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Toward fluid-structure-piezoelectric simulations
applied to flow-induced energy harvesters
C. HOAREAU, L. SHANG and A. ZILIAN
Institute of Computational Engineering, FSTC, University of Luxembourg , Luxembourg, [email protected]
Flow Induced Energy Harvesting Research objectives
Fluid Structure Piezo + circuit
High fidelity model+FEniCS Reduced Order Modelling
A priori approaches
Problems
Conclusion - Work in Progress
Work done Work in progress Outlook
References
1 2
3
4
5 4 6
4
7
Structure
Fluid
Piezoelectric patch
Circuit
Autonomous piezo-ceramic power generators
Energy
Harvester
Ambient fluid
flow energy Electrical
energy
Nonlinear dynamic behavior
Flutter induced vibrations (1) Frequency (Hz)
Elec. Power (mW/g)
Model and predict the nonlinear dynamic behavior (2) Quantify the sensibility under changing conditions
Allow just-in-time feedback : construct on-line approximations
Maximize the power output
Minimize the fatigue exposure
Error Model Experiment
Complex multiphysics phenomena Incresing number of parameters
Hypotheses
Fluid - Time integration Learn and use FEniCS
Solve Navier-Stokes problems
A posteriori reduction : POD
Example : Steady-Navier Stokes - 2D Lid driven cavity
A posteriori approaches
(ex: Steady Navier-Stokes)In vacuo eigenvalue problem
Induced fluid load approximation
Reduced problem with reduced added operators
Can we construct a Reduced Order Model
to approximate a nonlinear multiphysic problem ?
Methodology
Numerous fields (displacement, velocity, pressure, intensity ...) Nonlinear behaviors (geometrical nonlinearity, fluid dynamic)
displacement velocity acceleration
Example with hydroelasticity (beam in water)
Expensive
mathematical model
ROM
Modal Basis (6)
Example : Linearized hydroelasticity
Proper Generalized
Decomposition (5)
Example : Steady Navier-Stokes
Series of separated functions variables products
For each i-th modes : Greedy Algorithm (GA)
20 40 60 80 100
0.0 0.5 1.0 1.5 2.0 2.5
20 40 60 80 100
6 4 2 0 2
20 40 60 80 100
0 1 2 3 4 5
Strong form (velocity - pressure) in in on
Weak form (velocity - pressure)
Parameters
Quantity of interest
a posteriori
ROM
a priori Parameters
Approximation
Numerical tool (3)
Finite Element Method
Proper Orthogonal Decomposition (4)
Snapshots (velocity - pressure)
Incompressible
Newtonian
Homogeneous
Isotropic
Low Reynolds number
Moving mesh
? ?
1 50 100
Re
Discussions Discussions Discussions
Example with steady Navier-Stokes
Elastic
Homogeneous
Isotropic
Finite strain
0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Ux
0.0 0.2 0.4 0.6 0.8 1.0
y
Ux at x=0.5 and Re = 100
Fenics
Benchmark Ghia
400 P1- elements 400 P2- elements
Re Re Re
Spatial problem
Parametric problem GA
Satisfy Dirichlet BC
added stiffness added damping added mass
Incompressible
Potential flow
Homogeneous
Isotropic
Elastic
Homogeneous
Isotropic
Linearized
0 5 10 15 20 25 30
Num. singular value
10 12 10 10 108 106 104 102 100
Convergency at Re = 100
Singular values
0 2 4 6 8
Number of modes Number of modes
104 103 102 101
Velocity error
0 2 4 6 8
102
Pressure error f1 = 185 Hz
f1a = 95 Hz
f2 = 1163 Hz f2a = 605 Hz
Appropriate method / weakly nonlinear problems
Tests with moving meshes on-going
Subjected to the curse of dimensionality
Overcome the curse of dimensionality No convergency of the pressure for now Tests with time intergation on-going
Necessitates restriction of hypotheses Give valuable physical information
Linearization around flow on-going
Fluid - Structure : ALE formulation
Fluid - Structure : Aeroelasticity
A priori reduction : PGD Linearized hydroelasticity
Structure : Velocity formulation
Comparison between POD-PGD-MB
Add piezoelectricity equations
Investigate structural models (beam - plates)
Design an experimental set-up (7)
Machin Learning
(1) A. Kornecki & al., JSV, 1976
(2) S. Ravi & A. Zilian, IJNME 2017
(3) M. S. Alnaes & al., ANS, 2015
(4) G. Berkooz & al., ARFM, 1993
(5) F. Chinesta & al., ACME, 2011
(6) C. Hoareau, PhD thesis, 2019
(7) C. De Marqui & al., JFS, 2018
Generate a data-base of F-S-P simulations
