An Evaluation of The Quality of Hexahedral MeshesVia Modal Analysis

Xifeng GaoUniversity of Houston

Jin HuangZhejiang University

Siwang LiZhejiang University

Zhigang DengUniversity of Houston

Guoning Chen*University of Houston

AbstractHex-meshes have wide and important applica-tions in scientific computing. Various methodshave been proposed to tessellate a shape into ahex-mesh according to certain geometrical qual-ity criteria (e.g. scaled Jacobian), which is usu-ally not directly related to the downstream com-puting problem. This paper discusses such crite-ria on the elastic finite element analysis problemvia analyzing the result of Modal Analysis per-formed on different hex-meshes. With the iso-metric uniform material configuration, the largerthe scaled Jacobian (minimal/average) and thenumber of elements, the smaller are the eigen-values, which usually indicates better accuracyand stability (condition number). Besides theelement quality, singularity placement also af-fects the accuracy of the solution and is difficultto alter by simple subdivision. Results of thiswork may serve as a guide in evaluating hex-remeshing approaches and choosing appropriatehex-meshes for finite element analysis.Keywords: Finite element analysis, hexahedralmeshes, evaluation, singularity structure

1 Introduction

For many physically-based simulations involvedin a wide range of scientific and engineering ap-plications, volumetric representations, e.g., 3Dvolume tessellations, are required. Compareto the easily accessible tetrahedral meshes [1],hexahedral meshes are usually more attractivein their numerical property. But, generating

a suitable hexahedral mesh conforming to thegiven boundary is typically more challengingthan generating a tetrahedral one. In the pastdecades, various techniques for generating hexa-hedral meshes from the input boundary surfaceshave been proposed [2, 3, 4]. Up to date, themeshing community still largely relies on Jaco-bians of the elements to evaluate the quality ofthe obtained meshes. That is, the meshes withlarger average and minimum Jacobians are con-sidered better. Indeed, the local Jacobian hasbeen shown related to the local accuracy of thesimulation [5]. There exist a number of stud-ies on the benefits of conducting scientific sim-ulations on hexahedral meshes versus tetrahe-dron meshes [6, 7, 8]. However, there still lacksof a comprehensive evaluation and comparisonof the hex-meshes generated by recent methodsfrom the application perspective. Further, thereis little study on how different characteristics,such as the number of elements and differentsingularity structures of hex-meshes that are de-fined by non-valence-6 extraordinary nodes ofthe mesh (Figure 5), affect the simulations.

In this paper, we conduct a first systematicstudy on how various characteristics of the hex-meshes generated by a number of representa-tive techniques affect the accuracy and stabil-ity of elastic object simulation. Particularly,the mesh characteristics that are considered inour study include the Jacobians of elements(i.e., average/minimum Jacobians), numbers ofelements, and the structure of the hex-meshes(i.e., the numbers of singularities and the num-ber of the hexahedral components in the struc-ture). The hex-meshing methods that we are

considering include Volumetric PolyCubes [9],SRF [3], and L1-based PolyCubes [4]. By se-lecting the models that are used in all thesemethods, we specifically focus on the rockerarm model-representing a CAD object and thebunny model-an example of organic objects. Toevaluate the accuracy and stability of simula-tion run on different hex-meshes, Modal Analy-sis is employed. Especially, the eigenvalues andcondition numbers of the stiffness matrices gen-erated from the hex-meshes are measured andcompared. It is worth emphasizing that thiswork aims to study what influence of the char-acteristics of different hex-meshes can have tothe FEM simulations rather than identifying thebest hex-meshing technique. However, resultsreported in this work can in turn be used to helppractitioners select the proper mesh generationmethod given the metric that is sought by a spe-cific computation.

2 Related Work

Over the past decades, several studies have beenperformed to compare the convergence behav-ior between the FEM based simulations run ontetrahedral, hexahedral, and hybrid meshes. Ci-fuentes et al. [10] concluded that the resultsobtained with quadratic tetrahedral elements,compared to bilinear hexahedral elements, wereequivalent in terms of both accuracy of the simu-lated result and the computation time. The accu-racy of elastic and elasto-plastic FEA based sim-ulations of hexahedral and tetrahedral mesheswas compared by Benzley et. al [6]. It showeda preference for linear hexahedral mesh com-pared to linear tetrahedral mesh. By evalu-ating the FEA simulations on Femur meshes,Ramos et al. [7] observed that the accuracy ofthe simulation conducted on a linear tetrahe-dral mesh of the simplified bone is closer to thetheoretical ones, while the simulations on thequadratic hexahedral meshes are more stable tohex-meshes with higher resolution.

Up to date, there exists little study on compar-ing the accuracy properties of simulations com-puted on the hex-meshes generated with recenttechniques. Earlier, Muller et al.[11] investi-gated the effect of quality and the number of el-ements of hex-meshes on the simulation results,respectively. They concluded that the numberof elements of hex-meshes is important for thesimulation quality. However, their experimentalresults suggest that with the minimum element

quality above 0.1, average mesh quality doesnot has notable impact on the solution accuracy.Since they only used a very simple cylinder-likemodel throughout the whole experiments, it isdifficult to judge the completeness of their study.In this work, we propose to employ a few rela-tively complex models to verify their statements.Our study is also the first one that attempts to an-alyze the impact of the singularity structures ofhex-meshes on the simulation solutions.

3 Comparison Experiment Setup

The focus of this paper is to investigate the ac-curacy and convergence rate of simulations per-formed on the selected hex-meshes that have (1)varying local qualities (measured by Jacobians),(2) different numbers of elements, and (3) dif-ferent singularity structures (measured by differ-ent numbers). In order to conduct such a com-parison, several issues have to be addressed, in-cluding “which finite element method to use?”,“how to measure the dissimilarity between twosimulations?”, and “which 3D models to use?”.These topics are discussed in the following.

3.1 Modal Analysis of Elastic Problem

To understand the impact of different propertiesof the hex-meshes on the FEA based simula-tions, we make use of a well studied techniqueknown as Modal Analysis (MA). This sectionpresents a brief overview of modal analysis. Formore details, please refer to [12].

Giving an 3-dimensional linear elastic objectdiscretized by a mesh with n nodes, equationof motion is often formulated as Mu + Ku =f , where K,M ∈ R3n×3n, u, f ∈ R3n arethe stiffness matrix mass matrix, node displace-ment and external force respectively. ModalAnalysis computes the general eigen problemW TKW = Λ,W TMW = I , and then whichdiagonalizes the above equation into z+Λz = gvia substitution u = Wz, g = W T f . Sucha transform does not only boost the efficiencyof computation, but reveal an important prop-erty of the dynamics: the motion can be de-composed into a set of independent vibrationmodes (columns of W = (W1,W2, · · · ,Wn))at their stiffness (square of natural frequency)λi = Λii, λi ≤ λi+1.

Eigenvalue is a widely used indication forthe quality of discretization in terms of accu-racy and stability in computation. As explained

Min

Max

(a)0.916/0.209 (b)0.897/0.118 (c)0.860/0.100 (d)0.828/0.100Figure 1: Original rocker arm hex-mesh from [3] and the generated hex-meshes with varying quality.

in [6], the discretization introduces additionalnumerical stiffness because of the extra forceto restrict the deformation in a small subspace.The worse discretization is, the eigenvalue ismore overestimated, and the object is “stiffer”.Thus, smaller eigenvalue usually indicates bet-ter accuracy in discretization. In many cases,the condition number of the linear system in-volved in the simulation is nearly determinedby max{λi}. For example, an equation simi-lar to (M + h2K)∆v = hf should be solvedin each step of backward Euler method [13]for the change of velocity ∆v in time step h.The minimum and maximum eigenvalues of thesystem (M + h2K) (respect to W ) are 1 and1 + h2 max{λi} respectively, because the lead-ing 6 eigenvalues associated with rigid motionare all zero. Thus, the smaller the eigenvalue isand the smaller the condition number is, the bet-ter stability the simulation will have.

The discretization is characterized by themesh tessellating the object and the type of ba-sis applied to the elements in the mesh. In thispaper, we adopt 8-node trilinear basis functionfor all the hexahedral elements, and thus leavethe properties (e.g. Jacobian, density) of meshas the only factor related to the eigenvalues.

3.2 3D Model Data

Table 1 provides an overview the hex-meshesused for different comparison studies. Amongthem, different versions of Bunny1 frommethod [9] and Bunny2 from [3] are used toexamine the Modal Analysis properties of hex-meshes with various element numbers, whilethe first three versions of Rockerarm1 [4] andRockerarm2 [4] are employed for the analysison meshes with different scaled Jacobians (i.e.,Ave./ Min. Jacobians). Three hex-meshes, de-noted by Rockerarm1

3 [4], Rockerarm23 [4],

Table 1: Meshes used in the three Comparisons

Hex-Mesh V/H Ave./Min J.Bunny10 59887/54568 0.943/0.136Bunny11 49524/44909 0.941/0.128Bunny12 41593/37511 0.937/0.127Bunny13 35428/31759 0.940/0.125Bunny20 51175/47797 0.947/0.316Bunny21 32410/30119 0.948/0.322Bunny22 18654/17226 0.944/0.279

Rockerarm10 28397/24346 0.930/0.123

Rockerarm11 28397/24346 0.894/0.102

Rockerarm12 28397/24346 0.862/0.100

Rockerarm20 28814/24780 0.916/0.110

Rockerarm21 28814/24780 0.881/0.102

Rockerarm22 28814/24780 0.849/0.100

Rockerarm13 12919/10776 0.938/0.413

Rockerarm23 13014/10942 0.926/0.457

Rockerarm3 12751/10600 0.916/0.209

and Rockerarm3 [3], with distinct singularitystructures are also selected to investigate the in-fluence of the structure on the FEA. Note, thatthe sup-index of a mesh, e.g., Rockerarm3,represent one of the three methods [9, 4, 3] thatis used to generate the mesh, while sub-indices,e.g., Bunny10 and Bunny11 , indicate differentversions of the meshes. These hex-meshes areeither directly from the three work or generatedfrom the original hex-meshes using the followmethods.

To produce hex-meshes that have the samenumber of elements and singularity structures,but varying local quality, we follow the proce-dure in [11]. The difference of our approachis that for the vertex vi that is to be jittered, abounding surface is also constructed to constrainthe active space of vi. The bounding surface is

comprised of the quads of the one-ring hexahe-dra of vi that are not adjacent to vi. By addingthis constraint, the possibility of generating in-verted elements is excluded. Figure 1 shows thegenerated rocker arm meshes with different Ja-cobians.

Simplification technique of [14] is adapted togenerate meshes with different numbers of ele-ments . All simplified meshes are optimized byMesquite to achieve similar Jacobian quality. Toconsider the effect of different singularity struc-tures of the meshes to the Model Analysis, therocker arm meshes from two methods [4, 3] areemployed, which already have different struc-tures (Figure 5). We use the same approachabove to ensure they have similar numbers of el-ements and similar Jacobians.

4 Results

To evaluate the quality of hex-meshes gener-ated by different approaches, three types of com-parisons have been performed, including thecomparisons of hex-meshes with 1) differentAve./Min. Jacobians, 2) various numbers ofelements, and 3) distinct singularity-structures.The quality of the mesh is measured by the itheigenvalue λmi for mesh m.

In Figure 2, we investigate the meshes withthe same number of elements. To better vi-sualize the differences among the eigenvaluesderived from different meshes, we normalizeλmi into λmi = (Mλmi )/

∑Mj=1 λ

ji , where M is

the total number of meshes in the comparison.The smaller normalized value λmi is, the smallerthe eigenvalue λmi will be. Figure 2 demon-strates that as the average scaled Jacobians ofRockerarm1 (Figure 2(a)) and Rockerarm2

(Figure 2(b)) increase, the eigenvalues of thehex-meshes decrease. This observation con-firms that better Jacobians result in better accu-racy and condition number.

In Figure 3(a) and (b) we compare the mesheswith similar scaled Jacobians. As shown in Fig-ure 3(a), when the number of elements has nobig difference (in the range of 35k to 60k), theimprovement of accuracy by using more ele-ments is not significant. But when the densityincreases a lot (from 19k to 51k), the improve-ment becomes obvious.

As shown in Figures 4 and 5, the hex-meshes have very distinct singularity struc-tures. The distribution of the deformation dis-placements and the eigenvalues for different

modes are also different. From Figure 4, al-though Rockerarm1

3 and Rockerarm23 have

better Ave./Min. Jacobians and larger numberof elements, Rockerarm3 gives the best eigen-values for all modes. This may indicate that notonly the scaled jacobian and the element numberof a hex-mesh can affect its finite element anal-ysis, the singularity-structure of a hex-mesh canalso make an influence. Figures 4 shows that thenumbers of singularity nodes and hex-patches ofRockerarm3 are ranked in between of those ofRockerarm1

3 and Rockerarm23. By compar-

ing with their corresponding eigenvalues in Fig-ure 4, it implies that besides the numbers of sin-gularity nodes and hex-patches, there could beother properties of the singularity-structure thataffect its FEA behavior.

Figure 4: Comparisons among the eigenvaluescorresponding to various modes forthree rocker arm meshes.

5 Conclusion and Future Work

In this paper, we investigate how different char-acteristics of hex-meshes affect simulations con-ducted on these meshes. In particular, ModelAnalysis is performed on the stiffness matricescomputed on these meshes. Hex-meshes for arocker arm and a bunny with different character-istics (e.g., different numbers of elements, dif-ferent Jacobians, and different singularity struc-tures) are utilized for this study. Our initial re-sults indicate that when hex-meshes have thesame singularity-structure, the better behaviorof Modal Analysis is typically observed on hex-meshes with larger scaled Jacobians and morehex-elements. When the hex-meshes have dis-tinct structures but similar scaled Jacobians and

(a) (b)Figure 2: Comparisons among hex-meshes with various Jacobians denoted in the legends.

(a) (b)

Figure 3: Comparisons among hex-meshes with various numbers of elements denoted in the legends.

element numbers, the performances of Modalanalysis can still vary. In particular, goodsingularity-structures (e.g., with smaller numberof singularity nodes and hexahedral component)could compensate the shortness of smaller Jaco-bians or fewer elements.

Limitations and Future Work There aresome limitations in the current study. First, onlytwo objects are used in the experiments, whichmay not be convincing. Second, some proper-ties like the distribution of elements quality, andthe distribution of different scales of elementsare not covered in the current study. Third, howthe difference of singularity structures should bedefined? This is still an open problem. Also, theformal relation of the singularity structure withthe accuracy and performance of the simulationsneeds to be studied rigorously. Fourth, morehex-meshing approaches should be included. Fi-

nally, other simulations than elastic deforma-tion, such as Isogeometric Analysis (IGA) [15],should be examined in order to fully evaluate thebenefits of structured hex-meshes.

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