In this contribution, dual and primal domain decomposition techniques
are studied for the multiscale analysis of failure in quasi-brittle materials. The multiscale strategy essentially consists in decomposing the structure into a number of nonoverlapping domains and considering a refined spatial resolution where needed. In multiscale analysis of damage, the spatial refinement is performed where damage nucleation and propagation take place. The domain decomposition approach turns to be a computationally cheaper alternative to the direct numerical solution in which a fine scale model
is considered throughout the complete sample. Dual and primal domain decomposition techniques are appropriate for such concurrent multiscale analyses and provide identical results. Parallel scalability of the multiscale analysis is studied using a moderate number of processors and a parallel direct solver for the system obtained through the assembly of all domains.Postprint (published version

Everdij, Frank P. X.

Lloberas Valls, Oriol

Rixen, Daniel J.

Simone, Angelo

Sluys, Lambertus J.

English

UPCommons. Portal del coneixement obert de la UPC

11th World Congress on Computational Mechanics (WCCM XI)5th European Conference on Computational Mechanics (ECCM V)6th European Conference on Computational Fluid Dynamics (ECFD VI)E. On˜ate, J. Oliver and A. Huerta (Eds)MULTISCALE ANALYSIS OF DAMAGE USING DUAL ANDPRIMAL DOMAIN DECOMPOSITION TECHNIQUESORIOL LLOBERAS-VALLS1, FRANK P. X. EVERDIJ2,DANIEL J. RIXEN3, ANGELO SIMONE2 AND LAMBERTUS J. SLUYS21 International Center for Numerical Methods in Engineering (CIMNE)Campus Nord UPC, Edifici C-1, C/Jordi Girona 1-3, 08034 Barcelona, Spainolloberas@cimne.upc.edu2 Faculty of Civil Engineering and GeosciencesDelft University of TechnologyP.O. Box 5048, 2600 GA Delft, The Netherlands3 Faculty of Mechanical EngineeringTechnische Universita¨t Mu¨nchenBoltzmannstr. 15, D - 85748 Garching, GermanyKey words: Multiscale analysis, dual and primal domain decomposition methods, strainlocalization, strain softeningAbstract. In this contribution, dual and primal domain decomposition techniquesare studied for the multiscale analysis of failure in quasi-brittle materials. The multi-scale strategy essentially consists in decomposing the structure into a number of non-overlapping domains and considering a refined spatial resolution where needed. In mul-tiscale analysis of damage, the spatial refinement is performed where damage nucleationand propagation take place. The domain decomposition approach turns to be a compu-tationally cheaper alternative to the direct numerical solution in which a fine scale modelis considered throughout the complete sample. Dual and primal domain decompositiontechniques are appropriate for such concurrent multiscale analyses and provide identicalresults. Parallel scalability of the multiscale analysis is studied using a moderate numberof processors and a parallel direct solver for the system obtained through the assembly ofall domains.1 INTRODUCTIONMultiscale analysis of materials has gained momentum in the last years thanks to thebreakthroughs in computer technology. Many new numerical multiscale techniques haverecently emerged and the topic is in continuous evolution. An attempt to classify different1O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluysmultiscale methods for the simulation of constitutive behaviour of complex materials hasbeen done in [3] where the scale separation and structure of the computation serve asleading criteria for the classification. Accordingly one can distinguish between hierarchicaland concurrent techniques depending on the scale separation, and decoupled, weaklycoupled and strongly coupled micro-macro computations.The use of domain decomposition methods [8] for multiscale analysis is a good exam-ple of concurrent strongly coupled techniques and has captured the attention of manyresearchers due to its flexibility, generality and adequacy for the solution of large struc-tural problems. Examples of such approaches can be found in the work of of Amini etal. [9] where a FETI-DP framework is employed as a basis for the multiscale analysis, orin Ladeve`ze et al. [10] and Guidault et al. [11] where a more complex interface betweendomains is employed.The current contribution revises the main features of the multiscale framework based ondual domain decomposition methods [1] introduced in [3, 4], and presents the equivalentprimal formulation assessed in multiscale analysis of non-linear softening materials.2 DUAL AND PRIMAL MULTISCALE DOMAIN DECOMPOSITIONThe main idea behind a multiscale domain decomposition strategy is to lower theburden of a full fine scale approach by decomposing a structure into a number of domainsand consider a higher resolution where needed. The kernel of such an approach resides inthe strategy to link the different domains and the type of compatibility between coarseand fine scale substructures.2.1 General formulationConsider a homogeneous body Ω decomposed into a number Ns of domains Ω(s) (Fig-ure 1). In a dual assembly, the interface compatibility is enforced through the signedBoolean matrices B(s) asNs∑s=1B(s)u(s) = 0. (1)Consequently, the global systemKu = f (2)arising from the discretization and linearization of the equilibrium equations is re-castedinto the augmented systemK(1) 0 0 B(1)T0. . . 0...0 0 K(Ns) B(Ns)TB(1) . . . B(Ns) 0u(1)...u(Ns)λ =f (1)...f (Ns)0 , (3)where K, u, f and λ denote the global stiffness matrix, displacement vector, force vectorand Lagrange multipliers used to tie the different domains (bottom part of Figure 1).2O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. SluysΓI¯tΩΓΓu ΓtΓu Γt¯tDual assembly Primal assemblyΓ(s)IΩ(s)λΓI ΓIubλ : Lagrange multipliers ub : Boundary DOFFigure 1: Body Ω decomposed into a number of non-overlapping domains.In a primal assembly, the matching boundary nodes at the interface ub are assembledat the same position in the global system (bottom part of Figure 1) and therefore theinterface degrees of freedom (DOF) ub are uniquely defined. The local DOF are identifiedform the global set u asu(s) = L(s)u, (4)where L(s) contains identity coefficients which relate the internal DOF to the correspond-ing global DOF and Boolean coefficients expressing the assembly of the subdomains onthe interface. The global system (2) can be therefore re-cast into(Ns∑s=1L(s)TK(s)L(s))u =Ns∑s=1L(s)Tf (s). (5)It is important to realize thatBL = 0, (6)beingB =[B(1) . . . B(Ns)](7)L =[L(1) . . . L(Ns)]T. (8)For this reason one can obtain the localization operator L by solvingL = ker(B), (9)where ker denotes the kernel or null space of the matrix B.3O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluysub : Boundary DOFλ ,µ : Lagrange multipliersPrimal assemblyΓIubλDual assemblyµNs∑s=1¯B(s)u(s) = 0 u(s) = ¯L(s)uΓI¯L = ker( ¯B)Figure 2: Dual and primal assembly at the non-conforming interface ΓI.2.2 Algebraic multiscale extensionsThe main ingredient in the multiscale extension consists in enforcing a certain compat-ibility condition at a non-conforming interface between different resolution domains (toppart of Figure 2). Different types of compatibility have been recently reported in [4]. Afull collocation condition of the solution field u at the interface ΓI between fine f andcoarse c adjacent domains readsuf = uc at ΓI, (10)and can be enforced using the shape functions of the coarse domain restricted at theinterface. These conditions can be written using the compatibility matrices C(s) to obtainNs∑s=1C(s)u(s) = 0. (11)The previous relations can be simply enforced in a dual approach by adding the compat-ibility equations to the Boolean matrices as[B¯(1) . . . B¯(Ns)]=[B(1) . . . B(Ns)C(1) . . . C(Ns)]. (12)Consequently, the extended field of Lagrange multipliers readsΛ =[λµ](13)In a primal approach, the localization operators L(s) need to be modified in order toincorporate the new constraints arising from the coarse-to-fine displacement compatibility.The primal system (5) can be recovered by considering the modified localization operatoru(s) = L¯(s)u. (14)4O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. SluysZoom-indomain (2)Zoom-indomain (1)Domain (1) Domain (2)λ IIIλ Iµ IIConfiguration IConfiguration IIConfiguration IIIλ IIΛII = Rλ ,IIλ IΛIII = Rλ ,IIIλ II +Rµ,IIIµ IIΛI = λ IFigure 3: Force resolution migration in a multiscale dual approach.Other ingredients for an adaptive multiscale analysis consist in a criterion to triggerthe zoom-in events, the solution of local boundary value problems (BVP) and a set ofglobal relaxation steps as described in detail in [3, 4]. It is important to note that in adual multiscale approach one needs to keep track of the interface tying forces to set up theappropriate force field at the boundary of a newly refined domain. This force migrationscheme is accomplished through the resize matrices R (Figure 3) and is explained in moredetail in [3]. Such a procedure is obviously not needed in a primal multiscale approachdue to the absence of tying forces in the formulation.3 EXAMPLES3.1 Dual and primal multiscale analysisDual and primal multiscale analyses are performed on the L-shaped homogeneous speci-men depicted in Figure 4. The specimen is decomposed in 12 regular domains and zoom-inis performed at those domains affected by damage growth. A gradient enhanced dam-age model [7] is employed in the computations and the adopted material parameters are5O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluys0 1ω50 mm100 mm50mm100mmuxFigure 4: L-shaped specimen (left) and final damage contours of the adaptive multiscale analysis (right).0.10020.0996176175PrimalDualDNSDisplacement [mm]Force[N]10.750.50.250250200150100500efDisplacement [mm]Error(ef)10.750.50.2504e-113e-112e-111e-110Figure 5: Force-displacement plot for the dual and primal analyses (left) and force field relative error(right).identical to the ones reported in [3].The displacement versus reaction at the upper edge is plotted in Figure 5 (left) for theanalyses employing dual and primal approaches. The relative difference between reactionforces obtained with the dual and primal approaches is negligible and it is thereforeconcluded that both multiscale analyses provide identical results.3.2 Parallel performanceIn this example, the dual assembly is employed to perform a multiscale analysis sim-ulating a wedge split test of a concrete specimen. The test setup is sketched in Figure 6and the material parameters are adopted from [5, 6]. Three different decompositions with34, 68 and 136 domains are considered (Figure 7) and the global assembled system issolved using a parallel solver designed for the solution of large sparse linear systems based6O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. SluysΓnInterface ΓITraction free interface ΓnI60mm40mm20mm20 mm40 mmΩ(s)ux = 0.2 mmFigure 6: Boundary conditions (left) and domain decomposition (right) for the wedge split test.on direct methods. Automatic load-balancing and multi-threading are accounted for and,consequently, the performance of the solver is optimized according to the hardware used.All analyses have been performed on a Dell R710 PowerEdge machine (two X5570 quad-core Xeons, Core I7 Nehalem, clocked at 2.93 GHz with 24 GB of memory). Parallel tasksrelated to BVP solves of different zoomed-in domains and FE assemblies are distributedover one to eight cores using openMP. It is observed in Figure 8 that parallel scalability isfound for a low number of processors. Solution strategies for massively parallel comput-ers, typically employed in domain decomposition approaches, need robust iterative solversand are considered out of the scope of this contribution.3.3 3D opportunitiesTo illustrate the performance of the adaptive multiscale framework, a simple tensiletest is performed on a homogeneous bar with variable height (Figure 9). The sampleis decomposed into seven domains according to the planes x = 20, x = 40, x = 60,x = 80, x = 100 and x = 120 mm, respectively. The coarse domains are discretized withone 8-noded hexahedron while the fine domains contain 375 8-noded hexahedra. Theconstitutive model is linear elastic with Young’s modulus E = 35 GPa and Poisson’sratio ν = 0.2. Zoom-in is performed when the strain component εxx > 8.0× 10−4 at anyintegration point within the domain. Horizontal displacement contours on the deformedconfiguration at different loading stages are depicted in Figure 10.4 CONCLUSIONSThe performance of both dual and primal multiscale techniques is assessed for damageanalysis in quasi-brittle materials. Results turn to be identical and it is concluded thatboth formulations are equivalent when utilized in a multiscale context. Due to the lack of7O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluys0.0 1.0Damage34 domains (8704 Q4) 68 domains 136 domainsFigure 7: Final damage distributions for a decomposition using 34, 68 and 136 domains.136 domains68 domains34 domainsComputation time (t)Number of cores (p)t[sec]987654321048000400003200024000160008000136 domains68 domains34 domainsIdealSp =t1tpSpeedup (Sp)Number of cores (p)S p987654321108642136 domains68 domains34 domainsIdealEf =SppParallel efficiency (Ef)Number of cores (p)E f(%)987654321100806040200Figure 8: Computation times t, speed-up Sp, i.e. ratio between the time measured with one processort1 and p processors tp, and parallel efficiency Ef for up to eight cores.8O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluys140 mm20 mm yxz20 mm32 mmBoundary conditions (all units in mm)x = 0, y = 0, 0 ≥ z ≤ 20, uy = 0x = 0, y = 0, 0 ≥ z ≤ 20, uz = 0x = 140, y = 0, 0 ≥ z ≤ 20, uy = 0x = 140, y = 0, 0 ≥ z ≤ 20, uz = 0x = 140, 0 ≥ y ≤ 20, 0 ≥ z ≤ 20, ux = 0.2x = 0, 0 ≥ y ≤ 20, 0 ≥ z ≤ 20, ux = 0Figure 9: Geometry and boundary conditions for the three-dimensional testinterconnecting forces in the primal assembly there is no need for a force migration schemeas introduced in the dual framework. It is observed that parallel scalability is found for alow number of processors when parallel direct solvers are applied at the fully assembledsystem of equations. The proposed multiscale techniques show promising features forthe three dimensional analysis of large structures. However, further research is requiredin order to devise robust iterative solvers needed for the computations with massivelyparallel computers typically employed in domain decomposition approaches.5 ACKNOWLEDGEMENTSOriol Lloberas-Valls gratefully acknowledges the funding received from the MICINNof the Spanish Government through a ”Juan de la Cierva” Postdoctoral grant and par-tial funding from the European Research Council under the European Union’s SeventhFramework Programme (FP/2007-2013) / ERC Grant Agreement n. 320815 (ERC Ad-vanced Grant Project ”Advanced tools for computational design of engineering materials”COMP-DES-MAT).REFERENCES[1] D. J. Rixen. Parallel processing. In S. G. Braun, editor, Encyclopedia of Vibration,pages 990-1001. Elsevier, Oxford, 2001.[2] O. Lloberas-Valls, D. J. Rixen, A. Simone and L. J. Sluys: Domain Decompositiontechniques for the efficient modeling of brittle heterogeneous materials. ComputerMethods in Applied Mechanics and Engineering, 200(13–16):1577–1590, 2011.9O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluysux [mm] 0.20.0Final loading stageIntermediate loading stageInitial loading stageFigure 10: Horizontal displacement contours and deformed grids at different loading stages. 100×displacement magnification.10O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluys[3] O. Lloberas-Valls, D. J. Rixen, A. Simone and L. J. Sluys: Multiscale domain de-composition analysis of quasi-brittle heterogeneous materials. International Journalfor Numerical Methods in Engineering, 89(11):1337–1366, 2012.[4] O. Lloberas-Valls, D. J. Rixen, A. Simone and L. J. Sluys: On micro-to-macro con-nections in domain decomposition multiscale methods. Computer Methods in AppliedMechanics and Engineering, 225–228:177–196, 2012.[5] O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone and L. J. Sluys. Concur-rent multiscale analysis of heterogeneous materials. In J. Eberhardsteiner and H. J.Bo¨hm and F. G. Rammerstorfer, editors, Proceedings of the 6th European Congresson Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012),Vienna, Austria, 10 – 14 September, 2012.[6] O. Lloberas-Valls, F. P. X. Everdij, D. J. Rixen, A. Simone, and L. J. Sluys. Objectivemultiscale analysis of random heterogeneous materials. In E. On˜ate, D. R. J. Owen,D. Peric, and B. Sua´rez, editors, Proceedings of the XII International Conference onComputational Plasticity – COMPLAS XII, Barcelona, Spain, 3–5 September 2013.[7] R.H.J. Peerlings, R. de Borst, W.A.M. Brekelmans and J.H.P. de Vree: Gradient-enhanced damage for quasi-brittle materials. International Journal for NumericalMethods in Engineering, 39(19):3391-3403, 1996.[8] C. Farhat and F. X. Roux: A method of finite element tearing and interconnectingand its parallel solution algorithm. International Journal for Numerical Methods inEngineering, 32(7):1205-1227, 1991.[9] A. Mobasher Amini and D. Dureisseix and P. Cartraud: Multi-scale domain de-composition method for large-scale structural analysis with a zooming technique.International Journal for Numerical Methods in Engineering, 79(4):417–443, 2009.[10] P. Ladeve`ze and O. Loiseau and D. Dureisseix: A micro-macro and parallel com-putational strategy for highly heterogeneous structures. International Journal forNumerical Methods in Engineering, 52(12):121–138, 2001.[11] P. A. Guidault and O. Allix and L. Champaney and J. P. Navarro: A two-scaleapproach with homogenization for the computation of cracked structures. Computers& Structures, 85(17-18):1360–1371, 2007.11

Multiscale analysis of damage using dual and primal domain decomposition techniques

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Multiscale analysis of damage using dual and primal domain decomposition techniques

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