on the application of an element-by-element lanczos ... - Science Direct

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ON THE APPLICATION OF AN ELEMENT-BY-ELEMENT LANCZOS SOLVER TO LARGE OFFSHORE STRUCTURAL ENGINEERING PROBLEMS ALVARO L. G. A. GXJIINHO, Jo& L. D. ALWS, LUIZ LANDAU, EDISON C. P. LIMA and NEMN F. F. EBECKEN Civil Engineering Department, COPPE/UFRJ,

Caixa Postal 68506, 21945 Rio de Janeiro, Brasil

Abstract-Iterative methods for solving large sets of linear quations have been used as an alternative to direct methods of solution since the early beginning of numerical analysis. The conjugate gradient method (CGM), one of the most widely used, seeks a solution that minimizes the potential energy of the finite element assemblage. Recently, the use of Lanczos algorithm for the solution of large sets of linear equations has been reexamined. Lanczos biorthogonalization procedure is an oblique projection method that provides a solution approximation whose residual is orthogonal to a Krylov subspace. It has been shown that Lanczos and CGM share several properties but the former has the advantage of not being necessary to compute the approximated solution at each iteration. Jacobi preconditioning can also be employed in order to accelerate convergence. The Lanczos procedure was implemented using an element-by-element (EBE) scheme. The applications spread over typical offshore engineering problems encompassing regular and irregular meshes. These problems are normally ill-conditioned when compared with continuum problems. For all the analyses addressed the element-by-element Lanczos proccdures

presented outstanding computational

efficiency.

compute the approximated solution at each iteration. This method is also a powerful tool to solve large symmetric eigenvalue problems [6,7]. Furthermore, it is well known that Lanczos and conjugate gradients

INTRODUCTION methods for solving large systems of algebraic equations arising from finite element and finite difference discretizations of partial differential equations are currently subjected to a critical review facing the new generation of processors, whose main feature, is the possibility of parallelization. Generally these systems of equations can be expressed as Iterative

Au=b,

share the same properties in infinite precision arithmetic, the derivation of each method from the other being possible (see, for a good presentation [q). Therefore, Lanczos procedures may be used as an alternative to conjugate gradients as suggested by Parlett [8] and even for nonsymmetric systems of equations [ 1,9]. Successful applications of Lanczos procedures, in conjunction with Newton-type iterations, are given in [IO] for nonlinear optimization problems and in [1 1, 121 for nonlinear finite element equations. However, focusing on finite element systems of equations, the recent work of Hughes [2]and Carey [ 131 suggest element-by-element (EBE) schemes. The EBE approach seems to be extremely suitable for vector&d and parallel machines. Following these considerations, this paper presents an element-byelement Lanczos procedure (EBELAN, for short) for the solution of large systems of finite element equations. The EBELAN and the EBELAN with Jacobi preconditioning (EBEJLAN) were applied to the solution of large ill-conditioned, offshore structural engineering problems discretized by finite elements. The results were compared with those obtained by an element-by-element conjugate gra dient (EBECG) procedure, also with Jacobi preconditioning (EBEJCG), and when possible, with the Accelerated Viscous Relaxation introduced by Zienkiewicz[3], also implemented as an element-by element (EBEAVR) scheme.

(1)

where A is a large, sparse, positive definite and, restricting our discussion, symmetric matrix of order n, and II, b n-dimensional vectors, b being given. The solution sought, u, can be obtained by stationary iterative methods, such as SOR or SSOR methods [l], parabolic regularization methods [2,3] or variational methods such as the steepest descent method and conjugate gradient method, introduced by Hestenes and Stiefel [4]. In the latter case it can be shown that solving (1) is equivalent to the minimization of the corresponding quadratic functional f(u) = ;urAu - u%

(2)

Another alternative is the Lanczos biorthogonalization procedure, introduced by L.anczos[S] in the same year as conjugate gradients. Lanczos method is an oblique projection method that provides a solution approximation whose residual is orthogonal to a Krylov subspace. Contrary to CGM, Lanczos has the property of not being necessary to 21

ALVAROL. G. A. COUT~NHO e/ al

28

Table 1. Summary of Lanczos procedure

where

Q,‘Q, = 1,.

A - INITIALIZE

---__--___

v :- 0 ; r :-

It can be shown that there is no need to proceed with a complete Gram-Schmidt orthogonalization during each iteration [7-9,11,12], only the orthogonalization against the two previous vectors being necessary. Therefore, in the jth iteration, the weak form of eqn (1) can be written as

b ; p. :- (rTr)1’2 ; $* := 1.

B - LANCZOS MGORITUR _________________ for k

0.1.2.

l

. . . do

8.1 - Lancros step

aavc r ( Lancros vector

r:-r/p

Q; Au = Q,%

)

v := -pv @k:- r

and, introducing

the coordinate

v:=Ar+v euap v and r

U=

a:- v'r .f :- r -0s~ \:=

ei

(3:’

(r r)

QjX,

(6) transformation,

1

(7)

the weak form is given by

t

T,x, = B,e;,

l/2

B.2 - Evaluate

1 k - 1,2,

(8)

where residual

norm PO -PI

. . . ;oio ‘K,

go -PO )

yk := atant pk / 6~ k_l)

kk

(5)

:- rin ‘y, p

T, =

(9)

+ con YkKk

:- CO8 y,p

iD

:- sin

8k

y, 8k_l

is a tridiagonal matrix whose entries a, /l are the Gram-Schmidt orthogonalization parameters, e: is the first column of the identity matrix I,, and

:‘$k/&k

qk 8.3 -

Check

convergence

lf p$k < TOL then go’ to

1

C

q1=-

llre B.l

The matrix T, is the projection of A onto the Krylov subspace: (11) T, = Q: AQ,.

end 100~.

C - SOLVE( Gausslli8ination _____

1

D - CCMPUTE SOLVTION -____--.._--_--_u - Qa

( Q : Lanczor vectors stored in 8.1 1

LANCZOS ALGORITHM

=

SPAN (b, Ab, A2b, . . . , Ai- ‘b),

(3)

where j refers to the j th iteration, improving the Krylov sequence by Gram-Schmidt orthogonalization procedure, leading to the set of vectors

Q,= (q,

9

q2, .

3

q/l,

Therefore, like conjugate gradients convergence will be achieved theoretically in n iterations. However, in finite precision arithmetic this is not true and there will be convergence within a preset tolerance only after m iterations. In practice, the solution sought will be generally obtained for m +.iliJ,

(24)

i-l

where ii is an expanded vector with element related nodal quantities. The relation (21) is the essence of the simple element-by~lement scheme, as introduced by Hughes f2] and Carey [ 131. The outstanding characteristics of such a scheme can be readily seen, once nodal points ordering as well as elements ordering does not affect the number of required arithmetic operations while evaluating (24). In such a scheme the global coefficient matrix assemblage is not necessary, thus reducing the required storage area and, further, providing a good mean for the treatment of global matrix sparsity. For the Lanczos algorithm with Jacobi pr~on~tioning, the pr~onditioning matrix is assembled once and each element matrix is evaluated in the t~nsfo~~ form as A; = (C-i’2)T&;(C-1/2),

(25)

A: being the preconditioned expanded matrix for the ith finite element. However, it should be noted that the computations involved are performed at element level, the element matrices being sequentially stored. All in all, the EEE scheme is indeed a very suitable and natural choice for the solution of large sets of FEM equations on the recently developed generation of parallel and vector processing machines.

ALVAW L. G. A. Coun~~o

E = 210 GPa o= 0.3

t,b=0.9cm k = I.1 cm

et al.

a = l&km

c = 26cm

b=292cm

d =60&m

Fig. 1. Stiffened panel finite element mesh.

Table 3. Computer performance

This section presents numerical results obtained from the application of the element-by-element standard Lanczos procedure (EBELAN) and the Jacobi preconditioned procedure (EBEJLAN) to typical offshore structural en~nee~n~ problems. Results are compared with EBE conjugate gradient procedures (EBECG and EBEJCG), and, when possible, also with an EBE Accelerated Viscous Relaxation (EBEAVR) procedure. All these procedures were implemented by the authors [15-13 using standard FORTRAN and the exampies processed on the IBM 3081 computer. SIiffened panel The stiffened panel considered is a typical hull structural element of a self-elevating drilling unit designed for 1OOm water depth operation. The stiffened panel was discretized by 240 triangular flat-she11 finite elements and 143 nodal points considering the double symmetry of the panel. The resulting mesh, comprising 639 degrees of freedom and halfbandwidth 93, is shown in Fig. 1. Loading is a ~on~tudinal uniform compression of 20.6 MPa acting on the plate and stiffeners. Table 2 shows a comparison of the displacements in the load directian, at nodal point NP, for a residual norm tolerant vary ing from lo-’ to 10e6. As can be seen in Table 2, EBELAN and EBEJLAN leads to sbluiions with the same accuracy as the other methods. The necessary parameters of EBEAYR, the time step and acceleration parameter ratio, were taken, after a parametric

Method EBEJLAN EBEJCC EBELAN EBECG EBEAVR

Number of iterations

CPU time (W

40 55 271 154 447

17.77 20.89 81.82 49.30 134.24

Residual norm tolerance IO-‘. as 0.0582sec and 1.0, respectively. This was necessary because parameters for the application reported by Zenkiewicz 133were not suitable for this case. The computer performances of such solutions are shown in Tables 3-6. These tables present the number of iterations needed, the CPU times and the speed-up of EBEJLAN for the residual norm ranging from low (1W3) to high (10W6) accuracy. From the results shown in Tables 3-5, it can be verified that EBEJLAN achieved the best computer performance, presenting a speed-up against EBEJCG ranging from l&9%, for a low accuracy solution to 20.4%, for high precision (residual norm study,

Table 4. Computer performance Number of Method

EBEJCAN EBEJCG EBELAN EBECG EBEAVR

iterations

46 66 484 289 923

Residual norm tokrance

CPU time (*I 19.67 24.44 144.06 86.64 279.2 1 tOe4.

Table 2. Displacements in load direction ( x IO-’ m) Residual norm toterancg

EBE&.AN

EBELAN

EBUCG

EBECG

EBEAVR

10-L” 10-’ 10-J 10-e

-0.286885 -0.286502 -0.286424 -0.286433

-0.286502 -0.286447 -0.286435 -0.286435

-0.286376 -0.286433 -0.286439 -0.286433

-0.286515 -0.286486 -0.286455 -0.286447

-0.286417 -0.286433 t t

f No convergence.

Speed-up (%I 14.9 78.3 64.0 86.8

Speed-up (%I 19s 86.4 76.6

93.0

Lanczos method for large offshore structural engmeering problems Table 6. Computer performance

Table 5. Computer performance Method EBEJLAN EBEJCG EBELAN EBECG EBEAVRt

Number of iterations 71 99 936 420 3000

CPU time (se@ 27.01 33.93 271.24 123.90 899.72

Residual tolerance norm IO-‘. t No convergence.

31

Speed-up (%I 20.4 90.0 78.2 -

Method EBEJLAN EBEJCG EBELAN EBECG EBEAVRt

Number of iterations 122 113 1090 485 3141

CPU time @cJ 42.16 37.76 315.71 262.86 899.73

Residual norm tolerance IO-‘. t No convergence.

Fig. 2. Steel jacket platform finite element mesh.

Speed-up (%I -11.7 86.7 84.0 -

ALVARO L. G. A. COLJTINHO er a[.

32 Table 7.

Storm wave Height (m) 14.0

Table 9. Computer performance

Environmental actions

Period (xc)

Current velocity (m/sac) Bottom Surface

11.1

1.55

Number of iterations

Method EBEJLAN

149

107.46

EBEJCG EBELAN EBECG

249 615 921

204.86 419.68 607.51

0.09

tolerance = lO-5). However for the high accuracy solution (Table 6), EBEJCG had the better computer performance. It also should be noted in these tables that EBECG presented a better performance than EBELAN for all tolerances. Although using improved parameters, it should be observed that EB-

Method

EBELAN

EBECG

0.1474 -0.01573

0.1474 -0.01573

0.1474 -0.01573

(b) Deck rotations (x 10-r rad/sec) 0.1808 0.1806 RX -0.3921 -0.3898 RY

0.1806 -0.3905

0.1805 -0.3895

(a) Deck displacements (m) 0.1474 X -0.01571 2

47.5 74.4 82.3

EAVR did not produce cost-effective solutions. Also it should be mentioned that, employing a time step of 0.0233 set and an acceleration parameter ratio of

EBEJCG

EBEJLAN

SW-up (%)

Residual norm tolerance IO-‘.

Table 8. Deck displacements and rotations Global direction

CPU time (W

-cl-

EBEJLAN

_O-

EBEJCG

A

EBELAN

-+-

EBECG

-X-

EBEAVR

ITERATIONS

X IO’

Fig. 3. Residual norm evolution.

Lanczos method for large offshore structural engineering problems Table 10. Comparison bctwecn direct and iterative solutions Storage area CPU time

Method

(see)

in core disk accesses (KDP words) Number of

107.46 105.32

EBEJLAN Direct

12,457 56,520

3:

convergence was not achieved after 3000 iterations even for a residual norm tolerance of IO-‘.

0.05,

Steel jacket

plaflorm

The steel jacket platform studied is a conventional structure, designed for a I70 m water depth, similar to several platforms installed in Campos Basin, Rio de Janeiro, the main oil production site offshore Brasil. The structure was considered simply supported on seabed. It was discretizcd using 1097 space

(a)

LEG

CROSS

Dimensions

(b

1 JACKING Dimensions

frame elements, 432 nodal points, a total of 2463 degrees of freedom and half bandwidth 345. Figure 2 depicts a perspective of the resulting finite element mesh. The static equivalent nodal loads from the environmental actions corresponding to the storm wave and current shown in Table 7, were evaluated using Morison equation and Airy linear wave theory as given by Sarpkaya [18]. Wave and current were taken as acting in X global direction. For a residual norm tolerance of IO-‘, a linear static analysis was carried out, using the procedures presented herein. The obtained results for deck displacements and rotations are shown in Table 8. As can be observed in Table 8, the EBELAN and EBEJLAN solutions are as accurate as the EBECG and EBEJCG solutions. EBEAVR was also processed with the previous parameters, but no convergence was achieved after 3029 iterations. The

SECTION in mm

SYSTEM in

33

DETAIL

mm

Fig. 4. Self-elevating drilling unit leg structural arrangement. (a) Leg cross section. (b) Jacking system detail.

34

ALVARO L.

G. A.

computer performance of the obtained solutions are shown in Table 9, which presents the number of iterations needed, the CPU times and the speed-up of EBEJLAN. It should be observed in Table 9 that EBEJLAN presented a speed-up factor against EBEJCG of 47.5%. Further, unlike the previous application, the computational performance of EBELAN was also better than EBECG. The convergence of each solution can be visualized in Fig. 3. This figure shows a plot of the residual norm against the number of iterations. Although convergence was not achieved, the residual norm evolution of the EBEAVR solution is included in Fig. 3. Finally a direct solution (Gauss elimination) was carried out using a block band storage scheme. For a working area of 30 K double precision words, the stiffness matrix was partitioned into 39 blocks of 64 equations each. A comparison between the computer

GXJTINHO et

al.

performance of the direct and EBEJLAN solutions is given in Table 10. As can be seen in Table 10, both solutions required CPU times of the same order, but the number of disk accesses of the direct solution is about four times that of EBEJLAN. Regarding core usage, the comparison favours even more the iterative solution, once the direct solution requires six times more storage than EBEJLAN. Therefore, for the solution of this particular linear problem, the EBEJLAN procedure was more cost-effective than the direct solution. Tubular joint

This application is concerned with the evaluation of the stress distribution of an offshore tubular joint. This joint is a leg structural detail on a self-elevating drilling platform. Figure 4(a) shows the cross section of the platform leg and Fig. 4(b) depicts a side view, allowing the visualization of the jacking system. The tubular joint analyzed is also indicated in Fig. 4.

Fig. 5. Tubular joint finite element mesh.

35

Lanczos method for large offshore structural engjn~ring problems

’ 0.01

4.01

8.01

12.01 16.01 ITERATIONS x IO’ Fig.

20.01

24.01

6. Residual norm evolution.

Considering symmetry, lhc tubular joint was discretized through 3494 triangular fiat shell elements, 235 space frame elements and 1869 nodal points, comprising 10764 degrees of freedom and halfbandwidth 531. Figure 5 shows a perspective of the resulting finite element mesh. The space frame elements were employed in the simulation of three different situations. First, for the gear rack discretimtion along the main tube; next, for the domain extension up to the adjacent joints of the self elevating platform; and finally for the

transition between the two domains, that discretized by shell and space frame elements. For the latter situation, very large stiffness characteristics were assigned. Therefore, this practical problem can be considered, for the purposes of numerical analysis, very ill-conditioned. The actions are nodal forces which were evaluated in a global anaIysis for the whole self-elevating pIatform, and then applied in the ~~~imensiona~ joint model at the ends of the space frame elements used for shell domain extension, The tubular joint is

Table 11,Computer performance Method

Number of iterations

CPU time

Number of core usage disk accesses (KDP words)

EBEJLAN

1524

119min 1.84sec

396.902

25

EBEJCGt

3139

240 min

870,272

55

t No convergence.

28.01

ALVARO L. ,

G. A. C~UTINHOPX al. I

Lanczos method for large offshore structural engineering problems

considered simply supported at the nodal points corresponding to the jacking-system sprockets engaged with the rack, at the middle of a bay. A linear static analysis of the tubular joint was carried out using the EBEJLAN and the EBEJCG procedures. The residual norm tolerance was taken as 5 x JO-‘. The evolution of the residual norm, for both methods, is shown in Fig. 6. As can be observed in Fig. 6, convergence was only achieved for EBEJLAN. Although the EBEJCG solution was far from convergence, the computer performances of each method are given in Table 1i. As can be seen in Table If, the number of iterations needed by EBEJLAN to reach convergence was 1524, which is 14% of the total number of degrees of freedom. Moreover, it also should be noted that the core usage of EBEJLAN is almost one half that of EBEJCG. Finally, Fig. 7 presents the stress distribution on the main tube determined by EBEJLAN. The stress gradients around the braces and supports where the gear box is located can be observed in this figure. DZSCUSSIONAND CONCLUSIONS

The EBE Lanczos procedures presented herein, notably the Jacobi preconditioned procedure (EBEJLAN) achieved, without loss of accuracy, a better computer performance than the EBE procedures based upon conjugate gradients (EBECG and EBEJCG). The numerical applications addressed were, apart from mesh regularity reasons, noticeably ill-conditioned problems, once distinct physical nodal quantities are involved. Even in this circumstance, the EBJLAN was able to solve a particuiar linear problem presenting a better computer effectiveness than a direct solution. However, this behaviour cannot be generalized, and further experiments should be performed. Regarding the AVR solutions, it is the authors’ opinion that more detailed parametric studies should be carried out in order to try and achieve computer performances similar to those reported in the related reference [3]. Finally, successful applications of conjugate gradients in high-speed parallel environments have been reported by several authors (e.g. (19,201). The present authors believe that Lanczos based procedures offer an interesting alternative whose behaviour in such environments should be carefully examined. Acknowledgemenfs-The authors are indebted to IBM Brasil S.A. for the assistance in scientific research at the Civil Ennineerinn Denartment of COPPE/UF~. the Graduate C&e of the federal University of Rio de Janeiro, We would also like to thank MS W. M. Barros and Mr G. L.

37

Souza for their help in the preparation of the tinal manuscript.

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for non-linear transient heat conduction analysis employing element-by-element iterative strategies, Compuf. Me&. uppl. Mech. Engfzg5% Xl-815 (1985). 3. 0. C. Zienkiweicz and R. Liihner, Accelerated relaxation or direct solution? Future prospects for FEM. ht. 3. Namer. Meth. EnipnR21, i-11 (1985). 4. M. R. Hatenes and E. Stiefel, Method of conjugate gradients for solving linear systems. 1. Res. Nor. Bur. Srandardr, Seer. B 49, 409-436 (1952). 5. C. Lanczos, Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bur. Sfat&r& Serf, B 49, 33-53 (1952). 6. 8. N. Parlett, The Symmefric Eigeava~~ Problem. Prentice-Hail, Englewood Cliffs (1980). 7. J. K. Cullum and R. A. Willoughby, Luncros Afgorifkms Jar Large Symmefric Eigenvalue Computations. Vol. I Theory, Vol. II Programs. Birkhiiuxr, Boston (1985). 8. 8. N. Parlett, A new look at the Lanczos algorithm for s&in8 symmetric systems of hnear equations. Lin. Alg. Appt. 29, 323-346 (1980). 9. Y. Saad, The Lanaos biorthogonalization algorithm and other oblique projection methods for solving large unsvmmetric svstems. SIAM. J. Numer. Anal. 19. 485-506, (1982). 10. S. G. Nash, Newton-type minimization via the Lanczos method. SIAM, J. Namer. Anal. 21, 770-788 (1984). If. B. Nom-Omid, A Newton-Lanczos method for soiution of nonlinear finite element equations. Ph.D. Thesis, University of California, Berkeley (1981). 12. B. Nom-Omid, A Newton-Lanczos method for solution of non-linear finite element equations. Comput. Strucf. 16, 242-252 (1983). 13. G. F. Carey and B.-N. Jiang, Ei~ent-by-element linear and non-linear solution schemes, Comm. uppi. Numer. Mesh. 2, 145-154 (i986). 14. C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations. SIAM, J. Namer. Anal. 12, 617-629 (1975). 15. A. L. G. A. Coutinho, The Lanczos algorithm in static and dynamic structural analysis. D.&Z. Thesis, COPPE/UF~ 119871. 16. A. L. G. A. Coutinho, J. L. D. Atves and L. Landau, CONGR program: an EBE conjugate gradient solver for FEM equations. Res. Report, COPPE/UFRJ (1986) (in Portuguese). 17. A. L. G. A. Coutinho, J. L. D. Alves and L. Landau, AVR program: an EBE accelerated viscous relaxation solver for FEM equations. Res. Report, COPPE,WFR.i (i986) (in Portugu~). IS. T. Sarpkaya and M. Isaacson, Mechanics of Wuue Farces on Off-shore Sfrucrures. Van Nostrand Reinhold, New York (1981). 19. D. R. Kincaid, T. C. Oppe and D. M. Young, Vectorized iterative methods for partial differential equations. Comm. appi. Namer. Mesh. 2, 289-296 (1986). 20. M. K. Seager, Overhead ~~ide~tio~ for uaraflelizing conjugate gradient. Comm. Mesh. 2, 273-280 f f 986).