1. An Introduction to the Finite Element Analysis
Presented by
Niko Manopulo

2. Agenda PART I Introduction and Basic Concepts
Computational Methods
Idealization
Discretization
Solution
The Finite Elements Method
FEM Notation
Element Types
Mechanichal Approach
The Problem Setup
Strain Energy
External Energy
The Potential Energy Functional

3. Agenda PART II Mathematical Formulation
The Mathematics Behind the Method
Weighted Residual Methods
Approxiamting Functions
The Residual
Galerkin’s Method
The Weak Form
Solution Space
Linear System of Equations
Connection to the physical system

4. Agenda PART III Finite Element Discretization
The Trial Basis
Matrix Form of the Problem
Element Stiffness Matrix
Element Mass Matrix
External Work Integral
Assembling
Linear System of Equations
References
Question and Answers

5. Introduction and Basic Concepts
PART I
Introduction and Basic Concepts

6. 1.0 Computational Methods

7. 1.1 Idealization Mathematical Models Implicit vs. Explicit Modelling
“A model is a symbolic device built to simulate and predict aspects of behavior of a system.”
Abstraction of physical reality
Implicit vs. Explicit Modelling
Implicit modelling consists of using existent pieces of abstraction and fitting them into the particular situation (e.g. Using general purpose FEM programs)
Explicit modelling consists of building the model from scratch

8. 1.2 Dicretization Finite Difference Discretization
The solution is discretized
Stability Problems
Loss of physical meaning
Finite Element Discretization
The problem is discretized
Physical meaning is conserved on elements
Interpretation and Control is easier

9. 1.3 Solution Linear System Solution Algorithms
Gaussian Elimination
Fast Fourier Transform
Relaxation Techniques
Error Estimation and Convergence
Analysis

10. 2.0 Finite Element Method Two interpretations Physical Interpretation:
The continous physical model is divided into finite pieces called elements and laws of nature are applied on the generic element. The results are then recombined to represent the continuum.
Mathematical Interpretation:
The differetional equation reppresenting the system is converted into a variational form, which is approximated by the linear combination of a finite set of trial functions.

11. 2.1 FEM Notation Elements are defined by the following properties:
Dimensionality
Nodal Points
Geometry
Degrees of Freedom
Nodal Forces
(Non homogeneous RHS of the DE)

12. 2.2 Element Types

13. 3.0 Mechanical Approach Simple mechanical problem
Introduction of basic mechanical concepts
Introduction of governing equations
Mechanical concepts used in mathematical derivation

14. 3.1 The Problem Setup

15. 3.2 Strain Energy
Hooke’s Law:
where
Strain Energy Density:

16. 3.2 Strain Energy (cont’d)
Integrating over the Volume of the Bar:
All quantities may depend on x.

17. 3.3 External Energy Due to applied external loads
The distributed load q(x)
The point end load P. This can be included in q.
External Energy:

18. 3.4 The Total Potential Energy Functional
The unknown strain Function u is found by minimizing the TPE functional described below:

19. Mathematical Formulation
PART II
Mathematical Formulation

20. 4.0 Historical Background
Hrennikof and McHenry formulated a 2D structural problem as an assembly of bars and beams
Courant used a variational formulation to approximate PDE’s by linear interpolation over triangular elements
Turner wrote a seminal paper on how to solve one and two dimensional problems using structural elements or triangular and rectangular elements of continuum.

21. 4.1 Weighted Residual Methods
The class of differential equations containing also the one dimensional bar described above can be described as follows :

22. 4.1 Weighted Residual Methods
It follows that:
Multiplying this by a weight function v and integrating over the whole domain we obtain:
For the inner product to exist v must be “square integrable”
Therefore:
Equation (2) is called variational form

23. 4.2 Approximating Function
We can replace u and v in the formula with their approximation function i.e.
The functions fj and yj are of our choice and are meant to be suitable to the particular problem. For example the choice of sine and cosine functions satisfy boundary conditions hence it could be a good choice.

24. 4.2 Approximating Function
U is called trial function and V is called test function
As the differential operator L[u] is second order
Therefore we can see U as element of a finite-diemnsional subspace of the infinite-dimensional function space C2(0,1)
The same way

25. 4.3 The Residual
Replacing v and u with respectively V and U (2) becomes
r(x) is called the residual (as the name of the method suggests)
The vanishing inner product shows that the residual is orthogonal to all functions V in the test space.

26. 4.3 The Residual Substituting into
and exchanging summations and integrals we obtain
As the inner product equation is satisfied for all choices of V in SN the above equation has to be valid for all choices of dj which implies that

27. 4.4 Galerkin’s Method
One obvious choice of would be taking it equal to
This Choice leads to the Galerkin’s Method
This form of the problem is called the strong form of the problem. Because the so chosen test space has more continuity than necessary.
Therefore it is worthwile for this and other reasons to convert the problem into a more symmetrical form
This can be acheived by integrating by parts the initial strong form of the problem.

28. 4.5 The Weak Form Let us remember the initial form of the problem
Integrating by parts

29. 4.5 The Weak Form A(v,u) is called Strain Energy.
The problem can be rewritten as
where
The integration by parts eliminated the second derivatives from the problem making it possible less continouity than the previous form. This is why this form is called weak form of the problem.
A(v,u) is called Strain Energy.

30. 4.6 Solution Space
Now that derivative of v comes into the picture v needs to have more continoutiy than those in L2. As we want to keep symmetry its appropriate to choose functions that produce bounded values of
As p and z are necessarily smooth functions the following restriction is sufficient
Functions obeying this rule belong to the so called Sobolev Space and they are denoted by H1. We require v and u to satisfy boundary conditions so we denote the resulting space as

31. 4.7 Linear System of Equations
The solution now takes the form
Substituting the approximate solutions obtained earlier in the more general WRM we obtain
More explicitly substituting U and V (remember we chose them to have the same base) and swapping summations and integrals we obtain

32. 4.8 Connection to the Physical System
Mechanical Formulation Mathematical Formulation

33. Finite Element Discretization
PART III
Finite Element Discretization

34. 5.0 Finite Element Discretization
Let us take the initial value problem with constant coefficients
As a first step let us divide the domain in N subintervals with the following mesh
Each subinterval is called finite element.

35. 5.1 The Trial Basis
Next we select as a basis the so called “hat function”.

36. 5.1 The Trial Basis
With the basis in the previous slide we construct our approximate solution U(x)
It is interesting to note that the coefficients correspond to the values of U at the interior nodes

37. 5.1 The Trial Basis
The problem at this point can be easily solved using the previously derived Galerkin’s Method
A little more work is needed to convert this problem into matrix notation

38. 5.2 Matrix Form of the Problem
Restricting U over the typical finite element we can write
Which in turn can be written as
in the same way

39. 5.2 Matrix Form of the Problem
Taking the derivative
Derivative of V is analogus

40. 5.2 Matrix Form of the Problem
The variational formula can be elementwise defined as follows:

41. 4.3 The Element Stiffnes Matrix
Substituting U,V,U’ and V’ into these formulae we obtain

42. 4.4 The Element Mass Matrix
The same way

43. 4.5 External Work Integral
The external work integral cannot be evaluated for every function q(x)
We can consider a linear interpolant of q(x) for simplicity.
Substituting and evaluating the integral
Element load vector

44. 4.6 Assembling
Now the task is to assemble the elements into the whole system in fact we have to sum each integral over all the elements
For doing so we can extend the dimension of each element matrix to N and then put the 2x2 matrix at the appropriate position inside it
With all matrices and vectors having the same dimension the summation looks like

45. 4.6 Assembling
Doing the same for the Mass Matrix and for the Load Vector

46. 4.7 Linear System of Equations
Substituting this Matrix form of the expressions in
we obtain the following set of linear equations
This has to be satisfied for all choices of d therefore

47. References An Analysis of the Finite Element Method Carlos Felippa
Joseph E Flaherty,Amos Eaton Professor
Gilbert Strang, George J. Fix
An Analysis of the Finite Element Method
Prentice-Hall,1973