Nürnberg, Robert; Sacconi, Andrea An unfitted finite element method for the numerical approximation of void electromigration. (English) Zbl 1331.78027 J. Comput. Appl. Math. 270, 531-544 (2014). The electric contact between neighboring devices in integrated circuits is today mostly established by lines of aluminum that can contain small voids which can cause the loss of connections or failure of the circuit, especially because the width of metal interconnects is more and more reduced. Thus, the investigation of the mechanical failures in the lines induced by the motion of the voids is of great interest for the semiconductor industry.The interconnect line is assumed in this paper to be a rectangular solid in a two-dimensional model. The void can move through the conducting line as well as it can change its shape. A free boundary value problem is to solve. The authors consider two different phenomena that cause the motion of the voids: the surface tension and the electric field, summarized as electromigration. They introduce a new front-tracking finite element method for the computation of the electromigration. In contrast to earlier methods, a variational formulation of the bulk-interface coupled system is used and two independent meshes are applied for the finite element approximation of the bulk quantities and the parametric approximation of the moving boundary of the voids with the advantage that a remeshing at each time step is avoided, i.e., the method is unfitted. The moving interface is governed by a fourth-order geometric evolution equation. The electric potential in the bulk satisfies the Laplace equation with Dirichlet boundary conditions.The authors start with a review of the different numerical algorithms described in the literature for this problem. Then the finite element approximation of the moving boundary and of the bulk region is described. Initially, the interface is assumed to be a closed curve. Later, this case is expanded to multi-component interfaces. The existence and uniqueness of the solutions are proved. Additionally, a semidiscrete continuous-in-time approximation is considered.The results are validated by a number of numerical examples. The finite element approximation is implemented using the C\(^{++}\)-software DUNE. The linear systems of equations are solved using the sparse factorization package UMFPACK and the BiCGSTAB solver including preconditioning because the corresponding matrix is rather ill-conditioned.The mesh generation is based on triangulations. The adaptive mesh strategy results in a mesh that is fine around the interface and more coarse away from it. Topological aspects, such as the merging of two curves and the influence of the mesh configuration, are treated with the help of the package El-Topo. Reviewer: Georg Hebermehl (Berlin) Cited in 1 Document MSC: 78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 35R35 Free boundary problems for PDEs Keywords:unfitted finite element method; void electromigration; free boundary value problem; bulk-interface coupled system; meshing; Laplace equation; fourth-order geometric evolution equation Software:El-Topo; UMFPACK; DUNE; ALBERTA PDFBibTeX XMLCite \textit{R. Nürnberg} and \textit{A. Sacconi}, J. Comput. Appl. Math. 270, 531--544 (2014; Zbl 1331.78027) Full Text: DOI References: [1] Averbuch, A.; Israeli, M.; Ravve, I.; Yavneh, I., Computation for electromigration in interconnects of microelectronic devices, J. Comput. Phys., 167, 316-371 (2001) · Zbl 1116.78341 [2] Bänsch, E.; Morin, P.; Nochetto, R. H., A finite element method for surface diffusion: the parametric case, J. Comput. Phys., 203, 321-343 (2005) · Zbl 1070.65093 [3] Bänsch, E.; Schmidt, A., Simulation of dendritic crystal growth with thermal convection, Interfaces Free Bound., 2, 95-115 (2000) · Zbl 0960.35111 [4] Barrett, R.; Berry, M.; Chan, T. F.; Demmel, J.; Donato, J. M.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; Van der Vorst, H., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (1994), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0814.65030 [5] Barrett, J. W.; Elliott, C. M., A finite-element method for solving elliptic equations with Neumann data on a curved boundary using unfitted meshes, IMA J. Numer. Anal., 4, 309-325 (1984) · Zbl 0574.65121 [6] Barrett, J. W.; Garcke, H.; Nürnberg, R., A parametric finite element method for fourth order geometric evolution equations, J. Comput. Phys., 222, 441-467 (2007) · Zbl 1112.65093 [7] Barrett, J. W.; Garcke, H.; Nürnberg, R., A phase field model for the electromigration of intergranular voids, Interfaces Free Bound., 9, 171-210 (2007) · Zbl 1132.35082 [8] Barrett, J. W.; Garcke, H.; Nürnberg, R., On stable parametric finite element methods for the Stefan problem and the Mullins-Sekerka problem with applications to dendritic growth, J. Comput. Phys., 229, 6270-6299 (2010) · Zbl 1201.80075 [9] Barrett, J. W.; Garcke, H.; Nürnberg, R., On the parametric finite element approximation of evolving hypersurfaces in \(R^3\), J. Comput. Phys., 227, 4281-4307 (2008) · Zbl 1145.65068 [10] Barrett, J. W.; Garcke, H.; Nürnberg, R., Finite element approximation of one-sided Stefan problems with anisotropic, approximately crystalline, Gibbs-Thomson law, Adv. Differential Equations, 18, 383-432 (2013) · Zbl 1271.80005 [11] Barrett, J. W.; Nürnberg, R.; Styles, V., Finite element approximation of a phase field model for void electromigration, SIAM J. Numer. Anal., 42, 738-772 (2004) · Zbl 1076.78012 [12] Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Kornhuber, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in dune, Computing, 82, 121-138 (2008) · Zbl 1151.65088 [13] Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework, Computing, 82, 103-119 (2008) · Zbl 1151.65089 [14] Bhate, D. N.; Kumar, A.; Bower, A. F., Diffuse interface model for electromigration and stress voiding, J. Appl. Phys., 87, 1712-1721 (2000) [15] Bower, A.; Freund, L., Finite element analysis of electromigration and stress induced diffusion in deformable solids, MRS Proc., 391, 177-188 (1995) [16] Brochu, T.; Bridson, R., Robust topological operations for dynamic explicit surfaces, SIAM J. Sci. Comput., 31, 2472-2493 (2009) · Zbl 1195.65017 [17] Chen, X.; Toh, K. C.; Phoon, K. K., A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations, Internat. J. Numer. Methods Engrg., 65, 785-807 (2006) · Zbl 1114.74056 [18] Cummings, L. J.; Richardson, G.; Ben Amar, M., Models of void electromigration, European J. Appl. Math., 12, 97-134 (2001) · Zbl 0991.78003 [19] Davis, T. A., Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30, 196-199 (2004) · Zbl 1072.65037 [20] Deckelnick, K.; Dziuk, G.; Elliott, C. M., Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14, 139-232 (2005) · Zbl 1113.65097 [21] Elliott, C. M.; Garcke, H., Existence results for diffusive surface motion laws, Adv. Math. Sci. Appl., 7, 467-490 (1997) · Zbl 0876.35050 [22] Ho, P. S., Motion of inclusion induced by a direct current and a temperature gradient, J. Appl. Phys., 41, 64-68 (1970) [23] Juric, D.; Tryggvason, G., A front-tracking method for dendritic solidification, J. Comput. Phys., 123, 127-148 (1996) · Zbl 0843.65093 [24] Kraft, O.; Arzt, E., Electromigration mechanisms in conductor lines: void shape changes and slit-like failure, Acta Mater., 45, 1599-1611 (1997) [25] Li, Z.; Zhao, H.; Gao, H., A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid, J. Comput. Phys., 152, 281-304 (1999) · Zbl 0956.78016 [26] Mahadevan, M.; Bradley, R. M., Phase field model of surface electromigration in single crystal metal thin films, Physica D, 126, 201-213 (1999) [27] Mahadevan, M.; Bradley, R. M., Simulations and theory of electromigration-induced slit formation in unpassivated single-crystal metal lines, Phys. Rev. B, 59, 11037-11046 (1999) [28] Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12-49 (1988) · Zbl 0659.65132 [29] Schmidt, A.; Siebert, K. G., (Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA. Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, vol. 42 (2005), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1068.65138 [30] Sethian, J. A., (Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Monographs on Applied and Computational Mathematics, vol. 3 (1999), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0973.76003 [31] Xia, L.; Bower, A.; Suo, Z.; Shih, C., A finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion, J. Mech. Phys. Solids, 45, 1473-1493 (1997) · Zbl 0974.74566 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.