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Simulation of gravitational wave detectors

de Ronde, J.F.; van Albada, G.D.; Sloot, P.M.A.

Published in:Computers in Physics

DOI:10.1063/1.168591

Link to publication

Citation for published version (APA):de Ronde, J. F., van Albada, G. D., & Sloot, P. M. A. (1997). Simulation of gravitational wave detectors.Computers in Physics, 11(5), 484-497. https://doi.org/10.1063/1.168591

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https://doi.org/10.1063/1.168591

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https://doi.org/10.1063/1.168591

Simulation of gravitational wave detectorsJ. F. de Ronde,a! G. D. van Albada, and P. M. A. SlootFaculty of Mathematics, Computer Science, Physics, and Astronomy, University of Amsterdam,1098 SJ Amsterdam, The Netherlands

(Received 17 January 1997; accepted 11 June 1997)

A simulation program that provides insight into the vibrational properties of resonant massgravitational radiation antennas is developed from scratch. The requirements that are setnecessitate the use of an explicit finite element kernel. Since the computational complexity of thiskernel requires significant computing power, it is tailored for execution on parallel computersystems. After validating the physical correctness of the program as well as the performance ondistributed memory architectures, we present a number of ‘‘sample’’ simulation experiments toillustrate the simulation capabilities of the program. The development path of the code, consistingof problem definition, mathematical modeling, choosing an appropriate solution method,parallelization, physical validation, and performance validation, is argued to be typical for thedesign process of large-scale complex simulation codes. ©1997 American Institute of Physics.@S0894-1866~97!01805-1#

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INTRODUCTION

General background

Although the existence of gravitational radiation, predictby Einstein’s theory of general relativity,1 is unquestioned,its detection is a long-standing problem in experimenphysics. The aim of the GRAIL project2 is to realize aspherical resonant mass detector with a sensitivity thatfew orders of magnitude higher than the present generaof detectors, thus offering the possibility of validating thexistence of gravitational radiation that is emitted by astphysical sources.3 These sources include supernovae, slar collapses to black hole states, and coalescence of bineutron star systems.4

The quadrupole moment of a mass distributionr~x! isgiven by

Di j 5EVdVr~x!S xixj2

1

3d i j x

2D . ~1!

According to general relativity~GR!, an oscillation of thisquadrupole moment is the simplest mode of vibration tcan generate gravitational waves. The counterstatemenabsorption of gravitational waves is that in the simplcase it also takes place via the excitation of the quadrupmodes of vibration of a massive object. This principle is tmain argument for constructing a gravitational radiationtenna, which essentially is a large resonant mass thaforced into oscillation by impinging gravitational radiatio

In GRAIL, the antenna will consist of a spherical resnant mass of a Cu alloy, possibly CuAl, with a massabout 100,000 kg and a diameter of 3 m. It will bsuspended in vacuum inside a large cryostat in such athat the external vibrations at its resonant frequen

a!E-mail: [email protected]

484 COMPUTERS IN PHYSICS, VOL. 11, NO. 5, SEP/OCT 1997

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('700 Hz) are attenuated by at least a factor of 1016. Thesphere will be cooled to a temperature in the range10–20 mK in order to cancel out thermal noise. Transders will be attached to the surface of the sphere to detecvibrations induced by gravitational radiation. An analysof the electric signals from these transducers must giveformation ~such as source type and direction of incidenc!about the gravitational radiation that interacts with tsphere. The gravitational waves have extremely weak inaction with matter; the typical deformations induced onCu sphere of 3 m will be of the order of 10220 m, whichexplains why detection is such a difficult task and whyhas not yet been realized. Several arguments can be gfor using a spherically shaped antenna5–8 instead of, forexample, the cylinder or bar antennas in the pioneerdays of gravitational radiation detection9 and present-dayexperimental setups like the NAUTILUS detector at tFrascati National Laboratory in Rome.10

Since the estimated cost~' 45 million Dutch guilders!of the project is quite high, a pilot study has been fundedNWO11 to investigate the technological feasibility oGRAIL. In this project several critical aspects concernithe design of the antenna had to be addressed. The taour simulation group was to construct detailed modelsthe antenna that provide insight into its vibrational propties. A schematic design of the GRAIL antenna concepshown in Fig. 1.

The development and design of complex systemsbe significantly enhanced by means of simulation. A recexample in which simulation was used in the analysis ofbehavior of resonant mass antennas can be found in ReIn this work modal analysis was used to determine thequency spectrum of a truncated icosahedral antenna.accuracy of this simulation was fairly limited due to thlow model resolution. The main result was that one c

© 1997 AMERICAN INSTITUTE OF PHYSICS 0894-1866/97/11~5!/484/14/$10.00

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conclude that the behavior of a truncated icosahedraltenna is very similar to that of a sphere.

In this article we will describe the development ofhigh-resolution simulation system for resonant mass grtational radiation antennas that can be applied to a numof important design issues.

Design issues

The following questions need to be addressed by meanextensive simulation experiments.

~1! How does the suspension of the huge resonant maffect the eigenmodes and eigenfrequency of the fdamental quadrupole modes of a sphere?

~2! How will material inhomogeneities affect these eigemodes and eigenfrequencies?

~3! Given a perfect sphere, how will it deform under iown weight?

~4! Does the coupling between the suspension rod andsphere induce additional modes that will cause interence with the modes we are looking for?

~5! What is the effect of mounting transducers on thetenna on the frequency spectrum of the system?

~6! How will seismic noise, entering the sphere throughframework in which it is suspended, influence ttransducer signals?

~7! How will the fingerprint of typical gravitational wavesources be seen by the antenna?

~8! Can energy that has been deposited by cosmic rayparticles like muons induce system eigenmodes thatnot distinguishable from those induced by gravitationwaves?

We can identify two types of questions in this lisFirst, the design questions that need to be answered in oto tune the readout of the system, like the effects ofsuspension hole, suspension rod, or material inhomogities on the frequency spectrum of the antenna. Second

Figure 1. TheGRAIL antenna in a heat isolating vacuum chamber.

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have those questions that pertain to the various sourcevibration: How can we discriminate whether the vibratiothat we observe is the result of a gravitational wave orsomething else? Clearly a simulation that can help usanswering these questions can be of great value to thesign and development phase of the GRAIL project. Evetually, when the system is operational, a~well-defined!simulation system can even be used to analyze the trducer output by means of reverse engineering of the traducer signal towards the original source. Incorporationsimulation in the experimentation cycle is sometimesferred to as ‘‘living simulations’’ or ‘‘simulation in theloop.’’

The remainder of this article is structured as followSection I begins with a description of two solution methothat can be used to answer a few basic questions butare too restrictive for our overall goal. For this purposeresort to development of a simulation that is based onexplicit finite-element~FE! method. The simulation accuracy required necessitates high-resolution FE models.performance of the FE solver is therefore enhanced byploiting the models’ parallelism~Sec. II!. Since the simu-lation is developed from scratch, we can circumvent sevepitfalls that are known to hamper the migration of existisimulation codes to parallel computers.12–14 In Sec. III thephysical correctness of our simulation is validated bycomparison with an analytical solution as well as by expements that were carried out with a small prototype antenIt is further shown that we benefit greatly by exploitincomputational concurrency. In Sec. IV we present a ftypical simulation experiments that can be carried out wthe simulation program. Some concluding remarks regaing the computational science approach taken in our stare given in Sec. V.

I. SIMULATION METHODS

A. Analytical solution

The equation of motion for elastic objects, also knownthe Navier equation@Eq. ~2!# describes the variation in timeof the displacement field~u! in homogeneous isotropicelastic objects. The material of such objects is parameized by the two Lame´ parameters~l,m! and the materialdensity~r!.

mDu1~l1m!¹¹•u5ru. ~2!

Two parameters that are frequently used to descthe elastic properties of a material are the Poisson ratio~n!and Young’s modulus (E). Both parameters can be expressed in terms of the Lame´ parameters as follows:

n[l

2~l1m!, ~3!

E[m~3l12m!

l1m. ~4!

The general solution of Eq.~2! consists of two differ-ent contributions that correspond to divergence-free (¹•u50) and rotation-free (¹3u50) waves that propagatthrough the material with unique velocities.

COMPUTERS IN PHYSICS, VOL. 11, NO. 5, SEP/OCT 1997 485

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For both types of vibration a frequency equation cbe derived for a freely vibrating~traction-free! elasticsphere and can be solved numerically.15 The rotation-freevibrations are known asspheroidal eigenmodes. A generalspheroidal eigenmode,Cs, is given by Eq.~5!, which is thesuperposition of all particular rotation-free solutions of E~2! for a spherical object with traction-free boundary coditions; note that we use polar coordinates.

Cs5 (n50

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(m52n

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@an~r !er1bn~r !R¹#Ynm~u,f!, ~5!

with R the sphere radius ander the unit vector in the radiadirection;an(r ) andbn(r ) are dimensionless radial eigenfunctions determined by the boundary conditions, aYnm are spherical harmonics. For each value ofn the fre-quency equation contains infinitely many solutions. A geeral quadrupole mode of the sphere can be described asuperposition of the five modes withn52 (m522...2).Since ~according to GR! only these modes interact witgravitational radiation, they are the primary subject of oinvestigations. Figures 2 and 3 show the shape of one oquadrupole modes~denoted as thel 50 mode16! differingby half a period. This eigenmode oscillates between a plate and an oblate shape.

The applicability of this analytical model fails as sooas we perturb the sphere in any way. However, it provius with a useful gauge for a simulation program. First,take a look at an elegant numerical model that is lessstrictive than the analytical model described above.

Figure 2. The prolate stage of the l50 mode.


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B. Numerical method

If we write down the Lagrangian@Eq. ~6!# for a freelyvibrating elastic object~following Ref. 17! ~the summationconvention is used where appropriate!

L5EVS 1

2rv2uiui2

1

2ci jkl ui , juk,l DdV, ~6!

with ui the component of the displacement fieldu that isparallel to thei axis,ui , j the derivative ofui in the j direc-tion, ci jkl the elastic tensor~dependent onl andm!, andvthe eigen-~angular! frequency, and apply the principle oleast action, settingdL50, we get

dL5EV~rv2ui1ci jkl uk,l j !duidV

2ES~n j ci jkl uk,l !duidS

50, ~7!

where n j denotes the component along thej axis of theoutward-pointing normal vector on the surface of the oject. Equation~7! exactly expresses the elastic wave eqution ~2!, with traction-free boundary conditions, of Sec. I AIf we expand the displacement vector in a set of basis futions according to

u5(lmn

almnxlymzn, ~8!

and truncate the expansion to a certain polynomial ordEq. ~7! can be rewritten in terms of a generalized eigevalue problem~9!.

Figure 3. The oblate stage of the l50 mode.

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Table I. The analytical and ‘‘numerical’’ frequencies in Hz for a few spheroidal modes of a Cu sphere.The dashes denote that no frequencies in the same range were found for specificN. This is due to the factthat the polynomial expansion in question is not able to represent the corresponding eigenmode properly

nAnalyticalsolution

N55R5168

N56R5252

N57R5360

N58R5495

N59R5660

N510R5858

2 654.0828 654.090 654.090 654.085 654.085 654.085 654.01 891.7504 893.488 891.796 891.796 891.788 891.788 891.73 975.4565 988.374 975.550 975.550 975.464 975.464 975.44 1251.997 ••• 1281.98 1252.35 1252.35 1252.01 1252.012 1263.632 1270.78 1270.78 1263.78 1263.78 1263.68 1263.60 1395.698 1396.06 1396.06 1396.06 1396.06 1396.06 1396.05 1510.716 ••• ••• ••• 1511.6 1511.65 1510.833 1661.032 ••• 1681.20 1681.20 1661.6 1661.63 1661.046 1760.268 ••• ••• ••• ••• 1762.42 1762.421 1793.638 ••• 1843.70 1843.70 1795.91 1795.81 1793.67

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The matricesE andG can be computed by integrating E~7! properly. Solving forv provides us with estimates othe eigenfrequencies of the system in question.16,17

1. Validation

Table I shows several eigenfrequencies found with themerical method as well as the corresponding analyticalues for a Cu sphere~material parameters:r58933 kg/m3,E5129.8109 Pa, andn50.343!. N denotes the truncationorder of the basis function expansion, andR3R is the ma-trix size of the corresponding eigenvalue problem. We hused aLAPACK18 routine called DSYGV, which can be applied to symmetric generalized eigenvalue problemssolve the system.

2. Spheroidal perturbations

An ellipsoid is described by its three semi-axesdx , dy , anddz :

Figure 4. The fundamental quadrupole frequencies under variationdz . The spherical symmetry is broken in thez direction, resulting in athreefold splitting of the fivefold degenerate quadrupole frequency:doublets and one singlet. The doublets are denoted by 1c–and–1s and2c–and–2s and the singlet by 0.

x2

dx2 1

y2

dy2 1

z2

dz2 51. ~10!

For a sphere these axes have equal length. When we mone semi-axis longer and keep the other two constant,object is called a prolate spheroid. If we make one seaxis shorter than the other two, we have an oblate spherIn Fig. 4 the dependence of the lowest quadrupole fquency ondz is depicted@varied in the range~0.1, 5.0! m#,while the other two axes are kept at 1.5 m for a Cu spheWe can observe three lines that intersect atdz51.5. Sincethe spherical symmetry is broken in thez direction, thefivefold degeneracy is only partially removed. It splits intwo doublets and one singlet. Additional perturbationsthe x and y directions would break the symmetry completely, leaving us with five slightly different frequencies

C. A finite-element model

The numerical method of Sec. I B also was applied tosphere with a suspension hole. Again it can be obserthat the five frequencies are split up~see Fig. 5!. It leavesus with an uncomfortable feeling, however, since the shedges of the suspension hole possibly disrupt the nume

Figure 5. The fundamental quadrupole frequencies vs the radius ofsuspension hole. The spherical symmetry is broken in thez direction,resulting in a threefold splitting of the fivefold degenerate quadrupfrequency: two doublets and one singlet. The doublets are denote1c–and–1s and 2c–and–2s and the singlet by 0.


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process. Therefore, we do not know whether the frequesplitting is an artifact or is physically correct. Furthermowe are stuck with a variety of questions that cannotanswered by the analytical and numerical approacabove. Next we will consider the finite-element method athird alternative to answer our questions. We discretizeantenna model in first-order tetrahedral elements~each ele-ment has four nodal points!. As an example, Fig. 6 showthe hull of such a finite-element model, consisting39,684 tetrahedral elements. One approach to the probis by means of modal analysis, that is, by studyingfollowing eigenvalue problem:

Kd5v2Md, ~11!

with K the system’s stiffness matrix,M the mass matrix,dan eigenmode, andv an eigenfrequency of the system. Athough it is theoretically possible to investigate a linesystem such as ours on the basis of modal analysispractical realization is a different story. First of all, thmodel resolutions that we aim for are very high, necessiing solution of huge eigenvalue problems. These maysolvable using state-of-the-art numerical solvers. If we sceed in doing that, we have to cope with a storage problsince for each nodal point a superposition of at least o100 eigenmodes is necessary to characterize a generalfrequency solution of the system. In addition, it is difficuto model the effects that are due to coupling to exterforces that are aperiodic, like seismic noise entering viasuspension or stochastic sources.

On the other hand we can choose for explicit timintegration of the equation of motion~12!.

Ma1Cv1Kd1F50. ~12!

Figure 6. The hull of a finite-element antenna model with the suspenrod discretized into 39,684 tetrahedral elements.


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The matricesM , C, andK are constructed by assemblinthe local mass, the viscosity, and the stiffness matricesall individual finite elements in which the antenna modeldivided; a, v, andd are vectors that describe, respectivethe acceleration, velocity, and displacement fields, andF isan external force vector. In order to integrate this systexplicitly in time we adopt the following Newmarkscheme:19

dn115dn1vnDt1an

~Dt !2

2,

an115M 21K~dn11!, ~13!

vn115vn1~an1an11!Dt

2

with an , vn , anddn vectors that describe, respectively, thacceleration, the velocity, and the displacement fields inFE system at timen. Viscous damping (C) and externalforces ~F! can easily be added to this algorithm.19–21 Themasses are ‘‘lumped’’ at the nodal points, resulting indiagonal mass matrix. ThereforeM 21 can be calculatedquickly. For numerical stabilityDt must be less than theCourant value, since the time integration scheme is contionally stable. The choice of the algorithm is, on the ohand, motivated by the fact that it displays good numeriproperties~low numerical dispersion and damping!. Thiswas found through a number of exploratory experimentsa one-dimensional finite-element model~data not shown!.On the other hand, it is very well suited for exploitincomputational concurrency, as we will see below. Althouthis explicit approach has the drawback that we havesolve our system in the time domain and afterwards pform a spectral analysis, it is much more flexible for opurposes, and is not hampered by the restrictions that ato modal analysis. Furthermore, the simulation model cbe extended such that it can incorporate nonlinear effesuch asviolin modesin the suspension rod, that cannot brealized by modal analysis.

It was expected that the required simulation accuranecessitates the use of a very-high-resolution FE modelis computationally and memory intensive. The simulaticode was therefore designed and implemented for exetion on distributed-memory computers.

II. PARALLELIZATION

In good tradition, we have chosen to base our parallelition method on the decomposition of the FE mesh. Tconsequences of the decomposition method for the tintegration algorithm as well as the calculation of importasystem parameters~like energy! had to be considered in thdesign. In our discussion we will only consider the pufinite-element system. In the full-fledged simulation a vaety of phenomena can be taken into account, such as trducer mounting, suspension rod, and sources of vibratAt this point it suffices to say that the parallelizatiomethod chosen does not have major consequences fofull-fledged simulation, and therefore may be left out of thdiscussion. We focus on two important aspects of the pallelization process: parallel time integration~the kernel!and parallel calculation of the energy.

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The code was implemented in C with PVM22 messagepassing primitives according to the single-programultiple-data paradigm.23 The problem domain~the finite-element mesh consisting ofN tetrahedral elements! is par-titioned intoP subdomains. Each subdomain is assigneda single process and executes the same program appliits local data. The process with identifier 0 is used to rein the whole mesh; it supplies the otherP21 processeswith their locally essential data, and is used for input/out~I/O!, a primus inter paresor masteramong the paralleprocesses.

A. Energy

In order to check energy conservation, we have to monthe energy in the system. The total energyEtot is given by

Etot512~vTM v1dTKd!, ~14!

with v the velocity vector in which the individual velocitvectors of all nodal points are concatenated, andd the dis-placement vector, also formed by concatenating the fieat all nodal points. How do we calculate this in parallel?

The FE mesh is decomposed along the surfaces oftetrahedral elements that constitute the mesh. First, agraph of the mesh is constructed; this then can be ptioned using any graph-partitioning method.24 Nodal pointsthat lie on shared boundaries are replicated on each prowith this boundary in its local domain, whereas elemebelong to a unique process. This procedure is schematicdepicted in Fig. 7 for a simple quadrilateral mesh.

The global mass and stiffness matrices are construby assembling the individual contributions from each ement in a global matrix. If we denote the assembly of twelement stiffness matricesk1 andk2 by k1% k2 , K is con-structed~analogously for the mass matrix! as follows:

K5k1% k2% ...% kN , ~15!

with N the number of tetrahedral elements. If we monithe energy values during a simulation run~it would be niceif this could be done completely separate!, that is, for eachsubdomain (i ), the energy content (Etot

i ) is calculated, andonce in a while we assemble the subdomain energy cobutions and accumulate them on process 0. This candone as follows:

Etoti 5 1

2~viTMri vi1diTKid

i!, ~16!

vi denotes the velocity vector for the nodes in subdomaii ,di is the displacement field,Ki is the stiffness matrix of

Figure 7. Partitioning of a quadrilateral mesh into four parts. The nodpoints at the shared boundaries are replicated while elements aresigned to unique processes.

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subdomaini , which is simply the assembly of its locaelement stiffness matrices, and finallyMr

i is an adjustedversion of the~diagonal! mass matrix of subdomaini . The‘‘normal’’ mass matrix contains the mass for each nodpoint at the appropriate entry. In this adjusted versionmasses of the internal nodal points are untouched. Hever, for the shared nodal points this value is divided by‘‘redundancy’’ value, which denotes how many processhave a replica of it. For example, in Fig. 7 nodal point 5 ha redundancy value of 4. As a consequence, the cosponding entry in each local adjusted mass matrix, eitbeing on process 0, 1, 2, or 3, has to be divided by 4. Inway we can assure that we are not counting any enecontributions twice.

B. Time evolution

If we have constructed the mass and stiffness matriceseach subdomain~or process! correctly, we can parallelizethe Newmark scheme above as follows: Each process gunique process identifier~ID! equal to the subdomain identifier ( i ) above. Nodal points that are shared among twomore processes are considered to have one real owner~theowning process considers it as a node of type A!, while theother processes only have the replica of this node~consid-ering the node as type B!. Assigning a node to a specifiprocess is based on the process ID. The process withlowest ID considers a shared nodal point as A type, whthe other processes see it as B type. Consequently, pro0 can only have shared nodes of type A, while procesP21 ~assuming we haveP processes! can have only sharednodes of type B. The following construction parallelizes tNewmark scheme correctly:

dn115dn1vnDt1an(Dt)2/2,Senddn11 from A nodes to B nodes,Receivedn11 for B nodes from A nodes,an115M 21K(dn11),Sendan11 from B nodes to A nodes,Receivean11 for A nodes from B nodes,Add contributions toan11 from B nodes,vn115vn1(an1an11)Dt/2.

The A–B nodal point scheme was adopted from Lonsdet al.25

III. VALIDATION

Here we show a selection of experimental results that wproduced for validation of our parallel FE simulation. Firswe investigate whether the model is physically correNext, we measure the performance benefits that wefrom exploiting parallelism for different mesh resolutionMeshes are decomposed by two different graph-partitionmethods: recursive spectral bisection~RSB! and a general-ized version of orthogonal recursive bisection~ORB!,24

which we callgeneralizedORB ~GORB!. GORB operatesin such a way that consecutive partitionings are not necsarily orthogonal and that partitioning cardinalities are nnecessarily a power of 2~as is the case in ORB!. Figure 8shows partitioning in the case of GORB in four parwhere the consecutive partitionings are not performedthogonally but at an angle of 80°. GORB recursively b


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sects the geometrical domain in equal partitions~work loadbalancing!, in alternating directions, not paying any attetion to the mesh connectivity~communication!. RSB ap-plies a more complex, and consequently computationmore expensive, method to partition the mesh. In contwith ~G!ORB it also takes into account the mesh conntivity. It recursively bisects a finite-element mesh into equnumbers of elements~work load balancing!, while ~ap-proximately! minimizing the number of edges between tresulting subdomains~communication load balancing!. Fig-ures 8 and 9, respectively, show the partitioning incases of GORB and RSB into four parts, with both bistions applied to a FE sphere of 79,556 elements. Theference in partitioning quality between GORB and RSwill have to be determined by explicitly measuring the kenel execution times because it cannot be found by simvisual inspection.

The performance measurements are carried out ontarget platforms both installed at the IC3A in Amsterdam;26

a Parsytec PowerXplorer@32 PowerPCs 601, with each 3Mbytes of random access memory~RAM!# and a ParsytecCC ~40 PowerPCs 604, each having 96 Mbytes of RAM!.Our target material in the experiments below is the alCuAl ~90–10! ~with r57534.0 kg/m3, E5154.03109 Pa,and n50.3265!, and only considers spheres with radii1.5 m. All results with respect to the physical experimegiven below are independent of the number of process

Figure 8. The hulls of the four partitions obtained for a sphere consistof 79,556 tetrahedral elements obtained using theGORB partitioningmethod. Each of the four hulls has about 3140 triangular faces. Thebisection is along thex direction. Next, the bisection surface is rotate80° and the two subdomains are both bisected individually alongsecond direction.


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used, which indicates the correctness of our parallel impmentation.

A. Physical correctness

Let us consider simulations that are initiated with a puquadrupole mode, that is, each nodal point in the sysgets an initial velocityvT5h(x,2y,0), with h the timederivative ofh, a scale-free parameter denoting the defmation of space. Since the FE models are always sligasymmetric, the fivefold degeneracy will be removeTable II enumerates the average frequencies that are foaround the analytical eigenfrequencyf 0 ( f 05780.25 Hz)

Figure 9. The hulls of the four partitions obtained for the same spherein Fig. 8, but now using theRSBpartitioning method. Although the edgein this partitioning appear more ragged than those obtained usingORB,they have only about 3010 faces.RSB optimizes the connectivity withinpartitions and minimizes the communication between them.

Table II. The average frequency ^ f & and the approximated errors( f ) in % for the quadrupole principal mode for various modelresolutions.

n ^ f & s( f ) %

84 7.91e102 8.72e201124 7.92e102 1.70e100698 8.01e102 4.87e201

1086 7.97e102 4.44e2012724 7.90e102 3.23e2012851 7.96e102 3.62e2016114 7.86e102 2.25e2019412 7.85e102 3.01e201

51501 7.82e102 2.85e20179556 7.82e102 1.56e201

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and the deviation in % of the average value for 10 differFE models that consist ofn tetrahedral elements. Thesresults indicate that the dominating frequency for themodels approaches the analytical frequency value forcreasing model resolution~up to 0.16% from the analyticavalue for the highest resolution model!.

Figure 10 shows thex displacement of an arbitrarnodal point at the sphere surface in the same simulationthe highest-resolution model. At first glance, it seems tthe nodal point oscillates with a frequency of about 780 Hwith a signal of'70 Hz superimposed. However, Fourianalysis~data not shown! has pointed out that this signamainly results from superposition of the three oscillatiothat are most pronounced, that is, those oscillating at 781493, and 1499 Hz, respectively. Possible beats that refrom superposition of the principal quadrupole mod~around 780 Hz! are not visible in this picture.

We can also observe the system energy during silation. It can be derived that the analytical energy (Equad)for a sphere vibrating in a pure quadrupole mode canexpressed in terms of the radius (R) and the material density ~r! as follows:

Equad54

15 prR5h2. ~17!

For the CuAl sphere with a radius of 1.5 m it follows thEquad547,929.2 Nm. Within the finite-element simulationit was found that the energy approaches this value very wfor increasing model resolution~data not shown!. For in-stance, the total energy for the highest-resolution mode79,556 elements is approximately equal to 47,924.7, whapproaches the analytical value up to 0.01%.

Within the GRAIL project, a prototype spherical antenna of the same CuAl alloy as that above, but withradius of 0.075 m was fabricated. The following expement was carried out. The prototype is excited by a radimpulse. Next, the time series of radial vibrations arecorded at several~arbitrarily chosen! places on the antennsurface. From these time series we have derived the pospectrum of the antenna vibrations by alternately applyinfast Fourier transform~FFT! and a nonlinear least-square

Figure 10. The time trajectory in thex direction of an arbitrary nodalpoint at the surface of a FE model of 79,556 elements initiated with a pquadrupole velocity field.

r

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-

l

r

fitting procedure on the signal residue. Simultaneouslysimulation experiment was performed in which the saprocedure was carried out. That is, a radial displacemeninduced on an arbitrary surface point of a detailed finielement model consisting of 191,310 tetrahedral elemeThe model is realistic in the sense that it incorporatessuspension rod and the suspension hole. For several pat the model surface we record the time series, analogouthe real experiment, followed by the same procedureproduce a power spectrum. Table III lists a part of tfrequency spectrum that is obtained for a radial impulsethe prototype antenna, accompanied by the frequenfound from the finite element simulation. The third columin Table III corresponds to the ratio of model frequencyexperimental frequency. We can observe that the frequespectrum found in the simulation model is quite closeexperimental values. The residue of the experimental sigand the simulated signal turns out to be less than 0.1%

Table III. Matches between frequencies computed using the explicitfinite element code and experimental values for a 15 cm sphere that isexcited by a purely radial excitation. In both sets of data, mode exci-tations may have been below the detection threshold, explaining fre-quencies occurring in a single set only. The analytical frequencies thatare found for a pure sphere with the same material specifications are„within the same frequency range… 14830 „2,1…, 15,708„2,2…, 21,243„1,2…, 22,915 „3,1…, 23,407 „3,2…, 30,025 „4,2… Hz, with, for instance,14,830„2,1… denoting a mode vibrating with 14,830 Hz,n52 of toroi-dal type „1…, and 15,708„2,2… a mode vibrating with 15708 Hz,n52 ofspheroidal type „2….

Computed~Hz!

Measured~Hz!

Computed/measured

13,650 13,687 0.99713,740 13,720 1.001

13,79713,849 13,815 1.00214,391 14,424 0.99814,500 14,470 1.00214,626 14,587 1.00319,58019,689 19,690 1.00019,797 19,812 0.99921,04521,171 21,138 1.00221,207 21,289 0.99621,316 21,307 1.000

21,34321,371

21,623 21,647 0.99921,667

21,750 21,777 0.99921,840 21,809 1.00027,716 27,719 1.000

27,73927,802

27,843 27,863 0.99927,91927,95427,972

27,987 27,991 1.000


-R

492 COMPUTERS

Table IV. The kernel execution times in seconds on the PowerXplorer, for four different mesh sizes andtwo different decomposition methods„GORB denoted by suffix G and RSB denoted by suffix R… versusthe partitioning cardinality.

6114-G 6114-R 9412-G 9412-R 51501-G 51501-R 79556-G 79556

1 3.46e201 3.46e201 5.25e201 ••• ••• ••• •••

4 1.47e201 1.45e201 2.05e201 2.02e201 8.89e201 8.85e201 1.31e100 1.30e1006 1.58e201 ••• 2.02e201 ••• 6.95e201 ••• 1.01e100 •••

8 1.22e201 1.13e201 1.77e201 1.72e201 5.98e201 6.00e201 7.65e201 7.84e2019 1.52e201 ••• 2.17e201 ••• 5.99e201 ••• 7.80e201 •••

12 1.25e201 ••• 1.84e201 ••• 5.14e201 ••• 6.55e201 •••

16 1.10e201 1.04e201 1.64e201 1.63e201 5.00e201 4.53e201 6.34e201 6.41e20118 1.02e201 ••• 1.57e201 ••• 4.78e201 ••• 6.71e201 •••

24 7.99e202 ••• 1.31e201 ••• 4.61e201 ••• 6.93e201 •••

27 8.01e202 ••• 9.75e202 ••• 4.72e201 ••• 6.59e201 •••

32 7.00e202 8.58e202 9.07e202 1.21e201 4.42e201 4.59e201 6.11e201 6.36e201

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B. Parallel performance

Next we consider the parallel performance of the simution kernel. We apply GORB~partitioning cardinalities of2, 4, 6, 8, 9, 12, 16, 18, 24, 27, and 32! and RSB~parti-tioning cardinalities of 4, 8, 16, and 32! to four differentmeshes consisting of 6114, 9412, 51,501, and 79,556ments, respectively. Table IV displays the single-kernelecution times~one time step in the Newmark scheme! thatwere obtained for simulations on the PowerXplorer forvarying number of processors. Table V shows similarsults obtained on the Parsytec CC. The execution timesthe best case, that is, energy and external force calcula~modeling gravitational radiation sources! are left out.

In both Tables IV and V we can observe that the dference in execution speed between GORB and RSB issignificant. In fact GORB is often slightly better than RSfor partitioning cardinalities that are powers of 2. This cbe ascribed to the fact that both partitioning methods crecompact three-dimensional subdomains that do not conany disconnected parts, and this leads to an avesurface–volume ratio~or, equivalently, communication–calculation ratio! of these subdomains that is approximatethe same.

The working memory of the CC processors is mularger than that of the PowerXplorer. Therefore on the

IN PHYSICS, VOL. 11, NO. 5, SEP/OCT 1997

-

es

t

e

all problems can be executed on a single processor, in ctrast to the PowerXplorer, where single-processor permance cannot be determined for high-resolution probleThe CC communication network and CPU are significanfaster than those of the PowerXplorer, which accountsthe fact that the kernel execution time for each problemshorter on the CC.

For the PowerXplorer the kernel execution times anot significantly decreased for more than 16 partitiowhereas in case of the CC increasing partitioning cardinity consequently leads to faster execution times, at leasthigh-resolution models. Since we have considered theperformance, we can expect that the absolute performaof the simulation code in a more realistic simulation~takingother phenomena into account! will be degraded comparedto the best case and, consequently, result in a relativbetter ‘‘scalability’’ of the code, since the amount of worper subdomain increases whereas the communicationtern and size remain unchanged.

IV. ‘‘SAMPLE’’ SIMULATION EXPERIMENTS

Since the main theme of this article is the concept ocomplex parallel simulation program, we will not presedetailed quantitative results with respect to the feasibility

Table V. The kernel execution times in seconds on the CC for four different mesh sizes and two different decomposition methods„GORB denotedby suffix G and RSB denoted by suffix R… versus the partitioning cardinality.

6114-G 6114-R 9412-G 9412-R 51501-G 51501-R 79556-G 79556-R

1 1.38e201 1.38e201 2.10e201 2.10e201 1.12e100 1.12e100 1.71e100 1.71e1002 8.68e202 ••• 1.27e201 ••• 6.23e201 ••• 9.20e201 •••

4 5.41e202 5.34e202 7.63e202 7.48e202 3.47e201 3.39e201 4.99e201 4.93e2016 6.26e202 ••• 7.61e202 ••• 2.39e201 ••• 3.47e201 •••

8 5.49e202 5.40e202 7.94e202 7.84e202 2.23e201 2.26e201 2.92e201 2.93e2019 4.85e202 ••• 7.14e202 ••• 2.17e201 ••• 2.72e201 •••

12 4.26e202 ••• 5.93e202 ••• 1.83e201 ••• 2.72e201 •••

16 3.74e202 3.99e202 5.02e202 4.88e202 1.83e201 1.87e201 2.64e201 2.37e20118 3.66e202 ••• 4.84e202 ••• 1.64e201 ••• 2.30e201 •••

24 3.73e202 ••• 4.36e202 ••• 1.17e201 ••• 1.80e201 •••

27 3.75e202 ••• 4.36e202 ••• 1.24e201 ••• 1.79e201 •••

32 3.98e202 4.40e202 4.42e202 4.66e202 1.15e201 1.19e201 1.62e201 1.66e201

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building a gravitational wave detector. Nevertheless,will illustrate the capabilities of the FE simulation prograusing a few ‘‘sample’’ experiments.

All simulations are carried out with full-fledged antenna models of high resolution~approximately 130,000elements! on the two high-performance distributed-memoplatforms discussed in Sec. III B. The antenna materialsembles CuAl~90–10!. Six resonant mass transducers amounted on the antenna in somewhat of an arrangemthat places the transducers at the pentagons on the lhalf of a truncated icosahedron; this was also done inTIGA project.27 Each transducer consists of two massThe first mass ofM15100 kg is attached to the antennaa spring that is described by the diagonal stiffness maK15diag(2.273109,2.2731010,2.2731010) N/m. The sec-ond mass ofM250.385 kg is attached to the first mass byspring with stiffness matrix K25diag(8.703106,8.703107,8.703107) N/m.

Figure 11 schematically depicts a transducer thaattached to the antenna. The principal resonance frequf r of the transducer system is equal tof r5780 Hz, whichcoincides with the principal quadrupole frequency of tideal detector~see Sec. III A!. The two masses are chosesuch that the amplitude of vibration detected at the anna’s surface will be amplified by two orders of magnitudThe addition of the multimode resonant transducers resin a significant splitting of the resonant frequencies ofsystem~data not shown!. Although the transducer modethat we have implemented are rather simplistic~pointmasses coupled with ideal springs!, they have already givenquite satisfactory results in several applications~see, forexample, Sec. IV B!.

A. Gravity

Earth’s gravity will induce internal strain in the antennThe sphere that is suspended is expected to deform duthis strain. We would like to quantify the amount of defomation that results due to the presence of gravity. In orto do this we perform the following simulation. At timzero we ‘‘switch on’’ gravity (g59.8 m/s2), and let theantenna fall down. The suspension rod is fixed at the toprevent the sphere from dramatically accelerating towathe Earth’s surface. The velocities in the spherequenched by a small damping factor in order to get ridthe kinetic energy in the system. When the systemcome to rest, we take a look at the displacement and stdistribution within the sphere. Figure 12 displays a conto

Figure 11. The spherical antenna with one transducer system moun

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-

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plot of the displacement field in thez direction within theantenna, after this deformation process. Figure 13 zoomon the suspension region, again showing a contour plothe displacement in thez direction. Overall, the spheremoves about 2 mm downwards, which results from the fthat the suspension rod is elongated by this amount in thzdirection. The displacement values within the sphere vfrom about 231023 to 331024 m when we traverse thesphere from the outside towards the suspension rod.other interesting view is provided by the vertical stra

Figure 12. Contour plot of the displacement in thez direction of thesuspendedCuAl sphere due to gravity (g59.8m/s2). The displacementvalues of the various contours correspond to (1)20.35, (2)21.96, (3)22.18; further contours do not differ significantly from the value of cotour (3); all values are in 1023 m. The minus signs indicate displacements towards the Earth’s surface.


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(]uz /]z) within the antenna, which is shown in Fig. 14. Aexpected, the suspension rod is subjected to the higstrain, as is the region where the sphere and rod maketact.

A shortcoming of our simulation program is thatrelies on a linear finite element solver. As a consequenwe are not able to model nonlinear effects, such as so caviolin modesin the suspension rod, or show how the intenal strain of the sphere affects the eigenmodes. Incorpoing these into our simulation program is a future challen

Figure 13. The same contour plot as that displayed in Fig. 12, but zooin on the suspension region. The displacement values of the varioustours correspond to (1)21.60, (2)21.82, (3)22.16, (4)22.18; furthercontours do not differ significantly from the value of contour (4);values are in 1023 m. The minus signs indicate displacements towarthe Earth’s surface.


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-

B. Seismic noise

The suspension system of the antenna must attenuate vtions at the principal resonance frequency to a factor of 3dB.28 If we let seismic noise enter the system via the supension, how does it appear at the transducer readout?following experiment is carried out. The suspension poin~connections to the outside world! of the suspension rod aredriven by forced oscillation in thez direction. That is, thezcoordinate of the displacement fielduz(t) of each suspen-sion point oscillates as follows:

-

Figure 14. The vertical strain(]uz /]z) within the antenna. High strainvalues (dark areas) are found in the suspension rod, that is carryingentire sphere, and near the contact region of the sphere and the rod.strain in the sphere itself is much lower (light colored areas).

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@An cos~2p f nt !1Bn sin~2p f nt !#. ~18!

The amplitudesAn andBn and the frequency spectrumf nmodel the seismic noise. In the following experiment tnarrow-banded noise spectrum depicted in Fig. 15 wused. Thez displacement versus time, which is driving thsuspension points according to this spectrum, is displain Fig. 16.

Figures 17~inner mass! and 18~outer mass! show theradial displacement of the two transducer masses startint50 in one of the six transducers. The maximum amplituin the noise spectrum~Fig. 15! has the same order of magnitude as the maximum amplitude of the outer mass. Csequently, it is of paramount importance that all seismvibrations in the target frequency range are attenuatedlow the expected amplitude of vibrations that are inducby gravitational radiation in order to assure that we havdecent signal-to-noise ratio.

Figure 15. Narrow-banded seismic noise around the principal quadrupfrequency.

Figure 16. Thez displacement for the suspension point driven by tnoise spectrum of Fig. 15.

t

-

C. Chirps

A so-called gravitational radiation ‘‘chirp’’ will be emitted~expected theoretically! just before coalescence of a binaneutron-star system.29 The effects of this phenomenon othe detector can be modeled by applying a time-vary‘‘chirp force’’ f(t) on the antenna, which acts along a puquadrupole field, according to

fT~ t !}h~x,2y,0!, ~19!

with h the second-order time derivative of the deformatiof space, derived in a Newtonian approximation.

Let us consider a chirp that enters the antenna at tt520.04 s. Figure 19 models its time varying amplituh. The normalization of the chirp signal has an ordermagnitude that is representative for an event in the Vicluster. Since the antenna responds purely linearly,shape of the response signal resulting from interaction wany other identically shaped incoming signal will be eactly the same.

Figure 17. Time trajectory of the inner transducer mass on one traducer. The system is forced into oscillation by the seismic noise specof Fig. 15.

Figure 18. Time trajectory of the outer transducer mass on one traducer. The system is forced into oscillation by the seismic noise specof Fig. 15.


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Figure 20 shows the evolution of the energy in tantenna from the moment that the chirp enters until itpassed. The energy of the chirp gets deposited in thetenna. After the chirp has passed, the energy stabilizes~att50 s!.

The response to this chirp can again be measured.ure 21 shows the time trajectory in thex direction of oneof the outer transducer masses. We observe the('10219 m) displacements of the transducer masses.

V. CONCLUSIONS

In this article we have sketched the complete path alwhich a complex high-performance simulation was devoped. We were able to identify the problem definitiomodeling choices, the code design for parallel systevalidation, and experimentation. In order to realize sucsystem, it is mandatory that we have knowledge of a vety of disciplines, like computer science, numerical maematics, and~computational! physics. It is our feeling tha

Figure 19. The time dependent amplitude h¨ of a chirp initiating at timet520.04s. The chirp ends at t50.

Figure 20. The time evolution of the total energy deposited by the chirFig. 19. After t50 the energy stabilizes. Note that we have used ascale on the energy axis.


-

-

,

the whole process that is described above is typical fordevelopment of new simulation systems that utilize higperformance methodology and therefore can serve asexample ofcomputational science, which inherently is aninterdisciplinary research field. Mixing the best of seveworlds allowed us to realize a full-fledged simulation sytem that meets our purposes.

In the future we will focus on embedding the simultion program within the experimentation cycle of GRAILreferred to as ‘‘simulation in the loop’’ in the Introduction

ACKNOWLEDGMENTS

The work presented in this article was made possiblefunding from NWO of the GRAIL project pilot study. Thecontributions of J. de Rue to Sec. I B are gratefully aknowledged.

REFERENCES

1. C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravi-tation ~Freeman, San Francisco, 1973!.

2. http://qv3pluto.leidenuniv.nl/grail/grail.html. GRAILstands for ‘‘gravitational radiation antenna in Leiden

3. The GRAIL Project Team. GRAIL: A proposal forgravitational radiation antenna in The NetherlandTechnical Report No. NIKHEF/95-005, Nationaal Instituut voor Kernfysica en Hoge-Energie Fysica, 199

4. K. S. Thorn,300 Years of Gravitation, edited by S. W.Hawking and W. Israel~Cambridge University PressCambridge, 1987!.

5. N. Ashby and J. Dreitlein, Phys. Rev. D12, 336 ~July1975!.

6. J. A. Lobo, Phys. Rev. D52, 591 ~July 1995!.7. S. M. Merkowitz and W. W. Johnson, Phys. Rev. D51,

2546 ~March 1995!.8. S. M. Merkowitz, Ph.D. thesis, Lousiana State Unive

sity, 1995.9. J. Weber, Phys. Rev.117, 307 ~1960!.

10. http://www.roma1.infn.it/rog/rogmain.html. Romgravitational wave experiment, NAUTILUS, 1996.

11. Dutch organization of scientific research.

Figure 21. Thex displacement of one of the outer transducer masses.transducer already ‘‘notices’’ the chirp’s presence before t50.

tiveni-n,

.

s,

ry

96.

’s

r

d

R.ndry,

on,-

or

den-o-3.li-r-

tt.

.

12. P. M. A. Sloot and J. Reeve, Camas-tr-2.3.7 execureport on the Camas workbench; Technical report, Uversity of Amsterdam and University of SouthamptoOctober 1995.

13. B. D. Kandhai, P. M. A. Sloot, and J. P. Huot, inHigh-performance Computing and Networking, edited by B.Hertzberger, H. Liddel, A. Colbrook, and P. M. ASloot, No. 1067 in ISBN 3-540-61142-8~Springer,Berlin, 1996!, pp. 269–275.

14. P. M. A. Sloot, Eurosim ’95, Simulation CongresAmsterdam, The Netherlands, 1995, pp. 29–44.

15. A. E. H. Love,A Treatise on the Mathematical Theoof Elasticity ~Dover, New York, 1944!.

16. J. de Rue, MS. thesis, University of Amsterdam, 1917. T. M. Bell, W. M. Visscher, A. Migliori, and R. A.

Reinert, J. Acoust. Soc. Am.90, 2154~1991!.18. E. Andersen, Z. Bai, and C. Bischof, LAPACK User

Guide, 2nd ed. 1994.19. O. C. Zienkiewicz and R. L. Taylor,The Finite Ele-

ment Method~McGraw-Hill, New York, 1994!.20. T. J. R. Hughes,The Finite Element Method, Linea

and Dynamic Finite Element Analysis~Prentice-Hall,Englewood Cliffs, NJ, 1987!.

21. R. H. MacNeal,Finite Elements: Their Design an

Performance~Dekker, New York, 1994!.22. A. Geist, A. Beguelin, J. Dongarra, W. Jiang,

Manchek, and V. Sunderam, PVM 3 User’s Guide aReference Manual, Oak Ridge National LaboratoMay 1993.

23. G. Fox, M. Johnson, G. Lyzenga, S. Otto, J. Salmand D. Walker,Solving Problems on Concurrent Processors~Prentice-Hall, Englewood Cliffs, NJ, 1988!,Vol. 1.

24. H. D. Simon, Partitioning of unstructured problems fparallel processing. Comput. Syst. Engr.2, 135~1991!.

25. G. Lonsdale, J. Clinckemaillie, S. Vlachoutsis, J. F.Ronde, P. M. A. Sloot, N. Floros, and J. Reeve, Coference on Supercomputing Applications in the Autmotive Industries of the 26th ISATA, September 199

26. Interdisciplinary Center for Complex Computer Facities. http://www.wins.uva.nl/research/ic3a/. Amstedam, 1996.

27. W. W. Johnson and S. M. Merkowitz, Phys. Rev. Le70, 2367~April 1993!.

28. T. L. Aldcroft, Rev. Sci. Instrum.63, 3815~1992!.29. L. Blanchet, B. R. Iyer, C. M. Clifford, and A. G

Wiseman, Class. Quantum Grav.13, 575 ~1996!.


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