Parallel modeling of macroscopic surface flows

M. V. Avolio, G. M. Crisci, D. D’Ambrosio, S. Di Gregorio, G. Iovine, V. Lupiano, R. Rongo, W. Spataro, A. Trunfio

CAPI 2004Calcolo ad Alte Prestazioni in Italia

Milano, 24-25 novembre 2004

University of Calabria High Performance Computing Center

CNR - IRPI

E M P E D O C L E S R E S E A R C H G R O U PE M P E D O C L E S R E S E A R C H G R O U P

Contents

Cellular Automata (CA)

Modelling macroscopic surface flows by CA

The SCIARA, SCIDDICA and PYR models

Parallel Genetic Algorithms (PGAs)

HPC Resources & Performance

Discussion

Cellular Automata (CA) CA are a paradigm of parallel computing:

a space, partitioned in identical cells, each one embedding an identical finite automaton

The input of the finite automaton is given by the states of the “neighbouring” cells.

System’s evolution is given by local interactions among its constituent parts:the cells change state simultaneously according to the finite automaton transition function

Macroscopic surface flows

Phenomena involving flows can be modeled in terms of “local interactions” by CA: each cell must be related to a portion of the space on

which the phenomenon evolves the cell state must consider each characteristic, relevant to

the evolution of the system and relative to the space portion corresponding to the cell

the state variations must model the effect of mass exchanges among neighbouring cells

Hypothesis: debris, lava and pyroclastic flows are phenomena that can be approximately described in terms of local interactions

Modelling macroscopic surface flows by CA (1/5)

The set Q of the states of the cell is decomposed in “substates”:

Q=Q1Q2…Qn

Modelling macroscopic surface flows by CA (2/5)

The transition function, :QmQ, determines the change of the cell state

It is decomposed in “elementary processes” p :

The elementary processes are applied sequentially according a defined order. Different elementary processes may involve different neighbourhoods

Flows of some quantity, present in the cell and expressed as substate, must be described in terms of local interactions and minimize locally unbalance conditions

Modelling macroscopic surface flows by CA (3/5)

A set of physical and/or empirical parameters bound the model to the reality to be simulated; examples are: the length of the cell side (in case of square cell), the

length of the apothem (in case of hexagonal cells) the time corresponding to a CA step (the CA clock)

External influences account for phenomenon characteristics that cannot be modeled in terms of local rules; an example is the emission of lava from the volcano’s vents

Modelling macroscopic surface flows by CA (4/5) The minimization algorithm (Di Gregorio & Serra, 1999) is

the core of the models SCIARA and SCIDDICA. It computes the outflows from the central cell to the other

neighbouring cells in order to minimise the differences of a quantity q in the neighbouring cells:

i<j|qi–qj|

Modelling macroscopic surface flows by CA (5/5)

The minimization in pairs algorithm (Crisci & al., 2003) is the core of the PYR model, for pyroclastic flows simulation

SCIARA: Simulation by Cellular Interactive Automata of theRheology of Aetnean lava flows

SCIARA = <R, X, V, S, P, , >

• R = {(x, y)| x, y N, 0 x lx, 0 y ly} is the set of points with integer co-ordinates in the finite region, where the phenomenon evolves. N is the set of natural numbers.

• VR specifies the lava source cells: the vents.

• X = {(0,0), (0,1), (0,1), (1,0), (-1,0), (-1,1), (1,-1)} specifies the neighbourhood of a cell (the “central” cell). It identifies the geometrical pattern of the cells, which influence the cell state change and consists of the central cell (index 0) and its six adjacent cells (indexes 1, 2 … 6).

0,0

0,1-1,1

-1,0

0,-1

1,0

1,-1

0

2 1

4

5

3

6

The model SCIARA (1/4)

The model SCIARA (2/4) Q is the finite set of cell state

Qa is the altitude of the cell Qth is the lava thickness in the cell QT is the lava temperature QFth represents outflows Qx and Qy are the co-ordinates of the center of mass Qv is the velocity of the center of mass QFx and QFy are the outflows co-ordinates QFv is the outflow velocity

P is the set of parameters of the model pa is the apothem of the cell pt is the CA clock pc is the lava cooling parameter padh,v, padh,s, pT,v and pT,s are parameters also related to lava cooling

The model SCIARA (3/4) :Q7Q is the deterministic transition function:

I1 determination of the lava flows (by applying the minimization algorithm)

T1 determination of the lava flows displacement by applying the “velocity formulae”

T2 determination of the new values of thickness, temperature, velocity and co-ordinates of the center of mass of the lava inside the cell

T3 lava cooling by irradiation effect, and solidification

:VNQthQth specifies the emitted lava from the lava vents at each CA step (N is the set of natural number)

Simulation steps of 2002 Etnean lava flows in the NE flank, corresponding to the lava fields at the end of the first, fourth and ninth day; maximum extension was reached before the ninth day.

Fig.1 shows the real lava flow at the maximum extension. Fig.2 shows the corresponding simulation. Comparison between real and simulated event is satisfying, if we compare

involved areas, temperatures and lava thicknesses.

Fig.1

Fig.2

The model SCIARA (4/4)

The model SCIARA was developed for the simulation of type “aa” lava flows It has been applied to the simulation of many cases of

study occurred in the etnean area, since 1669!

The model SCIDDICA (1/2)

The model SCIDDICA was developed for the simulation of mud and debris flows It has been applied to

landslides of different type: Mt. Ontake (Japan, 1984), Tessina (Italy, 1992), Sarno (Italy, 1998) and Valle Caudina (Italy, 1998)

The model SCIDDICA (2/2)

The model PYR

The model PYR was developed for the simulation of pyroclastic flows It has been applied to the simulation of some cases of

study as the 1991 Pinatubo eruption (Philippines) and the May 1996 by the Soufriere Hills volcano in Monserrat (Lesser Antilles).

Real vs Simulated comparison

The application results of this PYR to the 1991 Pinatubo eruption, are very encouraging. If the real event (right) is compared with the simulated event (left), a good correspondence for the areas covered by the pyroclastic flow can be evidenced in the precision limits of the cell side.

Parallel Genetic Algorithms Above SCIARA, SCIDDICA and PYR simulation results have

been obtained after a “calibration phase” in which a “reliable” value was assigned to each model parameter

In the past, such a phase was performed manually by assigning iteratively reasonable values to the parameters and qualitatively evaluating the results

Subsequently Genetic Algorithms was employed as an optimization technique to improve the models calibration In a first attempt this was done on a sequential computer (Iovine

& al., 2003) , subsequently on parallel architectures (Spataro & al. 2004; D’Ambrosio & al, in press)

HPC Resources Beowulf Cluster

16 processors Pentium 4 (1.4GH)

Performance: unknown Nec TX7

16 processors Itanium (1Ghz). Performance: 64 GFLOPS

HP Sierra XC 600 24 processors Itanium2

(1.5Ghz) Performance: 144 GFLOPS

HP Alphaserver SC 64 processors Alpha (1.25Ghz) Performance: 160 GFLOPS

Master-Slave Genetic AlgorithmMaster-Slave GA {

[MASTER]t=0initiliaze population Pop(t)send n’/S individuals to each SLAVE

[SLAVE]receive n’/S individualsevaluate n’/S individualssend the n’/S computed fitness values to the MASTER

WHILE (NOT(STOPPING CRITERION)) {[MASTER] receive n’ computed fitness values from SLAVEst=t+1create Pop(t) by applying selection, crossover e mutationsend n’/S individuals to each SLAVE

[SLAVE]receive n’/S individualsevaluate n’/S individualssend the n’/S computed fitness values to the MASTER

}}

Performances (speed-up)

Speed Up (ft=0.1 secs)

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0 5 10 15

slave procs

spee

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ideal

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Speed Up (ft=1 secs)

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Speed Up (ft=0.001 secs)

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Speed Up (ft=0.01 secs)

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slave procs

spee

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Discussion Our interdisciplinary research group Empedocles

extended the classical notion of CA and developed an empirical approach for modelling and simulating complex macroscopic phenomena of type surface flows, which are very difficult to be managed with differential equation systems.

The response was successful for surface flows of very different nature: lava flow, debris flow and pyroclastic flows. The simulations capture the main features of real phenomenon.

The values of model parameters are very important, (little changes produce very different simulations, that is typical of non-linear systems), they cannot always be determined directly, e.g. by physical measures: they are commonly selected by comparing the results of simulations with the global behaviour of the real phenomenon

Discussion

A model cannot be considered reliable without a correct estimation of the values of the parameters

PGAs were utilized as optimization algorithms for the calibration phase with satisfying results both in terms of the comparison with the real events chosen as case of study and in terms of speed up

The definitive judgment on the validity of the model depends on the comparison between a large set of simulations and the real phenomenon,

When the model is validated, it may be used effectively for hazard forecasting purpose in similar cases.

Thanks for your attention