Finding the distance between the ellipsoid and the intersection of a linear manifold and ellipsoid

2015 ◽  
Author(s):  
Grigoriy Tamasyan ◽  
Andrew Chumakov
Keyword(s):  
Linear Manifold

scholarly journals Manifold reduction techniques for the comparison of crank angle-resolved particle image velocimetry (PIV) data and Reynolds-averaged Navier-Stokes (RANS) simulations in a spark ignition direct injection (SIDI) engine

2021 ◽  
pp. 146808742110131
Author(s):  
Xiaohang Fang ◽  
Li Shen ◽  
Christopher Willman ◽  
Rachel Magnanon ◽  
Giuseppe Virelli ◽  
...  
Keyword(s):  
Flow Field ◽  
Particle Image ◽  
Direct Injection ◽  
Linear Manifold ◽  
Main Flow ◽  
Navier Stokes ◽  
Reduction Techniques ◽  
Image Velocimetry ◽  
Manifold Reduction ◽  
Relevance Index

In this article, different manifold reduction techniques are implemented for the post-processing of Particle Image Velocimetry (PIV) images from a Spark Ignition Direct Injection (SIDI) engine. The methods are proposed to help make a more objective comparison between Reynolds-averaged Navier-Stokes (RANS) simulations and PIV experiments when Cycle-to-Cycle Variations (CCV) are present in the flow field. The two different methods used here are based on Singular Value Decomposition (SVD) principles where Proper Orthogonal Decomposition (POD) and Kernel Principal Component Analysis (KPCA) are used for representing linear and non-linear manifold reduction techniques. To the authors’ best knowledge, this is the first time a non-linear manifold reduction technique, such as KPCA, has ever been used in the study of in-cylinder flow fields. Both qualitative and quantitative studies are given to show the capability of each method in validating the simulation and incorporating CCV for each engine cycle. Traditional Relevance Index (RI) and two other previously developed novel indexes: the Weighted Relevance Index (WRI) and the Weighted Magnitude Index (WMI), are used for the quantitative study. The results indicate that both POD and KPCA show improvements in capturing the main flow field features compared to ensemble-averaged PIV experimental data and single cycle experimental flow fields while capturing CCV. Both methods present similar quantitative accuracy when using the three indexes. However, challenges were highlighted in the POD method for the selection of the number of POD modes needed for a representative reconstruction. When the flow field region presents a Gaussian distribution, the KPCA method is seen to provide a more objective numerical process as the reconstructed flow field will see convergence with an increasing number of modes due to its usage of Gaussian properties. No additional criterion is needed to determine how to reconstruct the main flow field feature. Using KPCA can, therefore, reduce the amount of analysis needed in the process of extracting the main flow field while incorporating CCV.

scholarly journals Exploiting Linear Manifold Features With Parts-Based Representation in Various Scenes

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 50045-50056
Author(s):  
Qiaoqin Li ◽  
Yongguo Liu ◽  
Shangming Yang
Keyword(s):  
Linear Manifold

Study of Fault Diagnosis Based on Manifold Learning

2013 ◽  
Vol 312 ◽  
pp. 650-654 ◽  
Author(s):  
Yi Lin He ◽  
Guang Bin Wang ◽  
Fu Ze Xu
Keyword(s):  
Fault Diagnosis ◽  
Manifold Learning ◽  
Linear Manifold ◽  
Rotating Machinery ◽  
Dimensional Manifold ◽  
Advantages And Disadvantages ◽  
Non Linear ◽  
Future Direction ◽  
Low Dimensional ◽  
Machinery Fault Diagnosis

Characteristic signals in rotating machinery fault diagnosis with the issues of complex and difficult to deal with, while the use of non-linear manifold learning method can effectively extract low-dimensional manifold characteristics embedded in the high-dimensional non-linear data. It greatly maintains the overall geometric structure of the signals and improves the efficiency and reliability of the rotating machinery fault diagnosis. According to the development prospects of manifold learning, this paper describes four classical manifold learning methods and each advantages and disadvantages. It reviews the research status and application of fault diagnosis based on manifold learning, as well as future direction of researches in the field of manifold learning fault diagnosis.

Orthogonal descent to a convex set in a linear manifold

1992 ◽  
Vol 28 (6) ◽  
pp. 861-867
Author(s):  
M. B. Shchepakin
Keyword(s):  
Convex Set ◽  
Linear Manifold
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