Model-reduction techniques aim to reduce the computational complexity of simulating dynamical systems by applying a (Petrov-)Galerkin projection process that enforces the dynamics to evolve in a low-dimensional subspace of the original state space. Frequently, the resulting reduced-order model (ROM) violates intrinsic physical properties of the original full-order model (e.g., global conservation, Lagrangian structure, state-variable bounds) because the projection process does not generally ensure preservation of these properties.However, in many applications, ensuring the ROM preserves such intrinsic properties can enable the ROM to retain physical meaning and lead to improved accuracy and stability properties. In this work, we present a general constrained-optimization formulation for projection-based model reduction that can be used as a template to enforce the ROM to satisfy specific properties on the kinematics and dynamics. We introduce constrained-optimization formulations at both the time-continuous (i.e., ODE) level, which leads to a constrained Galerkin projection, and at the time-discrete level, which leads to a least-squares Petrov-Galerkin projection, in the context of linear multistep schemes. We demonstrate the ability of the proposed formulations to equip ROMs with desired properties such as global energy conservation and bounds on the total variation.

This work proposes a framework for projection-based model order reduction (MOR) of computational models aiming at a mechanical analysis of abdominal aortic aneurysms (AAAs). The underlying full-order model (FOM) is patient-specific, stationary and nonlinear. The quantities of interest are the von Mises stress and the von Mises strain field in the AAA wall, which result from loading the structure to the level of diastolic blood pressure at a fixed, imaged geometry (prestressing stage) and subsequent loading to the level of systolic blood pressure with associated deformation of the structure (deformation stage). Prestressing is performed with the modified updated Lagrangian formulation (MULF) approach. The proposed framework aims at a reduction of the computational cost in a many-query context resulting from model uncertainties in two material and one geometric parameter. We apply projection-based MOR to the MULF prestressing stage, which has not been presented to date. Additionally, we propose a reduced-order basis construction technique combining the concept of subspace angles and greedy maximin distance sampling. To further achieve computational speedup, the reduced-order model (ROM) is equipped with the energy-conserving mesh sampling and weighting hyper reduction method. Accuracy of the ROM is numerically tested in terms of the quantities of interest within given bounds of the parameter domain and performance of the proposed ROM in the many-query context is demonstrated by comparing ROM and FOM statistics built from Monte Carlo sampling for three different patient-specific AAAs.