1. Introduction to Finite Element Method
Chapter 1
Introduction to Finite Element Method

2. Introduction to Finite Element Method
Course description
This course is an introduction to Finite Element Method(FEM) which is
an essential technology in engineering and mathematical physics.
FEM has been generally used for structural analyses such as simulation tool
of static or dynamic behavior of elastic or non-elastic solids and structures.
This method is now used to analyze fluid flow problems,
heat transfer phenomena and electromagnetic field,
geo-physics problems and etc..
This FEM class is designed to give the students the ability to program
their own simulation code for the specific research fields
based on what they learned in the class.
Course Website;
Course material
Lecture Note based on the book by
Becker, Carey and Oden,  Finite Elements - An Introduction Vol.1  1981.
Teaching Assistant
  Jung Eun Lee Tel. 7389)

3. Introduction to Finite Element Method
Course Introduction
Students will learn how to solve partial differential equations(PDEs) by numerical method in this course. Especially, the elliptic boundary value problems(BVPs) discretized by finite element technique by using the method of weighted residual and Galerkin approximation will be mainly considered. For thorough understanding on the Finite Element algorithm, students are required to program their own Finite Element code of one-dimensional BVPs according to the theories and algorithm they learned in the course. Direct time integration methods and eigen-value system solvers for the parabolic and the hyperbolic PDEs along with the semi-discretization are also taught briefly in this course. In addition to these classical theories, recent technologies including efficient algorithms and parallel computing techniques will be introduced. For further understanding of FEM, IPSAP, which is developed as an efficient parallel finite element program and DIAMOND which is a pre-post GUI program for IPSAP ( will be used to learn the convergence behaviour of the numerical solutions of partial differential equations upon mesh refinement. The homework and term project will be given to exercise the methods they learn

4. Introduction to Finite Element Method
References
- David S. Burnett, "Finite Element Analysis." Addison Wesley,
- Zienkiewitz and Taylor, "The Finite Element Method.", McGraw Hill, 2005
- Klaus J. Bathe, "Finite Element Procedures.", Prentice Hall, 1996.
- Thomas J. R. Hughes, "The Finite Element Method: Linear Static and Dynamic Finite Element Analysis ." , Prentice Hall, 2000
- R.D. Cook, D.S Malkus and M.E. Plesha, "Concepts and Applications
of Finite Element Analysis," 4th ed. Wiley, 2002
- IPSAP (Internet Parallel Structural Analysis Program), DIAMOND Manual.

5. 1.1 Sources of the Problems
1 - D  Heat Flow
     Energy conservation law ＋ Fourier's law
Section area A, length dx,             
By the energy conservation law,
From Fourier's law,
If is constant

6. 1.1 Sources of the Problems
2) Elastic Rod
    Force equilibrium ＋ Hooke's law     
σ : stress,     f(x) : body force
Hooke's law
;         : , young's modulus ,     : displacement
Or distributed load per unit length,
Essential BC(EBC) : u is given
Natural  BC(NBC) : σ is given

7. 1.1 Sources of the Problems
3) Cable deflection
     Balance of transverse force ＋ tension
F   : transverse component of tension , T
f(x) : distributed transverse force
w   : transverse displacement

8. 1.1 Sources of the Problems
If we assume that Θ is angle between the cable and the horizontal axis,
then the equilibrium gives the equation,
EBC : w       is given
NBC : T or f  is given

9. 1.2 General form of Two point Boundary value Problem
(1)
Jump Conditions
: x1 is material discontinuous point
but k, c, b are discontinuous
: point source   (2)
: a point of discontinuous distributed source
         is not to be defined
BC's
(3)

10. 1.3 Variational Formulation
Define the residual,
And then Eq (1) can be written by
(4)
And we have
Therefore,
(5)

11. 1.3 Variational Formulation
Also,
  Note that the set of eqs (1) (2) (3) is equivalent to the following statement,
      
Find          such that
(6)

12. 1.3 Variational Formulation
◉ What is the characteristics of Variational "Weak" Formulation(VWF) ?
  To find “ twice differentiable solution” to satisfy the second-order differential equation (1)
 to find the solutions from the square integrable “functions and their derivatives”
Therefore, the condition for the solutions becomes “weak”
Characteristics of VWF (6)
1)  Solutions which satisfy weaker conditions are acceptable.
This means that solutions can be found in wider range of function space,
that is,
2) Jump Condition and Boundary Condition are included in one integral equation.
3) If C=0, it becomes symmetric form
4) Various boundary conditions can be easily adopted

13. 1.4 Galerkin Approximation
Since the dimension of function space          is infinite, the functions can be written.
where, is a basis function of
But, we need to find a set of approximated solutions in the finite dimension to be solved by computer. The functions should be approximated as the functions in the finite dimension,
where, is basis functions of finite subspace,
And then, the problem (6) becomes the following statement,
Find       such that
(7)

14. 1.4 Galerkin Approximation
Or,
(8)
From the relations between and , the expression becomes

15. 1.4 Galerkin Approximation
By arrangement of the previous equation (8), we have the equation
(9)
Or, we can write
From the condition of , the eq. (9) becomes,
Finally, we obtain the N- dimensional simultaneous equation (10).
(10)

16. 1.4 Galerkin Approximation
where,
If we use the matrix form in direct notation, we write
Solutions could be obtained by solving the equation (10)
And, finally, the approximated solution for the set of equation (1) can be expressed in the form of the linear combination with the as
 We note that matrix K is symmetric if
여기서
Matrix Form으로는

17. 1.5 Symmetrization of the ODE
If , the matrix of the eq. (10) becomes nonsymmetric.
This nonsymmetry requires more memory space and the computation time in computation. It will be desirable if we can make the equation symmetric.
 We start from the given DE,
To change the equation into the symmetric form, we try to find some function
which satisfies the relation,
and then we have the symmetrized equation (11) through manipulation,
(11)

18. 1.6 Trial-Solution Method
1.6.1 General procedure and methods for the approximated solutions
: Trial Solution
This is generally expressed as Linear Combination(LC)
of known functions
General procedure of Trial Solution Methods
1. Construct the trial solution, , in terms of basis functions in the approximated space
2. Determine the method to obtain optimal solution
3. Predict the accuracy of

19. 1.6 Trial-Solution Method
( need be kept for boundary condition)
2. Method to obtain the Best possible solution
a. Method of weighted residual(MWR)
(1) Collocation method -- Collocation FEM
(2) Subdomain method Subdomain FEM
(3) Least-square method LS FEM
(4) Galerkin method Galerkin FEM
b. Ritz variation method(RVM) : For the Min/max problem
3. If we set as the point-wise error
the global error norm can be defined by

20. Example
For the generalized 1-D PDE in eqn (1), let us have an example problem with
the coefficients values of
Then, the equation is .

21. 1.6.2 Example
At first, we can consider polynomial forms as approximated solution,
If we take N=4, then the trial solution will be
With the BC, we have
Elimination of with above two equation gives the equation,
Now we have two undetermined coefficients,

22. 1.6.3 Approximate solutions by MWR
The residual is
MWR (Method of Weighted Residual) states
where, is weight function, or Test function

23. (1) The Collocation Method
Select number of node points same as the unknown number of
By using the definition of dirac delta,
, namely,
Here we have N=2 and select
And then the solutions are

24. (1) The Collocation Method
Accuracy Check

25. (2) Sub-Domain Method
We assume that he average of residual in each sub-domain be zero
This is called the sub-domain method. These domains could be overlapped. 1 2
For the previous example with
and then,

26. (2) Sub-Domain Method

27. (3) The Least-Square Method
Determine , to minimize the integral value of square of residual
in whole domain;
By differentiation,
Weight function becomes,
In the case of the previous example,
then, two equations are
By integral,

28. (3) The Least-Square Method
Therefore,

29. (4) Galerkin Method
Select the weight function from the same basis with Trial function
This means that ,
and then,
Applying to the previous example, we have

30. (4) Galerkin Method
Solutions :
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31. 1.6.4 Ritz Variational Method
(1) Calculus of Variation
We define a functional,
Now we introduce a class of minimization problems
as the following minimization problem,
“Find such that “
Let us use the variational method to obtain minimizer

32. Calculus of Variation Substitute the following function for in where,
could be obtained by
If , the extremum value could be found
The positiveness of the second variation means that minimize functional.
To have the non-negative value of
and the arbitrariness of forces the first variation being zero,
and then we obtain the equation, after the Integrated by Parts.

33. (2) Ritz approximation
For the previous minimization problem, we first select the proper trial-function and then substitute this to the functional to obtain unknowns minimizing the functional value. This method is called the Ritz approximation.
Let us select
And we substitute this into the functional,
Minimization with respect to means
Applying this method to the previous example, the functional for minimization becomes
The minimization of the approximated functional gives
Note that the equation is same as the one from Galerkin Method

34. Review
Galerkin

35. ◎ Formally Self Adjoint
With the definition of Inner Product
Let us visit the differential equation,
We define an operator A such that
and then,
We call this A as the “formally Self Adjoint operator”

36. 1.7 Finite Element basis function
(1) Basic principles of Finite Element Basis Function
a. Simple expression for easy numerical integration
b. Smoothness requirement of the basis function such the integral in the variational formulation should be meaningful,
c. At the ith node should satisfy
Basis functions which are easily generalized and formulated are necessary.
Finite element mesh: divide the given domain with finite elements.
Node: boundary nodal point constructing element.
This orthonormal condition in the finite dimensional space means

37. 1.7 Finite Element basis function
(2) Simple example of Finite Element basis function
Piecewise linear function
In the case of
it is called as Hat function

38. 1.7 Finite Element basis function

39. 1.7 Finite Element basis function
(3) Finite Element Interpolation
Lagrange Finite Element
a. Mapping function at ( ) to (-1,1),  
: master element
b.          , kth Lagrange polynomial

40. 1.7 Finite Element basis function
c.   first,
second,

41. 1.7 Finite Element basis function
third,

42. 1.7 Finite Element basis function
(4) Characteristics of Lagrange FE Basis function
   a. Slopes of solution will be discontinuous at the boundary node point
between elements since they are Lagrange functions in
b. All are 1, which means that the summation of all shape functions becomes 1.
c. Summation of all differentiated basis functions becomes zero.
The characteristics b. and c. could be used to verify shape functions when you do the Finite Element programming.

43. 1.8 Galerkin Finite Approximation
We now know that the system of differential equation (1) can be changed
by MWR and Galerkin approximation into a simultaneous equation,
(10)
where,
And then, the next step may be how to implement the concept of finite elements
to the equation (10).
Specially, convenient and efficient construction and operation of the K matrix
and F vector in the equation are very important for this method.

44. 1.8 Galerkin Finite Approximation
Therefore, we will consider the following items step by step in the following lectures
     1. How to construct and ?
       
 ․ Take advantage of the Finite Element Concept
      ․ Efficient numerical scheme to  obtain the values for the matrix and vector        
        ․ Element by Element construction of the matrix and vector
- Assembly of element matrix and vector from FE concept
     2. How to solve the simultaneous equations efficiently
- efficient solver to take advantage of the sparseness of FE matrix
        Band Solver            -----   Solving the band structure of FE matrix
        Skyline Solver        -----        Utilization of sparseness of FE matrix
        Frontal Solver         -----          Best utilization of FE concept
        Iterative method In the case of Large DOF problems
(For example, Pre-Conditioned Conjugate Gradient Method)    

45. 1.8 Galerkin Finite Approximation
Calculation process:
(1) Finite element in the range of , divide by
Node at to satisfy
Take one element ;
Weak form at Element can be written by

46. 1.8 Galerkin Finite Approximation
 By Galerkin approximation in element
where is number of nodes in
and is basis function in
Set

47. 1.8 Galerkin Finite Approximation
     
and then, we have equation in this element
where,

48. 1.8 Galerkin Finite Approximation
(2) Assemble
Let us look at three element in neighbor. # #
and then,

49. 1.8 Galerkin Finite Approximation
Since the flux should be in equilibrium, we have
and
Therefore Global Stiffness Matrix and Global Force Vector can be obtained

50. 1.8 Galerkin Finite Approximation
(3) Prescription Boundary and jump condition
Prescription of the boundary conditions with
will produce,

51. 1.8 Galerkin Finite Approximation
Jump condition;
Finally, we have the Global matrix and the Global force vector

52. 1.8 Galerkin Finite Approximation
Three kinds of Boundary conditions
(a)  Dirichle BC 
ⅰ) Eliminate unknown and , then solve the remain (N-2) dof problem
   
ⅱ) Apply       
This approach is an application of the Exterior penalty method.

53. 1.8 Galerkin Finite Approximation
It is given in the form of             ,
This form is same as the general boundary condition,
in the set of equation in (1).
Then an appropriate values of α, β, γ  will prescribe the boundary condition                                
(c) Mixed BC
(b) Neumann BC
The slope of the solution ( or flux ) will be prescribed
in both side of boundary points

54. 1.9 Error Estimation, Accuracy, Convergence
(a)  various error norm
All the definition of error norm should satisfy the following convergence property,

55. 1.9 Error Estimation, Accuracy, Convergence
(b)  a- priori error estimate
   The general error estimation is possible for the simple equation (1)
Let say,
then, the variational form is or

56. 1.9 Error Estimation, Accuracy, Convergence
From the variational statement in finite dimension,
The ② - ③ will result
then,
This means that the error and are orthogonal each other.
(w. r. t         norm )

57. 1.9 Error Estimation, Accuracy, Convergence
From the Cauchy - Schwartz  Inequality,
The equation ② can be written as
Therefore, we have an inequality equation,

58. 1.9 Error Estimation, Accuracy, Convergence
     If we assume  be a special FE weight function interpolating
Let us have a linear interpolation function such that
Three term Taylor series expansion of at is
where the last term in is the remainder, and

59. 1.9 Error Estimation, Accuracy, Convergence
Since , the equation will be

60. 1.9 Error Estimation, Accuracy, Convergence
If    has bounded second derivative, i.e., then e(x) can be written as
If we choose the point as the extreme point of
and note that, from the interpolation property,
Since and
and

61. 1.9 Error Estimation, Accuracy, Convergence
Then,
since the interpolation function in this example is linear polynomial.
Let say,
By the same procedure as the previous process,

62. 1.9 Error Estimation, Accuracy, Convergence
After some arrangements, we obtain ⑯
In summary from the previous results, the final error estimates are

63. 1.9 Error Estimation, Accuracy, Convergence
In general form, it could written by