Finite Elements E-Book

A. Baker

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 327

Veröffentlichungsjahr: 2012

Beschreibung

Approaches computational engineering sciences from the perspective of engineering applications Uniting theory with hands-on computer practice, this book gives readers a firm appreciation of the error mechanisms and control that underlie discrete approximation implementations in the engineering sciences. Key features: * Illustrative examples include heat conduction, structural mechanics, mechanical vibrations, heat transfer with convection and radiation, fluid mechanics and heat and mass transport * Takes a cross-discipline continuum mechanics viewpoint * Includes Matlab toolbox and .m data files on a companion website, immediately enabling hands-on computing in all covered disciplines * Website also features eight topical lectures from the author's own academic courses It provides a holistic view of the topic from covering the different engineering problems that can be solved using finite element to how each particular method can be implemented on a computer. Computational aspects of the method are provided on a companion website facilitating engineering implementation in an easy way.

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

Preface

About the Author

Notations

Chapter 1: The Computational Engineering Sciences: An Introduction

1.1 Engineering Simulation

1.2 A Problem-Solving Environment

1.3 Weak Formulation Essence

1.4 Decisions on Forming WSN

1.5 Discrete WSh Implementations

1.6 Chapter Summary

References

Chapter 2: Problem Statements: In the Engineering Sciences

2.1 Engineering Simulation

2.2 Continuum Mechanics Viewpoint

2.3 Continuum Conservation Principle Forms

2.4 Constitutive Closure for Conservation Principle PDEs

2.5 Engineering Science Continuum Mechanics

References

Chapter 3: Some Introductory Material: PDEs, BCs, Solutions, Discrete Concepts

3.1 Example Linear Heat Conduction Solutions

3.2 Multidimensional PDEs, Separation of Variables

3.3 Mathematical Foundation Essence for GWSN

3.4 A Legacy FD Construction

3.5 An FD Approximate Solution

3.6 Lagrange Interpolation Polynomials

3.7 Chapter Summary

Exercises

References

Chapter 4: Heat Conduction: An FE Weak Statement Tutorial

4.1 A Steady Heat Conduction Example

4.2 Weak Form Approximation, Error Extremization

4.3 GWSN Discrete Implementation, FE Trial Space Basis

4.4 Finite Element Matrix Statements

4.5 Assembly of {WS}e to form Algebraic GWSh

4.6 Solution Accuracy, Error Distribution

4.7 Convergence, Boundary Heat Flux

4.8 Chapter Summary

Exercises

Reference

Chapter 5: Steady Heat Transfer, n = 1: GWSh Implemented with {Nk(ζα)}, 1 ≤ k ≤ 3, Accuracy, Convergence, Nonlinearity, Templates

5.1 Introduction

5.2 Steady Heat Transfer, n = 1

5.3 FE k = 1 Trial Space Basis Matrix Library

5.4 Object-Oriented GWSh Programming

5.5 Higher Completeness Degree Trial Space Bases

5.6 Global Theory, Asymptotic Error Estimate

5.7 Nonsmooth Data, Theory Generalization

5.8 Temperature-Dependent Conductivity, Nonlinearity

5.9 Static Condensation, p-Elements

5.10 Chapter Summary

Exercises

Computer Labs

References

Chapter 6: Engineering Sciences, n = 1: GWSh {Nk(ζα)} Implementations in the Computational Engineering Sciences

6.1 Introduction

6.2 The Euler–Bernoulli Beam Equation

6.3 Euler–Bernoulli Beam Theory GWSh Reformulation

6.4 Timoshenko Beam Theory

6.5 Mechanical Vibrations of a Beam

6.6 Fluid Mechanics, Potential Flow

6.7 Electromagnetic Plane Wave Propagation

6.8 Convection–Radiation Finned Cylinder Heat Transfer

6.9 Chapter Summary

Exercises

Computer Labs

References

Chapter 7: Steady Heat Transfer, n > 1: n = 2, 3 GWSh for DE + BCs, FE Bases, Convergence, Error Mechanisms

7.1 Introduction

7.2 Multidimensional FE Bases and DOF

7.3 Multidimensional FE Operations for {Nk(ζα)}

7.4 The NCk = 1,2 Basis FE Matrix Library

7.5 NC Basis {WS}e Template, Accuracy, Convergence

7.6 The Tensor Product Basis Element Family

7.7 Gauss Numerical Quadrature, k = 1 TP Basis Library

7.8 Convection–Radiation BC GWSh Implementation

7.9 Linear Basis GWSh Template Unification

7.10 Accuracy, Convergence Revisited

7.11 Chapter Summary

Exercises

Computer Labs

References

Chapter 8: Finite Differences of Opinion: FE GWSh Connections to FD, FV Methods

8.1 The FD–FE Correlation

8.2 The FV–FE Correlation

8.3 Chapter Summary

Exercises

Chapter 9: Convection–Diffusion, n = 1: Unsteady Energy Transport, Accuracy/Convergence, Dispersion Error, Numerical Diffusion

9.1 Introduction

9.2 The Galerkin Weak Statement

9.3 GWSh Completion for Time Dependence

9.4 GWSh + θTS Algorithm Templates

9.5 GWSh + θTS Algorithm Asymptotic Error Estimates

9.6 Performance Verification Test Cases

9.7 Dispersive Error Characterization

9.8 A Modified Galerkin Weak Statement

9.9 Verification Problem Statements Revisited

9.10 Unsteady Heat Conduction

9.11 Chapter Summary

Exercises

Computer Labs

References

Chapter 10: Convection–Diffusion, n >1: n = 2, 3 GWSh/mGWSh + θTS, Stability, Error Characterization, Linear Algebra

10.1 The Problem Statement

10.2 GWSh +θ TS Formulation Reprise

10.3 Matrix Library Additions, Templates

10.4 mPDE Galerkin Weak Forms, Theoretical Analyses

10.5 Verification, Benchmarking, and Validation

10.6 Mass Transport, the Rotating Cone Verification

10.7 The Gaussian Plume Benchmark

10.8 Steady n-D Peclet Problem Verification

10.9 Mass Transport, a Validated n = 3 Experiment

10.10 Numerical Linear Algebra, Matrix Iteration

10.11 Newton and AF TP Jacobian Templates

Chapter Summary

Exercises

Computer Labs

References

Chapter 11: Engineering Sciences, n > 1: GWSh {Nk(ζα, ηj)} Implementations in the n-D Computational Engineering Sciences

11.1 Introduction

11.2 Structural Mechanics

11.3 Structural Mechanics, Virtual Work FE Implementation

11.4 Plane Stress/Strain GWSh Implementation

11.5 Plane Elasticity Computer Lab

11.6 Fluid Mechanics, Incompressible-Thermal Flow

11.7 Vorticity–Streamfunction GWSh + θTS Algorithm

11.8 An Isothermal INS Validation Experiment

11.9 Multimode Convection Heat Transfer

11.10 Mechanical Vibrations, Normal Mode GWSh

11.11 Normal Modes of a Vibrating Membrane

11.12 Multiphysics Solid–Fluid Mass Transport

11.13 Chapter Summary

11.14 Computer Labs

References

Chapter 12: Conclusion

Index

This edition first published 2012

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Library of Congress Cataloguing-in-Publication Data

Baker, A. J., 1936-

Finite elements: computational engineering sciences / A.J. Baker.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-119-94050-0 (cloth)

1. Finite element method. I. Title.

TA347.F5B3423 2012

620.001′51825–dc23

2012011745

A catalogue record for this book is available from the British Library.

ISBN: 9781119940500

Yogi Berra is quoted,“If you come to a fork in the road, take it,”so I did and ended up here.

Preface

The computer revolution has profoundly impacted how engineers and scientists conduct professional activities. In the early 1960s, a computer fully occupied and amply heated (!) a space the size of a classroom. The PC, introduced in the mid-1970s, was a “toy.” Yet by the millennia, Linux clusters of cheap gigahertz–gigabyte PCs could execute truly large-scale computational simulations. Indeed, the “desktop Cray,” fantasized in ~1980, was here and was truly inexpensive!

The companion maturation of theory and practice in the computational engineering sciences has been an evolutionary (not revolutionary!) process. It remains highly fragmented by discipline, even though computational fluid dynamics (CFD) and computational structural mechanics (CSM) emerged simultaneously from the research laboratory in the late 1950s. The former relied on finite difference (FD) methods to convert theory to computable form. Conversely, the latter's classical virtual work foundation enabled a calculus-based finite element (FE) theory implementation of the underlying variational principle extremum. Finally, in chemical engineering collocation methods were developed for process simulations, and at first glance these theories appear absolutely “linearly independent.”

Research now completed has proven that practically all developments supporting the computational engineering sciences can be formulated from the extremum of the mathematician's weak form theory termed a weak statement (WS). The weak form process enables theorization to be completed in the continuum, using calculus, vector field theory, and modern approximation concepts. When finished, the discrete implementation of the theory extremum can be formed using FE, FD, and/or finite volume (FV) procedures. The FE implementation is typically guaranteed optimal in its performance, that is, accuracy, asymptotic convergence rate, and so on. Furthermore, FE methodology leads to precise constructions devoid of heurism, since integral–differential calculus is used rather than difference algebra to generate the algebraic statement amenable to computing.

This text develops discrete implementations of WS theory for a diverse variety of problem statements in the computational engineering sciences. Unique to the FE discrete development, the resulting algorithms are immediately stated in computable form via a transparent, object-oriented programming syntax. The engineering science problem classes developed herein include

heat conductionstructural mechanicsmechanical vibrationsheat transfer, with convection and radiationfluid mechanicsheat/mass convective transport

The text is organized into twelve chapters. Following an introduction, and some very pertinent overview material, an elementary heat conduction tutorial clearly illustrates all element matrix constructs, the “famous” assembly algorithm and the concept of error estimation and measurement. Subsequent chapter pairs develop expository one-dimensional, then general n-dimensional FE WS implementations in each continuum engineering sciences discipline.

The sequence of developments serves to illustrate, examine, and generalize the available theoretical error estimates, with the concept of a norm central to this process. In moving to the convection–diffusion problem class, a sequence of Taylor series manipulations leads to modified conservation principle expressions, expressed in the continuum, which collectively improve asymptotic convergence rate coupled with annihilation of significant order discretization-induced phase lag and dispersive error mechanisms.

Incisive computer lab experiments complement each development, with principle focus to gain a firm usable understanding of approximation error mechanisms as influenced by data nonsmoothness, problem nonlinearity, stability, dispersion error and boundary conditions, each impacted by the selected FE basis completeness degree. The n-dimensional computer experiments focus on refinements for error nuances associated with nonconvex boundaries, phase lag, and artificial numerical diffusion. An intervening brief chapter clearly identifies the connections between FD, FV, and FE discrete implementations for a Poisson equation in n-dimensions.

Engineers are clearly of the opinion that, “theory is fine, but show me the numbers!,” which requires theory conversion to code practice. Since the FE-implemented WS theory is highly organized, the algorithm statement in any discipline ends up constituted of six, and only six, types of data to convert theory to practice. Capitalizing on object-oriented concepts, these six data types are organized into a template such that the computing statement, including explicit nonlinearity, is unambiguously expressible.

In summary, this text fully develops modern FE discrete algorithms for the computational engineering sciences with applications aimed to available and emergent problem solving environments (PSEs). Its organization and content has evolved from two decades of teaching the subject at UT. This text fully obsoletes the predecessor 1991 text Finite Elements 1-2-3, marketed with a “spaghetti” Fortran PC code on a 5.25 inch floppy disk.

All computer lab exercise MATLAB® .m files along with the specifically written MATLAB® toolbox FEmPSE are available for download from www.wiley.com/go/baker/finite. The .mph files for the COMSOL design studies may be downloaded from their user community web site www.comsol.com/community/exchange/?page=2. University faculty interested in presenting the internet-enabled academic course from which this text was generated will find complete support materials available at www.wiley.com/go/baker/finite.

Many colleagues and graduate students have contributed to the creation and refinement of text content. My thinking formality on the subject has benefited from a multidecade collegial association with Prof. J. Tinsley Oden. I owe a deep debt of gratitude to my Computational Mechanics Corp. co-founders Paul Manhardt, who invented the template concept, and Joe Orzechowski, who assimilated templates into reliable computational syntax for mainly CFD applications.

The dissertation research of Dr. Jin Kim, Dr. Subrata Roy, Dr. David Chaffin, Dr. Alexy Kolesnikov, and Dr. Sunil Sahu collectively formalized the improved theoretical and practical understanding of FE algorithm performance nuances detailed herein. Dr. Zac Chambers and Dr. Marcel Grubert along with Messrs. Mike Taylor and Shawn Ericson contributed significantly to polishing these fundamental underlying precepts to pedagogical acceptability.

A. J. BakerKnoxville, TNJanuary 2012

Note: All color originals are accessible at www.wiley.com/go/baker/finite.

About the Author

A. J. Baker, PhD, PE, left commercial aerospace research to join the University of Tennessee College of Engineering in 1975, to lead academic research in the exciting new field of CFD (computational fluid dynamics). Now Professor Emeritus and still Director, UT CFD Laboratory (http://cfdlab.utk.edu), his professional career started as a mechanical engineer with Union Carbide Corp. The challenges there prompted resigning after 5 years to enter graduate school full time in 1963 with the goal to “learn what a computer was and could do.” The introduction involved driving an IBM 1620 with 5 kB memory and no disk pack! A 1967 summer job with Bell Aerospace Company required assessing the first publication claiming unsteady heat conduction was amenable to finite element analysis. This led to the 1968 Bell Aerospace technical memorandum, “A Numerical Solution Technique for a Class of Two-dimensional Problems in Fluid Dynamics Formulated via Discrete Elements,” a truly pioneering expose in the fledgling FE CFD field. Finishing his dissertation in 1970, he joined Bell Aerospace as Principal Research Scientist to pursue full-time finite element methods in CFD. NASA Langley contracts with summer appointments at ICASE led to a visiting professorship at Old Dominion University, 1974–1975, from which he moved directly to UT forming Computational Mechanics Consultants, Inc., with two Bell colleagues, to assist converting academic FE CFD research progress into computing practice.

FE Computational Engineering Sciences with hands-on computing:

This is the first introductory level text to fully integrate the underlying theory with hands-on computer experiments supported by the MATLAB® and COMSOL® Problem Solving Environments (PSEs). You may download all .m and .mph files supporting each suggested computer experiment, also eight topical lectures for video-streaming on your PC available from www.wiley.com/go/baker/finite. The academic course engendering the text technical content became totally distance-enabled on Internet in 2005. Academics interested in presenting this course at their institution may acquire the complete academic support material at www.wiley.com/go/baker/finite.

Notations

aexpansion coefficientAplane area; one-dimensional FE matrix prefix; coefficientAgeneric square matrix[A]factored global matrixbcoefficient; boundary condition subscript;body force component, generic column matrix{b}global data matrixBtwo-dimensional FE matrix prefixBbody force, structural FE matrixccoefficient; specific heatCthree-dimensional FE matrix prefix, constant, Courant numberdcoefficient; FE matrix indicatorDdiagonal matrix, diffusion coefficient[DIFF]global diffusion matrixDOFapproximation degrees-of-freedomeelement-dependent; unit vector component, errore(·)error, a function of (·)eNapproximation errorehdiscrete approximation erroretajicoordinate transformation dataEenergy seminorm (subscript), elastic modulusEHooke's law matrixFradiation viewfactorFapplied force, flux on ∂Ωfkinetic flux vectorFDfinite differenceFEfinite elementFVfinite volume{F}homogeneous form of a discretized weak statementggravity magnitudeggravityGelastic shear modulus, amplification factor, Gebhart factorGrGrashoff numberGWSGalerkin weak statementhdiscretization (superscript), heat transfer coefficient, measureHGauss quadrature weight; Hilbert space[BC]boundary condition matrixisummation index, mesh node, imaginary unitunit vector parallel to xImoment of inertia; element matrix summation index[I]identity (diagonal) matrixjsummation index, mesh nodeunit vector parallel to yJtemplate summation index[J]coordinate transformation jacobian[JAC]jacobianelement of the [DIFF] and/or [STIFF] matrixkthermal conductivity, basis degree, index, diffusion coefficient,kspring constantaverage value of conductivityunit vector parallel to zKtemplate matrix summation index, viewfactor kernelelement length; summation indexdifferential equation on ∂ΩLdomain span, length measure, lower triangular matrix, lagrangiandifferential equation on Ωmintegermipoint massmGWSTaylor series-modified Galerkin weak statementmPDETaylor series-modified conservation principle PDEMelements in ; moment; matrix prefix; particle system massMiteration matrix[MASS]global mass matrixnindex; normal subscript; dimension of domain Ω; integers, normal coordinate, time index (subscript)outward pointing unit vector normal to ∂ΩNmatrix prefixNsummation termination; approximation (superscript), iteration matrixNCnatural coordinate basis{Nk}finite element basis of degree k non-D non-dimensionalpload (data); pressure, iteration indexPpoint load; Gauss quadrature order{P}computational matrix, distributed load DOFPanon-D parameter on ΩPbnon-D parameter on ∂ΩPrPrandtl numberqgeneralized dependent variableQdiscretized dependent variable; heat added{Q}:approximation DOF matrixrreference state subscript; radiusReReynolds numberR+the positive real axisEuclidean space{RES}global matrix statement residualssource term on Ω; heat added, tangent coordinatesunit vector tangent to ∂ΩSfinite element assembly operator; entropySORsuccessive over-relaxation{S}computational matrixttimeTtemperature, kinetic energyTcconvection heat transfer exchange temperatureTrradiation heat transfer exchange temperatureTsurface traction vectorTNapproximate temperature solutionTEtruncation errorTPtensor product basisTSTaylor seriesudisplacement vector; velocity vectorUupper triangular matrixuvelocity x component; speedUdiscretized speed DOF, phase velocity (speed)[VEL]global fluid convection matrixvvelocity y componentVshear force; volume; potential energyVvelocitywweight function; fin thickness; velocity z componentWweight; work done by systemWFweak formWSweak statementxgeneric unknownx, xicartesian coordinate, coordinate system 1 ≤ i ≤ ntransformed local coordinateXdiscrete cartesian coordinateydisplacement; cartesian coordinateYdiscrete cartesian coordinatezcartesian coordinateZthickness ratio; discrete cartesian coordinate(·)scalar (number){·}column matrix{·}Trow matrix[·]square matrixnorm∪union (non-overlapping sum)∩intersectiondet [·]matrix determinantsymsymmetricαcoefficientβcoefficientγshear strain, coefficientδijKronecker deltaδQiterateΔdiscrete incrementεnormal strain, emissivityϕelectric potential, flow potentialϕ(·)trial space function; potential functionpotential functiontest spacetrial spaceηcoordinate system in transform spaceηitensor product coordinate systemκthermal diffusivity, wave numberelement of a square matrixλLamé parameter, wavelengthμLamé parameter, dynamic viscosityPoisson ratio, kinematic viscosityO(·)order of (·)πpi (3.1415926...)θtime integration implicitness factorΘpotential temperatureρdensity, absorbtivitydσdifferential element on ∂Ωdτdifferential element on Ωτnormal stressfrequencyΩdomain of differential equationΩ:efinite element domaindiscretization of Ω∂Ωboundary of Ωnatural coordinate systemd(·)/dxordinary derivative∂(·)/∂xpartial derivativevector derivative2laplacian derivative operator

Chapter 1

The Computational Engineering Sciences: An Introduction

1.1 Engineering Simulation

The digital computer, coupled with engineering and computer science plus modern approximation theory, have coalesced to render computational simulation via math modeling an alternative modality supporting design optimization in engineering. Design has historically been conducted in the physical laboratory (Figure 1.1). The test device is a miniature of reality and the laboratory process sequence is:

model the geometry (similitude)determine desired data (cost)acquire the datainterpret the datadraw conclusions

The computational engineering sciences laboratory has emerged as the complement to, or replacement of, the legacy modality (Figure 1.2). The computational laboratory process sequence is:

Figure 1.1 Classic wind tunnel test

model the mathematics (fidelity)model the physics (cost)compute the datainterpret the datadraw conclusions

The first two components of the computational engineering sciences (CES) laboratory place a significant new burden on the engineer/scientist. Aspects of calculus and vector field theory, the language for expressing conservation principles in the engineering sciences, must be recalled. Additionally, dexterity with constitutive closure approximations, that is, the “physics model,” must be understood on a fidelity/mathematics as well as cost/benefit basis.

Figure 1.2 Cloud-computing visualization

The identical calculus and vector field topics underpin modern approximation theory guidance for generating a conservation principle approximate solution based on a weak formulation (WF) [1]. The mathematicians, in developing this approach to solution approximation, have endowed it with an elegant theory on optimal construction and error estimation. A WF, always completed in the continuum, theoretically transforms the solution of the partial differential equation (PDE) into a computable large-order algebraic equation system.

Once the continuum weak form theory is completed, the sole remaining decision is implementation. Herein this is accomplished by replacing the trial and test spaces with finite element (FE) trial/test space bases defined for a spatial discretization of the PDE domain of dependence. This identification directly enables WF integral evaluations using the calculus. Detailing this process for a diverse spectrum in the engineering sciences is the content of this text.

1.2 A Problem-Solving Environment

Historically, a frustrating aspect of computational simulation was interacting with the computer code! User interface and computer science issues dominate this facet, and the engineer/scientist interested in analysis is typically not well founded in the required skills. This issue is compounded by the tradition in olden times, that is, a decade or so ago, for the individual to code his/her own computer program.

This incredible dissipation of time and effort has been superceded by the emergence of component-based software leading to the concept of a problem-solving environment (PSE). Commercial code systems now exist throughout the engineering sciences possessing very powerful advances in user interfaces. Maturation of grid computing concepts will lead, in the not too distant future, to Internet-enabled just-in-time capabilities using remotely accessible high-performance computing and communications (HPCC) constructs [2].

Figure 1.3 illustrates this emergent scenario. The practicing design engineer possesses knowledge about his/her problem statement, and after absorbing this text's content will be thoroughly comfortable with the associated mathematics/physics issues with seeking an optimal approximate solution. From that point on only casual knowledge about the subsequent computer science issues will be required, as the Internet modality exists to complete the loop.

Figure 1.3 The problem-solving environment

An added historical aspect is that computational codes were irrevocably tailored to the specific discrete theory, for example, finite difference (FD), finite volume (FV), FE for a given engineering science problem class. This is now moot as completed research confirms these apparently very distinct computational constructs can be interpreted as specific decisions in implementing a weak statement (WS), the extremum of a WF for a PDE. Invariably the FE discrete implementation generates the optimal construction, the consequence of WF theory, and the use of calculus rather than difference algebra to form the algebraic statement.

The computational practice of FE methods is rapidly maturing, as academics in math, engineering, and computer science collectively resolve key theoretical issues. A by-product, developed in thoroughness in this text, is the object-oriented FE algorithm construct that directly communicates “compute desire” to a PSE via a template.

This approach recognizes a code is but a data-handling system, and the FE implementation of a WS generates only six data types for each and every (!) FE domain Ωe specifically including nonlinearity. The objects for all element-level matrix contributions {WS}e to a WS algebraic statement are thus organized as:

Coding of a FE WS discrete implementation is thus reduced to data identification in these six object categories.

Herein, the progression of a WS algorithm for an engineering science topic, FE discrete-implemented, leads to the object-oriented template transparently converting theory to executable code. Template generation occurs in a word-processing environment, and the result precisely encompasses all complexities, specifically including nonlinearity, in coupled PDE systems. The template-enabled computing PSE herein employs MATLAB® [3], via the specifically written FEmPSE toolbox for expository computing labs. Design-based computing experiments employ COMSOL [4], an FE-implemented multiphysics commercial PSE.

1.3 Weak Formulation Essence

An engineering design problem statement is invariably cast as a PDE written on the state variable (the dependent variable), herein labeled q = q(x) for the steady definition. The compact notation used in this text to denote a PDE is

(1.1)

In equation (1.1), is the PDE placeholder and its domain of influence is symbolized as Ω, a region lying on an n-dimensional euclidean space ℜn.

To “connect” the PDE to the specific problem statement requires boundary conditions (BCs) communicating this given information, that is, the data. The text-utilized BC compact notation is

(1.2)

where ∂Ω is the (n–1)-dimensional bounding enclosure of Ω. Figure 1.4 illustrates these formalisms.

Figure 1.4 Engineering problem statement notation

The exact solution q(x) satisfying a genuine problem equations (1.1) and (1.2) can never (!) be found analytically. Consequently, the key WF theory requirement is to formally define an (any!) approximation to q(x). Herein this requirement is expressed as

(1.3)

The assumption in equation (1.3) is that one can identify a suitable trial space Ψα(x), a set of functions on ℜn, to support any approximate solution. The summation therein couples each trial space member to an unknown expansion coefficient Qα, called a degree-of-freedom (DOF) of the approximation, the set of which is to become determined in the algebraic computing process.

Unless equations (1.1) and (1.2) define a trivial problem, qN cannot be identical with q. The difference between q and qN is the approximation error, herein denoted eN. Since everything is a function, obviously

(1.4)

The singular goal is to seek the best approximation qN, hence to constrain in some sense the “size” of eN(x). This is elegantly accomplished via the mathematicians' WF that requires the available measure of error

be made orthogonal (mathematically “perpendicular”) to an arbitrarily chosen test function w(x). The weak form (WFN) expression of this constraint on the approximation (1.3) is

(1.5)

The requirement of any function w(x) is cleanly handled via an interpolation, ([5], Chapter 2.2), followed by forming the extremum of rearranged equation (1.5) which identifies the test space Φβ(x) companion to the trial space Ψα(x). The scalar equation (1.5) thus becomes a set of equations of the order N defined in equation (1.3), which is termed the weak statement (WSN). All x-dependence vanishes in evaluating the defined integrals, hence WSN generates the algebraic equation system

(1.6)

As the final caveat, inserting equation (1.2) into equation (1.5) moves the BC-constrained DOF in the set Qα, into the data matrix {b} in equation (1.6). The remaining equation (1.2) -defined DOF populate the column matrix {Q} in equation (1.6), the exactly correct order algebraic equation system for determination of the unknown DOF defined in equation (1.3).

1.4 Decisions on Forming WSN

The key to weak form utility is the assumption that the integrals in WSN, equation (1.6), can be evaluated. This obviously centers on the functional form selected for the test and trial function spaces. These decisions in turn identify a specific algorithm from the wide range of WSN methods that can be derived. The following table provides a WSN summary essence categorized on these function sets.

Φβ(x),Ψα(x)ExamplesWSN LabelGlobalSine, cosine, Bessel, spherical harmonicsAnalytical methodology (separation of variables)Chebyshev polynomialsSpectral methodsGlobal–localChebyshev by blocksPseudospectral methodsLocalLagrange polynomialsSpatially discrete methods (FE, FV, FD)

For Φβ(x) and Ψα(x) spanning the entirety of Ω generates formulations closely associated with analytical PDE-solution methodology. However, this choice precludes geometric flexibility, as closures ∂Ω of the domain Ω must be coordinate surfaces. The singular key attribute is that these spaces contain functions that are indeed orthogonal on Ω. Hence, [Matrix] in equation (1.6) is typically diagonal, which renders the algebraic solution process trivial (recall separation of variables in your sophomore calculus class?).

Spectral methodology retains the definition and use of global span function spaces. Pseudospectral methods lie halfway between spectral and spatially discrete algorithms, and both typically inherit the liability that closure segments be coincident with global coordinate surfaces.

For domains with absolutely arbitrarily geometric closure, that is, essentially all practical problems, the FE discrete implementation of WSN, hereon denoted WSh, guarantees the extremum of WFN, equation (1.5), generates integrals that can be evaluated. This is accomplished by subdividing Ω into the union (nonoverlapping sum, symbol ) of small subdomains, see Figure 1.5. Each subdomain is called a FE, denoted Ωe, and their union can be manipulated to fit any geometrical shape of the domain closure ∂Ω.

Figure 1.5 Domain discretization

A WSN can be manipulated to interpret FD and FV methodology as will be illustrated. The formulation distinctions include integrals not being generated via calculus and the resultant algorithms are not predictable optimal, the key attribute of a specific FE implementation WSN, detailed shortly.

This process of subdividing Ω into the union of small subdomains is called spatial discretization, symbolized in the literature by superscript h. Unambiguously then

(1.7)

and the region is called a FE. The resultant FE solution approximation form (1.3) transitions to

(1.8)

Constructing the required discrete equivalents of Ψα(x) and , equations (1.3) and (1.5), generates the trial and test space basis functions. The theoretical foundation is typically Lagrange or Hermite interpolation polynomials, and FE basis functions are herein symbolized as the column matrix {N(x)}. Hence, for equation (1.8)

(1.9)

With equations (1.7)–(1.9) the WSh-generated [Matrix] in equation (1.6) is never diagonal. Hence, one must find an algebraic solution replacement for Cramer's rule, which introduces iterative matrix linear algebra methodology.

A fundamentally significant solution facet results upon making the discrete approximation decision. In the FE implementation, the DOF in the element-level approximation equation (1.9), that is, select DOF in equation (1.3), are usually generated only at mesh intersections on Ωh. Illustrated in Figure 1.5 as dots (•) they are called the nodes of the mesh.

The union of the local solutions qe(x) forms qh(x), equation (1.8), with the resultant spectral resolution controlled by node-separation distance. Specifically, for a mesh of measure Δx any 2Δx wavelength information cannot be resolved. This is clearly illustrated in Figure 1.6; on the left the DOF {Q} for the 2Δx sine waves are all zero (!) while those on the right are nonzero. Hence mesh resolution is central to accuracy; a too coarse mesh can produce totally wrong solutions, as will be illustrated.

Figure 1.6 Resolution illustrations on a mesh of measure Δ

x

1.5 Discrete WSh Implementations

Legacy FD and FV methods also employ a domain discretization Ωh = cΩc, where Ωc is a computational cell. Further, mathematicians and chemical engineers (in particular) have developed many node-based numerical methods, for example, collocation, least squares, weighted residuals. Do fundamental theory underpinnings exist for these apparently very diverse discrete procedures for PDEs?

The answer is a resounding YES!! Under the weak form umbrella, the distinctions reside strictly in the test and trial space basis functions chosen to form WSh. The following table summarizes algorithm decisions in the context of WSh.

NameTrial space, Ψα(x)Test space, Φβ(x)Galerkin (FE)Basis {N}Basis {N}CollocationBasis {N}Kronecker δFinite difference (FD)?NoneFinite volume (FV)?UnityLeast squaresBasis {N} ({N})Boundary element (BEM)Basis {N}Green's function

The fact that myriad choices exist, and have been computer implemented, immediately raises the fundamental question:

Does an optimal choice for the WSN trial and test space function sets Ψα(x) and Φβ(x) exist?

One must first define optimal to answer this. Mathematicians will work this to the point of distraction but engineers are not so burdened. Their obvious choice is the selection that produces the absolute minimum approximation error eN(x), equation (1.4).

Importantly, this answer must be and is absolutely independent of the particular choice for discrete implementation! For a wide range of PDEs describing problem statements in the engineering sciences, in the continuum the answer to the fundamental question is:

The WSN approximation error eN(x) is minimized, in a suitable norm, when the trial and test spaces are identical.

Now moving to the WSN spatially discrete implementation WSh, on a mesh Ωh, the approximation error becomes eh(x) ≡ q(x) − qh(x). Thereby, the WS discrete implementation corollary for optimal performance is:

The WSh approximation error eh(x) is minimized, in a suitable norm, when the trial and test space replacements contain the identical FE trial space basis functions.

The name historically attached to identical trial and test space functions is Galerkin; herein this form of WSN is denoted GWSN. The Galerkin criterion is optimal in theory and in the FE discrete implementation.

The vector “cartoon” in Figure 1.7 serves to illustrate that the exact solution q cannot lie in the “plane” containing the trial function set Ψα(x) supporting qN unless they are identical (not likely!). The “distance” between q and qN is the error eN and its “magnitude” is the smallest when eN is orthogonal (mathematically perpendicular) to the plane, as induced by the Galerkin criterion qN(GWSN). The solution qN(WSN) generated by any other trial/test function criterion produces the error which is not orthogonal to the trial/test function plane, hence the (dashed) error vector possesses a larger “magnitude.”

Figure 1.7 Vector cartoon illustrating GWSN optimality

1.6 Chapter Summary

The FE discrete implementation of GWSN, that is, GWSh, is the optimal decision for a wide class of problem statements in the CES. FE trial space basis availability guarantees accurate evaluation of the integrals defined in equation (1.5) for domains Ω enclosed by an arbitrarily nonregular shaped boundary ∂Ω. With this theory in place, the key topic becomes identification of the trial space basis functions spanning FE domains Ωe on n-dimensions. Their identification and performance quantification across the CES is the subject of this text.

In summary, the GWSN ⇒ GWSh process essence for designing optimally performing algorithms is:

References

1. Oden, J.T. and Reddy, J.N. (1976) An Introduction to the Mathematical Theory of Finite Elements, Wiley, New York.

2. Moore, S., Baker, A.J., Dongarra, J., Halloy, C. and Ng, C. (2002) Active Netlib: An Active mathematical software collection for inquiry-based computational science and engineering education. J. Digit. Inf., 2.

3.www.mathworks.com/Matlab.

4.www.comsol.com.

5. Baker, A.J. (2013) Optimal Modified Continuous Galerkin CFD, Wiley, London.

Chapter 2

Problem Statements: In the Engineering Sciences

2.1 Engineering Simulation

The engineer or scientist interested in computer simulations must acquire some dexterity with fundamental mechanics principles to be a cogent and competent analyst. This chapter provides a basic refresher on the range of pertinent academic materials traditionally included in undergraduate engineering curricula, which the author assumes is the reader's background.

Fundamentally, computational simulation involves seeking a solution to one or more usually nonlinear partial differential equations (PDEs) that stem from basic conservation principles. In the lagrangian point mass perspective, the mathematical statement of these mechanics principles is

(2.1)

(2.2)

(2.3)

(2.4)

In equations (2.1) and (2.2), mi denotes the ith point mass, M is total mass of a particle system, V the system velocity, and F denotes the applied (external) forces. Equations (2.3) and (2.4) state the first and second laws of thermodynamics where E is the total system energy, Q the heat added, W the work done by the system, and S the entropy. The undergraduate academic courses in statics, dynamics, strength of materials, thermodynamics, fluid mechanics, thoroughly cover the pertinent developments [1–3]. Do you recall your exposure to this material?

2.2 Continuum Mechanics Viewpoint

In practice, engineers almost never deal with the conservation principles written in the Lagrangian form. Instead, transition to the continuum (eulerian) description is made, wherein one assumes there are so many mass points per characteristic volume V (not boldface!) that a density function ρ can be defined, that is,

(2.5)

One then defines a control volume CV, with encompassing control surface CS, see Figure 2.1, and transforms the conservation principles from the lagrangian to the eulerian viewpoint via Reynolds transport theorem which states, [4]

(2.6)

Figure 2.1 Control volume for Reynolds transport theorem

The bracket (·) in equation (2.6) contains an appropriate conservation principle variable, V is the velocity vector field traversing CV, signifies the net efflux of (·) across CS, and dτ and dσ are the volume and surface differential elements.

One immediately notes the eulerian viewpoint involves transition to multidimensional vector differential and integral calculus! You will be called upon to regain dexterity with these fundamental undergraduate academic subjects.

2.3 Continuum Conservation Principle Forms