Baroli, Davide

Zilian, Andreas

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BAROLI, Davide

ZILIAN, Andreas

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Received: 31 May 2019 Accepted: 4 July 2019DOI: 10.1002/pamm.201900437Model order reduction applied to ALE-fluid dynamicsDavide Baroli1,∗ and Andreas Zilian11 University of Luxembourg, Faculty of Science, Technology and Communication, Institute of Computational Engineering,6 Avenue de la Fonte, L-4362 Esch-sur-Alzette, LuxembourgThe Arbitrary Lagrangian Eulerian formulation allows the description of fluid dynamics involving moving and deformingfluid domains as they are found in many fluid-structure interaction problems. This paper discusses a reduced order approachfor ALE incompressible Navier-Stokes flow formulated on a fixed reference configuration. Based on the proper orthogonaldecomposition (POD) for generation of the reduced basis, the supermizer technique is adapted to the formulation involving anon-affine term and used to ensure the fulfilment of the inf-sup condition among the velocity and pressure basis. The approachis studied for an extended version of the classical driven-cavity setup involving a non-affine parameterization of the prescribedmesh deformation and a time-dependent non-homogeneous lid-flow boundary condition.c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim1 ALE-Fluid DynamicsMany problems in fluid-structure interaction involve a fluid flow on a deforming domain due to a deformation of the structureor moving fluid boundaries. Energy harvesting devices driven by flow-induced vibrations [1] are an exemplary engineeringapplication. However, proper representation of the multi-physics problem requires high-fidelity unsteady simulations whichare computationally demanding. This is particularly the case when design studies require parameterization of the problem onthe material parameters, boundary conditions and geometry.An incompressible flow is considered, described by the Navier-Stokes equations in velocity-pressure form. Let Ωf be thespatial domain occupied by the current fluid body, the formulation in the ALE framework readsρ∂v∂t+ ((v −w) · ∇)v −∇ ·T = f , on Ωf × (0, T ) (1)∇ · v = 0 on Ωf × (0, T ) (2)where v is the velocity and ρ the density. The advective velocity v − w takes into account the relative motion of fluid anddiscretization, where w denotes the mesh velocity, which is assumed to be known in order to simplify the present study. Thestress tensor T of a Newtonian fluid is defined as T = −pI + 2µD(v) with D(u) = 12 (∇v + (∇v)T ), where p denotesthe pressure and µ the dynamic viscosity. The functional spaces for the state variables velocity and pressure are definedon the (fixed) reference domain Ωr := χ−1(Ωf ) by Vg = {v|v ∈ H1(Ωr),v = gv on Γru}; W = {w|w ∈ H1(Ωr),w =gw on Γw},Qg = {p|p ∈ L2(Ωr), p = δ(x−xc) on Γrp}. Introducing the weak parameterized formulation, the incompressibleNavier-Stokes in ALE formulation in the reference configuration reads: find (v, p) ∈ Vg ×Qg such that∫ΩrδvρJ · ∂v∂t· dx+∫Ωr∇δv : J(µ · (F−T · (∇v)T +∇v · F−1)) · F−T · dx (3)+∫Ωrδv · ρ J · ∇v · F−1 · (v −w) · dx+∫Ωrδp · ∇ · (JF−1v) · dx−∫Ωrδv · J · f · dx (4)+∫Ωr∇δv : J(−pI) · F−T dx = 0 (5)where ρ and µ are constant and the presence of J in terms of the prescribed mesh deformation leads to non-affinity of theparameterization. The mesh displacement d(x, t) provides mesh velocityw = ∂d∂t andF = (I+∇d) together with J = detF.2 Reduced order method: POD-EIMThe fundamental idea of projection-based reduced order method relies on the mapping of a high dimensional system to alow-dimensional subspace. Among different techniques described in [2], in this study the proper-orthogonal decompositionmethod (POD) is used to define the basis that span the sub-space manifold. The computational effort of the reduced ordermethod is separated in offline stage, where snapshots of velocity and pressure are collected and the POD basis is generated; andthe online stage where the main operations are the projection of pre-computed discrete full-order system and the reconstructionof the solution in the high-dimensional manifold.∗ Corresponding author: Andreas Zilian andreas.zilian@uni.luThis is an open access article under the terms of the Creative Commons Attribution License 4.0, which permits use,distribution and reproduction in any medium, provided the original work is properly cited.PAMM · Proc. Appl. Math. Mech. 2019;e201900437. www.gamm-proceedings.com 1 of 2https://doi.org/10.1002/pamm.201900437 c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim2 of 2 Section 7: Coupled problemsIn this study, the snapshot matrix has dimension (M×S)×N , withM the number of parameter realisations (e.g. Reynoldsnumber), S the number of time-steps and N the number of degree of freedom of full-order discretization. In addition, tosatisfy the inf-sup condition for the reduced velocity-pressure basis, the velocity POD basis is enriched by the supermizerapproach, introduced in [3], before the online projection. The enriched velocity space is defined as V = 〈{vi}Nvi=1〉 ⊕POD({Sµp(µi);Np}Npi=1) where the inner pressure supermizer operator is defined as(Sµq,v)H10 = J(d(x, t)) · (q,∇ · (J(d(x, t))F−T (d(x, t))v)) ∀v ∈ POD({vi}Nvi=1 ;Nv). (6)3 Numerical TestThe ALE fluid discussed here is investigated for the driven-cavity test-case to show the challenges associated with the PODapproach. The time-dependent displacement of the mesh d(x, t) = [0.1 · sin(4 · pi · x0) · sin(pi · t), 0] is prescribed at eachnode. The pressure level is prescribed at the upper left corner. An additional source of non-affine and non-linear term ariseswith the lid velocity v(t) = tanh(t) boundary condition. The development of the cavity vortex is studied in the time interval[0, 2] seconds for different Reynolds numbers (10-100) as problem parameterization. The implementation uses the open-sourcefinite element library FEniCS [4]; the reduced order method is implemented with the RBniCS package [5]. The number ofdegree of freedom of the full-order model is 20402 for velocity and 2601 for pressure.Fig. 1: Magnitude and vector field of the velocity basis (3 left) and enriching supermizer POD basis (2 right) for the driven cavity problem.A selection of basis vectors for the velocity state are given in Figure 1 together with the basis enrichment obtained with thesupermizer approach. The POD-generated velocity basis captures the complexity of the developing cavity vortex induced bythe lid velocity, while the supermizer basis contributes the complementary components required for stability of the discretizedvelocity-pressure fluid formulation. Figure 2 shows the relative error with respect to the number of modes used in the reducedmodel. The non-monotone pressure error is assumed to be caused by the mesh motion and requires further investigation.0 50 100 15010 1510 1010 5100105eigenvaluesvelocitysupermizerpressure0 5 1010 1relative error relative error velocityrelative error pressureFig. 2: POD eigenvalues and relative error in norm H10 and L2 for velocity and pressure with respect to the FEM solution.Acknowledgements We acknowledge the financial support provided through the FSI-HARVEST (EU 322151) and MACAFIVE projectand the support of F. Ballarin, developer of RBniCS.References[1] S. Ravi and A. Zilian. Simultaneous finite element analysis of circuit-integrated piezoelectric energy harvesting from fluid-structureinteraction. Mechanical Systems & Signal Processing, 114, 2019.[2] J. Hestaven, G. Rozza, B. Stamm, Certified reduced basis for parametrized partial differential equations, 2016, Springer.[3] G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains. CMAME 196,p. 1244–1260, 2007.[4] A. Logg, K.-A. Mardal, G. Wells, Automated solution of differential equation by the finite element method: the FEniCS book, 84, 2012,Springer Science & Business Media.[5] F. Ballarin, A. Sartori and G. Rozza, RBniCS-reduced order modelling in FEniCS. 2015. http://mathlab.sissa.it/rbnics, 2015.c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim www.gamm-proceedings.com

Model order reduction applied to ALE‐fluid dynamics

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Model order reduction applied to ALE‐fluid dynamics

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