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An hp DPG Method for Linear Elasticity with Symmetric Stresses

Thursday, March 31, 3:30PM – 5PM
ACES 6.304

Weifeng Qiu

In this research, we present two Discontinuous Petrov-Galerkin (DPG) finite element methods for linear elasticity. For the first method, we consider asymmetric test tensors for the constitutive equation and compute infinitessimal rotations, while in the second method we only use symmetric test tensors and therefore have fewer unknowns. We define optimal test functions which are shown to deliver the best approximation error if an optimal global test norm is used. To make the method practical, we show a localizable test norm is equivalent to the global optimal norm. The majority of this proof is the verification that the inf-sup condition holds for our DPG formulations using the localizable test space norm. From DPG theory, this proves our methods are quasi-optimal with constants independent of the mesh. We can then use results from approximation theory to show h and p convergence for both methods.

Since the quasi-optimal test space norm is localizable, we have implemented practical finite element codes that show h and p convergence of both methods at optimal rates. Additionally, the DPG framework provides an a priori error estimator determined by a local auxilliary variational problems. We use this estimator as the basis for various 'greedy' adaptive schemes. We test our adaptive algorithm using a manufactured smooth solution as well as a singular solution L-shape domain problem and observe adaptive h and hp convergence.

The principal contributions of this research are proving p convergence for the dual-mixed elasticity system, particularly without the need for a discrete exact sequence or commuting diagram, as well as a practical adaptive 2D elasticity code with a priori error estimation. We will present an overview of the theoretical DPG framework, the convergence proofs for both methods, and the numerical results for both a singular and smooth solution.

References

  • J. Bramwell, L. Demkowicz, and W. Qiu. Solution of Dual-Mixed Elasticity Equations using Arnold-Falk-Winther Element and Discontinuous Petrov-Galerkin Method, a Comparison. Technical Report 10-23, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, June 2010.

  • J. Bramwell, L. Demkowicz, J. Gopalakrishnan, and W. Qiu. An hp DPG Method for Linear Elasticity with Symmetric Stresses, in preparation.

Hosted by Leszek Demkowicz