Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Geometric Multigrid Methods for Flow Problems in Highly Heterogeneous Porous Media

In this dissertation, we develop geometric multigrid methods for the finite element approximation of flow problems (e:g:, Stokes, Darcy and Brinkman models) in highly heterogeneous porous media. Our method is based on H^(div)-conforming discontinuous Galerkin methods and the Arnold-Falk-Winther (AFW) type smoothers. The main advantage of using H^(div)-conforming elements is that the discrete velocity field will be globally divergence-free for incompressible fluid flows. Besides, the smoothers used are of overlapping domain decomposition Schwarz type and employ a local Helmholtz decomposition. Our flow solvers are the combination of multigrid preconditioners and classical iterative solvers. The proposed preconditioners act on the combined velocity and pressure space and thus does not need a Schur complement approximation. There are two main ingredients of our preconditioner: first, the AFW type smoothers can capture a meaningful basis on local divergence free subspace associated with each overlapping patch; second, the grid operator does not increase the divergence from the coarse divergence free subspace to the fine one as the divergence free spaces are nested. We present the convergence analysis for the Stokes' equations and Brinkman's equations ( with constant permeability field ), as well as extensive numerical experiments. Some of the numerical experiments are given to support the theoretical results. Even though we do not have analysis work for the highly heterogeneous and highly porous media cases, numerical evidence exhibits strong robustness, efficiency and unification of our algorithm

Multilevel Schwarz Methods for Incompressible Flow Problems

In this thesis, we address coupled incompressible flow problems with respect to their efficient numerical solutions. These problems are modeled by the Oseen equations, the Navier-Stokes equations and the Brinkman equations. For numerical approximations of these equations, we discretize these systems by Hdiv-conforming discontinuous Galerkin method which globally satisfy the divergence free velocity constraint on discrete level. The algebraic systems arising from discretizations are large in size and have poor spectral properties which makes it challenging to solve these linear systems efficiently. For efficient solution of these algebraic system, we develop our solvers based on classical iterative solvers preconditioned with multigrid preconditioners employing overlapping Schwarz smoothers of multiplicative type. Multigrid methods are well known for their robustness in context of self-adjoint problems. We present an overview of the convergence analysis of multigrid method for symmetric problems. However, we extend this method to non self-adjoint problems, like the Oseen equations, by incorporating the downwind ordering schemes of Bey and Hackbusch and we show the robustness of this method by empirical results. Furthermore, we extend this approach to non-linear problems, like the Navier-Stokes and the non-linear Brinkman equations, by using a Picard iteration scheme for linearization. We investigate extensively by performing numerical experiment for various examples of incompressible flow problems and show by empirical results that the multigrid method is efficient and robust with respect to the mesh size, the Reynolds number and the polynomial degree. We also observe from our numerical results that in case of highly heterogeneous media, multigrid method is robust with respect to a high contrast in permeability

Parallel finite element modeling of the hydrodynamics in agitated tanks

Mixing in the transition flow regime -- Technology to mix in transition flow regime -- Methods to characterize mixing hydrodynamics -- Challenges to numerically model transition flow regime in agited tanks -- Transition flow regime in agitated tanks -- Parallel computing -- Numerical modeling of the agitators motion -- Overall methodological approach -- Computational resources -- Program development strategy -- Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers -- Parallel numerical model -- Parallel implementation -- Three-dimensional benchmark cases -- A parallel finite element sliding mesh technique for the Navier-Stokes equation -- Numerical method -- Parallel implementation -- Numerical examples -- Parallel performance -- Finite element modeling of the laminar and transition flow of the Superblend dual shaft coaxial mixer on parallel computers -- Superblend coaxial mixer configuration -- Numerical model -- Hydrodynamics in Superblend coaxial mixer -- Mixing -- Mixing efficiency -- Parallel finite element solver -- Parallel sliding mesh technique -- Simulation of the hydrodynamics of a stirred tank in the transition regime -- Recommendations for future research -- Parallel algorithms -- Simulation of agited and the transition flow regime