An unfitted finite element method using level set functions for extrapolation into deformable diffuse interfaces. (English) Zbl 07525184
Summary: We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in embedded domains. We impose the interface conditions weakly and approximate surface integrals by volume integrals. The discretized weak form of the governing equation has the structure of an immersed boundary finite element method. That is, integration is performed over a fixed fictitious domain. Source terms are included to account for interface conditions and extend the boundary data into the complement of the embedded domain. The calculation of these extra terms requires (i) construction of an approximate delta function and (ii) extrapolation of embedded boundary data into quadrature points. We accomplish these tasks using a level set function, which is given analytically or evolved numerically. A globally defined averaged gradient of this approximate signed distance function is used to construct a simple map to the closest point on the interface. The normal and tangential derivatives of the numerical solution at that point are calculated using the interface conditions and/or interpolation on uniform stencils. Similarly to SBM, extrapolation of data back to the quadrature points is performed using Taylor expansions. Computations that require extrapolation are restricted to a narrow band around the interface. Numerical results are presented for elliptic, parabolic, and hyperbolic test problems, which are specifically designed to assess the error caused by the numerical treatment of interface conditions on fixed and moving boundaries in 2D.
MSC:
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
Keywords:
unfitted finite element method; level set algorithm; diffuse interface; regularized delta function; ghost penalty; extension velocityReferences:
| [1] | Adalsteinsson, D.; Sethian, J. A., A fast level set method for propagating interfaces, J. Comput. Phys., 118, 269-277 (1995) · Zbl 0823.65137 |
| [2] | Adalsteinsson, D.; Sethian, J. A., The fast construction of extension velocities in level set methods, J. Comput. Phys., 148, 2-22 (1999) · Zbl 0919.65074 |
| [3] | Anderson, R.; Barker, A.; Bramwell, J.; Camier, J.-S.; Cerveny, J.; Dobrev, V.; Dudouit, Y.; Fisher, A.; Kolev, Tz.; Pazner, W.; Stowell, M.; Tomov, V.; Dahm, J.; Medina, D.; Zampini, S., MFEM: a modular finite element library, Comput. Math. Appl., 81, 42-74 (2021) · Zbl 1524.65001 |
| [4] | Atallah, N. M.; Canuto, C.; Scovazzi, G., Analysis of the shifted boundary method for the Poisson problem in domains with corners, Math. Comput., 90, 2041-2069 (2021) · Zbl 1472.65140 |
| [5] | Brackbill, J. U.; Kothe, D. B.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354 (1992) · Zbl 0775.76110 |
| [6] | Burman, E., Ghost penalty, C. R. Math., 348, 1217-1220 (2010) · Zbl 1204.65142 |
| [7] | Burman, E.; Claus, S.; Hansbo, P.; Larson, M. G.; Massing, André, CutFEM: discretizing geometry and partial differential equations, Int. J. Numer. Methods Eng., 104, 472-501 (2015) · Zbl 1352.65604 |
| [8] | Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. Comput. Phys., 135, 8-29 (1997) · Zbl 0889.65133 |
| [9] | Chessa, J.; Smolinski, P.; Belytschko, T., The extended finite element method (XFEM) for solidification problems, Int. J. Numer. Methods Eng., 53, 1959-1977 (2002) · Zbl 1003.80004 |
| [10] | Engquist, B.; Tornberg, A. K.; Tsai, R., Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207, 28-51 (2005) · Zbl 1074.65025 |
| [11] | GlVis: OpenGL finite element visualization tool |
| [12] | Gorb, Yu.; Kurzanova, D.; Kuznetsov, Yu., A robust preconditioner for high-contrast problems, (Acu, B.; Danielli, D.; Lewicka, M.; Pati, A.; Saraswathy, R. V.; Teboh-Ewungkem, M., AWM Research Symposium. AWM Research Symposium, Houston, TX. AWM Research Symposium. AWM Research Symposium, Houston, TX, Advances in Mathematical Sciences, vol. 21 (2020), Springer), 289-310 · Zbl 1440.65040 |
| [13] | Hajduk, H.; Kuzmin, D.; Aizinger, V., New directional vector limiters for discontinuous Galerkin methods, J. Comput. Phys., 384, 308-325 (2019) · Zbl 1451.76069 |
| [14] | Hansbo, P.; Hansbo, A., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., 191, 5537-5552 (2002) · Zbl 1035.65125 |
| [15] | Hansbo, P.; Larson, M. G.; Zahedi, S., A cut finite element method for a Stokes interface problem, Appl. Numer. Math., 85, 90-114 (2014) · Zbl 1299.76136 |
| [16] | Hogea, C. S.; Murray, B. T.; Sethian, J. A., Simulating complex tumor dynamics from avascular to vascular growth using a general level-set method, J. Math. Biol., 53, 86-134 (2006) · Zbl 1100.92029 |
| [17] | Hysing, S., A new implicit surface tension implementation for interfacial flows, Int. J. Numer. Methods Fluids, 51, 659-672 (2006) · Zbl 1158.76350 |
| [18] | Kublik, C.; Tsai, R., Integration over curves and surfaces defined by the closest point mapping, Res. Math. Sci., 3, 1-17 (2016) · Zbl 1336.65021 |
| [19] | Kublik, C.; Tsai, R., An extrapolative approach to integration over hypersurfaces in the level set framework, Math. Comput., 87, 2365-2392 (2018) · Zbl 1391.65047 |
| [20] | Kuzmin, D., Algebraic flux correction I. Scalar conservation laws, (Kuzmin, D.; Löhner, R.; Turek, S., Flux-Corrected Transport: Principles, Algorithms, and Applications. Flux-Corrected Transport: Principles, Algorithms, and Applications, Scientific Computation (2012), Springer), 145-192 |
| [21] | Lehrenfeld, C.; Olshanskii, M., An Eulerian finite element method for PDEs in time-dependent domains, ESAIM: M2AN, 53, 585-614 (2019) · Zbl 1422.65223 |
| [22] | Li, K.; Atallah, N. M.; Main, A.; Scovazzi, G., The Shifted Interface Method: a flexible approach to embedded interface computations, Int. J. Numer. Methods Eng., 121, 492-518 (2020) · Zbl 07843207 |
| [23] | Li, X.; Lowengrub, J.; Rätz, A.; Voigt, A., Solving PDEs in complex geometries: a diffuse domain approach, Commun. Math. Sci., 7, 81-107 (2009) · Zbl 1178.35027 |
| [24] | Main, A.; Scovazzi, G., The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems, J. Comput. Phys., 372, 972-995 (2018) · Zbl 1415.76457 |
| [25] | Main, A.; Scovazzi, G., The shifted boundary method for embedded domain computations. Part II: Linear advection-diffusion and incompressible Navier-Stokes equations, J. Comput. Phys., 372, 996-1026 (2018) · Zbl 1415.76458 |
| [26] | May, S.; Berger, M., An explicit implicit scheme for cut cells in embedded boundary meshes, J. Sci. Comput., 71, 919-943 (2017) · Zbl 1372.65250 |
| [27] | Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 239-261 (2005) · Zbl 1117.76049 |
| [28] | Müller, B.; Kummer, F.; Oberlack, M., Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Int. J. Numer. Methods Eng., 96, 512-528 (2013) · Zbl 1352.65083 |
| [29] | Osher, S.; Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces (2003), Springer: Springer New York · Zbl 1026.76001 |
| [30] | Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517 (2003) · Zbl 1123.74309 |
| [31] | Quezada de Luna, M.; Kuzmin, D.; Kees, C. E., A monolithic conservative level set method with built-in redistancing, J. Comput. Phys., 379, 262-278 (2019) · Zbl 07581572 |
| [32] | Quezada de Luna, M.; Collins, J. H.; Kees, C. E., An unstructured finite element model for incompressible two-phase flow based on a monolithic conservative level set method, Int. J. Numer. Methods Fluids, 92, 1058-1080 (2020) |
| [33] | Sethian, J. A.; Smereka, P., Level set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35, 341-372 (2003) · Zbl 1041.76057 |
| [34] | Smereka, P., The numerical approximation of a delta function with application to level set methods, J. Comput. Phys., 211, 77-90 (2006) · Zbl 1086.65503 |
| [35] | Teigen, K. E.; Song, P.; Lowengrub, J.; Voigt, A., A diffuse-interface method for two-phase flows with soluble surfactant, J. Comput. Phys., 230, 375-393 (2011) · Zbl 1428.76210 |
| [36] | Teigen, K. E.; Wang, F.; Li, X.; Lowengrub, J.; Voigt, A., A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci., 7, 1009-1037 (2009) · Zbl 1186.35168 |
| [37] | Towers, J. D., Two methods for discretizing a delta function supported on a level set, J. Comput. Phys., 220, 915-931 (2007) · Zbl 1115.65028 |
| [38] | Utz, T.; Kummer, F., A high-order discontinuous Galerkin method for extension problems, Int. J. Numer. Methods Fluids, 86, 509-518 (2018) |
| [39] | Vermolen, F. J.; Javierre, E.; Vuik, C.; Zhao, L.; van der Zwaag, S., A three-dimensional model for particle dissolution in binary alloys, Comput. Mater. Sci., 39, 767-774 (2007) |
| [40] | Wadbro, E.; Zahedi, S.; Kreiss, G.; Berggren, M., A uniformly well-conditioned, unfitted Nitsche method for interface problems, BIT Numer. Math., 53, 791-820 (2013) · Zbl 1279.65134 |
| [41] | Zahedi, S.; Tornberg, A. K., Delta function approximations in level set methods by distance function extension, J. Comput. Phys., 229, 2199-2219 (2010) · Zbl 1186.65018 |
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