r3.06.10 Quadrangular element at one integration point, stabilized by the « Assumed Strain » method

Summary:

Despite all the potential of the linear isoparametric quadrilateral element for finite element calculation, the application of a classical bilinear formulation to define its field of displacement resulted in poor results. Although the sub-integration of the element makes it possible to improve its performance, it nevertheless reveals parasitic modes that make the calculations unstable. This document summarizes the main steps of a method for stabilizing calculations called « assumed strainmethod » and explains how it was implemented in the*Code_Aster* calculation code. Numerous results based on the stabilized element are compared and commented on in order to conclude on the performances of the method.

• 1. Defaulting the 2D element QUAD4:
  • 1.1. Test case no. 1
  • 1.2. Test case no. 2
• 2. Under integration of QUAD4
  • 2.1. Formulation
  • 2.2. Calculation failed in Code_Aster: parasitic modes
  • 2.3. Graphical interpretation of the problem related to underintegration
• 3. Stabilization of QUAD4 at a Gauss point
  • 3.1. Variational principle of the problem
  • 3.2. Enrichment of the discretized gradient operator
  • 3.3. Interpolation of the deformation field:
• 4. Integrating the element into Code_Aster
• 5. Highlighting the contribution of the QUAS4 element
  • 5.1. Test case no. 1 (SSLP106)
  • 5.2. Test case #2 (SSNP123)
  • 5.3. Comments
• 6. Conclusions
• 7. Bibliography
• 8. Description of document versions