Method of Imposing Local Fixed Constraints Exactly in Isogeometric Analysis

doi:10.12141/j.issn.1000-565X.230749

Abstract

Abstract:

Isogeometric analysis uses computer splines such as non-uniform rational B-splines as the basis functions. When the order of the basis function is 2 or greater, the control points do not coincide with the element nodes and the support domain of the basis function spans multiple elements, which makes it difficult to impose local fixed constraints precisely in isogeometric analysis. To solve this problem, this paper uses a step function to modify the displacement interpolation function of isogeometric analysis. The step function takes a value of 0 in the locally fixed constraint region and 1 in the other region, so that the displacement value in the fixed constraint region is forced to be 0, and the displacement interpolation function in other region is revert to the original form. In order to minimize the influence of step function on the analysis domain, the rising interval of the step function is set to be small. Meanwhile, the hierarchical spline is used to subdivide the elements in the rising interval locally, therefore, the Gaussian points of the subdivided elements fall into the rising interval of the step function as well as the step function has an effect on the stiffness matrix. In addition, the element subdivision also effectively improves the solution accuracy in the local constraint region where large strains are present. Finally, the method mentioned above is compared with analytical solution and the finite element method to verify its accuracy, flexibility and reliability, finding that the results of calculation coincide with the analytical solution. Finally, by considering the situations with different fixed constrains that vary in shape, area and location., the finite element method with coarse mesh and fine mesh are used to calculate the examples, finding that the displacement and stress obtained by the proposed method are closer to those obtained by the fine mesh finite element method, which illustrates that the solution accuracy can be achieved with fewer elements; and that the proposed method is of good accuracy, flexibility and reliability.

Key words: isogeometric analysis, fixed constraint, step function, hierarchical spline

CLC Number:

O302

WANG Yingjun, LI Jinghui. Method of Imposing Local Fixed Constraints Exactly in Isogeometric Analysis[J]. Journal of South China University of Technology(Natural Science Edition), 2024, 52(12): 65-78.

Figures/Tables 20

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References 35

1	乔发杰．基于等几何分析的大变形梁结构仿真分析算法［D］．大连：大连理工大学，2022．
2	郭玉杰，吴晗浪，李薇，等．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［J］．工程力学，2022，39（8）：31-48．
	GUO Yujie， WU Hanlang， LI Wei，et al ．Model order reduction for nonlinear dynamic analysis of parameterized curved beam structures based on isogeometric analysis［J］．Engineering Mechanics，2022，39（8）：31-48．
3	金灵智，王禹，郝鹏，等．加筋路径驱动的板壳自适应等几何屈曲分析［J］．力学学报，2023，55（5）：1151-1164．
	JIN Lingzhi， WANG Yu， HAO Peng，et al ．Adaptive isogeometric buckling analysis of stiffened panels driven by stiffener paths［J］．Chinese Journal of Theoretical and Applied Mechanics，2023，55（5）：1151-1164．
4	费建国，罗会信，左兵权，等．雷诺方程的数值计算方法概述［J］．润滑与密封，2020，45（4）：130-140．
	FEI Jianguo， LUO Huixin， ZUO Bingquan，et al ．An overview of numerical methods for Reynolds equation［J］．Lubrication Engineering，2020，45（4）：130-140．
5	于嘉瑞，岳宝增，李晓玉．燃料大幅晃动等几何分析仿真及航天器耦合动力学研究［J］．力学学报，2023，55（2）：476-486．
	YU Jiarui， YUE Baozeng， LI Xiaoyu ．Study on isogeometric analysis for large-amplitude propellant sloshing and spacecraft coupled dynamics［J］．Chinese Journal of Theoretical and Applied Mechanics，2023，55（2）：476-486．
6	PEROTTO S， BELLINI G， BALLARIN F，et al ．Isogeometric hierarchical model reduction for advection-diffusion process simulation in microchannels［M］∥CHINESTA F，CUETO E，PAYAN Y，et al eds．Reduced Order Models for the Biomechanics of Living Organs．Pittsburgh：Academic Press，2023：197-211．
7	张洪海，莫蓉，万能．应用等几何配点法求解电磁涡流场问题［J］．计算机辅助设计与图形学学报，2019，31（3）：496-503．
	ZHANG Honghai， MO Rong， WAN Neng ．Solving eddy current fields with isoeometric collocation method［J］．Journal of Computer-Aided Design & Computer Graphics，2019，31（3）：496-503．
8	ASHOUR M， VALIZADEH N， RABCZUK T ．Isogeometric analysis for a phase-field constrained optimization problem of morphological evolution of vesicles in electrical fields［J］．Computer Methods in Applied Mechanics and Engineering，2021，377：113669/1-27．
9	刘涛，张顺琦，刘庆运．热环境下压电功能梯度板的非线性等几何建模与主动控制研究［J］．振动与冲击，2023，42（8）：38-50．
	LIU Tao， ZHANG Shunqi， LIU Qingyun ．Nonlinear isogeometric modeling and active control of piezoelectric functionally graded plates in thermal environment［J］．Journal of Vibration and Shock，2023，42（8）：38-50．
10	高建伟，陈龙．基于等几何分析的股骨模型静力学分析［J］．计算机仿真，2015，32（5）：340-343．
	GAO Jianwei， CHEN Long ．Construction and static analysis of femur based on IGA［J］．Computer Simulation，2015，32（5）：340-343．
11	产启平，陈龙．基于等几何分析的非均质股骨近端模型的受力分析［J］．电子科技，2018，31（2）：15-19．
	CHAN Qiping， CHEN Long ．Stress analysis of heterogeneous femoral proximal model by using isogeometric analysis［J］．Electronic Science and Technology，2018，31（2）：15-19．
12	马书豪．心脏瓣膜流固耦合问题的等几何分析快速仿真方法［D］．杭州：杭州电子科技大学，2022．
13	COSTANTINI P， MANNI C， PELOSI F，et al ．Quasi-interpolation in isogeometric analysis based on generalized B-splines［J］．Computer Aided Geometric Design，2010，27（8）：656-668．
14	陈涛，莫蓉，万能．等几何分析中Dirichlet边界条件的配点施加方法［J］．机械工程学报，2012，48（5）：157-164．
	CHEN Tao， MO Rong， WAN Neng ．Imposing Dirichlet boundary conditions with point collocation method in isogeometric analysis［J］．Journal of Mechanical Engineering，2012，48（5）：157-164．
15	陈涛，莫蓉，万能，等．等几何分析中采用Nitsche法施加位移边界条件［J］．力学学报，2012，44（2）：371-381．
	CHEN Tao， MO Rong， WAN Neng，et al ．Imposing displacement boundary conditions with Nitsche’s method in isogeometric analysis［J］．Chinese Journal of Theoretical and Applied Mechanics，2012，44（2）：371-381．
16	HÖLLIG K， REIF U， WIPPER J ．Weighted extended B-spline approximation of Dirichlet problems［J］．SIAM Journal on Numerical Analysis，2001，39（2）：442-462．
17	SHAPIRO V ．Theory of R-functions and applications［R］．Ithaca：Cornell University，1988．
18	SHAPIRO V， TSUKANOV I ．Meshfree simulation of deforming domains［J］．Computer-Aided Design，1999，31（7）：459-471．
19	TSUKANOV I， SHAPIRO V， ZHANG S ．A meshfree method for incompressible fluid dynamics problems［J］．International Journal for Numerical Methods in Engineering，2003，58（1）：127-158．
20	SHAPIRO V， TSUKANOV I ．The architecture of SAGE - a meshfree system based on RFM［J］．Engineering with Computers，2002，18：295-311．
21	ZHANG W， ZHAO L， CAI S ．Shape optimization of Dirichlet boundaries based on weighted B-spline finite cell method and level-set function［J］．Computer Methods in Applied Mechanics and Engineering，2015，294：359-383．
22	VERSCHAEVE J C G ．A weighted extended B-spline solver for bending and buckling of stiffened plates［J］．Thin-Walled Structures，2016，107：580-596．
23	BURLA R K， KUMAR A V ．Implicit boundary method for analysis using uniform B‐spline basis and structured grid［J］．International Journal for Numerical Methods in Engineering，2008，76（13）：1993-2028．
24	KUMAR A V， PADMANABHAN S， BURLA R ．Implicit boundary method for finite element analysis using non‐conforming mesh or grid［J］．International Journal for Numerical Methods in Engineering，2008，74（9）：1421-1447．
25	COTTRELL J A， HUGHES T J R， REALI A ．Studies of refinement and continuity in isogeometric structural analysis［J］．Computer Methods in Applied Mechanics and Engineering，2007，196（41-44）：4160-4183．
26	SEDERBERG T W， ZHENG J， BAKENOV A，et al ．T-splines and T-NURCCs［J］．ACM Transactions on Graphics（TOG），2003，22（3）：477-484．
27	SEDERBERG T W， CARDON D L， FINNIGAN G T，et al ．T-spline simplification and local refinement［J］．ACM Transactions on Graphics（TOG），2004，23（3）：276-283．
28	BUFFA A， CHO D， SANGALLI G ．Linear independence of the T-spline blending functions associated with some particular T-meshes［J］．Computer Methods in Applied Mechanics and Engineering，2010，199（23/24）：1437-1445．
29	FORSEY D R， BARTELS R H ．Hierarchical B-spline refinement［C］∥Proceedings of the 15th Annual Conference on Computer Graphics and Interactive Techniques．New York：Association for Computing Machinery，1988：205-212．
30	FORSEY D R， BARTELS R H ．Surface fitting with hie-rarchical splines［J］．ACM Transactions on Graphics （TOG），1995，14（2）：134-161．
31	LI X， CHEN F， KANG H，et al ．A survey on the local refinable splines［J］．Science China Mathematics，2016，59（4）：617-644．
32	GARAU E M， VÁZQUEZ R ．Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines［J］．Applied Numerical Mathe-matics，2018，123：58-87．
33	KURU G， VERHOOSEL C V， VAN DER ZEE K G，et al ．Goal-adaptive isogeometric analysis with hierarchical splines［J］．Computer Methods in Applied Mechanics and Engineering，2014，270：270-292．
34	SCHMIDT M， NOËL L， DOBLE K，et al ．Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines［J］．Computational Mechanics，2023，71（6）：1179-1203．
35	KHOEI A R ．Extended finite element method：theory and applications［M］．Hoboken：John Wiley & Sons，2014．

算例	FEA单元数		IGA单元数
算例	粗网格	细网格	IGA单元数
算例1	4 180	18 424	4 332
算例2	7 678	15 946	7 727

算例	参数名称	符号及单位	粗网格FEA结果	细网格FEA结果	IGA结果	相对误差1/%	相对误差2/%
算例1	位移	s_x /m	2.01 × 10^-3	2.08 × 10^-3	2.07 × 10^-3	-3.37	-0.48
		s_y /m	-1.49 × 10^-4	-1.88 × 10^-4	-1.68 × 10^-4	-20.74	-10.64
		s_z /m	-1.54 × 10^-4	-1.87 × 10^-4	-1.68 × 10^-4	-17.65	-10.16
	应力	σ/Pa	6.27 × 10⁹	9.52 × 10⁹	9.45 × 10⁹	-34.14	-0.74
算例2	位移	s_x /m	3.52 × 10^-2	3.64 × 10^-2	3.66 × 10^-2	-3.30	0.55
		s_y /m	-2.21 × 10^-2	-2.30 × 10^-2	-2.31 × 10^-2	-3.91	0.43
		s_z /m	-2.21 × 10^-2	-2.29 × 10^-2	-2.31 × 10^-2	-3.49	0.87
	应力	σ/Pa	2.93 × 10¹⁰	4.51 × 10¹⁰	4.47 × 10¹⁰	-35.03	-0.89

算例	FEA单元数		IGA单元数
算例	粗网格	细网格	IGA单元数
算例3	2 391	13 589	2 064
算例4	3 032	14 459	2 995

算例	参数名称	符号及单位	粗网格FEA结果	细网格FEA结果	IGA结果	相对误差1/%	相对误差2/%
算例3	位移	s_x /m	1.81 × 10^-3	1.79 × 10^-3	1.74 × 10^-3	1.12	-2.79
		s_y /m	2.17 × 10^-4	1.56 × 10^-4	1.43 × 10^-4	39.10	-8.33
		s_z /m	-1.42 × 10^-4	-1.56 × 10^-4	-1.43 × 10^-4	-8.97	-8.33
	应力	σ/Pa	5.77 × 10⁹	8.46 × 10⁹	8.10 × 10⁹	-31.80	-4.26
算例4	位移	s_x /m	2.18 × 10^-2	2.36 × 10^-2	2.29 × 10^-2	-7.63	-2.97
		s_y /m	-1.35 × 10^-2	-1.46 × 10^-2	-1.42 × 10^-2	-7.53	-2.74
		s_z /m	-1.35 × 10^-2	-1.47 × 10^-2	-1.42 × 10^-2	-8.16	-3.40
	应力	σ/Pa	2.24 × 10¹⁰	4.06 × 10¹⁰	3.71 × 10¹⁰	-44.83	-8.62