Department of Engineering Mechanics – HCMUT 2016
Nguyễn Thanh Nhã
Email: nhanguyen@hcmut.edu.vn
Phone: 0908.56.81.81
Department of Engineering Mechanics – HCMUT 2016
1.1. Fundamentals of FEM for structure
1.2. Strong form of boundary value problem
1.3. Weak form of boundary value problem
Department of Engineering Mechanics – HCMUT 2016
1.1. Fundamentals of FEM for structure
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
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ain idea of FEM
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Main idea of FEM
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Real model
Discreted structure
Displacement field in element e
Displacement at nodes i, j k, l of element e
Vector of shape functions
“finite element”
Main idea of FEM
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
FEM in Modeling and Simulation
Main idea of FEM
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Truss, bar, link Beam Plate
Axisymmetric element
Solid 3D
Plane element
Shell
Main idea of FEM
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Main idea of FEM
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Main idea of FEM
1.1. Fundamentals of FEM for structure Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
CAD
PRE-PROCESSING
FEM
POST-PROCESSING
GOOD?
YES
NO
START
FINISH
Main idea of FEM
Department of Engineering Mechanics – HCMUT 2016
1.2. Strong form of boundary value problem
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Displacement – Strain relationship
1 2 3
11 22 33
1 2 3
1 2 1 3 2 3
12 21 13 31 23 32
2 1 3 1 3 2
; ;
1 1 1
; ; ;
2 2 2
u u u
X X X
u u u u u u
X X X X X X
Cauchy formulation:
1X 2X
3X
1u
2u
3u
O
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
1 2 3
11 22 33
1 2 3
1 2 1 3 2 3
12 21 13 31 23 32
2 1 3 1 3 2
; ;
1 1 1
; ; ;
2 2 2
u u u
X X X
u u u u u u
X X X X X X
Cauchy formulation
In tensor mode
1,1 1,2 2,1 1,3 3,1
1,2 2,1 2,2 2,3 3,2 , ,
1,3 3,1 2,3 3,2 3,3
1 1
( ) ( )
2 2
1 1 1
( ) ( ) ( )
2 2 2
1 1
( ) ( )
2 2
i j j i
u u u u u
u u u u u u u
u u u u u
Displacement – Strain relationship
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
11 22 33 12 23 132 2 2 T
In vector mode
1
11 2
22
1
333
2
12
3
2 123
13
3 2
3 1
0 0
0 0
0 0
2
0
2
2
0
0
X
X
u
X
u
uX X
X X
X X
11 12 13
22 23
33sym
1 2 3 Tu u u u
ε D u
: differential operatorD
sym ε u
Displacement – Strain relationship
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Momentum balance equations
11 12 13
1 1
1 2 3
21 22 23
2 2
1 2 3
31 32 33
3 3
1 2 3
b u
X X X
b u
X X X
b u
X X X
1 11,1 12,2 13,3 1
1 21,1 22,2 23,3 2
1 31,1 32,2 33,3 3
u b
u b
u b
In tensor mode
,i i ij i ij j
j
u b b
X
div u σ b 11,1 12,2 13,321,1 22,2 23,3 ,
31,1 32,2 33,3
ij jdiv
σ
Displacement – Strain relationship
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Momentum balance equations
11 22 33 12 23 13 T σ
In vector mode
11
221 2 3
1 1
33
2 2
122 1 3
3 3
23
3 2 1 13
0 0 0
0 0 0
0 0 0
X X Xu b
u b
X X X
u b
X X X
D σ b
11 12 13
22 23
33sym
σ
1 2 3 Tb b bb
: differential operatorD
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Stress – strain relationship (Hook’s law)
11 11
22 22
33 33
12 12
23 23
13 13
2 0 0 0
2 0 0 0
2 0 0 0
20 0
20
2sym
σ C ε
σ Cε
In vector mode
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
System of equations
div u σ b
σ Cε
sym ε u
( )symdiv u b C u
System of equations
Static problem
( )symdiv C u b 0
1.2. Strong form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Boundary conditions
*( , ) ( , )t t σ X n t X X
Neumann Boundary Condition (Stress)
Dirichlet Boundary Condition (Displacement)
*( , ) ( , ) ut t u X n u X X
*t
u
* 0u
Department of Engineering Mechanics – HCMUT 2016
1.3. Weak form of boundary value problem
1.3. Weak form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Principle of Virtual Work
Rewrite the strong form
( )symdiv C u b u
- In general, the solution of this differential equation is not possible
analytically
Some integral principles of mechanics
+ the principle of virtual displacements or the principle of virtual work
- Approximation methods (FEM and others) are used to find an approximate
solution.
- Numerical methods do not solve directly the strong form of differential
equation but solve its integral over the domain (weak form).
+ the principle of virtual forces
+ the principle of the minimum of total potential (static problem)
+ Hamilton's principle of continuum (dynamic problem)
1.3. Weak form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
- For structure problem, test function vector is chosen as the virtual
displacement
- Special test function has the following properties:
- In the Principle of Virtual Work, the strong form of the differential equation
as well as the static BCs are scalarly multiplied by a vector-valued test
function and integrated over the domain
u
+ satisfies the geometrical BCsu
0 uu X
+ satisfies the field conditions
sym u
+ is infinitesimal
+ is arbitrary
u
u
u
u
Principle of Virtual Work
1.3. Weak form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
Rewrite the balance momentum and the static BCs
* σ n t 0div u σ b 0
Multiplication by the test function , integration over the volume and
respectively over the Neumann boundary
u
*( ) ( ) 0dV div dV dA
u u b u σ u σ n t
Apply Gausses'
theorem:
( )div dV dA dA
u σ u σ n u σ n
*( ) : ( ) 0dV dV dA dA
u u b u σ u σ n u σ n t
Integral equation
Principle of Virtual Work
1.3. Weak form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
*( ) : ( ) 0dV dV dA dA
u u b u σ u σ n u σ n t
: : u σ ε σ
*:dV dV dV dA
u u ε σ u b u tWeak form equation
In energy conservation form intdyn extW W W
dynW dV
u u
int :
TW dV dV
ε σ u D CD u
*
extW dV dA
u b u t
Virtual work of
inertial forces
internal forces
external forces
Principle of Virtual Work
1.3. Weak form of boundary value problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016
- Dirichlet BCs are strongly fulfilled
- Neumann BCs and the equilibrium equation are fulfilled only weakly
- Principle of virtual work only gives approximate solutions
- The weak formulation forms the basis for FEM
Principle of Virtual Work
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