Post on 31-Oct-2019
transcript
© WZL/Fraunhofer IPT
Simulation Techniques in
Manufacturing Technology
Introduction
Laboratory for Machine Tools and Production Engineering
Chair of Manufacturing Technology
Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. Dr. h.c. F. Klocke
Seite 2 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 3 © WZL/Fraunhofer IPT
STMT-Lecture: Time schedule
Time:
Lecture (L): Fr, 10.00-11.30h
Exercise (E): Fr, 11.45-12.30h
Location: WZL, RWTH Aachen
Herwart-Opitz-Haus
Steinbachstr. 53
52074 Aachen
Room: 53C, 101
Contact: Dr.-Ing. M. Abouridouane
Tel. (0241) 80-28176
Date Typ Room No. Lecturer Topic
21. Okt L 53C/101 Abouridouane Introduction to STMT
53B, R312b
28. Okt L 53C/101 Ozhoga-Maslovskaja Forming technology basics
28. Okt E 53C/101 54A, R411
04. Nov L 53C/101 Ozhoga-Maslovskaja Actual simulation techniques in forming process
54A, R411
11. Nov. L 53C/101 Ozhoga-Maslovskaja Bulk metal forming
54A, R411
18. Nov L 53C/101 Ozhoga-Maslovskaja Sheet metal forming and separation
18. Nov E 53C/101 54A, R411
25. Nov L 53C/101 Abouridouane Principles of Cutting
25. Nov E 53C/101 53B, R312b
02. Dez L 53C101 Abouridouane Overview of the various cutting processes
02. Dez E 53C/101 53B, R312b
09. Dez L 53C/101 Abouridouane FE-Simulation of cutting processes
53B, R312b
16. Dez W 53B/101a Abouridouane FEM-Workshop (Abaqus and Deform)
16. Dez W 53B/101a Ozhoga-Maslovskaja
13. Jan L 53C101 Barth Cutting with geometrically undefined cutting edge I
13. Jan E 53C101 54A, R403
20. Jan L 53B/101a Barth Cutting with geometrically undefined cutting edge II
20. Jan E 53B/101a 54A, R403
27. Jan L 53B/101a Abouridouane Methods of validation and optimization techniques
27. Jan E 53B/101a 53B, R312b
03. Feb L 53C/101 Abouridouane Revision of contents
03. Feb E 53C/101 53B, R312b
Seite 4 © WZL/Fraunhofer IPT
STMT-Lecture: Exam, computer exercise, literature
Examination
– Type of examination: Oral
– Date of the examination: February, the xxth 2017
– Room, time, and distribution of groups will be given!
– Duration: 60 minutes for each group
Computer exercise
– Date for the FEM-Workshop: December, the 16th 2016
– Room: 53B/101a (WZL, Herwart-Opitz-Haus)
– Time: 10:00 to 18:00
Literature about metal machining (Emails list!)
– Manufacturing processes 1 - Cutting of Klocke
– Metal Machining (Theory and Applications) of Childs
– Machining Dynamics of Kai Cheng
Any other discussion points, comments or questions?
– Please contact Mr. Abouridouane (Tel.: +49 241 80-28176)
Seite 5 © WZL/Fraunhofer IPT
STMT-Lectutre: Supervisors
Cutting process
Forming process
Grinding process
Dr.-Ing. Mustapha Abouridouane
Herwart-Opitz-Haus 53B 312b
Tel.: +49 241 80-28176
Fax: +49 241 80-22293
m.abouridouane@wzl.rwth-aachen.de
Dr.-Ing. Oksana Ozhoga-Maslovskaja
Herwart-Opitz-Haus 54A 411
Tel.: +49 241 80-27428
Fax: +49 241 80-22293
o.ozhoga-maslovskaja@wzl.rwth-aachen.de
M.Sc. RWTH Sebastian Barth
Herwart-Opitz-Haus 54A 403
Tel.: +49 241 80-28183
Fax: +49 241 80-22293
s.barth@wzl.rwth-aachen.de
Seite 6 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 7 © WZL/Fraunhofer IPT
RWTH Aachen and Fraunhofer-Gesellschaft
RWTH Aachen University
Founded in 1870
40,375 students
Faculty of Mechanical Engineering
11,700 students
(incl. 2,700 first year students)
53 professors
2,600 employees
170 graduates per year
Fraunhofer-Gesellschaft
More than 65 institutes und facilities
at 40 locations in Germany
>23,000 employees
approx. € 2.0 billion research funds
per year, € 1.7 billion through research contracts
3 institutes in Aachen
Seite 8 © WZL/Fraunhofer IPT
Production Technology in Aachen
Laboratory for Machine Tools
and Production Engineering (WZL)
RWTH Aachen University institute
Founded in 1906
839 employees
16,000 m² offices and laboratories
Fraunhofer Institute for Production Technology IPT
Fraunhofer-Gesellschaft institute
Founded in 1980
450 employees
6300 m² offices and laboratories
Certified to DIN EN ISO 9001:2008
Seite 9 © WZL/Fraunhofer IPT
Budget 2013: WZL, Fraunhofer IPT, WZLforum, WZL Aachen GmbH
Industrial projects 41.70 %
Public funding* 33.57 %
Basic funding by 24.73 %
Fraunhofer-Gesellschaft and
RWTH Aachen University
Budget: 53.61 Mio €
* EU, AiF, BMBF, DFG
Seite 10 © WZL/Fraunhofer IPT
Two Institutes – One Philosophy
Process Technology
Production Machines
Mechatronic Systems Design
Production Quality and Metrology
Technology Management
Manufacturing Technology
Gearing Technology
Machine Tools
Metrology and Quality Management
Production Engineering and
Production Management
Seite 11 © WZL/Fraunhofer IPT
Our Focus
Process Technology
Machining and material removal
processes
Laser materials processing
Forming processes
CAx, Virtual Reality
Production and Machine Tools
Design of production machines
and components
Control technology and automation
Component and production machines
testing
Metrology
Tactile metrology
Optical metrology
Management
Business Engineering
Technology management
Innovation management
Production management
Quality management
Education
Professional training
Executive MBA for Technology Managers
Conferences, congresses, seminars
Gearing Technology
Gear manufacturing
Gear calculation and
investigation
Seite 12 © WZL/Fraunhofer IPT
Grinding and forming
Solid forming,
sheet metal
forming, hard
smooth rolling,
tribology
Turning,
milling,
drilling,
broaching
Cutting
technology
CAD/CAM
technologies
CAx
technologies
Process and Manufacturing Technology
Process and
product
monitoring
Material
removal
processes
Process moni-
toring systems
and strategies
Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. Dr. h.c. Fritz Klocke
Laser machining
Rapid
Manufacturing
Laser surface
treatment Joining,
cutting,
forming
Tool and die making
Precision and micro technology
Optics and optical systems
Plant engineering and construction
Automotive, aerospace, turbine construction
Grinding,
lapping,
polishing,
honing
Seite 13 © WZL/Fraunhofer IPT
Process and Manufacturing Technology
Manufacturing fundamentals
Machining with a geometrically defined cutting edge
Machining with a geometrically undefined cutting edge
Material removal processes
Forming processes
Laser machining
Gearwheel manufacture
Precision and ultra-precision processes
Process and product monitoring
Process simulations, methods and tools for technology planning and production design, virtual reality
Seite 14 © WZL/Fraunhofer IPT
Fundamentals of Cutting
Concept for
cooling
lubricants
Manufacturing Technology:
Group: Fundamentals of Cutting & Modeling and Evaluation
Modeling and Evaluation
Modelling and
model
development
Simulation of
tool wear
Machin-
ability
Analysis
of wear
Process
Design
Tool
development
Energy-saving
production
Technology
planning
Space- and aircraft-industry, turbine construction (machining of turbine disks and blades, structural components..)
Bild Bild Bild
Automotive industry (processing of crankcase, cylinder head, cylinder, camshaft, axle parts…)
Tool technology (wear analysis, tool layout, macro , micro geometry…)
Materials manufacturer (machinability, lead substitute …)
Resource efficiency (material, energy, auxiliaries…)
Seite 15 © WZL/Fraunhofer IPT
Virtuelle Realität
Analysis of the machinability
Material analysis of tools and
components, analysis of tool wear
Process development and process
optimisation in turning, milling,
drilling, broaching, tapping
Development of machining
strategies, HSC and HPC
machining, circular processes
Development and optimisation of
lubricooling strategies: dry, MQL,
conventional wet, high-pressure and
cryogenic
Tool development: substrate, macro
and micro geometry, cutting edge
preparation, coating, chip form
geometry
Development of environmentally
friendly and resource efficient
machining processes
Chair of Manufacturing Technology
Group Fundamentals of Cutting
Seite 16 © WZL/Fraunhofer IPT
Virtuelle Realität
Simulation of
cutting processes
Calculation of the
workpiece distortion
Energy and material
measurement and assessment
Energetic and environmental
evaluation of cutting processes
Chair of Manufacturing Technology –
Modelling and Evaluation of Cutting Processes
Selection
Modelling of cutting processes
Development of process models for
cutting technologies
Simulation of the thermo-mechanical
load spectrum during machining
Cutting tool design and optimization of
process parameters using FEM-
simulations
Evaluation of cutting processes
Acquiring and assessing of energy and
material consumptions of single
processes (indicators)
Evaluation and design of processes
and technology chains in respect to
energy and resource efficiency
Ecological life cycle management and
Life cycle assessment (LCA)
Makroskopisches
FEM-Modell
Berechnung des thermo-
mechanischen Bleastungskollektiv
Berechnung des Spanungsquerschnittes auf der Grundlage
der Durchdringung von Werkstück und Werkzeug
Methodik
InputWerkzeuggeometrie
Werkstückgeometrie
TCP, vc und f
Durchdringungsrechnung
Makrogeometrie nach der
spanenden Bearbeitung
Output Aktueller Spanungs-
querschnitt, vc und f
VW erkstü ck
W erk zeu gSp an
v c
Zerspankraft
Temperatur
T(t+t)
F(t+t)
Seite 17 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 18 © WZL/Fraunhofer IPT
Lecture objectives
Fundamentals and basic knowledge in manufacturing
technology for a better understanding of the mechanisms
during metal machining
Modelling approachs for simulation
Simulation techniques
Application of simulation in manufacturing technology
Challenges of simulation and future developments
Seite 19 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 20 © WZL/Fraunhofer IPT
Perception
Modelling – a human property
Experience Model vision
Seite 21 © WZL/Fraunhofer IPT
Sensorium
Modelling – a human property
IT-based converting
Modeling
Reality / Perception Mind
Seite 22 © WZL/Fraunhofer IPT
Basic of reasons
Deductive
reasoning
Modeling based on
theoretical derived
models
Inductive
reasoning
Modeling
based on
experience
The particular
Reality
The general
Model section
I n
d u
c t
i v
D e
d u
c t
i v
Seite 23 © WZL/Fraunhofer IPT
Simulation
A simulation is a replication of a dynamic process based on a model.
Model
A model is an abstract system that corresponds to a real system
and is used for expensive and/or impossible
- investigations,
- calculations and
- explanations or demonstration purposes.
It delivers general information about
- elements,
- structure and
- behavior
of a part of the reality.
Definitions
Seite 24 © WZL/Fraunhofer IPT
Continuously increasing demands of the market lead to
increasing requirements for manufacturing processes.
Motivation
Product
– innovative
– reliable
– cost-effective
Components
– high quality
– high precision
– long product life
Manufacturing process
– economic
– reproducible
– flexible
Seite 25 © WZL/Fraunhofer IPT
Why process modelling?
Car Body
Time Cost
Quality
Increase quality
Apply new materials
(Al, Mg, …)
Use material more
efficiently
Reduction of tool
cost
Reduction of lead
time
Reduction of time
required for training
Increase reliability
of production
Manufacture complex parts
Reduction of
pre production trials
Source: BMW
Seite 26 © WZL/Fraunhofer IPT
Integration of process modelling into the process chain
Source: BMW
Design
of car
exterior
Part
design
Part
production
Means of production Tool manu-
facturing
and testing Tool design Planning
Sheet metal forming simulation
Part evaluation Process optimisation
So
ftw
are
so
luti
on
A
pp
lic
ati
on
P
roc
es
s c
ha
in
Methods applied: - 2D modelling
- one-step modelling
- modelling with membrane elements Short computation time with sufficient
precision
Methods applied: - Simulation with membrane elements
- Simulation with shell elements
High Precision within acceptable
computation times
Seite 27 © WZL/Fraunhofer IPT
specification sheet
product
concept design and
choice of material
lay-out
manufacturing aspects
design
manufacturing planning
manufacturing
workpiece test
4
12
16
8
24
12
Manufacturing tests
FEM-Calculation
Today without process simulation
Modelling and Simulation: aims and requirements
Increase of the process
knowledge & comprehension
Prediction of the process
stability
Prediction of the component
characteristics
Reduction of planning and
development steps
Cost reduction
Goals
76 W
eeks
Seite 28 © WZL/Fraunhofer IPT
4
12
14
12
12
Future with process simulation
Modelling and Simulation: aims and requirements
Increase of the process
knowledge & comprehension
Prediction of the process
stability
Prediction of the component
characteristics
Reduction of planning and
development steps
Cost reduction
Goals
54 W
eeks
FEM
specification sheet
product
concept design and
choice of material
lay-out
design
manufacturing planning
manufacturing
workpiece test
Reduction of the cycle time by 30%
Simulation
Seite 29 © WZL/Fraunhofer IPT
High process reliability
High result quality
Realistic prediction of the
process results
Adaption of technological
innovations
Modelling and Simulation: aims and requirements
Increase of the process
knowledge & comprehension
Prediction of the process
stability
Prediction of the component
characteristics
Reduction of planning and
development steps
Cost reduction
Goals
Requirements
Seite 30 © WZL/Fraunhofer IPT
Partitioning of different model types from literature review
ANN
models
4 %
Rule & knowledge
models 6%
Overview article 2 %
Analytical models 38 %
kinematic geometrical
models 10 %
FEA - models
19 %
Basic & regression
models 20 %
MD - models 1%
Source: Heinzel 2009
Seite 31 © WZL/Fraunhofer IPT
The history of the development of cutting process models
Source: Ivester, 50th CIRP General Assembly, Sidney 2000; *Heinzel 2009
1937 1947 1957 1967 1977 1987 1997
2D, Thin Zone, Analytic
3D, Empirical, Mechanistic
3D, Analytic
2D, FEM
2D, Thick zone, Slip-line
The use*
Analytical models: 38%
Basic and regression models: 20%
FEA models: 19%
Kinematic geometrical models: 10%
Rule and knowledge models: 6%
ANN models: 4%
Overview article: 2%
MD models: 1%
Seite 32 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 33 © WZL/Fraunhofer IPT
Sheet Metal Forming Techniques
Ironing Bending
Stretch forming Spinning
Hydroforming
Deep drawing
Seite 34 © WZL/Fraunhofer IPT
Forming process: Extruding a transmission shaft
Seite 35 © WZL/Fraunhofer IPT
Face milling
Seite 36 © WZL/Fraunhofer IPT
Grinding process
Seite 37 © WZL/Fraunhofer IPT
Grinding process
Seite 38 © WZL/Fraunhofer IPT
Lattice Types of an Unit Cell
face-centred
cubic
(fcc)
body-centred
cubic
(bcc)
hexagonal
(hex)
examples:
sliding planes:
sliding directions:
sliding systems:
formability:
γ-Fe, Al, Cu
4
3
12
very good
α-Fe, Cr, Mo
6
2
12
good
Mg, Zn, Be
1
3
3
poor
Seite 39 © WZL/Fraunhofer IPT
tensile test compression test shear test
F
F
l0
A0
l1
A1
A0 F
F
A1
l1
l0
0A
F
0A
F
Stress Determination Depending on Load
0A
F
F
F a
l
q
A0
tensile stress compression stress shear stress
Seite 40 © WZL/Fraunhofer IPT
Strain Determination of an Idealized Upsetting Process
00
01
00
1
0
l
l
l
ll
l
dl
l
dld
l
l
xx
0
1
0
1
0
1 ln ;ln ;lnh
h
b
b
l
lzyx
0
1ln1
0l
l
l
dl
l
dld
l
l
engineering strain (elastic)
true strain (plastic)
including of volume constancy
)1( ln l
l
l
l ln
l
ll ln
l
ul ln
l
l ln
0
0
00
0
0
x0
0
1
xx
const. 111000 bhlbhl
0 zyx
connection between true strain - engineering strain
Seite 41 © WZL/Fraunhofer IPT
Stress-Strain Curve up to the Uniform Elongation
A
F
0
str
ess
strain
engineering stress: (related to starting section)
F
F
Rm
Re ,se
eel epl
l0
l
l
A0
A
load
relieving reload
A
F
true tensile stress: (related to real section)
σ‘ σ
Seite 42 © WZL/Fraunhofer IPT
Flow curve
Flo
w s
tress k
f
True strain
Required stress to overcome
strain hardening
Minimal required stress for
initial plastic deformation
Seite 43 © WZL/Fraunhofer IPT
Stress conditions with corresponding Mohr's stress circles
Uniaxial
Biaxial
Triaxial
Seite 44 © WZL/Fraunhofer IPT
Yield criteria
Shear stress hypothesis by Tresca
𝜎2 = 𝜎3 = 0
𝜎1 =𝐹
𝐴= 𝑘𝑓 = 2𝑘 k =
kf
2
with 𝜏𝑚𝑎𝑥 𝜎1 − 𝜎3 = 𝑘𝑓
Form change – Energy hypothesis by von
Mises
𝜏𝑚𝑎𝑥 = 𝑘
𝜎3 𝜎1
𝜏
𝜎
𝑘𝑓 =1
2𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎32 + 𝜎3 − 𝜎1
2
𝑘𝑓 =1
2max ( 𝜎1 − 𝜎2 , 𝜎2 − 𝜎3 , 𝜎3 − 𝜎1 )
Seite 45 © WZL/Fraunhofer IPT
The strain rate tensor and the deviatoric stress tensor are proportional to
each other (λ = proportionality factor).
Levy-Mises flow rule
Elastic Plastic
11
1
E
Hooke‘s law:
stress σ
strain εel
),(
fkf
Levy-Mises flow rule:
Mathematical dependence between yield stress and strain rate.
2
3
2
2
2
12
31
fk
m 11 m 22
m 33
Proportionality factor λ (not constant):
Mathematical dependence
between stress and strain.
Alternative form (division by dt):
Out of flow rule and v. Mises yield criterion follows:
mdd 22 mdd 11 mdd 33Flow rule:
yie
ld s
tress k
f
plastic strain φV
Seite 46 © WZL/Fraunhofer IPT
Yield stress and yield criterion
σr
σr
σz
σz
σt
σt
Assumption for plastic flow (v. Mises)
2
2132
32
2
21
fV k
σV
σV
Equivalent stress Yield criterion
σV = kf
F F
kf
Yield stress
kf
Real process
(multiaxial)
Determination of flow curves
(uniaxial)
F
Seite 47 © WZL/Fraunhofer IPT
Forming Property: Measuring Grid Technique
Deformation of the measuring grid because of tensile and compression stresses inside
the sheet metal while forming
The effective strain can be derived from the grid deformation = maximum deformation
(forming limit)
0
1b
d
bln
0
1l
d
lln
Seite 48 © WZL/Fraunhofer IPT
Forming Property: Forming Limit Curve
Determination of forming limit curve
to predict failure by using FEM
Quelle: ThyssenKrupp
Variable strip thickness to vary φ2 (one test
corresponds with one value of φ2)
Definition: φ1 > φ2
Strain φ2
Str
ain
φ1
Material: RR St 1403
Sheet thickness : 1 mm
“Well“
“Failure “
Test conditions:
deep drawing test with hemispherical
stamp and straight strip
Tenso-tenso
Tenso-
compressive
Seite 49 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 50 © WZL/Fraunhofer IPT
Considerations prior to a FE simulation study (Forming process)
Definition of the simulation problem
Objective of the simulation study
Relevant physical mechanisms:
– Mechanical, thermal, electro-magnetic…
Type of the problem:
– Linear
– Non-linear
Time dependency:
– Static
– Dynamic
Simulation software & hardware:
– Solvers for the intended objectives
– Element types
– Specific numerical technologies
Constituents of a model
Geometry
– Accurate form reproduction
– Stock or special FE mesh generator
– Critical areas, complex shapes
Material
– Material model formulation
– Elasticity and Poisson’s ratio
– Density, hardening
– Thermal properties
Boundary conditions
– Process parameters
– Process kinematics
– Process steps
Seite 51 © WZL/Fraunhofer IPT
Procedure of FE-Analysis
CAD-model
Idealization
Discretization
Boundary
conditions
FE-Analysis
Interpretation of
the results
Pre-processor
Solver
Post-processor
23
13
12
22
11
13
23
33
2212
1211
13
23
12
22
11
0000
0000
0000
000
000
G
G
G
GG
GG
Seite 52 © WZL/Fraunhofer IPT
FE Study process
CAD model
Idealization
Discretization
Material modeling
FE-Analyses
Evaluation
Boundary conditions
Geometry of a workpiece and a tool.
Often available as CAD Data.
Universal formats for 3D data (STEP, STP, STL…)
Simplification of the real geometry for a more structured mesh
Meshing of an object into discrete domains
Numerical reproduction of mechanic, kinematic, contact,
electro-magnetic, thermal conditions of a real process
Numerical formulation of relevant material properties
(elasticity, plasticity, shear etc.)
Calculation of elementary matrices, definition of the
system matrix and a vector of outer forces, solution of
linear equation systems for every integration point
Analysis of the results and answering the objective
of the study
Seite 53 © WZL/Fraunhofer IPT
Simulation of bulk metal forming processes
Chronology of FEM-Simulation: Material modelling
CAD model
Idealization
Discretization
Material modelling
FE-Analyses
Evaluation
Pre
pro
ce
ssor
So
lve
r P
ostp
ro.
Description of material behavior using
mathematical material models
Use of ideal-plastic material model is sufficient for
bulk metal forming processes
Use of elastoplastic material models for
simulation of sheet metal forming processes
Stress σ
Nominal strain ε
Elastic Ideal plastic
Plastic with
hardening
Elasto-plastic
with hardening
Boundary conditions Stress σ
Stress σ
Stress σ
Nominal strain ε
Nominal strain ε Nominal strain ε
Seite 54 © WZL/Fraunhofer IPT
Implicit solution method:
– Small number of time steps
(respectively long time increments)
– Higher effort for iterations compared to explicit
solution method
– Often less computation time then with explicit
solution method
– Applicable especially for static and
quasi-static problems
Explicit solution method:
– Length of increment depends on the speed of
sound c, Young‘s modulus E and material
density ρ; this requires a high number of
increments
– Longer computation time compared to implicit
solution method
– Applicable especially for highly dynamic
problems (e.g. crash-simulations)
Simulation of bulk metal forming processes
Chronology of FEM-Simulation: FE-Analysis
CAD model
Idealization
Discretization
Material modelling
FE-Analyses
Evaluation
Pre
pro
ce
ssor
So
lve
r P
ostp
ro.
Boundary conditions
Seite 55 © WZL/Fraunhofer IPT
Simulation of bulk metal forming processes
Movie: FEM-Simulation cross joint
Degree of damage Effective stress Mean stress
True strain Velocity field
Typical evaluation variables are stress-strain-profiles or
characteristic values such as the degree of damage.
CAD model
Idealization
Discretization
Material modelling
FE-Analyses
Evaluation
Boundary conditions
Seite 56 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 57 © WZL/Fraunhofer IPT
The great challenges of the cutting process
Source: Jaspers
Process Strain Strain rate / s-1 Thomolog
Extrusion 2 – 5 10-1 – 10-2 0.16 – 0.7
Forging /
Rolling 0.1 – 0.5 10 – 10+3 0.16 – 0.7
Sheet metal
forming 0.1 – 0.5 10 – 10+2 0.16 – 0.7
Cutting 1 – 5 10+3 – 10+6 0.16 – 0.9
Cutting process
Extreme conditions in the cutting process
Seite 58 © WZL/Fraunhofer IPT
Influence factors on the cutting process
Workpiece
µ-structure
Texture
Material
properties
Hardness
Residual stress
Cutting condition
Chip formation
mechanisms
Cooling lubricant
Cutting parameters
Contact
conditions
e. g.: Friction, heat
transfer, wear, etc.
Tool
Cutting material
Coating
Geometry
Tool holder
Machine
Machine design
Drive unit
Clamping device
Bild
eines
Prozesses
M
Seite 59 © WZL/Fraunhofer IPT
Input and output parameters of the cutting simulation
Tool
strain
stresses
temperatures
process forces
wear
Chip Formation
temperatures
stresses
deformations
strain rate
kind of chip
chip flow
chip breakage
Workpiece / Tool
geometries
material data
contact conditions
boundary conditions
cutting conditions
Workpiece
strain
temperatures
deformation
burr formation
distortion
prospective:
residual stresses,
surface qualities,
like: roughness,
dimensional- and formdeviation
Seite 60 © WZL/Fraunhofer IPT
Tool geometry modelling for a realistic tool CAD model
Macro-
geometry
4 mm
Drilling tool FE-CAD-Model Acquisition of tool geometry
Micro-
geometry
6 µm 6 µm
Real tool CAD-Model
Seite 61 © WZL/Fraunhofer IPT
Definition of element type
Seite 62 © WZL/Fraunhofer IPT
Material law to calculate stresses
i iB u
Seite 63 © WZL/Fraunhofer IPT
Thermo-mechanical behavior of material
Quelle: Diss-Abouridoaune
( , , )T
200
250
300
350
400
450
0 0.1 0.2 0.3 0.4
AA6063-T6
d/dt=10-3
s-1
& T=20°C
, -
,
M
Pa
300
350
400
450
10-3
10-1
101
103
AA6063-T6
=0.1 & T=20°C
d/dt , s-1
0
150
300
450
0 100 200 300 400 500
AA6063-T6
=0.1 & d/dt=1s-1
T , °C
Strain Rate HardeningStrain Hardening Thermal Softening
Source: Diss- Abouridouane
Seite 64 © WZL/Fraunhofer IPT
Empirical models: e.g. Johnson-Cook-Modell
Micro mechanical models: e.g. enhanced Macherauch-Vöhringer-Kocks-model
Semi-empirical models: e.g. Zerilli-Armstrong-model for bcc-materials
Constitutive material modelling for the FE cutting simulation
Source: Diss-Abouridouane
n -1/2
G 1 2 3 4 5σ = Δσ +C exp -C T +C Tln( ) +C +C L
Initial density
of dislocations Dislocation jam
Influence of temperature
and strain rate Influence of grain size
m
n r0
m r
T T(A B ) (1 Cln( / )) (1 )
T T
Strain hardening Strain rate sensitivity
Thermal softening
1/ p1/ q
* 0a 0
0
kT1 ·ln
G
Athermal processes Damping process
Thermal activated processes
Seite 65 © WZL/Fraunhofer IPT
Determination of High speed flow curves
Source: LFW
Split-Hopkinson-Pressure-Bar
AusgangsstabEingangsstab
Lager
Lager
Joch Projektil
Rohr
Zugprobe
Deckel mit
Luftanschluss
Pressluftbehälter mitSchnellöffnungsventil
AusgangsstabEingangsstab
Lager
Lager
Joch Projektil
Rohr
Zugprobe
Deckel mit
Luftanschluss
Pressluftbehälter mitSchnellöffnungsventil
Split-Hopkinson-Tension-Bar
Strain rate: 500 s-1 – 10000 s-1
Temperature range: 93 K – 1273 K
Projectile speed: 2,5 m/s – 50 m/s
Projectile mass: m = 3,15 kg
Joke Projectile Input rod Tensile specimen
Output rod
Tube Cover with air
connection
Air cylinder with
quick release valve
Lager LagerProbe
Temperier-
kammerAusgangsstabEingangsstab
Rohr
Preßluftbehälter
Projektil
AuffangbehälterCollection bag Sample
Input rod Output rod
Tube
Projectile
Air cylinder
Temperature
chamber
Bearing
Bearing Bearing
Bearing
Seite 66 © WZL/Fraunhofer IPT
Material law for high strain rate deformation
Quelle: Diss-Brodmann
dtBKa
BKk
n
n
adf
)(1
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
0 0 . 2 0 . 4 0 . 6 0 . 8
d / d t = 5 0 1 0 s
- 1
4 8 8 9 s - 1
4 3 5 0 s - 1
4 2 9 4 s - 1
3 4 5 0 s - 1
3 4 3 9 s - 1
2 5 5 8 s - 1
2 5 2 9 s - 1
0 . 0 0 1 s - 1
o : r
K = 9 6 0 M P a B = 0 . 0 3 1 n = 0 . 1 8 2
/ K = 6 . 2 5 · 1 0 - 6
s
A A 7 0 7 5 T 7 3 5 1 D r u c k v e r s u c h e
w
a h r e
S p a n n u n g , M
P a
True plastic strain, -
Lines: Calculation
Symbols: experiments
Pressure tests
Tru
e S
tress,
M
PA
Source: Diss- Brodmann
Seite 67 © WZL/Fraunhofer IPT
Material damage mechanisms
Quelle: Diss-Abouridouane
Type of load
Shear loading Tensile loading
TiAl6V4
20 µm 20 µm
Shear localisation model
(Imperfection theory)
Void growth model
(Hancock-Mackenzie)
Seite 68 © WZL/Fraunhofer IPT
Damage modelling for the FE cutting simulation (ductile failure)
Macromechanical damage models
– Effective stress / effective strain model: Dσ = σv,f / Dε = εv,f
– Gosh-Model: DGosh = (1+σ2/σ1) σ12
– Ayada-Model: dD = (σm/σv) dεv
Micromechanical damage models (Void expansion models)
– Hancock-Mackenzie-Model
– Gurson-Tveergard-Needleman-Model
– Johnson-Cook-Model
m
f 1 2 3 4 5
v 0 m
σ ε Tε = D + D exp -D 1+ D ln 1+ D
σ ε T
m
f n
v
σ3ε = ε + α exp -
2 σ
2
2V m1 1
V,M V,M
σ 3σ0 = + 2fq cosh - 1+ q f
σ 2σ
0
E
εf
Source: Diss-Abouridouane
Seite 69 © WZL/Fraunhofer IPT
Friction modelling for the FE cutting simulation
Thermal load on the tool-workpiece interface
1 primary shearing zone
2 secondary shearing zone of the face
3 seperative zone (stagnation point)
4 secondary shearing zone of the flank
5 preliminary deformation zone
workpiece structure
cut surface
turning tool
flank
rake face
shearing
plane chip
structure
2
vc
1
5
3
4
600
700 650
600
400
450
500
300 310
380 ºC
130
80 500
30
Workpiece
Chip
Tool
Workpiece material: Steel
Yield strength: kf = 850 N/mm2
Cutting material: HW-P20
Cutting speed: vc = 60 m/min
Uncut chip thickness: h = 0,32 mm
Rake angle: o= 10º
Temperature distribution on the contact zone
(by Kronenberg)
Seite 70 © WZL/Fraunhofer IPT
Friction modelling for the FE cutting simulation
Deformation on the tool-workpiece interface
turning tool
shearing zone
0,1 mm
1 primary shearing zone
2 secondary shearing zone of the face
3 seperative zone (stagnation point)
4 secondary shearing zone of the flank
5 preliminary deformation zone
workpiece material: C53E
cutting edge material: HW-P30
cutting speed: vc = 100 m/min
cross-section area of cut: ap x f = 2 x 0,315 mm2
cut surface
workpiece structure
cut surface
turning tool
flank
rake face
shearing
plane chip
structure
2
vc
1
5
3
4
Seite 71 © WZL/Fraunhofer IPT
Friction modelling for the FE cutting simulation
Mechanical load on the tool-workpiece interface
1 primary shearing zone
2 secondary shearing zone of the face
3 seperative zone (stagnation point)
4 secondary shearing zone of the flank
5 preliminary deformation zone
workpiece structure
cut surface
turning tool
flank
rake face
shearing
plane chip
structure
2
vc
1
5
3
4
Normal stress:
Shear stress:
Tool
by Oxley und Hatton
Contact zone
Seite 72 © WZL/Fraunhofer IPT
Coulomb friction model:
Shear friction model:
with
Transition from sliding friction (Coulomb) to
sticking friction (Shear):
Z.B.: Usui-Model
τR – Friction shear stress
N – Normal stress
k – Von Mises flow shear stress
kf – Von Mises flow stress
µ, m – Friction coefficients
kmR
NR
Friction modelling for the FE cutting simulation
R
3
fkk
Orowan / Özel
Usiu
Shaw /
Wanheim und Bay
Reality
N
Coulomb friction
Shear friction
Reality
N
R
Sticking Sliding
Seite 73 © WZL/Fraunhofer IPT
Wear modelling for the FE cutting simulation
Wear typs on the tool cutting edge and wear mechanisms
Sliding mechanisms No sliding mechanisms
Abrasion Adhesion Delamination Diffusion Electrochemical Oxidation
Tool cutting
edge
Cutting edge
breakouts
Crater wear
Flank wear
Oxidation
Flank wear
Crater wear
Built-Up-Edge
Tool
Flank face
Rake
face
Workpiece
Chip
Seite 74 © WZL/Fraunhofer IPT
Tool wear modelling
tool life by Taylor: tool life by Hasting:
B
AT
v
k
c CvT
T = tool life
= temperature k, A, B = constants
Cv = T for vc = 1 m/min
empirical
tool wear model
physical
tool wear model
tool wear modelling
tool wear rate models tool life equations
model by Usui: model by Takeyama: model by Archard:
H
SFK
dt
dV
3
)T
C(
1chn
2
eCvσdt
dV
R
E
eDvGdt
dVc
dV/dt = wear volume per time
H = hardness
F = mechanical load
S = cutting path
K, C1, C2, G, D = constants
n = normal pressure
vch = sliding velocity
= temperature
Abrasion + Diffusion Adhesion
Adhesion /
Abrasion
Seite 75 © WZL/Fraunhofer IPT
FEM Software Solution for FEM simulation of the Cutting Process
MSC.Marc
Seite 76 © WZL/Fraunhofer IPT
Criteria for the evaluation of FE software
Source: SIMULIA, ANSYS, LSTC, TWS, SFTC, COMSOL
Criteria
Program ABAQUS ANSYS/ LS-DYNA AdvantEdge DEFORM COMSOL
Creation of geometries Creation of geometries
and import of CAD data
Import of CAD data
Creation of simple
geometries and
import of CAD data
Creation of simple
geometries and
import of CAD data
Creation of simple
geometries and import
of CAD data
Material catalogue No, has to be defined Yes, expandable Yes, wide Yes, new catalogue
importable
yes
Element type Every type Every type tetrahedron,
rectangle
tetrahedron, rectangle Every type
Time integration Implicit / Explicit Implicit / Explicit Explicit Implicit Implicit
Remeshing routine none none yes yes yes
use general general Cutting process Deforming process general
Influence on simulation
computation
High, by Python Possible, by Fortran no High, by Fortran High, by Matlab
parallelization possible possible possible possible possible
Usage at the WZL Eigenfrequency analysis,
elast. Tool behavior,
elasto-plastic component
behavior
no no Cutting simulation Thermo-elastic
deformation
Seite 77 © WZL/Fraunhofer IPT
Cutting process simulation
Turning Drilling Milling
Calculation of the thermo-mechanical tool-load-collective
for an ideal dimensioning of the tools‘
micro- and macrogeometry
Seite 78 © WZL/Fraunhofer IPT
Simulation of the chip flow (turning)
Material:
C45E+N
Cutting material:
HC P25
Insert:
CNMG120408
Insert geometry:
Cutting velocity.:
vc = 300 m/min
Feed:
f = 0,1 mm
Depth of cut:
ap = 1 mm
Dry cutting
Chip breaker FN Chip breaker RN
0 0 S r
6° -6 ° 95°-6°
90°
0 0 S r
6° -6 ° 95°-6°
90°
Seite 79 © WZL/Fraunhofer IPT
First Contact Start of Shearing Crack Initiation Gliding
End of Gliding New Segmentation
Mate
rial S
pe
ed / m
/min
0
12,5
25
37,5
50
62,5
75
90
Segmented Chip Simulation reveals periodic sticking zone
Start of Shearing Crack initiation
strain rate
Seite 80 © WZL/Fraunhofer IPT
Verification of the mechanical load
Cutting tool material: HC-P25
Workpiece material: C45E+N
CL: dry
Process: turning
Ff
Fp
Fc
200
600
1000
1400
f = 0.1 mm f = 0.2 mm
sim
ula
tio
n
exp
erim
en
t
1800
Ff
Fp
Fc
constant: vc = 350 m/min; ap = 3 mm
200
600
1000
1400
vc = 250 m/min vc = 350 m/min
sim
ula
tio
n
exp
erim
en
t Ff
Fp
Fc
Pro
cess forc
es F
i/ N
Ff Fp
Fc
constant: f = 0,1 mm; ap = 3 mm
200
600
1000
1400
ap = 1 mm ap = 3 mm
sim
ula
tio
n
exp
erim
en
t
Ff Fp
Fc Ff
Fp
Fc
constant: vc = 350 m/min; f = 0.1 mm
Pro
cess forc
es F
i/ N
Pro
cess forc
es F
i/ N
Seite 81 © WZL/Fraunhofer IPT
533
3 µm 6 µm
TiN Al2O3
TiN
6 µm 0
510
520
530
540
550
560
570
Ca
lcu
late
d t
em
pe
ratu
re a
t th
e
chip
bottom
sid
e T
Sp / °
C
heat conductivity:
HW: 100 W/(mK)
TiN: 26,7 W/(mK)
Al2O3: 7,5 W/(mK)
material: C45E+N
tensile strength: Rm = 610 N/mm²
557
539
509
539
heat capacity:
HW: 3,5 J/(cm³K)
TiN: 3,2 J/(cm³K)
Al2O3: 3,5 J/(cm³K)
HW: HW-K10/20
TiN
3 µm
TiN
6 µm
HW TiN
6 µm
Al2O3
6 µm
coating
thickness
Tsp
FE simulation of turning considering coating
Seite 82 © WZL/Fraunhofer IPT
FE-Based Calibration process for the tool wear model
Verschleißmarkenbreite über die Schnittzeit
16MnCr5 (einsatzgehärtet), Stegbreite = 1 mm, cBN bestückte Einstechplatte der Sorte N151.2-600-50E-G
Schnittgeschwindigkeit vc = 150, 200, 300 m/min und Vorschub f = 0,06 mm
0
20
40
60
80
100
120
0 5 10 15 20 25 30 35 40
Schnittzeit t [min]
Ve
rsc
hle
ißm
ark
en
bre
ite
VB
[µ
m]
vc = 250 m/min vc = 200 m/min vc = 150 m/min
Cutting time t
To
ol-w
ear
VB
Wear curve
C2
lg C1
1/T
lg{w
/(n
VS)}
lg
{w/(
nV
S)}
Regression analysis FE-analysis
Temperature
Normal-
tension
Sliding speed
dW/dt
Machining experiments
Determination of the
specific material
parameters C1 and C2
)T
C(
1chn
2
eCvσdt
dW
t = 1 min
t = 4 min t = 6 min
t = 10 min
Modeling
Seite 83 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 84 © WZL/Fraunhofer IPT
Technical Sensors in Metal Cutting
Measuring platform Rotating cutting force
dynamometer
Acceleration sensor Force measuring pin
Load ring Pyrometer Process remarks
vc
Cutting force
Ac
ce
lera
tio
n
Source: Kistler Instrumente AG
Seite 85 © WZL/Fraunhofer IPT
Temperature Sensor
Thermo-element
Resistance thermometers
Two color pyrometer
Infrared camera
Seite 86 © WZL/Fraunhofer IPT
3D Coordinate Measuring
Seite 87 © WZL/Fraunhofer IPT
1 mm
vc= 20 m/min
f = 0,3 mm
Wo
rkpie
ce
hold
er
vc
wo
rkp
iece
Features and Technical Data of the Test Bench
Shaft holder max. 32x32
Grooving / Parting Tool holders for orthogonal cutting (Inclination angle = 0°, tool edge angle = 0°)
Phantom v7.3 High speed video camera, LED Illumination
Seite 88 © WZL/Fraunhofer IPT
Advanced Experimental Setup: Orthogonal Cut on Broaching Machine
High speed external broaching machine:
– Type: Forst RASX 8x2200x600 M/CNC
– Max. force: 80 kN
– Power: 40 kW
– Max. cutting speed: 150 m/min
– Tool fixed und workpiece moved
– Optimal filming of the cutting zone
High speed camera:
– Type: Vision Research Phantom v7.3
– Frame rate: 6.688 fps by 800 x 600 pixel
500.000 fps by 32 x 16 pixel
High speed IR camera:
– Type: FLIR SC7600
– Frame rate: 100 fps by 640 x 512 pixel
800 fps by 160 x 128 pixel
– Measurement range: -20°C - 3000 °C (±1°C)
wo
rkp
iece
wo
rkp
iece h
old
er
IR camera
tool
HS camera
tool holder
tool
measurement platform
wo
rkp
iece High-Speed Filming
High-Speed Thermography
wo
rkp
iece h
old
er
Seite 89 © WZL/Fraunhofer IPT
Turning: Comparison of Simulation and Real Chip Flow
CNMG120408
Chip breaker NF
HC-P15
r = 95°
n = -6°
s = -6°
C45E+N
ap = 1,9 mm
f = 0,25 mm
vc = 200 m/min
dry
vc vf
Seite 90 © WZL/Fraunhofer IPT
.
Full agreement
Experiment Simulation
FE simulation of face milling operation
Seite 91 © WZL/Fraunhofer IPT
FE computation of mechanical tool load and chip form in drilling
0,0
1,0
2,0
3,0
Exp
eri
men
t
Sim
ula
tio
n
Torque
Exp
erim
en
t
Sim
ula
tion
0
100
200
300
400
Exp
eri
men
t
Sim
ula
tio
n
Feed force
[Nm] [N]
4% Deviation 7% Deviation
Simulation
Experiment
Cutting speed: vc = 35 m/min Workpiece: C45E+N
Feed: f = 0.18 mm Cutting tool material: HW-K20
Drill diameter: d = 8 mm Cutting edge radius: rß = 60 µm
Seite 92 © WZL/Fraunhofer IPT
FE computation of the cutting temperature in drilling
Cutting speed: vc = 35 m/min Workpiece: C45E+N
Feed: f = 0,012 * d Cutting tool material: HW-K20
Coolant: none Cutting edge radius: rn = 4 µm
d = 3 mm
0
100
200
300
400
1 3 8 10
diameter d [mm]
Te
mp
era
ture
at th
e
ma
jor
cu
ttin
g e
dg
e T
[°C
]
Experiment
Simulation
Seite 93 © WZL/Fraunhofer IPT
High Speed Thermography During Chip Formation (vC = 150 m/min)
h = 0.10 mm h = 0.50 mm
Tool:
Carbide, uncoated
Sharp (rß 5 µm)
Workpiece:
AISI 1045 normalized
3.5 x 50 x 200 mm
600
400
350
300
250
200
50
0
°C
Seite 94 © WZL/Fraunhofer IPT
Material and Friction Laws Validation:
Chip Formation (Orthogonal Cut, vC = 150 m/min, AISI 1045)
h = 0.4 mm h = 0.5 mm h = 0.3 mm
FE simulation Experiment FE simulation Experiment FE simulation Experiment
h = 0.2 mm h = 0.04 mm h = 0.02 mm
FE simulation Experiment FE simulation Experiment FE simulation Experiment
Seite 95 © WZL/Fraunhofer IPT
Material and Friction Laws Validation:
FE Cutting Simulation (vC = 150 m/min, h = 0.50 mm, DEFORM)
Plastic strain[ - ]
[MPa]Effective stress
[ C]
Temperature
[1/s]Strain rate
Seite 96 © WZL/Fraunhofer IPT
Development of 3D FE computation model for macro twist drilling:
d = 8 mm, homogeneous microstructure, Deform 3D
Workpiece
Twist drill
Twist drill CAD model
Gühring KG
d = 8 mm
rß = 0 µm
RT100U_5510
Constitutive material law (WZL)
m
rm
r
0
n
TT
TT1
ε
εlnC1εBAσ
Cutting parameters definition
Boundary conditions adjustment
Twist drill: Rigid with mesh
d = 8 mm and rß = 30 µm
Workpiece: Visco-plastic
D x H = 12 x 4 mm
with heat dissipation
100,000 3D-Tetrahedron
Contact: Coulomb friction
law (µ = 0.30), heat transfer
(Conduction & Convection)
Workpiece material: 27R, 45R, 60R
Tool material: HW
Cutting speed: 120 m/min
Feed rate: 0.25 mm/rev
Seite 97 © WZL/Fraunhofer IPT
FE model results: Chip formation, temperature, computation time: 1 day
45R, vc = 120 m/min, f = 0.25 mm, d = 8 mm, dry
Seite 98 © WZL/Fraunhofer IPT
FE-Simulation of the drill entrance: Computation time: 5 days
45R, vc = 120 m/min, f = 0.25 mm, d = 8 mm, dry
Seite 99 © WZL/Fraunhofer IPT
Check of the optimized FE model: Feed force and torque
45R, vc = 120 m/min, f = 0.25 mm, d = 8 mm, dry
0
250
500
750
1000
1250
1500
0 10 20 30 40 50 60 70 800
2
4
6
8
10
12
14
16
Drilling time t / ms
Fe
ed f
orc
e F
z /
N
Torq
ue
Mz /
NmFeed force
Torque
with entrance without entrance
Seite 100 © WZL/Fraunhofer IPT
0
1
2
3
4
5
6
7
Feed force Torque Feed force Torque Feed force Torque
Test
FE-Simulation
FE model validation: Feed force and torque (deviation less than 15%)
vc = 120 m/min, f = 0.25 mm, d = 8 mm, dry
27MnCr5 C45E C60
kN
Nm
Seite 101 © WZL/Fraunhofer IPT
Optimization integration in the FEM 9
FE model validation 8
FE modelling for the cutting process 7
FE modelling for the forming process 6
Lecture topics and fundamental knowledge 5
Modelling and simulation: Definition, motivation and integration 4
Lecture objectives 3
Presentation of WZL 2
Lecture organisation 1
Outline
Seite 102 © WZL/Fraunhofer IPT
What is the Optimization Problem?
Source:Papalambros
F(x1,x2,…,xn) Minimize (or maximize)
an objective „performance“ function:
taking into account the constraints: Gi(x1,x2,…,xn) = 0, i=1,2,…,p
Hj(x1,x2,…,xn) < 0, j=1,2,…,q
(x1,x2,…,xn) are the n system variables
Gi(x1,x2,…,xn) are the p equality constraints
Hj(x1,x2,…,xn) are the q inequality constraints
Seite 103 © WZL/Fraunhofer IPT
System Example: Cantilever Beam
System variables: F(t), U(t), Mb(t), V(t)
System parameters: h, b, L
System constants: E, ρ
L
F(t)
Steel (E, ρ)
h
h
b U(t), Mb(t), V(t)
System U(t) F(t)
System F(t) U(t)
System V(t) h, b, L
E, ρ
System Mb(t) F(t)
Mathematical model:
3
3
3
33
hbE
LF4
12
bhE3
FL
EI3
FLU
Seite 104 © WZL/Fraunhofer IPT
Integration of the Optimization in the FEM
Physical problem
Mathematical model
Numerical model
Change physical
problem
Improve mathematical
model
Refine analysis
Process improvement and optimization
Does answer
make sense?
Yes!
No!
FEM Optimization
Seite 105 © WZL/Fraunhofer IPT
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