Automatic Thin Sheet Identification of Complex Model for Hexahedral Dominant Meshing

Liang Sun, Cecil G Armstrong, Trevor T Robinson, Christopher M Tierney.

School of Mechanical and Aerospace Engineering, Queen’s University Belfast, BT9 5AH, UK

Email: t.robinson@qub.ac.uk

Abstract

This paper describes an automatic method for identifying thin sheet regions in a complex solid model. The main purpose of thin sheet identification is to ease the current situation of hexahedral meshing for complex geometry in the aerospace industry, since swept hexahedral meshes can be easily applied to the identified thin sheet regions. A substantial reduction in degrees of freedom is achieved in the resulting thin sheet regions. The method is based on a face pairing technique, which matches bounding faces of a region satisfying the dis-tance and overlap criteria. Edges of the paired faces are discretized and imprinted to the mid-surface. They are represented in the UV parameter space of the mid-surface, transforming the problem from 3D to 2D. This en -ables 2D polygon intersection algorithms to be employed to obtain the critical points that determine the boundary of the thin sheet regions. Splitting faces are then generated based on the intersection result and ulti -mately used to isolate the thin sheet regions.

Key words: Thin sheet identification; Automatic decomposition; Hexahedral dominant meshing

1. Introduction

Finite Element Analysis (FEA), as a successful computational simulation approach, has gained wide application in numerous disciplines. Along with the exhaustive use of simulation during the product design process, the complexity of analysis models has also increased. A growing number of analyses are currently required to be performed at the as-sembly level in order to acquire more precise simu-lation results. Taking the aero engine for example, in order to gain an accurate assessment of the en-gine behaviour in extreme events like fan-blade-off or bird strike, it is essential to carry out analysis at the whole engine level using high quality meshes.

The rapid growth of computer power has offered objective foundations for these kinds of complex analyses. However, the increased computational capabilities do not permit arbitrary increase in the degrees of freedom (DOF) of the analysis models. Reducing the DOF of models under analysis to guarantee that the simulations are accomplished within a reasonable time range and with acceptable

accuracy is an ongoing challenge. Cost overheads are combined with the need to utilise specific finite element types to successfully capture the physics. Actual factors like run time and accuracy have to be dealt with carefully and sometimes compromised for each other. All these factors have restricted the practical analysis solution method as well as the mesh element type which can be used.

For example, in the area of nonlinear structural dy-namic analysis (e.g. using LS-Dyna), the explicit in-tegration method is normally used, especially for highly nonlinear event. Implicit integration methods are more computationally expensive when solving the stiffness matrix which involves matrix inversion and decomposition, although they have advantage in terms of stability. Implicit methods also suffer from the fact that there is no guarantee of convergence for highly nonlinear problems. The explicit method has no problem of convergence and with a carefully selected time step the true results can be properly approximated. As regards to the mesh element type in explicit method based analysis, the 3D hexahed-ral element is the preferred choice compared to tet-

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rahedral element. Conclusions about the advantages of using hexahedral elements can be easily found in book or papers [1-2]. Here two points are specially stressed. First, in the explicit integration method, the max allowable time step is dependent on the charac-teristic length of the smallest element. Hexahedral elements have larger time step than tetrahedral ele-ments. Second, 4-10 times more tetrahedral ele-ments are required to the mesh the same domain as hexahedral elements, which means a huge increase of DOF. Controlling the DOF is significantly im-portant in nonlinear dynamic analysis at a whole en-gine level since the number of DOF can reach tens or hundreds of millions. The contribution of using efficient mesh structures in time saving stands out if a large number of time steps or iterations are re-quired.

Unlike tetrahedral meshing, no robust automatic al-gorithms are currently available for all hexahedral meshing. In practice, in order to get hexahedral meshing with a good quality structure, geometry is not directly meshed. Rather it is decomposed manu-ally to a collection of sub-volumes to which existing meshing strategies such as mapping, sub-mapping or sweeping can be more readily applied. However the time and human effort taken for this tedious de-composition process often outweigh the need for ac-curacy and computational speed. The vital contribu-tion of the user is difficult to achieve automatically.

Since a total removal of user intervention is not pos-sible in the current stage of technology, the research here is focused on the automated identification of regions with special geometry characteristics, such as thin sheet regions. Hexahedral elements with good quality will be obtained automatically for these regions and only the complex residual regions will be manually worked on.

2. Related workDecomposition is an indispensable procedure for high quality hexahedral meshing. The purpose of decomposition is to obtain simple blocks which are hexahedral-meshable

Robinson et al. proposed a thick/thin decomposition process based on medial object (MO) [3] and this method is now available in the commercial software

CADfix [4]. MO is a skeleton representation formed by connecting the centre of the maximum circle/sphere as an inflatable circle/sphere moves within the original geometry [5]. A 3D MO of the geo-metry is first generated to decide the local thickness, followed by the creation of a 2D MO which is em-ployed to approximately indicate the lateral lengths. Areas with an aspect ratio (lateral length/local thick-ness) exceeding a specified value are identified as thin sheets, and it was shown that the reduction in DOF was proportional to (aspect ratio) 2. Yin et al. [6] proposed a method to isolate thin section using points on an approximate MO computed by an Octree-based algorithm. Surface triangles on oppos-ite faces of thin regions are associated with MO points. Full 3D p-version finite elements with low polynomial order through the thickness are applied on the thin domains. The MO based methods were restricted by the capabilities of a robust and efficient implementation of 3D MO. Moreover, sliver faces or edges exist in MO and need extra effort like ex-tending and trimming to deal with them.

Robinson et al. also predict that a further reduction of DOF proportional to the aspect ratio is possible if the so called long slender regions are identified and meshed with structured hexahedral elements swept along the length of the region [3]. The work of identifying thin sheet [3] has been extended by Makem et al. to find the long slender regions using a series of sizing measure methods [7]. Ellipsoids are generated based on the local measurement of edge mid-points and those with dimension of one axis much larger than that of the other two axes are treated as long slender regions. The work of Makem et al. is not based on MO and therefore can be ap-plied generally in any software. Mixed solid ele-ment finite element models are generated from the thin sheet and long slender decomposition. In addi-tion, work by Nolan [8] and Tierney [9] have util-ized simulation intent to demonstrate how the thin sheet and long slender decomposition can be used to automatically generate mixed dimensional analysis model.

F. Boussuge et al. [10] developed a method for identifying extrusion primitives in a model. A con-struction graph is generated during a recursive pro-cess to decompose a model into extrusion primit-

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ives. Several assumptions are made to simplify the process and the current range of shapes that can be robustly identified needs to be extended.

The thin sheet finding work in this paper is based on a face pairing technique. A similar application of face pairing in thin regions is to abstract the mid-surfaces of these regions for the purpose of dimen-sional reduction, which transformed 3D solid model into either a stiffened shell model or a mixed dimen-sional model [9-11]. Although the start point of identifying face pairs is the same, there are some differences between the problems. In the mid-sur-face abstraction, the pivotal issues are how to gener-ate mid-surfaces patches and how to connect (ex-tend/trim) them properly. In the thin sheet regions isolation, there is no need to generate mid-surfaces patches and the focus is how to decide the boundary of the target thin regions and how to create appro-priate faces to isolate these regions. Besides, mid-surface abstraction tools are available in many pack-ages while as far as the author is aware thin sheet identification based on face pairs is not offered in any mainstream modelling packages. The example below illustrates the different problems that are in-terested in thin sheet identification and mid-surfaces abstraction. A solid model is shown in Fig. 1 (a). For mid-surface abstraction, attention will be fo-cused on problems like how to generate a proper shell model such as is shown in Fig. 1 (b) instead of the one in Fig. 1 (c). For the thin sheet identifica-tion, the form of the mid-surfaces is irrelevant – the important issue is how to decompose the model into three parts such as in Fig. 1 (d).

Fig. 1 Comparison between the mid-surface abstraction and the thin sheet identification

3. Mid-surface abstraction in NXThe automatic thin sheet identifying method in this paper is implemented in Siemens NX 9.0 [13]. To isolate a thin sheet region it is first necessary to identify face pairs, which are the source and target faces bounding the thin sheet region. However, in NX this information can only be extracted after the mid-surfaces are explicitly generated. The quality of the actual mid-surfaces is not crucial provided the correct face pairs are returned. The face-pairing concept is simply reviewed below as well as the current mid-surface tools in NX.

3.1 Face pairing

Rezayat [12] initially proposed a technique to ab-stract the mid-surface from a solid model based on the idea of face pairing. Distance criteria and over-lap criteria are used when pairing faces to ensure valid faces are returned.

Let T be the distance between faces while L and H represent the maximum length and height of one face in the face pairs. Distance criteria means the formula below should be satisfied after X is input from the user.

Min(L ,H )/T> X (1)

At the same time, faces that are opposite to each other in a face pair should intersect if one of them is projected onto the other along the normal direction.

3.2 Mid-surface function in NX

A “T” shaped solid model with the mid-surface cre-ated in NX is shown in Fig. 2. For this model, two mid-surfaces are generated which are indicated in dotted lines in Fig. 2a. For each mid-surface, it has bounding faces on each side, based on which the mid-surface is created. These bounding faces are called face pairs. The two face pairs of this “T” shaped model are shown in Fig. 2b and Fig. 2c in solid lines. For each face pair, faces on one side of the mid-surface are named side 1 faces while faces on the other side are named side 2 faces. It is also worth noting that in the latest version of NX, a new method using the maximum diameter of a con-strained inflatable ball (shown in Fig. 2 in dot and dash line) is used to calculate the local thickness.

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Fig. 2 Illustration of the mid-surface function in NX

4. Thin sheet identification

4.1 Overview

This section shows the process of the automated thin thick decomposition for a 3D model. In a strict definition, a thin sheet region is defined as a section of a model where the lateral dimensions are much greater than the local thickness, i.e. a section with a high aspect ratio. In this paper, a region is treated as thin sheet if it is bounded by a valid face pair, which satisfies the distance criteria and overlap criteria. The maximum value X in equation 1 that the user

can define in NX is 3. The identification process of the thin sheet regions is shown in Fig. 3. The face pairs are first obtained through the mid-surface command in NX. The edges of side faces are dis-cretized and imprinted to the mid-surfaces. The im-printed edges are represented in the UV space of the mid-surfaces and the intersection regions of the im-printed edges are calculated through the 2D polygon Boolean operation. Cutting faces are created based on the intersection result and employed to isolate the thin sheet solid regions. Pseudo code for the method to identify thin sheet regions is shown in .

Fig. 3 Overview of the process of the thin sheet identification

Fig. 4 Algorithm for the thin sheet identification process

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4.2 Get face pairs and discretize the edges of the side faces

Face pairs are derived from the mid-surface func-tion in NX through NXOpen APIs using the C# lan-guage and the .NET framework. The faces in the face pairs are represented as 3D polygons. A poly-gon here means an ordered sequence of points which bounds a closed region (see Fig. 5). For a face with holes, it is represented as more than one polygon. Since the thin sheet regions are used for guidance of hexahedral meshing, points in the 3D polygon representation are created along the edges of the side faces with the intervals less than or equal to the user-defined target element length. In order to guarantee that the points are in the right order, e.g. points on the outer loop of the face are in counter clockwise (CCW) order and points on the inner loops (holes) are in clockwise (CW) order, the topo-logy information of the side face is interrogated to determine the orientation of an edge relative to the face it bounds. The algorithm of creating ordered points along the edges of a side face is shown in Fig. 6. The terminology used here is the same as those in Parasolid [14].

Fig. 5 Create the 3D polygon representation of a face (a) a solid face (b) the 3D polygon representation of the face, which is a

list of ordered points (counter clockwise for points on the outer boundary and clockwise for points on the holes)

Fig. 6 Algorithm for generating ordered points along the edges of a side face

4.3 Imprint points on side faces to the mid-sur-faces

The 3D polygon representations of side faces are imprinted to the mid-surfaces. In order to obtain the intersection regions of the imprinted polygons, the imprinted points are represented in the UV para-meter space of the mid-surfaces, transforming the problem from 3D to 2D. An example in Error: Ref-erence source not found shows the transformation of points in 3D Cartesian space to points in 2D UV space.

Fig. 7 Represent 3D polygons in the 2D UV space of mid-sur-faces (a) imprinted points in the 3D Cartesian space (b) points

represented in the2D UV space

4.4 Get the 2D polygon intersection and create cutting faces

The transformation from the 3D polygons to the 2D polygons benefits the use of 2D polygon Boolean operations to obtain the intersection region. Several algorithms [15] about polygon clipping are available among which Vatti’s algorithm has been marked as being efficient and supportive in both convex and concave polygons. This algorithm is employed in this paper and some ready-made library can be found online [16]. For the model shown in Fig. 7a (a thin block and a boss on each side), the top and bot-tom face of the thin region is identified as a face pair (Fig. 7b). Fig. 7c shows the 2D polygons in the UV space of the mid-surface. Points from different sides are represented by circles and crosses respect-ively. In this example, polygons from either side are comprised of an outer polygon (generated from the outer boundary of the face) and a hole polygon (generated from a hole). The intersection result is given in Fig. 7d using Vatti’s algorithm and based on the intersection result cutting faces are created as shown in Fig. 7f. After that, polygon triangulation [17] is performed for the intersection result and the centroid of the largest triangle is obtained (Fig. 7e). The xyz coordinates of the centroid is stored and

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linked to the cutting faces. This point will be used subsequently in the splitting body procedure.

Fig. 7 (a) A thin block with a boss on each side (b) The face pair for the model (c) The 2D polygons on the mid-surface. Points from dif-ferent sides are shown in circle and cross respectively (d) Intersection result of the 2D polygons. (e) Triangulation of the intersection res-

ult and the centroid point of the largest triangle. (f) The cutting faces for this model and the linked point

4.5 Split bodies

The cutting faces generated in the last procedure will be employed to partition the solid bodies. In a body with more than one thin region, after the first split operation it is necessary to determine which of the resulting bodies the second split operation needs

to be made for. It is also necessary to determine which body is a thin sheet region after splitting. For example in Fig. 8, body A is split into three bodies B, C, D and with body C identified as thin sheet (in step 1). Before the next splitting, it needs to be de-termined which body, B or D, will be split.

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Fig. 8 Identification of the body to split before splitting and thin sheet body after splitting; (a) The original solid body A (b) In step 1: three bodies (B, C, D) are generated after the first splitting and body B is identified as thin sheet (c) In step 2: the cutting faces for the

second splitting is highlighted in dashed line. Then Body B is determined to be split since it contains the linked point of the cutting faces. (d) In step 3: Body B is split into two bodies, E and F. Body E is identified as thin sheet body since it contains the linked point.

When a new splitting begins, the point that is linked to the current cutting faces will be interrogated first and the body that contains the xyz coordinates of this point will be passed to split. After splitting, the body that contains the xyz coordinates of this point will be identified as a thin sheet. Therefore, for the

model in Fig. 8, in step 2 if the highlighted dashed faces are the current cutting faces and the circle point is their linked point, body B will then be split and after splitting body E will be identified as thin sheet since it contains that point (in step 3).

5. Results and discussion

The method presented in this paper has been imple-mented using the C# language and .NET framework in Siemens NX 9.0. The input to the algorithm is a solid model and the whole process is automatic. This method is based on valid face pairs and this in-formation is interrogated from the mid surface func-tion in NX through APIs. Several simple models have been tested and the results are demonstrated to be correct.

To further test the proposed method, thin regions of a combustor casing model (half model) are identi-fied. The original and decomposed solid models are shown in Fig. 9 (a) and (b) respectively. Green bod-ies in Fig. 9 (b) represent thin bodies and those in yellow represent the residual thick bodies. The thin sheet identification process for this model takes 27 minutes on a 3.0GHz Intel Core(TM) 2 CPU ma-chine with 8GB RAM. 104 thin regions have been

identified for this model and occupy approximately 90% of the whole model’s volume.

The quality of the final result is related to the qual-ity of the face pairing results. An example of an in-correct face pairing leading to the failing of thin sheet identification is shown in Fig. 10. The original complete combustor casing model was partitioned into two half models before thin sheet identification using a plane about which the model is symmetric. This resulted in a large planar face at the cross-sec-tion on one side shown in Fig. 11 (a). The huge dif-ference in size between this face and the small face of the fin on the other side is believed to be the reason for the face pairing failing. Except for this failure, the quality of other face pairs is good and offers a good base for the subsequent operation.

Thin sheet regions are identified in this paper to re-duce time in hexahedral or hexahedral dominant meshing. These regions can be treated as 2.5D sub-domains and a hex mesh can be obtained by sweep-

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ing a quad mesh from one bounding face to the other. Since the face pairs information is acquired first, it is easy to keep track of the face relationship after splitting and automatically identify the oppos-ite boundary faces of thin regions, which can then be used as source and target faces in a sweeping

method to get hexahedral meshing. This process would gain further benefit if long slender regions are identified later using the method proposed and demonstrated in [7]. The use of hexahedral ele-ments, especially in thin regions will greatly de-crease the DOF of the analysis model [3].

Fig. 9 (a) The original casing model (b) Decomposed model: bodies in green are identified as thin sheet

Fig. 10 Missing of face pairs leads to failure of thin sheet identification

Fig. 11 Theoretical face pairs for the failure region shown in Fig. 10 (a) face on one side (b) face on the other side

6. Future Work This section illustrates another problem in the cas-ing model. As shown in Fig. 12, the yellow body in

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the square is small and has a very acute angle, which compromises the quality of the hex mesh. A better solution for regions like this is given in Fig.13 (the red lines indicate the position of the cutting

faces) and this requires merging of side1 and side2 edges whose imprints on the mid-surface are in close proximity. This will be solved in future work to generate a more appropriate cutting faces.

Fig. 12 The problem with generating sliver body

Fig. 13 An example to show creating better cutting faces: the red lines represent the position of cutting faces

7. ConclusionThis paper is based on the idea of automatically identifying thin sheet regions in a solid model to which good quality hexahedral meshes can be ap-plied. The idea aims at an automated solution so that useful reductions in the amount of manual effort can be accomplished. A method of automatically identi-fying thin sheet regions is proposed in this paper based on face pairing technology. The process is ac-complished in commercial software Siemens NX

using NXOpen APIs. The proposed method follows steps:

Create mid-surfaces and get the face pairs Discretize the edges of the side faces and rep-

resent the side faces as 3D polygons Imprint points in the 3D polygons representa-

tion of faces to the mid-surface Represent the imprinted points in the UV

space of the mid-surface and get 2D polygons Find the intersection region of the 2D poly-

gons and create cutting faces

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Split body and identify thin sheet bodies

An industry combustor casing model has been used to test the proposed algorithm and good perform-ance has been demonstrated in terms of both effect-iveness and quality. Some problems are summarized and problems like generating sliver bodies will be solved in the future work.

AcknowledgmentThis research is financially supported by Rolls-Royce through the TSB GHandI project. The au-thors would like to acknowledge the industrial part-ners for providing the model.

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