Aplicació instrument (checklist)

To demonstrate the power of the computational methods introduced in this chapter, we solve another practical engineering problem introduced in Chap-ter 3, the free, undamped vibration of structures. This is called normal modes analysis in the industry. This solution will be the foundation of the dynamic reduction technique of the next chapter. It is also very important in the dynamic analysis of global structures to avoid vibration conflicts with the en-vironment in which the structure is operating. The computational problem is

(K− λM)u = 0.

The goal of normal modes analysis is to find the natural frequencies (λ) and corresponding vibration shapes (u) of the structure. Mostly the lowest nat-ural frequencies of the structure are of particular interest to the engineer.

FIGURE 9.3 Trimmed car body model

Occasionally the natural frequencies in a certain range are needed, to assure the avoidance of resonance catastrophes or annoying vibrations in the audible range.

Let us consider a trimmed car body automobile model shown in Figure 9.3.

Such models have all major components of the car, such as wheels, shocks, win-dows, etc., incorporated and lead to large sparse eigenvalue problems solved in the industry mostly by the Lanczos method.

TABLE 9.1

Model statistics of trimmed car body

Model Number of Number of Number of Number of

data nodes shells solids rigids

380,007 361,249 3,762 9,056

Sizes g n f a

2,280,042 2,223,139 2,223,109 1,937,282

FIGURE 9.4 Speedup of parallel normal modes analysis

The statistics of Table 9.1 show the characteristics of such a finite element model. Note, that they are not the statistics of the illustration model. As it was shown earlier such models usually contain a variety of elements and a large amount of constraints and rigid components.

The task of finding the natural frequencies and corresponding mode shapes of such a model is truly an enormous one. The distributed parallel computa-tion of Seccomputa-tion 9.5 enables the feasible execucomputa-tion of this task.

The statistics of the computation encompassing about 900 modes up to 200 Hz are shown on Table 9.2. The analysis was executed on a cluster of 8 work-stations, each containing 8 processors with 1.5 GHz clock cycle. The cluster had a 1 Gigabit Ethernet connection.

The elapsed time utilizing 32 processors is already a feasible execution, considering the work environment and time schedule in typical automobile companies. The efficiency above that decreases, but the speedup is still in-creasing. It peaks at 56 processors, as it is shown in Figure 9.4 where the horizontal axis is the number of processors. More efficient implementation of the computational technique may overcome this limit. Furthermore, the

technique may scale to over 100 processors with a larger problem or wider frequency range.

TABLE 9.2

Distributed normal modes analysis statistics

I/O Elapsed Elapsed Number of GByte min:sec speedup processors 2,028.4 523:58 1.00 1

266.9 83:41 6.26 8

191.1 45:16 11.57 16

98.3 34:07 15.35 32

77.8 27:14 19.23 48

67.1 24:41 21.22 56

61.4 27:00 19.40 64

Another model is used to demonstrate the computational complexity of dense component models, also from the automobile industry. A complete engine model, such as shown for example in Figure 9.5 consisted of approxi-mately 12 million node points and 7.5 million elements.

Table 9.3 contains statistics of the matrices in the eigenvalue analysis. The max terms column indicates the maximum number of nonzero terms in the densest column of the matrices. The zero columns of the M matrix is the zero subspace of M . The max front is the maximum front size of the factor matrix.

TABLE 9.3

Normal modes analysis dense matrix statistics

K number of rows nonzero terms max terms matrix 35.7 million 1.38 billion 18,9571 M number of rows zero columns max terms matrix 35.7 million 5,317,732 11

Factor number of rows nonzero terms max front matrix 35.7 million 43.8 billion 30,310

The task of finding the natural frequencies and corresponding mode shapes was executed up to 200 Hz on a workstation with 8 (1.95 GHz) CPUs. The

FIGURE 9.5 Engine block model

computation required about 680 minutes elapsed and 100,000 seconds of CPU time using all 8 processors of the workstation in a shared memory fashion.

11.5 Terabytes of I/O was executed and 650 Gigabytes of disk footprint was required. The computational complexity of such industrial normal modes ap-plications is overwhelming.

References

[1] Cullum, J. K. and Willoughby, R. A.; Lanczos algorithms for large sym-metric eigenvalue computations, Birkhauser, Boston, 1985

[2] Francis, J. G. F.; The QR transformation I. and II., The Computer Journal, Vol. 4, pp. 265-271, Vol. 5, pp. 332-345, 1961, 1962

[3] Givens, W.; Numerical computation of the characteristic values of a real symmetric matrix, Report ORNL-1574, Oak Ridge National Laboratory,

1954

[4] Householder A. S.; Unitary triangularization of a non-symmetric matrix, J. Assoc. Comp. Mach., Vol 5. pp. 339-342, 1958

[5] Komzsik, L.; The Lanczos method: Evolution and Application, SIAM, Philadelphia, 2003

[6] Lanczos, C.; An iteration method for the solution of eigenvalue prob-lem of linear differential and integral operators, Journal of the National Bureau of Standards, Vol. 49, pp. 409-436, 1952

[7] Parlett, B. N.; The symmetric eigenvalue problem, Prentice-Hall, 1980 [8] Wilkinson, J. H.; The calculation of eigenvectors of co-diagonal matrices,

The Computer Journal, Vol. 1, pp 90-92, 1958

The damped vibration of structures is described by M ¨v + B ˙v + Kv = 0,

where B is the damping matrix, and ˙v refers to the velocity. Executing a Fourier transformation yields the following quadratic eigenvalue problem

(M λ²+ Bλ + K)φ = 0.

The matrices of this quadratic eigenvalue problem may be complex and the problem may also have a left-handed solution

ψ^H(M λ²+ Bλ + K) = 0

that is different from the right-hand solution. Here and in the following, ^H denotes complex conjugate transpose. The solution of this problem usually results in complex eigenvalues. In order to solve the quadratic eigenvalue problem, a transformation is executed to convert the original quadratic prob-lem to a linear probprob-lem of twice the size.