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Finite Element Methods
Max D. Gunzburger Janet S. Peterson
January 8, 2013
2
Chapter 1
Introduction
Many mathematical models of phenomena occurring in the universe involve dier- ential equations for which analytical solutions are not available. For this reason, we must consider numerical methods for approximating the solution of dieren- tial equations. The nite element method is one such technique which has gained widespread use in a diverse range of areas such as uid mechanics, structural me- chanics, biological science, chemistry, electromagnetism, nancial modeling, and superconductivity, to name a few. One can nd articles where nite element meth- ods have been employed to study everything from stress analysis of a human tooth to design of an airplane wing. Although the foundations for the nite element method were laid in the rst half of the twentieth century, it did not become widely used until much later. Struc- tural engineers were the rst to use the technique in the 1940's and 1950's; math- ematicians became interested in analyzing and implementing the method in the late 1960's. The rst symposium on the mathematical foundations of the nite element method was held in June of 1972 with over 250 participants and resulted in a now famous book by I. Babuska and A. Aziz. Prior to this symposium there had already been numerous national and international conferences held on the nite element method but mainly with an emphasis on engineering applications. In the following decades the nite element method has grown in popularity as a useful tool in design and application as well as a fertile area for mathematical analysis. This rst chapter is motivational in intent. We dene, in the simplest possible setting, a nite element method. We then make an attempt to analyze the method; this attempt fails to be rigorous because we do not have in hand the necessary mathematical tools. However, in making the attempt, we learn something about the nature of the tools that we need to acquire. We then compare and contrast nite element methods to the nite dierence approach and discuss some of the attractive features of nite element methods. 1
2 Chapter 1. Introduction
1.1 What are nite element methods?
Finite element methods are a class of methods for obtaining approximate solutions of dierential equations, especially partial dierential equations.
1As such, they can
be compared to other methods that are used for this purpose, e.g., nite dierence methods, nite volume methods or spectral methods. There are seemingly countless nite element methods in use, so that one cannot refer to any method asthenite element method any more that one can refer to any particular method as beingthe nite dierence method. In fact, there are numerous subclasses of nite element methods, each saddled with a modier, e.g.,Galerkin,mixed, orcollocationnite element methods. We draw distinctions between these dierent subclasses of nite element methods in later chapters. The nite element method is distinguished from other approaches to approxi- mating dierential equations by the combination of variational methods and piece- wise polynomial approximation. Piecewise polynomial approximation is very at- tractive due to the ease of use, its approximation properties, and the availability of bases which are locally supported; that is, bases that are nonzero over a small portion of the domain. Variational methods have their roots in the combination of partial dierential equations and the calculus of variations. The Rayleigh-Ritz Method, conceived individually by Lord Rayleigh and Walther Ritz, is a variational technique to nd the minimum of a functional dened on an appropriate space of functions as a linear combination of elements of that space. The variational aspect of the nite element method usually takes the form of a weak or variational prob- lem. In this and later chapters we see that some of the problems we consider are equivalent to an unconstrained minimization problem such as Rayleigh-Ritz. On the other hand, the variational principles that the nite element method encom- passes can handle problems which are related to constrained minimization and even those not related to optimization problems.
1.2 A Simple Example
In order to begin to understand the basic idea of the nite element method and the steps involved, we dene a nite element method for the very simple two-point boundary value problem u00(x) =f(x) 0< x <1;(1.1a) u(0) = 0;(1.1b) and u
0(1) = 0:(1.1c)
Here,f(x) is a given function dened forx2(0;1) andu(x) is the unknown function to be determined by solving (1.1). This boundary value problem can represent1 Finite element methods were not always thought of in this manner, at least in the structural mechanics community. In an alternate denition, structural systems are directly discretized into approximate submembers such as beams, plates, shells, etc., without any recourse to dierential equations. These submembers are then called \nite elements."
1.2. A Simple Example 3
a number of dierent physical situations; e.g., the temperature distribution in a uniform rod. It is important to note that this dierential equation arises from a steady-state problem, that is, one that does not result from the time evolution of some initial condition. The nite element approximationuh(x) to the solutionu(x) of (1.1) is dened to be the solution of the following problem: nduh(x)2Vhsuch thatZ 1 0du hdx dv hdx dx=Z 1 0 fvhdx8vh2Vh;(1.2) whereVhis a nite dimensional set (more precisely, a linear space2) of functions that vanish atx= 0 and are suciently smooth. Actually, Problem 1.2 denes a nite element method only if the approximating setVhis chosen to consist of piecewise polynomial functions. This choice of approximating functions, along with a judicious choice of basis forVh, is primarily responsible for the success of the nite element method as a computational method. We now ask ourselves what (1.2) has to do with the original problem (1.1). An obvious connection is that since functions belonging toVhvanish atx= 0 by denition, we have thatuh(x) satises the boundary condition (1.1b). To see further connections, consider the following problem which is analogous to (1.2) except it is posed over an innite dimensional vector spaceVinstead of the nite dimensional spaceVh: ndu(x)such thatu(0) = 0and Z 1 0 u0v0dx=Z 1 0 fv dx8v2V ;(1.3) where for eachv2V,v(0) = 0 andvis \suciently smooth". One can view Problem 1.2 as an approximation of Problem 1.3. Integrating the left-hand side of (1.3) by parts and using the fact thatv(0) = 0 allows us to write v(1)u0(1)Z 1
0u00(x) +f(x)v(x)dx= 0:(1.4)
Now the arbitrariness ofv(x) implies thatu(x) also satises (1.1a) and (1.1c). To see this, we rst choose an arbitraryv(x) that vanishes atx= 1 as well as atx= 0.
For all suchv(x), we have that
Z 1
0u00(x) +f(x)v(x)dx= 0;
so thatu(x) satises (1.1a). However, ifu(x) satises (1.1a), then (1.4) simplies to v(1)u0(1) = 0; where now againv(1) is arbitrary. Thus, we obtain (1.1c) as well. Hence we have demonstrated that ifu(x) is a suciently smooth solution of (1.3) then it also satises (1.1).2 Linear or vector spaces will be discussed in Chapter 2.
4 Chapter 1. Introduction
Now, let us reverse the above steps that took us from (1.3) to (1.1). Specif- ically, we require thatu(0) = 0 and we multiply (1.1a) by a suciently smooth functionv(x) that vanishes atx= 0 but is otherwise arbitrary. Then, we integrate the term involving the second derivative ofuby parts and use the boundary con- dition (1.1c) to obtain (1.3). In this manner, one can show
3that any solutionu(x)
of (1.1) is also a solution of the problem (1.3). Is the converse true? We have seen that the answer is yes only if the solution of (1.3) is suciently dierentiable so that substitution into (1.1a) makes sense. For this substitution to make sense,u(x) should be (at least) twice continuously dierentiable which, of course, requires that the given functionf(x) be continuous on (0;1). On the other hand, (1.3) may have solutions that cannot be substituted into (1.1a) because they are not suciently dierentiable. For example, we will see in later chapters that (1.3) has a solution for some functionsfthat are not continuous; these solutions cannot be solutions of (1.1).
1.2.1 Some Terminology
Let us now introduce some terminology that will be used throughout this book. We callu(x) aclassical solutionof (1.1) if, upon substitution into these relations, equality holds at every pointx2(0;1). We call solutions of (1.3) that are not classical solutions of (1.1)weak solutionsof the latter problem and (1.3) itself is referred to as aweak formulationof (1.1).4Analogously, problem (1.2) is termed a discrete weak problem. The functionsvhanduhin (1.2) are calledtestandtrialfunctions, respec- tively. The same terminology is used for the corresponding functionsvanduap- pearing in (1.3). Where do these names come from? Suppose someone gave us a functionuh(x) and claimed that it was a solution of the discrete weak problem (1.2). To verify the claim, we would put the functionuh(x) on \trial," i.e., we would determine if substituting it into (1.2) results in the left-hand side equal to the right-hand side for all possible test functionsvh(x)2Vh. The Dirichlet boundary condition (1.1b) and the Neumann boundary condi- tion (1.1c) are treated dierently within the framework of the weak formulation (1.3) or its approximation (1.2). First, we note that the Neumann boundary condi- tion (1.1c) is not imposed on the test or trial functions; however, we saw that ifu(x) satises the weak problem (1.3), then this Neumann boundary condition is indeed satised. Such boundary conditions, i.e., boundary conditions that are not required of the trial functions but are satised \naturally" by the weak formulation, are callednatural boundary conditions.On the other hand, nothing in the process we used to go from the weak problem (1.3) to the classical problem (1.1) implied that the Dirichlet boundary condition (1.1b) was satised. For this reason, we imposed the boundary condition as a constraint on the possible trial functions. Such bound-3
All the necessary steps can be made rigorous.
4The terminology about solutions is actually richer than we have indicated. There are also
solutions calledstrong solutionsintermediate between weak and classical solutions. We postpone further discussions of the dierent types of solutions until we have developed some additional mathematical background.
1.2. A Simple Example 5
ary conditions are calledessential boundary conditions.Note that for the discrete problem, the approximate solutionuh(x) satises (by construction) the essential boundary condition (1.1b) exactly, but that the natural boundary condition (1.1c) is only satised in a weak sense.
1.2.2 Polynomial Approximation
The two main components of the nite element method are its variational principles which take the form of weak problems and the use of piecewise polynomial approx- imation. In our example we use the discrete weak or variational formulation (1.2) to dene a nite element method but we have not used piecewise polynomials yet. In this example we choose the simple case of approximating with piecewise linear polynomials; that is, a polynomial which is linear when restricted to each subdivi- sion of the domain. To dene these piecewise polynomials, we rst discretize the domain [0;1] by lettingNbe a positive integer and setting thegrid pointsornodes fxjgNj=0so that 0 =x0< x1< x2<< xN1< xN= 1. Consequently we haveN+ 1 nodes andNelements. These nodes serve to dene a partition of the interval [0;1] into the subintervalsTi= [xi1;xi],i= 1;:::;N; note that we do not require the partition to be uniform. The subintervalsTiare the simplest examples ofnite elements.We choose the nite dimensional spaceVhin (1.2) to be the space of continuous piecewise linear polynomials over this partition of the given interval so that eachvhis a continuous piecewise linear polynomial. In particular, we denotevhi(x) =vh(x)jTifori= 1;:::;N; i.e.,vhi(x) is the restriction ofvh(x) to elementTi. For continuous piecewise linear polynomials, we formally dene the set of functionsVhas follows:vh(x)2Vhif (i)vhi(x) is a linear polynomial fori= 1;:::;N; (i)vhi(xi) =vhi+1(xi) fori= 1;:::;N1, and (iii)vh(x0) = 0:(1.5) Condition (i) of (1.5) guarantees that the functionvh(x) is a piecewise linear poly- nomial, Condition (ii) guarantees continuity and Condition (iii) guarantees thatvh vanishes atx= 0. With this choice forVh, (1.2) is called apiecewise linear nite element methodfor (1.1).
1.2.3 Connection with Optimization Problem
We note that the weak problems (1.2) and (1.3) can be associated with an opti- mization problem. For example, for a givenf(x) consider the functional
J(v;f) =12
Z 1 0 (v0)2dxZ 1 0 fv dx(1.6) and the unconstrained minimization problem: ndu(x)2Vsuch thatJ(u;f) J(v;f)8v2V ;
6 Chapter 1. Introduction
where the spaceVis dened as before. Using standard techniques of the calculus of variations, one can show that a necessary requirement for any minimizer of (1.6) is satisfying the weak problem (1.3). The converse is also true so that the two problems (1.6) and (1.3) are equivalent. In fact, in engineering applications this minimization approach is often used since it has the interpretation of minimizing an energy. However, not all weak problems have an equivalent minimization problem. We discuss this and its implications in later chapters.
1.3 How do you implement nite element methods?
We now translate the nite element method dened by (1.2) into something closer to what a computer can understand. To do this, we rst show that (1.2) is equivalent to a linear algebraic system once a basis forVhis chosen. Next we indicate how the entries in the matrix equation can be evaluated. Letfi(x)gNi=1be a basis forVh, i.e., a set of linearly independent functions such that any function belonging toVhcan be expressed as a linear combination of these basis functions. Note that we have assumed that the dimension ofVhis Nwhich is the case if we deneVhby (1.5). Thus, the setfi(x)gjNi=1has the property that it is linearly independent, i.e., N X i=1 ii(x) = 0 impliesi= 0 fori= 1;:::;N and it spans the space. That is, for eachwh2Vhthere exists real numberswi, i= 1;:::;N, such that w h(x) =NX i=1! ii(x): In the weak problem (1.2), the solutionuh(x) belongs toVhand the test function v h(x) is arbitrary inVh. Since the set spansVhwe can setuh=PN j=1jjand then express (1.2) in the following equivalent form: ndj2R1,j= 1;:::;N, such that Z 1 0ddx 0 NX j=1 jj(x)1 A ddx vhdx=Z 1 0 f(x)vhdx8vh2Vh: Since this equation must hold for each functionvh2Vhthen it is enough to test the equation for each element in the basis; that is, for eachi,i= 1;:::;N. Using this fact, the discrete problem is rewritten as ndj,j= 1;:::;N, such that NX j=1 Z1 0
0i(x)0j(x)dx
j=Z 1 0 f i(x)dxfori= 1;:::;N.(1.7)
1.3. How do you implement nite element methods? 7
Clearly (1.7) is a linear algebraic system ofNequations inNunknowns. Indeed, if the entries of the matrixKand the vectors~Uand~bare dened by K ij=Z 1 0
0i(x)0j(x)dx; Uj=j;andbj=Z
1 0 f(x)idxfori;j= 1;:::;N ; then, in matrix notation, (1.7) is given by K ~U=~b:(1.8) However, we have not yet completely formulated our problem so that it can be implemented on a computer. We rst need to choose a particular basis set and then the integrals appearing in the denition ofKand~bmust be evaluated or approximated. Clearly there are many choices for a basis for the space of continuous piecewise linear functions dened by (1.5). We will see in Section 1.6 that a judicious choice of the basis set will result in (1.8) being a tridiagonal system of equations and thus one which can be solved eciently inO(N) operations. For now, let's assume that we have chosen a specic basis and turn to the problem of evaluating or approximating the integrals appearing inKand~b. For a simple problem like ours we can often determine the integrals exactly; however, for a problem with variable coecients or one dened on a general polygonal domain inR2orR3this would not be practical. Even if we have software available that can perform the integrations, this would not lead to an ecient implementation of the nite element method. Thus to obtain a general procedure which would be viable for a wide range of problems, we approximate the integrals by a quadrature rule. For example, for the particular implementation we are developing here, we use the midpoint rule in each element to dene the composite rule Z 1 0 g(x)dx=NX k=1Z xk x k1g(x)dxNX k=1gxk1+xk2 (xkxk1): Using this rule for the integrals that appear in (1.8), we are led to the problem K h~Uh=~bh;(1.9) where the superscripthon the matrixKand the vector~bdenotes the fact that we have approximated the entries ofKand~bby using a quadrature rule to evaluate the integrals. Using the midpoint rule, the entries ofKhand~bhare given explicitly by K hij=NX k=1(xkxk1)0ixk1+xk2
0jxk1+xk2
;fori;j= 1;:::;N and b hi=NX k=1(xkxk1)fxk1+xk2 ixk1+xk2 ;fori= 1;:::;N :
8 Chapter 1. Introduction
In our example,Kh=K. To see this, recall that we have chosenVhas the space of continuous piecewise linear functions on our partition of [0;1] and thus the integrands inKare constant on each elementTi. The midpoint rule integrates constant functions exactly so even though we are implementing a quadrature rule, we have performed the integrations exactly. However, in general,~bh6=~bso that~Uh6=~U. Once the specic choice of a basis set forVhis made, the matrix problem (1.9) can be directly implemented on a computer. A standard linear systems solver can be used to obtain ~Uh. To eciently solve (1.9) the structure and properties of K hshould be taken into consideration. There are an innite number of possible basis sets for a nite element space. If the basis functions have global support, e.g., if they are nonzero over the whole interval (0;1), then, in general, the resulting discrete systems such as (1.8) or (1.9) will involve full matrices, i.e., matrices having possibly all nonzero entries. In order to achieve maximum sparsity in the discrete systems such as (1.8) or (1.9), the basis functions should be chosen to have local support, i.e., to be nonzero on as small a portion of the domain as possible. Typically the basis functions are required to havecompact support, that is, they are zero outside of a compact set; in nite elements the compact set consists of adjacent elements. In the one dimensional case we have considered here, the basis functions should be nonzero over as small a number of subintervals as possible. Such a basis set is provided by the \hat" functions dened by fori= 1;:::;N1,i(x) =8 >>>>>>:xxi1x ixi1forxi1xxi x i+1xx i+1xiforxixxi+1
0 otherwise(1.10)
and
N(x) =8
:xxN1x
NxN1forxN1xxN
0 otherwise.
(1.11) A sketch of these functions for the caseN= 4 on a nonuniform partition of [0;1] is given in Figure 1.3. Note that for alli= 1;:::;N,i(0) = 0,i(x) is continuous on [0;1], is a linear polynomial on each subinterval [xj1;xj],j= 1;:::;N, andi(x) is nonzero only in [xi1;xi+1]. It can be shown that the setfi(x)gNi=1given by (1.10) and (1.11) is linearly independent and forms a basis for the spaceVhdened by (1.5). Now let's examine the entries of the matricesKandKhappearing in the linear systems (1.8) or (1.9), respectively, for the basis functions dened in (1.10) and (1.11). It is easy to see that bothKij= 0 andKhij= 0 unlessjijj 1. Thus, for any number of elementsN, these matrices have nonzero entries only along the main diagonal and the rst upper and lower subdiagonals, i.e., they are
1.4. What is needed to analyze nite element methods? 9
Figure 1.1.Example of the hat basis functions for four intervals. tridiagonal matrices. This is the optimal sparsity achievable with piecewise linear nite elements. As a result, one can apply very inexpensive methods to solve the linear systems (1.8) or (1.9). 5
1.4 What is needed to analyze nite element
methods? In the previous section we saw how to implement the nite element method for a simple two point boundary value problem. In this section we turn to the question of determining how accurate the approximate solution is in our example. Specically, we want to derive an error estimate, i.e., a bound for the dierence between the nite element approximationuhsatisfying (1.2) and the weak solutionusatisfyingquotesdbs_dbs14.pdfusesText_20
[PDF] Finite Elements: Basis functions