Introduction to Modulef: simulation of a Quartz Crystal Microbalance displacement J.-M Friedt, L. Francis (November 13, 2001)
1
Definition of the structure
We define a 3D structure by first defining the 2D projection in the (X, Y ) plane of the structure using apnoxx, and expand the third dimension using apn3xx. We will first focus on the definition of the 2D structure, and later develop the use of apn3xx which allows defining the properties of each surface/node of the final volume.
1.1
2D definition using apnoxx
0.02s
221
POINTS
221
NOEUDS
408
ELEMENTS
408
TRIANGLES
27/10/01
0
le 27:10:1901 a 12h 21mn.
le 27:10:1901 a 12h 21mn.
T2IN.NOPO
0.03s
friedtj
Fortran 3D couleur 3.04
Fortran 3D couleur 3.04
MODULEF :
EXEMPLE 3D
TROU(S)
COIN BAS GAUCHE :
-2.7500E-03-3.7976E-03
COIN HAUT DROIT :
Utilisateur: friedtj
Utilisateur: friedtj
2.7500E-03 3.7976E-03
MODULEF :
friedtj
EXEMPLE 3D
27/10/01
T2OUT.NOPO
288
POINTS
288
NOEUDS
256
ELEMENTS
256
QUADRANGLES
1
TROU(S)
COIN BAS GAUCHE :
-7.7000E-03-1.0633E-02
COIN HAUT DROIT :
7.7000E-03 1.0633E-02
Dessin 1
Dessin 1
Figure 1: The two basic 2D structures required for defining the internal electrode coated part of the crystal and the external, non-coated part of the crystal resonator. The two 2D structures developed for representing the QCM are presented here (figure 1). We have chosen to divide the QCM in two concentric circular pieces in order to be able to differentiate the gold coated counter electrode from the gold-free part of the QCM and hence simulate the effect of a finite sized electrode. We first define one quarter (the bottom left) of the disc, rotate three times by 90o and paste the resulting pieces in order to build the disc. We have chosen to rotate and paste the pieces in apnoxx rather than in apn3xx in order to be able to use the sub-domain and element numbering functions of the sub-processor MA23 of apn3xx which would not have been available otherwise (since the rotation and gluing steps are made after the addition of the third dimension in apn3xx). One difficulty comes from the fact that the two elementary pieces do not have the same number of sides, and thus require different mesh base elements and thus different finite elements. The inner element has an odd number of sides and requires a triangular mesh, while the outer element has an even number of sides and require a mesh element with 4 sides.
1.2
3D definition using apn3xx
Our choice in the number of sub-parts for creating the final volume and the order in which we join them together (using apn3xx rather than in apnoxx) results from our requirement of defining
1
’EXEMPLE 3D ’ COURBES 1 $ IMPRE COURBE01(X,Y)= X**2+Y**2-0.0025**2; FIN ’POIN ’ 1 3 $ IMPRE NPOINT $ $ NOP NOREF(NOP) X(NOP). Y(NOP). $ 1 1 0.000000E+00 0.000000E+00 2 1 0.000000E+00 -.002500E+00 3 1 0.002500E+00 0.000000E+00 ’LIGN ’ 1 3 $ IMPRE NDLM $ $ NOLIG NOELIG NEXTR1 NEXTR2 NOREFL NFFRON RAISON $ 1 10 1 2 0 0 0.100000E+01 2 9 2 3 1 10 0.100000E+01 $ 5 -> 9 3 10 3 1 0 0 0.100000E+01 ’TRIH ’ 1 0 1 3 1 $ IMPRE NIVEAU NUDSD NBRELI NS1L $ LISTE DES LIGNES DU CONTOUR : 1 2 3 1 0 1 $ IMAX NQUAD 3
’REGU
’ 1
0
1
’ROTA 1 1 2 0 0 0.90000E+02 0.00000E+00 0.00000E+00 ’RECO’ 1 1 2 3 0.02000E-03 0 0 ’ROTA 1 3 4 0 0 0.18000E+03 0.00000E+00 0.00000E+00 ’RECO’ 1 3 4 5 0.02000E-03 0 0 ’RENC’ 1 5 6 ’SAUV 1 6 0 T2IN.NOPO $ NOM FICHIER ’FIN
’ $ IMPRE NIVEA1 NIVEA2 $ NBNNF NBNNSD $ TETA. X. Y. 0
$ IMP NIV1 NIV2 NIV3 EPS $ NBNNF NBNNSD ’ $ IMPRE NIVEA1 NIVEA2 $ NBNNF NBNNSD $ TETA. X. Y. 0
$ IMP NIV1 NIV2 NIV3 EPS $ NBNNF NBNNSD
’ $ IMPRE NINOPO NTNOPO
’
Table 1: The definition of the inner 2D structure (T2INNOPO.DATA) ’EXEMPLE 3D COURBES 1 $ IMPRE COURBE01(X,Y)= X**2+Y**2-0.0025**2; COURBE02(X,Y)= X**2+Y**2-0.007**2; FIN ’POIN 1 5 $ IMPRE NPOINT $ $ NOP NOREF(NOP) X(NOP). Y(NOP). $ 1 1 0.000000E+00 0.000000E+00 2 1 0.000000E+00 -.002500E+00 3 1 0.002500E+00 0.000000E+00 4 2 0.000000E+00 -.0070000E+00 5 2 0.007000E+00 0.000000E+00 ’LIGN 1 7 $ IMPRE NDLM $ $ NOLIG NOELIG NEXTR1 NEXTR2 NOREFL NFFRON RAISON $ 1 4 1 2 1 0 0.100000E+01 2 5 2 3 1 10 0.100000E+01 3 4 3 1 1 0 0.100000E+01 4 9 2 4 0 0 0.100000E+01 5 9 4 5 2 10 0.100000E+01 6 9 5 3 0 0 0.100000E+01 7 9 2 3 1 10 0.100000E+01
’
’QUAC 1 2 2 4 1 $ LISTE DES LIGNES DU CONTOUR : 4 5 6 7 9 1 ’ROTA 1 2 3 0 0 0.90000E+02 0.00000E+00 0.00000E+00 ’RECO’ 1 2 3 4 0.02000E-03 0 0 ’ROTA 1 4 5 0 0 0.18000E+03 0.00000E+00 0.00000E+00 ’RECO’ 1 4 5 6 0.02000E-03 0 0 ’RENC’ 1 6 7 ’SAUV 1 7 0 T2OUT.NOPO $ NOM FICHIER ’FIN
’
’
$ $ $ $
5 5 5 5
-> -> -> ->
9 9 9 9
’ $ IMPRE NIVEAU NUDSD NBRELI NS1L
$ 5 -> 9 ’ $ IMPRE NIVEA1 NIVEA2 $ NBNNF NBNNSD $ TETA. X. Y. 0
$ IMP NIV1 NIV2 NIV3 EPS $ NBNNF NBNNSD ’ $ IMPRE NIVEA1 NIVEA2 $ NBNNF NBNNSD $ TETA. X. Y. 0
$ IMP NIV1 NIV2 NIV3 EPS $ NBNNF NBNNSD
’ $ IMPRE NINOPO NTNOPO
’
Table 2: The definition of the outer 2D structure (T2OUTNOPO.DATA) a lower circular electrode with dimensions smaller than the total lower surface of the quartz resonator. The choice of the number of distinct sub-domains results from the necessity of using different mesh procedures depending on the region we consider (triangular mesh for a region defined by an odd number of sides, rectangular mesh for a region defined by an even number of sides). The definition of the various reference numbers of the different surfaces, and the translation from the references numbers defined in the 2D structures to the reference numbers in the 3D structures, is realized by the module ma23xx which is described in great details in the Modulef manual page 3-14, node 57 1 . Since we already rotated and pasted the pieces together in the previous apnoxx step, this apn3xx step is mainly focused with re-numbering the elements and the sub-domains, and pasting the final two pairs of 3D pieces together in order to define the QCM with a finite counter electrode (while the sensing surface is fully coated with the grounded electrode). The counter electrode is defined by reference number 4, the sensing electrode (covering the whole top surface of the QCM) as reference 3, and the other (non-coated) parts of the crystal are referred to by number 2 (reference number 1 disappeared when the two sub-elements were glued together). 1 we will refer to parts of the Modulef manual by their web page reference: 14/node57.html
2
3-14 here means Guide3-
MODULEF :
friedtj
MODULEF :
Fortran 3D couleur 3.04
Fortran 3D couleur 3.04
27/10/01 T3.NOPO
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 3 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 3 333 3 3 3 3 3 33 3 3 3 3 3 3 33 3 3 33 3 3 3 3 3 3 3 33 3 3 33 3 3 33 3 33 33 3 33 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 33 3 343 43 3 43 34 4 3 4 3 3 3 3 3 3 3 4 33 4 34 33 33 333 3 3 3 33 3 3 3 3 33 3 3 3 4 33 3 3 3 4 3 4 4 3 3 3 3 3 3 3 4 3 3 3 4 3 43 4 3 4 43 4 4 3 334 3 3 3 3 4 3 3 33 443 4 3 3 3 3 3 3 3 3 3 3 3 3 4 343 43 3 34 334433 4 43 43343 433 34 343 4 3 4 3 3 3 3 33 4 3 43 3 34 3 343 4 3 3 3 4 3 3443 3 43 4 3 3 3 3 3 4 434 33 3 44334 44 3 3 3433 34 3 3 3 3 3 3 4 4 4 3 4 3 3 33 3 33 3 3 3443 33 3 4334 34433 34 443 3 3 3 3 4 4 3 34 33 3 433 34 3 4 3 3 4 3434 34 433343 33 34 33 4 4 4 34 34 4 4 4 3 33 33 3 34 3 3 3 4433 433 434 343 43 3 3 4 34 43 4 4334 4344 33 4 34 3 4333 4 4 3 3 3 3 3 4 3 3 3 3 4 3 3 3 3 4 4 4 3 3 4 3 334 34 43 3 3 33 4 34 3 3 3 3 443 3 43444 33 4 43 43 4 3 4 3433 343 43 34443 43434 344343 4 3 3 34 34 3 3 3 3 4 3 4 3 3 33 3 4 33 34 44 43 4 3 434 3 4 3 3 43 3 3 4 333 434 3343 34 3 3 3 43 3 4 33 4 3 4333 443 343 3 344 3 343 4 344 4 34 3 3 3 3 43 3 4 4 3 334 343 34 433 3 3 343 3 3 43 3 343 3 4 3 344 3 3 4 334 3 44 34 343 434434334344 3 334 4 443434 34 3 34 3 4 33 3 34 44 4 34 3 4 43 4 33 4 33 4 3 3343 4 43 3 4 3 4 43 3 3 3 3 3 34343 43 4 3 3 4 3 34 33 44 34 4 3 4 4 44 3434 3 34343 43 3434 433 4 434 34 43 43 4 3 43443 3 3 443 43334 344 433 3 43 344 3 3 4 3 43 3 4 43 4 3 443 343 343 4 3 4 344 3 3 3 44 4 43 3 344 334 344 33 4 344 4343 344334 43 3 3 3 4 3 43 34 44 3 4 33 43 3 4 4 4 34 3 34434434344334 4 3 33 33 344 3434 3 434 4 4 3 4 3 4 4 3 3 33443 3 4 4 34 334 4 344 4 3 3434 43 4 4 3 3 44 3 4 3 3 43 434 3 3 3 4 3 4 3 3 3 4 4 4 4 4 4 43 43 34 443 33 3 4 4 3 3 43 43 34 3 4 3 4 4 4 4 3 3 4 4 3 43 4 4 43 34 34 3 43 4 3 4 34343 4 43 434 4 3 34 343 3 34 443 4 344 3 4 43 3 4 33 34 343 443 34 4 43 3333 4 3 33 4 334 34 3 34 3 4 4 43 3434 434 34 3 4344 334 4 3 343 444334 43 4 3434 34 4 334 3 3 4 3 4 4 3 4 3 34 3 3 4 3 3 4 3 4 334 4 43 4 4 34 3 3 4 33 4 3 44333 43 44 4 34 434 43 4 3 4 3434 3 3 43 334 434 3 443 4 34 4 3 4 3 3 43 4334 4 3 4343 3 433 3 4 4 3 43 434 4 4 3 4 44 34 34 43 3 434 43434 3 4 343434 43 434343443 3 3 4 3 34 3 3 4 4 4334 4 34 4333 344 3 4443444 44 3 3 3 3344 3 3 43 4 43 4 3 3 4 4 3 4 4 3 4 4 3 4 34 3 3 4 33 4 4 43 4 4 4 3 3 4 4 3 3 3 4 4 3 3 3 3 4 4 4 4 3 4 4 4 3 4 4 3 3 3 3 4 4 3 4 43 43 4 4 4 4 3 3 3 4 3 4434344 343 43 3 4 4 434 433 3434 34 3 4334343434 34 43 3 3 3434334 344 3 4 3 4 3 43 3434 34 343 3 4 33 4 4 334 4 3 4 4 3 443 4 434 343 434 4 4 3 443 43 3 4 4 3 43 4 4 3 4343 4 44 3 3 3 34 4 4343 3 3 4 4 4 3 4 3 43 3 4 3 343 3 3 3 4 43 4 4434 3 34 3 44 343 44 43 3 44 43 4 4 343 3 4 3 4 443 434 4343 4 4343 4 3343 3 4 3 434 43 434 3 3 3 33 4 4 4 4 3 34 4434443344 43 34 4 3 443 3 44 3 4 3 4 3 43 4 4 3 34 3 4 4 4 43 4 3 444 34 34 3 44 34 4 3 4 3434 43 4434 34 34 43 34 43 43 3 43 4343 4343434 3 34 43 343 4 3 34 4 4 4 3 4 34 34 3 4 333 4 34343 4 344334 434 4 3444 44 4 4 4 4 3 3 4 3 43 4 4 4 4 43 3 3 4 3 43 4 4 34 4 34 43 44 4 4 34 3 43 4 3 43 4 43 34 3 4 43 3 34 43 4 34 34 43 44 3 4 3 4 3 4 3 34 3 43 4 4 434 4 4 4 3 4 4344 44 34 4 3 44 3 3 3 4 3 3 44 4 3 4 43 4 4 44 4 4 4 4 34 4 4 4 3 4 43 34 4443 443 44 3 3 3 4 4 3 43 4343344 3 4 43 44 3 4 34 4 4 3 3 34 4 4 3 3 3 3 4 44 34 3 4 3 4 3 4 4 4 443 44 34 44 4 4 43 4 44 4 4 34 4 4 443 4 43 3 4 3 3434 3 3434 3 34 4 34 3 4 4 4 4 3 3 3 3 4 3 34 3 4 3 4 4 3 34 4 43 4 3 4 4 43 3 3 4 3 44 4 434 43 3 43 3 3 43434 4 3 43 43 3 4 4 43 434 43 4 4 3 4 4 4 43 43 43 4 44 4 444334 4 43 43 4 343 4 4 3 3 3 3 4 4 4 3 4 4 3 3 4 4 4 3 4 3 4 4 3 4 3 4 4 3 3 3 4 3 4 4 4 43 4 4 3 3 4 4 3 4 34 4 34 4 3 44 4 4 3 3 4 433 4 43 4 33 3 4 4 34 4 4 434 34 4 3 3 4 34 34 434 4 43 34 4 3 4 4 34 3 3 4 3 4 3 4 4 34 4 33 4 4 4 3 4 4 4 43 44 3 4 3 4 4 34 4 4 4 4 3 44 3 4 34 4 3 44 43 4 34 4 3 44 3 4 3 43 4 4 3 4 34 3 4 34 34 4 3 4 3 4 34 3 4 4 3 43 43 4 43 44 4 4 3 4 4 4 443 3 4 4 3 43 4 43 3 4 4 4 34 3 4 4 4 4 4 44 34 4 3 4 4 4 4 3 4 3 4 3 3 4 4 4 44 4 4 4 43 3 4 34 4 3 4 4 4 4 3 4 4 3 4 4 34 4 43 4 4 4 4 4 3 4 4 3 3 4 3 4 4 4 4 4 43 4 3 3 4 4 34 4 34 3 3 3 4 34 4 4 4 3 3 4 4 4 3 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3
3
3
3
3
POINTS
7880
FACES
2656
VOLUMES
1632
PENTAEDRES
1024
HEXAEDRES COMPOSANTE(S) O
1
COMPOSANTE(S) F
2 2 2 2
:
-7.000E-03-7.000E-03 POINT MAXIMAL
0.00 2
:
2 7.000E-03 7.000E-03 4.000E-04 2 2 2
OBSERVATEUR CARTESIEN :
2 1.069E-02 6.173E-03 7.328E-03 POINT REGARDE 0.00
2 2
2.000E-04
Utilisateur: friedtj
Utilisateur: friedtj
0.00
:
2
2 OBSERVATEUR SPHERIQUE 30.0
30.0
:
2
1.426E-02
2
OUVERTURE : 10.0 2 2 2 2
TOUTES ARETES DE LA PEAU
Dessin 1
Dessin 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 22 22 2 2 22 2 2 2 22 2 22 2 2 2 2 22 22 2 2 2 2 2 22 22 2 2 2 2 2 2 2 22 2 2 2 2 2 22 22 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 22 22 2 2 2 2 2 2 22 2 2 2 1 1 1 2 2 2 2 2 2 2 2 1 1 21 1 2 2 2 2 2 2 2 1 12 1 1 1 121 1112 1 1 2 2 2 1 2 2 2 1 2 1 1 1 2 1 21 2 2 2 2 2 2 11 1 1 11 1 1 1 1111 1111111 1 12 1 2 2 2 2 1 2 1 1 111 111 1 111 11111111 2 11 1 11 1 11 11 11 2 11 1 1 1 2 2 2 2 2 11 1 1 11 1 2 2 2 2 2 1111 11 1111111111 111 1 1 11 1 1111 1 1 1 1 1 1 2 2 1 12 2 1 11 1 111 11111 111111 111111 2 111 11111 1111 2 1 1 1 2 1 2 1 1 2 2 1 1 1 111 2 1 1 2 11 111111 111 111 1 1 11 1 1 111 1 1 2 2 11 1 11 2 2 1 1 111111 11111 1 11 1111 11 1111 111111111 11 1 11 1 11 1 11 111 11 1 2 11 1 11 11 1 1 2 2 2 1111111111 11 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 11 1 111 11 11 1 111 11 1 1 1 1 1 11 1 1 1 11 1 11 1111 111 11 22 2 2 22 2 2 11 1 1 2 2 2 22 1 11 1 1 11 1 1 1 11 1 11 1 111111111 1111111 11 1 11111 1 1 1 1 1111111 11 1 11 111 111 11 1 11 11 11 111 11 1 1 1111 11 1 1 1 22 2 11 1 2 1 1 111 111 1 1111111 111 1 111 11 2 1 22 2 2 2 22 1 111111111 1111 1 1111 111 1 1 11 1111111111111 2 1 11 2 1 11 1 1 1 2 11 1 1 1 1 1 1 111 2 111 111111111 1 11 11 111 111 11 1 1 1 1 1 1 1 22 1 1 11 1 1 1 11 11 1 11 22 1 111 11 1 11111 1111 111 11 111 1 22 1 1 1111 1 1 1 2 2 1 1 1 1 1 11 1 11 2 2 1 1 11 1 1 1 1 1 1 11 1 11 1 111111 1111 111 111 2 2 2 1 1 1 11 1 11 11 2 2 1 1 1 2 1 1 2 2 1 11 1 1 1 111 1 11 111 1 1 11 1 11 1 1 1 11 11 2 2 2 2 11 1 1 2 1 1 11 1 1 11 1 1 1 1 2 11 1 1 1 1 11 111 1 1 1 2 2 1 2 2 2 1 11 11 1 1 1 111 11 2 2 2 1 2 11 1 1 1 2 11 111 2 2 1 1 1 1 2 2 2 1 1 2 2 1 1 11 1 1 1 1 1 1 1 1 11 1 1 11 2 2 2 12 1 2 1 2 1 2 1 1 2 2 2 2 2 2 21 1 1 2 2 2 1 121 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 22 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
le 27:10:1901 a 12h 23mn.
le 27:10:1901 a 12h 23mn.
1
POINT MINIMAL
T3.NOPO
0.02s
0.02s
2385
REFERENCE (TOUT)
Z
O
friedtj
27/10/01
2385
POINTS
7880
FACES
2656
VOLUMES
1632
PENTAEDRES
1024
HEXAEDRES
1
COMPOSANTE(S) O
1
COMPOSANTE(S) F
POINT MINIMAL
:
-7.000E-03-7.000E-03 POINT MAXIMAL
0.00
:
7.000E-03 7.000E-03 4.000E-04
OBSERVATEUR CARTESIEN : 2.138E-02 1.235E-02 1.446E-02 POINT REGARDE 0.00
:
0.00
OBSERVATEUR SPHERIQUE 30.0
30.0
2.000E-04
: 2.851E-02
OUVERTURE : 10.0
TOUTES ARETES DE LA PEAU NUMERO SOUS DOMAINE
Z
O
Y
Y
X
X
Figure 2: The final 3D structure after expanding the 2D structures developed in the previous part, and joining them together, with the display of the two distinct sub-domains (right). ’SUITE EXEMPLE 3D ’ ’INTR ’ 1 0 $ IMPRE NINOPO ( SD EXTERIEURE ) T2IN.NOPO ’INTR ’ 1 1 $ IMPRE NINOPO ( SD EXTERIEURE ) T2OUT.NOPO $ NOM DU FICHIER STRUC IN DOWN ’MA23 ’ 1 0 7 $ IMPRE NIVO2D NIVO3D $ === DEFINITION DE LA FONCTION === TRAN 6 $ SECTION SUPERIEUR 0.0000000E+00 0.0000000E+00 0.0500000E-03 $ VECTEUR TRANSLATION BASE -0.0000000E+00 $ LA BASE FIN $ FIN DE LA DEFINITION DE LA FONCTION $ =========== LES OPTIONS ========= REF FAIN 4 0 0 0 SDSD 0 6 1 1 $ numero de sous domaine SDSD 0 6 2 2 $ numero de sous domaine F $ ======= APPEL DU MAILLEUR ======= GO $ STRUCT IN UP ’MA23 ’ 1 0 8 $ IMPRE NIVO2D NIVO3D $ === DEFINITION DE LA FONCTION === TRAN 2 $ SECTION SUPERIEUR 0.0000000E+00 0.0000000E+00 0.0500000E-03 $ VECTEUR TRANSLATION BASE 0.3000000E-03 $ LA BASE FIN $ FIN DE LA DEFINITION DE LA FONCTION $ =========== LES OPTIONS ========= REF FASU 3 0 0 0 $ reference face haut=electrode 3 SDSD 0 2 1 1 $ numero de sous domaine SDSD 0 2 2 2 $ numero de sous domaine F $ ======= APPEL DU MAILLEUR ======= GO $ STRUC OUT DOWN ’MA23 ’ 1 1 9 $ IMPRE NIVO2D NIVO3D $ === DEFINITION DE LA FONCTION ===
TRAN 6 $ SECTION SUPERIEUR 0.0000000E+00 0.0000000E+00 0.0500000E-03 $ VECTEUR TRANSLATION BASE -0.0000000E+00 $ LA BASE FIN $ FIN DE LA DEFINITION DE LA FONCTION $ =========== LES OPTIONS ========= REF SDSD 0 6 1 1 $ numero de sous domaine SDSD 0 6 2 2 $ numero de sous domaine F $ ======= APPEL DU MAILLEUR ======= GO $ STRUC OUT UP ’MA23 ’ 1 1 10 $ IMPRE NIVO2D NIVO3D $ === DEFINITION DE LA FONCTION === TRAN 2 $ SECTION SUPERIEUR 0.0000000E+00 0.0000000E+00 0.0500000E-03 $ VECTEUR TRANSLATION BASE 0.3000000E-03 $ LA BASE FIN $ FIN DE LA DEFINITION DE LA FONCTION $ =========== LES OPTIONS ========= REF FASU 3 0 0 0 $ reference face haut=electrode 3 ... FAIN 4 0 0 0 SDSD 0 2 1 1 $ numero de sous domaine SDSD 0 2 2 2 $ numero de sous domaine F $ ======= APPEL DU MAILLEUR ======= GO ’RECO’ 1 7 8 2 0.01000E-03 0 $ IMP NIV1 NIV2 NIV3 EPS IOPT 0 0 $ NBNNF NBNNSD ’RECO ’ 1 9 10 3 0.01000E-03 0 $ IMP NIV1 NIV2 NIV3 EPS IOPT 0 0 $ NBNNF NBNNSD ’RECO ’ 1 2 3 5 0.01000E-03 0 $ IMP NIV1 NIV2 NIV3 EPS IOPT 0 0 $ NBNNF NBNNSD ’RENE’ 1 5 6 ’SAUV’ 1 6 0 $ 6 T3.NOPO ’FIN’
Table 3: The definition of the 3D structure (T3NOPO.DATA)
2
Defining the interpolations using comaxx
The comaxx processor is used for assigning the required finite element to a given sub-domain. As mentioned earlier, our fundamental domains are distinguished by the number of side they are defined by in 2D (odd or even number of sides leading to triangular or rectangular finite elements). The piezoelectric finite element PENT PR1D is assigned to sub-domain 1 (odd number of sides) and HEXA PQ1D is assigned to sub-domain 2 (even number of sides). The resulting data arrays are stored in the output files mail and coor (these names are arbitrarily chosen by the user).
3
0 3 4 4
2 0 0
$ Y A T IL DES FONCTIONS INTERPRETEES $ NDIM NDSD NBSDC $ NNR NBLC
0
ELAS
$ $ $ $ $
1 PENT PR1D 0 1
HEXA PQ1D 0 T3.NOPO 0 mail 0 coor 0 0 0
NOM DE LA BIBLIOTHEQUE NTYED DU SD 1 LE NOM DES ELEMENTS DROITS NTYEC DU SD 1 NTYED DU SD 2
$ $ $ $ $ $ $ $ $
LE NOM DES ELEMENTS NTYEC DU SD 2 NOM DU FICHIER ET NIVEAU DE LA NOM DU FICHIER ET NIVEAU DE LA NOM DU FICHIER ET NIVEAU DE LA NTMAIL NTCOOR
DROITS
SD NOPO SD MAIL SD COOR
Table 4: The definition of the interpolation methods (T.COMAD)
3
Material definition using fomixx
Definition of the physical properties of the material and the physical conditions under which the experiment is performed (external forces/heat sources) is done by fomixx. This program helps defining the two arrays required for defining the physical parameters of our problem: FORC for the external forces, and MILLI for the material properties. ’ forc 1
1
mili 1 1 DONNEES RELATIVES A LA S.D. FORC for 5 24 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 ’ force surfaciques nulles sur ref 1 2 0 0 0 0 0 1 0 2 1 for 1 2 0 2 1 for 1 $ DONNEES RELATIVES A LA S.D. MILI mil 5 90 .867400000D+05 -.065824831D+05 .254824831D+05 .083533936D+05 .000000000D+00 .000000000D+00 .125255122D+06 .039350995D+05 -.100134939D+05 .000000000D+00 .000000000D+00 .846346784D+05 $
’ $ NOM DU FICHIER DE LA S.D FORC $ SON NIVEAU ET NB DE SES TAB. ASSOCIES $ NOM DU FICHIER DE LA S.D MILI $ SON NIVEAU ET NB DE SES TAB. ASSOCIES $ $ NOM TYPE NBREMOT $ for ( 1) $ for ( 2) $ for ( 3) $ for ( 4) $ for ( 5) $ for ( 6) $ for ( 7) $ for ( 8) $ for ( 9) $ for ( 10) $ for ( 11) $ for ( 12) 2 ’ $ CONTENU $ NDSM 0 0 $ NOSD NFRO NOPT ITRAIT (nopt=2=>notel) $ NTABL IADR $ NOSD NFRO NOPT ITRAIT (nopt=2=>notel) $ NTABL IADR $ $ NOM TYPE NBREMOT $ mil (C11) $ mil (C12) $ mil (C13) $ mil (C14) $ mil (C15) $ mil (C16) $ mil (C22) $ mil (C23) $ mil (C24) $ mil (C25) $ mil (C26) $ mil (C33)
.136073615D+05 .000000000D+00 .000000000D+00 .499650995D+05 .000000000D+00 .000000000D+00 .627244245D+05 .145412087D+05 .350905754D+05 .171000000D+03 -.094902059D+03 -.076097940D+03 .094181799D+03 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 .093952698D+03 -.037566503D+03 .000000000D+00 .000000000D+00 .000000000D+00 .000000000D+00 -.133433496D+03 .053352698D+03 -.392000000D+02 .000000000D+00 .000000000D+00 -.404033857D+02 -.008473235D+00 -.397966142D+02 ’ mat Quartz AT 2 0 0 0 1 0 2 1 mil 1 2 0 2 1 mil 1
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ 0
0
mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil mil
(C34) (C35) (C36) (C44) (C45) (C46) (C55) (C56) (C66) (e11) (e12) (e13) (e14) (e15) (e16) (e21) (e22) (e23) (e24) (e25) (e26) (e31) (e32) (e33) (e34) (e35) (e36) (eps11) (eps12) (eps13) (eps22) (eps23) (eps33)
’ $ CONTENU 0 0 $ NOSD NFRO NOPT ITRAIT (nopt=notel=2) $ NTABL IADR $ NOSD NFRO NOPT ITRAIT (nopt=notel=2) $ NTABL IADR
Table 5: The definition of the material properties and external forces (T.FOMID) The numerical values of the piezoelectric material for an AT-cut quartz crystal is obtained using the Matlab script and the numerical values given in table 6. The numerical values of the piezoelectric material for an AT-cut quartz
4
Assembling the data structures thelxx
The various data structures created in the previous steps are assembled using thelxx. As mentioned in the manual (Guide7-14/node174.html), the poba temporary data file is required and must be defined in the script used to run thelxx
4
function [M,Kp]=simul(a,K)
M(5,2) M(5,3) M(6,1) M(6,2) M(6,3)
%function SIMUL for trnasfer of rigidity matrix K %in rotated coordinates by matrix a %OUT : results are given in Kp (= K’) C = K(1:6,1:6) e = K(7:9,1:6) v = K(7:9,7:9)
a(3,2)*a(1,2) a(3,3)*a(1,3) a(1,1)*a(2,1) a(1,2)*a(2,2) a(1,3)*a(2,3)
%Lower Right part of M M(4,4) M(4,5) M(4,6) M(5,4) M(5,5) M(5,6) M(6,4) M(6,5) M(6,6)
%Upper Left part of M M(1:3,1:3) = a.^2 %Upper M(1,4) M(1,5) M(1,6) M(2,4) M(2,5) M(2,6) M(3,4) M(3,5) M(3,6)
right part of M = 2*a(1,2)*a(1,3) = 2*a(1,3)*a(1,1) = 2*a(1,1)*a(1,2) = 2*a(2,2)*a(2,3) = 2*a(2,3)*a(2,1) = 2*a(2,1)*a(2,2) = 2*a(3,2)*a(3,3) = 2*a(3,3)*a(3,1) = 2*a(3,1)*a(3,2)
%Lower M(4,1) M(4,2) M(4,3) M(5,1)
Left part of M = a(2,1)*a(3,1) = a(2,2)*a(3,2) = a(2,3)*a(3,3) = a(3,1)*a(1,1)
8.6740000e+10 6.9900000e+09 1.1910000e+10 -1.7910000e+10 0.0000000e+00 0.0000000e+00 1.7100000e-01 0.0000000e+00 0.0000000e+00
= = = = =
6.9900000e+09 8.6740000e+10 1.1910000e+10 1.7910000e+10 0.0000000e+00 0.0000000e+00 -1.7100000e-01 0.0000000e+00 0.0000000e+00
= = = = = = = = =
a(2,2)*a(3,3)+a(2,3)*a(3,2) a(2,1)*a(1,1)+a(2,3)*a(3,1) a(2,2)*a(3,1)+a(2,1)*a(3,2) a(1,2)*a(3,3)+a(1,3)*a(3,2) a(1,3)*a(3,1)+a(1,1)*a(3,3) a(1,1)*a(3,2)+a(1,2)*a(3,1) a(1,2)*a(2,3)+a(1,3)*a(2,2) a(1,3)*a(2,1)+a(1,1)*a(2,3) a(1,1)*a(2,2)+a(1,2)*a(2,1)
%Calcul de la transformee disp(’Transformee’) Cp = M*C*M’ ep = a*e*M’ vp = a*v*a’ Kp = [Cp ep’;ep vp] disp(’Fini, bon amusement !’) format long
1.1910000e+10 1.1910000e+10 1.0720000e+11 0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00
-1.7910000e+10 1.7910000e+10 0.0000000e+00 5.7940000e+10 0.0000000e+00 0.0000000e+00 -4.0600000e-02 0.0000000e+00 0.0000000e+00
0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 5.7940000e+10 -1.7910000e+10 0.0000000e+00 4.0600000e-02 0.0000000e+00
0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 -1.7910000e+10 3.9875000e+10 0.0000000e+00 -1.7100000e-01 0.0000000e+00
1.7100000e-01 -1.7100000e-01 0.0000000e+00 -4.0600000e-02 0.0000000e+00 0.0000000e+00 3.9200000e-11 0.0000000e+00 0.0000000e+00
0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 4.0600000e-02 -1.7100000e-01 0.0000000e+00 3.9200000e-11 0.0000000e+00
0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 0.0000000e+00 4.1000000e-11
Table 6: Matlab script for rotating by the appropriate angle (90 − 35.15o around the X axis) the parameters defining the piezoelectric properties of quartz, and matrix defining the piezoelectric properties of quartz used when executing this script. $ NOM DU FICHIER $ ET NIVEAU DE LA SD $ NOM DU FICHIER 1 $ ET NIVEAU DE LA SD tae $ NOM DU FICHIER 1 $ ET NIVEAU DE LA SD 0 $ NTTAE 1 $ 1 SI POBA EST UTILISEE /imec/users/friedtj/modulef99/linux/sta/etc/poba.direct $ NOM DU FICHIER POBA 1 $ 1 SI MILI EST UTILISEE
mili
mail
1
1 1
MAIL
coor
forc
COOR
1 2 0 0
TAE , 0 SINON :RIE VIDE
1
1
1
0 1
$ $ $ $ $ $ $ $ $ $
NOM DU FICHIER ET NIVEAU DE LA SD MILI 1 SI FORC EST UTILISEE , 0 SINON NOM DU FICHIER ET NIVEAU DE LA SD FORC NPROV NTHELA IOPT(*) NOM DU TABLEAU DES CL ET NOMBRE DE CL NOMCOU LVECT
, 0 SINON
Table 7: The definition of the data structures assembling methods (DATA.THELD)
5
Defining boundary parameters: cobdxx
The boundary conditions on each of the various surfaces defined by their respective reference numbers are defined using cobdxx. In our case, we define two sets of boundary conditions leading to 5 explicit boundary conditions: - we forbid any displacement of the sensing electrode (reference 3) by requiring that the 3 degrees of freedom (1, 2, 3) of the displacement variable (VN) are kept at 0 - we apply voltages to the electrodes: the sensing electrode (reference 3) is kept at ground level and the counter electrode is polarized at 0.5 V. These conditions are set by defining the electric potential degree of freedom (PHIE) to 0.0 value for faces referenced by number 3, and a value of 0.5E-3 kV for faces referenced by number 4 (counter electrode). The unit (kV for the electric potential) is chosen in order to be coherent with the other units used in defining the material properties, leading to a matrix filled with number closer to 1 than they would if S.I. units had been chosen (cf manual Guide7-17/node174.html).
5
mail 35 bdcl 35 0 1 3 3 3
5 1 2 3
5 VN VN VN
$ $ $ $ $ $ $ $ $
NOM DU FICHIER ET NIVEAU DE LA SD MAIL NOM DU FICHIER ET NIVEAU DE LA SD BDCL NTBDCL ICONST NBFR NTYP REF INC.VARIATIONNELLE MNEMO REF INC.VARIATIONNELLE MNEMO REF INC.VARIATIONNELLE MNEMO
3 4 PHIE 4 4 PHIE 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.5000000E-03 0 0
$ $ $ $ $ $ $ $ $
REF INC.VARIATIONNELLE MNEMO REF INC.VARIATIONNELLE MNEMO VAL: pas bouger dans les 3D sur la ... VAL ... face 1 VALEUR idem ... VALEUR potentiel electrode 3 VALEUR potentiel electrode 4 (kV) 1 SI SD NDL1 2 SI CL EN RL A LA MAIN ; -1 SI CL EN RL PAR SP
Table 8: The definition of the boundary conditions (DATA LIM.COBDD)
6
Solving the problem: crouxx
Once all the data arrays are defined (geometrical definition of the object, interpolation using the right finite elements, material properties, boundary conditions), the problem can be solved using crouxx. mail 33 1 tae 22 1
5
4
$ $ $ $ $ $
NOM DU FICHIER ET NIVEAU DE LA SD NDSM NTYP ND NOM DU FICHIER ET NIVEAU DE LA SD 1 SI BDCL EST UTILISE , 0 SINON
bdcl 15 0 sol.b 4 1
$ $ $ $ $ $
NOM DU FICHIER ET NIVEAU DE LA SD BDCL 1 SI CL. EN RL. EXISTE NOM DU FICHIER ET NIVEAU DE LA SD IMPREB
Table 9: The script for solving the problem DATA.CROUD)
7
Displaying the result: trc3xx
The results of the calculation are displaced using the trc3xx processor. The potential PHIE is displayed in figure 3 (left) while the displacement fields are displayed in figures 3 (right) and 4. The combination of the fields of the displacements in the three directions leads to a vector display as depicted in figure 5. MODULEF :
friedtj
MODULEF :
mail coor sol.b
friedtj
27/10/01
Fortran 3D couleur 3.04
Fortran 3D couleur 3.04
27/10/01
mail coor sol.b
NOEUDS
2385
NOEUDS
7880
FACES
7880
FACES
1632
PENTAEDRES
1632
PENTAEDRES
1024
HEXAEDRES
1024
HEXAEDRES
0.03s
0.02s
2385
30.
30.
:
0.57E-01
OUVERTURE : 10.
ISOVALEURS : INCONNUE :
20 4
MNEMO :PHIE
4.9750E-04 4.7367E-04 4.4735E-04 4.2102E-04 3.9469E-04 3.6837E-04 3.4204E-04 3.1571E-04 2.8939E-04 2.6306E-04 2.3673E-04 2.1041E-04 1.8408E-04 1.5775E-04 1.3143E-04 1.0510E-04 7.8775E-05 5.2448E-05 2.6122E-05 -2.0503E-07
30.
Dessin 1
Dessin 1
PEAU + ELIMINATION
Z
O
30.
:
0.57E-01
OUVERTURE : 10.
ISOVALEURS : INCONNUE : 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Utilisateur: friedtj
Utilisateur: friedtj
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
OBSERVATEUR SPHERIQUE
le 27:10:1901 a 12h 15mn.
le 27:10:1901 a 12h 18mn.
OBSERVATEUR SPHERIQUE
20 1
MNEMO :VN
1.0690E-06 1.0170E-06 9.5941E-07 9.0185E-07 8.4430E-07 7.8674E-07 7.2919E-07 6.7164E-07 6.1408E-07 5.5653E-07 4.9897E-07 4.4142E-07 3.8386E-07 3.2631E-07 2.6876E-07 2.1120E-07 1.5365E-07 9.6094E-08 3.8539E-08 -1.9015E-08
PEAU + ELIMINATION
Z
O Y
Y
X
X
Figure 3: Display of the potential (voltage) distribution and the amplitude of the displacement in the main displacement axis (X).
6
MODULEF :
friedtj
MODULEF :
mail coor sol.b
friedtj
27/10/01
Fortran 3D couleur 3.04
Fortran 3D couleur 3.04
27/10/01
mail coor sol.b
NOEUDS
2385
NOEUDS
7880
FACES
7880
FACES
1632
PENTAEDRES
1632
PENTAEDRES
1024
HEXAEDRES
1024
HEXAEDRES
0.02s
0.02s
2385
30.
30.
:
0.57E-01
OUVERTURE : 10.
ISOVALEURS :
20
INCONNUE :
MNEMO :VN
1.2502E-07 1.1272E-07 9.9141E-08 8.5558E-08 7.1975E-08 5.8392E-08 4.4809E-08 3.1226E-08 1.7643E-08 4.0596E-09 -9.5235E-09 -2.3107E-08 -3.6690E-08 -5.0273E-08 -6.3856E-08 -7.7439E-08 -9.1022E-08 -1.0461E-07 -1.1819E-07 -1.3177E-07
30.
Dessin 1
Dessin 1
PEAU + ELIMINATION
Z
O
30.
:
0.57E-01
OUVERTURE : 10.
ISOVALEURS :
20
INCONNUE : 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Utilisateur: friedtj
Utilisateur: friedtj
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
2
OBSERVATEUR SPHERIQUE
le 27:10:1901 a 12h 17mn.
le 27:10:1901 a 12h 17mn.
OBSERVATEUR SPHERIQUE
3
MNEMO :VN
1.0069E-07 9.1012E-08 8.0315E-08 6.9617E-08 5.8920E-08 4.8223E-08 3.7526E-08 2.6829E-08 1.6132E-08 5.4346E-09 -5.2626E-09 -1.5960E-08 -2.6657E-08 -3.7354E-08 -4.8051E-08 -5.8748E-08 -6.9445E-08 -8.0143E-08 -9.0840E-08 -1.0154E-07
PEAU + ELIMINATION
Z
O Y
Y
X
X
Figure 4: Display of the amplitude of the displacement in the two other displacement axis (Y and Z). MODULEF :
friedtj
MODULEF :
mail coor sol.b
NOEUDS
7880
FACES
1632
PENTAEDRES
1024
HEXAEDRES
30.
:
0.29E-01
OUVERTURE : 10.
VITESSES
:
VITESSE MAXI VITESSE MINI
coor sol.b
le 27:10:1901 a 12h 20mn.
le 27:10:1901 a 12h 19mn.
OBSERVATEUR SPHERIQUE 30.
mail
0.02s
0.03s
2385
0.11E-05 0.0
MAILLAGE SEMI-TRANSPARENT
friedtj
27/10/01
Fortran 3D couleur 3.04
Fortran 3D couleur 3.04
27/10/01
2385
NOEUDS
7880
FACES
1632
PENTAEDRES
1024
HEXAEDRES
OBSERVATEUR SPHERIQUE 30.
30.
:
0.14E-01
OUVERTURE : 10.
VITESSES
:
VITESSE MAXI VITESSE MINI
0.11E-05 0.0
MAILLAGE SEMI-TRANSPARENT
Utilisateur: friedtj
Utilisateur: friedtj
Dessin 1
Dessin 1
PEAU + ELIMINATION
Z
PEAU + ELIMINATION
Z
O
O Y
Y
X
X
Figure 5: Display of the velocity field, combining the informations of the 3 displacement matrices .
8
Conclusion
Finite element analysis leads to complementary results on the displacement of the QCM to the analytical analysis. The magnitude of the displacement is in agreement with the analytical solution [1, 2] with an amplitude in the pm range when a DC potential of 0.5 V is applied. Considering our QCMs, when oscillating in liquid, display a quality factor Q ' 1000 − 3000, we conclude that the amplitude of the oscillations are in the nm range. We also believe we can consider the relative displacement in the X direction and the Z direction to be correctly predicted by finite element analysis, and calibrate the displacement in the X direction from analytical resolution of the problem. In this particular case, finite element analysis allows us to estimate the displacement in the
7
Z direction, which cannot be predicted by analytical resolution since it is a side effect of the finite extension of the counter electrode. The analytical resolution of the problem of knowing the displacement of a piezoelectric substrate of infinite dimensions in the (X, Y ) plane, to which a DC potential is applied, predicts a displacement only in the X direction, which can be verified by finite element analysis (the main displacement component is also in that direction). In our case, the displacement of the QCM surface in the Z direction is of major importance as it leads to longitudinal waves in the liquid above the QCM surface, which is the source of frequency fluctuations as the resonator is disturbed by the standing wave. We conclude from finite element analysis that this component is about 1/10th of the X displacement and is thus not negligible in understanding the origin of longitudinal waves in the liquid which disturb the oscillation of the resonator and thus its frequency stability. all:sol.b
# echo "E T.FOMID 5 F" | fomixx xargs # echo "E TFORC.FOMID 5 F" | fomixx xargs
T2IN.NOPO: T2INNOPO.DATA echo "E T2INNOPO.DATA F" | apnoxx xargs
tae: DATA.THELD forc echo "E DATA.THELD 5 F" | thelxx xargs
T2OUT.NOPO: T2OUTNOPO.DATA echo "E T2OUTNOPO.DATA F" | apnoxx xargs T3.NOPO: T3NOPO.DATA T2IN.NOPO T2OUT.NOPO echo "E T3NOPO.DATA F" | apn3xx xargs
bdcl: DATA_LIM.COBDD tae echo "E DATA_LIM.COBDD 5 F" | cobdxx xargs # echo "E DATA_TMP.COBDD 5 F" | cobdxx xargs # echo "E DATA.COBDD 5 F" | cobdxx xargs
coor: T.COMAD T3.NOPO echo "E T.COMAD 5 F" | comaxx xargs
sol.b: DATA.CROUD bdcl echo "E DATA.CROUD 5 F" | ./crouxx xargs
forc: T.FOMID coor echo "E TRHO.FOMID 5 F" | fomixx xargs
clean: \rm T2IN.NOPO T2OUT.NOPO T3.NOPO coor forc mili tae mail bdcl sol.b core
Table 10: The Makefile for automatically running the whole simulation.
References [1] Bret A. Martin and Harold E. Hager. Velocity profile on quartz crystals oscillating in liquids. J. Appl. Phys., 65(7):2630–2635, 1989. 7 [2] K. Keiji Kanazawa. Mechanical behaviour of films on the quartz microbalance. Faraday Discuss., 107:77–90, 1997. 7
8
