Finite Element Method

1
Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 3

• THE FINITE ELEMENT
• METHOD

2
CONTENTS

• STRONG AND WEAK FORMS OF GOVERNING EQUATIONS
• HAMILTONS PRINCIPLE
• FEM PROCEDURE
• Domain discretization
• Displacement interpolation
• Formation of FE equation in local coordinate
  system
• Coordinate transformation
• Assembly of FE equations
• Imposition of displacement constraints
• Solving the FE equations
• STATIC ANALYSIS
• EIGENVALUE ANALYSIS
• TRANSIENT ANALYSIS
• REMARKS

3
STRONG AND WEAK FORMS OF GOVERNING EQUATIONS

• System equations strong form, difficult to
  solve.
• Weak form requires weaker continuity on the
  dependent variables (u, v, w in this case).
• Weak form is often preferred for obtaining an
  approximated solution.
• Formulation based on a weak form leads to a set
  of algebraic system equations FEM.
• FEM can be applied for practical problems with
  complex geometry and boundary conditions.

4
HAMILTONS PRINCIPLE

• Of all the admissible time histories of
  displacement the most accurate solution makes the
  Lagrangian functional a minimum.
• An admissible displacement must satisfy
• The compatibility equations
• The essential or the kinematic boundary
  conditions
• The conditions at initial (t1) and final time (t2)

5
HAMILTONS PRINCIPLE

• Mathematically

LT-PWf
where
(Kinetic energy)
(Potential energy)
(Work done by external forces)
6
FEM PROCEDURE

• Step 1 Domain discretization
• Step 2 Displacement interpolation
• Step 3 Formation of FE equation in local
  coordinate system
• Step 4 Coordinate transformation
• Step 5 Assembly of FE equations
• Step 6 Imposition of displacement constraints
• Step 7 Solving the FE equations

7
Step 1 Domain discretization

• The solid body is divided into Ne elements with
  proper connectivity compatibility.
• All the elements form the entire domain of the
  problem without any overlapping compatibility.
• There can be different types of element with
  different number of nodes.
• The density of the mesh depends upon the accuracy
  requirement of the analysis.
• The mesh is usually not uniform, and a finer mesh
  is often used in the area where the displacement
  gradient is larger.

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Step 2 Displacement interpolation

• Bases on local coordinate system, the
  displacement within element is interpolated using
  nodal displacements.

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Step 2 Displacement interpolation

• N is a matrix of shape functions

Shape function for each displacement component at
a node
where
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Displacement interpolation

• Constructing shape functions
• Consider constructing shape function for
• a single displacement component
• Approximate in the form

pT(x)1, x, x2, x3, x4,..., xp
(1D)
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Pascal triangle of monomials 2D
12
Pascal pyramid of monomials 3D
13
Displacement interpolation

• Enforce approximation to be equal to the nodal
  displacements at the nodes
• di pT(xi)? i 1, 2, 3,
  ,nd
• or
• deP ?
• where

,
14
Displacement interpolation

• The coefficients in ? can be found by

• Therefore, uh(x) N( x) de

15
Displacement interpolation

• Sufficient requirements for FEM shape functions

(Delta function property)
1.
(Partition of unity property rigid body
movement)
2.
(Linear field reproduction property)
3.
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Step 3 Formation of FE equations in local
coordinates
Since U Nde
Strain matrix
e LU
e L N de B de
Therefore,
?
or
where
(Stiffness matrix)
17
Step 3 Formation of FE equations in local
coordinates
Since U Nde
?
or
where
(Mass matrix)
18
Step 3 Formation of FE equations in local
coordinates
(Force vector)
19
Step 3 Formation of FE equations in local
coordinates
(Hamiltons principle)
?
FE Equation
20
Step 4 Coordinate transformation
(Local)
(Global)
where
,
,
21
Step 5 Assembly of FE equations

• Direct assembly method
• Adding up contributions made by elements sharing
  the node

(Static)
22
Step 6 Impose displacement constraints

• No constraints ? rigid body movement (meaningless
  for static analysis)
• Remove rows and columns corresponding to the
  degrees of freedom being constrained
• K is semi-positive definite

23
Step 7 Solve the FE equations

• Solve the FE equation,
• for the displacement at the nodes, D
• The strain and stress can be retrieved by using e
  LU and s c e with the interpolation, UNd

24
STATIC ANALYSIS

• Solve KDF for D
• Gauss elmination
• LU decomposition
• Etc.

25
EIGENVALUE ANALYSIS
(Homogeneous equation, F 0)
Assume
Let
?
(Roots of equation are the eigenvalues)
K - li M fi 0
(Eigenvector)
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EIGENVALUE ANALYSIS

• Methods of solving eigenvalue equation
• Jacobis method
• Givens method and Householders method
• The bisection method (Sturm sequences)
• Inverse iteration
• QR method
• Subspace iteration
• Lanczos method

27
TRANSIENT ANALYSIS

• Structure systems are very often subjected to
  transient excitation.
• A transient excitation is a highly dynamic time
  dependent force exerted on the structure, such as
  earthquake, impact, and shocks.
• The discrete governing equation system usually
  requires a different solver from that of
  eigenvalue analysis.
• The widely used method is the so-called direct
  integration method.

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TRANSIENT ANALYSIS

• The direct integration method is basically using
  the finite difference method for time stepping.
• There are mainly two types of direct integration
  method one is implicit and the other is
  explicit.
• Implicit method (e.g. Newmarks method) is more
  efficient for relatively slow phenomena
• Explicit method (e.g. central differencing
  method) is more efficient for very fast
  phenomena, such as impact and explosion.

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Newmarks method (Implicit)
Assume that
Substitute into
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Newmarks method (Implicit)
where
Therefore,
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Newmarks method (Implicit)
Start with D0 and
Obtain
using
March forward in time
using
Obtain
Obtain D?t and
using
32
Central difference method (explicit)
(Lumped mass no need to solve matrix equation)
33
Central difference method (explicit)
34
REMARKS

• In FEM, the displacement field U is expressed by
  displacements at nodes using shape functions N
  defined over elements.
• The strain matrix B is the key in developing the
  stiffness matrix.
• To develop FE equations for different types of
  structure components, all that is needed to do is
  define the shape function and then establish the
  strain matrix B.
• The rest of the procedure is very much the same
  for all types of elements.