An efficient methodology for complementary finite element approximations in three-dimensional elasticity

We present a methodology for the definition of the matrices involved in stress based finite element approximations in three-dimensional problems. In order to handle the increased complexity of the 3D case, a rotation of the global reference frame is used, which greatly simplifies the expressions involved. This rotation is also applicable to the 2D case. The main focus of this work concerns the matrices used in the implementation of the hybrid equilibrium approach. However, a similar complementary methodology (displacement based) is also presented, allowing for an efficient definition of the matrices required for dual analysis, wherein bounds of the solution error are obtained from pairs of complementary solutions, one compatible and the other equilibrated.

Automated summary

We propose a methodology to define the matrices used in stress based finite element approximations for three-dimensional problems. By rotating the global reference frame, the increased complexity of the 3D case is greatly simplified. This rotation can also be applied to the 2D case. The focus of this work is on the matrices used in the implementation of the hybrid equilibrium approach, but we also present a complementary displacement-based methodology for efficiently defining the matrices required for dual analysis.