基于等几何分析的参数化曲梁结构非线性动力学降阶模型研究

郭玉杰; 吴晗浪; 李薇; 李迎港; 吴剑晨

doi:10.6052/j.issn.1000-4750.2021.04.0285

基于等几何分析的参数化曲梁结构非线性动力学降阶模型研究

doi: 10.6052/j.issn.1000-4750.2021.04.0285

南京航空航天大学飞行器先进设计技术国防重点学科实验室，南京 210016

基金项目: 国家自然科学基金面上项目(11972187)

详细信息

作者简介:
吴晗浪(1993−)，男，贵州人，硕士生，主要从事等几何分析研究(E-mail: 473807649@qq.com)

李　薇(1998−)，女，河南人，硕士生，主要从事等几何结构动力学研究(E-mail: cyona@nuaa.edu.cn)

李迎港(1997−)，男，安徽人，硕士生，主要从事结构疲劳寿命研究(E-mail: liyinggang@nuaa.edu.cn)

吴剑晨(1997−)，男，安徽人，硕士生，主要从事航天器传热研究(E-mail: jianchenwu@nuaa.edu.cn)

通讯作者:
郭玉杰(1986−)，男，江苏人，副教授，博士，主要从事结构计算力学研究(E-mail: yujieguo@nuaa.edu.cn)

中图分类号: O342；TB122
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- 被引次数: 0
出版历程
- 收稿日期: 2021-04-16
- 修回日期: 2021-07-05
- 网络出版日期: 2021-08-06
- 刊出日期: 2022-08-01

MODEL ORDER REDUCTION FOR NONLINEAR DYNAMIC ANALYSIS OF PARAMETERIZED CURVED BEAM STRUCTURES BASED ON ISOGEOMETRIC ANALYSIS

Key Laboratory of Fundamental Science for National Defense-Advanced Design Technology of Flight Vehicle, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

摘要

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。
- 等几何分析 /
- 模型降阶 /
- 非线性动力学 /
- 离散经验插值 /
- 参数化模型
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.
- isogeometric analysis /
- model order reduction /
- nonlinear dynamic analysis /
- discrete empirical interpolation method /
- parameterized models

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图 1 NURBS曲线及NURBS基函数

Figure 1. NURBS curve and NURBS basis functions

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图 2 初始构型与当前构型

Figure 2. Reference and current configurations

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图 3 HHT-α时间积分算法流程图

Figure 3. HHT-α time integration scheme for nonlinear dynamic analysis

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图 4 两端固支半圆弧曲梁模型

Figure 4. Half-circle arc with clamped boundary conditions at both ends

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图 5 两端固支半圆弧曲梁位移收敛曲线(几何非线性)

Figure 5. Convergence studies of the half-circle arc beam(geometrically nonlinear)

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图 6 两端固支半圆弧曲梁奇异值

Figure 6. Singular values of displacement snapshots and stiffness matrix snapshots for polynomial degrees 2, 3, 4 for clamped half-circle arc

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图 7 两端固支半圆弧曲梁载荷-位移曲线

Figure 7. Load-displacement response of half-circle arc beam

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图 8 两端固支半圆弧曲梁中点轨迹对比图

Figure 8. Comparison of the paths traced by the mid-point of the half-circle arc beam obtained using the FOM and ROM

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图 9 两端固支半圆弧曲梁降阶与全阶模型位移差值图

Figure 9. Difference of mid-point displacement between FOM and ROM for half-circle arc beam

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图 10 两端固支半圆弧曲梁降阶、全阶模型计算时间对比

Figure 10. Comparison of computational time for different reduced order models of half-circle arc beam

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图 11 一端固支阿基米德螺旋梁

Figure 11. Archimedes spiral shaped beam with one end clamped

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图 12 一端固支阿基米德螺旋曲梁奇异值

Figure 12. Singular values of displacement snapshots and stiffness matrix snapshots for clamped Archimedes spiral beam

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图 13 一端固支阿基米德螺旋曲梁载荷位移曲线

Figure 13. Load-displacement curve of clamped Archimedes spiral arc beam

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图 14 一端固支阿基米德螺旋曲梁端点的轨迹对比图

Figure 14. Comparison of the paths traced by the end-point of the Archimedes spiral beam obtained using the FOM and ROM

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图 15 阿基米德螺旋曲梁端点降阶与全阶模型位移差值图

Figure 15. Difference of end-point displacement between FOM and ROM for the Archimedes spiral beam

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图 16 两端简支曲梁模型

Figure 16. Arc beam with simply supported boundary conditions at both ends

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图 17 两端简支曲梁奇异值

Figure 17. Singular values of displacement snapshots and stiffness matrix snapshots for simply supported arc beam

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图 18 两端简支曲梁载荷-位移曲线

Figure 18. Load-displacement curve of simply supported arc beam

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图 19 两端简支曲梁中点的轨迹对比图

Figure 19. Comparison of the paths traced by the mid-point of the arc beam obtained using the FOM and ROM

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图 20 两端简支曲梁降阶与全阶模型位移差值图

Figure 20. Difference of mid-point displacement between FOM and ROM for the arc beam

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图 21 两端简支曲梁频率响应对比图

Figure 21. Comparison of frequency response obtained using FOM and ROM

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图 22 两端简支曲梁载荷位移曲线($F = - 800\sin (36t)$)

Figure 22. Load-displacement curve of simply supported arc beam ($F = - 800\sin (36t)$)

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图 23 两端简支曲梁中点的轨迹对比图($F = - 800\sin (36t)$)

Figure 23. Comparison of the paths traced by the mid-point of the arc beam obtained using the FOM and ROM ($F = - 800\sin (36t)$)

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图 24 两端简支曲梁降阶与全阶模型位移差值图($F = - 800\sin (36t)$)

Figure 24. Difference of mid-point displacement between FOM and ROM for the arc beam ($F = - 800\sin (36t)$)

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图 25 两端简支曲梁频率响应对比图($F = - 800\sin (36t)$)

Figure 25. Comparison of frequency response obtained using FOM and ROM ($F = - 800\sin (36t)$)

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图 26 两端简支曲梁的参数化模型

Figure 26. Parametric model of simply supported curved beams

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图 27 控制点P₂抽样分布图

Figure 27. Sampling positions of the control point P₂

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图 28 参数化曲梁奇异值

Figure 28. Singular values of displacement snapshots and stiffness matrix snapshots for parameterized arc beam

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图 29 不同抽样方法所对应的位移载荷响应

Figure 29. Load-displacement response of half-circle arc beam

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图 30 参数化曲梁$\xi = 0.5$处的轨迹对比图

Figure 30. Comparison of the paths traced at the position of $\xi = 0.5$ for the parameterized arc beam obtained using the FOM and ROM

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图 31 参数化曲梁降阶与全阶模型位移差值图$(\xi = 0.5)$

Figure 31. Difference of the displacement between FOM and ROM for the parameterized arc beam $(\xi = 0.5)$

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图 32 参数化曲梁频率响应对比图

Figure 32. Comparison of frequency response obtained using FOM and ROM for the parameterized arc beam

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图 33 不同抽样方法所对应的位移载荷响应$(\xi = 0.5)$ ($F = - 800\sin (48t)$)

Figure 33. Comparisons of load-displacement histories of the parameterized arc beam at $(\xi = 0.5)$ ($F = - 800\sin (48t)$)

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图 34 不同抽样方法所对应的位移载荷响应$(\xi = 0.5)$ ($F = - 800\sin (48t)$, 时间步长：0.005 s)

Figure 34. Comparisons of load-displacement histories of the parameterized arc beam at $(\xi = 0.5)$($F = - 800\sin (48t)$, time step: 0.005 s)

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图 35 参数化曲梁$(\xi = 0.5)$处的轨迹对比图($F = - 800\sin (48t)$, 时间步长：0.005 s)

Figure 35. Comparison of the paths traced at the position of $(\xi = 0.5)$ for the parameterized arc beam obtained using the FOM and ROM ($F = - 800\sin (48t)$, time step: 0.005 s)

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图 36 参数化曲梁降阶与全阶模型位移差值图$(\xi = 0.5)$($F = - 800\sin (48t)$, 时间步长：0.005 s)

Figure 36. Difference of the displacement between FOM and ROM for the parameterized arc beam $(\xi = 0.5)$ ($F = - 800\sin (48t)$, time step: 0.005 s)

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图 37 参数化曲梁频率响应对比图($F = - 800\sin (48t)$, 时间步长：0.005 s)

Figure 37. Comparison of frequency response obtained using FOM and ROM for the parameterized arc beam ($F = - 800\sin (48t)$, time step: 0.005 s)

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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
计算模型	全阶模型	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

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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		−	−	−	−

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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	−	−	−	−

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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
	全阶模型	抽样1	抽样2	抽样1	抽样2
P₂ = (7.5,9.5) m F = −800 N	237.0	41.1	43.5	5.8	5.4
P₂ = (5.7,7.2) m F = −800 N	242.1	39.0	37.6	6.2	6.4
P₂ = (6,5.5) m F = −800sin(48t) N	402.0	78.1	71.6	5.1	5.6

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