Negative pivot in decomposition of correlation matrix

MarcosBeier — Thu, 24 Aug 2023 22:52:25 GMT

Current Revision posted to LARS | LEAP | OpenBridge | OpenTunnel | RM Wiki by MarcosBeier on 8/24/2023 10:52:25 PM

If the users now get the error message “Negative pivot in decomposition of correlation matrix” in CQC or CQCX mode superposition there are two conditions required and possible workarounds to still use the RM mode superposition method. The method of modifying the result vectors, or factorized natural modes, by multiplying them with the decomposed correlation matrix [a] is derived from the equation below.

[ρ] = [a]T*[a]

This is based on the following assumptions.

1) The structure of the correlation matrix is such that there exists a matrix [a] for which the equation [ρ] = [a]T*[a] is fulfilled

2) The matrix a is defined in the real space (no imaginary or complex terms)

This requires analyzing the nature of the correlation matrix [ρ]. As the CQC rule is the current standard and DSC is barely applied in practice, we concentrate on the following on the CQC rule using damping ratio for the mode and with a certain duration. The first condition that [ρ] = [a]T*[a] is generally fulfilled when we have a symmetric matrix with positive coefficients. By applying the above formula and setting ρj,i to ρi,j this condition is automatically given. Using the condition i > j and filling up the matrix with symmetric terms ρj,i is not explicitly mentioned in the paper below, but seems logic.

The second condition that the matrix is positive definite is usually fulfilled, when the off-diagonal terms are much smaller than the diagonal terms. This is apparently the case when the different Eigenfrequencies are very different (r = j/i << 1.0). However, if we have closely spaced natural frequencies the off-diagonal correlation terms may come near to 1.0 and the Cholesky decomposition might fail. The physical meaning of failure of the Cholesky decomposition is not yet clear and topic of further research work. We are currently trying to find an alternative method to offer a completely stable CQC procedure for TDV mode superposition.

Note: Previous program versions used a slightly different formula which was found in one of the used textbooks. Using r*ξi instead of r*ξi in the numerator seems to be a misprint in this textbook (all other papers conform to using r*ξj). However, the old formula seems to be more stable with respect to allowing for decomposition in real space. Therefore we get sometimes failure in the Cholesky decomposition in cases, where the decomposition of the old obsolete correlation matrix had no problem.

Workarounds:
Practical experience showed that the decomposition problem is mostly arising when a very high number of modes are considered. A possibility in practice would be to limit the number of considered modes if the high modes do not influence the result. The user can check the mass participation of the different modes and restrict the number of considered modes to those that still have relevant mass participation. EN for instance recommends considering modes up to the point where 90% of the total mass is covered (see EN 1998-2, 4.2.1.2 (2)). In certain cases, e.g. if a considerable part of defined masses are related to sub-structure, even a smaller mass coverage is allowed.
Also, the user can compare the maximum values (calculation without RM mode superposition) calculated with the full number of modes and with the reduced number of modes. If they are very similar, then there is no objective to use RM mode superposition with a reduced number of modes.
Another possibility is to use a higher duration of the earthquake. A high duration value has in fact the property that damping will be rather constant for all modes. For constant damping the matrix [ρ] seems to be less ill-conditioned and problems with decomposition should less often occur. The textbooks give a limit of 5 times the lowest vibration period for CQC to be reasonably accurate. However, RM Bridge uses 60 seconds as the default value; this is a fairly high value and further increasing it will normally not improve the situation.

Bibliography: A Replacement of the SRSS Method in Seismic Analysis, E.L. Wilson, A. Der Khiuregian and E.P. Bayo, Earthquake Engineering and Structural Dynamics, Vol. 9, 187-194, 1981