,同时也保留了原阶模型的主要信息从而保证了较高的计算精度。对于结构非线性以及参数化的模型降阶问题,常需要重复计算刚度矩阵等非线性以及参数依赖项,求解效率较低。此外,当参数化模型的几何形状改变时,往往需要重复进行CAD与有限元(FEA)模型的转换,这对于复杂结构较为耗时。等几何分析采用描述几何形状的非均匀有理B样条(NURBS)插值物理场,实现了CAD与FEA模型的统一,消除了两者之间繁琐的模型转换过程,其具有几何精确、高阶连续等优点,并且几何形状在细化过程中保持不变,非常适合于薄壁类结构的分析以及参数化表达。该研究结合等几何分析、特征正交分解(POD)以及离散经验插值方法(DEIM)研究参数化的平面曲梁结构的非线性动力学模型降阶问题。数值结果表明,基于等几何分析的POD-DEIM降阶模型能够显著提升平面曲梁结构的非线性动力学计算效率,并且该模型对于参数化以及变载荷等情形显示出了良好的适应性。Abstract: Model order reduction (MOR) approach generates lower dimensional approximations to the original system while preserving model's essential information and computational accuracy. For nonlinear and parameterized structural problems, where the stiffness matrix is configuration dependent, an iterative solution procedure is inevitable and a revisit to all the elements is essential for updating the stiffness matrix. Considering the difference between CAD and finite element models, a preprocessing step is necessary for each parameterized geometry, which is time consuming for complex geometries. Isogeometric analysis (IGA) utilizes non-uniform rational B-spline (NURBS) basis for both the geometry description and physical filed interpolation, thusly eliminates the time-consuming preprocessing step between CAD and finite element models. IGA is a perfect candidate for the analysis and geometry parameterization of thin-walled structures due to its geometrically exact and high order continuous properties. The nonlinear dynamics of the parameterized planar curved beams are studied based on the isogeometric analysis and their model order reductions are investigated based on the proper orthogonal decomposition and discrete empirical interpolation method (POD-DEIM). Numerical results show that IGA-based POD-DEIM method significantly improves the computational efficiency of the nonlinear dynamic analysis of planar curved beams. Additionally, the proposed method also applies to different external loadings and time step sizes, which demonstrate the adaptivity of the method.
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表 1 两端固支半圆弧曲梁降阶模型计算时间对比
Table 1. Comparisons of computational time for reduced order models of half-circle arc beam
计算模型 全阶
模型POD降阶模型 POD-MDEIM
降阶模型k = 2 k = 3 k = 5 k = 15 k = 20
m = 15计算时间/s 302.9 208.9 212.2 234.3 242.4 36.9 时间比 1 1.4 1.4 1.3 1.2 8.2 表 2 两端简支曲梁刚度矩阵MDEIM插值位置
Table 2. MDEIM Positions for stiffness matrix of simply supported arc beam
位置$ {\phi }_{i} $ 行(i) 列(j) 单元编号e − 位置$ {\phi }_{i} $ 行(i) 列(j) 单元编号e 223 1 3 1 7057 64 64 19,20,21,22 225 3 3 1 7615 67 69 20,21,22,23 2689 25 25 6,7,8,9 7953 72 72 33,34,35,36 3697 34 34 9,10,11,12 8961 81 81 24,25,26,27 3699 36 34 9,10,11,12 10081 91 91 28,29,30,31 4371 42 40 11,12,13,14 10753 97 97 30,31,32,33 5599 49 51 14,15,16,17 11091 102 100 31,32,33,34 6607 58 60 17,18,19,20 − − − − 表 3 两端简支曲梁刚度矩阵MDEIM插值位置
Table 3. MDEIM Positions for stiffness matrix of simply supported arc beam
位置$ {\phi }_{i} $ 行i 列j 单元编号e 位置$ {\phi }_{i} $ 行i 列j 单元编号e 243 3 3 1 8953 73 75 22,23,24,25 3752 32 32 8,9,10,11 11 738 98 98 30,31,32,33 4237 37 36 10,11,12 12 827 107 107 33,34,35,36 4841 41 41 11,12,13,14 13 795 115 115 36,37 5082 42 43 12,13,14 13 797 117 115 36,37 5814 54 49 15,16,17 14 037 117 117 36,37 7135 55 60 17,18,19 14 158 118 118 37 8108 68 68 20,21,22,23 − − − − 表 4 参数化曲梁降阶模型计算时间对比
Table 4. Comparisons of computational time for reduced order models of parameterized arc beam
控制点P2坐标 计算时间/s 速度提升率 全阶
模型POD-MDEIM
降阶模型抽样1 抽样2 抽样1 抽样2 P2 = (7.5,9.5) m
F = −800 N237.0 41.1 43.5 5.8 5.4 P2 = (5.7,7.2) m
F = −800 N242.1 39.0 37.6 6.2 6.4 P2 = (6,5.5) m
F = −800sin(48t) N402.0 78.1 71.6 5.1 5.6 -
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