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research-article

A Cartesian mesh approach to embedded interface problems using the virtual element method

Authors:

Sundararajan NatarajanAuthors Info & Claims

Volume 508, Issue C

https://doi.org/10.1016/j.jcp.2024.112997

Published: 01 July 2024 Publication History

Abstract

In this paper, we propose an elegant methodology to treat sharp interfaces that are implicitly defined which does not require (a) enrichment functions, (b) additional linear and bilinear terms such as the inter-element penalty terms as in Nitsche's method, or use of multipliers like Lagrange multiplier, in the weak form for enforcing the jump conditions across the interface, and (c) modification to the standard virtual element method solution space. The background mesh consists of structured quadrilateral elements with each element consisting of eight nodes, namely, the four vertices and the mid-points of four edges. A simple and efficient idea to generate an interface-fitted mesh is discussed where the number of nodes remains invariant, esp., for moving boundary problems. A linear virtual element method approximation is assumed on the fitted mesh. The efficiency and accuracy of the presented technique is demonstrated by solving and verifying the rate of convergence in both L 2 norm and H 1 semi-norm, for the benchmark problems with interfaces of various geometries and moving interfaces.

Highlights

•

Proposes an elegant way to treat embedded interface problems.

•

Does not need massive local remeshing.

•

Dirichlet boundary conditions can be imposed directly on the interface.

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Number of nodes remain constant even as the interface evolves.

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Less sensitive to mesh distortion.

References

[1]

Arvind Narayanaswamy, Ning Gu, Heat transfer from freely suspended bimaterial microcantilevers, J. Heat Transf. 133 (4) (2011).

[2]

Abdullah Alodhayb, Modeling of an optically heated mems-based micromechanical bimaterial sensor for heat capacitance measurements of single biological cells, Sensors 20 (1) (2020).

[3]

Mark Sussman, Peter Smereka, Stanley Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114 (1) (1994) 146–159.

Digital Library

[4]

R.W. Lewis, P.J. Roberts, B.A. Schrefler, Finite element modelling of two-phase heat and fluid flow in deforming porous media, Transp. Porous Media 4 (1989) 319–334.

[5]

Xiaobing Feng, Steven Wise, Analysis of a Darcy–Cahn–Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM J. Numer. Anal. 50 (3) (2012) 1320–1343.

[6]

Rafael Davalos, Yong Huang, Boris Rubinsky, Electroporation: bio-electrochemical mass transfer at the nano scale, Nanoscale Microscale Thermophys. Eng. 4 (3) (2000) 147–159.

[7]

J. Dutta, B. Dekha, N. Kumar, Finite element methods for the electric interface model: convergence analysis, Math. Methods Appl. Sci. 43 (2020) 4598–4613.

[8]

Martin K. Bernauer, Roland Herzog, Implementation of an X-FEM solver for the classical two-phase Stefan problem, J. Sci. Comput. 52 (2012) 271–293.

[9]

Chandrasekhar Annavarapu, Martin Hautefeuille, John E. Dolbow, A robust Nitsche's formulation for interface problems, Comput. Methods Appl. Mech. Eng. 225 (2012) 44–54.

[10]

Lucy Zhang, Axel Gerstenberger, Xiaodong Wang, Wing Kam Liu, Immersed finite element method, Comput. Methods Appl. Mech. Eng. 193 (21) (2004) 2051–2067.

[11]

Wing Kam Liu, Yaling Liu, David Farrell, Lucy Zhang, X. Sheldon Wang, Yoshio Fukui, Neelesh Patankar, Yongjie Zhang, Chandrajit Bajaj, Junghoon Lee, Juhee Hong, Xinyu Chen, Huayi Hsu, Immersed finite element method and its applications to biological systems, Comput. Methods Appl. Mech. Eng. 195 (13) (2006) 1722–1749.

[12]

R. Merle, J. Dolbow, Solving thermal and phase change problems with the eXtended finite element method, Comput. Mech. 28 (2002) 339–350.

[13]

Sining Yu, Yongcheng Zhou, G.W. Wei, Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces, J. Comput. Phys. 224 (2007) 729–756.

[14]

M.A. Dumett, J.P. Keener, An immersed interface method for solving anisotropic elliptic boundary value problems in three dimensions, SIAM J. Sci. Comput. 25 (2003) 348–367.

Digital Library

[15]

A.L. Fogelson, J.P. Keener, Immersed interface methods for Neumann and related problems in two and three dimensions, SIAM J. Sci. Comput. 22 (2000) 1630–1654.

Digital Library

[16]

Richard E. Ewing, Zhilin Li, Tao Lin, Yanping Lin, The immersed finite volume element methods for the elliptic interface problems, Math. Comput. Simul. 50 (1) (1999) 63–76.

[17]

Tianbing Chen, John Strain, Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems, J. Comput. Phys. 227 (2008) 7503–7542.

[18]

A. Main, G. Scovazzi, The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems, J. Comput. Phys. 372 (2018) 972–995.

Digital Library

[19]

Shuhao Cao, Long Chen, Ruchi Guo, Frank Lin, Immersed virtual element methods for elliptic interface problems in two dimensions, J. Sci. Comput. 93 (12) (2022).

[20]

Shuhao Cao, Long Chen, Ruchi Guo, Immersed virtual element methods for electromagnetic interface problems in three dimensions, Math. Models Methods Appl. Sci. 33 (3) (2023) 455–503.

[21]

J. Bramble, J. King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math. 6 (1996) 109–138.

[22]

I. Babuska, The finite element method for elliptic equations with discontinuous coefficients, Computing 5 (1970) 207–213.

[23]

Zhiqiang Cai, Xiu Ye, Shun Zhang, Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations, SIAM J. Numer. Anal. 49 (5) (2011) 1761–1787.

Digital Library

[24]

Andrea Cangiani, E.H. Georgoulis, Younis A. Sabawi, Adaptive discontinuous Galerkin methods for elliptic interface problems, Math. Comput. 87 (314) (2018) 2675–2707.

[25]

Lin Mu, Junping Wang, Xiu Ye, Shan Zhao, A new weak Galerkin finite element method for elliptic interface problems, J. Comput. Phys. 325 (2016) 157–173.

Digital Library

[26]

Bhupen Deka, Papri Roy, Weak Galerkin finite element methods for parabolic interface problems with nonhomogeneous jump conditions, Numer. Funct. Anal. Optim. 40 (3) (2019) 259–279.

[27]

Johnny Guzmán, Manuel A. Sánchez, Marcus Sarkis, Higher-order finite element methods for elliptic problems with interfaces, ESAIM: Math. Model. Numer. Anal. 50 (5) (2016) 1561–1583.

[28]

L. Beirao Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214.

[29]

L. Beirao Da Veiga, F. Brezzi, L.D. Marini, A. Russo, The hitchikkers guide to the virtual element method, Math. Models Methods Appl. Sci. 24 (2014) 1541–1574.

[30]

L. Beirao Da Veiga, F. Brezzi, L.D. Marini, A. Russo, Virtual element method for general second-order elliptic problems, Math. Models Methods Appl. Sci. 26 (4) (2016) 729–750.

[31]

Long Chen, Huayi Wei, Min Wen, An interface-fitted mesh generator and virtual element methods for elliptic interface problems, J. Comput. Phys. 334 (2017) 327–348.

[32]

Jai Tushar, Anil Kumar, Sarvesh Kumar, Virtual element methods for general linear elliptic interface problems on polygonal meshes with small edges, Comput. Math. Appl. 122 (2022) 61–75.

[33]

Shuhao Cao, Long Chen, Ruchi Guo, A virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh, Math. Models Methods Appl. Sci. 31 (14) (2021) 2907–2936.

[34]

E. Artioli, S. Marfia, E. Sacco, VEM-based tracking algorithm for cohesive/frictional 2D fracture, Comput. Methods Appl. Mech. Eng. 365 (2020).

[35]

Sonia Marfia, Elisabetta Monaldo, Elio Sacco, Cohesive fracture evolution within virtual element method, Eng. Fract. Mech. 269 (2022).

[36]

H. Choi, H. Chi, K. Park, Virtual element method for mixed-mode cohesive fracture simulation with element split and domain integral, Int. J. Fract. 240 (1) (2023) 51–70.

[37]

A. Hussein, F. Aldakheel, B. Hudobivnik, P. Wriggers, P.-A. Guidault, O. Allix, A computational framework for brittle crack-propagation based on efficient virtual element method, Finite Elem. Anal. Des. 159 (2019) 15–32.

Digital Library

[38]

F. Aldakheel, B. Hudobivnik, A. Hussein, P. Wriggers, Phase-field modeling of brittle fracture using an efficient virtual element scheme, Comput. Methods Appl. Mech. Eng. 341 (2018) 443–466.

[39]

F. Aldakheel, B. Hudobivnik, P. Wriggers, Virtual element formulation for phase-field modeling of ductile fracture, Int. J. Multiscale Comput. Eng. 17 (2) (2019) 181–200.

[40]

V.M. Nguyen-Thanh, X. Zhuang, H. Nguyen-Xuan, T. Rabczuk, P. Wriggers, A virtual element method for 2D linear elastic fracture analysis, Comput. Methods Appl. Mech. Eng. 340 (2018) 366–395.

[41]

B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini, A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391.

Digital Library

[42]

Andrea Cangiani, Gianmarco Manzini, Oliver J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems, IMA J. Numer. Anal. 37 (3) (2017) 1317–1354.

[43]

Stanley Osher, James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1) (1988) 12–49.

Digital Library

[44]

L. Beirao da Veiga, F. Brezzi, L.D. Marini, A. Russo, Serendipity nodal vem spaces, Comput. Fluids 141 (2016) 2–12.

[45]

L. Beirao Da Veiga, A. Russo, G. Vacca, The virtual element method with curved edges, ESAIM: M2AN 53 (2) (2019) 375–404.

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Information & Contributors

Information

Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 508, Issue C

Jul 2024

1037 pages

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Elsevier Inc.

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 01 July 2024

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