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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,[email protected]

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,

[email protected], 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


Faculty of Mathematics and Computer Science

P.O. Box 513, NL 5600 MB Eindhoven

phone +31-(0)40-2474847/2753, email [email protected]

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.


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 [email protected]

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).


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 [email protected]

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)


Kinetic Friction at Cryogenic Temperatures

E.G. de Vries and M.B. de Rooij

Phone +31-(0)53-3892440, e-mail [email protected] 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.


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 [email protected]

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


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: [email protected]

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


Strain Path Dependency in BCC Crystals

Tuncay Yalcinkaya, W.A.M. Brekelmans, M.G.D. Geers


Faculty of Mechanical Engineering (WH 4.115)

P. O. Box 513 , 5600 MB Eindhoven

phone +31 40 247 5392, e-mail [email protected]

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.


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, [email protected], 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)


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: [email protected]

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

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