finite element method by dr mirzaeid

7/27/2019 Finite Element method by Dr Mirzaeid

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Fini te Element Methods

Lecture Notes: 1Dr Majid Mirzaei, Dept. of Mechanical Eng., [email protected]://www.modares.ac.ir/eng/mmirzaei/Finite.htm

"Our minds are finite, and yet even in these circumstances of finitude we are surrounded

by possibilities that are infinite, and the purpose of human life is to grasp as much as we

can out of that infinitude."

Alfred North Whitehead, 1941

The Finite Element Method (FEM)One of the most important advances in applied mathematics in the 20th century has been

the development of the Finite Element Method as a general mathematical tool for

obtaining approximate solutions to boundary-value problems.

The theory of finite elements draws on almost every branch of mathematics and can beconsidered as one of the richest and most diverse bodies of the current mathematical

knowledge.

Mathematical Modeling of Physical SystemsThe field of Mechanics can be subdivided into three major areas: Theoretical, Applied,

and Computational.

Theoretical Mechanics deals with fundamental laws and principles of mechanics studied

for their intrinsic scientific value.

Applied Mechanics transfers this theoretical knowledge to scientific and engineeringapplications, especially through the construction of mathematical models of physical

phenomena.

Computational Mechanics solves specific problems by simulation through numerical

methodsimplemented on digital computers.

Due to the complexity of physical systems, some approximation must be made in the

process of turning physical reality into a mathematical model. It is important to decide at

what points in the modeling process these approximations are made. This, in turn,determines what type of analytical or computational scheme is required in the solution

process. Let us consider a diagram of the two common branchesof the general modeling-

solution process:
mailto:[email protected]://www.modares.ac.ir/eng/mmirzaei/Finite.htmhttp://www.modares.ac.ir/eng/mmirzaei/Finite.htmmailto:[email protected]


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Physical

Problem

Simplified ModelComplicated Model

Exact Solution

ForApproximate Model

Approximate SolutionFor

Exact Model

FEM APPROACH

Figure 1

For many real world problems the second approach is in fact the only possibility. For

instance suppose that the aim is to find the thermo-mechanical stresses in an air-cooledturbine blade depicted in Figure 2.

Figure 2


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The complex three-dimensional geometry of the blade along with the combined thermal

and mechanical loadings makes the analysis of the blade a formidable task. Nevertheless,many powerful commercial finite element packages are available that can be

implemented to perform this task with relative ease.

Figure 3

THE FEM ANALYSIS PROCESSA model-based simulation process using FEM consists of a sequence of steps. Thissequence takes two basic configurations depending on the environment in which FEM is

used. These are referred to as the Mathematical FEMand the Physical FEM.

The Mathematical FEMAs depicted in Figure 4, the centerpiece in the process steps of the Mathematical FEM is

the mathematical model which is often an ordinary or partial differential equation inspace and time. Using the methods provided by the Variational Calculus, a discrete finite

element model is generated from of the mathematical model. The resulting FEMequations are processed by an equation solver, which provides a discrete solution. In this

process we may also think of an ideal physical system, which may be regarded as a

realization of the mathematical model. For example, if the mathematical model is the

Poissons equation, realizations may be a heat conduction problem. In MathematicalFEM this step is unnecessary and indeed FEM discretizations may be constructed without

any reference to physics.

The concept oferrorarises when the discrete solution is substituted in the mathematical

and discrete models. This replacement is generically called verification. The solutionerroris the amount by which the discrete solution fails to satisfy the discrete equations.

This error is relatively unimportant when using computers. More relevant is the

discretization error, which is the amount by which the discrete solution fails to satisfy themathematical model.


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I deal

Physical

System

Mathematical

Model

Discrete Model

VerificationSolution error

VerificatioDiscretization + Solution error

Solution

Figure 4

The Physical FEMAs depicted in Figure 5, in the Physical FEM process the centerpiece is the physical

system to be modeled. The processes of idealization and discretization are carried out

concurrently to produce the discrete model. Indeed FEM discretizations may be

constructed and adjusted without reference to mathematical models, simply fromexperimental measurements. The concept of error arises in the physical FEM in two

ways, known as verification and validation, respectively. Verification is the same as in

the Mathematical FEM: the discrete solution is replaced into the discrete model to get thesolution error. As noted above, this error is not generally important.

Validation tries to compare the discrete solution against observationby computing the

simulation error, which combines modeling and solution errors. Since the latter istypically insignificant, the simulation error in practice can be identified with the

modeling error. Comparing the discrete solution with the ideal physical system would in

principle quantify the modeling errors.


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Physical

System

Ideal Mathematical

Model

Discrete Model

VERFICATION

Solution error

VALIDATION

Solution

Simulation Error = modeling + solution error

Discretization

Figure 5

MATHEMATICAL PRELIMINARIES

TENSORSIn science and engineering, we usually deal with physical quantities which are

independent of any particular coordinate system that may be used to describe them. At

the same time, these physical quantities are very often specified in a particular coordinatesystem by their components. We generally refer to theses quantities as tensors.

Specifying the components of a tensor in one coordinate system determines the

components in any other system. In practice, many physical laws are expressed by tensor

equations. Because tensor transformations are linear and homogeneous, such tensorequations, if they are valid in one coordinate system, are valid in any other coordinate

system. This invariance of tensor equations under a coordinate transformation is one of

the principal reasons for the usefulness of tensor methods. In a three-dimensional

Euclidean space, such as ordinary physical space, the number of components of a tensoris 3N, where N is the order of the tensor. Accordingly a tensor of order zero is specified

in any coordinate system in three-dimensional space by one component. Tensors of order

zero are called scalars. Physical quantities having magnitude only are represented by

scalars. Tensors of order one have three coordinate components in physical space and are


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known as vectors. Quanti ties possessing both magni tude and direction ar e represented

by vectors. Second-order tensors correspond to dyadics. Higher order tensors such astriadics, or tensors of order three, and tetradics, or tensors of order four can also be

defined.

Vector SpacesConsider the set V of all vectors a, b, on which there are defined two algebraic

operations of vector addition, and multiplication of vectors by scalars. Based on the

above consideration, we may define an algebraic structure called a realvector space (or

real linear space) which includes many sets (vectors, matrices, functions, etc.). Acomplex vector space is obtained if complex numbers are taken as scalars.

The concept of a vector space is basic in functional analysis, which has applications to

differential equations, numerical analysis, and other practical fields. In particular, we can

think of a function as being a vector with an infinity of components (an infinite

dimensional vector), the value of each component can be specified by a particular valueof function in some interval.

Spaces where vectors arefunctions are often termedfunction spaces. However, it should

be noted that in many aspects the individual functions are treated as vectors, so that muchof our geometric interpretations such as orthogonalityand distancecan be applied to

them. Consider the subset of the space of continuous functions given by

{u(x) : u(x) is continuous on [0, 1] and u(0) = 1}

The above is not a vector space, because nonzero vector addition and non-unit scalar

multiplication yield functions that do not satisfy the boundary condition at x = 0. For

example, ifu(x) and v(x) are members of this set, then u(x) + v(x) cannot be, since u(0) +v(0) = 21. The fundamental problem here is the fact that the non-homogenous condition

u(0) = 1precludes this set from containing the "zero vector" defined by the function u(x)

= 0 everywhere. This example indicates that vector spaces must contain the zero vector

as a member, which geometrically means that all vector spaces (including componentsubspaces) must contain the origin of the coordinate system.

In general, the precise definition of vector spaces does not admit such important concepts

as orthogonality of two vectors, or the length of a vector. These topics are topologicalconstructs, and not all vector spaces possess them.

Definition: The inner product( , ) is a real-valued function defined on a linear vectorspace Vthat satisfies the following criteria:

(1) (f,g) = (g,f)for all vectorsgandfin V

(2) (f,g+ h) = (f,g) + (f, h)for all vectorsf,gand h in V

(3) a (f,g) = (a f,g)for all vectorsfandgin Vand scalars a(4) (f,f) > 0 for all nonzero vectorsfin V


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Two functionf(x) andg(x) are said to be orthogonal on the interval [ ],a b if:

( ) ( ) 0b

af x g x dx =

Definition: A norm || || is a real-valued function defined on a linear vector space Vthatsatisfies the following criteria:

(1) ||f|| 0 for every vectorfin V(2) || af|| = | a | ||f|| for every vectorfin Vand each scalara

(3) ||f + g|| ||f|| + ||g|| for everyfandgin V(4) ||f|| = 0 only whenf = 0

The norm function generalizes the notion of the length of a vector. There are often many

choices of norms available for use with a given vector space.

If the norm function is defined via the inner product so that,

2( , ) ( )b

af f f f x dx= =

then we say that the norm is induced by the inner product. Complete vector spaces that

possess a norm induced by an inner product are termedHilbert Spaces, and many of the

solutions to boundary-value-problems in engineering and science reside in HilbertSpaces. The most important thing to keep in mind is that we are living in a Hilbert Space.

In fact the Hilbert Space is merely the generalization to the infinite-dimensional case of

the geometry that we are familiar with from the everyday world.

Vector spaces whose elements are solutions of differential equations commonly have aHilbert Space structure, and if we are interested not only in the solution u(x), but also its

derivatives, then the desired inner product may involve derivatives as well as functional

values. The presence of these derivatives in the inner product is the distinguishing feature

of a Sobolev Space. A Sobolev Space is merely a Hilbert Space where the inner productdepends upon derivatives of the underlying functions. Nearly all of the solution spaces

we will consider during our development of finite element models are Sobolev Spaces,

and this terminology is widely used in the literature.

Calculus of Variations

An alternative statement for a Boundary-Value-Problem can be found by casting the

problem in a variational form. This formulation may involve expressing the exactsolution of the problem as the minimizerof a scalar quantity that generally has a physical

interpretation of energy. The mathematical framework underlying the equivalence of the


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strong, weak, and functionalforms of the problem is termed the Calculus of Variations,

and constitutes one of the oldest and most diverse topics in mathematics.

Variational Formulation of Boundary-Value Problems

We may consider the FEM as a piecewise application of variational methods. In this

section we use variational formulation to develop an alternative integral framework for

solving physical problems that are usually modeled using differential equations.

Let us start with the following BVP:

2

2( ) ,

( ) ( ) 0

d uf x b x a

dx

u a u b

=

finite element method by dr mirzaeid

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