A Component-Based Reduced Basis Method for Many-Parameter Systems - PowerPoint PPT Presentation

A Component-Based Reduced Basis Method for Many-Parameter Systems David J. Knezevic Harvard University Institute for Applied Computational Science

Outline 1. Applications of the Reduced Basis Method ◮ Examples and Applications ◮ Parameter Estimation and Model Validation ◮ Toward More Parameters 2. Component-Based RB Method ◮ Motivation ◮ Formulation ◮ Akselos ◮ Numerical Results

1. Applications of the RB Method

Applications of the Reduced Basis Method ◮ Examples and Applications ◮ Parameter Estimation and Model Validation ◮ Toward More Parameters

Applications of the Reduced Basis Method ◮ Examples and Applications ◮ Parameter Estimation and Model Validation ◮ Toward More Parameters

Parametrized Partial Diff. Eqs. Let F be a PDE operator, models some physical system (e.g. fluids, solids, acoustics, electromagnetics, ...) F depends on parameter vector µ ∈ R P : µ characterizes system (e.g. material properties, boundary conditions, geometry, ...) Given µ ∈ D ⊂ R P Find u ( µ ) ∈ X such that F ( u ( µ ); µ ) = 0 , and output(s): ℓ ( · ; µ ) : X → R , s ( µ ) = ℓ ( u ( µ ); µ ) ∈ R

Offline Implementation RB for large-scale 3D PDEs requires high-performance implementation of “Offline stage” All results in this talk are based on: ◮ libMesh : Open source C++ parallel finite element library ◮ rbOOmit : Reduced Basis functionality within libMesh ◮ PETSc + SLEPc : Parallel linear algebra, eigensolver

Online Implementation Once the Offline stage is complete, we can: 1. evaluate the RB model at any parameter value in the pre-defined parameter range ( µ ∈ D ) 2. evaluate error bounds with respect to the underlying “truth” finite element discretization 3. evaluate output quantities of interest 4. visualize solution field (linear combination of RB basis functions) These Online computations are “lightweight” ◮ 1, 2, 3 are independent of FE discretization ◮ 4 depends on FE mesh, but still much cheaper than an FE solve

Online Implementation: Smartphone App Developed (with Phuong Huynh) open source RB smartphone app for Android to demonstrate efficiency of Online stage

Online Implementation: Smartphone App

Footbridge Example

4 E sp 000 (N/m (N/m 2 2 ) ) ( E sp , 1 , , ρ ρ sp sp , , t t d , 000 ≤ 1 , ≤ 7 (GPa) a) 740 740 ≤ ≤ ρ ρ sp 7 , 800 σ d 800 (kg/m (kg/m 3 3 ) ) 20 20 ≤ ≤ σ d , σ E sp ) t( x x ; ; P P aramete arameter r) P emen P aramete arameter r D D omain: omain: P P = = 4 ent( acem σ d ( P d ) (m) O u utpu tput t( P aramete Displac arameter r) ) F F ield ield ≡ ≡ Displ 200 (GP ≤ 200 ≤ E ) E ) g/m ) (kg/m 0 (k 740 = 74 sp ) ood wo ρ w ρ a) ρ sp Pa) (GP 11 (G = 11 t d od oo wo E w E 3 3 E steel = 200 d 2 σ d 11 ≤ 11 m) (cm) 20 (c ≤ 20 d ( E t d ≤ t 2 ≤ ) , ρ (kg/m ) 800 (kg/m , 800 7 , = 7 E sp sp , ρ steel ρ a) Pa) (GP 200 (G 3 Footbridge Example Motivation: ◮ Develop an RB model for a Bridges to Prosperity bridge design ◮ RB model can be evaluated on a smartphone “in the field” guardrails deck stiffener Los Montes Simple Span Bridge springer (I-beam) d = d = steel = steel = u d ( ) (m) d ≤ sp ≤ wood sp ≤ d ≤

“Truth” Finite element approximation Starting point for RB: Introduce high-fidelity (“truth”) FE space X N ⊂ X , dim( X N ) = N For this footbridge mesh, N ≈ 1 . 5 × 10 6

Footbridge Example: Offline stage Construct RB space X N ⊂ X N , N ≪ N , from solution snapshots at “greedily selected” parameters ξ 1 ξ 2 ξ 3 . . . N = 33: Offline requires ≈ 2 hours (for 33 truth solves + extra preprocessing) on 64 processors

Footbridge Example: Online Stage Then using RB model, footbridge problem can be solved (with error bounds) on smartphone in real-time Plot shows parameter-dependent output: Vertical displacement at bridge midpoint as function of bridge deck thickness ( t d )

Footbridge Example: Online Stage We can also plot 3D solution fields on the phone 1 1 Uses Android’s implementation of OpenGL ES

Footbridge Example: Online Stage Can also model dynamic response to harmonic forcing by solving parametrized Helmholtz equation, resonant frequency at ≈ 13Hz

CFD Example

CFD Example: Natural convection We can also solve nonlinear PDEs with RB 2 , e.g. Boussinesq equations: Find u , p , T satisfying √ √ ∂ u 1 Gr Pr ∇ p − ∇ 2 u + √ ∂ t + u · ∇ u + Gr Pr T ˆ g = 0 2 Gr Pr ∂ T 1 u · ∇ T − 1 Pr ∇ 2 T = 0 √ ∂ t + 2 Gr Pr ∇ · u = 0 Parameters: ◮ φ : direction of gravity, ˆ g ≡ ( − sin φ, 0 , − cos φ ) ◮ Gr: Grashof number (ratio of buoyancy to thermal diffusion) ◮ Pr = 0 . 71: Prandtl number (for air) 2 Knezevic & Peterson, CMAME, 2011; Recent work on space-time formulation for Boussinesq: Yano and Patera

CFD Example: Natural convection 3D domain: cross-section at y = 2 . 5 of the computational domain, three output regions Outputs: Average temperatures on D 1 , D 2 , D 3

CFD Example: Natural convection Finite element mesh: N = 241 , 520 degrees of freedom

Another example: Natural convection FE solve requires 21.7 minutes on 128 processors, too long for many-query or real-time context x = 2 . 5 y = 2 . 5 ( Gr , φ ) = (6000 , 0 . 2), t = 0 . 16 RB Offline stage: 42 hours on 128 processors, N = 90

CFD Example: Natural convection RB Online on laptop: ◮ Output evaluation: 0.9 seconds ◮ Error bounds 3 : 18 seconds 0.25 0.2 0.15 0.1 0.05 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 (Gr , φ ) = (6000 , 0 . 2) 3 Computational cost of error bounds scales as O ( N 4 ) for quadratic nonlinearities

LabVIEW Example

Collaboration with National Instruments Worked with engineers at National Instruments (NI) to build an RB plug-in for LabVIEW Goal: Evaluate RB models in LabVIEW on NI hardware for real-time control and data acquistion NI CompactRIO

Collaboration with National Instruments RB model for 6-parameter thermal problem to simulate a 4-core CPU with thermal fin = ⇒ real-time PDE solves on CompactRIO

Applications of the Reduced Basis Method ◮ Offline/Online Implementations ◮ Parameter Estimation and Model Validation ◮ Toward More Parameters

Experimental Parameter Estimation

X Y Z Z Y Experimental Parameter Estimation Consider the parametrized physical system (transient heat transfer): Macor top layer Cu leads Artic Ag Thermal Grease NiCr Wire Macor bottom layer Parameters: µ 1 : thickness of top Macor layer µ 2 : thermal conductivity of Arctic Ag Output: Temperature above NiCr on top surface

Experimental parameter estimation Prof. Ian Hunter’s group (MIT MechE) implemented this experimental setup

Experimental parameter estimation Goal: Determine µ 1 (thickness of “top layer”) by fitting output of parametrized PDE to experimental measurements Nonlinear-least squares problem ( µ → output is a nonlinear mapping), use Levenberg-Marquardt algorithm to find best fit Real-time response is desirable = ⇒ employ RB method

Experimental parameter estimation Develop RB model for the system, transient solve requires approx. 0 . 01 seconds 35 30 25 20 15 10 5 0 0 50 100 150 200 RB temperature field Surface temp. output

T ฀ ฀ T 0 T 0 Experimental parameter estimation 240 301.5 240 325 240 330 220 220 220 301 325 200 200 320 200 300.5 180 180 180 320 160 300 160 160 315 140 140 140 299.5 315 120 120 120 299 310 100 100 100 310 298.5 80 80 80 305 60 298 60 60 305 40 40 40 297.5 300 20 20 20 300 297 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 t = 0 s t = 100 s t = 200 s 35 30 25 20 15 10 Run 1 5 Run 2 Best fit 0 0 50 100 150 200 time (s) Best fit for µ 1 : 1.66mm (correct to three significant digits!), requires � 1 second

In Situ Model Validation

In Situ Model Validation Suppose we have a Physical System (PS) which provides (noisy) measurement data at times t m , 1 ≤ m ≤ M Suppose we have a proposed parametrized PDE model that represents PS Model Validation Question: For which set of parameters A ⊂ D is the proposed PDE consistent with PS? s PS ( t m ) + noise PS ˆ µ ∈ A Yes Accept PDE( ˆ µ )? No ˆ PDE µ �∈ A s n ( t m ;ˆ µ ) ˆ µ ∈ D (RB) ∆ s n ( t m ;ˆ µ )

In Situ Model Validation We employ a frequentistic approach: 4 Perform an independent µ ∈ A hypothesis test at each candidate ˆ µ to determine if ˆ ◮ Hypothesis test (introduce null hypothesis, use confidence µ ) misfit ≥ ∆ s regions, etc): Reject ˆ µ if RB(ˆ µ ) vs. PS(ˆ N (ˆ µ ) ◮ If hypothesis test with confidence level γ rejects ˆ µ , then probability that PDE(ˆ µ ) is consistent with PS is ≤ (1 − γ ) ◮ If we reject all parameters, then we conclude that the PDE model is invalid and should be rejected 4 Huynh, Knezevic, Patera, CMAME 2012; Related prior work: Balci:1982, McFarland:2008

In Situ Model Validation: FRP Example application: ◮ Fiber Reinforced Polymer (FRP) is attached to concrete to provide structural reinforcement ◮ Important to detect FRP debonding cracks, which can compromise structural integrity