A numerical solution of the Navier-Stokes equations using the finite element technique

Reengineering of the aircraft structural life prediction process to fully exploit advances in very high performance digital computing is proposed. The proposed process utilizes an ultrahigh fidelity model of individual aircraft by tail number, a Digital Twin, to integrate computation of structural deflections and temperatures in response to flight conditions, with resulting local damage and material state evolution. A conceptual model of how the Digital Twin can be used for predicting the life of aircraft structure and assuring its structural integrity is presented. The technical challenges to developing and deploying a Digital Twin are discussed in detail.

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Reengineering Aircraft Structural Life Prediction Using a Digital Twin

Space-time finite element methods are developed for classical elastodynamics. The approach employs the discontinuous Galerkin method in time and incorporates stabilizing terms of least-squares type. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Optimal error estimates are predicted, and confirmed numerically, for arbitrary combinations of displacement and velocity interpolations. The procedures developed are easily generalized to structural dynamics and a wide class of second-order hyperbolic problems.

Space-time finite element methods for elastodynamics: formulations and error estimates

Tensor Analysis of Networks. By Gabriel Kron. Pp. xxix, 635. 70s. 1965. (Macdonald)

Finite Elements, An Introduction

SUMMARY In Part I of the this paper. topological properties of finite element models of functions defined on spaces of finite dimension were examined. In this part, a number of applications of the general theory are presented. These include the generation of finite element models in the time domain and certain problems in wave propagation, kinetic theory of gases, non-linear partial differential equations, non-linear continuum mechanics, and fluid dynamics.

A general theory of finite elements ii. applications

General tinite-element models of compre ol>e local approximations of the 'elocit~· field, the density. and the tempemture for compressible fluids and the ,'e1ocity. temperature. and prl'S'i,re ror incompressihlc tluids. Thcuries or local solenoidal approximl.tions and mixed tiniteclement models ror coml,n'ssihle film lIn' lIerh·ed. A numhl'r or cumpulallonal s..ht'mes are lIe"elopt'll ror Ihe uumerical solulion or hoth lransient and stl':IlI~'nOllllniformtlow prohlems illml\'illg illcol1lpressihlelIuids. Numericlll resulls ohtllinro rrom seH~rallest problel1lsllre gh'el1. It is Shllllil Ihat Ihe finile elel1lenl ml,thod ha" grelll llolenlial for U'ie ill tlo" prohll'I1I".and represenls II p,,"erful new 1001 for Ihe llllalysis of I'iwous flows. ..

Analysis of Flow of Viscous Fluids by the Finite-Element Method

This paper presents a general theory of finite elements. The concept of finite elements is cast in a general topological framework valid for spaces of finite dimension. It is shown that the idea of finite element models can be developed in higher-dimensional spaces, independent of specific co-ordinate systems, for any type of continuous abstract function defined on the space. Generalizations of the familiar Lagrange and Hermite interpolation functions are presented as well as a general statement of the notion of generalized variables and conjugate fields. It is also shown that admissible finite elements can be developed for non-Euclidean spaces of finite dimension. Topological properties of finite element models are examined in Part I of the paper. Part II is devoted to certain applications.

A general theory of finite elements. I. Topological considerations

The purpose of this note is to present brief derivations of the finite element equations describing a discrete model of compressible and incompressible Stokesian fluids. The ideas presented represent extensions and elaborations of a similar procedure described elsewhere.

Finite-Element Analogue of Navier-Stokes Equation

In this note, the application of the finite-element method to a class of problems in fluid dynamics is considered Such applications involve several pecularities not encountered in the usual struct

J. Tinsley Oden

Papers

A general theory of finite elements ii. applications

Analysis of Flow of Viscous Fluids by the Finite-Element Method

A general theory of finite elements. I. Topological considerations

Finite-Element Analogue of Navier-Stokes Equation

Finite-Element Applications in Fluid Dynamics