Automatic Thin Sheet Identification of Complex Model for Hexahedral Dominant Meshing

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:

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 distance 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 enables 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 ultimately used to isolate the thin sheet regions.

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

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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 assembly level in order to acquire more precise simulation results. Taking the aero engine for example, in order to gain an accurate assessment of the engine 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 dynamic analysis (e.g. using LS-Dyna), the explicit integration 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 hexahedral element is the preferred choice compared to tetrahedral 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 characteristic length of the smallest element. Hexahedral elements have larger time step than tetrahedral elements. Second, 4-10 times more tetrahedral elements are required to the mesh the same domain as hexahedral elements, which means a huge increase of DOF. Controlling the DOF is significantly important in nonlinear dynamic analysis at a whole engine 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 required.

Unlike tetrahedral meshing, no robust automatic algorithms 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 manually 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 decomposition process often outweigh the need for accuracy and computational speed. The vital contribution of the user is difficult to achieve automatically.

Since a total removal of user intervention is not possible 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 work

Decomposition 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 geometry is first generated to decide the local thickness, followed by the creation of a 2D MO which is employed to approximately indicate the lateral lengths. Areas with an aspect ratio (lateral length/local thickness) 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 opposite 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 extending 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 applied generally in any software. Mixed solid element finite element models are generated from the thin sheet and long slender decomposition. In addition, work by Nolan [8] and Tierney [9] have utilized 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 construction graph is generated during a recursive process to decompose a model into extrusion primitives. 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 dimensional reduction, which transformed 3D solid model into either a stiffened shell model or a mixed dimensional model [9-11]. Although the start point of identifying face pairs is the same, there are some differences between the problems. In the mid-surface abstraction, the pivotal issues are how to generate mid-surfaces patches and how to connect (extend/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 appropriate faces to isolate these regions. Besides, mid-surface abstraction tools are available in many packages 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 interested in thin sheet identification and mid-surfaces abstraction. A solid model is shown in Fig. 1 (a). For mid-surface abstraction, attention will be focused 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 identification, 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 NX

The 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 abstract the mid-surface from a solid model based on the idea of face pairing. Distance criteria and overlap 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 created 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 constrained 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

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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 discretized and imprinted to the mid-surfaces. The imprinted edges are represented in the UV space of the mid-surfaces and the intersection regions of the imprinted 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. 4.

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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 function in NX through NXOpen APIs using the C# language and the .NET framework. The faces in the face pairs are represented as 3D polygons. A polygon 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 topology 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)