scholarly journals Enhancing the accuracy of deep learning based FE$^2$ algorithm using proper orthogonal decomposition

2021 ◽  
Author(s):  
Saumik Dana
Keyword(s):  
Deep Learning ◽  
Proper Orthogonal Decomposition ◽  
Computational Cost ◽  
Orthogonal Decomposition ◽  
Neural Nets ◽  
Neural Net ◽  
Input Output ◽  
Elastic Solids ◽  
Scale Modeling ◽  
Proper Orthogonal

The deep learning leveraged FE$^2$ algorithm for two-scale modeling of elastic solids eliminates the need to solve the RVE problem on-the-fly by replacing the effective input-output causality by a neural network. This potentially reduces the computational cost of the FE$^2$ algorithm significantly. In this work, we put forth the use of snapshot proper orthogonal decomposition to improve the accuracy of the machine learning leveraged algorithm. Instead of training one neural net, multiple neural nets are trained with the coefficients of the basis of the snapshot matrix as the target.

MODEL REDUCTION FOR FLUIDS, USING BALANCED PROPER ORTHOGONAL DECOMPOSITION

2005 ◽  
Vol 15 (03) ◽  
pp. 997-1013 ◽  
Author(s):  
C. W. ROWLEY
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Model Reduction ◽  
Projection Method ◽  
Computational Cost ◽  
Plane Channel ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
High Dimensional ◽  
Balanced Truncation ◽  
Proper Orthogonal

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.

Prediction of Transient Thermal Behavior of Planar Interconnect Architecture Using Proper Orthogonal Decomposition Method

2011 ◽  
Author(s):  
Banafsheh Barabadi ◽  
Yogendra K. Joshi ◽  
Satish Kumar
Keyword(s):  
Thermal Behavior ◽  
Proper Orthogonal Decomposition ◽  
Joule Heating ◽  
Computational Cost ◽  
Orthogonal Decomposition ◽  
Galerkin Projection ◽  
Fe Model ◽  
Inhomogeneous System ◽  
Transient Thermal ◽  
Proper Orthogonal

A major challenge in maintaining quality and reliability in today’s microelectronics devices comes from the ever increasing level of integration in the device fabrication as well as the high level of current densities that are carried through the microchip during operation. Cyclic thermal events during operation, stemming from Joule heating of the metal lines, can lead to fatigue failure due to the varying thermal expansion coefficients of the different materials that compose the microchip package. To aid in the avoidance of such device failures, it is imperative to develop a predictive capability for the thermal response of micro-electronic circuits. This work studied the problem of transient Joule heating in interconnects in a two-dimensional (2D) inhomogeneous system using a reduced order modeling approach of the Proper Orthogonal Decomposition (POD) method and Galerkin Projection Technique. This study considers an interconnect structure embedded in the bulk of a microelectronic device. The effect of different types of current pulses, pulse duration, and pulse amplitude were investigated. By using a representative step function as the heat source, the model predicted the exact transient thermal behavior of the system for all other cases without generating any new observations, using just a few POD modes. To validate this unique capability, the result of the POD model was compared with a finite element (FE) model developed in LS-DYNA®. The behaviors of the POD models were in good agreements with the corresponding FE models. This close correlation provides the capability of predicting other cases based on a smaller sample set which can significantly decrease the computational cost.

A Reduced-Order Model for Turbomachinery Flows Using Proper Orthogonal Decomposition

2013 ◽  
Author(s):  
Thomas A. Brenner ◽  
Forrest L. Carpenter ◽  
Brian A. Freno ◽  
Paul G. A. Cizmas
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Computational Cost ◽  
Wheel Speed ◽  
Orthogonal Decomposition ◽  
Reduced Order Model ◽  
Computational Time ◽  
Order Model ◽  
Reduced Order ◽  
Proper Orthogonal

This paper presents the development of a reduced-order model based on the proper orthogonal decomposition (POD) method. The POD method has been developed to predict turbomachinery flows modeled by the Reynolds-averaged Navier–Stokes equations. The purpose of using a POD-based reduced-order model is to decrease the computational cost of turbomachinery flows. The POD model has been tested for two configurations: a canonical channel with a bump case and the transonic NASA Rotor 67 case. The Rotor 67 case has been simulated at design wheel speed and at three off-design conditions: 70, 80, and 90% of the wheel speed. The results of the POD-based reduced-order model where in excellent agreement with the full-order model results. The computational time of the reduced-order model was approximately one order of magnitude smaller than that of the full-order model.

Verification of Nonlinear Proper Orthogonal Decomposition Reduced Order Modeling for BWR Fuel Assemblies

2013 ◽  
Author(s):  
Dennis P. Prill ◽  
Andreas G. Class
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Power Distribution ◽  
Computational Cost ◽  
Orthogonal Decomposition ◽  
Low Flow ◽  
Reduced Order ◽  
Linear Character ◽  
Operation Conditions ◽  
Non Linear ◽  
Proper Orthogonal

Thermal-hydraulic coupling between power, flow rate and density, intensified by neutronics feedback are the main drivers of boiling water reactor (BWR) stability behavior. Studying potential power oscillations require focusing on BWR operation at high-power low-flow conditions interacting with unfavorable power distribution. Current design rules assure admissible operation conditions by exclusion regions determined by numerical calculations and analytical methods. Analyzing an exhaustive parameter space of the non-linear BWR system becomes feasible with methodologies based on reduced order models (ROMs) saving computational cost and improving the physical understanding. A general reduction technique is given by the proper orthogonal decomposition (POD). Model-specific options and aspects of the POD-ROM-methodology are considered. A first verification is illustrated by means of a chemical tubular reactor (TR) setup. Experimental and analytical results for natural convection in a closed circuit (NCC) [1, 2] serve as a second verification example. This setup shows a strongly non-linear character. The implemented model is validated by means of a linear stability map. Transient behavior of the NCC-POD-ROM can not only reproduce the input data but rather predict different states.

scholarly journals Spectral proper orthogonal decomposition

2016 ◽  
Vol 792 ◽  
pp. 798-828 ◽  
Author(s):  
Moritz Sieber ◽  
C. Oliver Paschereit ◽  
Kilian Oberleithner
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Coherent Structures ◽  
Computational Cost ◽  
Orthogonal Decomposition ◽  
New Method ◽  
Dynamic Mode ◽  
Additional Parameter ◽  
Fourier Decomposition ◽  
Multiple Frequencies ◽  
Proper Orthogonal

The identification of coherent structures from experimental or numerical data is an essential task when conducting research in fluid dynamics. This typically involves the construction of an empirical mode base that appropriately captures the dominant flow structures. The most prominent candidates are the energy-ranked proper orthogonal decomposition (POD) and the frequency-ranked Fourier decomposition and dynamic mode decomposition (DMD). However, these methods are not suitable when the relevant coherent structures occur at low energies or at multiple frequencies, which is often the case. To overcome the deficit of these ‘rigid’ approaches, we propose a new method termed spectral proper orthogonal decomposition (SPOD). It is based on classical POD and it can be applied to spatially and temporally resolved data. The new method involves an additional temporal constraint that enables a clear separation of phenomena that occur at multiple frequencies and energies. SPOD allows for a continuous shifting from the energetically optimal POD to the spectrally pure Fourier decomposition by changing a single parameter. In this article, SPOD is motivated from phenomenological considerations of the POD autocorrelation matrix and justified from dynamical systems theory. The new method is further applied to three sets of PIV measurements of flows from very different engineering problems. We consider the flow of a swirl-stabilized combustor, the wake of an airfoil with a Gurney flap and the flow field of the sweeping jet behind a fluidic oscillator. For these examples, the commonly used methods fail to assign the relevant coherent structures to single modes. The SPOD, however, achieves a proper separation of spatially and temporally coherent structures, which are either hidden in stochastic turbulent fluctuations or spread over a wide frequency range. The SPOD requires only one additional parameter, which can be estimated from the basic time scales of the flow. In spite of all these benefits, the algorithmic complexity and computational cost of the SPOD are only marginally greater than those of the snapshot POD.

scholarly journals A scheme for comprehensive computational cost reduction in proper orthogonal decomposition

2018 ◽  
Vol 69 (4) ◽  
pp. 279-285 ◽  
Author(s):  
Satyavir Singh ◽  
M Abid Bazaz ◽  
Shahkar Ahmad Nahvi
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Cost Reduction ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Computational Cost ◽  
Nonlinear Function ◽  
Orthogonal Decomposition ◽  
Order System ◽  
Nonlinear Dynamical ◽  
Proper Orthogonal

Abstract This paper addresses the issue of offline and online computational cost reduction of the proper orthogonal decomposition (POD) which is a popular nonlinear model order reduction (MOR) technique. Online computational cost is reduced by using the discrete empirical interpolation method (DEIM), which reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD: this is the POD-DEIM approach. Offline computational cost is reduced by generating an approximate snapshot-ensemble of the nonlinear dynamical system, consequently, completely avoiding the need to simulate the full-order system. Two snapshot ensembles: one of the states and the other of the nonlinear function are obtained by simulating the successive linearization of the original nonlinear system. The proposed technique is applied to two benchmark large-scale nonlinear dynamical systems and clearly demonstrates comprehensive savings in computational cost and time with insignificant or no deterioration in performance.

Sign in / Sign up

Export Citation Format

Share Document