Imperial College London

Transcription

1 F. Fang 1, C.C. Pan 1, I.M. Navon 2, M.D. Pggott 1, G.J. Gorman 1, P.A. Allson 1 and A.J.H. Goddard 1 1 Appled Modellng and Computaton Group Department of Earth Scence and Engneerng Imperal College London, K nder fundng from NE/C52101X/1 Page 1 2 School of Computatonal Scence and Department of Mathematcs Florda State nversty

2 Outlne of talk Ams and Objectves Proper Orthogonal Decomposton (POD) n ICOM Mesh adaptvty wth POD Goal-based error measurement Test cases Concluson Page 2

3 Intro. to ICOM e.g. Lock exchange problem The two colours represent the dfferent densty. The heaver one snks and runs under Page 3 the lghter. Kelvn-Helmholtz bllows are created at the nterface due to shear.

4 Flow past a cylnder (Re = 100, Mnmum mesh sze= 0.04, Maxmum mesh sze =1) forward (top), POD (bottom) (20 bass functons, 41 snapshots) Page 4

5 Ams and Objectves ----Develop a reduced order POD controller for a novel advanced mesh adaptve fnte element model whch ncludes many recent developments n ocean modellng. In partcular, our am s to develop a new goal-based approach to: gude the mesh adaptvty and nverson; estmate error and optmse the POD bases. Page 5

6 What s Proper Orthogonal Decomposton (POD)? POD- A numercal procedure that can be used to extract a bass for a model decomposton from an ensemble of sgnals. -- orgnally proposed by Kosamb, Loeve and Karhunen; -- also known as Prncpal Components analyss (PCA) n statstcs; Emprcal Orthogonal Functon (EOF) n oceanography and meteorology. The varables can be expressed as an expanson n POD ( t, x, y, z) = ( x, y, z) + α M m= 1 m Φ k ( t) Φ m ( x, y, z) The orgnal PDE The reduced order ODE Page 6

7 Page 7 POD-based reduced order model POD-based reduced order model Defne the mean of varables = = K K 1 1 ) :,,, ( p w v u = = Φ + = M m m m POD z y x t z y x z y x t 1 ),, ( ) ( ),, ( ),,, ( α The varables can be expressed as an expanson n Φ k Hv λv = Solve the egenvalue problem: ; ) )( (, Ω Ω = Η d j j The spatal correlaton matrx K j 1, = = Φ K m m 1 ), ( ) (ν The POD bass functons: M m,..., 1 = = = = K M M I 1 1 ) ( λ λ Energy:

8 Mesh adaptvty n the POD model Challenge: The snapshots can be of dfferent length at dfferent tme levels Interpolaton Reference mesh Adaptve mesh Page 8

9 A functonal or goal A functonal can be defned as the model reducton error or soluton whch s of nterest n a target regon. The functonal s used to optmse uncertantes (nverson problem) n models; determne the error measure for mesh adaptvty; optmse the POD bases. Page 9

10 The goal-based error measure approach for mesh adaptvty To satsfy the goal, the mnmal ellpsod s obtaned by superscrbng both ellpses and used for mesh adaptvty Page 10

11 Dual-weght POD approach A dual weghted method s developed to analyse the error of models and fnd an optmal POD bass. To maxmse the accuracy of the functonal, a weghted dagonal Matrx s ntroduced to the snapshots Page 11

12 Case: Gyre (Re = 400) Computatonal doman: 1000 km x 1000 km 11 ρ =1000 β = τ = Assmlaton perod: 200 days Tme step: 3 hrs Am: to fnd an optmsed mesh for the reduced forward and adjont models Run the reduced forward and adjont models to fnd Page 12

13 Optmsed adaptve mesh for the reduced order forward and adjont models Inverson Vortcty Page 13

14 Optmsed adaptve mesh for the reduced order forward and adjont models Page 14

15 Case 4: Gyre nverson of ntal condtons Objectve Functon used n Inverson Spn-up perod: 200days Smulaton perod: [200, 400] day Tme step: 6hrs Pseudo-observatons are u and v, whch are avalable at t= 300 and 350 days. Page 15

16 250 day 300 day 350 day Page 16

17 Concluson The advantages of the POD model developed here over exstng POD approaches are the ablty to: Introduce adaptve meshes nto the POD model; The goal-based error measure approach developed here can be used to (1) gude the mesh adaptvty and nverson; (2) optmse the POD bases (weght the snapshots). Page 17

18 Page 18

19 Case: Gyre nverson of ntal condtons Full model Inverson model Page 19

20 The goal-based error measure approach for mesh adaptvty b Page 20