Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T20:07:20.491Z Has data issue: false hasContentIssue false
  • English
  • Français

Application of Craig-Bampton Reduction Technique and 2D Dynamic Infinite Element Modeling Approach to Membrane Vibration Problems

Published online by Cambridge University Press:  26 December 2018

D. S. Liu  and
Y. W. Chen
D. S. Liu
Affiliation:
Department of Mechanical Engineering and Advanced Institute of Manufacturing with High-Tech Innovations National Chung Cheng University Chiayi, Taiwan
Y. W. Chen*
Affiliation:
Department of Mechanical Engineering and Advanced Institute of Manufacturing with High-Tech Innovations National Chung Cheng University Chiayi, Taiwan
*
* Corresponding author (ywc0314@gmail.com)
Get access

Abstract

An approach is presented for solving membrane vibration problems using an integrated scheme consisting of the Craig-Bampton (CB) reduction technique and a 2D dynamic infinite element modeling (DIEM) method. In the proposed CB-DIEM scheme, the substructure domain is partitioned into multiple layers of geometrically-similar infinite elements (IEs) which use only the data of the boundary nodes. A convergence criterion based on the first invariant of the DIEM mass matrix is used to determine the optimal parameters of the CB-DIEM scheme, namely the proportionality ratio and number of layers in the DIEM partitioning process and the number of retained frequency modes in the CB reduction method. Furthermore, in implementing the CB method, the inversion of the global stiffness matrix is calculated using only the stiffness matrix of the first element layer. Having reduced the DIEM model, a coupled DIE-FE algorithm is employed to model the dynamic problems of the full structure, which removes the respective methods disadvantages while keeping their advantages. The validity and performance of the proposed CB-DIEM method are investigated by means of three illustrative problems.

Keywords

Craig-Bampton methodDynamic infinite element modelMembrane vibrationEigenvalue problem
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 4 , August 2019 , pp. 513 - 525
Copyright
© The Society of Theoretical and Applied Mechanics 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Xu, R., Li, D. X., Jiang, J. P. and Liu, W., “Nonlinear Vibration Analysis of Membrane SAR Antenna Structure Adopting a Vector Form Intrinsic Finite Element,” Journal of Mechanics, 31, pp. 269–277 (2015).CrossRefGoogle Scholar
Xu, R., Li, D. X., Jiang, J. P. and Liu, W., “Adaptive Fuzzy Vibration Control of Smart Structure with VFIFE Modeling,” Journal of Mechanics, 31, pp. 671–682 (2015).CrossRefGoogle Scholar
João, L. S., Carvalho, R. and Fangueiro, R., “A Study on the Durability Properties of Textile Membranes for Architectural Purposes,” Procedia Engineering, 155, pp. 230–237 (2016).CrossRefGoogle Scholar
Long, Z. et al., “Micro-Membrane Electrode Assembly Design to Precisely Measure the in Situ Activity of Oxygen Reduction Reaction Electrocatalysts for PEMFC,” Analytical Chemistry, 89, pp. 6309–6313 (2017).CrossRefGoogle ScholarPubMed
Houmat, A., “A Sector Fourier P-Element for Free Vibration Analysis of Sectorial Membranes,” Computers & Structures, 79, pp. 1147–1152 (2001).CrossRefGoogle Scholar
Houmat, A., “Free Vibration Analysis of Membranes Using the H-P Version of the Finite Element Method,” Journal of Sound and Vibration, 282, pp. 401–410 (2005).CrossRefGoogle Scholar
Houmat, A., “Free Vibration Analysis of Arbitrarily Shaped Membranes Using the Trigonometric P-Version of the Finite-Element Method,” Thin-Walled Structures, 44, pp. 943–951 (2006).CrossRefGoogle Scholar
Thatcher, R. W., “Singularities in the Solution of Laplace’s Equation in Two Dimensions,” Journal of the Institute of Mathematics and Its Applications, 16, pp. 303–319 (1975).CrossRefGoogle Scholar
Thatcher, R. W., “On the Finite Element Method for Unbounded Region,” SIAM Journal on Numerical Analysis, 15, pp. 466–477 (1978).CrossRefGoogle Scholar
Ying, L. A., “The Infinite Similar Element Method for Calculating Stress Intensity Factors,” Scientia Sinica, 21, pp. 19–43 (1978).Google Scholar
Han, H. D. and Ying, L. A., “An Iterative Method in the Finite Element,” Mathematica Numerica Sinica, 1, pp. 91–99 (1979).Google Scholar
Go, C. G. and Lin, Y. S., “Infinitely Small Element for the Problem of Stress Singularity,” Computers & Structures, 37, pp. 547–551 (1991).Google Scholar
Go, C. G. and Chen, G. C., “On the Use of an Infinitely Small Element for the Three-Dimensional Problem of Stress Singularity,” Computers & Structures, 45, pp. 25–30 (1992).CrossRefGoogle Scholar
Ying, L. A., “An Introduction to the Infinite Element Method,” Mathematics in Practice Theory, 2, pp. 69–78 (1992).Google Scholar
Leung, A. Y. T. and Su, R. K. L., “Mode I Crack Problems by Fractal Two Level Finite Element Methods,” Engineering Fracture Mechanics, 48, pp. 847–859 (1994).CrossRefGoogle Scholar
Leung, A. Y. T. and Su, R. K. L., “Mixed-Mode Two-Dimensional Crack Problem by Fractal Two Level Finite Element Method,” Engineering Fracture Mechanics, 51, pp. 889–895 (1995).CrossRefGoogle Scholar
Leung, A. Y. T. and Su, R. K. L., “Fractal Two-Level Finite Element Method for Cracked Kirchhoff’s Plates using DKT Elements,” Engineering Fracture Mechanics, 54, pp. 703–711 (1996).CrossRefGoogle Scholar
Reddy, R. M. and Rao, B. N., “Fractal Finite Element Method Based Shape Sensitivity Analysis of Mixed-mode Fracture,” Finite Elements in Analysis and Design, 44, pp. 875–888 (2008).CrossRefGoogle Scholar
Reddy, R. M. and Rao, B. N., “Continuum Shape Sensitivity Analysis of Mixed-Mode Fracture Using Fractal Finite Element Method,” Engineering Fracture Mechanics, 75, pp. 2860–2906 (2008).CrossRefGoogle Scholar
Rao, B. N. and Reddy, R. M., “Fractal Finite Element Method Based Shape Sensitivity Analysis of Multiple Crack System,” Engineering Fracture Mechanics, 76, pp. 1636–1657 (2009).CrossRefGoogle Scholar
Dong, C. Y., Lo, S. H. and Cheung, Y. K., “Numerical Solution for Elastic Half-Plane Inclusion Problems by Different Integral Equation Approaches,” Engineering Analysis with Boundary Elements, 28, pp. 123–140 (2004).CrossRefGoogle Scholar
Chen, J. T., Shen, W. C. and Chen, P. Y., “Analysis of Circular Torsion Bar with Circular Holes Using Null-Field Approach,” CMES: Computer Modeling in Engineering & Sciences, 12, pp. 109–119 (2006).Google Scholar
Chen, J. T., Hsiao, C. C. and Leu, S. Y., “Null-field Integral Equation Approach for Plate Problems with Circular Boundaries,” ASME Journal of Applied Mechanics, 73, pp. 679–693 (2006).CrossRefGoogle Scholar
Shiah, Y. C. and Chong, J. Y., “Boundary Element Analysis of Interior Thermoelastic Stresses in Three-Dimensional Generally Anisotropic Bodies,” Journal of Mechanics, 32, pp. 725–735 (2016).CrossRefGoogle Scholar
Shiah, Y. C. and Chang, L. D., “Boundary Element Calculation of Near-Boundary Solutions in 3D Generally Anisotropic Solids by the Self-Regularization Scheme,” Journal of Mechanics, 33, pp. 813–820 (2017).CrossRefGoogle Scholar
Liu, D. S. and Chiou, D. Y., “A Coupled IEM/FEM Approach for Solving the Elastic Problems with Multiple Cracks,” International Journal of Solids and Structures, 40, pp. 1973–1993 (2003).CrossRefGoogle Scholar
Liu, D. S. and Chiou, D. Y., “3D IEM Formulation with an IEM/FEM Coupling Scheme for Solving Elastostatic Problems,” Advances in Engineering Software, 34, pp. 309–320 (2003).CrossRefGoogle Scholar
Liu, D. S. and Chiou, D. Y., “Modeling of Inclusion with Interphases in Heterogeneous Material Using the Infinite Element Method,” Computational Materials Science, 31, pp. 405–420 (2004).CrossRefGoogle Scholar
Liu, D. S., Chen, C. Y. and Chiou, D. Y., “3D Modeling of a Composite Material Reinforced with Multiple Thickly Coated Particles Using the Infinite Element Method,” CMES: Computer Modeling in Engineering & Sciences, 9, pp. 179–191 (2005).Google Scholar
Liu, D. S., Zhuang, Z. W. and Chung, C. L., “Modeling of Moisture Diffusion in Heterogeneous Epoxy Resin Containing Multiple Randomly Distributed Particles Using Hybrid Moisture Element Method,” CMC: Computers Materials & Continua, 13, pp. 89–113 (2009).Google Scholar
Liu, D. S., Zhuang, Z. W., Lyu, S. R., Chung, C. L. and Lin, P. C., “Modeling of Moisture Diffusion in Permeable Fiber-Reinforced Polymer Composites Using Heterogeneous Hybrid Moisture Element Method,” CMC: Computers Materials & Continua, 26, pp. 111–136 (2011).Google Scholar
Liu, D. S., Fong, Z. H., Lin, I. H. and Zhuang, Z. W., “Modeling of Moisture Diffusion in Permeable Particle-Reinforced Epoxy Resins Using Three-Dimensional Heterogeneous Hybrid Moisture Element Method,” CMES: Computer Modeling in Engineering & Sciences, 93, pp. 441–468 (2013).Google Scholar
Liu, D. S., Tu, C. Y. and Chung, C. L., “Eigenvalue Analysis of Mems Components with Multi-Defect Using Infinite Element Method Algorithm,” CMC: Computers Materials & Continua, 28, pp. 97–120 (2012).Google Scholar
Liu, D. S., Tu, C. Y. and Chung, C. L., “Coupled PIEM/FEM Algorithm Based on Mindlin-Reissner Plate Theory for Bending Analysis of Plates with Through-Thickness Hole,” CMES: Computer Modeling in Engineering & Sciences, 92, pp. 573–594 (2013).Google Scholar
Liu, D. S. and Lin, Y. H., “Vibration Analysis of the Multiple-Hole Membrane by Using the Coupled DIEM-FE Scheme,” Journal of Mechanics, 32, pp. 163–173 (2016).CrossRefGoogle Scholar
Jain, J., Li, H., Cauley, S., Koh, C. K. and Balakrishnan, V., “Numerically Stable Algorithms for Inversion of Block Tridiagonal and Banded Matrices,” Purdue University TR ECE, pp. 357 (2007).Google Scholar