Variational multiscale methods to embed the macromechanical continuum formulation with fine-scale strain gradient theories

Variational multiscale methods to embed the macromechanical continuum formulation with ne-scale strain gradient theories

A variational basis is presented to link ne-scale theories of material behaviour with the classical, macromechanical continuum theory. The approach is based on the weak form of the linear momentum balance equations, and a separation of the weighting function and displacement elds into coarse and ne-scale components. Coarse and ne-scale weak forms are de ned. The latter is used to introduce a st...

Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods

We derive an explicit formula for the fine-scale Green’s function arising in variational multiscale analysis. The formula is expressed in terms of the classical Green’s function and a projector which defines the decomposition of the solution into coarse and fine scales. The theory is presented in an abstract operator format and subsequently specialized for the advectiondiffusion equation. It is...

The Fine-scale Field in the Multiscale Discontinuous Galerkin Method: Implications for Stabilized and Variational Multiscale Methods

In Hughes et al. [1], a new procedure was developed with the expressed intent of making the discontinuous Galerkin method (DG) more computationally tractable. Specifically, local, element-wise problems were solved in order to develop a transformation between the degrees-of-freedom of DG and those of a related continuous Galerkin method (CG), for which there were many fewer degrees-of-freedom. T...

Geometric, Variational Discretization of Continuum Theories

This study derives geometric, variational discretizations of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomor...

Variational Formulation of Problems and Variational Methods