The Sub-Region Variational Principles

  • Yu-Qiu Long
  • Zhi-Fei Long
  • Song Cen


This chapter focuses on the developments of the variational principles which are usually considered as the theoretical basis for the finite element method. In this chapter, we will discuss the sub-region variational principles which are the results by the combination of the variational principles and the concept of sub-region interpolation. Following the introduction, the sub-region variational principles for various structural forms, i.e., 3D elastic body, thin plate, thick plate and shallow shell, are presented respectively. Finally, a sub-region mixed energy partial derivative theorem is also given.


variational principle sub-region variational principle sub-region mixed energy partial derivative theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Washizu K (1968, 1975, 1982) Variational methods in elasticity and plasticity. 1st edn, 2nd edn, 3rd edn, Pergamon Press, OxfordzbMATHGoogle Scholar
  2. [2]
    Chien WZ (1980) Calculus of variations and finite elements (Vol. 1). Science Press, Beijing (in Chinese)Google Scholar
  3. [3]
    Hu HC (1984) Variational principles of theory of elasticity with applications. Science Press, BeijingGoogle Scholar
  4. [4]
    Finlayson BA (1972) The method of weighted residuals and variational principles. Academic Press, New YorkzbMATHGoogle Scholar
  5. [5]
    Pian THH (1964) Derivation of element stiffness matrices by assumed stress distributions. AIAA Journal, 2: 1333–1335CrossRefGoogle Scholar
  6. [6]
    Atluri SN, Gallagher RH, Zienkiewicz OC (1983) (eds). Hybrid and mixed finite element method. Wiley, ChichesterGoogle Scholar
  7. [7]
    Zienkiewicz OC (1983) The generalized finite element method—state of the art and future directions. Journal of Applied Mechanics (50th anniversary issue), 50: 1210–1217CrossRefMathSciNetzbMATHGoogle Scholar
  8. [8]
    Long YQ (1985) Advances in variational principles in China. In: Zhao C et al (eds). Proceedings of the Second International Conference on Computing in Civil Engineering. Elsevier Science Publishers, Hangzhou, pp1207–1215Google Scholar
  9. [9]
    Long YQ (1981) Piecewise generalized variational principles in elasticity. Shanghai Mechanics, 2(2): 1–9 (in Chinese)Google Scholar
  10. [10]
    Long YQ, Zhi BC, Yuan S (1982) Sub-region, sub-item and sub-layer generalized variational principles in elasticity. In: He GQ et al (eds). Proceedings of international conference on FEM. Science Press, Shanghai, pp607–609Google Scholar
  11. [11]
    Long YQ (1987) Sub-region generalized variational principles and sub-region mixed finite element method. In: Chien WZ (eds). The advances of applied mathematics and mechanics in China. China Academic Publishers, Beijing, 157–179Google Scholar
  12. [12]
    Long YQ (1987) Sub-region generalized variational principles in elastic thin plates. In: Yeh KY eds. Progress in Applied Mechanics. Martinus Nijhoff Publishers, Dordrecht, Netherlands, pp121–134Google Scholar
  13. [13]
    Long YQ (1983) Sub-region generalized variational principles for elastic thick plates. Applied Mathematics and Mechanics (English Edition) 4(2): 175–184CrossRefMathSciNetzbMATHGoogle Scholar
  14. [14]
    Long YQ, Long ZF, Xu Y (1996) Sub-region generalized variational principles in shallow shells and applications. In: Zhong WX, Cheng GD and Li XK (eds). The Advances in computational mechanics. International Academic Publishers, Beijing, pp69–77Google Scholar
  15. [15]
    Long YQ (1995) Sub-region mixed energy partial derivative theorem. In: Long YQ (ed). Proceedings of the Fourth National Conference on Structural Engineering. Quangzhou, pp188–194 (in Chinese)Google Scholar
  16. [16]
    Prager W (1968) Variational principles of clastic plates with relaxed continuity requirements. International Journal of Solids and Structures 4(9): 837–844CrossRefzbMATHGoogle Scholar

Copyright information

© Tsinghua University Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg 2009

Authors and Affiliations

  • Yu-Qiu Long
    • 1
  • Zhi-Fei Long
    • 2
  • Song Cen
    • 3
  1. 1.Department of Civil Engineering, School of Civil EngineeringTsinghua UniversityBeijingChina
  2. 2.School of Mechanics & Civil EngineeringChina University of Mining & TechnologyBeijingChina
  3. 3.Department of Engineering Mechanics, School of AerospaceTsinghua UniversityBeijingChina

Personalised recommendations