This paper describes the use of a quadratic manifold for the model order reduction of structural dynamics problems featuring geometric nonlinearities. The manifold is tangent to a subspace spanned by the most relevant vibration modes, and its curvature is provided by modal derivatives obtained by sensitivity analysis of the eigenvalue problem, or its static approximation, along the vibration modes. The construction of the quadratic manifold requires minimal computational effort once the vibration modes are known. The reduced order model is then obtained by Galerkin projection, where the configurationdependent tangent space of the manifold is used to project the discretized equations of motion.

In this paper, a generalization of a quadratic manifold approach for the reduction of geometrically nonlinear structural dynamics problems is presented. This generalization is constructed by a linearization of the static force with respect to the generalized coordinates, resulting in a shift of the quadratic behavior from the force to the manifold. In this framework, static derivatives emerge as natural extensions to modal derivatives for displacement fields other than the vibration modes, such as the Krylov subspace vectors. Here the dynamic problem is projected onto the tangent space of the quadratic manifold, allowing for a much less number of generalized coordinates compared to linear basis methods. The potential of the quadratic manifold approach is investigated in a numerical study, where several variations of the approach are compared on different examples, indicating a clear pattern where the proposed approach is applicable.

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Investigations on gas-liquid flows in horizontal pipes are of immanent importance for Reactor Safety Research. In case of a breakage of the main cooling circuit of a Pressurized Water Reactor (PWR), the pressure losses of the gas-liquid flow significantly govern the loss of coolant rate. The flow regime is largely determined by liquid and gas superficial velocities and contains slug flow that causes high-pressure pulsations to the infrastructure of the main cooling circuit. Experimental and numerical investigations on adiabatic slug flow of a water-air system were carried out in a horizontal pipe of about 10 m length and 54 mm diameter at atmospheric pressure and room temperature. Stereoscopic high-speed Particle Image Velocimetry in combination with Laser Induced Fluorescence was successfully applied on round pipe geometry to determine instantaneous three-dimensional water velocity fields of slug flows. After grid independence studies, numerical simulations were run with the open-source CFD program OpenFOAM. The solver uses the VOF method (Volume of Fluid) with phase-fraction interface capturing approach based on interface compression. It provides mesh refinement at the interfacial area to improve resolution of the interface between the two phases. Furthermore, standard k-ε turbulence model was applied in an unsteady Reynolds averaged Navier Stokes (URANS) model to resolve self-induced slug formation. The aim of this work is to present the feasibility of both relatively novel possibilities of determining two-phase slug flows in pipes. Experimental and numerical results allow the comparison of the slug initiation and expansion process with respect to their axial velocities and cross-sectional void fractions.

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