Speaker
Description
Numerical solutions of, e.g. elasticity or heat conduction suffer from the deterioration of the overall accuracy when re-entrant corners, internal-respective boundary layers, or shock-like fronts are present [1]. A-posteriori error estimates offer a systematic way of retaining accuracy by localised mesh refinement. Following the seminal idea of Prager and Synge [2], such error estimates can be constructed based on the comparison of the discontinuous dual quantity, calculated from the primal approximation, and any H(div) conforming function, satisfying the so-called equilibrium conditions. When it comes to elasticity, the distinct symmetry of the stress tensor has to be considered (see e.g. [3]).
Besides the construction of adaptive solution procedures, the equilibrated dual quantity offers an increased accuracy, especially on boundaries with prescribed Dirichlet conditions. Within this contribution, we discuss an efficient implementation of flux and stress equilibration within the finite element framework FEniCSx. Based on structures with an auxetic sub-structure, the advantages of adaptive solution algorithms regarding efficiency and approximation quality are discussed.
References:
[1] Verfürth, R. (1994). A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math., 50, p. 67-83.
[2] Prager, W. \& Synge, J. L. (1947). Approximations in elasticity based on the concept of function space. Quart. Appl. Math., 5, p. 241–269.
[3] Bertrand, F. et al. (2021). Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity. Numer. Methods Partial Differ. Equ., 4, p. 2783-2802
