Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective-diffusive context

摘要

This paper presents the construction of novel stabilized finite elementmethods in the convective-diffusive context that exhibit correct-energybehavior. Classical stabilized formulations can create unwanted artificialenergy. Our contribution corrects this undesired property by employing theconcepts of dynamic as well as orthogonal small-scales within the variationalmultiscale framework (VMS). The desire for correct energy indicates that thelarge- and small-scales should be $H_0^1$-orthogonal. Using this orthogonalitythe VMS method can be converted into the streamline-upwind Petrov-Galerkin(SUPG) or the Galerkin/least-squares (GLS) method. Incorporating both large-and small-scales in the energy definition asks for dynamic behavior of thesmall-scales. Therefore, the large- and small-scales are treated as separateequations. Two consistent variational formulations which depict correct-energy behaviorare proposed: (i) the Galerkin/least-squares method with dynamic small-scales(GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods arepresented in combination with an energy-decaying generalized-$\alpha$time-integrator. Numerical verification shows that dissipation due to thesmall-scales in classical stabilized methods can become negative, both on alocal and global scale. The results show that without loss of accuracy thecorrect-energy behavior can be recovered by the proposed methods. Thecomputations employ NURBS-based isogeometric analysis for the spatialdiscretization.