Available online at www.sciencedirect.com

Procedia Engineering 00 (2017) 000–000www.elsevier.com/locate/procedia

26th International Meshing Roundtable, IMR26, 18-21 September 2017, Barcelona, Spain

Feature moving operation of hexahedral meshChun Shena, Rui Wanga, Shuming Gaoa,∗, Hiroki Maehamab

aState Key Lab of CAD&CG, Zhejiang University, Hangzhou 310058, PR ChinabGraduate school of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan

Abstract

In the product design process, the finite element mesh regeneration is an inevitable step, but it is time consuming. The mesh editingis a kind of an effective mesh regeneration technique. In this paper, a method for widely moving features of hexahedral mesh isproposed to satisfy application requirement. After the feature moving operation is applied to CAD model, the mesh deformationbased on the mean value coordinates is carried out in the corresponding deformed region of the hex mesh to decrease the possibleinverted elements caused by a wide range deformation. Then, the parameters of generating the density field are extracted from theinitial hexahedral mesh, and used to calculate the local target density field by considering the geometric and topological informationon the deformed region. Finally, based on difference between current density field and target density field of deformed region, thelocal refining and coarsening operations are devised and conducted to make the deformed region compatible with the target densityfield. Experimental results for the mechanical parts verify the effectiveness of proposed method.c© 2017 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of the scientific committee of the 26th International Meshing Roundtable.

Keywords: hex mesh; mesh deformation; feature moving; adaptive meshing.

1. Introduction

Mesh editing refers to the method for generation of deformed mesh model by modifying the initial mesh after theCAD model is modified. In current product design, the design model needs to be constantly modified in order to meetthe requirements of analysis. Therefore, we constantly need to generate a new mesh model according to the modifiedgeometric model. The trial-and-error process is time consuming and it requires a large number of manual operations,so it greatly increases the cost of product design. An effective solution is mesh editing which reuses the initial meshmodel to improve the efficiency.

There are many studies on 3D mesh editing. The mesh deformation [1–3] is an effective method to solve the meshediting tasks. This method is efficient and maintains the affine invariance. However, this method only changes theposition of points, and keeps the topology unchanged. Therefore, many twisted or even inverted elements could appearin mesh deformation. Maehama et al. [4] proposed a quality improvement method for deformed tetrahedral meshbased on Optimal Delaunay Triangulation(ODT) smoothing. In their method, element shape qualities are improvedfrom the mesh boundary to the inside of the mesh, and original mesh properties are recovered in deformed mesh.

∗ Corresponding author. Tel.: +86 571 88206681 ; fax: +86 571 88206680.E-mail address: smgao@cad.zju.edu.cn

1877-7058 c© 2017 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of the scientific committee of the 26th International Meshing Roundtable.

2 Chun Shen et al. / Procedia Engineering 00 (2017) 000–000

Zhu et al. [5] proposed a direct editing method for hexahedral mesh after the CAD model is manually modified. Themethod automatically optimizes the quality of associated hexahedral mesh model through dual operations. Sheffer etal. [6] proposed a method for re-meshing CAD models after parameter modifications to maintain the required sizingand quality for tetrahedral and hexahedral meshes. For deformed tetrahedral mesh, authors used an adaptive methodbased on the ball filling method. On the other hand, for deformed hexahedral mesh, the density of mesh was adjustedby dual operation. Borden et al. [7] proposed the Mesh Cutting algorithm to cut a simple mesh model with the givengeometric model as a reference to construct the complex negative features. However, it is difficult to deal with a widerange feature moving using the existing hexahedral mesh editing methods, which results in generation of poor-qualityelements and may not meet the required mesh density. In addition, in the actual design process, such a large changein model is inevitable, so this paper presents a method for wide range feature moving of hexahedral mesh.

During the mesh editing process, in order to ensure the quality and size of edited mesh model, we need to recal-culate the target density field [8]. The commonly used method for, which defines size functions using a backgroundoverlay grid, was proposed by Zhu et al. [9]. Moreover, this method defines the size function based on curvature andproximity of geometric entities in model. Afterwards, Zhu et al. [10] extended the method of size field generationto consider the pre-meshed geometric entities. Quadros et al. [11] proposed a skeleton-based computational methodto generate a mesh sizing function. The method proposed in [11] adequately analyzes the geometric complexity ofmodel to calculate the local thickness and uses the octree as a background mesh to save memory and to improve queryprocessing.

The adaptive adjustment of mesh model is a key step of mesh editing which tends to achieve the target density. Therefinement template [12,13] is often used to refine a given region of structured hexahedral mesh, but local refinementdomain determination and mesh size gradient control cause problems. In order to solve these problems, Bordenet al. [14] and Tchon et al. [15] developed a pillowing-based hexahedral refinement method that can be applied tounstructured mesh. Through iteratively pillowing operations, the required density field is met. The hexahedral meshcoarsening can be achieved by sheet extraction operation, but if we extract the entire sheet, that will influence theentire mesh extensively. Shepherd et al. [16] developed a localized sheet generation method in order to extract sheetin a local domain.

A method capable for widely moving features of hexahedral mesh is presented in this paper. Its objective is togenerate a high quality hex mesh automatically and efficiently after certain features of the original hex mesh arewidely moved due to requirement. In order to achieve this goal, a mesh deformation based on mean value coordinatesis firstly conducted in the deformed region, which is the region affected by feature moving. Then, the target densityfield of deformed region is calculated according to the geometric and topological information on the initial hex mesh.Finally, an adaptive local adjustment of hex mesh is performed to make the deformed region meet the target densityrequirement.

2. Algorithm Overview

In this paper, we present a method of moving a feature on a plane for hexahedral mesh. The input of the algorithmconsists of hexahedral mesh model, geometric model, specified feature, the features moving direction and distance.The output of the algorithm is the hexahedral mesh, which is locally regenerated after feature moving.

In order to enable the algorithm to handle the case of wide range feature moving, three critical problems need tobe addressed: 1. Reducetion of inverted elements during deformation of deformed region; 2. Reasonable calculationof target density field of deformed region; 3. Local adjustment of mesh density of deformed region to meet therequirement. With the aim to solve the first problem, we use the mean value coordinates to achieve mesh deformationby reasonable construction of control mesh of the deformed region and to avoid the inverted elements. In orderto solve the second problem, we use the parameters of generating the size field extracted from original mesh andlocal geometric and topological information to calculate the target density field of deformed region. Lastly, in order tosolve the third problem, those regions that need to be refined or coarsened are accurately determined by comparing thecurrent density field with the target density field, and they are adjusted by well devised local refining and coarseningoperations.

As it is illustrated in Fig. 1,the proposed method mainly consists of three following steps:

Chun Shen et al. / Procedia Engineering 00 (2017) 000–000 3

Target mesh density field

Mesh model after mesh deformation

Input: original mesh model

Calculation of density field of Deformed

region

Mesh Deformation based on Mean value

coordinates

feature moving

Local mesh refinement

Local mesh coarsening

Output: Improved mesh model

Two different sheets after chord collapse

One sheet for local refinement

Mesh model after mesh optimization

Deformed meshCross section of deformed

region

Fig. 1: The flowchart of feature moving of hexahedral mesh.

1). Mesh deformation based on mean value coordinates;2). Calculation of target density field of deformed region;3). Local mesh coarsening and refinement.

3. Mesh deformation based on mean value coordinates

In order to re-mesh the hexahedral mesh locally after feature moving, we limit the geometric and topologicaladjustment in the deformed region. Namely, we firstly perform mesh deformation in the deformed region in order tokeep the consistency between the mesh and the geometric model.

In general mesh deformation case, we move the vertices associated with feature followed by feature movement andconduct a geometric optimization in the deformed region. However, this scenario leads to many distorted and invertedelements, which make a subsequent adjustment of mesh density difficult.

In order to solve this problem, the paper presents a method for mesh deformation based on mean value coordinates(MVC) in the deformed region. The MVC maintains the affine invariance and it can keep the elements in the deformedregion un-inverted. Therefore, the key point is to construct a resonable control mesh of MVC. Our strategy is togenerate control mesh by considering feature shape and editing information.

As it is illustrated in Fig. 2, the mesh deformation consists of two parts:

1). Cage generation;2). Mesh deformation in the deformed region.

3.1. Cage generation

In order to conduct the mesh deformation in the deformed region by MVC, we first need to generate a cage. Sinceit is difficult to determine the precise cage at the beginning, a cuboid is constructed as an initial cage to estimate the

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Intersection Operation

Initial cage Final cage

Cage generation

Cage deformation

Control mesh before deformation

Control mesh after deformation

Hex mesh before deformation

Hex mesh after deformation

Mesh deformation

Fig. 2: The flowchart of mesh deformation in the deformed region.

deformed region. Then, the initial cage is modified to conform to the CAD model in order to obtain the final cage,which represents the real deformed region.

3.1.1. Initial cage generationAfter the user selects the feature F and moving distance D in ~x direction as shown in Fig. 3(a), we determine the

initial cage using feature orientation, length and location. The specific steps are as follows:

(1) Determine three perpendicular directions ~x, ~y,~z of the cuboid. ~x is the feature moving direction, ~y is the negativenormal direction of the plane where feature is located and ~z is the cross product of ~x and ~y.

(2) Determine the length in all three directions. The cuboid generation has three requirements: First, it should bebig enough to embrance the feature in three directions; Second, it should be big enough to avoid that elementsin the cage are heavily compressed; Third, it should be kept away from the boundary of feature that is helpfulto local refinement. The lengths Cx,Cy,Cz in three directions are calculated as follows. In ~x direction: Cx =

2D + S average + Lx; in ~y direction: Cy = D f + D if the feature is concave, Cy = D if the feature is convex; andin ~z direction: Cz = 2D + Lz; wherein S average is the average size of the original mesh which can help to find agood quad set to perform mesh refinement presented in Section 5, Lx is the feature length in ~x direction. D f isthe depth of feature. Lz is the feature length in ~z direction.

(3) Determine the cuboid location. Asumming that node O is the farthest position from the feature on the plane in ~xdirection, according to the requirements in (2), we can define a front face of the cuboid by three points A, B,C,which are calculated by:

A = O − ~x · S average +~z · (Cz/2)B = O − ~x · S average −~z · (Cz/2)

C = B + ~y ·Cy

(1)

Then the initial cage can be determined by sweeping the front face Cx in ~x direction.

3.1.2. Final cage generationIn section 3.1.1, the geometric model and context of moved feature are not taken into account, so a further process-

ing is needed. First of all, the intersection between initial cage and original CAD model is conducted to keep the mesh

Chun Shen et al. / Procedia Engineering 00 (2017) 000–000 5

(a)

Front face

x

(b)

x

y

z

Feature F

DSaverage

O

A

B

C

Front face

LxLz

(c)

Fig. 3: Cage generation: (a) initial cage generation; (b) the nodes of control mesh to be deformed; (c) the control mesh after deformation.

vertices inside the model during mesh deformation. Otherwise there will be many inverted elements in mesh. Then,we check the intersection result to discard the parts unrelated to the moved feature in order to increase efficiency.

3.2. Mesh deformation in the variational area

A mean value coordinate(MVC) in 3D space expresses a point in a closed triangle mesh as a combination of thevertex positions of the closed mesh [18]. The closed triangle mesh of mean value coordinates is called a control mesh.The point inside a control mesh can be represented as a weighted sum of vertex positions of control mesh. Oncethe mean value coordinate is determined, the positions of vertices in control mesh can be changed by deforming thecontrol mesh. The point pεR3 inside a control mesh can be represented as a weighted sum of the vertex positions {vi}

of control mesh [2]:

p =∑m

i=1 wivi (∑m

i=1 wi = 1) (2)

The control mesh of MVC is generated by trianglating the cage. The elements inside the cage can be kept un-inverted due to the affine invariance of MVC during mesh deformation.

Hence, the key issue is to deform the cage after feature movement. Consequently, the front face can be movedfor distance D along ~x direction to deform the cage. In this process, some nodes of control mesh should be movedsimultaneously. These nodes are located between the front face before moving and the front face after moving (seered points in Fig. 3(b)). We deform the control mesh by moving these nodes for distance D along the ~x direction(see inFig. 3(c)). After changing the shape of control mesh, the new position of vertices in the cage is recalculated. Althoughthere are few distorted elements around the cage after mesh deformation, the elements in the cage are kept uninverted.

4. Calculation of the target density field of deformed region

After mesh deformation based on MVC, the density field of deformed region is not reasonable, so we need todetermine the reasonable density distribution in deformed region which is called the target density field.

Generally speaking, the density field of 3D model is directly related to the density on the surface of the model.That means that density on the surface of deformed region has an important impact on calculation of the size in theentire region. Therefore, we firstly need to determine the density field of the surface of deformed region and then tocalculate the density inside deformed region.

4.1. Calculation of the density on the surface of deformed region

To recover the sizing properties of the original mesh model, we should extract the parameters of generating thedensity field and use the information to recalculate the target density field. The specific steps are as follows:

(1) Restore the density information and extract the parameters from the original mesh model. The octree is chosen asa background grid because of efficiency. If the point is in the octree grid, then the size of the point is calculated by

6 Chun Shen et al. / Procedia Engineering 00 (2017) 000–000

tri-linear interpolation function [17]. In addition, the parameters of generating the original density field should beextracted, including maximal spanning angle θ, maximal size S max, minimal size S min, global mesh size S global.θ can be calculated by θ = max{2 ∗ arcsin( s∗k

2 )}, where (k is the curvature of curve at the position of vertex, s isthe size of vertex). For each vertex, we calculate the average length of its adjacent edges. Moreover, S max, S min

and S global are set as the maximal size, the minimal size and the highest frequency size, respectively.(2) Get sample points on the surface and calculate the size. Firstly, we need to triangulate the surface of the cage to

get the sample points. The sample points on the surface can be divided into two categories: points on the originalCAD model surface, and points inside the original CAD model.For the first category, we recalculate the size according to the parameters extracted from original model. Wecalculate the size of the cage edge, and if the sample point is on the curve edge, the size is 2 ∗ sin(θ/2)/k (k isthe curvature of curve at the position of vertex), and if the sample point is on the straight edge, the size is equalto S global. Since the influence zone of the curve edge is not limited to the curve edge itself, we spread the size ofcurve edge to the area which is found by extending the adjacent node for three times. The size of each layer iscalculated as follows:

s = s0 ∗ gn(n = 1, 2, 3) (3)

where s0 is the size of the curve edge, g is the gradient for smooth transition (g = 1.5), and n is the topologylayer during extention. The size of other sample points is equal to S global.For the second chategory, we calculate the density according to the original mesh model because there are nogeometric entities available for considering.

4.2. Calculation of the density inside the deformed region

Compared with the tetrahedral mesh, the hexahedral mesh has a global structure. Therefore, the hexahedral meshgenerated by existing methods such as sweeping method and block decomposition method might have layered struc-ture. Thus, the density field has the same property, which can ensure the quality of mesh. Therefore, we need toidentify the region with layered structure so that the existing layered structure can be maintained. The region withlayered structure denotes the structure wherein hexahedral elements are structurally arranged in a certain direction.The method for calculation of density inside the deformed region is as follows:

(1) Determine the region with layered structure in three steps:

a. Find the singular edge strings which are parallel to the normal of the plane. A singluar edge is the edgeinterior to the mesh adjacent to something other than four hexahedra [24]. Since the direction of layeredstructure can be determined by singular edges in the deformed region, we need to find singular edges first.However, we only consider the direction perpendicular to the plane where the moving feature is locatedbecause that direction is influenced by the feature movement. So, all the singular edges with an angle of notmore than 25◦ from the normal of plane are found and sorted into sets form singular edge strings. However,a singular edge string may not simply-connected, that is to say, all nodes on any singular edge string is notadjacent to either one or two singular edges. In order to get simply-connected singular edge strings (see inFig. 4(b)), we split at the node which is adjacent more than two singular edges.

b. Find the region with layered structure. The regions with layered structure are formed by collecting hexa-hedra adjacent to each singular edge string, and iteratively expanding until other singular edges appear (seeFig. 4(c)). While expanding, checks are required to maintain a layered structure in each region. In order toguide the generation of target density field, we move the regions with layered structure for distance D in ~xdirection. The new area will keep the layer structure in the density field.

(2) Calculate the density field inside the deformed region. Namely, different strategies are used based on whetherthe region has layered structure or not. If there is a layered structure region, the density field inside the deformedregion will be diffused from the surface according to the layer direction. If two sides of layer direction haveconflict, then we choose the smaller size as the size in that direction. If the region does not have layered structure,the inside size of deformed region is calculated by interpolating from the size on the surface because the size in

Chun Shen et al. / Procedia Engineering 00 (2017) 000–000 7

(a) (b) (c)

Fig. 4: The determination of layered structure regions: (a) all the hexahedral elements in deformed region; (b) all the singular edges parallel to thedirection perpendicular to the plane; (c) a region with complex layered structures(different layered structures with different colors).

0

0.5

0.7

1.3

1.5

>1.5

ratio

Fig. 5: The ratio between current density and target density in the deformed region.

the volumetric mesh elements mostly dependes on the size of boundary mesh elements. Finally, the size of thewhole deformed region is smoothed in order to get a smooth target density field.

5. Local mesh coarsening and refinement

In order to achieve the ideal size in the deformed region, size adjustment is carried out according to the ratiobetween current density field and target density field of the deformed region, which is defined by:

ratio =current densitytarget density

The current density is the density of hex mesh after mesh deformation. The target density is calculated based ontarget density field in Section 4. The size ratio greater than 1.3 indicates that the element in the grid is too largeand should be refined. The size ratio less than 0.7 indicates that the element in the grid is too small and should becoarsened as shown in Fig. 5. The local mesh coarsening and refinement are conducted iteratively until the mesh sizemeets the requirement. In each iterations, the threshold is changed from high level to low level such that the areas areadjusted according to the priority.The local mesh refinement is conducted first because the elements in one side of themovement after deformation are too coarse.

8 Chun Shen et al. / Procedia Engineering 00 (2017) 000–000

(a) (b) (c) (d)

Fig. 6: The process of sheet inflation (a) hex mesh before sheet inflation; (b) quad set for sheet inflation; (c) new sheet created by inflationg thequad set; (d) hex mesh after sheet inflation.

5.1. Local mesh refinement

The local refinement tends to locally increase the size of stretched area. We use sheet inflation and pillowingoperation to achieve the goal of mesh refinement. Sheet inflation takes a continuous quad set as input and generatesa new sheet by inflating the quad set. The process of the sheet inflation in Fig. 6. shows the quad set is critical forthe sheet inflation because it determines the position, shape and topology of the sheet [17]. Pillowing is a simpleand rough strategy compared with sheet inflation. So we prefer to use sheet inflation and the determination of quadset is the key to the problem. After the local region which should be refined is determined, the difficulty is how toconstruct a qualified quad set under the given constraints. Since the region is heavily stretched after a wide rangefeature movement, we need to identify the quad sets that are perpendicular to the stretched direction for sheet inflation[22,23]. If the region which should be refined is near the model boundary, the outline of the good quad set for inflationshould be on the mesh boundary (see Fig. 7(c)). If the region is almost inside the model, then we perform the pillowingoperation on region locally.

It is very difficult to find the effective quad set perpendicular to the stretched region, so we first find the outline ofquad set on the boundary and then find the quad set. Firstly, the heavily stretched elements on one feature side areidentified as shown in Fig. 7(a). Secondly, to avoid the difficulty of determing a qualified inflatable quad set directly,we first find the boundary outline of the quad set by path searching. The boundary edges should meet the followingrequirements: 1) the angle between edge and deformation direction should be between 50◦ and 125◦; 2) the differencein size of two edge sides should greater than (D − S global) (see red lines in Fig. 7(b)). Thirdly, to get a continousboundary ouline of the quad set, we use path searching [19] by connecting the boundary edges found in 1) (see yellowlines in Fig. 7(b) and Fig. 7(c)). Fourthly, we use the max-flow-min-cut algorithm on a directed graph to construct thevalid quad set to meet the boundary constraints(see in Fig. 7(d)). If we construct a graph where the nodes stand for aquad or a quad set and the edges stand for the adjacency relationship between the quads or quad sets, then to do localseparation is similar to find a cut set on this graph. Hence, the max-flow-min-cut algorithm can be used to help us findnecessary edges to make the boundary outline satisfy the local separation [19]. Finally, the region is initially refinedby repeating the sheet inflation for D/S global times(see in Fig. 7(e)).

After the heavily stretched region is refined, due to the feature moving, we need to handle general cases. Therefinement region, which is determined by ratio in descending order can be refined by pillowing operation. Therefinement operation is terminated when the percentage of elements whose ratio is above 1.3 is less than 8%.

5.2. Local mesh coarsening

The goal of local mesh coarsening is to increase the size by deleting compressed elements from the mesh. If thepercentage of compressed elements in the entire sheet is greater than 70%, it is considered as completely compressedand the sheet is extracted. On the other hand, if the proportion of compressed elements is too small, it is consideredas partially compressed. Shepherd et al. [16] proposed a local mesh coarsening method to generate a local sheet in

Chun Shen et al. / Procedia Engineering 00 (2017) 000–000 9

(a) (b) (c)

(d) (e)

Fig. 7: The refinement process on one side of moved feature: (a) refinement region; (b) initial boundary edges near the vertical to the movingdirection(in red), other boundary edges found by A* algorithm; (c) two complete loops; (d) two quad sets; (e) new insertion sheet for refinement.

(a) (b) (c)

Fig. 8: Local sheet extraction: (a) Two sheets for chord collapse; (b) Two different sheets after chord collapse; (c) The sheet in green is extractedand the sheet in red is left.

a specific region for extraction. We used the coarsening method proposed by Zhu et al. [5] as shown in Fig. 8. Thecoarsening operation is terminated when the percentage of elements whose ratio is below 0.7 becomes less than 8%.

6. Experimental results

The proposed method was implemented using C++ based on ACIS geometric engine. The preliminary experi-mental results are illustrated in Fig. 1, Fig. 9, Fig. 10 and Fig. 11. The number of hex elements, scaled Jacobian andrunning time are presented in Tab. 1. The adaptation results of Example 1,2,3 and 4 are represented in Tab. 2, Tab. 3,Tab. 4 and Tab. 5 repectively. The comparison of different target density fields of Example 1 is shown in Fig. 12. Thetime was measured on a PC with Intel Core i7 3.60GHz CPU and 8GB RAM.

The original adaptive hexahedral mesh generated by Abaqus is presented in Fig. 1 and Fig. 10. The hex mesh pre-sented in Fig.9 is an anisotroic mesh to the certain extent and it was generated by the block decomposition method [21].Two concave features were moved for 80cm simultaneously. The union of each cage was used for mesh deformation.The hex mesh presented in Fig. 11 is generated by the grafting method. The boss is not perpendicular to the plane andthere is no layered structure in deformed region.

10 Chun Shen et al. / Procedia Engineering 00 (2017) 000–000

(a) (b) (c)

(d) (e) (f)

Fig. 9: The major process of Example 2: (a) input hex mesh; (b) mesh after mesh deformation; (c) target density field; (d) mesh after refining theheavily stretched region; (e) deformed mesh; (f) cross section of mesh in deformed region.

Even though the quality of deformed mesh is not ideal because of chord collapse, but most of elements satisfy therequirement of the target density field from Tab. 2, Tab. 3 and Tab. 4. In Fig. 12, the target density field in Fig. 12(b)which we calculate is denser than the density field in Fig. 12(a) from hex mesh generated from Abaqus. In ourmethod, we determine the gradient is 1.5 around the curve edge, so the size is smaller than the size of mesh generatedby Abaqus. Besides, the density field in Fig. 12(c) is more reasonable than density field in Fig. 12(d), so we prefer toconsider the layered structure in the deformed region when it is possible.

7. Conclusion and future work

This paper presents a novel approach to feature moving of hex mesh, which is capable of dealing with a wide rangefeature movement. Compared to the existing methods, the proposed approach has the following advantages:

1) By constructing a proper cage according to the editing information, a mesh deformation based on the mean valuecoordinates remarkably reduces the number of inverted elements caused by deformation in the deformed region.

Table 1: Scaled Jacobian and running time of results.

Hex Num. (Before\After)Scaled Jacobian of

original mesh (Mean\Min)Scaled Jacobian of

deformed mesh (Mean\Min)Running time(s)

Example 1 6189\8583 0.916\0.548 0.850\0.148 21.69Example 2 30668\37112 0.981\0.252 0.877\0.104 152.39Example 3 19655\22003 0.921\0.374 0.887\0.109 122.74Example 4 2596\5370 0.936\0.561 0.938\0.337 11.72

Table 2: Adaptation results of Example 1.

Size ratio Percentage of total Size ratio Percentage of total<0.5 0% 1.0-1.3 51.50%

0.5-0.7 1.27% 1.3-1.5 7.29%0.7-1.0 39.94% >1.5 0%

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Table 3: Adaptation results of Example 2.

Size ratio Percentage of total Size ratio Percentage of total<0.5 0% 1.0-1.3 40.86%

0.5-0.7 7.98% 1.3-1.5 7.24%0.7-1.0 42.02% >1.5 1.9%

Table 4: Adaptation results of Example 3.

Size ratio Percentage of total Size ratio Percentage of total<0.5 0.5% 1.0-1.3 49.86%

0.5-0.7 3.43% 1.3-1.5 2.84%0.7-1.0 56.25% >1.5 0.13%

Table 5: Adaptation results of Example 4.

Size ratio Percentage of total Size ratio Percentage of total<0.5 0% 1.0-1.3 80.02%

0.5-0.7 0% 1.3-1.5 7.58%0.7-1.0 12.41% >1.5 0%

(a)(b) (c)

(e) (g)(f)

(d)

(h)

Fig. 10: The major process of Example 3: (a) input hex mesh; (b) mesh after mesh deformation; (c) the cage before deformation; (d) cage afterdeformation; (e) target density field; (f) mesh after refining the heavily stretched region; (g) deformed mesh; (h) cross section of mesh in deformedregion.

2) By analyzing the sizing properties of original mesh, the calculated density field of local deformed region becomesmore precise.

3) By comparing the current density field with the target density field, the determination of regions that shouldbe coarsened and refined is more accurate. Moreover, high quality local refining and coarsening operations areconstructed and used to achieve the hex mesh with target mesh size.

There are several limitations of our approach that need to be overcome in the future. Firstly, the moved featureis currently limited to the feature on the plane, therefore we need to consider other complex situations in the future.Secondly, the method is limited to the feature moving operation, thus we need to extend it to other types of featureediting operations including the parameter modification of feature, feature deletion, etc. Thirdly, we will work ondevelopment of a more robust method for refining and coarsening of the local region in order to adjust the mesh sizemore efficiently. Finally, since our current method is hard to achieve anisotropic sizes, we will study the method ofcapturing anisotropy sizes.

12 Chun Shen et al. / Procedia Engineering 00 (2017) 000–000

(a)(b) (c)

(d) (f)(e)

(a)

Fig. 11: The major process of Example 4: (a) input hex mesh; (b) mesh after mesh deformation; (c) target density field; (d) mesh after localrefinement; (e) deformed mesh; (f) cross section of mesh in deformed region.

(a) (b) (c) (d)

Fig. 12: The comparison of different target density fields: (a) our calculated target density field; (b) density field extracted from the mesh modelgenerated by Abaqus; (c) cross section of our target calculated density field; (d) cross section of target density field calculated by interpolating fromthe size on the surface.

Acknowledgements

The authors are very grateful to the financial supports from NSF of China (61572432).

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