Many problems in engineering and sciences involve multiple simultaneous physical phenomena, each governed by their own balance laws and constitutive models, and therefore associated material, spatial and temporal scales. In some problems hierarchical PDEs are operational concurrently over the spatial domains, while in others cases different PDEs govern adjoining subdomains. Open issues in the modeling of this class of problems relate to lack of variationally consistent coupling techniques for multiple PDEs so that the resulting numerical methods are stable and convergent. This talk presents new developments in multi-model coupling methods by embedding Discontinuous Galerkin (DG) ideas in the Continuous Galerkin (CG) framework within the context of Stabilized Methods. The first part of the talk discusses ideas in the context of fluid mechanics problems and presents a heterogeneous modeling method with embedded interfaces for porous media flows where the global physics is governed by Darcy equation while the local model is Darcy-Stokes equation. A multi-model interface operator is developed for the common inter-domain boundary along which the global and local models commute. The interface operator finds roots in local application of DG ideas along the common interface and this approach relies on applying concepts from the VMS method locally that also helps accommodate jump conditions between subdomains governed by different mathematical models. The second part of the talk presents this multi-model interface framework in the context of solid mechanics problems while accommodating a range of interface kinematics. Emphasis is placed throughout on unifying mathematical framework that enables the representation of various continuity conditions in a variationally consistent manner. Method is extended to large deformations and is applied to model engineered materials with anisotropic properties. Representative model problems are presented to show how these building blocks result in a powerful framework for application to additive manufacturing.

Arif Masud received Ph.D. in Computational Mechanics from Stanford University in April 1993. He is Professor of Engineering Mechanics in the Department of Civil and Environmental Engineering, and the Department of Aerospace Engineering at the University of Illinois at Urbana-Champaign. Dr. Masud has made fundamental and pioneering contributions to the development of Stabilized and Multiscale Methods for fluid and solid mechanics, residual-based Turbulence models, non-Newtonian fluids, and chemically reacting porous media flows. He has authored or co-authored over 130 refereed journal and conference papers, and is co-editor of the book Finite Element Method: 1970's and Beyond. He is past Chair of the Fluid Mechanics Committee of ASME, and the Computational Mechanics Committee of ASCE. He has served as an Associate Editor of the ASCE Journal of Engineering Mechanics, ASME Journal of Applied Mechanics, and ASCE Journal of Nanomecahnics and Micromechanics. Dr. Masud was the General Conference Chair for the ASME Applied Mechanics and Materials Conference (McMAT) 2011, and he is elected General Conference Chair for the Finite Element in Flows Conference (FEF-2019) in April 2019, and for the US Association of Computational Mechanics Conference (USACM-2021) in July 2021. Prof. Masud is Associate Fellow of AIAA, and Fellow of USACM, IACM, ASME, EMI, and AAM. In 2010 he was named Robert H. Dodds Faculty Scholar in the Department of Civil and Environmental Engineering at the University of Illinois.