AMS 529: Finite Element Methods:Fundamentals, Applications, and New Trends

Lecture 1: Course Overview;The Past and Present of FEM

Xiangmin Jiao

Stony Brook University

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Outline

1 Course Overview

2 Overview of FEM

3 FEA Software

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Course Overview

This is an intermediate-level course on the finite element methods(FEM) for solving partial differential equations. It will introduce themathematical formulation, numerical analysis, and computationalaspects of FEM, applications in solid mechanics, fluid mechanics, andmultiphysics phenomena, as well as the recent trends in improving theirstability, accuracy, efficiency, and generality. Computing projects willinvolve programming in Python and MATLAB/Octave, as well as usingsoftware FEniCS and ANSYS for understanding the typical workflow offinite element analysis for solving real-world problems.

Course syllabusCourse website:http://www.ams.sunysb.edu/~jiao/teaching/ams529

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Where Does FEM Fit in Science/Engineering?

Mechanics,

electromagnatics,

di�. geometry,

physics, etc.

Applied

Math. &

Numerical

Analysis

Computer

Science &

Computer

Aided

Design

Finite

Element

Method

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Example Team Project Ideas

An advanced simulation using ANSYS or another software (e.g.,nonlinear elasticity, fractures, multiphase flow, multiphysics problem,topology-optimization problem, etc.)Implementation of FEM for linear elasticity or heat equationParallelization of FEM codeImplementing a new feature in FEniCSImplementing multigrid methodMultiphysics coupling methodologyOther project of your research interest

All team project will include:a proposal with problem statement and methodologya detailed report on verification and validation, code or lessonsa 20–30 minute presentation

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Outline

1 Course Overview

2 Overview of FEM

3 FEA Software

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What is FEM

The finite element method (FEM) is a numerical method for solvingdifferential equations (typically boundary-value problems) inengineering and mathematical physicsIt typically involves subdividing a domain into smaller components,called elementsPiecewise smooth approximations are defined based on elements toconvert differential equations into algebraic equationsSolution of algebraic equations gives approximate values of theunknowns at discrete points over the domain, which is theninterpolated to define solution over domainTypical problem areas of interest include structural analysis, heattransfer, fluid flow, mass transport, electromagnetics, etc.

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Examples FEM Mesh and Solution in 2-D

Sources: particleincell.com

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Typical Processes of Finite Element Analysis

Source: http://www.colorado.edu/engineering/cas/courses.d/IFEM.d

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Some Examples of FEM

Sources: wikiversity.org, stressebook.com, machinedesign.com, andtransmittingscience.org

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Two Interpretations of FEM

Physical InterpretationBreakdown of structuralsystem into components(elements) andreconstruction by theassembly processFocus of Part II

Mathematical InterpretationNumerical approximation ofa boundary value problem bya variational formulationwith piecewise basis and testfunctions of local supportFocus of Part I

Most engineering textbooks favor the former, and applied math.textbooks favor the latterWe refer to physical interpretation as finite element analysis (FEA)Computer scientists are also concerned with their implementations,mesh generation, etc.

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Physical Interpretations of FEM (FEA)Chapter 1: OVERVIEW

Physical system

simulation error: modeling & solution errorsolution error

Discretemodel

Discretesolution

VALIDATION

VERIFICATION

FEM

CONTINUIFICATION

Ideal Mathematical

model

IDEALIZATION &DISCRETIZATION

SOLUTION

occasionallyrelevant

Figure 1.3. The Physical FEM. The physical system (left box) is the sourceof the simulation process. The ideal mathematical model (should one go to the

trouble of constructing it) is inessential.

The concept of error arises in the Physical FEM in two ways. These are known as verification andvalidation, respectively.9 Verification is done by replacing the discrete solution into the discretemodel to get the solution error. This error is not generally important. Substitution in the idealmathematical model in principle provides the discretization error. This step is rarely useful incomplex engineering systems, however, because there is no reason to expect that the continuummodel exists, and even if it does, that it is more physically relevant than the discrete model.Validation tries to compare the discrete solution against observation by computing the simulationerror, which combines modeling and solution errors. As the latter is typically unimportant, thesimulation error in practice can be identified with the modeling error. In real-life applications thiserror overwhelms the others.10

Oneway to adjust the discrete model so that it represents the physics better is calledmodel updating.The discrete model is given free parameters. These are determined by comparing the discretesolution against experiments, as illustrated in Figure 1.4. Inasmuch as the minimization conditionsare generally nonlinear (even if the model is linear) the updating process is inherently iterative.

Physical system

simulation error

Parametrizeddiscretemodel

Experimentaldatabase

Discretesolution

FEM

EXPERIMENTS

Figure 1.4. Model updating process in the Physical FEM.

9 Programming analogs: static and dynamic testing are called verification and validation, respectively. Static testing iscarried at the source level (e.g., code walkthroughs, compilation) whereas dynamic testing is done by running the code.

10 “All models are wrong; some are useful” (George Box)

1–10

Source: http://www.colorado.edu/engineering/cas/courses.d/IFEM.d

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Mathematical Interpretations of FEM§1.4 THE FEM ANALYSIS PROCESS

Discretization & solution error

REALIZATIONIDEALIZATION

solution error

Discretemodel

Discretesolution

VERIFICATION

VERIFICATIONFEM

IDEALIZATION &DISCRETIZATION

SOLUTION

Idealphysical system

Mathematicalmodel

ocassionally relevant

Figure 1.5. The Physical FEM. The physical system (left box) is the sourceof the simulation process. The ideal mathematical model (should one go to the

trouble of constructing it) is inessential.

§1.4.2. The Mathematical FEM

The other canonical way of using FEM focuses on themathematics. The process steps are illustratedin Figure 1.5. The spotlight now falls on the mathematical model. This is often an ordinarydifferential equation (ODE), or a partial differential equation (PDE) in space and time. A discretefinite element model is generated from a variational or weak form of the mathematical model.11This is the discretization step. The FEM equations are solved as described for the Physical FEM.On the left, Figure 1.5 shows an ideal physical system. This may be presented as a realization ofthe mathematical model. Conversely, the mathematical model is said to be an idealization of thissystem. E.g., if the mathematical model is the Poisson’s PDE, realizations may be heat conductionor an electrostatic charge-distribution problem. This step is inessential and may be left out. IndeedMathematical FEM discretizations may be constructed without any reference to physics.

The concept of error arises when the discrete solution is substituted in the “model” boxes. Thisreplacement is generically called verification. As in the Physical FEM, the solution error is theamount by which the discrete solution fails to satisfy the discrete equations. This error is relativelyunimportant when using computers, and in particular direct linear equation solvers, for the solutionstep. More relevant is the discretization error, which is the amount by which the discrete solutionfails to satisfy themathematicalmodel.12 Replacing into the ideal physical systemwould in principlequantify modeling errors. In the Mathematical FEM this is largely irrelevant, however, because theideal physical system is merely that: a figment of the imagination.

§1.4.3. Synergy of Physical and Mathematical FEM

The foregoing canonical sequences are not exclusive but complementary. This synergy13 is one of

11 The distinction between strong, weak and variational forms is discussed in advanced FEM courses. In the present booksuch forms will be largely stated (and used) as recipes.

12 This error can be computed in several ways, the details of which are of no importance here.13 Such interplay is not exactly a new idea: “The men of experiment are like the ant, they only collect and use; the reasonersresemble spiders, who make cobwebs out of their own substance. But the bee takes the middle course: it gathers itsmaterial from the flowers of the garden and field, but transforms and digests it by a power of its own.” (Francis Bacon).

1–11

Source: http://www.colorado.edu/engineering/cas/courses.d/IFEM.d

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Synergy of Physical and Mathematical ViewsChapter 1: OVERVIEW

FEM Library

Component

discrete

model

Component

equations

Physical

system

System

discrete

model

Complete

solution

Mathematical

model

SYSTEM

LEVEL

COMPONENT

LEVEL

Figure 1.6. Combining physical and mathematical modeling through multilevelFEM. Only two levels (system and component) are shown for simplicity.

the reasons behind the power and acceptance of the method. Historically the Physical FEMwas thefirst one to be developed to model complex physical systems such as aircraft, as narrated in §1.7.The Mathematical FEM came later and, among other things, provided the necessary theoreticalunderpinnings to extend FEM beyond structural analysis.Aglance at the schematics of a commercial jet aircraftmakes obvious the reasons behind thePhysicalFEM. There is no simple differential equation that captures, at a continuum mechanics level,14 thestructure, avionics, fuel, propulsion, cargo, and passengers eating dinner. There is no reason fordespair, however. The time honored divide and conquer strategy, coupled with abstraction, comesto the rescue.First, separate the structure out and view the rest as masses and forces. Second, consider the aircraftstructure as built up of substructures (a part of a structure devoted to a specific function): wings,fuselage, stabilizers, engines, landing gears, and so on.Take each substructure, and continue to break it down into components: rings, ribs, spars, coverplates, actuators, etc. Continue through as many levels as necessary. Eventually those componentsbecome sufficiently simple in geometry and connectivity that they can be reasonably well describedby the mathematical models provided, for instance, by Mechanics of Materials or the Theory ofElasticity. At that point, stop. The component level discrete equations are obtained from a FEMlibrary based on the mathematical model.

14 Of course at the (sub)atomic level quantum mechanics works for everything, from landing gears to passengers. Butit would be slightly impractical to represent the aircraft by, say, 1036 interacting particles modeled by the Schrodingerequations. More seriously, Truesdell and Toupin correctly note that “Newtonian mechanics, while not appropriate to thecorpuscles making up a body, agrees with experience when applied to the body as a whole, except for certain phenomenaof astronomical scale” [828, p. 228].

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Source: http://www.colorado.edu/engineering/cas/courses.d/IFEM.dXiangmin Jiao Finite Element Methods 14 / 20

A Brief History of FEMOrigin of FEM

I 1851: K. Schellbach, Probleme der Variationsrechnung, J. ReineAngew. Math.

I 1941: A. Hrennikoff, Solution of problems of elasticity by theframework method. Journal of Applied Mechanics

I 1943: R. Courant, Variational methods for the solution of problems ofequilibrium and vibrations, Bulletin of Amer. Math. Soc.

I 1960s: J.T. Oden; Zienkiewicz and Chang; Pestel; ...Mathematical foundation of FEM

I 1965: FENG Kang, A Difference Formulation Based on the VariationalPrinciple (in Chinese), Appl. Mathematics and Comp. Mathematics(first paper proving convergence but unknown to western world)

I 1968: M.W. Johnson Jr. and R.W. McClay; M. Zlamal; P.G. Ciarlet;A.E. Oliveira (error estimations and convergence)

I 1972: I. Babuska and A.K. Aziz; G. Strang and G. Fix; P.G. Ciarlet andP.A. Raviart; ...

1968: NASA released open-source software NASTRAN,Source: J. Tinsley Oden, Historical Comments on Finite Elements, in AHistory of Scientific Computing, ACM, 1990.

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Primary References for This Class

Mathematics-oriented books:M.G. Larson and F. Bengzon, The Finite Element Method: Theory,Implementation, and Applications, Springer, 2013.G. Strang, Computational Science and Engineering, Chapters 3, 6, &7, Wellesley-Cambridge Press, 2007.S. H. Lui, Numerical Analysis of Partial Differential Equations,Chapters 3 & 7, Wiley, 2011.

Application-oriented reference:C. Felippa, lecture notes on Introduction to Finite Element Methodsat University of Colorado.

Software-oriented references:H. P Langtangen and A. Logg, Solving PDE in Python: The FEniCSTutorial, Springer, 2016.ANSYS and FLUENT Learning Modules at Cornell University

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Related Methods

Other popular numerical methods for discretizing PDEs include:Finite Difference Methods

I Like FEM, also solve unknowns at nodesI Traditionally limited to structured (logically Cartesian) meshes, but

there are generalized FDM on unstructured meshesI Most popular for initial value problems

Finite Volume MethodsI Unknowns are at cell centers (typically average quantities over control

volumes)I Only for hyperbolic conservation laws and fluid flows

Spectral MethodsI Use trigonometric basis functions with global supportI Works best for periodic problems

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Various Generalizations of FEM

There are also various generalizations of FEMDiscontinuous GalerkinExtended/generalized FEMSpectral element methodsMeshless methodsIsogeometric analysisNURBS-enhanced FEMLeast squares FEMAdaptive Extended Stencil FEM (AES-FEM)and others

Understanding the fundamentals of FEM can also help you understandthese generalizations

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Outline

1 Course Overview

2 Overview of FEM

3 FEA Software

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Some Popular FEM Software

ProprietaryI ABAQUSI ADINAI ANSYSI COMSOLI ...

Open-source softwareI CalculiXI Code_AsterI FEniCSI ElmerI ...

See http://feacompare.com/ for a comparison

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