A multi-scale finite element model for inelasticbehavior of heterogeneous structures and its parallelcomputing implementationDamijan Markovica,b, Adnan Ibrahimbegovica,Rainer Niekampcand Hermann MatthiescaEcole Normale Sup´erieure de Cachan, LMT-Cachan,61 avenue du pr´esident Wilson, 94235 Cachan, Francee-mail: [email protected]: +33(0)1 47 40 22 40bFaculty of Science and Technology,University of Ljubljana, SloveniacInstitute of Scientific Computing,Technical University of Braunschweig, GermanyApril 8, 2004AbstractIn this work we consider a strongly coupled multi-scale problem in the contextof the inelastic structural mechanics. Supposing the significant but not neglect-ing ratio between scales, we replace the somehow more standard phenomenologicaland analytical homogenization approaches by a lower level numerical descriptionof the micro-structural behavior. More particularly, we take continuum damageand plasticity finite element method model to describe the matrix-inclusion typeof micro-structure (e.g. porous or hard inclusion composite). The micro FEMmodel is then coupled to the macro FEM model through the localized Lagrangemultiplier approach as introduced in [5]. This multi-scale strategy being very welladapted to parallel computing, we parallelized the algorithm using the Commu-nication Template Library CTL, a recent development of the institute of ScientifcComputing in Braunschweig, Germany. The efficiency of the implementation isshown on numerical large scale examples.1 IntroductionComposite material have always had an important role in structural engi-neering. Either involuntarily, because the material processing cannot always1avoid some final defaults like porosity or voluntarily because one wants toimprove material resistance, like in hard inclusion composites (e.g. metalcomposites, concrete). Unfortunately, heterogeneous nature of materials in-duces more significant heterogeneous field solutions in a structural loadingframework and consequently makes modeling in general more difficult.The homogenization techniques, which are able to model the compositematerials can be divided into two groups: analytical and numerical. Inthe first approach one does some physically justified assumptions on theheterogeneity of the micro deformation and stress fields and defines themacroscopic response through the constituent behaviors analytically (seee.g. [1], [2] or [8]). On the other hand, in numerical homogenization, themicro response is obtained by a numerical method, like the finite elementmethod (FEM). When the scales are separated enough the micro-level FEcomputation are carried out for each integration point of the macro FEcomputation, in order to obtain constitutive stress-strain relation (e.g. see[3]).Since in many civil and mechanical engineering applications the scalesare strongly coupled, we have proposed a multi-scale strategy, where the finemicrostructure representation is not introduced at the level of the Gausspoint, but on the level of the whole element (see [4] and [5]). We applythe localized Lagrangian multiplier method to couple the two scales ([7] and[6]). In this context, the macro scale is the frame situated between differentsub-domain, which can be interpreted as micro scale, because of its muchfiner FE representation. Thus, for each macro element, we have a choiceusing either a macroscopic model, obtained in the most convenient way, ora very fine FE model, which replaces the less exact constitutive relations.An important characteristics of our approach (see [4] and [5]) is the localnature of micro calculations. The microstructure representing FE modelscommunicate between them exclusively by the macro degrees of freedom.Hence, the substructuring does by no means affect the macro resolutionprocedure. In addition, the independency of micro calculations renders themicro-macro strategy easily parallelizable.2 Formulation of the strong coupling multi-scaleinterface2.1 Concept of strongly coupled scalesWe consider a general class of heterogeneous structure problems submittedto an arbitrary loading and obeying non-linear physical laws resulting inphenomena like damage and plasticity. It is assumed that the scales arestrongly coupled and that their evolution has to be calculated simultane-ously. The FEM has shown undoubted efficiency in solid and structural2analyses and we thus decide to use it on both scales. Therefore, the struc-ture is meshed with a macro FE mesh and the microstructure representingvolume similarly with a micro FE mesh (see Figure 1).rrr rrrrrrrrrrrrs ss suMum,e’e’Figure 1: Micro-macro finite element model of a simple structure. Eachmacro FE equally represents a meso scale window containing the microstruc-ture information, which is again modeled by the FE meshBy the appropriate formulation the macro mesh element quantities (resid-ual, stiffness etc.) are obtained from the micro FE calculation, replacing inthis way the macroscopic constitutive law at the finite element level ratherthan at the level of a Gauss numerical integration point. The microstructurewindow (see Figure 1) is chosen such that its dimensions match those of thecorresponding macro finite element.2.2 Variational formulationThe coupling of the scales in our multi-scale FE model is obtained throughthe framework of localized Lagrange multipliers (e.g. see [7] and [6]). Themacro mesh plays the role of a frame which is connected to the micro meshthrough the Lagrange multipliers.We can write the elastic potential, whose stationary value will lead tothe solution of the problem, as following:Π =ZΩmΨm(², ξk) dV +ZΓλ(um−uM) dS−Z∂σΩuMt dS−ZΩumf dV 7→ stat.,(1)where uMis macro displacement field (coarse discretization), umis microdisplacement field (fine discretization). Accordingly, Ψmis micro scale freeenergy, ² is the deformation field at the micro scale, ξkare the internalvariables at micro scale, and λ is the Lagrange multiplier field glueing twodifferent scales. We also note that Ω defines the total domain, ∂σΩ is thepart of the boundary, where tractions are applied and Γ denotes the totalinterface surface between the two scales. Using the stationarity condition ofthe potential,δΠ =∂Π∂umδum+∂Π∂uMδuM+∂Π∂λδλ = 0; ∀δum, δuM, δλ (2)3we can get the weak formulation in terms of three equations: macro equilib-rium (δuM), micro equilibrium (δum) and micro-macro compatibility (δλ).From the present variational formulation