[[wiki-architecture]] · [[Architectural Concepts and History]] · [[ARCHITECTURE]] · [[000]] # Proper orthogonal decomposition The proper orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational fluid dynamics and structural analysis (like crash simulations). Typically in fluid dynamics and turbulences analysis, it is used to replace the Navier–Stokes equations by simpler models to solve. Proper orthogonal decomposition is associated with model order reduction. The orthogonally decomposed model can be characterized as a surrogate model; to this end, the method is also associated with the field of machine learning. == POD and PCA == The main use of POD is to decompose a physical field (like pressure, temperature in fluid dynamics or stress and deformation in structural analysis), depending on the different variables that influence its physical behaviors. As its name hints, it's operating an Orthogonal Decomposition along with the Principal Components of the field. As such it is assimilated with the principal component analysis from Pearson in the field of statistics, or the singular value decomposition in linear algebra because it refers to eigenvalues and eigenvectors of a physical field. In those domains, it is associated with the research of Karhunen and Loève, and their Karhunen–Loève theorem. == Mathematical expression == The first idea behind the Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector field u(x, t) into a set of deterministic spatial functions Φk(x) modulated by random time coefficients ak(t) so that: u ( x , t ) = ∑ k = 1 ∞ a k ( t ) ϕ k ( x ) {\displaystyle u(x,t)=\sum _{k=1}^{\infty }a_{k}(t)\phi _{k}(x)} The first step is to sample the vector field over a period of time in what we call snapshots (as display in the image of the POD snapshots). This snapshot method is averaging the samples over the space dimension n, and correlating them with each other along the time samples p: U = ( u ( x 1 , t 1 ) ⋯ u ( x n , t 1 ) ⋮ ⋮ u ( x 1 , t p ) ⋯ u ( x n , t p ) ) {\displaystyle U={\begin{pmatrix}u(x_{1},t_{1})&\cdots &u(x_{n},t_{1})\\\vdots &&\vdots \\u(x_{1},t_{p})&\cdots &u(x_{n},t_{p})\end{pmatrix}}} with n spatial elements, and p time samples The next step is to compute the covariance matrix C C = 1 ( p − 1 ) U T U {\displaystyle C={\frac {1}{(p-1)}}U^{T}U} We then compute the eigenvalues and eigenvectors of C and we order them from the largest eigenvalue to the smallest. We obtain n eigenvalues λ1,...,λn and a set of n eigenvectors arranged as columns in an n × n matrix Φ: ϕ = ( ϕ 1 , 1 ⋯ ϕ 1 , n ⋮ ⋮ ϕ n , 1 ⋯ ϕ n , n ) {\displaystyle \phi ={\begin{pmatrix}\phi _{1,1}&\cdots &\phi _{1,n}\\\vdots &&\vdots \\\phi _{n,1}&\cdots &\phi _{n,n}\end{pmatrix}}} == References == - [[Professional Practice/Construction Management/Project Planning]] - [[Building Services/Building Automation]] - [[Professional Practice/Codes & Standards/National Building Code of India/Part 06 - Structural Design/Section 3B - Bamboo]] - [[Environmental Design/Strong Foundations for Sustainable Constructions/satellites]] - [[Design/Building Typologies/Cultural and Religious]] - [[Professional Practice/Codes & Standards]] - [[Professional Practice/Client Management]] - [[Professional Practice/Codes & Standards/National Building Code of India/Part 09 - Plumbing Services/Section 2 - Drainage and Sanitation]] - [[Urban and Planning/Urban Design Principles]] - [[Interior Architecture/Renovation and Conservation/Restoration Techniques]] == External links == MIT: http://web.mit.edu/6.242/www/images/lec6_6242_2004.pdf Stanford University - Charbel Farhat & David Amsallem https://web.stanford.edu/group/frg/course_work/CME345/CA-AA216-CME345-Ch4.pdf Weiss, Julien: A Tutorial on the Proper Orthogonal Decomposition. In: 2019 AIAA Aviation Forum. 17–21 June 2019, Dallas, Texas, United States. French course from CNRS https://www.math.u-bordeaux.fr/~mbergman/PDF/OuvrageSynthese/OCET06.pdf Applications of the Proper Orthogonal Decomposition Method http://www.cerfacs.fr/~cfdbib/repository/WN_CFD_07_97.pdf