×

zbMATH — the first resource for mathematics

Domain decomposition methods for domain composition purpose: chimera, overset, gluing and sliding mesh methods. (English) Zbl 1437.65220
Summary: Domain composition methods (DCM) consist in obtaining a solution to a problem, from the formulations of the same problem expressed on various subdomains. These methods have therefore the opposite objective of domain decomposition methods (DDM). Indeed, in contrast to DCM, these last techniques are usually applied to matching meshes as their purpose consists mainly in distributing the work in parallel environments. However, they are sometimes based on the same methodology as after decomposing, DDM have to recompose. As a consequence, in the literature, the term DDM has many times substituted DCM. DCM are powerful techniques that can be used for different purposes: to simplify the meshing of a complex geometry by decomposing it into different meshable pieces; to perform local refinement to adapt to local mesh requirements; to treat subdomains in relative motion (Chimera, sliding mesh); to solve multiphysics or multiscale problems, etc. The term DCM is generic and does not give any clue about how the fragmented solutions on the different subdomains are composed into a global one. In the literature, many methodologies have been proposed: they are mesh-based, equation-based, or algebraic-based. In mesh-based formulations, the coupling is achieved at the mesh level, before the governing equations are assembled into an algebraic system (mesh conforming, Shear-Slip Mesh Update, HERMESH). The equation-based counterpart recomposes the solution from the strong or weak formulation itself, and are implemented during the assembly of the algebraic system on the subdomain meshes. The different coupling techniques can be formulated for the strong formulation at the continuous level, for the weak formulation either at the continuous or at the discrete level (iteration-by-subdomains, mortar element, mesh free interpolation). Although the different methods usually lead to the same solutions at the continuous level, which usually coincide with the solution of the problem on the original domain, they have very different behaviors at the discrete level and can be implemented in many different ways. Eventually, algebraic-based formulations treat the composition of the solutions directly on the matrix and right-hand side of the individual subdomain algebraic systems. The present work introduces mesh-based, equation-based and algebraic-based DCM. It however focusses on algebraic-based domain composition methods, which have many advantages with respect to the others: they are relatively problem independent; their implicit implementation can be hidden in the iterative solver operations, which enables one to avoid intensive code rewriting; they can be implemented in a multi-code environment.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
Software:
Alya; CUBIT; METIS; TetGen
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alonso, A; Trotta, R; Valli, A, Coercive domain decomposition algorithms for advection-diffusion equations and systems, J Comput Appl Math, 96, 51-76, (1998) · Zbl 0935.65137
[2] Auge, A; Kapurkin, A; Lube, G; Otto, FC; Espedal, M (ed.); Keyes, D (ed.); Bjørstad, PE (ed.), A note on domain decomposition of singularly perturbed elliptic problems, 163-170, (1998), Bergen
[3] Avila, M; Folch, A; Houzeaux, G; Eguzkitza, B; Prieto, L; Cabezón, D, A parallel CFD model for wind farms, Proc Comput Sci, 18, 2157-2166, (2013)
[4] Beckert, A, Coupling fluid (CFD) and structural (FE) models using finite interpolation elements, Aerosp Sci Technol, 4, 13-22, (2000) · Zbl 0999.74106
[5] Beckert, A; Wendland, H, Multivariate interpolation for fluid-structure-interaction problems using radial basis functions, Aerosp Sci Technol, 5, 125-134, (2001) · Zbl 1034.74018
[6] Behr, M; Tezduyar, T, The shear-slip mesh update method, Comput Methods Appl Mech Eng, 174, 261-274, (1999) · Zbl 0959.76037
[7] Belytschko, T; Krongauz, Y; Organ, D; Fleming, M; Krysl, P, Meshless methods: an overview and recent developments, Comput Methods Appl Mech Eng, 139, 3-47, (1996) · Zbl 0891.73075
[8] Belytschko, T; Organ, D; Krongauz, Y, A coupled finite element-element-free Galerkin method, Comput Mech, 17, 186-195, (1995) · Zbl 0840.73058
[9] Ben Belgacem, F; Maday, Y, The mortar element method for three-dimensional finite elements, RAIRO Modél Math Anal Numér, 31, 289-302, (1997) · Zbl 0868.65082
[10] Bernardi, C; Maday, Y; Patera, A; Kaper, H (ed.); Garbey, M (ed.), Domain decomposition by the mortar element method, No. 384, 269-286, (1993), Dordrecht · Zbl 0799.65124
[11] Bernardi, C; Maday, Y; Rapetti, F, Basics and some applications of the mortar element method, GAMM-Mitt, 28, 97-123, (2005) · Zbl 1177.65178
[12] Biotto C, Peiró J (2010) A zonal Euler/viscous solver for compressible flows. In: Pereira J, Sequeira A (eds) Proceedings of the V European conference on computational fluid dynamics, ECCOMAS CFD 2010
[13] Bjørstad, PE; Widlund, OB, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J Numer Anal, 23, 1097-1120, (1986) · Zbl 0615.65113
[14] Blacker T, Bohnhoff W, Edwards T (1994) Cubit mesh generation environment. Volume 1: users manual. Tech. rep., Sandia National Labs., Albuquerque, NM (United States)
[15] Boer, A; Zuijlen, A; Bijl, H, Review of coupling methods for non-matching meshes, Comput Methods Appl Mech Eng, 196, 1515-1525, (2007) · Zbl 1173.74485
[16] Bourgat, JF; Glowinski, R; Tallec, P; Vidrascu, M; Chan, T (ed.); Glowinski, R (ed.); Périaux, J (ed.); Widlund, O (ed.), Variational formulation and algorithm for trace operator in domain decomposition calculations, 3-16, (1989), Philadelphia
[17] Brezzi, F; Canuto, C; Russo, A, A self-adaptive formulation for the Euler/Navier-Stokes coupling, Comput Methods Appl Mech Eng, 73, 317-330, (1989) · Zbl 0688.76024
[18] Brezzi F, Marini L (1994) A three-field domain decomposition method. In: Domain decomposition methods in science and engineering, the sixth international conference on domain decomposition, contemporary mathematics. American Mathematical Society, Como (I), vol 157, pp 27-34 · Zbl 0801.65116
[19] Carlenzoli C, Quarteroni A (1995) Adaptive domain decomposition methods for advection-diffusion problems. In: Babuška I (ed) Modeling, mesh generation, and adaptive numerical methods for partial differential equations, IMA volumes in mathematics and its applications, vol 75. Springer, Berlin, pp 165-186 · Zbl 0831.65129
[20] Casoni, E; Jérusalem, A; Samaniego, C; Eguzkitza, B; Lafortune, P; Tjahjanto, D; Sáez, X; Houzeaux, G; Vźquez, M, Alya: computational solid mechanics for supercomputers, Arch Comput Methods Eng, 22, 557-576, (2014) · Zbl 1348.74007
[21] Cebral, J; Löhner, R; Choyke, P; Yim, P, Merging of intersecting triangulations for finite element modeling, J Biomech, 34, 815-819, (2001)
[22] Cebral, JR; Löhner, R, Conservative load projection and tracking for fluid-structure problems, AIAA J, 35, 687-692, (1997) · Zbl 0895.73077
[23] Ciccoli, MC, Adaptive domain decomposition algorithms and finite volume/finite element approximation for advection-diffusion equations, J Sci Comput, 11, 229-341, (1996) · Zbl 0890.76057
[24] Ciccoli, MC; Trotta, R; Bjørstad, PE (ed.); Espedal, M (ed.); Keyes, D (ed.), Multidomain finite elements and finite volumes for advection-diffusion equations, 540-547, (1998), Bergen
[25] Clark B, Hanks B, Ernst C (2008) Conformal assembly meshing with tolerant imprinting. In: Proceedings of the 17th international meshing roundtable. Springer, Berlin, pp 267-280
[26] Cleijne, J, Results of sexbierum wind farm, Report, 1, 92-388, (1992)
[27] Codina R, Houzeaux G (2006) Verification and validation methods for challenging multiphysics problems. In: Implementation aspects of coupled problems in CFD involving time dependent domains. Theory and Engineering Applications of Computational Methods. CIMNE, pp 99-123
[28] Deutsch CV (2002) Geostatistical reservoir modeling. Oxford University Press, Oxford
[29] Discacciati, M; Gervasio, P; Giacomini, A; Quarteroni, A, The interface control domain decomposition method for Stokes-Darcy coupling, SIAM J Numer Anal, 54, 1039-1068, (2016) · Zbl 1337.49056
[30] Discacciati M, Gervasio P, Quarteroni A (2012) Heterogeneous mathematical models in fluid dynamics and associated solution algorithms. In: Multiscale and adaptivity: modeling, numerics and applications, Lecture Notes in Math., vol 2040. Springer, Heidelberg, pp 57-123 · Zbl 1426.76527
[31] Discacciati, M; Gervasio, P; Quarteroni, A, The interface control domain decomposition (ICDD) method for elliptic problems, SIAM J Control Optim, 51, 3434-3458, (2013) · Zbl 1284.65180
[32] Discacciati, M; Gervasio, P; Quarteroni, A, Interface control domain decomposition methods for heterogeneous problems, Int J Numer Methods Fluids, 76, 471-496, (2014)
[33] Eguzkitza B (2014) HERMESH: a geometrical domain composition method in computational mechanics. Ph.D. Thesis, Universitat Politècnica de Catalunya, Barcelona (Spain) · Zbl 1158.76405
[34] Eguzkitza, B; Houzeaux, G; Aubry, R; Vázquez, M, A parallel coupling strategy for the Chimera and domain decomposition methods in computational mechanics, Comput Fluids, 80, 128-141, (2013) · Zbl 1284.65166
[35] Eguzkitza, B; Houzeaux, G; Calmet, H; Vázquez, M; Soni, B; Aliabadi, S; Bates, A; Doorly, D, A gluing method for non-matching meshes, Comput Fluids, 110, 159-168, (2013) · Zbl 1390.65041
[36] Farhat, C; Roux, F, A method of finite element tearing and interconnecting and its parallel solution algorithm, Int J Numer Methods Eng, 32, 1205-1227, (1991) · Zbl 0758.65075
[37] Funaro, D; Quarteroni, A; Zanolli, P, An iterative procedure with interface relaxation for domain decomposition methods, SIAM J Numer Anal, 25, 1213-1236, (1988) · Zbl 0678.65082
[38] Gander, M; Halpern, L; Japhet, C; Martin, V; Widlund, O (ed.); Keyes, D (ed.), Advection diffusion problems with pure advection approximation in subregions, 239-246, (2007), Berlin
[39] Gander, M; Wanner, G; Erhel, J (ed.); Gander, M (ed.); Halpern, L (ed.); Pichot, G (ed.); Sassi, T (ed.); Widlund, O (ed.), The origins of the alternating Schwarz method, 415-422, (2013), Berlin
[40] Gastaldi, F; Gastaldi, L, On a domain decomposition for the transport equation: theory and finite element approximation, IMA J Numer Anal, 14, 111-135, (1993) · Zbl 0792.65071
[41] Gastaldi, F; Gastaldi, L; Quarteroni, A; Bjørstad, PE (ed.); Espedal, M (ed.); Keyes, D (ed.), ADN and ARN domain decomposition methods for advection-diffusion equations, 334-341, (1998), Bergen
[42] Gastaldi, F; Quarteroni, A, On the coupling of hyperbolic and parabolic systems: analytical and numerical approach, Appl Numer Math, 6, 3-31, (1989) · Zbl 0686.65084
[43] Gervasio, P; Lions, JL; Quarteroni, A, Heterogeneous coupling by virtual control methods, Numer Math, 90, 241-264, (2001) · Zbl 1002.65133
[44] Glowinski, R; Dinh, Q; Périaux, J, Domain decomposition methods for nonlinear problems in fluid dynamics, Comput Methods Appl Mech Eng, 40, 27-109, (1983) · Zbl 0505.76068
[45] Glowinski, R; Tallec, P; Chan, TF (ed.); Glowinski, R (ed.); Périaux, J (ed.); Widlund, OB (ed.), Augmented Lagrangian interpretation of the nonoverlapping Schwarz alternating method, 224-231, (1990), Philadelphia
[46] Hoppe, R; Iliash, Y; Kuznetsov, Y; Vassilevski, Y; Wohlmuth, B, Analysis and parallel implementation of adaptive mortar element methods, East-West J Numer Math, 6, 223-248, (1998) · Zbl 0913.65098
[47] Houzeaux G (2002) A geometrical domain decomposition method in computational fluid dynamics. Ph.D. Thesis, Universitat Politècnica de Catalunya, Barcelona (Spain) · Zbl 1118.76002
[48] Houzeaux, G; Cajas, J; Eguzkitza, B; Vázquez, M; Iványi, P (ed.); Topping, BHV (ed.), Techniques for parallel, distributed and cloud computing in engineering, computational science, engineering and technology series, No. 36, 61-92, (2015), Stirling
[49] Houzeaux, G; Codina, R, Transmission conditions with constraints in finite element domain decomposition method for flow problems, Commun Numer Methods Eng, 17, 179-190, (2001) · Zbl 1033.76024
[50] Houzeaux, G; Codina, R, A Chimera method based on a Dirichlet/Neumann(Robin) coupling for the Navier-Stokes equations, Comput Methods Appl Mech Eng, 192, 3343-3377, (2003) · Zbl 1054.76049
[51] Houzeaux, G; Codina, R, An iteration-by-subdomain overlapping Dirichlet/Robin domain decomposition method for advection-diffusion problems, J Comput Appl Math, 158, 243-276, (2003) · Zbl 1033.65111
[52] Houzeaux, G; Eguzkitza, B; Aubry, R; Owen, H; Vázquez, M, A Chimera method for the Navier-Stokes equations, Int J Numer Methods Fluids, 75, 155-183, (2014) · Zbl 1284.65166
[53] Huerta, A; Fernández-Méndez, S, Enrichment and coupling of the finite element and meshless methods, Int J Numer Methods Eng, 50, 507-524, (2000) · Zbl 0976.74067
[54] Jaiman, R; Jiao, X; Geubelle, P; Loth, E, Conservative load transfer along curved fluid-solid interface with non-matching meshes, J Comput Phys, 218, 372-397, (2006) · Zbl 1158.76405
[55] Kim, H, Interface element method (IEM) for a partitioned system with non-matching interfaces, Comput Methods Appl Mech Eng, 191, 3165-3194, (2002) · Zbl 1101.74365
[56] Tallec, P; Tidriri, M, Convergence analysis of domain decomposition algorithms with full overlapping for the advection diffusion problems, Math Comput, 68, 585-606, (1999) · Zbl 1043.65112
[57] Li, S; Liu, W, Meshfree and particle methods and their applications, Appl Mech Rev, 55, 1-34, (2002)
[58] Lions, JL; Pironneau, O, Algorithmes parallèles pour la solution de problèmes aux limites. C. R. acad, Sci Paris Série I, 947, 327-352, (1998)
[59] Lions, PL; Glowinski, R (ed.); Golub, GH (ed.); Meurant, GA (ed.); Périaux, J (ed.), On the Schwarz alternating method I, 1-42, (1988), Philadelphia
[60] Lions, PL; Chan, T (ed.); Glowinski, R (ed.); Périaux, J (ed.); Widlund, O (ed.), On the Schwarz alternating method II, 47-70, (1989), Philadelphia
[61] Lions, PL; Chan, TF (ed.); Glowinski, R (ed.); Périaux, J (ed.); Widlund, OB (ed.), On the Schwarz alternating method III: a variant for nonoverlapping subdomains, 202-223, (1990), Philadelphia
[62] Liou M, Kao K (1994) Progress in grid generation: from Chimera to DRAGON grids. Tech. Rep. NASA Technical Memorandum 106709, ICOMP-94-19, NASA
[63] Liu, W; Uras, R; Chen, Y, Enrichment of the finite element method with the reproducing kernel particle method, J Appl Mech, 64, 861-870, (1997) · Zbl 0920.73366
[64] Lo, S; Wang, W, A fast robust algorithm for the intersection of triangulated surfaces, Eng Comput, 20, 11-21, (2004)
[65] Maday, Y; Magoulès, F, Absorbing interface conditions for domain decomposition methods: a general presentation, Comput Methods Appl Mech Eng, 195, 3880-3900, (2006) · Zbl 1168.65423
[66] Magoulès F, Roux FX, Houzeaux G (2015) Parallel scientific computing, Computer Engineering Series. Wiley-ISTE, London
[67] Marini, L; Quarteroni, A; Glowinski, R (ed.); Golub, GH (ed.); Meurant, GA (ed.); Périaux, J (ed.), An iterative procedure for domain decomposition methods: a finite element approach, 129-143, (1988), Philadelphia
[68] METIS, family of multilevel partitioning algorithms. http://glaros.dtc.umn.edu/gkhome/views/metis
[69] Nataf F (1993) On the use of open boundary conditions in block Gauss-Seidel methods for the convection-diffusion equation. Tech. Rep. RI284, Centre de Mathématiques Appliquées, Ecole Polytechnique · Zbl 0688.76024
[70] Nataf, F; Nier, F, Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains, Numer Math, 75, 357-377, (1997) · Zbl 0873.65108
[71] Owen, SJ; Saigal, S, Formation of pyramid elements for hexahedra to tetrahedra transitions, Comput Methods Appl Mech Eng, 190, 4505-4518, (2001)
[72] Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. In: Numerical mathematics and scientific computation. The Clarendon Press/Oxford University Press, New York · Zbl 0931.65118
[73] Rabczuk, T; Xiao, S; Sauer, M, Coupling of mesh-free methods with finite elements: basic concepts and test results, Commun Numer Methods Eng, 22, 1031-1065, (2006) · Zbl 1109.65082
[74] Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia · Zbl 1031.65046
[75] Code_Saturne. http://code-saturne.org/cms · Zbl 0799.65124
[76] Si, H, Tetgen, a Delaunay-based quality tetrahedral mesh generator, ACM Trans Math Softw, 41, 111-1136, (2015) · Zbl 1369.65157
[77] Smith BF, Bjørstad PE, Gropp WD (1996) Domain decomposition. Cambridge University Press, Cambridge · Zbl 0857.65126
[78] Staten, ML; Shepherd, JF; Ledoux, F; Shimada, K, Hexahedral mesh matching: converting non-conforming hexahedral-to-hexahedral interfaces into conforming interfaces, Int J Numer Methods Eng, 82, 1475-1509, (2010) · Zbl 1188.74095
[79] Tian, R; Yagawa, G, Non-matching mesh gluing by meshless interpolationan alternative to Lagrange multipliers, Int J Numer Methods Eng, 71, 473-503, (2007) · Zbl 1194.74478
[80] Toselli A, Widlund O (2004) Domain decomposition methods: algorithms and theory, Springer Series in Computational Mathematics. Springer, Berlin · Zbl 1069.65138
[81] Trotta, R, Multidomain finite elements for advection-diffusion equations, Appl Numer Math, 21, 91-118, (1996) · Zbl 0853.65119
[82] Vázquez, M; Houzeaux, G; Koric, S; Artigues, A; Aguado-Sierra, J; Arís, R; Mira, D; Calmet, H; Cucchietti, F; Owen, H; Taha, A; Burness, ED; Cela, JM; Valero, M, Alya: multiphysics engineering simulation towards exascale, J Comput Sci, 14, 15-27, (2016)
[83] White, D; Saigal, S; Owen, S, An overset-grid method for 3D unsteady incompressible flows, Int J Numer Methods Eng, 59, 1839-1860, (2004) · Zbl 1060.74682
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.