scholarly journals Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy

2020 ◽  
Vol 476 (2243) ◽  
pp. 20200589
Author(s):  
Evangelia Nicolaidou ◽  
Thomas L. Hill ◽  
Simon A. Neild
Keyword(s):  
Kinetic Energy ◽  
Boundary Conditions ◽  
Nonlinear Dynamic Analysis ◽  
Static Solution ◽  
Thin Plates ◽  
Fe Model ◽  
Engineering Structures ◽  
Reduced Order ◽  
Reduced Order Modelling ◽  
Nonlinear Manifolds

Nonlinear dynamic analysis of complex engineering structures modelled using commercial finite element (FE) software is computationally expensive. Indirect reduced-order modelling strategies alleviate this cost by constructing low-dimensional models using a static solution dataset from the FE model. The applicability of such methods is typically limited to structures in which (a) the main source of nonlinearity is the quasi-static coupling between transverse and in-plane modes (i.e. membrane stretching); and (b) the amount of in-plane displacement is limited. We show that the second requirement arises from the fact that, in existing methods, in-plane kinetic energy is assumed to be negligible. For structures such as thin plates and slender beams with fixed/pinned boundary conditions, this is often reasonable, but in structures with free boundary conditions (e.g. cantilever beams), this assumption is violated. Here, we exploit the concept of nonlinear manifolds to show how the in-plane kinetic energy can be accounted for in the reduced dynamics, without requiring any additional information from the FE model. This new insight enables indirect reduction methods to be applied to a far wider range of structures while maintaining accuracy to higher deflection amplitudes. The accuracy of the proposed method is validated using an FE model of a cantilever beam.

scholarly journals Accounting for Quasi-Static Coupling in Nonlinear Dynamic Reduced-Order Models

2020 ◽  
Vol 15 (7) ◽  
Author(s):  
Evangelia Nicolaidou ◽  
Venkata R. Melanthuru ◽  
Thomas L. Hill ◽  
Simon A. Neild
Keyword(s):  
High Frequency ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Analysis ◽  
Low Frequency ◽  
Nonlinear Effects ◽  
Fe Model ◽  
Engineering Structures ◽  
Reduced Order ◽  
Modal Coupling ◽  
High Frequency Modes

Abstract Engineering structures are often designed using detailed finite element (FE) models. Although these models can capture nonlinear effects, performing nonlinear dynamic analysis using FE models is often prohibitively computationally expensive. Nonlinear reduced-order modeling provides a means of capturing the principal dynamics of an FE model in a smaller, computationally cheaper reduced-order model (ROM). One challenge in formulating nonlinear ROMs is the strong coupling between low- and high-frequency modes, a feature we term quasi-static coupling. An example of this is the coupling between bending and axial modes of beams. Some methods for formulating ROMs require that these high-frequency modes are included in the ROM, thus increasing its size and adding computational expense. Other methods can implicitly capture the effects of the high-frequency modes within the retained low-frequency modes; however, the resulting ROMs are normally sensitive to the scaling used to calibrate them, which may introduce errors. In this paper, quasi-static coupling is first investigated using a simple oscillator with nonlinearities up to the cubic order. ROMs typically include quadratic and cubic nonlinear terms, however here it is demonstrated mathematically that the ROM describing the oscillator requires higher-order nonlinear terms to capture the modal coupling. Novel ROMs, with high-order nonlinear terms, are then shown to be more accurate, and significantly more robust to scaling, than standard ROMs developed using existing approaches. The robustness of these novel ROMs is further demonstrated using a clamped–clamped beam, modeled using commercial FE software.

Continuous Mode Interpolation between Multiple Operating and Boundary Conditions for Reduced Order Modelling of the Flow

2011 ◽  
Author(s):  
Witold Stankiewicz ◽  
Robert Roszak ◽  
Marek Morzyński ◽  
Bernd R. Noack ◽  
Gilead Tadmor ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Continuous Mode ◽  
Reduced Order ◽  
Reduced Order Modelling

Reduced order models for nonlinear dynamic analysis of structures with intermittent contacts

2017 ◽  
Vol 24 (12) ◽  
pp. 2591-2604 ◽  
Author(s):  
Stefano Zucca ◽  
Bogdan I Epureanu
Keyword(s):  
Boundary Conditions ◽  
Contact Area ◽  
Complex Geometry ◽  
Numerical Test ◽  
Normal Modes ◽  
Nonlinear Dynamic Analysis ◽  
Forced Response ◽  
Reduced Order Models ◽  
Frequency Range ◽  
Reduced Order

The forced response of structures with complex geometry and intermittent contacts is nonlinear due to contact breathing phenomena that occur during vibration. Therefore, calculation times to predict such responses can be extremely long especially because highly refined finite element models are necessary to properly model geometrically complicated structures. To alleviate this issue, reduced order models can be very beneficial as they can dramatically speed up the analysis process by reducing the size of the system and thus the calculation times. In this paper, a reduced order modeling method for the forced response of structures with intermittent contacts is developed. The proposed approach assumes that the dynamics of the nonlinear system in the frequency range of interest is spatially correlated. The spatial correlation can be dominated by normal modes of the open (or sliding) linear system. Nonetheless, the boundary conditions of a vibrating structure with intermittent frictionless contacts vary in time (i.e. at any time t the contacts are partly open and closed), and the actual extent of the contact area changes over time. Here, this observation is complemented by the assumption that, given the frequency range of the harmonic excitation force, the system dynamics is dominated by one of the modes of either the open or the sliding system, and thus the time evolution of the contact area can be estimated by knowing (a) the normal penetration at the contact node pairs at rest due to pre-stress, and (b) the vector of normal relative displacements at the contact nodes of that dominant mode. As a result, a set of normal modes – referred to as piecewise linear modes – is computed, by imposing special boundary conditions at the nodes lying on the contact surfaces, and a reduction basis is selected and used to reduce the size of the model of the system. Two numerical test cases, specifically a cracked plate and two coaxial cylinders are used for validation. Results show that the proposed method allows to accurately compute the nonlinear forced response of structures with intermittent contacts both in case of zero gap and in case of initial pre-stress at the contacts.

Cyclic Breathing Simulations in Large Scale Models of the Lung Airway From the Oronasal Opening to the Terminal Bronchioles

2012 ◽  
Author(s):  
D. Keith Walters ◽  
Greg W. Burgreen ◽  
Robert L. Hester ◽  
David S. Thompson ◽  
David M. Lavallee ◽  
...  
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Large Scale ◽  
Whole Body ◽  
Patient Specific ◽  
Cfd Simulations ◽  
Reduced Order ◽  
Scale Models ◽  
Steady State Simulation ◽  
Conducting Zone

Computational fluid dynamics (CFD) simulations were performed for unsteady periodic breathing conditions, using large-scale models of the human lung airway. The computational domain included fully coupled representations of the orotracheal region and large conducting zone up to generation four (G4) obtained from patient-specific CT data, and the small conducting zone (to G16) obtained from a stochastically generated airway tree with statistically realistic geometrical characteristics. A reduced-order geometry was used, in which several airway branches in each generation were truncated, and only select flow paths were retained to G16. The inlet and outlet flow boundaries corresponded to the oronasal opening (superior), the inlet/outlet planes in terminal bronchioles (distal), and the unresolved airway boundaries arising from the truncation procedure (intermediate). The cyclic flow was specified according to the predicted ventilation patterns for a healthy adult male at three different activity levels, supplied by the whole-body modeling software HumMod. The CFD simulations were performed using Ansys FLUENT. The mass flow distribution at the distal boundaries was prescribed using a previously documented methodology, in which the percentage of the total flow for each boundary was first determined from a steady-state simulation with an applied flow rate equal to the average during the inhalation phase of the breathing cycle. The distal pressure boundary conditions for the steady-state simulation were set using a stochastic coupling procedure to ensure physiologically realistic flow conditions. The results show that: 1) physiologically realistic flow is obtained in the model, in terms of cyclic mass conservation and approximately uniform pressure distribution in the distal airways; 2) the predicted alveolar pressure is in good agreement with previously documented values; and 3) the use of reduced-order geometry modeling allows accurate and efficient simulation of large-scale breathing lung flow, provided care is taken to use a physiologically realistic geometry and to properly address the unsteady boundary conditions.

The Quasi-Static Analysis for Two-Dimensional Thin Plates

2011 ◽  
Vol 255-260 ◽  
pp. 166-169
Author(s):  
Li Chen ◽  
Yang Bai
Keyword(s):  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Solution Space ◽  
Expansion Method ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Elastic Plates ◽  
Various Boundary Conditions ◽  
Eigenfunction Expansion Method ◽  
Concentration Phenomena

The eigenfunction expansion method is introduced into the numerical calculations of elastic plates. Based on the variational method, all the fundamental solutions of the governing equations are obtained directly. Using eigenfunction expansion method, various boundary conditions can be conveniently described by the combination of the eigenfunctions due to the completeness of the solution space. The coefficients of the combination are determined by the boundary conditions. In the numerical example, the stress concentration phenomena produced by the restriction of displacement conditions is discussed in detail.

scholarly journals Model reduction methods based on Krylov subspaces

Acta Numerica ◽  
2003 ◽  
Vol 12 ◽  
pp. 267-319 ◽  
Author(s):  
Roland W. Freund
Keyword(s):  
Large Scale ◽  
Krylov Subspace ◽  
Circuit Simulation ◽  
Krylov Subspaces ◽  
Lanczos Algorithm ◽  
Linear Dynamical Systems ◽  
Reduced Order ◽  
Reduced Order Modelling ◽  
Reduction Methods ◽  
Modelling Techniques

In recent years, reduced-order modelling techniques based on Krylov-subspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools for tackling the large-scale time-invariant linear dynamical systems that arise in the simulation of electronic circuits. This paper reviews the main ideas of reduced-order modelling techniques based on Krylov subspaces and describes some applications of reduced-order modelling in circuit simulation.

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