Shape optimisation for faster washout in recirculating flows. (English) Zbl 1461.76583

Summary: How to design an optimal biomedical flow device to minimise trapping of undesirable biological solutes/debris and/or enhance their washout is a pertinent but complex question. While biomedical devices often utilise externally driven flows to enhance washout, the presence of vortices – arising as a result of fluid flows within cavities – hinder washout by trapping debris. Motivated by this, we solve the steady, incompressible Navier-Stokes equations for flow through channels into and out of a two-dimensional cavity. In endourology, the presence of vortices – enhanced by flow symmetry breaking – has been linked to long washout times of kidney stone dust in the renal pelvis cavity, with dust transport modelled via advection and diffusion of a passive tracer [J. G. Williams et al., ibid. 902, Paper No. A16, 26 p. (2020; Zbl 1460.76960)]. Here, we determine the inflow and outflow channel geometries that minimise washout times. For a given flow field \(\boldsymbol{u}\), vortices are characterised by regions where \(\det \nabla \boldsymbol{u} >0\) [J. Jeong and F. Hussain, ibid. 285, 69–94 (1995; Zbl 0847.76007)]. Integrating a smooth form of \(\max (0, \det \boldsymbol{\nabla }\boldsymbol{u})\) over the domain provides an objective to minimise recirculation zones [H. Kasumba and K. Kunisch, Comput. Optim. Appl. 52, No. 3, 691–717 (2012; Zbl 1258.49070)]. We employ adjoint-based shape optimisation to identify inflow and outflow channel geometries that reduce this objective. We show that a reduction in the vortex objective correlates with reduced washout times. We additionally show how multiple solutions to the flow equations lead to solution branch switching during the optimisation routine by characterising the change in solution bifurcation structure with the change in inflow/outflow channel geometry.


76Z05 Physiological flows
76D55 Flow control and optimization for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics


Fireshape; COFFEE; TSFC; PETSc
Full Text: DOI


[1] Abergel, F. & Temam, R.1990On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn.1 (6), 303-325. · Zbl 0708.76106
[2] Alnæs, M.S., Logg, A., Ølgaard, K.B., Rognes, M.E. & Wells, G.N.2014Unified Form Language: a domain-specific language for weak formulations of partial differential equations. ACM Trans. Math. Softw.40 (2), 9:1-9:37. · Zbl 1308.65175
[3] Amestoy, P.R., Duff, I.S., L’Excellent, J.-Y. & Koster, J.2000 MUMPS: a general purpose distributed memory sparse solver. In International Workshop on Applied Parallel Computing, pp. 121-130. Springer.
[4] Balay, S.et al.. 2018 PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.10. Argonne National Laboratory.
[5] Balay, S.et al.. 2016 PETSc web page.
[6] Balay, S., Gropp, W.D., Mcinnes, L.C. & Smith, B.F.1997 Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing, pp. 163-202. Birkhäuser. · Zbl 0882.65154
[7] Brenner, S.C. & Sung, L.-Y.\(2005C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput.22-23 (1-3), 83-118. · Zbl 1071.65151
[8] Clavica, F., Zhao, X., Elmahdy, M., Drake, M.J., Zhang, X. & Carugo, D.2014Investigating the flow dynamics in the obstructed and stented ureter by means of a biomimetic artificial model. PLoS ONE9 (2), e87433.
[9] Coppola, G. & Caro, C.2009Arterial geometry, flow pattern, wall shear and mass transport: potential physiological significance. J. R. Soc. Intl6, 519-528.
[10] Dalcin, L.D., Paz, R.R., Kler, P.A., Cosimo, A.2011Parallel distributed computing using Python. Adv. Water Resour.34 (9), 1124-1139. New Computational Methods and Software Tools.
[11] Delfour, M.C. & Zolésio, J.-P.2011Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. Society for Industrial and Applied Mathematics.
[12] Farrell, P.E., Birkisson, Á. & Funke, S.W.2015Deflation techniques for finding distinct solutions of nonlinear partial differential equations. SIAM J. Sci. Comput.37 (4), A2026-A2045. · Zbl 1327.65237
[13] Franca, L.P., Frey, S.L. & Hughes, T.J.R.1992Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Meth. Appl. Mech. Engng95 (2), 253-276. · Zbl 0759.76040
[14] Geuzaine, C. & Remacle, J.-F.2009Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Intl J. Numer. Meth. Engng79 (11), 1309-1331. · Zbl 1176.74181
[15] Guzmán, J. & Scott, L.R.2018The Scott-Vogelius finite elements revisited. Maths Comput.88 (316), 515-529. · Zbl 1405.65150
[16] Haller, G.2005An objective definition of a vortex. J. Fluid Mech.525, 1-26. · Zbl 1065.76031
[17] Ham, D.A., Mitchell, L., Paganini, A. & Wechsung, F.2019Automated shape differentiation in the Unified Form Language. Struct. Multidiscipl. Optim.60, 1813-1820.
[18] Homolya, M., Kirby, R.C. & Ham, D.A.2017 Exposing and exploiting structure: optimal code generation for high-order finite element methods. ACM Trans. Math. Softw. (in press), arXiv:1711.02473.
[19] Homolya, M., Mitchell, L., Luporini, F. & Ham, D.A.2018 TSFC: a structure-preserving form compiler. SIAM J. Sci. Comput.40 (3), C401-C428. · Zbl 1388.68020
[20] Jeong, J. & Hussain, F.1995On the identification of a vortex. J. Fluid Mech.285, 69-94. · Zbl 0847.76007
[21] Jiménez, J.M. & Davies, P.F.2009Hemodynamically driven stent strut design. Ann. Biomed. Engng37 (8), 1-24.
[22] John, V., Linke, A., Merdon, C., Neilan, M. & Rebholz, L.G.2017On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev.59 (3), 492-544. · Zbl 1426.76275
[23] Kasumba, H. & Kunisch, K.2010 Shape design optimization for viscous flows in a channel with a bump and an obstacle. In 15th International Conference on Methods and Models in Automation and Robotics, MMAR 2010, pp. 284-289.
[24] Kasumba, H. & Kunisch, K.2012Vortex control in channel flows using translational invariant cost functionals. Comput. Optim. Appl.52, 691-717. · Zbl 1258.49070
[25] Luporini, F., Ham, D.A. & Kelly, P.H.J.2017An algorithm for the optimization of finite element integration loops. ACM Trans. Math. Softw.44, 3:1-3:26. · Zbl 1380.65381
[26] Micheletti, A.M.1972Metrica per familglie di domini limitati a proprietá generiche degli autovalori. Ann. Scu. Norm. Super. Pisa26 (3), 683-694. · Zbl 0255.35028
[27] Min, D., Fischer, P.F. & Pearlstein, A.J.2020High Schmidt number ‘washout’ of a soluble contaminant downstream of a backward facing step. Intl J. Heat Mass Transfer159, 119740.
[28] Murat, F. & Simon, J.1976 Etude de problemes d’optimal design. In Optimization Techniques Modeling and Optimization in the Service of Man Part 2, pp. 54-62. Springer. · Zbl 0334.49013
[29] Nocedal, J. & Wright, S.J.2006Numerical Optimization. Springer.
[30] Paganini, A. & Wechsung, F.2020 Fireshape: a shape optimization toolbox for Firedrake. Struct. Multidiscipl. Optim. (submitted), arXiv:2005.07264.
[31] Paster, A., Shlain, S., Barghouthy, Y., Liberzon, A., Aviram, G. & Sofer, M.2019Assessing the influence of irrigation flows on clearance of calculi fragments during percutaneous nephrolithotomy: a numerical and physical model study. Urology124, 46-51.
[32] Qin, J.1994 On the convergence of some low order mixed finite elements for incompressible fluids. PhD thesis, Pennsylvania State University.
[33] Rathgeber, F., Ham, D.A., Mitchell, L., Lange, M., Luporini, F., Mcrae, A.T.T., Bercea, G.-T., Markall, G.R. & Kelly, P.H.J.2016Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Softw.43 (3), 24:1-24:27. · Zbl 1396.65144
[34] Rhines, B. & Young, W.R.1983How rapidly is a passive scalar mixed within closed streamlines?J. Fluid Mech.133, 133-145. · Zbl 0576.76088
[35] Ridzal, D. & Kouri, D.2014 Rapid Optimization Library. Software, US Department of Energy. Available at: https://www.osti.gov//servlets/purl/1232084.
[36] Silvester, D., Elman, H. & Wathen, A.2014Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press. · Zbl 1304.76002
[37] Simon, J.1980Differentiation with respect to the domain in boundary value problems. Numer. Func. Anal. Opt.2 (7-8), 649-687. · Zbl 0471.35077
[38] Wechsung, F.2019 Shape optimisation and robust solvers for incompressible flow. PhD thesis, University of Oxford.
[39] Wells, G.N., Kuhl, E. & Garikipati, K.2006A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Phys.218 (2), 860-877. · Zbl 1106.65086
[40] Williams, J.G., Castrejon-Pita, A.A, Turney, B.W., Farrell, P.E., Tavener, S.J., Moulton, D.E. & Waters, S.L.2020Cavity flow characteristics and applications to kidney stone removal. J. Fluid Mech.902, A16. · Zbl 1460.76960
[41] Xhang, P.H., Tkatch, C., Newman, R., Grimme, W., Vainchtein, D. & Kresh, J.Y.2019The mechanics of spiral flow: enhanced washout and transport. Artif. Organs43, 1144-1153.
[42] 2020 Software used in ‘Shape optimisation for faster washout in recirculating flows’. Available at: https://doi.org/10.5281/zenodo.4075330.
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