Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport

ESAIM: Mathematical Modelling and Numerical Analysis

Volumn 45, Issue 3, 2011, Pages 447-476

  D'Angelo, Carlo ^a   Zunino, Paolo ^a  

^a POLITECNICO DI MILANO   (Italy)

Author keywords

biological flows and mass transfer; cardiovascular applications; Coupled Stokes Darcy's problem; finite element approximation; interior penalty method; iterative splitting strategy; optimal preconditioning

Indexed keywords

BLOOD; BLOOD VESSELS; CONSTRAINED OPTIMIZATION; HEMODYNAMICS; INCOMPRESSIBLE FLOW; ITERATIVE METHODS; MASS TRANSFER; NAVIER STOKES EQUATIONS; NUMERICAL METHODS; VISCOSITY;

BIOLOGICAL FLOWS; CARDIOVASCULAR APPLICATIONS; COUPLED STOKES/DARCY'S PROBLEM; FINITE ELEMENT APPROXIMATIONS; INTERIOR PENALTY METHOD; OPTIMAL PRECONDITIONING; SPLITTING STRATEGIES;

FINITE ELEMENT METHOD;

EID: 79952544088     PISSN: 28227840     EISSN: 28047214     Source Type: Journal    
DOI: 10.1051/m2an/2010062     Document Type: Article

References (44)

1. S. Badia and R. Codina, Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47 (2009) 1971-2000.
2. B. Balakrishnan, A.R. Tzafriri, P. Seifert, A. Groothuis, C. Rogers and E.R. Edelman, Strut position, blood flow, and drug deposition. Implications for single and overlapping drug-eluting stents. Circulation 111 (2005) 2958-2965. (Pubitemid 40814189)
3. R. Balossino, F. Gervaso, F. Migliavacca and G. Dubini, Effects of different stent designs on local hemodynamics in stented arteries. J. Biom. 41 (2008) 1053-1061.
4. J.M. Boland and R.A. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731.
5. J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the L2 projection in H1(O). Math. Comp. 71 (2002) 147-156.
6. E. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem. Numer. Meth. Partial Diff. Equ. 24 (2008) 127-143.
7. E. Burman and P. Hansbo, A unified stabilized method for Stokes' and Darcy's equations. J. Comput. Appl. Math. 198 (2007) 35-51. (Pubitemid 44345052)
8. E. Burman and P. Zunino, A domain decomposition method based on weighted interior penalties for advection-diffusionreaction problems. SIAM J. Numer. Anal. 44 (2006) 1612-1638. (Pubitemid 47264162)
9. E. Burman, M.A. Fernández and P. Hansbo, Continuous interior penalty finite element method for Oseen's equations. SIAM J. Numer. Anal. 44 (2006) 1248-1274. (Pubitemid 47135922)
10. J. Cahouet and J.-P. Chabard, Some fast 3D finite element solvers for the generalized Stokes problem. Int. J. Numer. Methods Fluids 8 (1988) 869-895. (Pubitemid 18649331)
11. C. Calgaro, P. Deuring and D. Jennequin, A preconditioner for generalized saddle point problems: Application to 3D stationary Navier-Stokes equations. Numer. Methods Partial Diff. Equ. 22 (2006) 1289-1313. (Pubitemid 44678763)
12. C. D'Angelo and A. Quarteroni, On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18 (2008) 1481-1504.
13. C. D'Angelo and P. Zunino, A numerical study of the interaction of blood flow and drug release from cardiovascular stents, in Numerical Mathematics and Advanced Applications - Proceedings of ENUMATH 2007, Springer, Berlin (2008) 75-82.
14. C. D'Angelo and P. Zunino, A finite element method based on weighted interior penalties for heterogeneous incompressible flows. SIAM J. Numer. Anal. 47 (2009) 3990-4020.
15. C. D'Angelo and P. Zunino, Multiscale models of drug delivery by thin implantable devices, in Applied and industrial mathematics in Italy III, Ser. Adv. Math. Appl. Sci. 82, World Sci. Publ. (2009).
16. C. D'Angelo and P. Zunino, Numerical approximation with Nitsche's coupling of transient Stokes'/Darcy's flow problems applied to hemodynamics. Technical report, MOX, Department of Mathematics, Politecnico di Milano (submitted).
17. M.C. Delfour, A. Garon and V. Longo, Modeling and design of coated stents to optimize the effect of the dose. SIAM J. Appl. Math. 65 (2005) 858-881. (Pubitemid 40892129)
18. M. Discacciati, A. Quarteroni and A. Valli, Robin-Robin domain decomposition methods for the Stokes Darcy coupling. SIAM J. Numer. Anal. 45 (2007) 1246-1268.
19. H.C. Elman, D.J. Silvester and A.J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation, Oxford University Press, New York (2005).
20. L. Formaggia, A. Quarteroni and A. Veneziani Eds., Cardiovascular mathematics - Modeling and simulation of the circulatory system, MS& A Modeling, Simulation and Applications 1. Springer-Verlag Italia, Milan (2009).
21. L. Formaggia, S. Minisini and P. Zunino, Modeling erosion controlled drug release and transport phenomena in the arterial tissue. Math. Models Methods Appl. Sci. (to appear).
22. R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press, Cambridge (1990), Corrected reprint of the 1985 original.
23. A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term. SIAM J. Sci. Comput. 19 (1998) 172-184. Special issue on iterative methods (Copper Mountain, CO, 1996). (Pubitemid 128617506)
24. A. Klawonn and G. Starke, Block triangular preconditioners for nonsymmetric saddle point problems: Field-of-values analysis. Numer. Math. 81 (1999) 577-594.
25. V.B. Kolachalama, A.R. Tzafriri, D.Y. Arifin and E.R. Edelman, Luminal flow patterns dictate arterial drug deposition in stent-based delivery. J. Control Release 133 (2009) 24-30.
26. W.J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195-2218.
27. S.L. Lee and C.W. Gear, Second-order accurate projective integrators for multiscale problems. J. Comput. Appl. Math. 201 (2007) 258-274. (Pubitemid 46084980)
28. D. Loghin and A.J.Wathen, Analysis of preconditioners for saddle-point problems. SIAM J. Sci. Comput. 25 (2004) 2029-2049.
29. M.A. Lovich and E.R. Edelman, Mechanisms of transmural heparin transport in the rat abdominal aorta after local vascular delivery. Circ. Res. 77 (1995) 1143-1150.
30. F. Migliavacca, L. Petrini, M. Colombo, F. Auricchio and R. Pietrabissa, Mechanical behavior of coronary stents investigated through the finite element method. J. Biomech. 35 (2002) 803-811. (Pubitemid 34518378)
31. L. Petrini, F. Migliavacca, F. Auricchio and G. Dubini, Numerical investigation of the intravascular coronary stent flexibility. J. Biomech. 37 (2004) 495-501. (Pubitemid 38292988)
32. G. Pontrelli and F. de Monte, Mass diffusion through two-layer porous media: An application to the drug-eluting stent. Int. J. Heat Mass Transfer 50 (2007) 3658-3669. (Pubitemid 46680313)
33. K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17 (2007) 215-252.
34. P. Saffman, On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 292-315.
35. D.V. Sakharov, L.V. Kalachev and D.C. Rijken, Numerical simulation of local pharmacokinetics of a drug after intravascular delivery with an eluting stent. J. Drug Targ. 10 (2002) 507-513. (Pubitemid 35230201)
36. Ch. Schwab, p- and hp-Finite Element Methods - Theory and applications in solid and fluid mechanics. Numerical Mathematics and Scientific Computation, Oxford University Press, New York (1998).
37. D. Silvester and A. Wathen, Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners. SIAM J. Numer. Anal. 31 (1994) 1352-1367.
38. J.S. Soares and P. Zunino, A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks. Biomaterials 31 (2010) 3032-3042.
39. G. Starke, Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems. Numer. Math. 78 (1997) 103-117. (Pubitemid 127375363)
40. S. Tada and J.M. Tarbell, Internal elastic lamina affects the distribution of macromolecules in the arterial wall: A computational study. Am. J. Physiol. Heart Circ. Physiol. 287 (2004) H905-H913. (Pubitemid 38955329)
41. A.R. Tzafriri and E.R. Edelman, On the validity of the quasi-steady state approximation of bimolecular reactions in solution. J. Theor. Biol. 233 (2005) 343-350. (Pubitemid 40093593)
42. C. Vergara and P. Zunino, Multiscale boundary conditions for drug release from cardiovascular stents. Multiscale Model. Simul. 7 (2008) 565-588.
43. A. Wathen and D. Silvester, Fast iterative solution of stabilised Stokes systems. I. Using simple diagonal preconditioners. SIAM J. Numer. Anal. 30 (1993) 630-649. (Pubitemid 23682561)
44. P. Zunino, C. D'Angelo, L. Petrini, C. Vergara, C. Capelli and F. Migliavacca, Numerical simulation of drug eluting coronary stents: Mechanics, fluid dynamics and drug release. Comput. Methods Appl. Mech. Eng. 198 (2009) 3633-3644.