J.N. Reddy
Advanced Computational Mechanics Laboratory
Department of Mechanical Engineering
Texas A & M University, College Station, TX 77843-3123
e-mail: jnreddy [at] tamu [dot] edu; http://mechanics.tamu.edu
ABSTRACT
Finite element formulations based on the weak-form Galerkin method and symmetric bilinear forms in structural mechanics resulted in enormous success. However, extension of these concepts to fluid mechanics and other areas of mechanics where the differential operators are either non-self adjoint or non-linear have met with mixed success. Numerical schemes such as modified weight functions, modified quadrature rule, optimal upwinding and so on have been presented in the literature to alleviate problems encountered with weak-form Galerkin procedures in solving non-self adjoint and nonlinear problems outside of solid mechanics.
The lecture presents the formulation and application of the least-squares finite element formulations to the numerical solution of the Navier-Stokes equations governing two-dimensional flows. Finite element models of the vorticity-based or velocity gradients-based Navier-Stokes equations are developed using the least-squares technique. The use of least-squares principles leads to a symmetric and positive-definite system of algebraic equations that allow the use of iterative methods for the solution of resulting algebraic equations. Exponentially fast decay of the least-squares functional, which is constructed using the L2 norms of the residuals in the governing equations, is verified for increasing order of the spectral approximations of the field variables. Numerical results are presented for the flow over a backward-facing step, lid-driven cavity flow, and flow around cylinders to demonstrate the predictive capability and robustness of the least-squares based finite element models low to moderate Reynolds numbers.
A brief discussion of finite volume method will be presented and compared with the finite element method to bring out the salient features of the two techniques.
BIOGRAPHICAL SKETCH OF J. N. REDDY
Dr. Reddy, the Oscar S Wyatt Endowed Chair Professor, Distinguished Professor, and Regents Professor of Mechanical Engineering at Texas A&M University, is a ISI highly-cited researcher, author of 21 textbooks and over 650 journal papers, and a leader in the applied and computational mechanics field for more than 40 years.
Professor Reddy is known worldwide for his significant contributions to the field of applied mechanics through the authorship of widely used textbooks on the linear and nonlinear finite element analysis, variational methods, composite materials and structures, and continuum mechanics. His pioneering works on the development of shear deformation theories (that bear his name in the literature as the Reddy third-order plate theory and the Reddy layerwise theory) have had a major impact and have led to new research developments and applications. Some of his ideas on shear deformation theories and penalty finite element models of fluid flows have been implemented into commercial finite element computer programs like ABAQUS, NISA, and HyperXtrude.
His earlier research focused primarily on mathematics of finite elements, variational principles of mechanics, shear deformation and layerwise theories of laminated composite plates and shells, analysis of bimodular materials, modeling of geological and geophysical phenomena, penalty finite elements for flows of viscous incompressible fluids, least-squares finite element models of fluid flows and solid continua. In recent years, Reddy’s research deals with 7- and 12-parameter shell theories, nonlocal and non-classical continuum mechanics problems, and problems involving couple stresses, surface stress effects, discrete fracture and flow, micropolar cohesive damage, and continuum plasticity of metals from considerations of non-equilibrium thermodynamics.
Dr. Reddy earned many honors and awards. Recent honors and awards include: 2016 Prager Medal, Society of Engineering Science, 2016 Thomson Reuters IP and Science’s Web of Science Highly Cited Researchers – Most Influential Minds, and the 2016 ASME Medal from the American Society of Mechanical Engineers, the 2017 John von Neumann Medal from the US Association of Computational Mechanics, and the 2018 von Karman Medal from the American Society of Civil Engineers. He is a member US National Academy of Engineering and foreign fellow of Indian National Academy of Engineering, the Canadian Academy of Engineering, and the Brazilian National Academy of Engineering.