Accelerating Construction of Non-Intrusive Nonlinear Structural Dynamics Reduced Order Models through Hyperreduction

Abstract

We present a novel technique to significantly reduce the offline cost associated to non-intrusive nonlinear tensors identification in reduced order models (ROMs) of geometrically nonlinear, finite elements (FE)-discretized structural dynamics problems. The ROM is obtained by Galerkin-projection of the governing equations on a reduction basis (RB) of Vibration Modes (VMs) and Static Modal Derivatives (SMDs), resulting in reduced internal forces that are cubic polynomial in the reduced coordinates. The unknown coefficients of the nonlinear tensors associated with this polynomial representation are identified using a modified version of Enhanced Enforced Displacement (EED) method which leverages Energy Conserving Sampling and Weighting (ECSW) as hyperreduction technique for efficiency improvement. Specifically, ECSW is employed to accelerate the evaluations of the nonlinear reduced tangent stiffness matrix that are required within EED. Simulation-free training sets of forces for ECSW are obtained from displacements corresponding to quasi-random samples of a nonlinear second order static displacement manifold. The proposed approach is beneficial for the investigation of the dynamic response of structures subjected to acoustic loading, where multiple VMs must be added in the RB, resulting in expensive nonlinear tensor identification. Superiority of the novel method over standard EED is demonstrated on FE models of a shallow curved clamped panel and of a nine-bay aeronautical reinforced panel modelled, using the commercial finite element program Abaqus.

Figures (10)

• Figure 1: Geometry (a) and FE mesh (b) of the curved panel.
• Figure 2: Time history (a) and PSD (b) of the uniform in space pressure applied to the curved panel. PSD was obtained with Welch's method. In (c) static modal participation factor of the pressure load, while in (d) natural frequencies of the curved panel. VMs before $f_{\text{coff}}$ are plotted in red, the one after in blue.
• Figure 3: Reduced mesh for the curved panel. Color intensity maps to ECSW element's weight. The number of active elements is $\tilde{N}_e = 73$, corresponding to $4.71\%$ of the total number of elements in the original FE mesh.
• Figure 4: Time domain displacements of node A $(x_A = 0.5\cdot l , y_A = 0.516\cdot w, z_A = h)$ and their PSDs. In (a) and (b) out of plane displacements, while in (c) and (d) in plane displacements. Displacements in (a) and (c) are normalized with respect to the thickness of the plate. The four lines in each plot correspond to solutions obtained using Abaqus HFM (Abaqus), the ROM with tensors identified through EED (ROM EED), the ROM with tensors identified using EED-ECSW (ROM EED-ECSW) and the linear ROM (ROM Lin). The RB used in the nonlinear ROMs consists of 7 VMs and 28 SMDs.
• Figure 5: Time domain displacements of node B $(x_B = 0.34\cdot l, y_B = 0.322 \cdot w, z_B =0.90 \cdot h)$ and their PSDs. In (a) and (b) out of plane displacements, while in (c) and (d) in plane displacements. Displacements in (a) and (c) are normalized with respect to the thickness of the plate. The four lines in each plot correspond to solutions obtained using Abaqus HFM (Abaqus), the ROM with tensors identified through EED (ROM EED), the ROM with tensors identified using EED-ECSW (ROM EED-ECSW) and the linear ROM (ROM Lin). The RB used in the nonlinear ROMs consists of 7 VMs and 28 SMDs.