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Distributed optimization for microsystem design
S. Tosserams, L.F.P. Etman, J.E. Rooda
Eindhoven University of Technology
Department of Mechanical Engineering
P.O. Box 513, 5600 MB Eindhoven, NL
s.tosserams,l.f.p.etman,j.e.rooda@tue.nl
Introduction: Microsystem design
Microsystem design is a large-scale engineering
problem since it typically involves (Fig. 1):
• multiple components and/or
• multiple physics
a) Multidisciplinary b) Multi-level
Figure 1: Microsystem decomposition types
Traditionally, the subsystem designs are sequenced
in an “over-the-wall” fashion with limited feedback
(Fig. 2a). An optimal system can only be obtained
when the interactions between the subsystems are
addressed more systematically: A coordination
approach to system optimization is required (Fig. 2b).
Coordination approach
The developed coordination approach [1] has a
number of benefits:
• design subproblems remain autonomous
• freedom to select local design/analysis tools
• flexibility in setting up coordination
• mathematical rigor of coordination
Each subsystem is represented by an optimization
problem (Fig. 3). Interaction between subsystems is
coordinated using an augmented Lagrangian penalty
p on shared variables and system-wide functions.
a) Over the wall b) Distributed coordination
Figure 2: System design approaches
Figure 3: Subsystem design optimization problem
An iterative algorithm exchanges optimal subproblem
solutions until convergence in an inner loop, and
updates the penalty function in an outer loop.
Case: Micro-accelerometer design
An ADXL150-type lateral capacitive accelerometer
design problem (Fig. 4) is developed and will be
used as a demonstrator case for the coordination
approach. The developed models are analytical and
therefore reproducible and suitable to test the new
coordination method.
Figure 4: Micro-accelerometer case study
Outlook
As a next step, we desire to further develop the case
study such that it may be used as a benchmark
problem in multidisciplinary design optimization.
References
1. Tosserams, S., Etman, L.F.P., and Rooda, J.E. (2007):
Augm. Lagr. coordination for distr. optimal design in
MDO, IJNME, in press. DOI:10.1002/nme.2158
Tenth Engineering Mechanics Symposium P-71
The HYDRA control study
Chris Valentin, Daniel Rixen and Paul van Woerkom
Delft University of Technology, Mekelweg 2, 2628 CD Delft,
c.l.valentin@tudelft.nl, tel. 015 278 6508
Introduction
At the European Space Research and Technology
Center (ESTEC), the large HYDRA multi-axis vibra-
tion test facility is situated. With this facility, it is possi-
ble to perform dynamic testing of launcher payloads.
In specific, the facility performs base driven tests of
satellites to check if they can withstand launch con-
ditions. HYDRA is superior to conventional installa-
tions because it is able to test heavy payloads (up to
20 tons) in a wide frequency range (1 to 100 Hz) and
it can test them in any direction without changing the
test setup [2]. The facility has been successfully used
in the Envisat, Herschel and ATV testing campaigns.
Payload
Table
Servo-
actuator
Accu-
mulator
Seismic
mass
Hydraulic
power unit
Control
room
Figure 1 : The HYDRA shaker facility at ESTEC.
Problem statement
Testing of large payloads on HYDRA does involve sig-
nificant challenges. Because hydraulic cylinders are
used to excite the payload base, eigenfrequencies at-
tributed to the oil column stiffness are present in the
frequency range of testing. When a flexible payload
is tested on HYDRA, interaction between the payload
structural dynamics and test facility dynamics come
into existence. These phenomena can cause diffi-
culties when satellites must be qualified for launch,
resulting in additionally required simulations [1].
Research project
In 2007, the HYDRA control study was initiated by
ESTEC in partnership with Delft University of Tech-
nology (TU Delft). The objective of the study is to
analyze potential upgrades of the control algorithms
of HYDRA in order to improve the dynamics, fully uti-
lize its 6 DOF capabilities and extend the benefits of
the HYDRA facility to its customers.
Plan of action
In order to achieve the study goals, the following anal-
ysis steps have been defined:
• analysis of the present HYDRA facility,
• literature study on control strategies of other
shaker facilities,
• research into new and innovative control strate-
gies for large shaker facilities,
The applicability of potential control strategies are in-
vestigated with the HYDRA Simulink R© simulator [1].
Figure 2 : Herschel test on HYDRA and FE Model.
References1. Appolloni, M., and Cozzani, A. Test predictions by an ad-
vanced numerical simulation tool for the hydra facility. In
Proc. of 23rd Aerospace Testing Seminar (Manhattan
Beach, Los Angeles, CA, USA, 10-12 October 2006).
2. Brunner, O. The hydra multi-axis vibration test facility at es-
tec. In Proc. of 4th International Symposium on Environ-
mental Testing for Space Programmes (Palais de Congres,
Lieges, Belgium, 2001).
P-72 Tenth Engineering Mechanics Symposium
Automatic partitioning and multirating
applied to IC transient analysis
A. Verhoeven, B. Tasic, E.J.W. ter Maten, R.M.M. Mattheij
Eindhoven University of Technology
Faculty of Mathematics and Computer Science
P.O. Box 513, NL 5600 MB Eindhoven
phone +31-(0)40-2474847/2753, email averhoev@win.tue.nl
Multirate transient simulation
The design of Integrated Circuits requires the time-
integration of the following (large set of) differential-
algebraic equations:
d
dt
[
q(t,x)]
+ j(t,x) = 0, x ∈ Rd, (1)
where the unknown state vector x consists of voltages
and currents. Often, electrical circuits consist of parts
with completely different activity levels. For that case
multirate schemes have been developed.
0 2 4 6 8 10
x 10−9
0
1
2
3
4
5 Singlerate xL
Multirate xL
0 2 4 6 8 10
x 10−9
0
1
2
3
4
5 Singlerate xA
Multirate xA
Figure 1 : Example of multirate behaviour.
Multirate schemes solve the slow and fast parts of (1)
on a coarse and fine time-grid, respectively. Assume
that the circuit model can be partitioned as follows.
d
dt
[
qA(t,xA,xL)]
+ jA(t,xA,xL) = 0, xA ∈ RdA,(2)
d
dt
[
qL(t,xA,xL)]
+ jL(t,xA,xL) = 0, xL ∈ RdL .(3)
We use a multirate version of the Backward Differ-
ence Formula (BDF) scheme which first computes
(2),(3) at the coarse grid. Afterwards the fast equa-
tion (2) is integrated at the fine time-grid. Finally the
solution at the coarse time-grid is corrected by the re-
finement results. In [3] we showed that this method is
absolutely stable if both parts are stable and weakly
coupled. The local error can be controlled by adap-
tive timestepping of the coarse and fine timesteps [2].
The method has been applied successfully to several
circuit models of NXP Semiconductors.
Automatic partitioning
The speed-up factor of a multirate method with
coarse/refined stepsizes H,h, multirate factor m =Hh
> 1 and workload ratio E < 1 satisfies
S =Wsing
Wmult≈
1
1
m+ E
. (4)
The multirate factor m and the workload ratio E typ-
ically depend on the used partition. If e is the esti-
mated local error vector and p is the order of the used
BDF scheme, then m can be estimated by
m =
(‖eA‖
‖eL‖
) 1p+1
. (5)
The workload ratio E can be approximated by
E =
(dA
d
)α
, (6)
where α ∈ (1,3) and d = dA + dL . Thus the optimal
partition satisfies:
maxindA, indL
S ≈ maxindA,indL
1
1
m+ E
︸ ︷︷ ︸
S
. (7)
We developed several algorithms which are able to
solve this optimization problem [1]. An extended ver-
sion is also able to update the partition during the
time-integration such that S stays sufficiently large.
References
1. Verhoeven, A. et al (2007) Automatic partitioning for multi-
rate methods, SCEE (Sinaia, Romania), pp. 229–236.
2. Verhoeven, A. et al (2007), BDF Compound-Fast multirate
transient analysis with adaptive stepsize control, submitted
to JNAIAM, Tech. Report CASA 07-05.
3. Verhoeven, A. et al (2007), Stability analysis of the BDF
slowest fi rst multirate methods, Int.J. of Comp.Math., 84,
pp. 895–923.
Tenth Engineering Mechanics Symposium P-73
Multi-scale modelling of fracture in
piezoelectric ceramics
C.V. Verhoosel
Delft University of Technology
Faculty of Aerospace Engineering
P.O.Box 5058, NL 2600 GB Delft
phone +31-(0)15-2781528, email c.v.verhoosel@tudelft.nl
Introduction
Piezoelectric ceramics exhibit relatively large defor-
mations through application of electric fields and
vice versa. This electromechanical coupling makes
piezoelectrics suitable for many applications, such
as MEMS (micro electromechanical systems). The
brittleness of piezoelectric ceramics makes the mate-
rial, however, sensitive to damage. A hierarchic multi-
scale method [1] is developed to obtain an accurate
and robust numerical model for assessing this dam-
age.
Figure 1 : A hierarchic multi-scale method. Left: Parti-
tion of unity finite element model. Right: Polycrystaline
micro models in the interfaces.
Implemented models
• On the macro-scale the partition of unity method
(PUM) is used to model crack propagation. Dis-
crete displacement and potential jumps caused
by cracks can be modelled appropriately by
enriching the continuous finite element shape
functions with discontinuous Heaviside functions.
The cohesive behaviour is obtained from the ho-
mogenised response of a microscopic finite ele-
ment model.
• On the micro-scale a polycrystal is consid-
ered. Interface elements are inserted in the grain
boundaries to model intergranular fracture, which
is assumed to be the dominant fracture mode
in the case of relatively small (a few microns)
grains. An electromechanical cohesive law is ob-
tained by complementing a mechanical law with
electrical relations derived for a parallel plate ca-
pacitor.
Micro-Macro coupling
In a hierarchic multi-scale approach it is of crucial im-
portance that the scales are coupled correctly. This is
done by requiring that:
• The macroscopic jumps correspond to volume
averages of the microscopic displacement and
potential fields.
• The traction and surface charge density are de-
fined such that the coupling neither removes, nor
adds energy to the system.
Numerical results
Numerical experiments have been performed to test
the developed method. The three-point bending test
with off-centered crack (Figure 2) demonstrates the
appropriateness of the method. A jump in the electric
potential upon crack opening is observed, which is in
accordance with experimental observations.
Microscopic imperfections can, especially in the case
of MEMS, significantly influence the response of a
system. Determination of the most important imper-
fections and their influence on the reliability of MEMS
is the primary goal of this project.
Figure 2 : Electric potential field for the three-point
bending test with off-centered crack, loaded by a me-
chanical force and an electric voltage.
Bibliography
[1] V. Kouznetsova, M.G.D. Geers and W.A.M. Brekelmans,
Multi-scale constitutive modeling of heterogeneous mate-
rial with a gradient-enhanced computational homogenization
scheme. Int. J. Numer. Meth. Engng., 54, 1235, (2002).
P-74 Tenth Engineering Mechanics Symposium
A new engineering approach to
predict the long-term hydrostatic
strength of uPVC pipes
H.A. Visser, T.A.P. Engels, L.E. Govaert, T.C. Bor
Institute of Mechanics, Processing and Control - Twente
University of Twente
P.O. Box 217, 7500 AE Enschede, The Netherlands
phone +31 (0)53 4894346, email h.a.visser@ctw.utwente.nl
Introduction
Extruded unplasticized Poly(Vinyl Chloride) (uPVC)
pipes are certified for use in e.g. water distribution
systems using pressurised pipe tests. During these
tests the pipes are subjected to a certain temperature
and internal pressure. At the same time the time-to-
failure, the time at which the internal pressure drops
due to rupture or fracture, is measured. These tests
are time consuming and are therefore costly. To
reduce both the costs and the required testing time, a
model-based approach is proposed which can predict
the time-to-failure of pressurized pipes based on only
one (short term) test.
Initiation of failure
Although uPVC pipes can fail in a ductile, a semi-
ductile or a brittle manner (Fig. 1), it is our
hypothesis that plastic deformation initiates failure for
all three failure modes. Upon loading a polymer,
plastic deformation will accumulate continuously. At
a certain (plastic) strain the polymer enters its
softening region and fails due to localization of strain.
Observations of our experimental data shows that, for
uPVC, this critical strain (γcr) is constant for a wide
range of applied stresses and temperatures.
Figure 1 : Three different modes of fracture of uPVC
subjected to an internal pressure [1].
Model
From our hypothesis follows that the time-to-failure of
a polymer under static load and isothermal conditions
can be calculated with:
tf =γcr
˙γ (T, p, τ). (1)
The equivalent plastic deformation rate ( ˙γ) resulting
from the applied load and temperature can be
equated with a pressure modified Eyring relation. The
values for the material parameters in this model can
be obtained by uniaxial tensile and creep testing. In
addition a reference point or e.g. a tensile test is
needed to determine the thermodynamic state of the
polymer.
102
104
106
108
0
5
10
15
20
25
30
35
40
Time−to−failure [s]
Equiv
ale
nt str
ess [M
Pa]
Ref. point
20°C
40°C
60°C
Figure 2 : Typical result of pressurized pipe test
on uPVC (markers) [2] with predictions using the
presented model (lines). The blue, green and orange
markers represent ductile, semi-ductile and brittle
failure respectively.
The model predictions agree quantitatively with data
from Niklas et al. [2]. Both the slope and temperature
dependence are predicted accurately for all failure
modes. This supports the hypothesis that plastic
deformation up to a critical level of strain, initiated
failure for all three failure modes within the range of
stresses and temperatures studied here. This makes
the presented approach a promising tool for reducing
certification costs significantly.
ACKNOWLEDGEMENTS This work was performedwith the support of Cogas Netbeheer, Continuon, Eneco
Netbeheer and Essent Netwerk and partly carried out at
Eindhoven University of Technology.
1 H. Niklas, et al.; Kunststoffe, 49:109113 (1959)
2 H. Niklas, et al.; Kunststoffe, 53:886891 (1963)
Tenth Engineering Mechanics Symposium P-75
Kinetic Friction at Cryogenic Temperatures
E.G. de Vries and M.B. de Rooij
Phone +31-(0)53-3892440, e-mail e.g.devries@ctw.utwente.nlUniversity of Twente, Faculty of Engineering Technology,
Section of Surface Technology and Tribology, P.O. Box 217, NL 7500 AE Enschede
CryogenicsCryogenics is the science of producing andstudying low-temperature conditions. Itcomes from the Greek word Cryos,meaning “cold” and the shortened Englishverb “to generate”. It applies totemperatures from approximately -100°Cdown to absolute zero (-273°C = 0K)
In the 19th century the development of air-cycle refrigerators to preserve food evolvedinto the liquefaction of permanent gases byscientists. The last element to be liquefiedwas Helium at a boiling point of 4.2 K.The mechanical properties of manymaterials change dramatically when cooledto 100 K or lower. For example plastics andrubber become brittle and metals lose theirelectrical resistance (superconductivity).
ApplicationsNew technologies in cryogenic applicationslike superconductors as well as practicalapplications such as transportation,astronomy (see fig.1), metallurgy andmedicine lead to a growing interest inresearch for tribological properties at lowtemperatures.
Figure1: Very large telescope interferometer(ESO)
These conditions of ultrahigh vacuum (<10-7
mbar) and cryogenic temperatures result insome unique tribological problems.At low temperatures liquid lubricants willfreeze or become too viscous. Furthermore,protective oxide layers formed underambient conditions will not be restored.
ExperimentsIn order to measure material properties andfriction in a cryogenic environment a frictiontester is developed which makes it possibleto measure friction forces, hardness andadhesion in vacuum at temperatures downto 60 K.
Figure 2: Cryogenic-vacuum tribotester
ModelThe existing friction models describing therole of adhesion, deformation and shear ofinterfacial layer will be extended to includethe influence of low temperatures andvacuum.
References[1] Th.M. Flynn, Cryogenic
Engineering, Marcel Dekker, 2005.[2] Ye.L.Ostrovskaya, Low temperature
tribology at the B. Verkin Institute forLow temperature Physics &Engineering, Tribology International34, 2001.
P-76 Tenth Engineering Mechanics Symposium
Vibration Measurements using
Sound
J.W. Wind, Y.H. Wijnant and A. de Boer
Institute of Mechanics, Processes and Control
Chair of Structural Dynamics and Acoustics
University of Twente
P.O. Box 217, 7500 AE Enschede, The Netherlands
phone +31-(0)53-4893605, email j.w.wind@ctw.utwente.nl
Introduction
An attractive way to perform vibration measurements
is to measure the sound near the structure
surface. However, the acoustical pressure is
often overshadowed by background noise. Theory
suggests that the fluid velocity does not have this
drawback. We present new measurement results to
confirm this claim.
Theory
V
P
xx
xx
P
V
(a) (b)
Figure 1: Acoustical response to (a) structural vibration
and (b) incoming noise
In a 1D model, the pressure and velocity respond
equally to structural vibration (see figure 1(a)).
An interference pattern occurs for incoming noise
because the sound reflects from the structure (see
figure 1(b)). Near the structure surface, the pressure
is at a maximum but the velocity tends to zero. By
superimposing signal and ambient noise, it becomes
clear that ambient noise hardly contributes to the
velocity measurements near the structure surface:
only structural vibration is measured.
Experiments
Figure 2: aluminium box, incoming noise source and
sensor
Experiments are performed in an anechoic chamber.
Structural vibration is created by a loudspeaker in a
stiff aluminium box covered with a 1mm plate (see
figure 2) and the incoming noise is represented by a
loudspeaker in the far field of the structure. Pressure
and fluid velocity are measured at several distances
from the structure. Results are depicted in figure 3.
Conclusion
Both the structural behavior and the distance to
the structure surface have significant impact on the
sensitivity to background noise but off-resonance,
velocity measurements tend to be10dB less sensitive
to noise than pressure measurements near the
structure surface.
0 50 100 150 200 250 300−20
−15
−10
−5
0
5
distance from plate [mm]
pre
ssu
re [
dB
(Pa
/Pa
)]
Model 750 Hz
Model 1000 Hz
Model 1250 HzExperiment 750 Hz
Experiment 1000 Hz
Experiment 1250 Hz
0 50 100 150 200 250 300−90
−85
−80
−75
−70
−65
−60
−55
−50
distance from plate [mm]
pa
rtic
le v
elo
city [
dB
(m/s
/Pa
)]
Model 750 Hz
Model 1000 Hz
Model 1250 HzExperiment 750 Hz
Experiment 1000 Hz
Experiment 1250 Hz
Figure 3: Experimental results. Left: Pressure, Right: fluid velocity
Tenth Engineering Mechanics Symposium P-77
Multi-level Optimization of Composite Materials A.J. de Wit, and F. van Keulen
Department of Precision and Microsystem Engineering, Mekelweg 2, 2628 CD Delft
Delft University of Technology phone +31-(0)15-2786506, e-mail: a.j.dewit@tudelft.nl
Introduction Multi-level design optimization techniques rely on a decomposition of the optimization problem into separate levels or subsystems. These techniques incorporate design variables, objectives and constraints originating from different levels into the design.
ObjectiveThe project concentrates on the development of a multi-level optimization framework for composite structures. Here both the non-linear mechanical behaviour as well as the optimization covers multiple levels.
DecompositionFig. 1 shows how a structure consists of a composition of smaller subsystems. Two different patterns may be distinguished: hierarchic and non-hierarchic.
Fig. 1 Identifying a multi-level hierarchy in the design of an aeroplane.
Different compatibility constraints between the subsystems are possible. They are made visible in the optimization problem matrix. For example see Fig. 2.
Fig. 2 Ideal problem matrix, all the subsystems from the optimization problem are independent.
There are four different patterns of the problem matrix which dictate a type of multi-level optimization.
CoordinationThe information that is necessary for the optimization of the decoupled subsystems needs to be coordinated in such a way that the entire system converges to a solution. One such approach is to add optimum sensitivity information to account for changes in lower levels due to changes in higher level subsystems. However, many other techniques are available depending on the decomposition used and pattern identified in the problem matrix.
0 0
1
0
1r h
1 1
0
1
0r h
1v
0v Level-0
Level-1
xFig. 3(a) Optimization of two coupled subsystems, coordination of the information flow is necessary.
1 2 3
Fig. 3(b) Coordinating two levels by means of adding sensitivity information of lower level subsystems to higher level subsystems.
DiscussionWe have identified a number of steps which allow a designer to choose a decomposition approach and a coordination strategy depending on the design problem or user preference. Furthermore, we have classified
1 a
number of main stream multi-level optimization approaches recorded in literature.
Acknowledgement This project is funded by the technology foundation STW, the Dutch organisation NWO and the German Research Foundation.
References1. de Wit and van Keulen (2007), 3
rd MDOSC, April 23-26
0
fv
0 0,g hv v
0 0x, r
1 1x, r
1 1,g hv v
1
fv
x x x
1v
1
0
0, ,v h
Optimum Sensitivity
Analysis
Compute total Derivatives
Extrapolate to upper level10 0
1 0, ,hh x
0 0
1
0
1r h
1
0
0,h x
1 1,x v
0
cv
0
cv 1 1D , Dh hx
1Dhv
1 1 1
0
1
0c r h
2,1 2,2 2,3xx x
2,2,1 2,2,2 2,2,3x x x
P-78 Tenth Engineering Mechanics Symposium
Strain Path Dependency in BCC Crystals
Tuncay Yalcinkaya, W.A.M. Brekelmans, M.G.D. Geers
Eindhoven University of Technology
Faculty of Mechanical Engineering (WH 4.115)
P. O. Box 513 , 5600 MB Eindhoven
phone +31 40 247 5392, e-mail t.yalcinkaya@tue.nl
Introduction
Materials experience a complex deformation history
during sheet metal forming processes, which usually
is characterized by a non-uniform strain path.
Figure 1 : Strain path change during deep drawing process.
Objective
Changes in the strain path
directions result in tran-
sient hardening and soft-
ening of the materials. The
aim of the present work is
to develop a constitutive
sb
r
s
s
eepre
x
model that predicts the plastic anisotropy in BCC
structured metals, by concentrating on the effect of
the microstructure evolution during the load path
changes.
Modeling strategy
Modeling starts with a proper description of disloca-
tion movement. A crystal plasticity framework is im-
plemented and validated for this purpose. Peculiar
features of BCC crystals are integrated in the model.
GRAINS
DISLOCATION CELLS
CONSTITUTIVE MODELING OF DISLOCATION MOVEMENT
DISLOCATION CELL BLOCKS
Figure 2 : Global modeling strategy including the bridges be-
tween micro, meso and macro levels.
Currently, incorporation of a cell model into a crystal
plasticity framework is being carried out.
Composite Cell Model
Strain path change effects
physically originate from a
complex microstructure evolu-
tion. The present work deals
with the contribution of the
evolution of dislocation cell
structures to these effects.
The material with embedded cells is modelled to be-
have like a composite consisting of hard cell walls and
soft cell interiors. Evolution of internal variables, ρw,
ρc, w and r reflects the changes in the microstructure
(see figure 3).
D
D
cell walls
cell interior
cr sc ,
wr sw ,
r
w
Figure 3 : Composite cell structure representation.
The macroscopic response of the model is obtained
by applying Taylor averaging and the rule of mixtures:
σ = fσw + (1− f)σc
where f is the volume fraction of the cell walls.
Future work
The composite cell
model gives reasonable
results in a von Mises
plasticity framework for
the basic strain path
changes such as, cross
and Bauschinger tests. Strain
Str
ess
Monotonic test
Cross test
Bauschinger test
Future goal is to simulate the mentioned anisotropic
stress-strain responses of BCC crystals in the crystal
plasticity framework.
Tenth Engineering Mechanics Symposium P-79
Goal-oriented adaptivity for
steady fluid-structure interaction
K.G. van der Zee, E.H. van Brummelen, R. de Borst
Delft University of Technology, Faculty of Aerospace Engineering
Engineering Mechanics, PO Box 5058, 2600 GB Delft, The Netherlands
+31 (0)15 278 1089, K.G.vanderZee@tudelft.nl, www.lr.tudelft.nl/em
Introduction
Numerical simulations of fluid-structure interaction
typically requires vast computational resources.
Adaptive finite-element techniques could offer a sub-
stantial improvement in the efficiency of such simula-
tions.
Model problem We consider Stokes flow in a flexi-
ble tube Ωα, where the vertical displacement α of the
upper wall Γα satisfies a string equation:
Find (u, p, α) such that
−∆u + ∇p = 0
−∇ · u = 0
in Ωα
− α′′ = (−∇u · n + p n) · (0, 1) on Γα
Γα
Goal-oriented error estimation
Goal-oriented adaptive finite-element methods rely
on a posteriori error estimates of quantities of inter-
est. For this a well-established framework exists for
generic boundary value problems [1,2]. However, the
free-boundary character of fluid-structure interaction
forms a fundamental complication in obtaining such
error estimates.
Reference domain method By introducing an α-
dependent map Tα : Ωαh → Ωα, we essentially trans-
port our problem from the variable domain Ωα to the
fixed domain Ωαh . By linearizing the involved func-
tionals, this yields an appropriate dual problem in the
complementing dual variables (z, q, ζ) [3].
Dual problem boundary condition The dual struc-
ture displacement ζ acts as a Dirichlet boundary con-
dition for the dual fluid velocity z, i.e., primal roles
have switched. An example dual velocity solution is
depicted in the figure below, where the quantity of in-
terest is the average structure displacement
∫
Γ0
α.
Numerical results
We discretize the Stokes problem with Taylor-Hood
elements and the string problem with linear finite ele-
ments. We solve the corresponding dual problem on
the approximate domain Ωαh using the same mesh
but the dual shapefunctions are of one order higher.
Error estimate The figure below shows the conver-
gence of the error estimate (green) and the true error
(red) versus the number of degrees of freedom.
10
10
10
10 101010
-1
-2
-3
-442 3
Adaptive refinement The goal-oriented error esti-
mate can straightforwardly be introduced in an adap-
tive refinement procedure. The figures below show
sample meshes obtained using adaptive refinement
and uniform refinement and illustrate the efficiency
gained for this problem.
Elements = 175, Error in quantity = 0.000988
Elements = 224, Error in quantity = 0.002628
References1. R. Becker and R. Rannacher, An optimal control approach to
a posteriori error estimation in finte element methods, Acta
Numerica, (2001), pp. 1–102.
2. S. Prudhomme and J.T. Oden, On goal-oriented error es-
timation for elliptic problems: application to the control of
pointwise errors, Comput. Methods Appl. Mech. Engrg. 176
(1999), pp. 313–331.
3. K.G. van der Zee, E.H. van Brummelen and R. de Borst,
Goal-oriented error estimation for Stokes flow interacting
with a flexible channel, Int. J. Num. Meth. Fluids (to ap-
pear)
P-80 Tenth Engineering Mechanics Symposium
Quantitative fault discontinuity modelingusing the Partition of Unity Method
G.J. van Zwieten, M.A. Gutierrez, R.F. Hanssen
Delft University of TechnologyFaculty of Aerospace Engineering
tel: +31(0)152781829, email: g.j.vanzwieten@tudelft.nl
Introduction
Accurate quantitative knowledge of the current state ofstress in fault areas is of key importance in the assessmentof seismic hazard. Regions of high Coulomb stress havebeen shown to correlate closely with the location and ori-entation of subsequent seismic events [1]. Modeling thesestresses, however, involves computational complexities dueto the discontinuous nature of fault systems. The Partitionof Unity Method has been developed in the field of frac-ture mechanics to deal with problems of similar kind, andis expected to proof its value in geophysics as well.
Partition of Unity Method
To solve a differential equation Lφ = 0, the conventionalFinite Element approach is to narrow the solution space bydividing the domain Ω into a finite number of subdomains –the elements, and defining on those a number of continuousshape functions hi. The Finite Element solution φ is theunique combination of shape functions that satisfies theGalerkin orthogonality condition,
φ : Ω →
∑
i
hiφi, ∀j
∫
Ω
hjLφ dΩ = 0.
The narrowed solution space thus defined as the span ofcontinuous shape functions contains continuous solutionsonly. In situations where the solution can be expected tocontain or develop discontinuities this subspace does notsuffice. The Partition of Unity Method is an enhancementto standard Finite Element Methods to make it suitable forproblems with embedded discontinuities. Unlike most com-peting methods it does not rely on expensive remeshing;instead it uses the unmodified, continuous shape functionsto represent a discontinuous field. This is achieved by incor-porating a Heaviside function that takes on different valueson either side of the discontinuity. Galerkin’s orthogonalitycondition – the integral, separates in two sub-integrals overthe disjoint subdomains,
φ :Ω+
Ω−
→
∑
i
hi
(
φi + Hφi
)
, ∀j
∫
Ω+
hjLφ dΩ = 0∫
Ω−
hjLφ dΩ = 0.
In practical implementation, the difference with conven-tional Finite Element Methods shows only at elements thatare intersected. These elements can be thought of being du-plicated, with different integration schemes acting on bothsides of the crack. This is visualized in Figure 1; integrationtakes place only over the coloured areas.
Figure 1: Schematic representation of a PUM enhanced mesh.
Initial results
Figure 2 shows a plane of constant dislocation cutting theelements under a small angle, and the stress response ina direction parallel to that plane. Interpreted as a faultsystem this is a strike-slip fault, and the Coulomb stress tothe right can be related to aftershock risk. Future work willfocus on applying the Partition of Unity Method to morerealistic, three-dimensional fault systems.
Figure 2, from left to right: shear, normal and Coulomb stress.
References
[1] King et al., 1994. Static stress changes and the triggering of
earthquakes. Bulletin of the Seismological Society of Amer-ica, 84:935-953
Tenth Engineeering Mechanics Symposium P-81
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