Towards Compact Structural Bases for Coupled Structural-Thermal Nonlinear Reduced Order Modeling 199 (a) (b) (c) Fig. 2 Three temperature fields in the two-temperature-field scenario. (a) Quarter heating, (b) center heating, and (c) a linear combination of the quarter and the center heating Using a Galerkin approach, the governing equations of the coupled structural-thermal NLROM have been derived based on thermoelasticity theory, see [1] for details. In particular, when the structural properties are independent of the temperature, the governing equations for the structural generalized coordinates qn(t) are Mij ¨qj +Dij ˙qj + K (1) ij −K (th) ijl τl qj +K (2) ijl qj ql +K (3) ijlp qj ql qp =Fi +F (th) il τl (3) The thermal effects on the structural deformation are twofolds: one of them is the change of the linear stiffness of the structure through the termK (th) ijl τl on the left hand side (LHS) of Eq. (3), usually inducing a softening effect responsible for thermal buckling. The other effect is the appearance of the pseudo force F (th) il τl on the right hand side (RHS) of Eq. (3), which typically induces notable in-plane deformations at the contrary of mechanical loads, e.g., pressure, which are typically transverse dominated. The in-plane deformations induced by the pseudo force are different from the membrane stretching effects due to the geometric nonlinear effect of large deformation, as will be shown later. The current investigation focuses on the construction of the structural basis φ(n) which must be able to capture the LHS and the RHS thermal effects, in addition to the nonlinear geometric effects due to large structural deformations. The strategy employed here for this construction is an “enrichment” approach, i.e., starting from the isothermal (cold) structural basis, additional modes (enrichments) are sought to capture the thermal effects. Various enrichment options have been defined in
RkJQdWJsaXNoZXIy MTMzNzEzMQ==