A novel surface-derivative-free of jumps AIIM with triangulated surfaces for 3D Helmholtz interface problems. (English) Zbl 07843843
Summary: Triangular surface-based 3D IIM (Immersed Interface Method) algorithms face major challenges due to the need to calculate surface derivative of jumps. This paper proposes a fast, easy-to-implement, surface-derivative-free of jumps, augmented IIM (AIIM) with triangulated surfaces for 3D Helmholtz interface problems for the first time, which combines the simplified AIIM with domain decomposed and embedding techniques. The computational domain is divided into sub-domains along the interface and the solutions of sub-domains are continuously extended into larger regular domains by embedding. The jumps in normal derivative of solution along the interfaces in the extended domains are introduced as unknowns to impose the original jump relations. The original problem is simplified into Helmholtz interface problems with constant coefficients by coupling them with the augmented equation, which is then solved using fast simplified AIIM. This approach eliminates the need to compute surface derivatives of jumps, making implementation of 3D IIM based on triangulated surfaces fairly simple. Numerical results demonstrate that the algorithm is efficient and can achieve the overall second-order accuracy.
MSC:
| 76Mxx | Basic methods in fluid mechanics |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
| 76Dxx | Incompressible viscous fluids |
Keywords:
simplified AIIM; domain decomposed method; triangular meshes; surface derivative; Helmholtz/Poisson equations; two-phase Stokes flowReferences:
| [1] | Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159, 1994 · Zbl 0808.76077 |
| [2] | Yakonov, Y. G.D., Introduction to the theory of difference schemes, USSR Comput. Math. Math. Phys., 4, 259-261, 1964 |
| [3] | Shubin, G. R.; Bell, J. B., An analysis of the grid orientation effect in numerical simulation of miscible displacement, Comput. Methods Appl. Mech. Eng., 47, 47-71, 1984 · Zbl 0535.76117 |
| [4] | Zhu, S.; Yuan, G.; Sun, W., Convergence and stability of explicit/implicit schemes for parabolic equations with discontinuous coefficients, Int. J. Numer. Anal. Model., 1, 131-145, 2004 · Zbl 1075.65116 |
| [5] | Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 220-252, 1977 · Zbl 0403.76100 |
| [6] | Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517, 2002 · Zbl 1123.74309 |
| [7] | Leveque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019-1044, 1994 · Zbl 0811.65083 |
| [8] | Li, Z.; Ito, K., Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23, 339-361, 2001 · Zbl 1001.65115 |
| [9] | Li, Z.; Ji, H.; Chen, X., Accurate solution and gradient computation for elliptic interface problems with variable coefficients, SIAM J. Numer. Anal., 55, 570-597, 2017 · Zbl 1362.76037 |
| [10] | Li, Z.; Ito, K., The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, vol. 33, 2006, SIAM · Zbl 1122.65096 |
| [11] | Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35, 230-254, 1998 · Zbl 0915.65121 |
| [12] | Rutka, V.; Li, Z., An explicit jump immersed interface method for two-phase Navier-Stokes equations with interfaces, Comput. Methods Appl. Mech. Eng., 197, 2317-2328, 2008 · Zbl 1158.76410 |
| [13] | Deng, S.; Ito, K.; Li, Z., Three-dimensional elliptic solvers for interface problems and applications, J. Comput. Phys., 184, 215-243, 2003 · Zbl 1016.65072 |
| [14] | Lai, M.-C.; Tseng, H.-C., A simple implementation of the immersed interface methods for Stokes flows with singular forces, Comput. Fluids, 37, 99-106, 2008 · Zbl 1194.76179 |
| [15] | Huang, W.-X.; Chang, C. B.; Sung, H. J., Three-dimensional simulation of elastic capsules in shear flow by the penalty immersed boundary method, J. Comput. Phys., 231, 3340-3364, 2012 · Zbl 1404.74159 |
| [16] | Hu, W.-F.; Lai, M.-C.; Seol, Y.; Young, Y.-N., Vesicle electrohydrodynamic simulations by coupling immersed boundary and immersed interface method, J. Comput. Phys., 317, 66-81, 2016 · Zbl 1349.76938 |
| [17] | Xu, J.-J.; Shi, W.; Hu, W.-F.; Huang, J.-J., A level-set immersed interface method for simulating the electrohydrodynamics, J. Comput. Phys., 400, Article 108956 pp., 2020 · Zbl 1453.76144 |
| [18] | Hu, W.-F.; Lai, M.-C.; Young, Y.-N., A hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations, J. Comput. Phys., 282, 47-61, 2015 · Zbl 1351.76165 |
| [19] | Medina Dorantes, F.; Itzá Balam, R.; Uh Zapata, M., The immersed interface method for Helmholtz equations with degenerate diffusion, Math. Comput. Simul., 190, 280-302, 2021 · Zbl 1540.65458 |
| [20] | Itza Balam, R.; Uh Zapata, M., A fourth-order compact implicit immersed interface method for 2D Poisson interface problems, Comput. Math. Appl., 119, 257-277, 2022 · Zbl 1524.65713 |
| [21] | Colnago, M.; Casaca, W.; de Souza, L. F., A high-order immersed interface method free of derivative jump conditions for Poisson equations on irregular domains, J. Comput. Phys., 423, Article 109791 pp., 2020 · Zbl 07508410 |
| [22] | Barthès-Biesel, D.; Rallison, J. M., The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech., 113, 251, 1981 · Zbl 0493.76129 |
| [23] | Ramanujan, S.; Pozrikidis, C., Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities, J. Fluid Mech., 361, 117-144, 1998 · Zbl 0921.76058 |
| [24] | Floryan, J. M.; Rasmussen, H., Numerical methods for viscous flows with moving boundaries, Appl. Mech. Rev., 42, 323-341, 1989 |
| [25] | Hayashi, M.; Hatanaka, K.; Kawahara, M., Lagrangian finite element method for free surface Navier-Stokes flow using fractional step methods, Int. J. Numer. Methods Fluids, 13, 805-840, 2010 · Zbl 0741.76037 |
| [26] | Tan, Z.; Le, D. V.; Lim, K. M.; Khoo, B. C., An immersed interface method for the incompressible Navier-Stokes equations with discontinuous viscosity across the interface, SIAM J. Sci. Comput., 31, 1798-1819, 2009 · Zbl 1317.76065 |
| [27] | Okamoto, T.; Kawahara, M., Two-dimensional sloshing analysis by Lagrangian finite element method, Int. J. Numer. Methods Fluids, 11, 453-477, 2010 · Zbl 0711.76008 |
| [28] | Kolahdouz, E. M.; Bhalla, A. P.S.; Craven, B. A.; Griffith, B. E., An immersed interface method for discrete surfaces, J. Comput. Phys., 400, 2020 · Zbl 1453.76075 |
| [29] | Xu, S.; Pearson, G. D., Computing jump conditions for the immersed interface method using triangular meshes, J. Comput. Phys., 302, 59-67, 2015 · Zbl 1349.76656 |
| [30] | Xu, S., A boundary condition capturing immersed interface method for 3D rigid objects in a flow, J. Comput. Phys., 230, 7176-7190, 2011 · Zbl 1408.76172 |
| [31] | Xu, S.; Wang, Z. J., A 3D immersed interface method for fluid-solid interaction, Comput. Methods Appl. Mech. Eng., 197, 2068-2086, 2008 · Zbl 1158.74540 |
| [32] | Hui, Z.; Yin, C.; Zhang, W.; Liu, Y.; Xiong, B., 3D MCSEM modeling based on domain-decomposition finite-element method, J. Appl. Geophys., 207, Article 104847 pp., 2022 |
| [33] | Peng, N.; Wang, C.; An, J., An efficient finite-element method and error analysis for the fourth-order elliptic equation in a circular domain, Int. J. Comput. Math., 99, 1785-1802, 2022 · Zbl 1513.65469 |
| [34] | Xu, W.; Yang, H.; Yang, Y.; Wang, Y.; Zhou, K., Stress-aware large-scale mesh editing using a domain-decomposed multigrid solver, Comput. Aided Geom. Des., 63, 17-30, 2018 · Zbl 1441.65027 |
| [35] | Xie, Y.; Wu, J.; Xue, Y.; Xie, C.; Ji, H., A domain decomposed finite element method for solving Darcian velocity in heterogeneous porous media, J. Hydrol., 554, 32-49, 2017 |
| [36] | Jia, P.-H.; Liu, Q. H.; Chen, Y.; Huang, W.-F.; Li, X.; Nie, Z.; Hu, J., Hybrid FEM-DDM and BEM-BoR for the analysis of multiscale composite structures, IEEE Trans. Antennas Propag., 68, 4753-4763, 2020 |
| [37] | Wang, W.; Tan, Z., A simple 3D immersed interface method for Stokes flow with singular forces on staggered grids, Commun. Comput. Phys., 30, 227-254, 2021 · Zbl 1473.65123 |
| [38] | Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869, 1986 · Zbl 0599.65018 |
| [39] | Wang, W.; Tan, Z., A simple augmented IIM for 3D incompressible two-phase Stokes flows with interfaces and singular forces, Comput. Phys. Commun., 270, Article 108154 pp., 2022 · Zbl 1516.76058 |
| [40] | Adams, J. C.; Swarztrauber, P. N.; Sweet, R., FISHPACK: efficient FORTRAN subprograms for the solution of separable elliptic partial differential equations, (Astrophysics Source Code Library, 2016) |
| [41] | Li, Z.; Ito, K.; Lai, M.-C., An augmented approach for Stokes equations with a discontinuous viscosity and singular forces, Comput. Fluids, 36, 622-635, 2007 · Zbl 1177.76291 |
| [42] | Pozrikidis, C., A Practical Guide to Boundary Element Methods with the Software Library BEMLIB, 2002 · Zbl 1019.65097 |
| [43] | Xu, J.-J.; Zhao, H.-K., An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput., 19, 573-594, 2003 · Zbl 1081.76579 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
