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) – I
Miroslav Halilovič, 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.
𝛀 …+ 𝚪 … =0
What is FEM?
5
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎0𝜎0 𝑥
𝑦
What is FEM?
6
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎0𝜎0 𝑥
𝑦
𝜎𝑥𝑥𝜎𝑥𝑥
What is FEM?
7
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎0𝜎0
𝜎0
3𝜎0
𝑥
𝑦
𝜎𝑥𝑥𝜎𝑥𝑥
What is FEM?
8
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝜎0𝜎0
𝜎0
3𝜎0
𝑥
𝑦
𝜎𝑥𝑥𝜎𝑥𝑥
What is FEM?
9
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝜎𝑖𝑗 ∝ 𝛻u
𝜎0𝜎0
𝜎0
3𝜎0
𝑥
𝑦
𝜎𝑥𝑥𝜎𝑥𝑥𝑐0 + 𝑐1𝑥 + 𝑐2𝑦 𝑐0 + 𝑐1𝑥 + 𝑐2𝑦 + 𝑐3 𝑥 𝑦
What is FEM?
10
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝜎𝑖𝑗 ∝ 𝛻u
𝜎0 𝜎0
𝑐0 + 𝑐1𝑥 + 𝑐2𝑦 + 𝑐3 𝑥 𝑦𝑐0 + 𝑐1𝑥 + 𝑐2𝑦
What is FEM?
11
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝐾 . 𝑢 = {𝐹}… written in terms of 𝑢
𝜎𝑖𝑗 ∝ 𝛻uk k k
k
𝜎0 𝜎0
What is FEM?
12
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝐾 . 𝑢 = {𝐹}… written in terms of 𝑢
𝐾𝑔𝑙𝑜𝑏 . 𝑢𝑔𝑙𝑜𝑏 = {𝐹𝑔𝑙𝑜𝑏}… solving for 𝑢𝑔𝑙𝑜𝑏
𝜎𝑖𝑗 ∝ 𝛻uk k k
k
𝜎0 𝜎0
What is FEM?
13
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝐾 . 𝑢 = {𝐹}
𝐾𝑔𝑙𝑜𝑏 . 𝑢𝑔𝑙𝑜𝑏 = {𝐹𝑔𝑙𝑜𝑏}
… written in terms of 𝑢
… solving for 𝑢𝑔𝑙𝑜𝑏
𝜎𝑖𝑗 ∝ 𝛻u
𝑢𝑔𝑙𝑜𝑏 → 𝜖𝑖𝑗 → 𝜎𝑖𝑗
k k k
k
k
k
𝜎0 𝜎0
What is FEM?
14
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝐾 . 𝑢 = {𝐹}
𝐾𝑔𝑙𝑜𝑏 . 𝑢𝑔𝑙𝑜𝑏 = {𝐹𝑔𝑙𝑜𝑏}
… written in terms of 𝑢
… solving for 𝑢𝑔𝑙𝑜𝑏
𝜎𝑖𝑗 ∝ 𝛻uk k k
k
𝑢𝑔𝑙𝑜𝑏 → 𝜖𝑖𝑗 → 𝜎𝑖𝑗k k
𝜎0 𝜎0
𝜎0
3𝜎0
𝜎𝑥𝑥
What is FEM?
15
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝐾 . 𝑢 = {𝐹}
𝐾𝑔𝑙𝑜𝑏 . 𝑢𝑔𝑙𝑜𝑏 = {𝐹𝑔𝑙𝑜𝑏}
… written in terms of 𝑢
… solving for 𝑢𝑔𝑙𝑜𝑏
𝜎𝑖𝑗 ∝ 𝛻uk k k
k
𝑢𝑔𝑙𝑜𝑏 → 𝜖𝑖𝑗 → 𝜎𝑖𝑗k k
𝜎0 𝜎0
𝜎0
3𝜎0
𝜎𝑥𝑥
What is FEM?
16
Governing
Diff. Eq.
𝛀 …+ 𝚪 … =0
𝜎𝑖𝑗 = 𝜎𝑖𝑗(u)
… searching for u(x,y)
𝐾 . 𝑢 = {𝐹}
𝐾𝑔𝑙𝑜𝑏 . 𝑢𝑔𝑙𝑜𝑏 = {𝐹𝑔𝑙𝑜𝑏}
… written in terms of 𝑢
… solving for 𝑢𝑔𝑙𝑜𝑏
𝜎𝑖𝑗 ∝ 𝛻uk k k
k
𝑢𝑔𝑙𝑜𝑏 → 𝜖𝑖𝑗 → 𝜎𝑖𝑗k k
𝜎0 𝜎0
𝜎0
3𝜎0
𝜎𝑥𝑥
GIGO
17
[1] https://www.r-bloggers.com/2019/08/new-course-learn-advanced-data-cleaning-in-r
[1]
Common FEM applications
18
• Mechanical/Aerospace/Civil/Automotive Engineering
• Structural/Stress Analysis
- Static/Dynamic
- Linear/Nonlinear
• Fluid Flow
• Heat Transfer
• Electromagnetic Fields
• Soil Mechanics
• Biomechanics
Common FEM applications
19
• Mechanical/Aerospace/Civil/Automotive Engineering
• Structural/Stress Analysis
- Static/Dynamic
- Linear/Nonlinear
• Fluid Flow
• Heat Transfer
• Electromagnetic Fields
• Soil Mechanics
• Biomechanics
STATIC Stress analysis
20
𝜎𝑖𝑗
𝑥
𝜎𝑥𝑥
𝑧
𝑦𝜎𝑦𝑦
𝜎𝑧𝑧𝜎𝑥𝑦
𝜎𝑥𝑧
𝜎𝑦𝑧 𝜎𝑥𝑦𝜎𝑦𝑧
𝜎𝑥𝑧
Common FEM applications
𝜎𝑒𝑞
fixed
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠
Modal analysis
21
Common FEM applications
𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠𝑚𝑜𝑑𝑒 𝑠ℎ𝑎𝑝𝑒𝑠
supported
supported
Nozzle loads
𝑢𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒
22
Common FEM applications
Heat transfer
ℎ𝑒𝑎𝑡 𝑓𝑙𝑢𝑥
𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒ℎ𝑒𝑎𝑡 𝑓𝑙𝑢𝑥
23
Common FEM applications
Transient thermo-hydraulic simulation (Fluid dynamics)
𝑊𝐴𝑇𝐸𝑅 𝑑𝑜𝑚𝑎𝑖𝑛
𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑊𝐴𝑇𝐸𝑅
𝐴𝐼𝑅
24
Common FEM applications
Coupled problems: Fluid-Structure Interaction (FSI)
inflow
t t
p
outflow
v
𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑜𝑓 𝑠𝑜𝑙𝑖𝑑𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑
25
simulation
FEM simulation steps
26
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
27
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• 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. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
29
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
30
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• 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. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
32
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
33
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
34
• Static
• Dynamic
• Implicit
• Explicit
• Visco
• Heat transfer• Steady state
• Transient
• Coupled temperature-displacement
• Buckling
• Electromagnetism
• Fluid Flow
- Linear
- Nonlinear
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• 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 – Learning from Engineering Examples, Springer Science, doi: 10.1007/978-1-4939-2101-0
[1]
1. Geometry• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
FEM simulation steps
36
𝜎𝑒𝑞1. Geometry
• geometrical simplifications• reduction of dimensions
2. Physical properties• material properties
• structural properties
3. Geometrical discretization• element type
• meshing
4. Type of analysis
5. Loading and Boundary/Initial conditions
6. Presentation and analysis of results
Defining a simulation
37
• Geometry
• Sets
• Material behaviour
• Type of analysis
• Solver type
• Loading, Boundary/Initial conditions
• Output
Defining a simulation
38
• Geometry
• Sets
• Material behaviour
• Type of analysis
• Solver type
• Loading, Boundary/Initial conditions
• Output
m n
o p
k
Defining a simulation
39
• Geometry
• Sets
• Material behaviour
• Type of analysis
• Solver type
• Loading, Boundary/Initial conditions
• Output
m n
o p
k
Defining a simulation
40
• Geometry
• Sets
• Material behaviour
• Type of analysis
• Solver type
• Loading, Boundary/Initial conditions
• Output
Defining a simulation
41
• Geometry
• Sets
• Material behaviour
• Type of analysis
• Solver type
• Loading, Boundary/Initial conditions
• Output
Static, Dynamic (Implicit, Explicit),Visco, Thermal, Coupled thermal-displacement,Linear/Nonlinear…
Equation solverSolution TechniquesIncrementationConvergence tolerances…
Defining a simulation
42
• Geometry
• Sets
• Material behaviour
• Type of analysis
• Solver type
• Loading, Boundary/Initial conditions
• Output
Defining a simulation
43
• Geometry
• Sets
• Material behaviour
• Type of analysis
• Solver type
• Loading, Boundary/Initial conditions
• 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.
