Towards Compact Structural Bases for Coupled Structural-Thermal Nonlinear Reduced Order Modeling 203 Fig. 5 Means and standard deviations of the representation errors of the nonlinear static responses to the set of random temperature fields vs. the number of modes in the 37-mode basis with pPOD enrichments. (a) Transverse, Ty; (b) in-plane, Tx; and (c) in-plane, Tz 4.2 Method of Static Condensation The other strategy investigated to reduce the size of the basis is to perform an approximate static condensation of the enrichments. There are significant assumptions associated with this process. Specifically, the enrichments must be very strongly in-plane dominant so that (1) the natural frequencies to which they are associated are much larger than those of the transverse modes. It is also required that (2) the nonlinear restoring forces present in the enrichment equations be essentially linear with respect to the enrichment generalized coordinates and involve a coupling with the transverse modes that is only through the quadratic terms of their generalized coordinates. Effectively, the equations for the enrichments generalized coordinates must be approximated in the form K(1) ee qe −K (th) eel qeτl +K (1) et qt −K (th) etl qt τl +K (2) ett q 2 t =Fe +F (th) el τl, (11) where the subscript e and t refer to the enrichments and to the rest of the basis, respectively. To maintain a cubic stiffness for the generalized coordinates of the remaining modes, it is further assumed that these have the following form before condensation K (1) tt qt −K (th) ttl qt τl +K (1) te qe −K (th) tel qeτl +K (2) ttt q 2 t +K (2) tte qt qe +K (3) tttt q 3 t =Ft +F (th) tl τl, (12) Even then, for the structural-thermal coupled ROM, the static condensation is more complicated since the structuralthermal coupling coefficients also need to be considered. The process is exemplified below with a 2 mode model with 1 such enrichment (1e mode and1t mode). Note that the linear coupling stiffness coefficients and the in-plane static forces, which are usually ignored in the static condensation, are considered in the present study. From Eq. (11), qe = Fe +F (th) el τl − K (1) et qt −K (th) etl qt τl +K (2) ett q 2 t K (1) ee −K (th) eel τl . (13)
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