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On the Use of the Subspace Iteration Method

The solution of frequencies and mode shapes of large structural systems is now common practice, and the subspace iteration method is a technique widely used for such solutions.

When, some decades ago, the method was proposed in Ref. [1], a large problem was a structural model of 10,000 degrees of freedom for which the lowest 20 frequencies and mode shapes had to be computed. Since then, because of its reliability and effectiveness, the subspace iteration method has been used widely on structural and finite element models with now millions of degrees of freedom and hundreds of frequencies and mode shapes, and has also been used in entirely new fields of applications.

The basic equations of the subspace iteration method are, see Ref. [2]



As discussed in Refs. [1] and [2], the method combines inverse iterations on a subspace – hence the name 'subspace iteration method' – with Rayleigh-Ritz analysis to extract the best eigenvalues and eigenvectors from each subspace. The key is to establish a 'good' and adequate set of starting iteration vectors. The solution procedure can also be accelerated through shifting.

The importance of an adequate set of starting iteration vectors for fast convergence has been studied, for example, in Ref. [3], where the method has been used in the solution of frequencies and mode shapes of proteins; Figure 1 shows a typical protein. The use of an optimal number of iteration vectors significantly reduced the number of required iterations, and the computational time increased linearly with the number of eigenpairs sought. Another important application in protein analysis is the calculation of frequencies and mode shapes in conformational change pathways. Here the eigensolutions are sought in numerous neighboring conformations and, as indicated in Figure 2, the subspace iteration method is particularly well suited for this computational task. More details on these applications are given in the Ref. [3].




Figure 1  A typical protein





Figure 2  Solution time per conformation using the subspace iteration method, see Ref. [3]


In this News we focus on another important item, namely the use of parallel processing in the subspace iteration method. Considering the basic equations given above, it can be seen that the method entails an initial LDLT factorization of the K-matrix, and then forward-reductions and back-substitutions of many iteration vectors. While the factorization of the coefficient matrix can be performed to some extent in SMP and DMP, i.e. shared and distributed parallel-processing, the forward reduction and back-substitution of the iteration vectors particularly lends itself to parallelization because each vector is calculated independently from all others. Hence, blocks of vectors can be assigned to different nodes of a machine and can be processed simultaneously.

ADINA offers SMP and DMP for the subspace iteration method and the method can be used in linear and nonlinear analyses including large deformations and contact.

Figure 3 shows the finite element model of a coil and support structure of a plasma fusion device. We have used this model in earlier studies and the model contains many contact surfaces. The movie above shows its sixth mode shape of vibration. The frequencies were solved on an HP cluster consisting of 4 nodes, each with dual quad-core processors running at 2.66 GHz connected by InfiniBand, and the solution times given in Figure 4 have been obtained.




Figure 3  Finite element model of a coil and support structure of a plasma fusion device; about 6 million degrees of freedom & 120,000 contact segments





Figure 4  Solution times for frequencies and mode shapes of the plasma fusion device model


Figure 5 shows the model of a bolted wheel structure that we also considered in earlier studies. The frequencies and mode shapes were calculated using the same machine with the solution times given in Figure 6. It is interesting to note that in this case the solution time is increasing almost linearly with the number of eigenpairs sought, whereas the solution time in Figure 4 is increasing to a higher degree but not by much. We did not, in either case, try to optimize the computational times, and did not apply any acceleration techniques such as shifting.




Figure 5  Finite element model of a bolted wheel assembly; about 2.5 million degrees of freedom & 5,000 contact segments




Figure 6  Solution time for frequencies and mode shapes of the bolted wheel assembly model


These are some experiences regarding the application of the subspace iteration method for the calculation of frequencies and mode shapes – the parallelization is clearly of general interest in linear and nonlinear structural analysis but also in other fields, like in the normal mode solutions of proteins.

References

  1. K.J. Bathe, "Solution Methods for Large Generalized Eigenvalue Problems in Structural Engineering", Report UCSESM 71-20, Department of Civil Engineering, University of California, Berkeley, November 1971.

  2. K.J. Bathe, Finite Element Procedures, Prentice Hall, 1996.

  3. R.S. Sedeh, M. Bathe, and K.J. Bathe, "The Subspace Iteration Method in Protein Normal Mode Analysis", J. Computational Chemistry, 31:66-74, 2009.


Keywords:
Subspace iteration method, structures, frequency, mode shape, parallel-processing, shared memory, distributed memory, proteins

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