scholarly journals Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Imran Akhtar ◽  
Jeff Borggaard ◽  
Alexander Hay
Keyword(s):  
Finite Difference ◽  
Standard Method ◽  
Elliptic Cylinder ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Shape Sensitivity ◽  
Reduced Order ◽  
Flow Models ◽  
Low Dimensional ◽  
Parameter Values

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.

scholarly journals GENETIC ALGORITHM-BASED CALIBRATION OF REDUCED ORDER GALERKIN MODELS

2011 ◽  
Vol 16 (1) ◽  
pp. 233-247 ◽  
Author(s):  
Witold Stankiewicz ◽  
Robert Roszaka ◽  
Marek Morzyńskia
Keyword(s):  
Genetic Algorithm ◽  
Bluff Body ◽  
Calibration Procedure ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Data Set ◽  
Reduced Order ◽  
Low Dimensional ◽  
Fluid Behaviour ◽  
Proper Orthogonal

Low-dimensional models, allowing quick prediction of fluid behaviour, are key enablers of closed-loop flow control. Reduction of the model's dimension and inconsistency of high-fidelity data set and the reduced-order formulation lead to the decrease of accuracy. The quality of Reduced-Order Models might be improved by a calibration procedure. It leads to global optimization problem which consist in minimizing objective function like the prediction error of the model. In this paper, Reduced-Order Models of an incompressible flow around a bluff body are constructed, basing on Galerkin Projection of governing equations onto a space spanned by the most dominant eigenmodes of the Proper Orthogonal Decomposition (POD). Calibration of such models is done by adding to Galerkin System some linear and quadratic terms, which coefficients are estimated using Genetic Algorithm.

scholarly journals Reduced order method for finite difference modeling of cardiac propagation

2020 ◽  
Vol 6 (3) ◽  
pp. 107-110
Author(s):  
Riasat Khan ◽  
Shahebul Hasan Shahebul Hasan ◽  
Kwong Ng
Keyword(s):  
Finite Difference ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Dimensional System ◽  
Bidomain Model ◽  
Reduced Order ◽  
Bidomain Equations ◽  
Low Dimensional

AbstractEfficient numerical simulation of cardiac electrophysiology is crucial for studying the electrical properties of the heart tissue. The cardiac bidomain model is the most widely accepted representation of the electrical behaviour of the heart muscle. The bidomain model offers fast cardiac potential variation, which can lead to high computational cost due to the required large grid sizes and small time steps. In this paper, the complexity of the finite difference approximation of the bidomain equations is reduced with the model order reduction technique. Proper orthogonal decomposition, a projection-based algorithm, is used to efficiently approximate the original high fidelity cardiac bidomain model with a low-dimensional system of equations. The low-dimensional basis functions are computed first from the ‘snapshots,’ which contain the solutions of the full-order system for different temporal and spatial parameters. Galerkin projection of the original cardiac bidomain system onto the subspace of the reduced order basis functions reduces the size of the linear system. Numerical results confirm the efficiency of the proposed reduced order modeling technique, reducing the simulation time by a factor of 9.54, while maintaining an RMS error of 0.769 mV between the original full order solution and the reduced order POD solution.

scholarly journals Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Zhe Bai ◽  
Liqian Peng
Keyword(s):  
Machine Learning ◽  
Nonlinear Dynamical Systems ◽  
Reduced Order Models ◽  
Simulation Code ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Nonlinear Model Reduction ◽  
Regression Techniques ◽  
Low Dimensional ◽  
Modern Machine

AbstractAlthough projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $$10^3{\times }$$ 10 3 × in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

Anisothermal Flow Control by Using Reduced-Order Models

2012 ◽  
Author(s):  
Alexandra Tallet ◽  
Cédric Leblond ◽  
Cyrille Allery
Keyword(s):  
Fluid Flow ◽  
Flow Control ◽  
Projection Method ◽  
Control Strategies ◽  
Pollutant Concentration ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Optimal Projection ◽  
Fluid Flow Control

Despite constantly improving computer capabilities, classical numerical methods (DNS, LES,…) are still out of reach in fluid flow control strategies. To make this problem tractable almost in real-time, reduced-order models are used here. The spatial basis is obtained by POD (Proper Orthogonal Decomposition), which is the most commonly used technique in fluid mechanics. The advantage of the POD basis is its energetic optimality: few modes contain almost the totality of energy. The ROM is achieved with the recent developed optimal projection [1], unlike classical methods which use Galerkin projection. This projection method is based on the minimization of the residual equations in order to have a stabilizing effect. It enables moreover to access pressure field. Here, the projection method is slightly different from [1]: a formulation without the Poisson equation is proposed and developed. Then, the ROM obtained by optimal projection is introduced within an optimal control loop. The flow control strategy is illustrated on an isothermal square lid-driven cavity and an anisothermal square ventilated cavity. The aim is to reach a target temperature (or target pollutant concentration) in the cavity, with an interior initial temperature (or initial pollutant concentration), by adjusting the inlet fluid flow rate.

scholarly journals NEW REGULARIZATION METHOD FOR CALIBRATED POD REDUCED-ORDER MODELS

2016 ◽  
Vol 21 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Badr Abou El Majd ◽  
Laurent Cordier
Keyword(s):  
Regularization Method ◽  
Regularization Parameter ◽  
Weighted Average ◽  
Calibration Method ◽  
Wake Flow ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Normalized Error ◽  
Curve Method

Reduced-order models based on Proper orthogonal decomposition are known to suffer from a lack of accuracy due to the truncation effect introduced by keeping only the most energetic modes. In this paper, we propose a new regularized calibration method aiming at minimizing a weighted average of normalized error, and a term measuring the change of the coefficients from their value obtained by Galerkin projection. We also determine the optimal value of the regularization parameter by analogy of the L-curve method. This paper is a sequel of [8] in which we compared various methods of calibration and introduced a Tikhonov-based regularization method. The proposed approach is assessed for a two dimensional wake flow around a cylinder, characteristic of the configurations of interest.

Reduced-Order Modeling for Compositional Simulation by Use of Trajectory Piecewise Linearization

SPE Journal ◽  
2014 ◽  
Vol 19 (05) ◽  
pp. 858-872 ◽  
Author(s):  
Jincong He ◽  
Louis J. Durlofsky
Keyword(s):  
Oil And Gas ◽  
Dimensional Subspace ◽  
Reduced Order Modeling ◽  
Galerkin Projection ◽  
Compositional Simulation ◽  
Piecewise Linearization ◽  
Reduced Order ◽  
Computational Optimization ◽  
Trajectory Piecewise Linearization ◽  
Low Dimensional

Summary Compositional simulation can be very demanding computationally as a result of the potentially large number of system unknowns and the intrinsic nonlinearity of typical problems. In this work, we develop a reduced-order modeling procedure for compositional simulation. The technique combines trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) to provide a highly efficient surrogate model. The compositional POD-TPWL method expresses new solutions in terms of linearizations around states generated (and saved) during previously simulated “training” runs. High-dimensional states are projected (optimally) into a low-dimensional subspace by use of POD. The compositional POD-TPWL model is based on a molar formulation that uses pressure and overall component mole fractions as the primary unknowns. Several new POD-TPWL treatments, including the use of a Petrov-Galerkin projection to reduce the number of equations (rather than the Galerkin projection, which was applied previously), and a new procedure for determining which saved state to use for linearization are incorporated into the method. Results are presented for heterogeneous 3D reservoir models containing oil and gas phases with up to six hydrocarbon components. Reasonably close agreement between full-order reference solutions and compositional POD-TPWL simulations is demonstrated for the cases considered. Construction of the POD-TPWL model requires preprocessing overhead computations equivalent to approximately three or four full-order runs. Runtime speedups by use of POD-TPWL are, however, very significant—up to a factor of 800 for the cases considered. The POD-TPWL model is thus well suited for use in computational optimization, in which many simulations must be performed, and we present an example demonstrating its application for such a problem.

Performance of Reduced Order Models of Moving Heat Source Deposition Problems for Efficient Inverse Analysis

2014 ◽  
Author(s):  
John G. Michopoulos ◽  
Brian Dennis ◽  
Foteini Komninelli ◽  
Athanasios Iliopoulos ◽  
Ashkan Akbariyeh
Keyword(s):  
Heat Source ◽  
Inverse Analysis ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Moving Heat Source ◽  
Reduced Order ◽  
Computational Implementation ◽  
Model Size ◽  
Transfer Problems ◽  
Proper Orthogonal

In order to reduce the demanding computational requirements for the numerical solution of problems involving heat transfer problems of moving heat source deposition, we present an approach utilizing reduced order models based on proper orthogonal decomposition and associated Galerkin projection. We subsequently describe the finite element implementation of solution methodology for both the full order and the reduced order models, as well as the respective computational implementation details. Using this methodology, we performed a sensitivity analysis for a problem of a moving heat source to investigate the performance characteristics of the relevant reduced order model size and present the efficiency of the approach. We demonstrated the efficiency of the reduced models for performing inverse analysis.

scholarly journals Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators

2021 ◽  
Author(s):  
Zhe Bai ◽  
Liqian Peng
Keyword(s):  
Machine Learning ◽  
Nonlinear Dynamical Systems ◽  
Reduced Order Models ◽  
Simulation Code ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Nonlinear Model Reduction ◽  
Regression Techniques ◽  
Low Dimensional ◽  
Modern Machine

Abstract Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $10^3\times$ in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs.

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