Geophys. J. Int. (2001) 145, 679–688
Finite element three-dimensional direct current resistivity modelling: accuracy and efficiency considerations Zhou Bing1 and S. A. Greenhalgh2 1 2
Department of Geotechnology, Institute of Technology, Lund University, Box 118, S-221 00, Lund, Sweden. E-mail:
[email protected] Department of Geology and Geophysics, The University of Adelaide, SA 5005, Australia
Accepted 2000 December 19. Received 2000 November 17; in original form 1999 November 24
SUMMARY The finite element method is a powerful tool for 3-D DC resistivity modelling and inversion. The solution accuracy and computational efficiency are critical factors in using the method in 3-D resistivity imaging. This paper investigates the solution accuracy and the computational efficiency of two common element-type schemes: trilinear interpolation within a regular 8-node solid parallelepiped, and linear interpolations within six tetrahedral bricks within the same 8-node solid block. Four iterative solvers based on the pre-conditioned conjugate gradient method (SCG, TRIDCG, SORCG and ICCG), and one elimination solver called the banded Choleski factorization are employed for the solutions. The comparisons of the element schemes and solvers were made by means of numerical experiments using three synthetic models. The results show that the tetrahedron element scheme is far superior to the parallelepiped element scheme, both in accuracy and computational efficiency. The tetrahedron element scheme may save 43 per cent storage for an iterative solver, and achieve an accuracy of the maximum relative error of < + ðp+GÞ ¼ ÿdðr ÿ rc Þ , r, rc [ ) ; (1) LG > : þ lG ¼ 0 , r [ L) , Ln where G is the Green’s function of the electric potential due to a unit current source I=1 (magnitude) at position rc. The quantity s is the conductivity of the medium whose spatial range and boundary are denoted by V and hV. In general, the conductivity is a function of the spatial coordinates r=(x, y, z). Here n is a coefficient due to the artificial boundary (Dey & Morrison 1979), and it may be written in general form when considering a buried source: l¼
r0 2 cos h þ r2 cos h0 , r0 rðr þ r0 Þ
(2)
where r=|~ r | and rk=|~ rk| are the distances from the current source and its image position (due to the Earth’s surface) to the boundary; and h and hk are the angles between the normal vector of the boundary and the vectors ~ r and ~ rk, respectively. On the Earth’s surface, we set n=0 because of hG/hn=0. Accordingly, we have the following expressions for the potential, apparent resistivity and Fre´chet derivative based on the Green’s function (Zhou & Greenhalgh 1999): Uðrc , rÞ ¼ IGðrc , rÞ ,
(3)
oa ¼ K½dGMN ðrA Þ ÿ dGMN ðrB Þ , ð Loa ¼ ÿK ½+dGAB ðrÞ . +dGMN ðrÞd) , Lpe )e
(4) (5)
where dGjg(r)=G(rj, r)xG(rg, r), K is the geometry factor, which can be calculated from the positions of the four electrodes rA, rB, rM and rN in a configuration (i.e. pole–pole, pole–bipole, bipole–pole and bipole–bipole), and se is the conductivity of a model element Ve. In eqs (4) and (5), if a remote electrode (infinite location r?) is involved in the computation, we set G(r, r?)=0. Eqs (4) and (5) are thus available for any electrode configuration. Eqs (3), (4) and (5) show that in theory the potential, the apparent resistivity and the Fre´chet derivative can be calculated by the Green’s function, which can be obtained by solving eq. (1). The FEM can be used in solving eq. (1). According to finite element theory, two principles, namely the variational principle (Pridmore et al. 1981) and Galerkin’s criterion (Zienkiewicz 1971), are available for various applications. It has been shown that, if the same shape function is employed in the two schemes, they lead to the same linear equation system. Galerkin’s criterion, however, gives a simpler formulation and is easier to understand. In the following section, we adopt Galerkin’s criterion for the FEM formulation. #
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FINITE ELEMENT SOLUTION
Parallelepiped element
According to Galerkin’s criterion, the solution of eq. (1) reduces to solving the following integral equation: ð Wj ½+ . ðp+GÞ þ dðrc ÿ rÞd) ¼ 0 , VWj [ H 1 ð)Þ , (6)
A parallelepiped element is shown in Fig. 1(a), simply represented by eight nodes {I1, I2, . . . , I8}. If trilinear interpolation is used within the element, each node has the following shape function (Schwarz 1988):
)
where Wj is a weighting function that belongs to Hilbert space H1(V). After carrying out integration by parts and substituting for the boundary condition in eq. (1), eq. (6) becomes ð ð pð+Wj . +GÞd) þ pWj ðlGÞd! ¼ Wj ðrc Þ : (7) )
L)
By discretizing the 3-D region into finite elements, for example V=SeVe and hV=SeCe, in which the Green’s function can be expressed as a combination of the shape functions Nie(r), i.e. GðrÞ ¼
n X
Nie ðrÞGi ,
r [ )e ,
(8)
i¼1
)e
i¼1
þ
n ð X i¼1
) pWj ðlNi Þd! Gi ¼ Wj ðrc Þ ,
(9)
!e
The weighting function in Ve can be chosen to be the same as the shape function, that is Wje(r)=Nje(r), j=1, 2, . . . , n, so that eq. (9) can be written in the matrix form ~ ¼ bs : MG
(10)
~ is the vector of Green’s functions, bs=(0, 0, . . . , 1, . . . , 0) Here, G is the source vector with only one non-zero component, at the current injection position, and M is an nrn square matrix assembled from the individual element matrices, namely ð ð Mije ¼ p+Ni . +Nj d) þ plNi Nj d! , i, j ¼ 1, 2, . . . , n , )e
!e
(11) which depends on the element integral over Ve and the boundary integral on Ce. Obviously, the element matrix Mije is symmetric. Once the shape functions [Ni(r), i=1, 2, . . . , n] are chosen and the conductivity s is given, the element matrix can be calculated using eq. (11), and one can obtain the Green’s function vector ~ by solving eq. (10). The electric potential or the apparent G resistivity at any observational point can be calculated with the Green’s function with expressions (3) or (4). Therefore, the FEM for resistivity modelling reduces to calculating the matrix M and solving the linear equation system (10).
#
2001 RAS, GJI 145, 679–688
ði ¼ I1 , I2 , . . . , I8 Þ ,
where m¼
2 ½x ÿ ðx1 þ x2 Þ=2 , *x
g¼
2 ½ y ÿ ðy1 þ y2 Þ=2 *y
and 2 ½z ÿ ðz1 þ z2 Þ=2 : *z
The coordinates (x1, x2), (y1, y2) and (z1, z2) represent the range of the parallelepiped element. Substituting eq. (12) into eq. (11), the element integral becomes ð gi gj mi mj fi fj *Ve pe 1 þ p+Ni . +Nj d) ¼ 1 þ 16 *x2 3 3 )e gi gj mi mj fi fj þ 2 1þ 1þ *y 3 3 gi gj fi fj mi mj þ 2 1þ 1þ , *z 3 3 ði, j ¼ I1 , I2 , . . . , I8 Þ ,
(13)
from which one can see that, due to the symmetry between i and j, only 36 independent non-zero components need to be calculated for each parallelepiped element. To calculate the boundary integral in eq. (11), one may suppose that the global boundary consists of the Earth’s surface and five subsurface planes—the left, the right, the front, the back and the bottom boundaries. On the Earth’s surface the integral vanishes due to n=0. On the others it gives rise to five boundary integrals to be calculated. As an example,
(b)
(a) x I2
I6 I5
m
I4 I3
I1
ELEMENT MATRIX CALCULATIONS In order to calculate the element matrix (see eq. 11), one must choose the elements {Ve}, which together compose the whole model range V. For 3-D problems, two common element types—parallelepiped and tetrahedron elements—are often employed for the computation of the element matrix.
1 ð1 þ mmi Þð1 þ ggi Þð1 þ ffi Þ 8
(12)
f¼
where n is the total number of shape functions, one obtains the discrete form of eq. (7): ( n ð X X pð+Wj . +Nie Þd) Gi e
Ni ðm, g, fÞ ¼
y I8
I7
l
i j
z Figure 1. Two common element specifications in the 3-D finite element method: (a) parallelepiped element; (b) tetrahedron element.
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the calculation for the left boundary is given. Setting g=0 in eq. (12) and substituting into the boundary integral of eq. (11), we have ð mi mj fi fj *x*z le ð1 ÿ gi Þð1 ÿ gj Þ 1 þ 1þ , plNi Nj d! ¼ 64 3 3 !e ði, j ¼ I1 , I2 , I5 , I6 Þ ,
(14)
from which one can see that only 10 components need to be calculated, as a result of the symmetry. The contributions from the other boundaries can be obtained in a similar way. Tetrahedron element
Diagonal band elimination
A tetrahedron element is shown in Fig. 1(b), simply represented by four nodes (i, j, l, m). Pridmore et al. (1981) and Sasaki (1994) presented a tetrahedron element scheme in which the 3-D region is divided into a large number of bricks and each brick is assembled from five tetrahedron elements. Zhou & Zhong (1984) presented another tetrahedron element scheme in which each of the bricks is assembled from six tetrahedron elements. Both use a linear shape function within the tetrahedron for 3-D resistivity modelling. Here, we use the sixtetrahedron element scheme and briefly outline the procedure. The six tetrahedrons can be denoted by their nodes (see Fig. 1): (I1, I2, I3, I5), (I2, I3, I5, I6), (I3, I5, I6, I7), (I2, I3, I4, I6), (I3, I4, I6, I7) and (I4, I6, I7, I8). So, from eq. (11) the integral over the brick is calculated by the summation of the six tetrahedron integrals: ! ð ð 6 X e . Mij ¼ p+Ni +Nj d) þ lNi Nj d! , (15) k¼1
*ke
!e*
where CDe represents the side of the tetrahedron element that coincides with the artificial boundary. For simplicity, in the following text we use (i, j, l, m) to represent the four nodes of a tetrahedron element, and their shape functions can be written as follows: Np ðx, y, zÞ ¼
large, sparse, symmetric and positive-definite matrix. One can employ a direct matrix method or an iterative solver, depending on the available computer resources and the task of the computation. This section is concerned with two kinds of solvers applied to our case. One is called diagonal band elimination, specifically the banded Choleski factorization method, and the other is the pre-conditioned conjugate gradient method (PCG) with different pre-conditioners. They each use different amounts of computer resources and offer different efficiencies for 3-D resistivity modelling.
1 ðap þ bp x þ cp y þ dp zÞ , 6Ve
ð p ¼ i, j, l, mÞ , (16)
which satisfies Np(xq, yq, zq)=dpq (p, q=i, j, l, m). Here Ve is the bulk of the tetrahedron element, and ap, bp, cp and dp are constants which can be calculated from the node coordinates {(xp, yp, zp), p=i, j, l, m}. Substituting (16) into the bulk integral of eq. (15), we have ð p p+Np . +Nq d) ¼ ðbp bq þ cp cq þ dp dq Þ , 36Ve )*e ð p, q ¼ i, j, l, mÞ ,
The attractive feature of diagonal band elimination is that the solving procedure is conducted only once for all the source vectors (different vectors bs in eq. 10). This advantage suggests that one should apply a band elimination method to solve eq. (10) for a large number of source vectors. In our case, it may be an option for 3-D resistivity inversions, because the Green’s functions of all the current and potential electrodes in a 3-D measurement must be calculated for the Fre´chet derivative (see eq. 5). A 3-D resistivity measurement may employ hundreds of electrodes. This means that, if applying an iterative solver to eq. (10), the iterative procedure has to be repeated hundreds of times for all the Green’s functions. It may be that the computer time required for implementing the band elimination once is less than that required when employing an iterative solver for a large number of electrodes. Considering the various algorithms of band elimination methods, the banded Choleski factorization algorithm is a suitable choice for this case, because it can deal with only the lower half of the symmetric band matrix M, thus saving a significant amount of memory (Schwarz 1988). The banded Choleski factorization algorithm implements the factorization M=LLT, where L is a lower-triangular matrix but has the same bandwidth m as M (Mij=0, Lij=0 for |ixj|>m). Therefore, only one array, with size (n, m+1), is required for storage, and then by performing forward- and back-substitution one can obtain all the solutions for the different vectors bs. The amount of calculation necessary for the banded Choleski factorization is directly proportional to the order n and to the square of the bandwidth m (Schwarz 1988); that is, Computer time ! nm2 :
(18)
It is therefore particularly important that the bandwidth m should be kept as small as possible for lower storage requirements and minimal computation.
(17)
which is much simpler than eq. (13). The boundary integral in eq. (15) can be calculated in a similar way to in the parallelepiped element scheme, but using the shape function given by eq. (16) (see Zhou & Zhang 1984).
LINEAR EQUATION SYSTEM SOLVERS The final step of the FEM is to solve the linear equation system (10). From the previous section, the coefficient matrix M is a
Pre-conditioned conjugate gradient method (PCG) ~x1 (M ~ is The basic idea of the PCG is to multiply a matrix M called a pre-conditioner) with the linear equation system (10) so that the resultant coefficient matrix is close to the identity matrix ~x1M#I or M ~ #M and the CG algorithm has a fast conM ~x1bs. ~ x1Mx=M vergence rate for the linear equation system M Applying this idea to the CG algorithm, we write the PCG algorithm as follows: #
2001 RAS, GJI 145, 679–688
FEM for 3-D DC resistivity modelling ~ r0=bxMx0 and setting p0=r0; (1) initialization: solving M (2) loop for i=0, 1, 2, 3, . . . . ~ i ¼ Mpi , Mq xiþ1 ¼ xi þ ai pi , bi ¼
ðriþ1 , riþ1 Þ , ðri , ri Þ
riþ1 ¼ ri ÿ ai qi
(19)
~ from which one can find that the pre-conditioner M should be close to M and has some specific properties ~ qi = Mpi. for efficiently solving the linear equation M ~ may be the diagonal matrix of Simple choices for M ~ =diag(M11, M22, . . . , Mnn) or the tridiagonal partition M: M ~={M11, M12, M21, M22, M23, . . . , M(nx1)n, Mnn}. The of M: M former is actually the scaled conjugate gradient method (SCG), and the latter (called TRIDCG) has more components than the diagonal choice; with both of them it is easy to obtain the qi ~ are the in the algorithm (19). A further two choices for M symmetric successive over-relaxation matrix method (SSORCG; see Axelsson 1984; Spitzer 1995) and the incomplete Cholesky decomposition of M (ICCG; see David 1978; Ajiz & Jennings 1984; Manolis & Michael 1991; Zhang et al. 1995). Following the SSORCG, we rewrite the pre-conditioner in the following factorization form: ~ ¼ H1 H2 ¼ ðD þ wEÞðI þ wDÿ1 E T Þ M
(20)
where D and E are the diagonal and the lower-triangular matrices that consist of the diagonal and the lower-triangular components of M, respectively (M=D+E+ ET), w is called the relaxation parameter, and F(w) represents the two w-dependent terms on the right-hand side of eq. (20). Eq. (20) shows that the ~ is not necessarily symmetric (in the SSORCG M ~ conditioner M is made symmetric by pre-scaling), and the factors H1=D+wE and H2=I+wDx1ET are directly assembled from the lowerand upper-triangular matrices M. With such factors, the vector qi can be obtained just by forward- and back-substitution in the PCG. This method may be called SORCG. Moreover, eq. (20) shows that the optimal relaxation parameter w is the value that minimizes the norm of the matrix F(w). It has been shown that w depends on the eigenvalue of the multiplication of the ~ x1M (Axelsson 1984). In general, it is difficult to matrices M find the optimal value, but it may be chosen by trial and error in different cases. The ICCG solver, based on the factorization M=LLT, defines the pre-conditioner with an incomplete Choleski factorization— performing the Choleski factorization with rejection by magnitude or by position (David 1978). Manolis & Michael (1991) pointed out that the incomplete factorization based on rejection by magnitude has the disadvantage that the storage size and pattern are not known at the start and the cost in computer space may become high since some auxiliary vectors will be needed for storage of the pre-conditioning matrix, but the factorization with ~ keeps the same sparsity pattern as the rejection by position (M matrix M) prescribes the additional storage beforehand and the computing times are very competitive. Following Manolis and Michael’s method (1991), we rewrite ICCG based on rejection by 2001 RAS, GJI 145, 679–688
lki lkj ,
8 qffiffiffiffiffiffiffiffi > 1 > Mii1 > 0 , > Mij , > > < vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lii ¼ u n X u > ðkÞ > > tMii1 þ cii , Mii1 ƒ0 , > > : k¼1,k=i
i, j [ PðMij Þ and j§i ,
lij ¼ Mij1 =lii (21) where P(Mij) represents the sparsity pattern of M and calculated by the following equations: ii ¼ M 1 þ M ii
jÿ1 X
ðkÞ
cii ,
k¼1,k=i ð jÞ ii =M jj Þ1=2 jM 1 j , cii ¼ ðM ij
jj ¼ Mjj þ M
iÿ1 X
c(k) ii
is
ðkÞ
cjj
k¼1
(22)
ðiÞ jj =M ii Þ1=2 jM 1 j , cjj ¼ ðM ij
The diagonal modification for the case M*ii j0 in eq. (21) was suggested by Ajiz & Jennings (1984) to retain the stability of ICCG. COMPUTER IMPLEMENTATION
¼ ðD þ E þ E T Þ þ ðw ÿ 1ÞðE T þ EÞ þ w2 EDÿ1 E T
#
iÿ1 X k¼1
piþ1 ¼ riþ1 þ bi pi ,
¼ M þ F ðwÞ ,
position as follows: Mij1 ¼ Mij ÿ
ðri , ri Þ , ai ¼ ð pi , qi Þ
683
As presented above, the parallelepiped and tetrahedron element schemes lead to different coefficient matrices M, but they result in the same number of nodes in a 3-D discretization. The different solvers mentioned above can be applied to eq. (10) for the 3-D solution. It will be seen that with the two element schemes one can obtain different solution accuracies and with the different solvers one can obtain different computing efficiencies, because these schemes and solvers have quite different storage-size requirements and computer-time consumption. Storage size of matrix M With the parallelepiped approximation each node has a maximum of 26 adjacent nodes around it (see Fig. 2). This means that the coefficient matrix M generally has 27 non-zero components in each row. Considering the symmetric band, the computation and the compact storage may only be that required to calculate and store the lower or upper band; that is, 14 nonzero elements are calculated in each row of M. As an example, if NX, NY and NZ (n=NXrNYrNZ) are the total numbers of the nodes in the x-, y- and z-directions, and NXY (=NXrNY) is the number of the nodes in the xy-plane, and the condition NXYjNXZjNYZ (NXZ=NXrNZ, NYZ=NYrNZ) holds, the 14 non-zero elements and their node numbers are shown in Fig. 2. These node numbers are also indicators of their position in M. So, the parallelepiped element scheme requires a storage size of at least M(n, 14) for any iterative solver and M(n, NXY+NX+2) for the banded Choleski factorization (Jennings & Mckeown 1992). With the tetrahedron element scheme, however, the storage size may reduce to M(n, 8) for any iterative solver and M(n, NXY+1) for the banded Choleski factorization (see Fig. 2). A great deal of computer memory (about 43 per cent for an iterative method) is saved. In general, the
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2 i+1
1 i
i+NX+1
4 i+NX
3 i+NX-1
6 5
i+NXY-NX+1
i+NXY-NX i+NXY-NX-1
7
i+NXY+1
8 i+NXY
i+NXY-1
i+NXY+NX+1 i+NXY+NX
i+NXY+NX-1
Figure 2. The adjacent nodes of the ith node in a 3-D discretization. The nodes with a number are the non-zero components of the ith row in the coefficient matrix M when the tetrahedron element is used. The nodes with a subscript yield the non-zero entries in the ith row in the coefficient matrix M when the parallelepiped element is employed.
tetrahedron element scheme needs a storage size M(n, min(NXY, NXZ, NYZ)+1) for the band elimination algorithm. Obviously, for 3-D modelling the storage size is still significant: a 128 MB PC can only handle a 3-D grid size not larger than 31r31r31 with the banded Choleski factorization, while with an iterative method one may have a wide range of grid sizes for different applications. Programming of the PCG From algorithm (19), one finds that the only difference between the PCG and the standard CG than solving the linear equation ~qi=Mpi (if M ~=I, the PCG becomes the normal system M CG). The crucial point is the choice of the pre-conditioner ~ according to M. Matrix M ~ should (1) achieve rapid M convergence, (2) have an additional storage size as small as possible, and (3) obtain qi efficiently. In fact, the four schemes of the PCG (SCG, TRIDCG, SORCG and ICCG) referred to above have no problem with the third criterion, because these pre-conditioners are diagonal, tridiagonal, lower- and uppertriangular matrices by which qi can be efficiently obtained by just performing the forward- and back-substitution. Therefore, only the convergence and the storage size of the pre-conditioners need to be examined against the parallelepiped and tetrahedron element schemes for 3-D resistivity modelling. Based on the above compact storage of M, it is not difficult to write the segments of the program to obtain qi for SCG, TRIDCG and SORCG. In fact, these three pre-conditioners are implicitly contained in M, so it is unnecessary to prescribe ~. As an example, with the tetrahedron additional storage for M element the segment of the FORTRAN code for the SORCG may be written as follows according to eq. (20): Part 1: DO I=1,N T=0.0 CALL CMJ(I,MJ) DO 5 J=1,7 K=MJ(J) IF(K.LE.0) GO TO 5 T=T+w*M(K,9-J)*V(K) 5 CONTINUE V(I)=(B(I)-T)/M(I,1) END DO
Part 2: DO I=N,1,-1 CALL CMJ(I,MJ) T=0.0 DO 12 J=9,15 K=MJ(J) IF(K.LE.0) GO TO 12 T=T+w*A(I,J-7)*Q(K)/M(I,1) 12 CONTINUE Q(I)=V(I)-T END DO
where CMJ(I,MJ) is a subroutine to find the node numbers MJ(15) for the 15 adjacent nodes of the ith node (they indicate the positions of the non-zero elements of the ith row in M), w is the relaxation parameter, M(n, 8) is the compact storage of M, and B(n)=Mpi. Part 1 is the implementation of the forward-substitution with H1=D+wE, and Part 2 obtains the vector qi through back-substitution with H2=I+wDx1ET. From this code, one finds that only one vector V(n) needs to be prescribed, and the other array MJ(15) is very small in size (15 elements). For ICCG, Ajiz & Jennings (1984) gave the code for the incomplete Choleski factorization based on rejection by magnitude. Manolis & Michael (1991) improved the scheme and gave another program for general structural analysis. In our case, we found that with the compact storage M(n, 14) (parallelepiped element) or M(n, 8) (tetrahedron element), the algorithm based on rejection by position becomes very simple; that is, using the tetrahedron element scheme, the major computation of ix1 Sk=1 lki lkj in eq. (21) is fulfilled by the following segment: SUM=0.0 DO 5 KK1=1,7 K1=MI(KK1) IF(K1.LE.0)GO TO 5 DO 3 KK2=1,7 K2=MJ(KK2) IF(K2.LE.0)GO TO 3 IF(K1.NE.K2)GO TO 3 SUM=SUM+L(K1,9-KK1)*L(K2,9-KK2) 3 CONTINUE 3 CONTINUE where MI(15) and MJ(15) are two integer arrays for the positions of the non-zero elements of the ith and jth rows, and L(n, 8) is the incomplete Choleski factorization. From this algorithm one can see that the main additional storage is L(n, 8), which has the same size as M(n, 8). After the factorization, the same code as for SORCG can be used for ICCG by replacing M(n, 8) with L(n, 8) and setting w=1.
NUMERICAL EXPERIMENTS In order to examine the efficiency and accuracy of the FEM schemes (parallelepiped and tetrahedron elements) and the above linear equation solvers, we conducted numerical experiments with three synthetic models, namely homogeneous, twolayered and vertical contact, whose analytic solutions are easily obtained for comparison. The computer used for the experiments was a Pentium-II, 128 MB PC running at 450 MHz. We recorded the storage sizes and computing times of these solvers and the accuracies of the parallelepiped and tetrahedron element schemes. One can compare these records and determine an accurate and efficient element scheme and solver for 3-D DC resistivity solutions. Figs 3, 4 and 5 show the solutions with the tetrahedron element scheme for the three models. They depict equipotential contours in the horizontal and vertical planes, as well as the voltage versus depth profile for model 1 (see Fig. 3), apparent resistivity sounding with a dipole–dipole array for model 2 (see Fig. 4), and voltage change along the x-axis for model 3 (see Fig. 5). The grid size of the FEM was 119 r 119 r 80 (Dx=Dy=Dz=0.5 m), and the SORCG was employed for the #
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Figure 3. FEM result of a buried current electrode in a homogenous half-space. The left diagram is the equipotential contour volume and the right diagram is the voltage versus depth profile in the borehole sketched in the left diagram. The exact analytic solution is also shown for comparison.
solutions. The time required was about 45 min for each model. Fig. 6 is a comparison of the accuracy of the parallelepiped and tetrahedron element schemes. These results show that the tetrahedron element scheme is much better than the parallelepiped element scheme, not only in the storage size (savings of 43 per cent—see the previous section) but also in the accuracy (the maximum relative errors are j4 per cent and j1 per cent for element sizes of 1 m and 0.5 m respectively). As mentioned above, with the SORCG solver one has to predetermine the relaxation parameter w in applications (see eq. 20). In order to detect the sensitivity of w to the convergence
of SORCG, we repeated the above three simulations with four relaxation parameters (w=0.5, 1.0, 1.5, 2.0) and recorded the computer times for the same degree of convergence (10x10). We found that with both the parallelepiped and tetrahedron element schemes the optimal value of the relaxation parameter was w=1.5 for the three models (see Fig. 7). This value is very close to Spitzer’s (1995) result in finite difference experiments. These experiments show that the optimal value is relatively stable with the two element schemes and the three models. Therefore, it can be confidently applied to 3-D resistivity computations.
Figure 4. FEM result of a dipole–dipole survey in a two-layer model. A and B are current electrodes, M and N are potential electrodes. The left diagram shows equipotential contours in the horizontal and vertical planes. The right diagram is the apparent resistivity graph for the FEM. The exact (analytic) solution is shown for comparison. #
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Figure 5. FEM result of a pole–pole survey near a vertical contact. A and M represent the current injection and potential point. The equipotential diagram and the voltage versus AM-separation curve are shown.
than SCG and TRIDCG, and ICCG has the best convergence speed of the four solvers. These results indicate that SORCG is a suitable option for large-scale computations, but if the computer memory is available for additional storage L(n, 8), ICCG is the best choice of the iterative solvers. Both of them may save on computer time by at least y50–70 per cent. To investigate the computing efficiency of a band elimination solver, we applied the banded Choleski factorization method to the three models. Owing to the limitation of the PC memory, only the tetrahedron element scheme and two grid sizes, n=17r17r14 and n=33r33r27, could be employed for the experiments. The second grid has nearly double the density of the first one. Although the two grid sizes are not large enough for a real case, their storage sizes and the computer times may
A further comparison of the computing efficiency of the four PCG algorithms (SCG, TRIDCG, SORCG and ICCG) was made with the two element schemes and three models. Table 1 gives the details of the comparison using the tetrahedron element scheme, and shows the differences in additional work, additional storage size, number of iterations and computer times to obtain the Green’s function of one source vector. The parallelepiped element scheme has similar results to those in Table 1. From this table one can see that (1) SCG and TRIDCG do not require any additional storage, SORCG has one more vector, but ICCG needs much more computer memory [nearly double the storage size of others, because L(n, 8) is the same size as M]; and (2) TRIDCG has little improvement in convergence rate over SCG, SORCG converges much faster
~ =bs. Table 1. Comparison of PCG algorithms for solving MG PCG
SCG
TRIDCG
SORCG
ICCG
Additional work
Dqi=pki
Tqi=pki
H1H2qi=pki
MFLLT LLTqi=pki
Additional storage size
0
0
V(n) for H2qi
L(n, 8) and V(n) for LTqi
Iteration No. (N) and computer time (T) for one bs
N=1253, 1257, 1261 T=8.1, 8.2, 8.3 min (for three models)
N=1122, 1125, 1125 T=8.0, 8.1, 8.2 min (for three models)
N=264, 265, 265 T=3.6, 3.7, 3.8 min (for three models)
N=172, 174, 175 T=2.3, 2.4, 2.5 min (for three models)
pki=Mpi, D=diag(M), T=tridiag(M), H1=D+1.5E, H2=D+1.5Dx1ET, E=lower triangular portion of M, L=incomplete Choleski factorization, n=dimension of M (57r57r49). Three synthetic models: homogenous, two-layered and vertical contact. #
2001 RAS, GJI 145, 679–688
FEM for 3-D DC resistivity modelling parallelepiped tetrahedron
(a) 12
687
12.29 %
11.56%
10.69 %
8 3.72 %
4
2.57 %
1.82 %
0 homogeneous
two-layered vertical-contact
Synthetic model parallelepiped tetrahedron
(b)
5.5 %
5.02 %
Figure 7. Effect of the relaxation parameter of SORCG for three synthetic models. Computing time is least for a value of w=1.5. (a) Parallelepiped; (b) tetrahedron.
4.5 2.78 %
2.5 0.98 %
0.82 %
0.69 %
0.5 homogeneous
two-layered vertical-contact
Synthetic model Figure 6. Comparison of solution accuracy between the parallelepiped and tetrahedron element schemes. (a) 3-D grid size is 57r57r49 (element size=1 m); (b) 3-D grid size is 119r 119r80 (element size=0.5 m).
help us to estimate the computing efficiency for a larger-size simulation and inversion. According to the storage requirement of the tetrahedron element scheme (M[n, min{NX*NY, NX*NZ, NY*NZ}+1]), the second grid size requires at least 128 MB of memory for the banded Choleski factorization. This is the maximum size we can handle with our PC. The results of the experiments (omitted here) showed that the two grid sizes hardly attain reasonable levels of accuracy for the three models; the first grid size took about 20 s and the second grid size spent over one hour solving eq. (10). The experiments proved that a coarse grid cannot give satisfactory solutions, and a fine grid, for example just doubling the density in the three directions of the coarse grid, will cost over 150 times as much computer time as a coarse grid. In fact, the time consumption of the factorzation method can be estimated using the relation (18), from which we have the following computer-time for the Choleski elimination method for two grid sizes: n2 m22 T2 ¼ T1 n1 m21 ¼
#
NX2 |NY2 |NZ2 NX1 |NY1 |NZ1
NX2 |NY2 þ 1 2 T1 , NX1 |NY1 þ 1
2001 RAS, GJI 145, 679–688
(23)
where T1 and T2 are assumed to be the computing times for the banded Choleski factorization using grid 1 and grid 2, whose dimensions in the x-, y- and z-directions are (NX1, NY1, NZ1) and (NX2, NY2, NZ2), respectively. From this expression one can see that if grid 2 has double the size of grid 1, the computing time is at least 27=128 times that of grid 1. Our experiments showed that the actual computing time is greater than the estimate, because eq. (23) is just for the banded Choleski factorization. Such an increase in the computing time should be compared with the iterative solver SORCG or ICCG, even for many source vectors, otherwise the procedure still costs more computing time than the iterative solvers. Consequently, the efficiency of the elimination method is only achieved when the following inequality is true: TCF < NRD TSORG or TCF < NRD TICCG , where TCF, TSORG and TICCG stand for the computing times required to solve eq. (10) using the Choleski factorization, SORCG and ICCG solvers, respectively. NRD is the total number of electrodes.
CONCLUSIONS For 3-D DC resistivity modelling and inversion, trilinear interpolation within an 8-node solid parallelepiped and linear interpolations within six tetrahedral bricks within the same 8-node solid block may be employed in the FEM, but the latter is far superior to the former in both accuracy and computational efficiency. With the compact storage, the tetrahedron element scheme saves at least 43 per cent in computer memory for an iterative solver and can produce much better simulating accuracy than the parallelepiped element scheme with the same grid size. The pre-conditioned conjugate gradient method is a suitable option as an iterative solver for 3-D FEM DC resistivity
688
Z. Bing and S. A. Greenhalgh
solutions, specifically the versions SORCG and ICCG. Our numerical experiments show that both have a much better convergence rate than SCG and TRICG. Applying the tetrahedron element scheme and ICCG, one can obtain the best convergence speed of all the iterative solvers. With the compact storage M(n, 8) in the tetrahedron element scheme, the main programming segments for SORCG and ICCG become very simple and it is shown that SORCG has only one additional vector storage V(n), but that ICCG needs nearly double the storage size of the others for the incomplete Choleski factorization L(n, 8). The banded Choleski factorization algorithm may be an alternative to a standard elimination solver for a 3-D resistivity inversion that involves hundreds of electrodes for the measurement, but it must be noted that the minimum storage size needs a 2-D array M(n, min(NXY, NXZ, NYZ)+1), and for a fine grid size which is half of a coarse grid size in each direction, the computing time becomes over 27=128 times that required for the coarse grid size. Such an increase in the computing time should be compared with SORCG or ICCG so as to achieve the maximum computing efficiency.
ACKNOWLEDGMENTS The authors are grateful to the Department of Primary Industries and Resources, South Australia and to the CSIRO Center for Groundwater Studies for funding the project. They also thank the anonymous reviewers for their comments.
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