fem presentation
TRANSCRIPT
An Overview of the FINITE ELEMENT METHOD
(FEM)
By
VIJAY G. S.
Senior Lecturer, Dept. of Mechanical Engg.,
NMAM Institute of Technology,
Nitte
IntroductionFor studying physical phenomena, engineers and scientists are involved with two major tasks:
Mathematical formulation of the physical problem –behaviour/governing equations
Numerical analysis of the mathematical model – numerical method & computer
Problems in engineering may be:
1. Boundary Value Problems
2. Initial Value Problems
3. Boundary and Initial Value
Problems
4. Eigen Value Problems
1. Boundary Value Problems – values of dependent variables (and its derivatives) are known on the continuum (boundary) of the problem.
Example:
01
0 ,)0( gdx
duadu
x
10
xforfdx
dua
dx
dis the governing equation
are the boundary conditions
2. Initial Value Problems – values of the dependent variables (and its derivatives) are known at an initial instant (i.e., at t=0). These are time dependent problems.
Example:
is the governing equation02
2
0 ttforfaudt
ud
00
0 ,)0( vdt
duuu
t
are the boundary conditions
3. Boundary and Initial Value Problems – values of the dependent variables (and its derivatives) are known on the boundary at specific time instants.Example:
00
10),(
tt
xfortxf
t
u
x
ua
x
)()0,(),(),(),0( 001
0 xuxutgx
uatdtu
x
is the governing equation
are the boundary conditions
4. Eigen Value Problems – the problem of determining value of the constant λ such that:
10
xforudx
dua
dx
d
0,0)0(1
xdx
duau
is called the eigen value problem for the above differential equation.
In all the above problems of engineering, we may require to find the value of the dependent variable at any specified point in the continuum.
For this, the above governing differential equations must be solved to get the value of the dependent variable.
But, in actual practice, engineering problems involve complicated geometries (continuums), loadings and varying material properties.
Due to this, it may be impossible to specify the boundary conditions, consider material properties and solve the governing differential equation.
In such a situation, we go for numerical methods such as “Finite Element Method (FEM)” to get approximate but acceptable solutions.
The Finite Element Method is a numerical method for solving problems of engineering and mathematical physics where their behaviour/governing equations are expressed by integral or differential equations.
The FEM formulation of the problem results in a set of simultaneous algebraic equations for solution, instead of requiring the solution of the governing differential equation
This yields approximate values of the variables at discrete points in the continuum
Irregular Geometry
Area = Length Breadth
Regular Geometry
Area = (area of sub divisions)
Basic Concept of FEM
General Steps of the Finite Element Method
1. Select Element type and discretize the continuum
2. Select a Displacement Function3. Define the Strain/Displacement and
Stress/Strain Relationships4. Derive the Element Stiffness Matrix
and Element Equations5. Assemble the Element Equations to
obtain Global Equations6. Apply Boundary Conditions and
modify the Global Equations7. Solve for unknown variables8. Solve for Element Strains and
Stresses9. Interpret the results
Some terminologies used in FEMDiscretization (Meshing): The process of dividing the model of the problem continuum into a finite number of regular subdivisions
Elements: Each subdivision is called an “Element”
Nodes: The grid (connection) points at which the elements meet each other are called “nodes”
Degrees of Freedom (DOF): The total number of variables (displacements) that are associated with each node
Boundary Conditions:Known values of the variable at the continuum boundary
Displacement Function:It is an assumed polynomial expression which closely represents the anticipated variation of the unknown variable over the element domain.
Discretization
Form the element
equations
{F}e=[k]e{q}e1. Direct stiffness method
2. Energy method
3. Weighted residual method
Assemble the element
equations to form global equations
{F}g=[K]g{Q}g
Apply Boundary Conditions
Solve the set of global simultaneous
equations
{Q}g=Inv([K]g){F}g
Obtain the stress and strain in each
element
PQ
• One Dimensional Problems: Variable along only one coordinate axis is required to
represent the problem behaviour Line Elements are used to model 1D problems
• Two Dimensional Problems:
Variable components along two coordinate axes are required to represent the problem behaviour Area Elements are used to model 2D problems
• Three Dimensional Problems:
Variable components along three coordinate axes are
required to represent the problem behaviour Volume Elements are used to model 2D problems
Problem Dimensions
2D Elements (Area Elements) 3D Elements (Volume Elements)
1D Element (Line Element)
BASIC
ELEMENT
TYPES
IN
FEM
Triangular Element
Quadrilateral Element
Brick Element
Tetrahedron Element
Convergence: The monotonic approach of the FEM Solution to the Actual Solution
Coarse MeshNo. of Elements=n1
Element Size = h1
Medium MeshNo. of Elements=n2
Element Size = h2
Fine MeshNo. of Elements=n3
Element Size = h3
n1 < n2 <n3
h1 > h2 > h3
No. of Elements
Sol
uti
on E
rror
FEM Solution
Basic form of the Element Equations:
ee
Q
QK
F
F
2
1
2
1
11
11
eee QKF
where, {F}e is called the ELEMENT FORCE VECTOR
[K]e is called the ELEMENT STIFFNESS MATRIX
{Q}e is called the ELEMENT DISPLACEMENT VECTOR
Element equation form for a Structural Problem
eee
Q
Q
L
AE
F
F
2
1
2
1
11
11
1 m 0.6 m
20 kN 10 kN
1 2 31 2
eee QKF
Element equation form for a Conduction Heat Transfer Problem
e
e
ee
eo
T
T
L
kA
Q
QLAq
2
1
2
1
11
11
1
1
2
T1T2 T3 T4
1 2 3Inside
temperature T0
Outside temperature
T5
5 cm 3 cm 1 cm
k1k2 k3
1 2 3 4
eee QKF
Element equation form for a Hydraulic Network Problem
128
,11
11
2
14
2
1
cwhereP
P
L
dc
Q
Qe
e
e
e
Q1
2 3
4P = 0
R1R2
R4
R3
R5
4
32
1
1
2
4
3
5
eee QKF
Element equation form for a DC Electric Network ProblemR3
R1
R2 R4
R5
E1E2
2
1
34
e
e
e
V
V
RI
I
2
1
2
1
11
111
eee QKF
20 kN 10 kNA1, E1, L1
A2, E2, L2
A3, E3, L3
1 2 3
1 2 3 4
Assembly of Element Equations to form Global Equations
Example: Consider the structural problem
1
1
111 L
EAK
21
2
222 L
EAK
2 32 3 43
3
333 L
EAK
12
11
11
11
12
11
Q
Q
KK
KK
F
F1 2
1
2
23
22
22
22
23
22
Q
Q
KK
KK
F
F2 3
2
3
34
33
33
33
34
33
Q
Q
KK
KK
F
F3 4
3
4
ggg QKF
Assembled Force Vector Assembled Stiffness Matrix
Assembled Displacement
Vector
34
33
23
22
12
11
33
3322
2211
11
34
33
23
22
12
11
00
0
0
00
Q
Q
KK
KKKK
KKKK
KK
F
FF
FF
F2
2
1
1 3
3
4
4
Basic form of the Global Equations:
nnnnn
n
n
n Q
Q
Q
KKK
KKK
KKK
F
F
F
..
..
....
..........
..........
....
....
..
..2
1
21
22221
11211
2
1
where, {F}g is called the GLOBAL FORCE VECTOR
[K]g is called the GLOBAL STIFFNESS MATRIX
the elements of [K]g (K11, K12,….Knn) are called INFLUENCE
COEFFICIENTS
{Q}g is called the DISPLACEMENT VECTOR
ggg QKF
Application of FEMStructural: Stress analysis of truss and frame, stress
concentration problems Buckling problems Vibration analysis
Non Structural: Heat Transfer Fluid flow, including seepage through porous media Distribution of electric or magnetic potential Acoustics
Others: Biomedical engineering problems – analysis of
human spine, skull, hip joints, jaw/gum tooth implants, heart and eyes.
Advantages of FEM Model irregularly shaped bodies quite easily. Handle general load conditions without
difficulty. Model bodies composed of different materials
because their element equations are evaluated individually.
Handle unlimited numbers and kinds of Boundary Conditions
Vary the size of the elements to make it possible to use small elements where ever necessary.
Alter the FEM Model relatively easily & cheaply Include dynamic effects Handle nonlinear behaviour existing with large
deformations and non linear materials
PRE PROCESSOR
Model the continuum (Coordinate data, constants & material properties)Select Element Type & Discretize(Mesh) the continuumStore all the input data
PROCESSOR
Compute element coefficient matrices and column vectorsAssemble element equationsImpose Boundary ConditionsSolve equations
POST PROCESSOR
Compute solution at points other than at the nodesLists the resultsPlots the resultsProvides simulation
Module 1
Module 2
Module 3
General Structure of Commercially available FEM software
Use of a computer in FEM:
To provide a User Interactive Graphics Environment while
modeling
the problem continuum.
To create a data base of the input data.
To evaluate the element equations and store them in matrix
form.
To assemble the element equations (by superposition) and
form the
global equations.
To solve the very large set of simultaneous algebraic
equations by
tools like ‘Gauss Elimination Method’.
To evaluate the unknown element parameters.
To create a data base of the results.
To graphically display the results
To provide the simulation.
20
20
60
80
R = 10
R = 50
R = 30