Poster Presentations I - the home page of EM

Transcription

4
SURVEY
of
POSTER PRESENTATIONS
This section contains a survey of poster presentations of actual PhD-projects within the
Graduate School Engineering Mechanics. Individual poster presentations are collected in a
separate report, which will be supplied at the start of the symposium and can be obtained
from the Secretariat of the Graduate School. Furthermore, poster presentations are
available through:
http://www.em.tue.nl
Survey of Poster Presentations
Nr
1
2
3
4
5
6
7
8
9
Name
N.P. van der Aa
B.J. Aalderink
A. Andreykiv
A. Balmachnov
R.A. van de Berg
A.W. Blom
S.H.A. Boers
M.H.A. Bonte
M.J. v.d.Bosch
Uni
TU/e
UT
TUD
TU/e
TU/e
TUD
TU/e
UT
TU/e
10
11
12
13
14
15
16
17
18
19
20
P. Broomans
I.A. Burchitz
M.V. Cid Alfaro
N.E. Conza
W. Dijkstra
N.T. Dung
M.E. Erinç
R.P.H. Faassen
M.W. de Graaf
W.B.J. Hakvoort
M.H.C. Hannink
TUD
UT
TUD
TUD
TU/e
TUD
TU/e
TU/e
UT
UT
UT
22
23
24
25
26
27
28
29
30
31
32
33
34
35
P.J. Heres
B.D. Heru Utomo
T.S. Hille
T. Hofman
M. Hrapko
P.J.M. Janssen
A.J.J. Koopman
M.G.M.M. v. Kraaij
F. Kraaijeveld
M. Langelaar
G.K. Lau
G. van der Linde
O. Lloberas
J.M. López de la
Cruz
X. Ma
N.J. Mallon
R.M.C. Mestrom
M.J.J. Nijhof
E.J. Oosterhuis
P. Owczarek
I. Özdemir
M.J.P.D. Patricio
R.R. Pedersen
E.S. Perdahcioglu
T. Rahman
M. van Riel
M.K. Saraswat
R. Scholte
E. Shcherbakov
TU/e
TUD
TUD
TU/e
TU/e
TU/e
UT
TU/e
TU/e
TUD
TUD
UT
TUD
TUD
Title Poster
Sensitivity theory of the Rigorous Coupled-Wave Analysis
Shedding Light on Laser Welding
Numerical modelling of electrostatic-structural coupling
Tailoring of processable metastable steels (1)
Convengence of switched systems
Mechanics of Thin-Walled, Variable-Stiffness Shells made by Tow-Placed Composites
Optimum path and discrete 3D forming
Optimisation of Forming Processes
A combined experimental-numerical approach to characterize delamination in polymer
coated steel
Orientation and Position of the Glenoid Component
Improving numerical predictability of Springback
Mode I Crack Tunneling in Fibre-Metal Laminates
Frequency dependent properties in modal parameter identification
Condition number of the boundary element method matrices
Discontinuous Galerkin methods for structural systems
Fatigue Analysis of Lead-free Solder Interconnect
Identification of Spindle Dynamics in High Speed Milling
Real-time trajectory generation for sensor-guided robotic laser welding
Improved tip tracking for an industrial robot using Iterative Learning Control
Reduction of sound transmission through panels by means of tuned acoustic
resonators
Robust Model Order Reduction by Krylov subspace methods
High-speed impact modelling and testing of composite structures
Design and development of high temperature coating system for engines
Design of a Hybrid Vehicle: From generic to specific design
The Mechanical Behaviour of Brain Tissue
Grain statistics: an experimental investigation
Flow front tracking in aluminium extrusion dies by means of particle trajectories
A more Rigorous Coupled-Wave Analysis
A singularity solution of shear faulting in swelling ionised porous media
Topology Optimization of Shape Memory Alloy Thermal Actuators
Electrostatic Actuators Using Elastomers
Galling mechanism in deep drawing processes
First-order multi-scale framework for discontinous modelling of failure
Simulation of Environmentally assisted failures of metals
TUD
TU/e
TU/e
UT
UT
UT
TU/e
TU/e
TUD
UT
TUD
UT
TUD
TU/e
TU/e
Effect of Moisture on the Viscoeleasticity of Thermosetting IC Packaging Polymers
Dynamic stability of a base excited thin beam with top mass
Dynamics and multiphysics interaction in Microsystems
Optimization of folded resonators for broadband reduction of computer fan noise
Inverse dynamics for durability testing
Design rules for close tolerance and lubricant free piston compressors
Multiscale Modeling of Thermal Shock: From Microstructure to Failure
Elastic stationary analysis of a cracked plate
Computational study of impact fracture of concrete structures
Constitutive modeling of metastable austenitic stainless steel
Fast tools for Multi-Fidelity Nonlinear Finite Element Analysis of Structures
Strain path dependent Material models for forming and crash
Cure Shrinkage Monitoring in Thermoset Resins
High Resolution Planar Near-field Acoustic Holography
Development of a Maxwell's Solver
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Eighth Engineering Mechanics Symposium, Nov. 2005
39
Nr
51
Name
J. Shi
52
53
P.J. Sloetjes
Q.H.C. Snippe
54
55
56
N.J. Suman
P. Tiso
S. Tosserams
57
58
59
60
61
62
63
64
A.A. van Veggel
I. Vegt
A. Verhoeven
C.V. Verhoosel
L. Wang
J.W. Wind
A.J. de Wit
D.G. Yang
65
K.G. van der Zee
Uni Title poster
TUD The competition between solid-state phase transformations and plactic deformation:
discrete interfaces and discrete dislocations
UT Balancing and stabilizing flexible shafts with piezoelectric materials
UT Design and optimisation of vertex detector foils as a function of the deformation
process, in
particular superplastic forming
TUD Effect of resin chemistry on thermoset visoelasticity
TUD Finite Element Based Asymptotic Postbuckling Analysis of General Structures
TU/e An Augmented Lagrangian Relaxation for Analytical Target Cascading using the
Alternating
Directions Method of Multipliers
TU/e Silicon Carbide optical bench for stability measurements
TUD Experimental study to the impact behaviour of concrete
TU/e Hierarchical multirate BDF methods for IC transient simulation
TUD Spectral solution of the random eigenvalue problem
TUD Flexibility Study of Ultra-thin Substrate
UT Inverse acoustics IBEM vs. PNAH
TUD Multilevel Optimization of Composite Structures
TUD Micromechanical Modeling of Cure-Induced Stresses in a Particle-Filled Electronic
Packaging Polymer
TUD A Posteriori Error Estimation for Free-Boundary Problems
40
Sensitivity theory of the Rigorous
Coupled-Wave Analysis
N.P. van der Aa
Eindhoven University of Technology
Faculty of Mathematics and Computing Science
P.O. Box 513, NL 5600 MB Eindhoven
phone +31-(0)40-2475518, email [email protected]
Introduction
Sensitivity theory
Diffraction of light plays an important role in various
metrology applications. Periodic structures, called
gratings, cause the monochromatic incident light to
be diffracted (see figure). This diffracted field gives
an accurate measurement of the position of the grating, which in general has a period in the same order
of magnitude as the field’s wavelength. Since the
estimate of the grating shape can never be 100 %
accurate, it is crucial to know how the diffracted field
behaves when small deviations occur in the grating
shape.
Two methods are under consideration:
• Finite Difference Technique (FDT)
The FDT is a general method to compute sensitivity. It recomputes the field for a small change
in the shape parameter under consideration, because the difference between new and old field
gives the derivative information.
• Analytical approach
Straightforward differentiation of the RCWA
equations requires eigenvalue and eigenvector
derivatives. Fortunately, these derivatives can
be computed without solving additional eigensystems according to Murthy and Haftka [2].
The analytical approach has two advantages: no additional eigenvalue systems need to be solved (faster)
and no additional approximations are made besides
the ones already made by RCWA (more accurate).
Figure 1 : Diffraction by a one-dimensional grating.
Objective
RCWA
Before the field behaviour with respect to grating
shape parameters can be computed, the field itself
should be available for a given grating profile. A powerful method to find to field is the Rigorous CoupledWave Analysis (RCWA) [1]. Its key features are:
• Division of the grating structure in thin layers;
• Approximation of the material properties by a
piecewise constant function;
• Introduction of Fourier expansions of both field
and material properties inside each layer;
As a result, Maxwell’s equations inside each layer are
reduced to one eigenvalue problem. The fields inside
each layer are matched by means of the boundary
conditions. The accuracy of RCWA depends on the
number of layers and the number of terms in the
Fourier expansions.
Results
The field derivatives for a binary grating can be computed by both methods. The difference between them
will not be the accuracy, but the computation speed.
As predicted and illustrated in the figure, the analytical approach is much faster than FDT.
FDT
analytical approach
time
The goal of this project is to find the shape parameter
sensitivity of the diffracted field in an accurate, but
fast way for any type of diffraction grating.
truncation number
Figure 3 : Computation speed of the sensitivity theories.
Conclusion
A faster theory than FDT is developed to compute the
field sensitivity with respect to the grating shape.
References
1.
Figure 2 : Actual (left) and approximated grating (right)
Moharam M.G. et al. (1995) Formulation for stable and efficient implementation of the rigorous coupled-wave analysis
of binary gratings, J. Opt. Soc. Am. A 12(5).
2. Murthy, D.V. and Haftka, R.T. (1988) Derivatives of Eigenvalues and Eigenvectors of a General Complex Matrix, Int. J.
Num. Methods in Engineering 26.
1
Shedding Light on Laser Welding
B.J. Aalderink*, R.G.K.M. Aarts, J.B. Jonker and J. Meijer
*Netherlands Institute for Metals Research
University of Twente, Laboratory of Mechanical Automation
Institute of Mechanics, Processes and Control - Twente
P.O. Box 217, 7500 AE Enschede, The Netherlands
+31-(0)53-4894846, [email protected]
Introduction
Performance
Nd:YAG laser welding is often used in industry to
obtain high quality joints. This however does not
mean that monitoring or control of this process is
common practice. A few commercial products are
available but none of these systems can be used for
monitoring the laser welding process of aluminium.
Within the NIMR project Multivariable melt
pool control for double spot laser welding a
monitoring system is developed based on a CMOS
camera which is suited for the observation of
Nd:YAG laser welding of various materials,
including aluminium, under different conditions.
The current system uses a 10 W, 810 nm diode laser
and a narrowband optical filter to achieve a
sufficiently large S/N ratio.
In figure 2 some examples of the resulting
images using this monitoring system are displayed,
together with an image without illumination.
The images illustrate that the system is
applicable for the observation of single and double
spot laser welding of aluminium, with and without
cold wire feeding, using both a standard CMOS and
a high speed camera. Spectroscopic measurements
show that an optimized system will also function for
laser welding of steel although this has not yet been
tested. Current investigations also focus on the
applicability for laser/MIG hybrid welding.
High power laser
beam
Optical fiber
Coaxial CMOS
camera
Image quality standard
CMOS camera without
illumination
(single spot laser welding
AA5182).
Welding head
Narrow band
optical filter
Image quality new
monitoring system with
high speed camera
(single spot laser welding
AA5182).
Illumination diode
laser light
Weld plume
Image quality new
monitoring system with
standard CMOS camera
Workpiece
(twin spot laser welding
AA5182, with cold wire feeding).
Figure 1: Overview of the new monitoring system.
Traditional Monitoring System
Image is mixture of reflected laser light and
temperature radiation, making it difficult to interpret.
Only indirect information on welding process and
Figure 2: Image quality of the monitoring system
using different configurations compared to typical
traditional monitoring system images.
quality.
Information depends strongly on material and
process parameters.
New Monitoring System
Image contains only geometrical information,
simplifying interpretation and FE model verification.
References
1.
2.
B.J. Aalderink, R.G.K.M. Aarts, J.B. Jonker, J. Meijer.
Weld Plume Emissions During Nd:YAG Laser Welding,
Proceedings of LIM '05, 2005.
B.J. Aalderink, R.G.K.M. Aarts, J.B. Jonker, J. Meijer.
Experimental Observations of the Laser Keyhole Welding
Process of AA5182, Proceedings of ICALEO '05, 2005.
Direct information on the welding process and
quality.
Information independent from material and welding
parameters.
2
Numerical modelling of electrostaticstructural coupling
A. Andreykiv, D. J. Rixen, F. van Keulen
Department of Precision and Microsystems Engineering
Faculty 3ME
Delft University of Technology
Mekelweg 2, 2628 CD Delft, The Netherlands
Phone +31 (0)15 2786818, E-mail [email protected]
Introduction
Results
Fig. 1. Example of an electrostatic actuation
Fig. 2 Electric potential around two beams with
applied voltage.
Voltage, V
Coulomb force is a basic principle of many
MEMS actuators. Previously, calculation of
electrostatic structural coupling was mainly
performed using a staggered method. However,
it can not be applied in case of a strong coupling
or a calculation of a snap back behaviour. An
earlier introduced monolithic formulation [1] was
able to resolve these issues, but
poor
conditioning of the resulting system required an
additional step for the electrostatic domain (“air”)
mesh update. This potentially could result in
severe distortions of the elements on the
structure-“air” interface.
Alternative formulation
Similarly to [1], electrostatic and mechanical
fields are solved simultaneously in the same
variational formulation. Strong form of the
formulation is obtained by taking a derivative of
the energy of the system. The energy of the
system is taken as the sum of the mechanical
and electrical energies. The source of instability
in [1] was a non-linear effect that appeared
when the electrostatic force was applied to the
nodes of the “air” domain. In this work we
propose an alternative formulation, where the
structure is considered independent of the
electrostatic field, while the electrostatic field is
updated with the motion of the structure. The
calculated electrostatic force is then applied only
to the interface between the structure and the
“air”. The electrostatic force is considered as a
follower force, so the appropriate modifications
to the stiffness matrix of the interface “air”
elements are made, making it fully consistent.
0
0
Displacement, m
0.00359
Fig. 3 Displacement of the left beam tip versus
the applied voltage. A snap “through behaviour”
calculated with an Arc-length method allows
determination of a critical (“pull-in”) voltage.
Discussion
The proposed formulation allows a single step
solution of the electrostatic-structural problems
which can improve the stability and convergence
of the problem comparatively to the two-step
approach. The formulation can also be easily
implemented in a commercial FE code.
References
1.
V. Rochus, D. J. Rixen, J.-C. Golinval. Int. J. Numer.
Meth. Engng (2005) In Press
3
Tailoring of processable metastable steels (1)
A. Balmachnov♮ , V.G. Kouznetsova, and M.G.D. Geers
Faculty of Mechanical Engineering
♮
phone: +31-(0)40-2472054, ♮ e-mail: [email protected]
Results
The lamellae model was successfully implemented
within 2D non-linear FE-framework and tested on various plain strain homogeneous loading cases (see
Fig. 2, 3: Plain strain tensile test).
0
MACRO
MICRO
austenite (grains)
Engineering level
Lamellae model
martensite
FE
F
Solving boundary
value problem
1
ξ
~
N
FM,PM
Hydrostatic pressure, GPa
−0.2
−0.3
−0.4
−0.5
0.08
0.06
0.04
0.02
Equivalent Green−Lagrange strain
0
(b)
0.4
MESO
RVE level
0.4
0.2
−0.1
0.08
0.06
0.04
0.02
Figure 2 : (a) Equivalent Cauchy stress and (b) hydrostatic pressure vs. equivalent Green-Lagrange strain
Martensite volume fraction
This project focuses on multi-scale computational modeling and microstructural optimization of
metastable austenitic steels within the multi-scale
computational framework (Fig. 1).
0.8
0.6
0
0
(a)
Method
1
(a)
0.4
Martensite volume fraction
Thanks to their properties, advanced materials
like steels with transforming metastable phases
are used for great number of technological applications. Design of products with certain properties/specifications relies
on properties of employed
materials. However, for metastable steels material design can not be done independently from the
processing route during which the material will be
formed and shaped towards a product with optimal
performance.
Employed interdisciplinary approach aims to bridge
materials science (micromechanics) with materials
engineering (process & product engineering).
• Averaging rules
• Constitutive equations for each phase
• Interface interaction relations
• Transformation criterion
Further on, averaging over all 24 martensitic variants
is performed to capture behavior of a transforming
austenitic grain.
Equiv. Cauchy stress, GPa
Introduction
0.3
0.2
0.1
0
0
0.08
0.06
0.04
0.02
(b)
0.3
1, 2, 5, 6
3, 4, 7, 8
9, 10, 13, 14
11, 12, 15, 16
17, 18, 23, 24
19, 20, 21, 22
0.2
0.1
0
0
0.08
0.06
0.04
0.02
martensite
austenite
F A, P A
PE
P, x
Figure 3 : Martensite volume fraction vs. equivalent
Green-Lagrange strain: (a) averaged (b) per variant
Future Work
Figure 1 : General multi-scale framework for metastable
austenitic steel
Micromechanical Model
On the micro-level, a lamellae model is employed to
compute the evolution of martensitic volume fraction ξ
and the behavior of one transforming martensitic vari¯.
ant for a given total gradient deformation tensor F
The model consists of
• Extension of the implementation to the 3D nonlinear FE setting
• Comparison of simulations with experiments
References
1. Chen, S. P., Kouznetsova, V., and Geers, M. G. D. (2005),
Modelling martensitic transformation induced plasticity at finite strains, submitted
4
Convergence of switched systems
R.A. van den Berg, A.Yu. Pogromsky, J.E. Rooda
Department of Mechanical Engineering
Introduction
Example
It is well known that LTI systems with bounded input
have a unique limit solution that only depends on its
input. Nonlinear or hybrid systems that also show
this desirable property are referred to as convergent.
Since the solution of a convergent system does not
depend on its initial conditions (after a transient period), simulation can be used for performance evaluation.
We consider the class of switched linear systems:
Consider the switched system in Fig. 1 that can be
written in the form of (1), with
ẋ = Ai x + Bi w, i = 1, . . . , k
y = Cx,
(1)
where x(t) ∈ Rn is the state, input w(t) is piecewise
continuous and bounded for all t ∈ R, and i denotes
the operation mode. Furthermore, all system matrices Ai are Hurwitz.



0.6
−1.6 1 0
A1 =  −2.5 0 1  , B1 =  1.5  ,
0.7
−0.7 0 0




1.6
−2.6 1 0
A2 =  −3.3 0 1  , B2 =  2.3  , C = [ 1 0 0 ] .
1.2
−1.2 0 0

For this system a common Lyapunov matrix P can be
found such that the conditions of Theorem 1 are satisfied. For the convergence of the transient part of the
solution the following bound can be determined:
|x1 (t) − x2 (t)| ≤ 2.666 |x1 (0) − x2 (0)| e−0.205t
Objective
The general problem is to find a switching rule such
that the closed-loop system (1) is convergent. Here
we focus on a switching rule that is based on static
state feedback, i.e. i = σ(x).
For the limit solution a Bode-like plot is drawn (Fig. 2).
Here we used w(t) = sin(bt) with b ∈ [10−2 , 102 ], and
we compared the switched system with the separate
PID controllers.
PID1
SWITCH
-
linear
system
y(t)
PID2
||w − y||2 /||w||2
w(t)
+
1
0.8
PID1
PID2
Switched
0.6
0.4
0.2
Figure 1 : Switched system
Exponential convergence
0
10−2
10−1
100
b
101
102
Figure 2 : Performance evaluation
Theorem 1 Let P = P T > 0 satisfy
ATi P + P Ai < 0,
P (Ai − Aj ) − (Ai − Aj )T P = 0 ∀i, j ≤ k.
Then, the following switching rule
σ(x, w) = arg min{xT (ATi P + P Ai )x + 4xT P Bi w}
i
makes system (1) exponentially convergent.
Discussion
Once convergence is obtained for a certain
(sub)class of nonlinear/hybrid systems, e.g. by defining a specific switching rule, simulation can be used
for performance evaluation and optimization of the
system.
5
Mechanics of Thin-Walled, Variable-Stiffness
Shells made by Tow-Placed Composites
A.W. Blom
Faculty of Aerospace Engineering
Kluyverweg 1, 2629 HS Delft
Introduction
Fiber-reinforced composites are gaining ground on
metals in aerospace industry, largely due to intelligent
use of the directional properties of fibers. With towplacement machines these directional properties can
be used even more efficiently, due to the capability to
steer the fibers within the plane of a laminate. By fiber
steering, the local stiffness properties of a laminate
can be varied. Earlier research on variable stiffness
flat plates by Gürdal and Tatting demonstrated that
buckling load may be increased by a factor of 2 to 3,
while keeping the panel weight approximately same
as the traditional laminates 1 .
manufacturing constraints. Arbitrary combinations of
paths can be used to construct a laminate. The variable stiffness properties are implemented in a finite
element program, such that any finite element analysis can be carried out.
Figure 2 : Surface plot of the number of layers of a
variable-stiffness cone made of a ± overlap laminate
Discussion
Figure 1 : A variable-stiffness overlap panel
Objective
The objective of this research is to improve the structural response of thin-walled cylindrical and conical
shells through local stiffness tailoring achieved by judicious selection of fiber paths.
Different cone geometries under different loading
conditions are being studied to investigate the influence of path parameters on buckling and vibration
performance. Furthermore, configurations are being
designed for experimental validation of the concept.
Finally, cost effective analytical solution techniques
are being considered for three-dimensional shell configurations.
Methods
General path definitions along which the fibers can be
steered are developed, taking into account the limitations of the tow-placement machine. Using these definitions and the shell geometry, a program is written
that provides a finite element analysis with the local
stiffness properties. The capability allows parametric
study of improved lay-ups as well as determination of
best design using optimization.
Results
Four consistent path definitions are developed for
general conical shells, while taking into account the
Figure 3 : Apache tailboom, possible application of a
variable-stiffness shell
References
1.
Tatting, B.F., and Gürdal, Z. ”Design and Manufacture of
Elastically Tailored Tow Placed Plates,” NASA/CR 2002211919, August 2002
6
Optimum path and discrete 3D forming
S.H.A. Boers, P.J.G. Schreurs, M.G.D. Geers
Introduction
Results
Discrete die forming is a useful concept in a small-lot
production environment because different products can
be made with the same reconfigurable die and many resources are saved. From a scientific point of view: a
discrete die allows a changing strain path during forming;
products with an optimized internal strain distribution can
be produced, e.g. the lifetime of parts carrying cyclic (thermal) loading can be enhanced considerably.
Using the discrete die and photogrammetry technique, two
geometrically equal products are produced with different
strain distributions.
Conventional die
Discrete die
rubber pad
Pyramid−parabolic vs. Parabolic geometry, major strain %
20
10
10
0
0
−10
−10
−20
−20
Discrete die with 'interpolator'
rubber pad
sheet metal
Initial set-up
20
rubber interpolator
discrete die
die
−30
−30
discrete die
−20
−10
−4
−2
0
10
0
−30
−30
20
2
4
6
−20
−10
8
0
10
10
20
12
Forming step
Final part
smooth surface
dimpled surface
smooth surface
Figure 1 : Rubber pad forming with conventional die, reconfigurable die and interpolator
Figure 3 : Distribution of major plastic strain for two geometrically identical products
Using the flexible mould, a product without defects is produced with an intermediate forming step.
Objective:
A numerical tool to calculate the boundary conditions of
a deformation process that yields geometrically identical
products with different internal strain distributions. Validation will be done by using an experimental set-up. The
numerical tool controls the internal strain distribution by
means of non-proportional forming-limit diagrams.
Figure 4 : Multi-step forming experiment with discrete die
Discussion
Figure 2 : Prototype of discrete die, surface dimensions:
20x30 mm, forming pressures up to 5 ton
Methods
¯ Experimental deformation process with discrete die
prototype in which the strain path is variable.
¯ Determination of strain distribution and product geometry with photogrammetry technique.
¯ Numerical algorithm, providing boundary conditions
for an optimal deformation process.
¯ Non-proportional loading tests must be done for
model input.
¯ Implement numerical tool within FEM framework.
¯ Criterion must be formulated to determine optimal
strain distribution.
¯ Automated control of discrete die is preferable.
References
1.
2.
S. Li, E. Hoferlin, A. v. Bael, P. v. Houtte and C. Teodosiu (2001),
Finite element modeling of plastic anisotropy induced by texture and
strain-path change,International Journal of Plasticity 19 (2003), 647674
C. Zhongyi and L.Mingzhe, Optimum path forming technique for
sheet metal and its realization in multi-point forming, Journal of Materials Processing Technology, Volume 110, Issue 2, 19 March 2001,
Pages 136-141
7
Optimisation of Forming Processes
M.H.A. Bonte
University of Twente
Introduction
Cost saving and product improvement have always
been important goals in the metal forming industry. To
achieve these goals, metal forming processes need
to be optimised. During the last decades, simulation
software based on the Finite Element Method (FEM)
has contributed significantly to designing feasible
processes more easily. More recently, coupling FEM
to mathematical optimisation algorithms is offering a
very promising opportunity to design optimal metal
forming processes instead of only feasible ones.
Objective
The aim of the project is the design of an optimisation
strategy for metal forming processes. An optimisation
strategy consists of the modelling and the solving of
a mathematical optimisation problem [1]. This poster
addresses the solving part: an optimisation algorithm
is proposed and its high potential is demonstrated by
optimising a forging process.
Figure 2 : A forging process: (a) Preform; (b) Gear
Optimising a forging process The SAO algorithm
is applied to optimise the forging process of a gear.
Figure 2 shows the two stages of the process:
producing a preform and forging this preform into the
final product. The objective is to minimise (i) defects
in the final part; and (ii) the energy consumption
during forging. This is achieved by influencing the
design variables that describe the preform shape as
shown in Figure 3(a). Figure 3(b) shows the preform
shape proposed by the gear manufacturer.
Results and Discussion
The optimised preform shape is presented in Figure
3(c). The optimised forging process resulted in
a 10% reduction of both part defects and energy
consumption. These good results demonstrate the
potential of the SAO algorithm for forming processes.
References
1.
Papalambros, P.Y. et al. (2000) Principles of Optimal Design,
2nd ed., Cambridge (UK), Cambridge University Press.
Figure 1 : Sequential Approximate Optimisation
Methods
The optimisation algorithm
A Sequential
Approximate Optimisation (SAO) algorithm is
proposed for optimising metal forming processes
using time-consuming FEM simulations.
The
algorithm is presented in Figure 1. It includes a
Design Of Experiments (DOE) strategy, running
a number of parallel FEM calculations, fitting a
metamodel and optimising this metamodel. The
algorithm allows for sequential improvement to obtain
more accurate results.
Figure 3 : (a) Design variables; (b) Initial preform shape;
(c) Optimised preform shape
8
A combined experimental-numerical
approach to characterize delamination in
polymer coated steel
M.J. van den Bosch, P.J.G. Schreurs, M.G.D. Geers
Netherlands Institute for Metals Research
Introduction
Results
Polymer coated metal sheet is developed by Corus
to reduce production costs of e.g. aerosol and food
cans.
Delamination experiments are conducted inside a
Scanning Electron Microscope (SEM). The delamination front is observed (see figure 3) and the force is
measured.
Figure 1 : Examples of applications for polymer coated
metal sheets.
During forming processes the coated sheets may be
damaged, as shown in figure 2:
Figure 3 : Micrograph of a delamination experiment in
the SEM.
The interface parameters are determined by simulating the experiments and fit the results (see figure 4).
This approach requires accurate material models and
parameter values for both the polymer and steel.
Figure 2 : Delamination, cracking and surface roughening of the polymer layer after a forming process.
Objective
Predict the occurrence of coating delamination during
industrial forming processes.
Figure 4 : Schematics of a zero degree delamination
experiment with the result of a simulation. The colors
indicate the equivalent von Mises stress [Nmm−2 ].
Methods
A Finite Element model is used to simulate industrial forming processes, such as bending and deepdrawing. Between the polymer layer and the metal
substrate interface elements are present to simulate
the delamination of the polymer layers. Experiments
are needed to determine parameters for the simulations.
Discussion
The interface parameters can be determined with the
combined experimental/numerical procedure. In the
future the influence of pre-deformation on the interface parameters will be investigated.
9
Orientation and Position of the
Glenoid Component
P. Broomans, C.W. Oosterlee, F. van Keulen and F.J. Vermolen
Faculty of Mechanical Engineering & Marine Technology
Mekelweg 2, 2628 CD Delft
Introduction
energy density (SED) U remains below
In contrast to total hip and total knee arthroplasty, replacement of the shoulder joint is faced with much more problems
post-operatively and revisions. Component loosening has
been reported in 44% to 59% of the implanted glenoids.
It has been reported that malalignment of an implant can be
a cause of or contribute to implant failure1 . Alignment of the
glenoid component has shown to be of influence on component loosening, humeral head subluxation, cement mantle
stresses and shoulder muscle forces, but investigations focused on the mechanical environment in the bone adjacent
to the implant have not been so extensive.
(1 − s)Uref ,
where s, the remodeling threshold, is set to 0.1 and Uref is
the reference SED value.
Results
Initial results show that bone resorption mainly occurs on
the lower side of the bone-implant interface (Figure 2). The
highest percentages of bone resorption occur in the configuration with the backward inclined implant. For the forward
and downward (not shown!) inclined implants nearly no resorption occurs.
Figure 1 : Cross section of the
glenoid part of the 3D model showing the different materials. (red polyethylene; blue - CoCrMo metal;
yellow - porous CoCrMo metal;
green - bone). The interface is defined as the line between metal and
bone.
Objective
The goal is to determine the influence of the orientation of
the glenoid component on bone resorption and the possibility of bone ingrowth.
Methods
A 3D material model of the scapula, acquired through data
from a CT scan, is available for use in a FE analysis.
Five alignments of the implant are simulated, one in a ’central’ alignment and four each 10◦ inclined in either forward,
downward, backward and upward direction with respect to
the central alignment
From the abduction of the upper arm six static load cases
have been derived with the aid of the Delft Shoulder and
Elbow Model2 .
A configuration without prosthesis is used as a reference
to determine in which parts of the bone-implant interface
bone resorption occurs. At a specific location bone resorption within the model occurs if for all load cases the strain
Figure 2 : Percentages of the particular areas of the boneimplant interface where bone resorption will occur (left is back
side) for three alignments. The hatched area is the location
of the screw.
Hopkins et al.3 reported that the lowest cement mantle
stresses are calculated for the central alignment and the
highest values appear for downward and upward directed inclinations. For the latter configuration, however, our results
suggest that also areas with significant lower stress values
are observed.
Future Work
An extension of the model to incorporate the process of
bone ingrowth is the next goal of this project. Modelling
the influence of locally applied growth factors, e.g. within
coatings of metal-backed implants, can be of interest for our
model.
References
1.
2.
3.
Hasan, S.S. et al. (2002) Characteristics of unsatisfactory shoulder
arthroplasties, J Shoulder Elbow Surg 11: 431-441.
Van der Helm, F.C.T. (1994) Analysis of the kinematic and dynamic
behavior of the shoulder mechanism, J Biomech, 27: 527-550.
Hopkins, A.R. et al. (2004) The effects of glenoid component alignment
variations on cement mantle stresses in total shoulder arthroplasty, J
Shoulder Elbow Surg 13: 668-675.
10
Improving numerical predictability of
Springback
I.Burchitz, T. Meinders, J. Huétink
Faculty of Engineering Technology, NIMR - University of Twente
phone: +31-(0)53-4894069, email: [email protected]
Introduction
Finite element software is used in the design process
of new sheet metal parts (figure 1). During the
process the amount of springback (elastically-driven
change of product shape) is numerically predicted.
This information, being used in tools design phase,
ensures that the desired product shape will be
reached. Current accuracy of numerical prediction of
springback is insufficient. Required surfaces of tools
can only be obtained after employing the extensive
experimental trial and error process.
Figure 2 : Characteristic component 3
Solver type. Sensitivity analysis on component 3
(figure 2) and simulations of U-bending test showed
that an iterative solver can deliver an inaccurate
solution. Depending on its parameters and values of
global convergence criteria the product shape after
springback may be completely unrealistic (figure 3).
Guidelines are needed that define a selection of
iterative solver parameters for springback analysis.
Figure 1 : Schematic of the design process
Objective
The major goal of the project is to improve
the numerical predictability of springback to meet
industrial requirements.
Methods
Additional analysis of sensitivity of springback to
various physical and numerical parameters showed
that:
• increasing coefficient of friction does not
necessarily decrease springback, i.e.
in a
situation when a change of shape is dominated
by a relaxation of membrane stresses;
• springback is highly influenced by a decrease of
the apparent unloading modulus.
Results
Mesh density. Recommendations, available in the
literature, were tested using the U-bending problem.
Results showed that an optimal discretisation level
strongly depends on in-plane tension, R/t ratio and
material properties. To understand this dependency
a simple model of a beam under combined bending
moment and tension was built.
The model is
used to develop practical guidelines that define an
appropriate mesh density to assure the required
accuracy of springback prediction.
Figure 3 : Influence of solver type on springback
Discussion
Authors in [1] showed that depending on material and
process parameters 30-68 integration points through
the thickness are required to reach 1% of springback
accuracy. Accurate analysis of a plastically deforming
sheet material requires that an integration point
lies on each surface where yielding begins. An
attractive approach is to perform a through-thickness
integration by an algorithm that adapts sampling
points to the stress situation.
The adaptability
may include changing a location of integrations
points and/or changing their absolute number. This
adaptive integration algorithm can help to achieve 1%
accuracy of springback prediction at minimal costs.
References
1.
Wagoner, R. H. and Li, M. ”Advances in springback.” in proc.
Numisheet 2005. Detroit, MI, U.S.A., p.209-214.
11
Mode I Crack Tunneling in
Fibre-Metal Laminates
M. Cid Alfaro
P.O.Box 5058, NL 2600 GB Delft
Introduction
Recently, steady-state crack tunneling and planestrain delamination have been studied in laminates
of alternating layers of two dissimilar but isotropic
elastic, brittle solids, subjected to a remote uniform
tensile strain within a two-dimensional finite element
framework [1]. In this work three fracture mechanisms were distinguished, which are (i) the tunneling
of a mode I crack without delamination, (ii) the tunneling of an H-shape crack with constant delamination
length, and (iii) the unstable delamination growth in
all directions, see Figure 1.
Mechanism 1
Mechanism 2
and fibre-reinforced prepeg layers are modelled using
continuum solid like shell elements, combined with
an isotropic elastic model. The assumption of elastic
isotropic layers may be acceptable for laminates with
fibres in multiple directions.
Preliminary Results
The traction profile in the tunneling direction is shown
in Figure 2. The crack initiates from a small imperfection and induces a traction profile that is approximately uniform over the specimen width in the tunneling direction. This is, because the fracture process
zone is of a similar size as the specimen width.
Mechanism3
Figure 1 : Three possible failure mechanisms for a laminate of two dissimilar, isotropic materials.
Figure 2 : Traction profile in the tunneling direction.
Left: hardening branch; Right: softening branch.
Objective
Analyse Mode I crack propagation in a Fibre-Metal
Laminate comprising two layers of aluminium and one
fibre-reinforced epoxy layer.
Numerical Models
Three-dimensional numerical simulations of a tunneling mode I crack without delamination have been
performed. An interface damage model [2] has
been implemented to describe the onset and growth
of mode I cracking in the aluminium sheets. Interface elements furnished with the interface damage
model were placed where cracking is expected to occur. The interface layer is endowed with a high stiffness in order to suppress elastic deformations prior
to the onset of cracking. The individual aluminium
Future Work
Examine size effects due to ratio between fracture
process zone and specimen width, study effect of
plasticity in aluminium layers, investigate the effects
of delamination at the layer interfaces and anisotropy
in the fibre-reinforced epoxy layers.
References
[1] A.S.J. Suiker & N.A. Fleck, Crack tunneling and planestrain delamination in layered solids. International Journal
of Fracture 125: 1-32, 2004.
[2] A. Turon, P.P. Camanho, J. Costa & C.G. Davila An interface damage model for the simulation of delamination under variable-mode ratio in composite materials. NASA/TM2004-213277.
12
Frequency dependent properties
in modal parameter identification
N. E. Conza and D. J. Rixen
Faculty of Mechanical, Maritime and Materials Engineering
Engineering Dynamics
TU Delft, Mekelweg 2, 2628 CD Delft, The Netherlands
Tel: ++31 15 278 6508 e-mail: [email protected]
Introduction
The assumption underlying our research is
that abnormal biomechanical properties of
the sacroiliac joints in the pelvis might
provoke low back pain. Up to now constant
mechanical properties have been assumed
in the pelvis; however there are indications
that stiffness and damping might depend on
the excitation frequency. This fact might
impair the identification starting from
measured Frequency Response Functions.
Objective
The objective of this study is to investigate
the impact that frequency dependency in the
system matrices might have on the quality
of
the
system
identification.
Two
commercially available identification tools
have been tested (SDT and ME'scope).
Method
Data generation
The model use to generate data consists of
three rigid bodies, twelve linear springs,
and damping matrix constructed from preset
damping ratios.
Frequency
dependency
has
been
introduced in the stiffness and damping
matrices according to the formula:
( −ω2 M + i ωC + iD + K + ωK' ) z = s
ω
with D = f C max
3
and K' = f K
2 ω max
The factor f has been set at different values
between 0 and 1 to investigate the effects at
different level of dependency.
Identification tools evaluation
The 18 identified resonance frequencies,
modal damping ratios and mode shapes are
compared to the theoretical values
computed by means of the model.
Results
Resonance frequencies: very well for both
tools (errors < 1% for all factors f).
Damping ratio: ok for both tools (errors
ranging between -5% to +20% at f = 1).
Mode shapes: very well for SDT (all good
except mode 18), a little worse for ME'scope
(difficulties in modes 9, 15, 16, 17, 18).
Discussion
The better mode shape estimation of SDT is
probably due to its fitting algorithm, which
optimizes poles and residues at the same
time [1]. ME'scope, on the other hand,
optimizes first the poles, and then the
residues [2].
Conclusion
Results show that standard identification
tools assuming constant system matrices
are still able to correctly identify the
frequencies, while damping and mode
shapes estimation present more difficulties.
References
[1] Balmès E, Frequency Domain Identification of
Structural Dynamics Using the Pole/Residue
Parametrization,
International
Modal
Analysis
Conference XIV, Dearborn, Michigan, 2006.
[2] Richardson MH, Formenti DL, Global Curve Fitting
of Frequency Response Measurements using the
Rational Fraction Polynomial Method, International
Modal Analysis Conference III, Orlando, Florida, 1985.
13
Condition number of the boundary element
method matrices
tu
W. Dijkstra
Faculty of Mathematics
Introduction
Dirichlet problem (m = N )
We focus on the Laplace problem on a circular domain with mixed boundary conditions.
Partial
tion

 ∆u
u

q
differential equa-
= 0,
x ∈ Ω,
= ũ,
x ∈ Γu ,
∂u
:= ∂n
= q̃, x ∈ Γq ,
Unknowns: u and q .
Γq
R
W
G
u
G
π .
N|
In the figures below we show the behavior of the condition numbers.
100
50
N=4
N=8
N=12
N=16
45
40
10
35
30
25
10
q
5
1
1
0
2
5
R
~u
H
~
=
~q
Neuman problem (m = 0)
π
π + tan N
cond(A) =
|π − (N − 1) tan
20
Algebraic equations
First m elements have Dirichlet boundary conditions.
1
Gq = I + H u =: H̃u,
2
To construct the system matrix A we select a block
from G and from H̃.
G
15
Γu
q
1
2 , | log R| .
1
N , | log R|
cond(A)
Equations
max
cond(A) =
min
cond(G)
The boundary element method (BEM) transforms a
partial differential equation into a boundary integral
equation, and after discretisation of the boundary of
the domain into a set of linear algebraic equations.
The matrix of this set of equations has a special structure, and we investigate the condition number of the
matrix.
Dirichlet problem
10
15
20
25
30
N
35
40
45
50
Neuman problem
Mixed boundary conditions (0 < m < N )
We use a decomposition A = F∗ DQU, with F and Q
unitary, D diagonal and U block diagonal. The condition number of A is estimated by
cond(A) ≤ cond(D)cond(U).
5
u
10
estimate
exact
4
x
B
~u
3
10
2
~q
f
10
1
10
0.5
This yields Ax = f. We investigate the condition
number of the matrix A.
Results
For m = N and m = 0 explicit expressions for the
condition number:
1
1.5
2
2.5
3
3.5
4
R
{
A
=
u
{
A
q
cond(A)
10
Mixed problem
Discussion
We want to extend the research to a square domain.
After analysing the circular and square domain we
hope to draw conclusions for more general domains.
14
Discontinuous Galerkin methods for
structural systems
N.T. Dung and G.N. Wells
Faculty of Civil Engineering and Geosciences
P.O. Box 5048, 2600 GA Delft
phone +31 15 278 85710, email: [email protected]
phone +31 15 278 87922, email: [email protected]
Introduction
When solving structural systems such as thin plates
or shells with the Continuous Galerkin (CG) method
using C1 continuous interpolation functions, challenges derive from the fact that it is not easy to fulfill all the requirements of continuity. A preferred approach is using C 0 elements combined with ReissnerMindlin theory, but other difficulties appear, such as
shear-locking problems. When the Discontinuous
Galerkin (DG) method is used, only C 0 elements are
needed, no shear-locking difficulty appears, and the
above problems can be avoided.
fracture problems and delamination of structures, the
methods have obvious advantages since the discontinuities on the interior edges between elements are
naturally existent. Although FFC is still a program in
the early-stage of its development, in the near future
it might be very powerful for solving large structural
mechanical problems. The code shown in Box 1 is an
example for future applications of FFC.
(a)
Objective
In this work, various Discontinuous Galerkin (DG)
and Continuous/Discontinuous Galerkin (C/DG) formulations are implemented to analyse thin plate and
shell structures. The two important aspects that have
been studied are the stability condition and the convergence rate.
(b)
Methods
For solving second-order problems (e.g. membrane
plate), DG formulations are used; while for fourthorder problems (e.g. bending plate), C/DG formulations seem to be more convenient.
The FEniCS Form Compiler (FFC) program (see
www.fenics.org for more details) has been used in order to make multi-forms in finite element processes
automatically by describing only simple notations.
Results
The numerical analyses for the elastic PoissonKirchhoff plate have been made. In Fig 1, one can
see the discontinuities of displacements (Fig 1.b) or of
the slopes (Fig 1.c) across interior boundaries. When
penalty parameters are chosen large enough, the stability condition is fulfilled and the results are converging to the exact solution rapidly.
Future research
Applying Discontinuous Galerkin methods for nonlinear problems should be considered. For solving
(c)
Fig 1: Illustration of solving plate structure:
(a) Model, (b) Deformation for membrane load
case, (c) Deformation for bending load case.
e = FiniteElement("Discontinuous Lagrange", "tetrahedron", 2)
v = BasisFunction(e) # test function
u = BasisFunction(e) # trial function
f = Function(e) # source term
b = Constant() # penalty term
a = dot(grad(v),grad(u))*dx - dot(jump(v),ave(gradn(u)))*ds \
- dot(ave(gradn(v)),jump(u))*ds + (b/h)*dot(jum(v),jump(u))*ds
L = dot(v,f)*dx
Box 1: Illustration of FFC code, in which the
Interior Penalty formulation will be applied for
the Poisson equation using 3D quadratic
elements
Acknowledgments
Financial support from the Vietnamese Ministry of Education and
Training (VMOET) and the Netherlands Technology Foundation
(STW) is gratefully acknowledged.
15
Fatigue Analysis of Lead-free Solder
Interconnect
M. Erinç, P.Schreurs, GQ. Zhang and M.Geers
Introduction
Fatigue Tests and Modeling
In the industrial search for a lead-free solder, Sn4Ag-0.5Cu alloy is currently the most promising alternative. However, there is an urgent need for
characterization of material and mechanical properties, the most important being thermo-mechanical
creep&fatigue behavior. In BGA assemblies, failure is
caused by thermo-mechanical fatigue and generally
occurs at the bump/pad connection. Thus the solder/metallization interface is of crucial importance in
solder joint reliability, which is determined by the metallurgical reactions between the solder, SnAgCu, and
the metallization, Ni/Au.
Pure tensile and pure shear experiments are set
up, which are used to fit the normal and tangential
springs of the cohesive zones. Specimens are cyclically loaded at various peak strain levels.
Figure 3 : Soldered joints of Au coated Ni plates.
In the cohesive zone formulation, traction is a linear
function of the separation. A damage variable is introduced to account for the accumulation of damage
throughout the cycling process(Abdul-Baqi et al):
Tα = kα (1 − Dα )∆α
˙ α | (1 − Dα + r)m
Ḋα = cα |∆
Figure 1 : BGA layout and a failed solder bump.
Objective
Under mechanical loading, the crack path between
Ni/Au substrate and SnAgCu follows pre-known interfaces and the propagation can be split into normal
and tangential components (Erinç et al). Thus the
cohesive zone approach fits well to the experimental
observations.
Figure 2 : (a)Crack paths under normal and tangential
loading, (b)Cohesive zone element.
The objective of this study is to conduct dedicated
fatigue tests with Ni/Au-SnAgCu soldered specimens
and use the experimental data to develop a cohesive
zone based solder joint fatigue model.
(1)
|Tα |
− σf
1 − Dα
(2)
Individual fatigue tests are simulated with finite elements under plane stress conditions. Material parameters are gathered experimentally. A thin layer of cohesive zones is placed between the solder and the
intermetallic layer, since this is the region where failure is observed. After a number of cycles, damage
evolves at the bonding interfaces, thus the reaction
force decreases at successive cycles.
Figure 4 : (a)Damage at cohesive zones after 500 cycles, (b)Reaction force drop due fatigue damage.
Conclusions
The solder/metallization interface is characterized under fatigue loading. Using a cohesive zone approach
to simulate fatigue damage in solder interconnects is
promising.
16
Identification of Spindle Dynamics in High
Speed Milling
R.P.H. Faassen, N. van de Wouw, J.A.J. Oosterling, H. Nijmeijer
phone +31-(0)40-247 5730, email [email protected]
Introduction
Results
To obtain the maximum material removal rate in highspeed milling [2], an accurate model is required, see figure 1. This model will be used to suggest a proper set of
machining conditions (e.g. spindle speed) to the machineoperator.
Identification of the machine-tool dynamics is performed
with cutting tests. Measurements of the force by the dynamometer (F2 ) and the displacement by the eddy current sensor (x2 ) are analysed with N 4 SID [3] and by using
Welch’s averaged periodogram method (WAPM), see figure
3. For comparison, also the FRF using the hammer tests
are shown Fx11 . Here, besides using WAPM, a model is fitted
using the RFP method [1].
Delay
Static
chip
thickness
+
Cutting
Force
Machine
Displacement
−
+
+
−6
10
Dynamic
chip
thickness
Trigonometric relations
In this model, the machine-tool dynamics are represented
by a linear state-space model (the block Machine). Usually, the necessary parameters for this model are obtained
by performing impulse hammer tests on a non-rotating
tool. However, in practice, it would be more beneficial if
it is possible to identify the machine-tool dynamics by performing cutting tests.
Magnitude [m/N]
Figure 1: Block diagram of the milling process.
−7
10
−8
10
Hammer Fx11 WAPM
Cutting Fx22 WAPM
Hammer Fx11 RFP
Cutting Fx22 N 4 SID
Objective
−9
The goal of this research is to obtain the machine-tool dynamics by performing cutting tests.
10
2000
3000
4000
5000
6000
7000
8000
9000 10000
Figure 3: Frequency Response Functions.
In order to imitate the impact of the hammer by performing
cutting tests, a small strip is cut, see figure 2. This results
in a short impulse-like force to act on the cutter. After the
very short contact time, the tool can freely vibrate until the
next tooth enters the cut. For comparison, also standard
impulse hammer tests have been performed.
Spindle
Accelerometer
Toolholder
Fy , xy
Mounting device
x2
Feed
x1 , F1
Tool
Eddy current sensor
Workpiece
Dynamometer F2
(a) Top view
1000
Frequency [Hz]
Methods
Ω
0
Accelerometer
The benefits (X) and drawbacks (✘) of the method are presented below:
X In-process excitation
X Easy to do in a machine shop environment
X Spindle-speed dependency can be modelled
X Broadband frequency excitation
X Resonance frequencies can be found
✘ No mode shape information
✘ Poor sensor placement
✘ Unknown number of modes
Future work
Future topics include improving the sensor position and/or
the data processing and validating the model.
References
[1]
[2]
(b) Front view
Figure 2: Experimental set-up.
[3]
D.J. Ewins. 2000, Modal Testing, Research Studies Press.
R.P.H. Faassen, N. van de Wouw, J.A.J. Oosterling, H. Nijmeijer.
2003, Prediction of regenerative chatter by modelling and
analysis of high-speed milling Int. J. Mach. Tool. & Manu.,
43:1437-1446.
L. Ljung. 1999, System Identification , Prentice Hall.
17
Real-time trajectory generation for
sensor-guided robotic laser welding
Menno de Graaf, Ronald Aarts, Ben Jonker
+31-(0)53-4895442/2502, [email protected]
Introduction
Results
Robotic laser welding imposes high demands on the used
manipulator as high accuracies (down to 0.1 mm) have to
be reached at high velocities (up to 250 mm/s). To meet
these specifications with industrial robots, a sensor measuring at the robot tip needs to be applied.
A corner trajectory (Figure 2) has been replayed with a
maximum linear velocity of 100 mm/s and rotational velocity of 60 deg/s.
Motion
descriptor
Motion
Location
Buffer
Reference
joint position
Cartesian
setpoints
Setpoint
Generator
Inverse
Kinematics
+
Robot
Joint Motion
Controller
-
Joint velocity [deg/s]
Integration of an optical seam-tracking sensor in a robotic
laser welding cell for:
• Sensor-guided teaching of the 3D seam trajectory.
• Increasing the positional accuracy during welding.
Joint position [deg]
Objectives
150
150
100
50
0
−50
−100
0
100
50
0
−50
−100
1
2
Time [s]
−150
0
3
1
2
Time [s]
3
Figure 3 : Reference joint position and velocity (Joint 1, Joint 2,
Joint 3, Joint 4, Joint 5, Joint 6). Both the reference joint position
and velocity are smooth.
Measured joint position
100
80
80
Tip velocity
To use sensor information obtained during the robot motion
a real-time trajectory generator (Figure 1) based on cubic interpolation of both position and orientation has been
implemented on an industrial Stäubli robot. It calculates
setpoints for the robot joint motion controller every 4ms.
Features of the trajectory generator are:
• The setpoints are calculated on-the-fly, which allows
the addition of cartesian locations to the Motion Location Buffer during the robot motion.
• Seam locations obtained from the robot and seamtracking sensor can be added to the Motion Location
Buffer after proper filtering. Such seam locations are
e.g. computed from synchronised data of robot joint
angles and sensor measurement
100
60
40
20
0
0
1
2
Time [s]
Trajectory
1
r=50 mm
5
4
2
3
0.1
0
−0.1
0
x
2
Time [s]
−0.2
6
y
1
0.2
d=50 mm
Current
location
0
0
3
Figure 4 : Reference and measured tip velocity (Linear [mm/s],
Rotational [deg/s]). The linear velocity decreases at the corner,
because of the bounds in rotational velocity.
7
Setpoint
60
40
20
Tip position error [mm]
Methods
Tip velocity
Figure 1 : Real-time control architecture
3
Figure 2 : Corner trajectory
1
2
Time [s]
3
Figure 5 : Measured tip position error (X, Y, Z) based on joint
measurements. The tip position error is largest at the acceleration/deceleration phases and when the joint velocity changes
sign (joint friction).
References
1.
M.W. de Graaf et al, Real-time trajectory generation for sensorguided robotic lase welding, Submitted to the IFAC Syroco 2006 conference, Bologna, 2006
18
Improved tip tracking for an industrial robot using Iterative Learning Control
W.B.J. Hakvoort*, R.G.K.M. Aarts, J.B. Jonker
* Netherlands Institute for Metals Research
phone: +31-(0)53-4895442, email: [email protected]
Introduction
The laser welding process puts high demands on the
manipulator that moves the laser beam with respect
to the weld seam. Typically an accuracy of about
0.1 mm is required at speeds beyond 100 mm/s.
From an industrial perspective the use of six-axes industrial robots is attractive as these can access complex three dimensional seam geometries. However,
using standard industrial controllers the tracking accuracy of these robots is insufficient for laser welding.
Objective
The goal of this project is to improve the tracking accuracy at the tip of an industrial robot (Figure 1).
welding head
camera
laser diode
weld seam
product
Figure 2 : Seam tracking sensor
Results
-660
seam
0.1mm boundary [x25]
trial 0 [x25]
trial 5 [x25]
y-axis [mm]
-680
-700
-720
-740
-760
450
500
550
600
x-axis [mm]
Figure 1 : The six-axes industrial Stäubli RX130 robot
Methods
The tip tracking accuracy is improved with Iterative
Learning Control [1].
• An initial reference trajectory for the robot joints
is obtained from CAD-data of the seam and a
kinematic model of the robot.
• The robot joints track the reference trajectory,
while an optical seam tracking sensor (Figure 2)
measures the tip tracking error.
• Using a straightforward model of the robot the
reference trajectory for the joints is updated to
compensate for the measured tip tracking error.
• The last two steps are repeated until the tip tracking error converges to a steady value.
Figure 3 : Tracking error perpendicular to weld seam
Discussion
Iterative Learning Control can reduce the tracking error at the tip of an industrial robot, measured with an
optical seam tracking sensor, close to 0.1 mm. The
remaining error is mainly due to resonance vibrations
of the robot. Further reduction of the error requires
either a better robot model or smooth trajectory generation.
References
1.
Hakvoort, W.B.J., R.G.K.M. Aarts, J. van
Dijk, J.B. Jonker, Iterative Learning Control for Improved End-effector Accuracy
of an Industrial Robot., submitted for the
8th IFAC Symposium on Robot Control,
September 2006, Bologna, Italy.
19
Reduction of sound transmission through panels
by means of tuned acoustic resonators
M.H.C. Hannink, Y.H. Wijnant and A. de Boer
Chair of Structural Dynamics and Acoustics, University of Twente
Introduction
When a panel is acoustically excited by a sound
source, it will vibrate (Figure 1). Due to this vibration,
sound is radiated to the other side of the panel.
This phenomenon is called sound transmission and
is mostly unwanted. The radiated sound can be
reduced by means of acoustic resonators (Figure 2),
which are tuned in such a way that the volume flow
at the entrance of the resonators is opposite to the
volume flow at the surface of the panel (Figure 3).
Sound source
Vibrating panel
Incident
Radiated
Reflected
Resonator
Panel
Figure 3 : Normal incidence transmission of sound
through a characteristic area
Results
The sound transmission loss T L is a measure for the
sound reduction, and is defined as the ratio of the
incident and the radiated sound power:
T L = 10 log10 |pA /pC |2
Receiving room
Figure 4 shows the sound transmission loss for
panels with different porosities Ω and a resonator
length of 0.11 m.
Transmission loss [dB]
120
Figure 1 : Sound transmission between two rooms
Objective
The reduction of sound transmission through panels
by the application of tuned acoustic resonators.
Characteristic area
Ω=0
Ω = 0.25
Ω = 0.40
Ω = 0.48
100
80
60
40
20
0
5⋅102
Resonator
Figure 2 : Part of a panel with acoustic resonators
Methods
The harmonically vibrating panel is divided into a
number of identical characteristic areas (Figure 2).
The effect of the acoustic resonators is studied with
a one-dimensional model of such a characteristic
area (Figure 3). By solving the one-dimensional
wave equations, the radiated sound pressure pC is
calculated. The radiated sound is minimised by
tuning the resonator length and the porosity of the
panel. These parameters determine the frequency
range in which the radiated sound is reduced and the
shape of the spectrum, respectively.
2⋅103
3⋅103
Frequency [Hz]
Panel
3
10
Figure 4 : Normal incidence sound transmission loss for
different porosities
Discussion
Panels with acoustic resonators show a large
reduction of the radiated sound over a broad
frequency range, compared to a panel of the same
mass without acoustic resonators (–). The next step
is to study the effect of acoustic resonators on a large
scale, both numerically and experimentally.
Reference
1.
Hannink, M.H.C. et al. (2005) Application of acoustically
tuned resonators for the improvement of sound insulation in
aircraft, Internoise, Rio de Janeiro, Brazil.
20
Multivariable Frequency Response Functions Estimation for Industrial Robots
T. Hardeman, R.G.K.M. Aarts and J.B. Jonker
+31-(0)53-4892567, [email protected]
Introduction
shows that the square wave is effective in preventing unwanted velocity reversals, leading to complicated friction
behavior.
ė3
(m)
(rad/s)
The accuracy of industrial robots limits its applicability
for high demanding processes, like robotised laser welding. We are working on a nonlinear exible model of
0.1
the robot manipulator to predict these inaccuracies. This
0
poster presents the experimental results on estimating
−0.1
PSfrag replacements
the Multivariable Frequency Response Functions (MFRF)
5
4
3
2
1
0
time (s)
¤
of the Staubli
RX90 robot depicted in gure 1. Future work
will be the parametrisation of the frequency response funcFigure 3 : Resulting velocity joint 3
tions based on physical models.
9
8
7
6
MFRF estimation
Figure 1 : Stäubli RX90 robot with 3D acceleration sensor
Closed loop robot system
For stability and safety reasons, the experiments will be
carried out in closed loop; see gure 2. The robot controller will drive the robot such that the joint angles e(m)
and velocities ė(m) are in agreement with the reference
trajectory r and ṙ. The driving torques τ (m) are perturbed
with feedforward torques τ (f f ) having a frequency spectrum above the bandwidth of the closed loop system. The
(m)
outputs
lacements(m)of the system are the joint angles e and velocities ė . Furthermore a 3D acceleration sensor measures
the accelerations of the tip in horizontal s(h) and vertical
s(v) direction.
+ -
Controller
+
e(m), ė(m)
Figure 2 : Closed loop system
where G is the MFRF. To be able to extract G from data
m different (independent) experiments are needed, were
m is the number of inputs. The data vectors from different
experiments can than be collected into matrices U and Y,
where each column corresponds to one experiment. An
estimate of G can be formed as
provided matrix U has full rank.
Results
(m)
s(h,v)
Robot
The feedforward signal τ (f f ) is a multi sine containing 100
frequencies in the range from 10 to 100 Hz. Crest factor
optimisation of this signal has improved the signal to noise
ratio of the measurements by a factor 2. The reference trajectory ṙ for the joint velocity is a square wave. In gure
3 the resulting velocity of joint three is given. The gure
(m)
The estimated MFRF from τ2
and τ3
to the outputs
(m)
(m)
(v)
(h)
ë2 , ë3 , s and s
are given in gure 4. The gure
shows that although the excitation is only in the vertical
plane (see gure 1), the resulting motion is a complicated
3D motion.
PSfrag replacements
time (s)
(m)
ė3 (rad/s)
Experiment Design
(2)
Ĝ(ωk ) = Y(ωk )U−1 (ωk ),
1
10
Ĝτ (m) ,ë(m)
2
amplitude
r, ṙ
τ (ff )
+ τ (m)
Let U (ωk ) be the Discreet Fourier Transform (DFT) of the
input signals consisting of the driving torques τ m at frequency ωk and let Y (ωk ) be the output vector consisting
of the DFT of the joint accelerations ë(m) and the accelerations s(h,v) . For periodic signals the following linear mapping holds
Y (ωk ) = G(ωk )U (ωk ),
(1)
2
Ĝτ (m) ,ë(m)
2
3
Ĝτ (m) ,s(h)
2
Ĝτ (m) ,s(v)
−1
10
2
Ĝτ (m) ,ë(m)
3
2
Ĝτ (m) ,ë(m)
3
3
Ĝτ (m) ,s(h)
3
Ĝτ (m) ,s(v)
−3
10
0
3
25
50
75
frequency (Hz)
100
Figure 4 : Frequency Response Functions in (rad/Nms2 ) and
(m/Nms2 )
21
Robust Model Order Reduction by Krylov
subspace methods
Pieter Heres, Wil Schilders
Center for Analysis, Scientific computing and Applications
Large linear time-invariant models can very well be replaced by smaller equivalents. Examples of such models
can be found in many areas, as for instance in CFD, Structural Mechanics, circuit simulation and electromagnetism.
The smaller models have the same behaviour as the large
models and the reduction techniques preserve the stability of the models. In this research we investigated a robust and efficient algorithm for Krylov subspace methods
for Model Order Reduction (MOR).
We here show a model from the SLICOT benchmark library [1], in our opinion the one which is most hard to approximate. The original model (size 480) and its apprximation of size 80 are shown in the picture below. In the next
picture we zoomed in.
0
−50
Magnitude transfer function
Introduction
Objective
From the host of MOR methods, Krylov space methods are
popular since they are relatively cheap and can be generally applied. The methods can be applied to any system of
ODE’s or DAE’s:
C
−100
−150
−200
−250
−300
−1
10
dx(t)
= −Gx(t) + Bu(t)
dt
y(t) = BT x(t)
1
0
10
Frequency (Hz)
10
3
2
10
10
−30
To ensure robustness, the formulation of the system and
the numerical process of orthogonalisation should be handled with special care. Preferable C and G are positive
definite and positive real, respectively.
Magnitude transfer function
−40
Methods
The basic idea is to project the large system onto a smaller
space. Accordingly, the following Krylov space is generated:
Kq (B, A) = [B, AB, . . . , Aq B]
−50
−60
−70
−80
−90
−0.6
10
−0.4
10
−0.2
10
Frequency (Hz)
0
10
Let V be the orthonormal basis of this space, then the matrices of the reduced system are formed as:
e = VT GV
G
e = VT CV
C
e i = V T Bi
B
e o = V T Bo
B
Results
With a robust implementation of the aforementioned methods models can be fairly reduced, while preserving their
behavior. The largest model we reduced was a dense
model of size 5200, which had 4 ports. It was reduced
to size 40, which gave an excellent approximation of the
model up to the maximum frequency of 30 Ghz.
Discussion
Krylov subspace methods are able to preserve the stability of the model. Moreover, a stable reduced model can be
formulated in terms of positive RLC-components. Consequently, the compact model can be analysed quickly, by a
tool tailored for circuits.
References
1.
2.
3.
Y. Chahlaoui and P. Van Dooren, 2002.
Available at:
www.win.tue.nl/niconet/NIC2/benchmodred.html
Check www.win.tue.nl/smurf for more references
This research was granted by NWO, grant number 635.000.010.
22
High-speed impact modelling and
testing of composite structures
B.D. Heru Utomo
Faculty of Mechanical, Maritime and Material Engineering
Mechanics of Materials
TU Delft, Mekelweg 2, NL 2628 CD Delft
Tel: +31-(0)15-278 6512, e-mail: [email protected]
Introduction
Methods
Composite structures are nowadays more
and more applied in many defence
applications, such as in vehicle and
personal protection (figure 1).
To obtain a thorough understanding of the
failure mechanisms that occur, techniques
like microscopy (figure 2) are used.
Ballistic experiments will be performed.
During these experiments, the material
behaviour will be recorded with equipment
such
as
high-speed
(video)cameras
(figure 3) and Doppler radars. The development of a predictive tool will be done in
parallel to this process.
Figure 1: Personal protective equipment.
Helmet (left), bullet proof vests (right)
Therefore, the availability of (computer)
tools that are able to predict their behaviour
under ballistic loading has become
indispensable to guarantee a continuous
improvement of quality and performance.
Figure 3: High speed camera images of
ballistic impact
Results
Objectives
The objectives of the present research are
to obtain a thorough understanding of the
failure events that occur in composite
structures that are loaded by ballistic impact
and to create a tool that is able to predict
the behaviour of composite structures
subjected to such loading.
Predicting the behaviour of ballistically
loaded composites seems promising
(figure 4). For a good prediction however,
more information on the failure mechanisms
is still required.
Figure 4: Simulation of projectile
impact on composite target plate
Discussion
Figure 2: SEM image of composite
plate loaded by a fragment
The (prediction of) failure mechanisms of
composites under ballistic loading are complicated. The question is, on which level
(macro, meso or micro)
the
predictions
should be done such that they are still
accurate enough.
23
Design and development of high temperature
coating system for engines
Thomas S. Hille
P.O.Box 5058, NL 2600 GB Delft
Introduction
The efficiency of high temperature engines, such as
airplane jet-turbines, increases with service temperature, whereas the mechanical strength of the employed structural components (substrate) generally
decreases. To allow for higher service temperatures,
the substrate is protected against melting and oxidation by a thermal barrier coating (TBC) system. This
system provides a temperature difference of about
200 ◦ C between the external surface and the substrate surface.
s
o
m
e
mm
R = some mm
Figure 2 : A schematization of the TBC-substrate system corresponding to the blade geometry.
Preliminary Results
Hot gas 1200°C
Te
m
pe
ra
tu
re
gr
ad
ie
nt
300 μm
T
B
C
L
a
y
e
r
TGO
B
C
Substrate cooling system 1000°C
Figure 1 : A micrograph section of a TBC-system showing the coloumnar structure of the TBC-layer on top, the
bond coating (BC) adjacent to the substrate and the
thin thermally grown oxid (TGO) in between.
Due to a mismatch in the thermal expansion coefficients of the coating components, stresses evolve
when the temperature distribution through the TBC
departs from the stress-free reference state. A
thermo-mechanical model that has been coded in
C++ computes these stresses for a deviation of about
1200 ◦ C from the reference state. Locally, the von
Mises stress exceeds the yield strength of the BC
significantly, which motivates the implementation of a
plasticity model for this component.
von Mises stress [Pa]
2.87e9
Objective
Development of a numerical tool to simulate the thermal, chemical and mechanical processes occuring in
the TBC-system during life time. Special attention is
dedicated to the simulation of spallation mechanisms,
which lead to failure of the coating system.
1.0e8
Figure 3 : A contour plot of the von Mises stress. The
yield strength of the BC relates to light yellow.
Future Work
Required Models
•
•
•
•
TBC-layer: Transversely isotropic.
TGO: Thermodynamic kinetic growth law.
BC: J2 -plasticity / single-crystal plasticity.
Fracture mechanics: Cohesive zones
combined with the partition of unity method.
• Multi-scale approach for different length scales.
• Fatigue model for cyclic loading characteristics.
A J2 -plasticity model for the BC, using a closest point
return-mapping algorithm, has already been coded
and will be integrated in the simulation. Attention will
be given to the transversely isotropic TBC-layer.
Acknowledgement
The Netherlands Institute for Metals Research is
gratefully acknowledged for financing this project.
24
Design of a Hybrid Vehicle:
From generic to specific design
Theo Hofman
Introduction
A schematic overview is given of different applicable
Drivetrain hybridization implies adding a Secondary power technologies for the S (Battery, Flywheel and Super
source (S) to a Primary power source (P) in order to Capacitor storage systems with energy conversion and
improve driving functions, i.e., fuel economy, emissions, transmission components) and the T in Figure 3.
driveability, comfort and safety. Designing a hybrid
$$
drivetrain fulfilling the required vehicle driving functions is a
$- . *
+
complex task, due to unknown sensitivity of vehicle
!"
!"
!
performance to system component specifications, the
/
01 2
34 *
+
/$
!
defined interaction between system components and the
ability to operate the system components at different
!"
set-points at any time (see Figure 1).
!"
/
/
,
/
&
Hybrid functions, e.g., Start-Stop, Brake Energy Recovery
(BER), Power Boost, Hybrid and fully Electric propulsion of
the vehicle increase with system voltage specification of
t h e S (se e Fi g u re 2 ). Th e sh o rt a n d lo n g -t e r m
e n e r g y/ p o we r r e q u i r e m e n t s , e . g . , a c c e l e r a t i o n
performance, gradeability, towing capacity, electric driving,
determine the size of the P and S. The ability to operate
the components at optimal operation points and the
maximum power throughput determine the design
specifications for the transmission technology (T).
" !
!
"
!
## !
$
$
!"
% &
'
,
6"1 7 "
#/
$
6 /$ 7
!
"
'( ))
*
+
*
+
1
6!" 7
Development of the design method
%
,
!
5
"
!
#
!"
Research steps for developing a method are: (i) Defining
order of design steps, (ii) Sensitivity analysis of design
specs for P and S to targets. Selecting and designing of
new technologies for P, S and T. (iii) Sequentially
optimization of system components (S, T) and the vehicle
system control (C) to targets. (iv) Finally, designing an
optimal hybrid drivetrain according to specs. A cascaded
design process is proposed (see Figure 4). The design
order is defined by the sensitivity of the total system
efficiency to component efficiency and the decrease in
functional constraints of the sub-system components.
!!
!!
*+
//
$$
-
→
$$
33
$%& !
% '(
//
.*!
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*+
→
,
!!
!
-
!!
"
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+
$
,
//
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$$
&& '(
3
33
→
)& '(
33
!!
//
33
Conclusion
$$
*+
,
//
8
!
//
33
'
3
#
/
!
A design process for designing a hybrid vehicle is
discussed. Currently, the proposed design method and the
developed design optimization tools are evaluated on the
design of a hybrid drivetrain for a mid-sized passenger car
(1134 kg) focused on significant fuel use and emissions
reduction (>50%) on a representative drive cycle.
25
The Mechanical Behaviour of Brain Tissue
M. Hrapko, J.A.W. van Dommelen, G.W.M. Peters,
and J.S.H.M. Wismans
P.O. Box 513, 5600 MB Eindhoven
Introduction
Model
Brain injury is the major cause of fatalities in traffic accidents. Finite Element (FE) models are being
developed (Figure 1, right), in order to predict the mechanical response of the contents of the head during
impact. However, they still lack an accurate descriptions of the mechanical behaviour of brain tissue.
Objective
The purpose of this study was to experimentally characterise the mechanical behaviour of brain tissue and
to develop a constitutive model that is able to describe
the behaviour of brain tissue in:
• large strain behaviour (up to a strain of 20%),
• complex loading paths (loading/unloading),
• different deformation modes (shear - compression).
Based on the stress relaxation measurements, a nonlinear viscoelastic model was developed to describe
the mechanical behavior of brain tissue (Figure 1,
middle). The viscoelastic modes σ dve were modeled
by an elastic Mooney-Rivlin model and a Ellis model
was chosen to describe the stress-dependence of the
viscosity. The elastic part σ de was described by a NeoHookean model modified with a damping function.
h
i
p
σ de = Ge (1 − A)exp −C I1 − 3 + A B d
Results
In the following figure experimental results (black
line), the model fit (red line) as well as model predictions (blue line) are shown.
200
80
protocol A
150
40
τ [Pa]
100
τ [Pa]
protocol B
60
50
20
0
0
−20
−50
−40
0.2
0.1
0
γ [−]
0.3
0.4
0.5
20
15
10
5
0
25
time [s]
80
150
Figure 1 Schematical representation of FE modeling.
−σ [Pa]
40
20
0
An improved eccentric sample placement was used in
shear experiments on an ARES II rotational rheometer. Shear measurements consist of:
• loading/unloading cycles with a constant shear
rate and increasing strain levels (Figure 2 left),
• stress relaxation tests with constant shear rate
and increasing strain levels (Figure 2 right).
protocol A
strain
strain
100
τ [Pa]
Methods
time
protocol B
−σ [Pa]
protocol B
60
−20
50
100
0
−100
0
0.05
0.1
−ε [−]
0
−50
−40
−100
0
0.05
0.1
γ [−]
0.15
0.2
0
10
30
20
40
time [s]
Figure 3 Top & bottom left: shear tests; bottom right:
compression tests.
No significant immediate change in mechanical behaviour due to previous deformation was found (protocol A).
Conclusions
time
Figure 2 Testing protocols.
Compression experiments were performed on an
MTS device and consist of stress relaxation tests with
a constant shear rate.
The model showed:
• a good prediction in the loading phase of shear
deformation,
• a lack of accuracy in the unloading phase,
• a partially qualitative description of the behaviour
in compression.
26
Grain statistics: an experimental investigation
P.J.M. Janssen1,2 , Th.H. de Keijser1 , J.P.M. Hoefnagels2 , M.G.D. Geers2
1
Netherlands Institute for Metals Research
2
Introduction
Due to miniaturisation ever thinner metal sheets are
being processed (figure 1). Consequently, only a few
crystals may be present in the sheet. As a result, the
properties of the individual grains become more important; grain statistics effects occur.
Figure 1: (a) Industrial applications (b) Micro-parts.
The obtained grain structure is rather homogeneous
and the specimens have a pronounced Cube texture
(figure 3). The average grain size measured on the
specimen surface is about 800 µm.
Figure 3: Microstructure and {001} pole figure of a typical
recrystallised specimen.
Objective
Investigation of grain statistics effects, by analysis of
the mechanical behaviour of thin Al sheet (320 µm)
with through thickness grains. The number of grains
in the specimens is altered by changing the specimen
geometry.
Mechanical behaviour
The mechanical properties are analysed in uniaxial
tension. Two specimen widths and lengths have been
analysed, preliminary results are shown in figure 4.
40
Microstructure
b
35
30
30
25
25
σtrue (MPa)
σtrue (MPa)
A strain-anneal protocol is followed to produce material with a reproducible microstructure. From the asreceived sheets, strips are cut parallel to the rolling
direction. These strips are recovered at 200 o C for
one hour to relieve internal stresses. As expected,
there are no significant differences in the grain structure and orientation (figure 2).
40
a
35
20
15
10
10
5
0
0
20
15
w = 9.8 mm
w = 2.1 mm
0.05
0.1
0.2
0.15
εtrue
0.25
0.3
0.35
5
0
0
l = 10 mm
l = 16 mm
0.05
0.1
0.2
0.15
εtrue
0.25
0.3
0.35
Figure 4: Stress-strain curves, (a) varying width and constant length (b) varying length and constant width.
Decreasing the specimen width:
• the spread in stress-strain behaviour increases
• small steps in the tensile curves become visible
• explanation: increasing individuality of the grains
Increasing the specimen length:
• the average fracture strain decreases
• the spread in fracture strain increases
• explanation: increasing probability of the presence of weaker grains
Figure 2: Microstructures and {001} pole figures of (a) asreceived material (b) recovered material (grain colour indicates crystal direction parallel to the specimen normal).
The recovered strips are strained 5.5 % in uniaxial
tension and recrystallised for 30 minutes at 600 o C.
Future work
Further exploration of the mechanical behaviour and
the local deformation (using digital image correlation)
of specimens with known microstructure.
27
Flow front tracking in aluminium extrusion
dies by means of particle trajectories
A.J. Koopman, H.J.M. Geijselaers, J. Hu étink
University of Twente
Introduction
Methods
Increasingly tighter requirements on complexity and
geometric tolerances raise the demand for more
insight in the aluminium extrusion process. While
an optimized extrusion process should create not
more then 15% scrap, it is in everyday practice
not uncommon to have a scrap rates over 25%.
A consistent application of design rules developed
either by simulations or by experience shows a
dramatical increase in the performance of extrusion
dies.
In 1986 Thompson [1] introduced a pseudoconcentration function C(X, t) to track the flow front.
Objective
Billet to billet extrusion is not a continuous process.
Temperature, stresses and deformation rates change
during the process.
Figure 2 : Extrusion force from Benchmark Zurich
The discontiniuty of the pseudo-concentration
function around the flow front introduces inaccuracy
during the convection through the FEM mesh. To
overcome this problem we introduced a smooth
function, the original coordinate function (figure 3)
The original coordinates are transported through the
mesh and every step the stiffness matrix is adjusted
based on the original coordinates. .
Figure 1 : Filling of the extrusion dies
However, the aluminium flow through the die after
initial start-up can be regarded as a semi-stationary
process. Shown in figure 2, after the first 50 mm
the extrusion force is decreasing linearly with the
ram displacement, in literature this effect is attributed
to the decreasing friction between the billet and
the container. Due to the stationary nature of the
aluminium flow an Eulerian FEM formulation is used
to obtain the results. Using an Eulerian description
the high deformations in the extrusion dies can be
simulated without the need for constant remeshing.
In the extrusion practice the suspicion is raised
that the die deflection during the filling affects the
aluminium flow in the semi-stationary part.
To
simulate this effect the aluminium flow front has to be
tracked in the Eulerian domain.
Figure 3 : Pseudo-concentration function versus
original coordinate function
Discussion
Simulations using this method will help with the
development of new design rules for extrusion dies.
References
1.
Thompson, E. (1986). ”Use of pseudo-concentrations to
follow creeping viscous flows during transient analysis.” Int.
J. Num. Meth. Fluids 6(10): 749-761.
28
A more Rigorous Coupled-Wave Analysis
M.G.M.M. van Kraaij
Technische Universiteit Eindhoven
Department of Mathematics and Computer Science
Introduction
Analysis
The semiconductor industry uses lithography systems for manufacturing complex integrated circuits
(also called ICs) onto wafers. Because the complexity of producing ICs with more functionality increases,
good models and simulation tools are needed.
Because the grating is infinitely periodic, the computational domain can be reduced to one unit-cell. For
each layer i the following equations are solved:
Here we focus on
a small part of the
chip making process
that deals with measuring the position of
a wafer (’alignment’).
Today, small gratings
Lithography system
on the wafer are used
for this alignment step. Gratings are periodic structures printed on the wafer and are even smaller than
ICs. These gratings are illuminated with a laser beam
and by measuring the diffracted light one can get information on the position of a wafer. Here Maxwell’s
equations are the starting point and an algorithm
known as Rigorous Coupled-Wave Analysis (RCWA)
is then used to calculate the diffracted field [1].
TE polarization
∂2
∂x2
Ei,y +
∂2
∂z 2
Ei,y + k02 εri (x)Ei,y = 0,
TM polarization
∂
1
∂
r
εi (x) ∂x εr (x) ∂x Hi,y +
i
∂2
∂z 2
Hi,y + k02 εri (x)Hi,y = 0.
Here Ei,y and Hi,y represent the y-components of the
electromagnetic field. The complex relative permittivity is denoted with εri (x) and k0 is the wave number of
free space. The following steps are taken to calculate
the scattered field:
• Fourier expansions for permittivity [2].
• Pseudo-periodic Fourier expansions for field inside grating layers.
• Rayleigh expansions for scattered field.
• Stable enhanced transmittance matrix approach.
Objective
Computational results
Get a better understanding of the optical diffraction
model and RCWA algorithm. Improving the convergence and robustness of the algorithm.
For a grating an angle-resolved spectrum is calculated with the RCWA-algorithm and compared with a
measurement. Both figures show the relative amount
of energy that is reflected and captured by a sensor.
Assumptions
0.3
0.3
0.25
0.25
Simplifications of Maxwell’s equations are based on
the following assumptions:
0.2
0.2
0.15
0.15
• Incident field is an arbitrary linearly polarized
monochromatic plane wave.
• Electromagnetic fields are time-harmonic.
• Media are linear, homogeneous and isotropic.
• Gratings are infinitely periodic and are approximated with a layered structure:
0.1
0.1
0.05
0.05
0
Measured image
0
Calculated image
References
1.
re -
y x
Real trapezoidal grating
?
z
Approximated layered grating
Moharam, M.G. et al. Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary
gratings. J. Opt. Soc. Am. A, 12(5):1068-1076, May 1995.
2. Li, L. Use of Fourier series in the analysis of discontinuous
periodic structures. J. Opt. Soc. Am. A, 13(9):1870-1876,
September 1996.
29
A singularity solution of shear faulting in
swelling ionised porous media
F. Kraaijeveld
Introduction
Results and discussion
Shales, clays, gels and biological tissues all endure swelling associated with ionisation of the porous
medium [2.].
Phenomena like crack formation
and crack propagation during shrinkage are studied
through numerical simulations [4.,6.]. Analytical solutions are a must for validation of numerical codes.
Shear stress The shear stress shows an high order
singularity in x (O ( r12 )). Shear stress at t = 0 is the result of initial incompressible behaviour of the medium.
Fluid flow results in relaxation of the stresses. This is
consolidation. Shear stress at equilibrium decreases
with decreasing osmotic pressure. (fig 2)
0
Objective
Shearstress for x = 2.40e−001 with cfc =−2.00e−004
−6
−3.2
x 10
−3.4
A 2D analytical solution for a dislocation in a swelling
ionised porous medium.
shearstress (Gpa)
−3.6
Methods
−3.8
−4
−4.2
−4.4
−4.6
−4.8
Model Lanir’s small deformation theory [3] describes
the saturated porous medium by assuming incompressible constituents and infinitely fast ion flow leaving a strongly coupled, biphasic model. Perturbation on a homogeneous situation ( = tr ()RT C1 ),
pre-stressed or not, gives:
~ tr() r
~ f = 0 ;
r2~u + ( + + RT C1 )r
tr()
K r2 f = 0:
t
c = 1.50e−004
ex
c = 3.50e−004
ex
c = 5.50e−004
−5
ex
−5.2
0
100
300
200
600
500
400
700
800
900
1000
time (s)
Figure 2: Shear stress in crack: consolidation.
Chemical potential At compression areas (xz > 0)
chemical potential is high and fluid flow is initiated
to lower areas (xz < 0) . Chemical potential rapidly
decreases further from the crack. (fig 3)
=−2.00e−004 at t=7.40e+001
muf(x,z) for cex = 1.50e−004 with cfc
0
−6
x 10
2
Shear faulting A dislocation on an existing crack is
described by a displacement jump between z = 0+
and z = 0 [1.] (fig 1).
3
1.5
z (mm)
2
1
1
0.5
0
0
−0.5
−1
−1
−2
−1.5
−3
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
x (mm)
Figure 1 : Shear fault: u
=
f (x )
H(t)
Derivation solution Decoupling the systems of
equations by means of stress functions [5.] results
in
r
r S = 0;
K r E = 0:
2
2 E
t
Figure 3: Chemical potential: fluid transport
Black line denotes crack
References
4
The new system is solved using linear transformations (Fourier and Laplace). The shear stress is calculated at the crack surface. Using Simpson rule
chemical potential away from the surface is computed.
1.
2.
3.
4.
5.
6.
J.R. Booker, J. of Geophys. Res., 1974
J.M. Huyghe et al., Int. J. of Eng. Science, 1997
Y. Lanir, Biorheology, 1987
R. van Loon et al., Int. J. for Num. Meth. in Eng., 2003
J. McNamee and R.E.Gibson, Quart. J. Mech & Appl. Math.,
1960
S. Wognum et al., Spine, in press
30
Topology Optimization of Shape Memory
Alloy Thermal Actuators
M. Langelaar1 , G.H. Yoon2 , Y.Y. Kim2 and F. van Keulen1
1 Delft University of Technology,
Faculty of Mechanical, Maritime and Materials Engineering, Mekelweg 2, 2628 CD Delft
phone +31-(0)15-2786506, e-mail [email protected]
2 Seoul National University,
National Creative Research Initiatives Center for Multiscale Design, Seoul, Korea
Introduction
Results
Shape memory alloys (SMA’s) are excellent materials for
miniature actuation applications involving large forces and
displacements, because of their unique properties. However, the complexity of their constitutive behavior complicates the design of effective two- and three-dimensional
SMA actuators. Improvements are possible through the
use of structured design techniques combined with computational modeling of SMA behavior.
Topology optimization has successfully been performed on
a number of thermal actuator design problems, using the
MMA optimizer [4]. Objective of the optimization is to maximize the stroke of the SMA actuator over a given temperature range. No volume constraint is used. Starting from
a uniform material distribution, functional SMA actuators
were obtained for various loadcases (see e.g. Fig. 2).
Objective
The aim of this research is to develop robust and effective
topology optimization techniques for the design of SMA actuators. Initially, the focus is on planar SMA structures in
plane stress configurations, acting against a constant load,
and actuated by uniform changes in temperature.
Methods
The conventional approach in topology optimization is the
so-called density-based method, where material properties of finite elements are functions of the associated design variables [1]. This approach is popular for linear
problems, but it has several disadvantages for physically
and geometrically nonlinear problems, such as the present
SMA actuator case:
Figure 2 : Topology optimization problem definition, final structure
and evolution during optimization.
Furthermore, also the redesign of existing structures has
been studied. Fig. 3 shows the result of applying the developed ECP-based topology optimization technique to an
SMA thermal actuator problem starting from a given baseline design. An improvement of the output stroke of more
than 100% has been obtained.
1. excessive distortion of low-density elements, resulting in lack of convergence,
2. influential arbitrariness in choosing material property
interpolation functions,
3. and the need to perform full sensitivity analysis of
complex material models.
These disadvantages have lead to the development of an
alternative formulation of the topology optimization problem, the Element Connectivity Parameterization [2,3]. In
this ECP approach not the material properties of elements,
but rather the stiffness of zero-length links connecting the
elements are controlled by the design variables (Fig. 1).
Figure 3 : Baseline and optimized SMA thermal actuator design,
and a performance comparison.
Discussion
Topology optimization of SMA thermal actuators has been
performed for the first time. Crucial to this result is the use
of the ECP formulation instead of the conventional densitybased approach. Next to the automatic generation of efficient actuator designs, the developed technique is also
capable of significantly improving existing SMA designs.
References
Figure 1 : Mesh layout, element-link patch and zero-length link
used in the ECP approach.
[1] Bendsoe, M.P. and Sigmund, O., Topology Optimization - Theory, Methods and
Applications, Springer-Verlag, Berlin, 2003.
[2] Yoon, G.H. and Kim, Y.Y., Element connectivity parameterization for topology optimization of geometrically nonlinear structures, International Journal of
Solids and Structures 42(7) 1983–2009, 2005.
[3] Langelaar, M., Yoon, G.H., Kim, Y.Y. and Van Keulen, F., Topology optimization
of shape memory alloy actuators using element connectivity parameterization,
6th World Congress on Structural and Multidisciplinary Optimization, Rio de
Janeiro, Brazil, May 30 - June 3, 2005.
[4] Svanberg, K., MMA - Method of moving asymptotes - a new method for structural optimization, International Journal for Numerical Methods in Engineering
24 359–373, 1987.
31

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