A New Class of Numerical Methods with Embedded Computational Intelligence for Coupled Field Problems and Discrepancy Modeling

Abstract:

This talk presents a systematic procedure for building computational intelligence in the modeling and simulation methods for multifield problems in science and engineering. A multiscale decomposition of the unknown fields into coarse and fine scales leads to two coupled system of equations that describe physics at the global and the local levels. Employing discontinuous functions in this framework results in Variational Multiscale Discontinuous Galerkin (VMDG) class of methods for mathematically non-smooth problems with weak and strong discontinuities and internal constraints. The fine-scale equations in the VMDG method are endowed with adjoint-based error estimation feature which concurrently quantifies the unresolved part of physics as a function of the residual of the Euler-Lagrange equations. Fine-scale models for this missing part of physics are derived and variationally embedded in the coarse-scale equations. This modelling step builds computational intelligence in the numerical method that adapts locally in space and time to yield solutions that possess enhanced stability and accuracy. Mathematical structure of the VMDG formalism is exploited for discrepancy modeling wherein physics-based models are augmented via variationally derived loss functions that penalize the difference between the computed quantities and the measured sensor data. The structure of the derived kernel functions provides ideas for the integration of machine learning approaches in the modeling methods.

The VMDG method is extended to material and geometric nonlinearity and applied to layered additive manufacturing. A thermodynamically consistent constitutive model that emanates from mixture theory is employed for 3D printing with materials that undergo thermo-chemo-mechanical curing. One-sided VMDG formalism leads to Immersed Boundary Method for weak enforcement of Dirichlet boundary conditions on immersed surfaces that traverse through the computational grids. These ideas are further explored in the context of fluid mechanics to show the generality of the proposed methods and their range of application to problems of contemporary interest in science and engineering.

Bio:

Arif Masud is John and Eileen Blumenschein Professor of Mechanics and Computations in the Department of Civil and Environmental Engineering, and the Department of Aerospace Engineering, at the University of Illinois at Urbana-Champaign. He also holds joint appointment as Professor of Biomedical and Translational Sciences in the Carle-Illinois College of Medicine. Dr. Masud has made fundamental and pioneering contributions to the development of Variational Multiscale (VMS) Methods for fluid and solid mechanics. He is the President of the Society of Engineering Science (SES) for 2023 and is the Vice-President of the Engineering Mechanics Institute (EMI) of ASCE. He has served as the Associate Editor (AE) of the ASCE Journal of Engineering Mechanics, and AE of the ASME Journal of Applied Mechanics. Dr. Masud was Chair of the Computational Mechanics Committee of ASCE, and Chair of the Fluid Mechanics Committee of ASME. He is an Associate Fellow of AIAA, and Fellow of USACM, IACM, AAM, ASME, EMI, and SES. Prof Masud was awarded the 2019 G.I. Taylor Medal by SES, and the 2022 Ted Belytschko Applied Mechanics Award by AMD-ASME for fundamental contributions to the Theory of Stabilized and Variational Multiscale Methods in Computational Mechanics.