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A mesh-geometry-based solution to mixed-dimensional coupling

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Computer-Aided Design 42 (2010) 509–522
Contents lists available at ScienceDirect
Computer-Aided Design
journal homepage: www.elsevier.com/locate/cad
A mesh-geometry-based solution to mixed-dimensional coupling
Jean-Christophe Cuillière
∗
, Sylvain Bournival, Vincent François
Department of Mechanical Engineering, Université du Québec à Trois-Rivières, 3351 Boulevard des Forges, CP 500, Trois-Rivières, Québec G9A 5H7, Canada
a r t i c l e i n f o
Article history:
Received 23 April 2009Accepted 18 January 2010
Keywords:
Mixed-dimensional analysisFEA analysisBeam elementsShell elementsReduced-dimensional analysis
a b s t r a c t
When conducting a finite element analysis, the total number of degrees of freedom can be dramaticallydecreased using finite elements such as beams and shells. Because of geometric complexities, entiremodels (or portions of models) must be meshed using volume elements in order to obtain accuratesimulation results. If however some parts of these models fit the description of shells or beams, thena mixed dimensional model containing shell, beam and volume elements side by side can be used. Thisapproach,referredtoasamixed-dimensionalanalysis(MDA)cansignificantlyreducethetimeneededtomesh and solve the system. Unfortunately, problems arise when trying to connect the different types of element in part due to incompatibilities between the degrees of freedom, and due to the discontinuitiesbetweenmeshesgeneratedindependently.Thispaperpresentsasolutiontotheseproblemsbasedonthegeneration of a compatible mesh composed solely of standard finite elements and without requiring theuseofconstraintequations.ThismeshcanthenbeexportedtoastandardFEsolverwithoutusingspecificconnection elements.Crown Copyright
©
2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction
When conducting a finite element analysis (FEA) the engineeris often confronted with long calculation times or limits on thenumberoffiniteelementsandnodes.Dimensionalreductionisoneof the techniques employed to reduce the computing time and/orthe number of elements. Using plate, shell or beam elements canconsiderably decrease the total number of degrees of freedom(DOF) without a loss of accuracy. Not all models, however, aresuitable for a dimensional reduction. With some models, oftenstructures, some portions of their geometry can be modeledusing reduced-dimensional elements (typically beams and shells),while other portions must be modelled using volume elements.Examples of these types of model can be found in [1,2]. In these
situations it is possible to use mixed-dimensional FEA models,which combine several types of finite element (beam, shell andvolume elements, for example). This is usually referred to as
mixed-dimensional analysis
(MDA).MDA combines the advantages of using each type of element.It also reduces the computing time and the number of DOF, andit is able to take complex geometries into account by meshingwith volume elements. In fact, computer systems and processinglimitations mean that many complex models cannot be analysed
∗
Corresponding author. Tel.: +1 819 376 5011x3920; fax: +1 819 376 5152.
E-mail addresses:
Jean-Christophe.Cuilliere@uqtr.ca (J.-C. Cuillière),Sylvain.Bournival@uqtr.ca (S. Bournival), Vincent.Francois@uqtr.ca (V. François).
except by using mixed-dimensional models. Another advantage of MDA is the ability to easily modify the FE model when evaluatingdesign scenarios. For example, if a shell thickness has to bemodified, the modification can be done by changing only that oneparameter(theshellthickness);themeshcanbekeptasisanddoesnotneedtobere-generated.Ifthesameanalysisisperformedusingafull3Dmodel,theCADmodelmustbemodifiedandthestructureusually needs to be completely re-meshed from scratch, which isobviously a great waste of time. It is an even greater waste of timewhen the analysis requires significant adaptive mesh refinement;repeating these tasks (mesh generation and adaptive refinement)for every new design alternative makes the design process veryexpensive in terms of processing time.A mechanical/structural engineer facing a problem requiringMDA, either because of the structure’s complexity or timeconstraints, must follow specific steps to obtain a viable model.Fig. 1 illustrates these steps when the mixed-dimensional processis performed without using the automation tools proposed in thispaper. Some of these steps are specific to MDA and others arestandard FEA procedures.The input is a part or assembly model built in a CAD system.This CAD model is constructed in great detail, which is usefulin the product development process (drawings, documentation,manufacturing, etc.) but which over-constrains the FEA process.Thus, for FEA, this product development model has to besimplified.Thedecisionwhethertoconsiderortosimplifyfeaturesof the product development model, is usually taken by themechanical/structuralengineer,basedonhisorherknowledgeandexperience of the problem. In this context, an ongoing research
0010-4485/$ – see front matter Crown Copyright
©
2010 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2010.01.007
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J.-C. Cuillière et al. / Computer-Aided Design 42 (2010) 509–522
Fig. 1.
Basic steps in a MDA without using the proposed automated tools.
aims at the automation of detailed CAD model simplification forthe purposes of FEA [3,4]. The next step in the overall process is
the idealisation of the 3D geometry, which is specific to mixed-dimensional analyses. The analyst decides which portions of themodel, or which components of an assembly, can be modelledusing reduced dimensional elements (typically the beam and shellelements). Topological adaptation is used to ensure that commonnodesareproperlylocatedatadimensionalinterfacebetweentwoentitiesanditisfollowedbyautomaticmeshgeneration[5].When
usingstandardautomaticmeshgenerationproceduresonamixed-dimensional model, each entity of the geometric model is meshedseparately and consequently, further processing is required inorder to connect them together before being transferred to thesolver along with the boundary conditions.ThefinalstepintheMDAprocessissolvingtheFEAmodel[6,7].
The type of solver used may depend on the types of elementinvolved in the FEA model. For example, the use of specificconnection elements, requires the use of specific solvers which isa problem for interoperability.The paper is organised as follows. Section 2 lists the mainproblems related to the use of MDA and presents related work,followed in Section 3 by an outline of our approach. Section 4
details a mesh-geometry based solution to the connectionsbetween the edges. Section 5 presents a solution to the edge–faceconnections while Section 6 demonstrates how that solution canbe extended to the face–face connections. Section 7 presents thepractical implementation of these concepts, the results obtainedand the comparisons with other approaches.
2. Problem statement and related work
2.1. Automation of mixed-dimensional analysis
Performed without automation tools, the mixed-dimensionalprocess is long, tedious, and prone to all sorts of errors. Forexample, if a rigid (bolted or welded) beam–volume connectionis not processed properly, it results in a ball-joint type of connection. Solving mixed-dimensional models is much fasterthansolvingfull3Dmodels,butthetimerequiredtoprocessthesemixed-dimensional models without automation tools eliminatesthat advantage. Also, for problems where MDA is the onlysolution (typically, for complex structures), FEA becomes amajor bottleneck in the design process. In that situation, thedesign of specific automation tools specific to MDA could inducemajor gains. Unfortunately, the automation of MDA faces majorproblems, which are detailed in the following section. Theseproblems are mainly related to the finite element modelling of theinterface between elements featuring different dimensions.
Definition 1.
In this paper, a zone where 1D (beams), 2D (shells)and 3D (volume) elements join is referred to as a
dimensionalinterface
.
2.2. Deriving and modelling idealised geometry
The first problem encountered when trying to automatethe MDA process is deriving and modelling the idealised ge-ometry. Methods aimed at the automation of the idealisation
J.-C. Cuillière et al. / Computer-Aided Design 42 (2010) 509–522
511
Fig. 2.
The adapted BREP structure.
process [2,8–10] are under development. Using the medial axis
transform (MAT) or similar representations of 3D objects seemspromising [8,10,11] but unfortunately, the idealisation is far from
fully automated and much research remains to be done. It shouldbe noted that in the work presented here, the idealised models arebuilt by the designer in a CAD system, which means they will beconsidered as inputs in the following sections.Standard CAD models have to be adapted in the contextof mixing 1D, 2D and 3D geometry. In this project, we usean adapted B-Rep structure to handle the reduced dimensionalentities such as shells (2D) and beams (1D). Another issue ismodelling the dimensional interfaces explicitly. In our work, thedimensional interfaces are derived explicitly and automaticallythroughaface–splittingprocess.Thefacesplittingprocessconsistsofdividingtheconnectionfaceswithrespecttotheactualshapeof thedimensionalinterfaces[12].Aswewillseeinthenextsections,
this face–splitting process is fundamental in our approach to theautomation of MDA as it also contributes to overcoming the otherproblems related to MDA, which are insuring mesh continuity anddealing with incompatible degrees of freedom. The introduction,of beams and shells requires an adaption of the classic B-Repstructure as illustrated in Fig. 2. Classic B-Rep entities have been
enhanced with topologic entries for shells and beams. These newentries are referred to as
Open Skin
(for shells) and
Open Loop
(for beams). Properties are attached to these new B-Rep entitiesas required to describe the thicknesses for shells and the sectionproperties for beams.Fig. 3 illustrates, on a very simple example, this adapted B-Repstructure before face–splitting and it can be observed that there isno topologic link between the solid and the connecting beam andshell, while in Fig. 4, the introduction of face–splitting induces a
natural topologic link between associated B-Rep entities.
2.3. Mesh discontinuity
The second problem encountered when trying to automate theMDA process is ensuring mesh continuity. For example, when abeam has to be connected with a shell (see Fig. 5), modellingthe continuity of the structure requires a common node at theintersection point.A typical automatic mesh generation procedure would han-dle this beam–shell connection by meshing both entities indepen-dently. This means that the shell would be meshed without anyconcern for the beam’s mesh. Thus, if no specific processing is ap-plied, there will be no common nodes at the intersection and thetwo components will not be bonded properly.
2.4. DOF compatibility
The third major problem that arises when using MDA is theincompatibledegreesoffreedom(DOF)atadimensionalinterface.For example, typical 3D solid elements (such as structuraltetrahedrons or hexahedra) feature only 3 DOF (3 translations)per node while the structural beam and shell elements typicallyfeaturefiveorsixDOFpernode(3translationsand2or3rotations).Consequently, when coupling solid and beam (or shell) elements,even if the nodes of both connecting parts are merged properly,connectinganodefeaturing6DOF(forexampleforbeamandshellelements) with a node featuring only 3 DOF (for example for solidelements) is akin to a ball-joint connection and not to what isrequired in most cases. If the intersection between a shell and avolume is a straight line, the resulting effect is akin to a hinge.In the next section, we discuss how these two problems (meshcontinuity and DOF compatibility) have been addressed in theliterature.
2.5. Related work
PreviousresearchinthefieldofautomatingMDAhasfocusedonthedevelopmentofspecificfiniteelementsandonspecificsolvers.A team at the University of Belfast introduced a method tocouple different types of finite element based on the use of constraint equations. In this type of approach, the DOF of nodeslocated on both sides of a given dimensional interface are linkedbyasetof equationsthatequalsthemechanicalworkoneachsideof the interface. The method has been applied to many types of connection and the results are reported in [8,13,14].
Fig. 3.
An adapted BREP structure before face splitting.
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Fig. 4.
An adapted BREP structure after face splitting.
Fig. 5.
Example of incompatible mesh.
Another approach to the problem of incompatible DOF is theuseofspecificfiniteelements(typically,withadditionalrotationalDOF). The principle is based on Allman’s triangle [15], a quadraticfinite element triangle featuring only 3 nodes, each with 4 DOF.For each node of Allman’s triangle, the fourth DOF is to be seenas a pseudo-rotation. The principles on which Allman’s triangle isbasedhavebeenextendedtotetrahedrons[16].Severalothertypes
of element based on the same concepts can also be found in [17].Although these elements have not been specifically designed toovercome the DOF inconsistency problems, they can be applied inthis context. Tests have been performed to couple beam elementsand 3D elements with additional DOF but stress results weredisappointing.Two other alternatives proposed by Craveur [18] are illustratedon a beam–shell connection in Fig. 6. The beam connects to
the shell on a common node, referred to as the
intersectionnode
. The first alternative (Fig. 6a) is independent of the beam
section’s shape. Four extra beam elements are added to the shellelements edges connecting with the intersection node. Problemsof the DOF inconsistency are avoided because all the force andmoment components are transmitted between the beam and theshell through these extra beam elements. The ball-joint effectmentioned earlier is therefore averted. The length and location of the extra beam elements are independent of the beam’s sectionbecause they are directly based on the underlying surface mesh.Craveur’ssecondproposedalternative(Fig.6b)istousespecific
rigidconnectionelements.Theseadaptedconnectionelementsareof infinite stiffness, are specific to the type of connection, featurespecific connecting nodes, and are specific to the shape of thebeam’s section. In the example shown in Fig. 6b, the use of thespecific connection element induces a rigid connection betweenthe beam and shell elements. The ball-joint effect mentionedearlieristhusalsoavoidedhereandtheadaptedelementtransmitsboth forces and moments through the dimensional interface.It must be emphasised here that these proposals have beenintroducedbyCraveurin[18]asanalysisalternativesinthecontextof good practice when using FEA in mechanical engineering. Thisis not a research work because these proposals have not beenstudied thoroughly and have not been quantified with regard tothe accuracy if compared with a full 3D model.AnothertechniquethatcouldpotentiallybeusedistheArlequinmethod [19]. The method is designed to combine two different
meshes, usually a coarse one for the whole domain and a finer onelocatedaroundsensitivedetails.Basically,theprincipleunderlyingthe Arlequin method is using a weighted mean between thestresses obtained from both meshes. The method can be adaptedto deal with meshes involving a mix between different types of element and consequently can be used in the context of MDA.In this context, the dimensional interface is processed using anoverlay of two meshes featuring two different types of element.It must be mentioned that several commercial FEA sys-tems recently introduced operators aimed at modelling mixed-dimensional interfaces and consequently at speeding up theprocess of mixed dimensional analysis. These operators are eitherbased on mesh mating, which relies on constrained meshing or ontheadditionofconstraintsequations,whichhasalreadybeenmen-tioned earlier. Both of these solutions will generate a connection,but will not reproduce accurately the shape of the connection, andconsequently it will not reproduce accurately the mechanical be-haviour of the connection. Basically, this is due to the fact that, us-ingtheseapproaches,theconnectionismadethroughsetsofnodes(a single node in some cases) that are inconsistent with the actualconnectionshape.Ithasbeenshownthatthishas,ingeneral,aneg-ativeeffectonthestressaroundandacrosstheinterfacesandthat,in some cases it also has a negative effect on the displacement re-sults far away from these interfaces.
2.6. Objectives
The approaches reviewed above either produce inaccurateresults (because the shape and behaviour of the interface is notprecisely modelled) or require the use of specific elements (or
J.-C. Cuillière et al. / Computer-Aided Design 42 (2010) 509–522
513
a b
12s34
Fig. 6.
Craveur’s suggestion: (a) Section independent method (b) Adapted element method.
Fig. 7.
Concept of the mesh-geometry based approach: (a) Edge–face connections (b) Face–face connections.
specific constraint equations) requiring the use of very specificmesh generation procedures and of specific FEA solvers. Inthe following sections, we will explain how it is possible toautomaticallysolvetheproblemsofmeshandDOFincompatibility,while obtaining accurate results at the dimensional interfacesusing standard CAD processing and specific arrangements of standard finite elements. This mesh-geometry-based solutioncan be applied easily and efficiently using any CAD/meshgeneration/FEA package.Consequently, the objectives are as follows:
•
the process should be fully automated, starting from anidealised model along with the relevant technical data (shellthicknesses and beam section properties, boundary conditions,material data and element size constraints) to analysis results.
•
theprocessisintendedtomodelthebehaviourofinterfaces(asidealised) as accurately as possible (with regard to the strainand stress distribution), which means as in the case of fullymodelling the interface with a refined mesh only featuringvolume finite elements.
•
theprocessshouldnotrequiretheuseofspecificfiniteelementsor constraints equations and should be based on the use of standard beam, shell and volume finite elements (Bernoullibeam elements, triangular thin or thick shell elements, andbasic tetrahedrons). Consequently, the process should be ableto use any FEA solver.
•
the process should be able to use any standard automatic meshgeneration system.
3. The proposed solutions
3.1. A solution to mesh discontinuities
Standardautomaticmeshgenerationprocedures[5]areusuallyled through the following basic pattern. Vertices are meshed first,then the edges, followed by the faces, and finally the volumes.Several alternatives can guarantee that nodes will be generated ata given location.
•
adding a node at the intersection point and remeshing locally
•
movingtheclosestnodetotheintersectionlocation,whichmayalso require, in some cases, a local remeshing
•
using constrained mesh generation
•
face–splitting.We chose face splitting as the solution to mesh discontinuitiesbecause:
•
it allows to model explicitly the dimensional interfaces(as mentioned in Section 2.2). Indeed, one of the major
contributions of the work presented here is that the facesplitting is performed with respect to the actual shape of the dimensional interface (the beam section for a connectioninvolving a beam and the shell thickness for a connectioninvolving a shell).
•
it also allows an accurate and efficient solution to the problemof DOF incompatibility.For example, if a single node is required at a given point on aface belonging to a volume, and no vertex is located at that point,then introducing a vertex, through face splitting, at that locationis one solution as illustrated in Fig. 7a. This face–splitting conceptcan be easily extended to other types of intersection (see Fig. 7b).
3.2. A solution to DOF incompatibility
Our solution to DOF incompatibility is inspired by Craveur’sidea to add extra beams in order to avoid the ball jointeffect. Moreover, our solution allows reproducing accurately themechanical behaviour at the dimensional interface, which is not

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