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

TL;DR

This paper introduces novel stabilized finite element methods that ensure correct energy behavior in convective-diffusive problems by employing dynamic and orthogonal small-scales within the VMS framework, improving upon classical methods.

Contribution

It develops two new stabilized formulations, GLSD and DO, that incorporate dynamic and orthogonal small-scales to achieve correct-energy evolution in finite element analysis.

Findings

Classical stabilized methods can produce negative energy dissipation.

The proposed methods recover correct-energy behavior without losing accuracy.

Numerical results confirm the effectiveness of the new formulations with isogeometric analysis.

Abstract

This paper presents the construction of novel stabilized finite element methods in the convective-diffusive context that exhibit correct-energy behavior. Classical stabilized formulations can create unwanted artificial energy. Our contribution corrects this undesired property by employing the concepts of dynamic as well as orthogonal small-scales within the variational multiscale framework (VMS). The desire for correct energy indicates that the large- and small-scales should be $H_0^1$-orthogonal. Using this orthogonality the 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 the small-scales. Therefore, the large- and small-scales are treated as separate equations. Two consistent variational formulations which depict correct-energy behavior are proposed: (i) the Galerkin/least-squares method with dynamic small-scales (GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods are presented in combination with an energy-decaying generalized-$\alpha$ time-integrator. Numerical verification shows that dissipation due to the small-scales in classical stabilized methods can become negative, both on a local and global scale. The results show that without loss of accuracy the correct-energy behavior can be recovered by the proposed methods. The computations employ NURBS-based isogeometric analysis for the spatial discretization.