Computational and Applied Mathematics

, Volume 36, Issue 2, pp 859–875 | Cite as

Incompatible modes with Cartesian coordinates and application in quadrilateral finite element formulation

  • Yang Xia
  • Guojun Zheng
  • Ping Hu


Incompatible modes with parametric coordinates are widely used in finite element method to develop low-order elements with high accuracy. But it leads to unstable results and must be used along with the constant Jacobian matrix technique to assure convergence. The incompatible modes with Cartesian coordinates are proposed as an alternative. The advantage is that the present method can improve the accuracy, assure the convergence of the elements without additional correction technique and greatly reduce the amount of calculation. With this method, a new quadrilateral element IMQ6 is formulated within the quasi-conforming scheme. Both theoretical and numerical analyses are conducted and it is proved that the present incompatible modes improve the element’s performance in both precision and efficiency.


Incompatible modes Cartesian coordinate Quasi-conforming Convergence Patch test Quadrilateral element Finite element 

Mathematics Subject Classification




This work was funded by the Educational Department of Liaoning Province (No. L2014031), the Project of the National Natural Science Foundation of China (No. 11272075), and the Fundamental Research Funds for the Central Universities (No. DUT13RC(3)50, DUTI5RC(5)44). These supports are gratefully acknowledged. We would like to thank Dr. John Paul Villforth for the editorial help. Many thanks are due to the referees for their valuable comments.


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© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015

Authors and Affiliations

  1. 1.School of Automotive Engineering, State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianPeople’s Republic of China

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