Interdisciplinary Center for Applied Mathematics (ICAM)

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Browsing Interdisciplinary Center for Applied Mathematics (ICAM) by Author "Borggaard, Jeffrey T."

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    Computing Functional Gains for Designing More Energy-Efficient Buildings Using a Model Reduction Framework
    Akhtar, Imran; Borggaard, Jeffrey T.; Burns, John A. (MDPI, 2018-11-23)
    We discuss developing efficient reduced-order models (ROM) for designing energy-efficient buildings using computational fluid dynamics (CFD) simulations. This is often the first step in the reduce-then-control technique employed for flow control in various industrial and engineering problems. This approach computes the proper orthogonal decomposition (POD) eigenfunctions from high-fidelity simulations data and then forms a ROM by projecting the Navier-Stokes equations onto these basic functions. In this study, we develop a linear quadratic regulator (LQR) control based on the ROM of flow in a room. We demonstrate these approaches on a one-room model, serving as a basic unit in a building. Furthermore, the ROM is used to compute feedback functional gains. These gains are in fact the spatial representation of the feedback control. Insight of these functional gains can be used for effective placement of sensors in the room. This research can further lead to developing mathematical tools for efficient design, optimization, and control in building management systems.
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    Learning-based Robust Stabilization for Reduced-Order Models of 2D and 3D Boussinesq Equations
    Benosman, Mouhacine; Borggaard, Jeffrey T.; San, Omer; Kramer, Boris (2017-09)
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    Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition
    Hay, Alexander; Borggaard, Jeffrey T.; Pelletier, Dominique (Cambridge University Press, 2009-06)
    The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied 'off-design'. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier-Stokes equations for large parameter changes.
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    Model Reduction for DAEs with an Application to Flow Control
    Borggaard, Jeffrey T.; Gugercin, Serkan (Springer-Verlag Berlin, 2015-01-01)
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    On efficient solutions to the continuous sensitivity equation using automatic differentiation
    Borggaard, Jeffrey T.; Verma, Arun (Siam Publications, 2000-06)
    Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.
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    Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach
    Akhtar, Imran; Borggaard, Jeffrey T.; Hay, Alexander (Hindawi Publishing Corporation, 2010)
    Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used "off-design" (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.