Thomas J. R. Hughes

Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations

Abstract A new finite element formulation for convection dominated flows is developed. The basis of the formulation is the streamline upwind concept, which provides an accurate multidimensional generalization of optimal one-dimensional upwind schemes. When implemented as a consistent Petrov-Galerkin weighted residual method, it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods. The accuracy of the streamline upwind/Petrov-Galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples. The formulation is extended to the incompressible Navier-Stokes equations. An efficient implicit pressure/explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step. The effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a Reynolds number of 100.

Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods

Abstract An approach is developed for deriving variational methods capable of representing multiscale phenomena. The ideas are first illustrated on the exterior problem for the Helmholtz equation. This leads to the well-known Dirichlet-to-Neumann formulation. Next, a class of subgrid scale models is developed and the relationships to ‘bubble function’ methods and stabilized methods are established. It is shown that both the latter methods are approximate subgrid scale models. The identification for stabilized methods leads to an analytical formula for τ, the ‘intrinsic time scale’, whose origins have been a mystery heretofore.

A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of

Abstract A new Petrov-Galerkin formulation of the Stokes problem is proposed. The new formulation possesses better stability properties than the classical Galerkin/variational method. An error analysis is performed for the case in which both the velocity and pressure are approximated by C 0 interpolations. Combinations of C 0 interpolations which are unstable according to the Babuska-Brezzi condition (e.g., equal-order interpolations) are shown to be stable and convergent within the present framework. Calculations exhibiting the good behavior of the methodology are presented.

A new finite element formulation for computational fluid dynamics: VIII. The galerkin/least-squares method for advective-diffusive equations☆

Galerkin/least-squares finite-element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of streamline-upwind Petrov-Galerkin methods, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented. Some numerical results for compressible Navier-Stokes flows are presented.

A new finite element formulation for computational fluid dynamics: II. Beyond SUPG

Abstract A discontinuity-capturing term is added to the streamline-upwind/Petrov-Galerkin weighting function for the scalar advection-diffusion equation. The additional term enhances the ability of the method to produce smooth yet crisp approximations to internal and boundary layers.

Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆

Abstract A transient, finite element formulation is given for incompressible viscous flows in an arbitrarily mixed Lagrangian-Eulerian description. The procedures developed are appropriate for modeling the fluid subdomain of many fluid-solid interaction, and free-surface problems.

The variational multiscale method—a paradigm for computational mechanics

Abstract We present a general treatment of the variational multiscale method in the context of an abstract Dirichlet problem. We show how the exact theory represents a paradigm for subgrid-scale models and a posteriori error estimation. We examine hierarchical p -methods and bubbles in order to understand and, ultimately, approximate the ‘fine-scale Greens function’ which appears in the theory. We review relationships between residual-free bubbles, element Greens functions and stabilized methods. These suggest the applicability of the methodology to physically interesting problems in fluid mechanics, acoustics and electromagnetics.

Mixed finite element methods—reduced and selective integration techniques: a unification of concepts

The equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established. This enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method. Applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles. These include thin beams and plates, and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics.

Finite element modeling of blood flow in arteries

Abstract A comprehensive finite element framework to enable the conduct of computational vascular research is described. The software system developed provides an integrated set of tools to solve clinically relevant blood flow problems and test hypotheses regarding hemodynamic (blood flow) factors in vascular adaptation and disease. The validity of the computational method was established by comparing the numerical results to an analytic solution for pulsatile flow as well as to published experimental flow data. The application of the finite element method to qualitatively and quantitatively assess the blood flow field in a number of clinically relevant models is described.

Computational Methods for Transient Analysis

Preface. 1 . An Overview of Semidiscretization and Time Integration Procedures (T. Belytschko). 2 . Analysis of Transient Algorithms with Particular Reference to Stability Behavior (T.J.R. Hughes). 3 . Partitioned Analysis of Coupled Systems (K.C. Park and C.A. Felippa). 4 . Boundary-Element Methods for Transient Response Analysis (T.L. Geers). 5 . Dynamic Relaxation (P. Underwood). 6 . Dispersion of Semidiscretized and Fully Discretized Systems (H.L. Schreyer). 7 . Silent Boundary Methods for Transient Analysis (M. Cohen and P.C. Jennings). 8 . Explicit Lagrangian Finite-Difference Methods (W. Hermann and L.D. Bertholf). 9 . Implicit Finite Element Methods (M. Geradin, M. Hogge and S. Idelsohn). 10 . Arbitrary Lagrangian-Eulerian Finite Element Methods (J. Donea). Indices.