## NUMERICAL APPROXIMATION OF ONE-FLUID PROBLEMS

*Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre*

### in Mathematical Methods for the Magnetohydrodynamics of Liquid Metals

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.003.0003
- Subject:
- Mathematics, Mathematical Physics

This chapter addresses the discretization of the one-fluid magnetohydrodynamics equations. At the numerical level, the difficulties due to nonlinearities (coming both from the hydrodynamics equations ... More

## Topics Concerning Finite Elements

*Peter Monk*

### in Finite Element Methods for Maxwell's Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0008
- Subject:
- Mathematics, Numerical Analysis

Besides the first family of edge elements discussed in Chapter 5, there is a second family of elements of full polynomial degree at each order, also due to Nedelec. These have the potential advantage ... More

## An Introduction to Nonlinear Finite Element Analysis

*J. N. Reddy*

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This book presents the theory and computer implementation of the finite element method as applied to nonlinear problems of heat transfer and similar field problems, fluid mechanics (flows of ... More

## The Finite Element Method: A Review

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter reviews the basic steps involved in the development of the finite element method as applied to one- and two-dimensional problems described by typical second-order differential equations. ... More

## Flows of Viscous Incompressible Fluids

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Fluid mechanics is concerned with the motion of gases and liquids and their interaction with the surroundings. A fluid can either be inviscid (the viscosity is assumed to be zero) or incompressible ... More

## Application of Weighted Residual Methods to Dynamic Economic Models

*Ellen R. McGrattan*

### in Computational Methods for the Study of Dynamic Economies

- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780199248278
- eISBN:
- 9780191596605
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199248273.003.0006
- Subject:
- Economics and Finance, Macro- and Monetary Economics

Many problems in economics require the solution to a functional equation as an intermediate step, and typically, decision functions are sought that satisfy a set of Euler conditions or a value ... More

## INTRODUCTION

*George Em Karniadakis and Spencer J. Sherwin*

### in Spectral/hp Element Methods for Computational Fluid Dynamics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528692
- eISBN:
- 9780191713491
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528692.003.0001
- Subject:
- Mathematics, Numerical Analysis

This chapter presents reduced models of the compressible and incompressible Navier-Stokes equations which are used in the various discretization concepts discussed in the rest of the book. The ... More

## Mixed Multiscale Finite Element Methods on Adaptive Unstructured Grids Using Limited Global Information

*J. E. Aarnes, Y. Efendiev, T.Y. Hou, and L. Jiang*

### in Multiscale Methods: Bridging the Scales in Science and Engineering

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199233854
- eISBN:
- 9780191715532
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233854.003.0001
- Subject:
- Mathematics, Applied Mathematics

We study mixed multiscale finite element methods (MsFEM) on unstructured coarse grids. Unstructured grids are often used when highly heterogeneous reservoirs are discretized via irregular anisotropic ... More

## Nonlinear Bending of Elastic Plates and Shells

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Simple two-dimensional plate and shell theories can be developed to study the deformation and stresses in plate structures undergoing small strains, small to moderate rotations, and large ... More

## Coarse-Grained Molecular Dynamics: Concurrent Multiscale Simulation at Finite Temperature Protect 1

*Robert E. Rudd*

### in Multiscale Methods: Bridging the Scales in Science and Engineering

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199233854
- eISBN:
- 9780191715532
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233854.003.0005
- Subject:
- Mathematics, Applied Mathematics

Coarse-grained molecular dynamics (CGMD) is a computer modeling technique that couples conventional molecular dynamics (MD) in some spatial regions of the simulation to a more coarse-grained ... More

## Finite Element Formulations of Solid Continua

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In the linear description of the motion of solid bodies, the displacements and strains are assumed to be very small and the material is linearly elastic. In addition, the equilibrium equations are ... More

## Some numerical methods for fluids

*Bijan Mohammadi and Olivier Pironneau*

### in Applied Shape Optimization for Fluids

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199546909
- eISBN:
- 9780191720482
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199546909.003.0004
- Subject:
- Mathematics, Mathematical Physics

This chapter deals with the numerical methods that will be used for the flow solvers. As in most commercial and industrial packages, unstructured meshes with automatic mesh generation and adaptation ... More

## Heat Transfer and Other Field Problems in One Dimension

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers the use of the finite element method in analysing heat transfer and other field problems in one dimension. A model nonlinear differential equation governing one-dimensional ... More

## Nonlinear Analysis of Time-Dependent Problems

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this chapter, finite element models are used to solve time-dependent problems with nonlinearities and some standard time approximation schemes are described. All classes of nonlinear problems ... More

## Material Nonlinearities and Coupled Problems

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Nonlinearities arise from two independent sources: nonlinearity due to changes in the geometry or position of the material particles of a continuum, which is called the geometric nonlinearity, and ... More

## Nonlinear Bending of Straight Beams

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers the application of the finite element method to the analysis of a slightly more complicated one-dimensional nonlinear problem: the bending of straight beams. When the applied ... More

## Introduction

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

One of the most important thing engineers and scientists do is to model natural phenomena. They develop conceptual and mathematical models to simulate physical events, whether they are aerospace, ... More

## Finite Element Formulations of Solid Continua

*J. N. Reddy*

### in An Introduction to Nonlinear Finite Element Analysis, 2nd Edn: with applications to heat transfer, fluid mechanics, and solid mechanics

- Published in print:
- 2014
- Published Online:
- June 2015
- ISBN:
- 9780199641758
- eISBN:
- 9780191789557
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641758.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter discusses the nonlinear continuum finite element formulation of elastic solids and presents updated and total Lagrangian finite element formulations using the principle of virtual ... More

## Heat Transfer and Other Field Problems in Two Dimensions

*J. N. REDDY*

### in An Introduction to Nonlinear Finite Element Analysis

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers the use of the finite element method in analysing problems in which the coefficients of the differential equation are possibly functions of the dependent variable and its ... More

## The Finite-Element Method

*Heiner Igel*

### in Computational Seismology: A Practical Introduction

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198717409
- eISBN:
- 9780191835070
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198717409.003.0006
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

The finite-element method is introduced first for the static elastic problem. The weak form of the wave equation and the Galerkin principle are discussed with linear basis functions. The solution for ... More