Scholarly Works, Interdisciplinary Center for Applied Mathematics (ICAM)
http://hdl.handle.net/10919/47108
Sat, 08 May 2021 01:53:29 GMT2021-05-08T01:53:29ZComputing Functional Gains for Designing More Energy-Efficient Buildings Using a Model Reduction Framework
http://hdl.handle.net/10919/86494
Akhtar, Imran; Borggaard, Jeffrey T.; Burns, John
Fri, 23 Nov 2018 00:00:00 GMThttp://hdl.handle.net/10919/864942018-11-23T00:00:00ZWe 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.Model Reduction for DAEs with an Application to Flow Control
http://hdl.handle.net/10919/81922
Borggaard, Jeffrey T.; Gugercin, Serkan
King, R.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10919/819222015-01-01T00:00:00ZLearning-based Robust Stabilization for Reduced-Order Models of 2D and 3D Boussinesq Equations
http://hdl.handle.net/10919/77671
Benosman, M.; Borggaard, Jeffrey T.; San, O.; Kramer, B.
http://hdl.handle.net/10919/77671A new wavelet family based on second-order LTI-systems
http://hdl.handle.net/10919/75188
Abuhamdia, Tariq; Taheri, Saied; Burns, John A.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10919/751882016-01-01T00:00:00ZIn this paper, a new family of wavelets derived from the underdamped response of second-order Linear-Time-Invariant (LTI) systems is introduced. The most important criteria for a function or signal to be a wavelet is the ability to recover the original signal back from its continuous wavelet transform. We show that it is possible to recover back the original signal once the Second-Order Underdamped LTI (SOULTI) wavelet is applied to decompose the signal. It is found that the SOULTI wavelet transform of a signal satisfies a linear differential equation called the reconstructing differential equation, which is closely related to the differential equation that produces the wavelet. Moreover, a time-frequency resolution is defined based on two different approaches. The new transform has useful properties; a direct relation between the scale and the frequency, unique transform formulas that can be easily obtained for most elementary signals such as unit step, sinusoids, polynomials, and decaying harmonic signals, and linear relations between the wavelet transform of signals and the wavelet transform of their derivatives and integrals. The results obtained are presented with analytical and numerical examples. Signals with constant harmonics and signals with time-varying frequencies are analyzed, and their evolutionary spectrum is obtained. Contour mapping of the transform in the time-scale and the time-frequency domains clearly detects the change of the frequency content of the analyzed signals with respect to time. The results are compared with other wavelets results and with the short-time fourier analysis spectrograms. At the end, we propose the method of reverse wavelet transform to mitigate the edge effect.Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition
http://hdl.handle.net/10919/49634
Hay, A.; Borggaard, Jeffrey T.; Pelletier, D.
Mon, 01 Jun 2009 00:00:00 GMThttp://hdl.handle.net/10919/496342009-06-01T00:00:00ZThe 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.Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach
http://hdl.handle.net/10919/48916
Akhtar, Imran; Borggaard, Jeffrey T.; Hay, Alexander
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10919/489162010-01-01T00:00:00ZReduced-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.Issues related to least-squares finite element methods for the stokes equations
http://hdl.handle.net/10919/48158
Deang, J. M.; Gunzburger, M. D.
Thu, 01 Oct 1998 00:00:00 GMThttp://hdl.handle.net/10919/481581998-10-01T00:00:00ZLeast-squares finite element methods have become increasingly popular for the approximate solution of first-order systems of partial differential equations. Here, after a brief review of some existing theories, a number of issues connected with the use of such methods for the velocity-vorticity-pressure formulation of the Stokes equations in two dimensions in realistic settings are studied through a series of computational experiments. Finite element spaces that are not covered by existing theories are considered; included in these are piecewise linear approximations for the velocity. Mixed boundary conditions, which are also not covered by existing theories, are also considered, as is enhancing mass conservation. Next, problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved. A conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh-dependent weights in the least-squares functional almost always improves the accuracy of the approximations. Concluding remarks concerning three-dimensional problems, the nonlinear Navier-Stokes equations, and the conditioning of the discrete systems are provided.On efficient solutions to the continuous sensitivity equation using automatic differentiation
http://hdl.handle.net/10919/48156
Borggaard, Jeffrey T.; Verma, A.
Thu, 01 Jun 2000 00:00:00 GMThttp://hdl.handle.net/10919/481562000-06-01T00:00:00ZShape 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.Inexact Kleinman-Newton method for Riccati equations
http://hdl.handle.net/10919/48144
Feitzinger, F.; Hylla, T.; Sachs, E. W.
Sun, 01 Mar 2009 00:00:00 GMThttp://hdl.handle.net/10919/481442009-03-01T00:00:00ZIn this paper we consider the numerical solution of the algebraic Riccati equation using Newton's method. We propose an inexact variant which allows one control the number of the inner iterates used in an iterative solver for each Newton step. Conditions are given under which the monotonicity and global convergence result of Kleinman also hold for the inexact Newton iterates. Numerical results illustrate the efficiency of this method.Mesh independence of Kleinman-Newton iterations for Riccati equations in Hilbert space
http://hdl.handle.net/10919/48137
Burns, J. A.; Sachs, E. W.; Zietsman, L.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10919/481372008-01-01T00:00:00ZIn this paper we consider the convergence of the infinite dimensional version of the Kleinman-Newton algorithm for solving the algebraic Riccati operator equation associated with the linear quadratic regulator problem in a Hilbert space. We establish mesh independence for this algorithm and apply the result to systems governed by delay equations. Numerical examples are presented to illustrate the results.