Equation-Free/Galerkin-Free Reduced-Order Modeling of the Shallow Water Equations Based on Proper Orthogonal Decomposition

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
Vahid Esfahanian ◽  
Khosro Ashrafi
Keyword(s):  
Shallow Water ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Scale Model ◽  
Fine Scale ◽  
Coarse Scale ◽  
Reduced Order ◽  
Shallow Water Flows ◽  
The One ◽  
Proper Orthogonal

In this paper, two categories of reduced-order modeling (ROM) of the shallow water equations (SWEs) based on the proper orthogonal decomposition (POD) are presented. First, the traditional Galerkin-projection POD/ROM is applied to the one-dimensional (1D) SWEs. The result indicates that although the Galerkin-projection POD/ROM is suitable for describing the physical properties of flows (during the POD basis functions’ construction time), it cannot predict that the dynamics of the shallow water flows properly as it was expected, especially with complex initial conditions. Then, the study is extended to applying the equation-free/Galerkin-free POD/ROM to both 1D and 2D SWEs. In the equation-free/Galerkin-free framework, the numerical simulation switches between a fine-scale model, which provides data for construction of the POD basis functions, and a coarse-scale model, which is designed for the coarse-grained computational study of complex, multiscale problems like SWEs. In the present work, the Beam & Warming and semi-implicit time integration schemes are applied to the 1D and 2D SWEs, respectively, as fine-scale models and the coefficients of a few POD basis functions (reduced-order model) are considered as a coarse-scale model. Projective integration is applied to the coarse-scale model in an equation-free framework with a time step grater than the one used for a fine-scale model. It is demonstrated that equation-free/Galerkin-free POD/ROM can resolve the dynamics of the complex shallow water flows. Moreover, the computational cost of the approach is less than the one for a fine-scale model.

scholarly journals A higher-order parametric nonlinear reduced-order model for imperfect structures using Neumann expansion

2021 ◽  
Author(s):  
J. Marconi ◽  
P. Tiso ◽  
D. E. Quadrelli ◽  
F. Braghin
Keyword(s):  
Reduced Order Model ◽  
Galerkin Projection ◽  
Order Model ◽  
Displacement Fields ◽  
Reduced Order ◽  
Total Displacement ◽  
Elastic Forces ◽  
Strain Formulation ◽  
The One ◽  
Definition Of

AbstractWe present an enhanced version of the parametric nonlinear reduced-order model for shape imperfections in structural dynamics we studied in a previous work. In this model, the total displacement is split between the one due to the presence of a shape defect and the one due to the motion of the structure. This allows to expand the two fields independently using different bases. The defected geometry is described by some user-defined displacement fields which can be embedded in the strain formulation. This way, a polynomial function of both the defect field and actual displacement field provides the nonlinear internal elastic forces. The latter can be thus expressed using tensors, and owning the reduction in size of the model given by a Galerkin projection, high simulation speedups can be achieved. We show that the adopted deformation framework, exploiting Neumann expansion in the definition of the strains, leads to better accuracy as compared to the previous work. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

scholarly journals A New Well-Balanced Reconstruction Technique for the Numerical Simulation of Shallow Water Flows with Wet/Dry Fronts and Complex Topography

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1661 ◽  
Author(s):  
Zhengtao Zhu ◽  
Zhonghua Yang ◽  
Fengpeng Bai ◽  
Ruidong An
Keyword(s):  
Shallow Water ◽  
Water System ◽  
Reconstruction Technique ◽  
Test Cases ◽  
Bed Topography ◽  
Surface Gradient ◽  
Shallow Water Flows ◽  
Shallow Water System ◽  
The One ◽  
Flow Variables

This study develops a new well-balanced scheme for the one-dimensional shallow water system over irregular bed topographies with wet/dry fronts, in a Godunov-type finite volume framework. A new reconstruction technique that includes flooded cells and partially flooded cells and preserves the non-negative values of water depth is proposed. For the wet cell, a modified revised surface gradient method is presented assuming that the bed topography is irregular in the cell. For the case that the cell is partially flooded, this paper proposes a special reconstruction of flow variables that assumes that the bottom function is linear in the cell. The Harten–Lax–van Leer approximate Riemann solver is applied to evaluate the flux at cell faces. The numerical results show good agreement with analytical solutions to a set of test cases and experimental results.

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 Proper orthogonal decomposition investigation in fluid structure interaction

2007 ◽  
pp. 401-418
Author(s):  
Erwan Liberge ◽  
Mustapha Benaouicha ◽  
Aziz Hamdouni
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Fluid Structure Interaction ◽  
Orthogonal Decomposition ◽  
Fluid Structure ◽  
Reduced Order ◽  
One Dimensional ◽  
Structure Interaction ◽  
Moving Domain ◽  
The One ◽  
Proper Orthogonal

This paper describes Reduced Order Modeling (ROM) in Fluid Structure Interaction (FSI) and discusses Proper Orthogonal Decomposition (POD) utilization. The ROM method was selected because its performance in fluid mechanics. The principal problems of its application in FSI are due the space character of the modes resulting from the POD whereas domains are mobile. To use POD in moving domain, a charateristic function of fluid is introduced in order to work on a fixed rigid domain, and the global velocity (fluid and structure) is studied. The POD modes efficiency is tested to reconstruct velocity field in one and two-dimensional FSI case. Then reducing dynamic system using POD is introduced in moving boundaries problem. In addition, the one dimensional case of Burgers equation coupled with spring equation is tested.

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 An Evolve-Then-Correct Reduced Order Model for Hidden Fluid Dynamics

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 570 ◽  
Author(s):  
Suraj Pawar ◽  
Shady E. Ahmed ◽  
Omer San ◽  
Adil Rasheed
Keyword(s):  
Process Model ◽  
Correction Term ◽  
A Priori ◽  
Basis Functions ◽  
Least Squares Regression ◽  
Physical Processes ◽  
Order Model ◽  
Reduced Order ◽  
Real Time Simulations ◽  
Proper Orthogonal

In this paper, we put forth an evolve-then-correct reduced order modeling approach that combines intrusive and nonintrusive models to take hidden physical processes into account. Specifically, we split the underlying dynamics into known and unknown components. In the known part, we first utilize an intrusive Galerkin method projected on a set of basis functions obtained by proper orthogonal decomposition. We then present two variants of correction formula based on the assumption that the observed data are a manifestation of all relevant processes. The first method uses a standard least-squares regression with a quadratic approximation and requires solving a rank-deficient linear system, while the second approach employs a recurrent neural network emulator to account for the correction term. We further enhance our approach by using an orthonormality conforming basis interpolation approach on a Grassmannian manifold to address off-design conditions. The proposed framework is illustrated here with the application of two-dimensional co-rotating vortex simulations under modeling uncertainty. The results demonstrate highly accurate predictions underlining the effectiveness of the evolve-then-correct approach toward real-time simulations, where the full process model is not known a priori.

Analytical Upgridding Method To Preserve Dynamic Flow Behavior

2010 ◽  
Vol 13 (03) ◽  
pp. 473-484 ◽  
Author(s):  
Seyyed Abolfazl Hosseini ◽  
Mohan Kelkar
Keyword(s):  
Single Phase ◽  
Flow Behavior ◽  
Pressure Profile ◽  
Scale Model ◽  
Optimum Number ◽  
Fine Scale ◽  
Coarse Scale ◽  
Flow Process ◽  
Number Of Layers ◽  
Pressure Profiles

Summary A geocellular model contains millions of gridblocks and needs to be upscaled before the model can be used as an input for flow simulation. Available techniques for upgridding vary from simple methods such as proportional fractioning to more complicated methods such as maintaining heterogeneities through variance calculations. All these methods are independent of the flow process for which simulation is going to be used, and are independent of well configuration. We propose a new upgridding method that preserves the pressure profile at the upscaled level. It is well established that the more complex the flow process, the more detailed the level of heterogeneity needed in the simulation model. In general, ideal upscaling is the process that preserves the "pressure profile" from the fine-scale model under the applicable flow process. In our method, we upgrid the geological model using simple flow equations in porous media. However, it should be remembered that to obtain a better match between fine scale and coarse scale, we also need to use appropriate upscaling of the reservoir properties. The new method is currently developed for single-phase flow; however, we used it for both single-phase and two-phase flows for 2D and 3D cases. The method differs fundamentally from the other methods that try to preserve heterogeneities. In those methods, gridblocks are combined that have similar velocities (or other properties) by assuming constant pressure drop across the blocks. Instead, we combine the gridblocks that have similar pressure profiles, although to release some of our assumptions such as having constant velocities in gridblocks, we balance our equation with the K2 term. The procedure is analytical and, hence, very efficient, but preserves the pressure profile in the reservoir. The gridblocks (or layers) are combined in a way so that the difference between fine- and coarse-scale pressure profiles is minimized. In addition, we also propose two new criteria that allow us to choose the optimum number of layers more accurately so that a critical level of heterogeneity is preserved. These criteria provide insight into the overall level of heterogeneity in the reservoir and the effectiveness of the layering design. We compare the results of our method with proportional layering and the King et al. method (King et al. 2006) and show that, for the same number of layers, the proposed method captures the results of the fine-scale model better. We show that the layer merging not only depends on the variation in the permeability between the gridblocks (K2 term), but also on the relative magnitude of the permeability values by combining 1/K2 and K2 terms.

A Cantilever Conveying Fluid: Coherent Modes Versus Beam Modes

2002 ◽  
Author(s):  
A. Sarkar ◽  
M. P. Paidoussis
Keyword(s):  
Eigenvalue Problem ◽  
Cantilever Beam ◽  
Large Scale ◽  
Galerkin Projection ◽  
Reduced Order ◽  
Non Linear ◽  
Proper Orthogonal Modes ◽  
Linear Oscillation ◽  
Proper Orthogonal ◽  
Conveying Fluid

A semianalytical approach to obtain the proper orthogonal modes (POMs) is described for the non-linear oscillation of a cantilevered pipe conveying fluid. Theoretically, while the spatial coherent structures are the eigenfunctions of the time-averaged spatial autocorrelation functions, it emerges that once the Galerkin projection of the proper orthogonal modes is carried out using the uniform cantilever-beam modes, the spatial dependency of the integral eigenvalue problem can be eliminated by analytical manipulation which avoids any spatial discretization error. As the solution of the integral equation is obtained semianalytically by linearly projecting the proper orthogonal modes on the cantilever-beam modes, any linear or non-linear operation can be carried out analytically on the proper orthogonal modes. Futhermore, the reduced-order eigenvalue problem minimizes the numerical pollution which leads to spurious eigenvectors, as may arise in the case of a large-scale eigenvalue problem (without the Galerkin projection of the eigenvectors on the cantilever-beam modes). This methodology can conveniently be used to study the convergence of the numerically calculated proper or-thogonal modes obtained from the full-scale eigenvalue problem.

Sign in / Sign up

Export Citation Format

Share Document