In continuum mechanics, the strong enforcement of the symmetry of the stress tensor represents the pointwise fulfillment of the balance of angular momentum. It is well-known that in the context of equal-order polynomial approximations, the hybridizable discontinuous Galerkin (HDG) method suffers from suboptimal convergence when low-order discretizations of symmetric second-order tensors are involved. Exploiting Voigt notation to strongly enforce the symmetry of the stress tensor by storing solel...
In continuum mechanics, the strong enforcement of the symmetry of the stress tensor represents the pointwise fulfillment of the balance of angular momentum. It is well-known that in the context of equal-order polynomial approximations, the hybridizable discontinuous Galerkin (HDG) method suffers from suboptimal convergence when low-order discretizations of symmetric second-order tensors are involved. Exploiting Voigt notation to strongly enforce the symmetry of the stress tensor by storing solely its non-redundant components, the HDG-Voigt formulation of incompressible flow problems is able to retrieve optimal convergence order of the discretized strain rate tensor, even for low-order approximations. Moreover, a novel local postprocessing procedure is discussed to construct a superconvergent velocity field element-by-element [1]. This information is furtherly employed to compute an inexpensive error indicator and construct non-uniform high-order discretizations via a local degree adaptivity strategy [2]. Eventually, for problems of industrial interest, the HDG framework is also exploited to construct an efficient low-order approximation, the so-called face-centered finite volume (FCFV) method. FCFV is robust to mesh distortion and stretching, which are usually responsible for the degradation of classical finite volume solutions, and provides efficient optimally-convergent solutions of large-scale flow problems [3].
[1] M. Giacomini, A. Karkoulias, R. Sevilla, and A. Huerta. “A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor” J. Sci. Comput. 77:1679-1702 (2018).
[2] R. Sevilla, and A. Huerta. “HDG-NEFEM with degree adaptivity for Stokes flows” J. Sci. Comput. 77:1953-1980 (2018).
[3] R. Sevilla, M. Giacomini, and A. Huerta. “A face-centred finite volume method for second-order elliptic problems” Int. J. Numer. Methods Eng. 115(8):986-1014 (2018).