This project has been funded with support from the European Commission.
This publication [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which
may be made of the information contained therein.
Introduction to the Finite Element Method (FEM) β I
Miroslav HaliloviΔ, Bojan Starman, Janez Urevc, Nikolaj Mole
Faculty of Mechanical Engineering, University of Ljubljana 06/2021
What is FEM?
2
[1] https://manilsuri.umbc.edu/what-are-finite-elements/[2] https://www.simscale.com/blog/2016/10/what-is-finite-element-method/
[2]
[1]
What is FEM?
Finite Element Method:
- is a procedure for obtaining numerical approximation to the solution of a boundary value problem.
3
What is FEM?
4
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
What is FEM?
5
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
π0π0 π₯
π¦
What is FEM?
6
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
π0π0 π₯
π¦
ππ₯π₯ππ₯π₯
What is FEM?
7
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
π0π0
π0
3π0
π₯
π¦
ππ₯π₯ππ₯π₯
What is FEM?
8
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
π0π0
π0
3π0
π₯
π¦
ππ₯π₯ππ₯π₯
What is FEM?
9
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πππ β π»u
π0π0
π0
3π0
π₯
π¦
ππ₯π₯ππ₯π₯π0 + π1π₯ + π2π¦ π0 + π1π₯ + π2π¦ + π3 π₯ π¦
What is FEM?
10
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πππ β π»u
π0 π0
π0 + π1π₯ + π2π¦ + π3 π₯ π¦π0 + π1π₯ + π2π¦
What is FEM?
11
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πΎ . π’ = {πΉ}β¦ written in terms of π’
πππ β π»uk k k
k
π0 π0
What is FEM?
12
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πΎ . π’ = {πΉ}β¦ written in terms of π’
πΎππππ . π’ππππ = {πΉππππ}β¦ solving for π’ππππ
πππ β π»uk k k
k
π0 π0
What is FEM?
13
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πΎ . π’ = {πΉ}
πΎππππ . π’ππππ = {πΉππππ}
β¦ written in terms of π’
β¦ solving for π’ππππ
πππ β π»u
π’ππππ β πππ β πππ
k k k
k
k
k
π0 π0
What is FEM?
14
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πΎ . π’ = {πΉ}
πΎππππ . π’ππππ = {πΉππππ}
β¦ written in terms of π’
β¦ solving for π’ππππ
πππ β π»uk k k
k
π’ππππ β πππ β πππk k
π0 π0
π0
3π0
ππ₯π₯
What is FEM?
15
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πΎ . π’ = {πΉ}
πΎππππ . π’ππππ = {πΉππππ}
β¦ written in terms of π’
β¦ solving for π’ππππ
πππ β π»uk k k
k
π’ππππ β πππ β πππk k
π0 π0
π0
3π0
ππ₯π₯
What is FEM?
16
Governing
Diff. Eq.
π β¦+ πͺ β¦ =0
πππ = πππ(u)
β¦ searching for u(x,y)
πΎ . π’ = {πΉ}
πΎππππ . π’ππππ = {πΉππππ}
β¦ written in terms of π’
β¦ solving for π’ππππ
πππ β π»uk k k
k
π’ππππ β πππ β πππk k
π0 π0
π0
3π0
ππ₯π₯
GIGO
17
[1] https://www.r-bloggers.com/2019/08/new-course-learn-advanced-data-cleaning-in-r
[1]
Common FEM applications
18
β’ Mechanical/Aerospace/Civil/Automotive Engineering
β’ Structural/Stress Analysis
- Static/Dynamic
- Linear/Nonlinear
β’ Fluid Flow
β’ Heat Transfer
β’ Electromagnetic Fields
β’ Soil Mechanics
β’ Biomechanics
Common FEM applications
19
β’ Mechanical/Aerospace/Civil/Automotive Engineering
β’ Structural/Stress Analysis
- Static/Dynamic
- Linear/Nonlinear
β’ Fluid Flow
β’ Heat Transfer
β’ Electromagnetic Fields
β’ Soil Mechanics
β’ Biomechanics
STATIC Stress analysis
20
πππ
π₯
ππ₯π₯
π§
π¦ππ¦π¦
ππ§π§ππ₯π¦
ππ₯π§
ππ¦π§ ππ₯π¦ππ¦π§
ππ₯π§
Common FEM applications
πππ
fixed
πππ πππππππππ‘π π π‘πππ π ππ
Modal analysis
21
Common FEM applications
πππ‘π’πππ πππππ’ππππππ ππππ π βππππ
supported
supported
Nozzle loads
π’ππππππ‘π’ππ
22
Common FEM applications
Heat transfer
βπππ‘ πππ’π₯
π‘πππππππ‘π’ππβπππ‘ πππ’π₯
23
Common FEM applications
Transient thermo-hydraulic simulation (Fluid dynamics)
ππ΄ππΈπ ππππππ
ππππππππ‘π’ππ
π£ππππππ‘π¦
ππ΄ππΈπ
π΄πΌπ
24
Common FEM applications
Coupled problems: Fluid-Structure Interaction (FSI)
inflow
t t
p
outflow
v
πππ ππππ π ππ π πππππππ ππππ π ππ πππ’ππ
25
simulation
FEM simulation steps
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1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
27
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
28
[1] Carlos A. Felippa, 2004, Introduction to Finite Element Methods. Available at: https://vulcanhammernet.files.wordpress.com/2017/01/ifem.pdf (06/2021)
[1]
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
29
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
30
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
y
z
y
z
y
z
y
z
y
z
n
FEM simulation steps
31
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
32
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
33
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
34
β’ Static
β’ Dynamic
β’ Implicit
β’ Explicit
β’ Visco
β’ Heat transferβ’ Steady state
β’ Transient
β’ Coupled temperature-displacement
β’ Buckling
β’ Electromagnetism
β’ Fluid Flow
- Linear
- Nonlinear
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
35
[1] Emri, I., Voloshin, A., 2016, Statics β Learning from Engineering Examples, Springer Science, doi: 10.1007/978-1-4939-2101-0
[1]
1. Geometryβ’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
36
πππ1. Geometry
β’ geometrical simplificationsβ’ reduction of dimensions
2. Physical propertiesβ’ material properties
β’ structural properties
3. Geometrical discretizationβ’ element type
β’ meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
Defining a simulation
37
β’ Geometry
β’ Sets
β’ Material behaviour
β’ Type of analysis
β’ Solver type
β’ Loading, Boundary/Initial conditions
β’ Output
Defining a simulation
38
β’ Geometry
β’ Sets
β’ Material behaviour
β’ Type of analysis
β’ Solver type
β’ Loading, Boundary/Initial conditions
β’ Output
m n
o p
k
Defining a simulation
39
β’ Geometry
β’ Sets
β’ Material behaviour
β’ Type of analysis
β’ Solver type
β’ Loading, Boundary/Initial conditions
β’ Output
m n
o p
k
Defining a simulation
40
β’ Geometry
β’ Sets
β’ Material behaviour
β’ Type of analysis
β’ Solver type
β’ Loading, Boundary/Initial conditions
β’ Output
Defining a simulation
41
β’ Geometry
β’ Sets
β’ Material behaviour
β’ Type of analysis
β’ Solver type
β’ Loading, Boundary/Initial conditions
β’ Output
Static, Dynamic (Implicit, Explicit),Visco, Thermal, Coupled thermal-displacement,Linear/Nonlinearβ¦
Equation solverSolution TechniquesIncrementationConvergence tolerancesβ¦
Defining a simulation
42
β’ Geometry
β’ Sets
β’ Material behaviour
β’ Type of analysis
β’ Solver type
β’ Loading, Boundary/Initial conditions
β’ Output
Defining a simulation
43
β’ Geometry
β’ Sets
β’ Material behaviour
β’ Type of analysis
β’ Solver type
β’ Loading, Boundary/Initial conditions
β’ Output
Thank you for your attention!
http://sctrain.eu/
This project has been funded with support from the European Commission.
This publication [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which
may be made of the information contained therein.
