Hexahedral Mesh Quality Improvement with Geometric Constraints

Hexahedral mesh is of great value in the analysis of mechanical structure, and the mesh quality has an important impact on the efficiency and accuracy of the analysis. This paper presents a quality improvement method for hexahedral meshes, which consists of node classification, geometric constraints based single hexahedron regularization and local hexahedral mesh stitching. The nodes are divided into different types and the corresponding geometric constraints are established in single hexahedron regularization to keep the geometric shapes of original mesh. In contrast to the global optimization strategies, we perform the hexahedral mesh stitching operation within a few local regions surrounding elements with undesired quality, which can effectively improve the quality of the mesh with less consuming time. Several mesh quality improvements for hexahedral meshes generated by a variety of methods are introduced to demonstrate the effectiveness of our method.


Introduction
Finite element analysis has been widely used in the field of engineering applications, such as manufacturing process optimization, structural strength analysis and biomechanical simulation [1].The Hexahedral element is the primary element in 3D finite element simulation of bulk forming process [2], which is often used in process simulation of casting and forging.Its mesh quality directly affects the efficiency and accuracy of numerical analysis results.Therefore, the optimization of element quality is an issue that must be thoroughly studied in the field of finite element analysis.After nearly thirty years of development, mesh generation algorithm has been greatly developed, which can generate surface meshes and tetrahedral meshes with high quality.However, there are still many challenges in the generation of hexahedral meshes for complex solids, especially the quality of generated hexahedral meshes is still far from the requirements of finite element simulation [3].Therefore, the emphasis of this paper is to improve the mesh quality of hexahedral meshes.
In recent years, a variety of methods on hexahedral mesh smoothing have been developed.Some of the existing methods include: (1) improving quality by changing the mesh connectivity, (2) mesh smoothing by geometric operations, (3) mesh optimization by various energy functions.Gridbased method is a fully automated hexahedral mesh generation method, but the quality of surface elements is poor.To solve this problem, Sun et al. [4] proposed a grid-based quality improvement algorithm, which combines curvaturebased Laplacian node smoothing and topological optimization.A fundamental hexahedral mesh is introduced in paper [5] to convert boundary sheets of existing hexahedral meshes to fundamental sheets by flipping operations.Re-meshing [6] is also a method to improve the quality of meshes, which improves the quality of local meshes through boundary feature recognition, node smoothing and topology-based mesh repair operations.Wang et al. [7] presented a novel topological optimization method for hexahedral mesh by sheet adjustment guided by frame filed.The implementation of topology based hexahedral mesh quality improvement is very complex and difficult because of its specific connection.The most common used technology in geometry-based methods is Laplace smoothing [8], in which nodes are iteratively moved to the geometric center of their neighborhood.This method is easy to implement and has high efficiency.However, for some complex models, especially when the surface has local concave features, it is difficult to guarantee the mesh quality and easy to generate invalid elements.In order to ensure the validity of the smoothed elements, Freitag [9] proposes a smart Laplace method, which only displaces nodes that can improve mesh quality.Leng et.al.[10] presented a geometric flow-based mesh smoothing algorithm for segmented quadrilateral/hexahedral meshes, in which different types of nodes are optimized via different geometric flow driven strategies.The geometric element transformation method (GETMe) proposed by Vartziotis et al. [11,12] has been proved to be effective for hexahedral mesh smoothing, which used a simple element regularizing transformation to achieve mesh optimization.While geometry-based mesh quality improvement methods are simple and easy to complement, they do not consider the integrity of the mesh, and ignore the shape features of the surface mesh, which weakens the effect of mesh quality improvement.
Optimization based methods perform the quality improvement by minimizing an energy function defined by different quality evaluation criteria.Knupp [13] uses the condition number of Jacobian matrix as the quality metric to create global smoothed cost function, and obtains the global optimization mesh by conjugate gradient and line-search method.With the same objective function, particle swarm approach [14] is used to optimization the global mesh quality.Since then, the local Gauss-Seidel iteration [15] and some nonlinear methods [16] are introduced to untangle and smooth the hexahedral meshes.Livesu et al. [17] presented an untangling method by minimizing an objective function defined in terms of the edge cone descriptor that indirectly measures the distortion of the hexahedral element.With this descriptor, an optimization strategy [18] by minimizing an angle-based energy function is performed on local regions with poor quality hexahedral elements.Vartziotis and Bohnet [19] used a damped oscillation system to perform discrete geometric triangle transformation to relocate nodes.Peng et al. [20] proposed a local-to-global mesh quality improvement method by a combination of a geometric transformation based local regularization and a volumetric Laplacian based global optimization.While the optimization-based methods and local-to-global smoothing algorithm are effective for most hexahedral meshes, the solution of linear equations in the global optimization process is very time-consuming, especially for the meshes with a large number of nodes, even some calculation errors may occur.
Previous local-to-global mesh quality improvement method considers the geometric characteristics of the mesh when optimizing the mesh as a whole, and cannot perform the mesh optimization locally, resulting in the need to solve a very large energy matrix.The limitations of previous hexahedral mesh optimization algorithms mainly fall into three categories: easy to produce distorted elements, ignoring surface features, and time-consuming global optimization.This paper proposes an optimization strategy from element to local area and then to the global.Four node types are defined to accurately describe the surface geometric features.And then, the energy equation based on geometric constraints is solved to obtain high-quality regular hexahedron elements.The improved algorithm consists of the following steps: (1) Node classification: classified all nodes into four categories (internal nodes, surface smooth nodes, surface crease nodes and surface corner nodes) by its normal voting tensor; (2) Element clustering: hexahedral elements with quality lower than the user-specified threshold (i.e., elements are connected with each other) are clustered into several local regions; (3) Single element regularization: Convert each hexahedral element to the optimal shape without changing the geometric features by using a geometric constraint-based element regularization algorithm; (4) Global optimization for local regions: to accelerate the computation, we perform the global optimization only in those local regions with element whose quality is lower than the user-specified threshold.Contributions of this paper are summarized as follows: • A progressive mesh quality improvement strategy from single element, local region to global model is proposed, which avoids the global Equation solving of large-scale grids and speeds up the calculation.• Four types of geometric constraints are defined, and a geometric constraints-based regularization algorithm for single hexahedron is presented to improve the quality of a single element.
The remainder of this paper is organized as follows.In Sect.2, we introduce the details of the hexahedral mesh quality improvement algorithm.Several mesh-smoothing examples are given in Sect.3, followed by our conclusions in Sect. 4.

Hexahedral Mesh Quality Improvement
The goal of hexahedral mesh quality improvement is to produce a more regular mesh in the premise of ensuring the shape of the original geometry.In order to preserve the features of the original hexahedral mesh, each node can be classified into either an internal node, a surface smooth node, a surface crease node or a surface corner node.Different from the previous local-to-global mesh optimization algorithm, our method adopts different optimization strategies for different regions.For most regions with relatively highquality hexahedral elements, we use the local hexahedral regularization method based on geometric constraints to keep the geometric features of the original model and avoid the high time-consuming caused by the global optimization operation.For local regions with low-quality hexahedral elements, the global operation is added to ensure the validity of the improved elements.For most regions with relatively high-quality hexahedral elements, only the local hexahedral regularization method is needed to accelerate the calculation speed of mesh quality improvement.The process of our algorithm is listed in Algorithm 1.

Node Classification
The premise of mesh quality optimization is that the shape of the original model cannot be changed, so the displacement of the nodes needs to be constrained.As shown in Fig. 1, we classify the nodes into four categories: (1) Internal nodes inside the hexahedral mesh model, whose displacement is not limited.(2) Surface smooth nodes are nodes with lower curvature on the surface of hexahedral mesh, and can only move on its tangent plane.(3) Surface crease nodes are those with high curvature in only one direction on the surface of hexahedral mesh, and they are allowed to move only along the crease direction.
(4) Surface corner nodes are the intersections of at least three sharp edges that are confined to the initial position.In this study, the types of nodes are identified by the normal tensor voting theory [21,22].Let v i represents a surface node of the hexahedral mesh and N f (v i ) denotes its adjacent surface elements, the normal tensor T is defined by weighted unit normal vectors of its neighbor surface elements: here n l is the unit normal vector of its neighbor surface element t i,l in N t (v i ), and ω l is a weight defined by: where A(t i,l ): area of element t i,l , A max : maximum element area among N f (v i ), g l : distance from v i to the barycenter of element t i,l , σ: edge length of a cube that encircles the adjacent elements of each vertex.
The symmetric positive semidefinite matrix T vi can be decomposed into the following form: where λ 1 , λ 2 and λ 3 are eigenvalues of T vi and d 1 , d 2 and d 3 are the corresponding unit eigenvectors of λ 1 , λ 2 and λ 3 respectively.We can classify surface vertices into smoothing, crease and corner nodes by relationships of the eigenvalues, described as: (1) Smooth node: λ 1 is dominant, λ 2 and λ 3 are close to 0.

Fig. 1 Node classification
Here, the parameters ε and η are used to balance the influence of noise on feature detection, which are both set to 4.0 in our experimental tests for good classification results.

Geometric Constraints-Based Element Regularization
In the previously proposed local-to-global mesh optimization algorithm, each single element is converted to its regular form by using single geometric transformation.However, the constraints of geometric features are not taken into account in the local regularization operation, which results in the inappropriateness of this method in some regions where global optimization is not required.We propose a hexahedral mesh regularization algorithm based on geometric constraints.The first step is to best align an equilateral hexahedral element of equal volume to a given element.Then, according to the type of nodes, corresponding geometric constraint equations are established, including point constraints, plane constraints and vector constraints.Finally, element regularization is expressed as a quadratic optimization problem, in which the minimum energy consists of two parts: the edge direction constraints of the equilateral element and geometric constraints established according to the type of nodes, and a linear solution can be used to obtain a regularized hexahedral mesh.Given a hexahedral element E = (v 1 , …, v 8 ), v i = (x i , y i , z i ) ∈ R 3 is the nodes of the given element, an equal volume of equilateral hexahedral element E′ can be constructed.Our goal is to find a transformation that best superposes the equilateral hexahedral elements E′ with E. The iterative closest point (ICP) algorithm [23] is adopted to perform the alignment, in which the source equilateral hexahedral elements E′ is transformed to target hexahedral element E by minimizing the Euclidean distance between the source element and their corresponding target element.The mean square objective function to be minimized is: where R is the rotation matrix and T is the translation matrix to transform the equilateral hexahedral elements E′ to E (see Fig. 2b).In order to obtain the minimum value of equation [4], scholars have proposed a variety of non-iterative optimization methods, such as singular value decomposition method, quaternion method, etc.Here we use singular value decomposition method to get the rigid transformation matrix R and T.
In order to keep the shape of the original hexahedral mesh and improve the quality of hexahedral element as much as possible, the equilateral hexahedral elements E′ is adjusted to E″ by minimizing an energy function so that the position of node satisfies the geometric constraints.The first term of the energy function is the edge direction constraints of the equilateral element, ensuring the high quality of the regularized element E″.For each edge e″ k,j = v″ k − v″ j of the regularized element E″, it is parallel to the edge e′ k,j = v′ k − v′ j of the equilateral element E′, where v″ i = (x″ i , y″ i , z″ i ) ∈ R 3 is the nodes of the regularized element E″.It can be formulized as: where e′ k,j = (e x , e y , e z ) can be easily calculated by the equilateral element E′.Equation ( 5) can be written as the following equations: Hence, the simultaneous equations of all edges can be obtained, which is equivalent to the matrix form: where K is a 36 × 24 matrix; X E = {x″ 1 , y″ 1 , z″ 1 , …, x″ 8 , y″ 8 , z″ 8 } T is a 24 × 1 column vector of the node positions of the element E″.
In addition to preserving the directions of equilateral element, we should add geometry constraints to keep the geometric shapes and features of original hexahedral mesh.For a vertex v″ i , a linear vertex constraint can be developed to penalize the displacement of vertices from their original positions, can be illustrated as below: (1) The geometric constraint for an internal node is The surface smooth nodes are constrained to lie on a plane, formulized as: ax″ i + by″ i + cz″ i + d = 0; (3) The surface crease nodes are needed lying on a straight line, the geometric constraint is: (4) The surface corner nodes are restricted to their original location: v″ iv i = 0; Thus, we try to minimize the following energy function: where θ K is the weight of the edge direction constraints; θ In , θ Ss , θ So and θ Sc are contributions of internal nodes, surface smooth nodes, surface corner nodes and surface crease nodes respectively, while n In , n Ss , n So and n Sc are number of corresponding nodes.After many tests and adjustments, θ K , θ In , θ Ss , θ So and θ Sc are set to be 100, 10, 200, 200, and 200 respectively in this study.We can obtain the set of coordinates of by minimizing the above function, which results in a sparse linear system as the following: where A is a (36 + 3n In + n Ss + 3n So + 2n Sc ) × 24 matrix; B is a 24 × 1 column vector.The sparse linear system in Eq. ( 9) can be solved in a least square sense as: demonstrates the regularization process of single hexahedron with a given initial element E show in Fig. 2a.The nodes are first classified to surface smooth nodes (v 0 ,v 1 ,v 2 ,v 3 ), surface crease nodes (v 4 ) and internal nodes (v 5 ,v 6 ,v 7 ).An equal volume of equilateral hexahedron E′ is drawn with red lines in Fig. 2b by minimizing the Euclidean distance between equilateral hexahedron E′ and initial element E, in which e 4,0 = v′ 4 − v′ 0 , e 1,0 = v′ 1 − v′ 0 and e 3,0 = v′ 3 − v′ 0 are edge directions to be maintained.The positions of the final regularization hexahedron can be founded by solving Eq. ( 9) with θ K = 100, θ In = 10, θ Ss = 200, θ Sc = 200 and θ So = 200, plotted by black lines in Fig. 2c.After regularization operation, the scaled Jacobian value is updated from 0.39 to 0.92.We can see that our method obtains a very high-quality regular element, while the geometric features of initial mesh remain unchanged.

Mesh Quality Improvement
The global optimization process in previous local-to-global algorithm is to optimize all hexahedral meshes as a whole, ( 10) while the global optimization process in this study only performs in local regions.The recognition of local regions consists of the following steps: (1) When a seed hexahedral element with its scaled Jacobian lower than the user-specified threshold q e (we set q e to 1/4 of the maximum scaled Jacobian of all hexahedral elements) is encountered, a new local region is created, containing this hexahedral element, associated with a new label.The scaled Jacobian [24], as the main mesh quality standard, is used to measure whether the element is qualified.For a node v i and its three connected nodes {v i,1 , v i,2 , v i,3 } belonging to a hexahedron E, a 3 × 3 matrix can be formed by three vectors . By normalizing the column vectors of matrix U(v i ), the scaled Jacobian value of node v i can be defined as J (v i ) = det (U(v i )).The minimal scaled Jacobian among all nodes in hexahedral element is the scaled Jacobian of element E, expressed as J(E).
(2) Then a recursive process extends this region: for each hexahedral element with quality lower than the userspecified threshold in the local region, the adjacent elements are integrated into the local region, associated with new labels.The termination condition of this recursive process is that there is no element lower than the user-specified threshold at the periphery of the local region.
(3) The recognition is repeated for every other hexahedral element with quality lower than the user-specified threshold and still unlabeled.

Global Optimization for Local Regions
To prevent the generation of invalid elements, a linear elastic energy-based optimization is used to relocate the nodes belonging to the local regions.The process of local hexahedral regularization can be seen as a process in which the initial hexahedral element is elastically deformed into a regularized hexahedral mesh, and the internal force of the node is bound to occur during the deformation.Thus, the global optimization can be recast into that of solving the deformation of hexahedral meshes under the internal force.
For an initial hexahedron 3 and its corresponding regularized hexahedron E″ = (v″ 1 , …,v″ 8 ), v″ i = (x″ i , y″ i , z″ i ) ∈ R 3 , the displacement of hexa- hedron is q e = {△x 0 , △y 0 , △z 0 ,…, △x 7 , △y 7 , △z 7 }, can be calculated as follows: △x i = x″ i − x i , △y i = y″ i − y i , △z j = z″ i − z i , (i = 0,…,7).Assume that regular hexahedron E″ is obtained by linear elastic deformation of initial hexahedron E, the eight-node hexahedron isoparametric element is used to calculate the internal forces: where matrix k e is the stiffness matrix of hexahedral element, which consists of the strain matrix B and the linear elastic matrix D of the hexahedral element [25].After calculating the internal forces of each element by formula [11], suppose that L(i) = {l 1 ,…, l m(i) } denotes m(i) hexahedrons adjacent to the node v i .The internal force of node v i is obtained by average of its adjacent elements: The energy equation under node load is as follows: where K is the global stiffness matrix; ΔX = {△x 0 , △y 0 , △z 0 , …, △x n−1 , △y n−1 , △z n−1 } is the displacement vectors of all nodes in a local region.The energy equation [13] is quadratic and the displacements can be calculated by solving a sparse linear system:

Update Node Location
The new mesh node positions are determined by different strategies.For nodes in local regions, linear elastic energybased optimization is used to compute the displacement of the nodes simultaneously, and the optimized node location is v oi = v i + △X i .Otherwise, the new mesh node position will be directly derived from the following weighted averages as: where L(i) represents all elements surrounded to node v i ; v″ i is the node position of local regularized hexahedral element E″; and μ i is the weight of node v″ i , defined as μ i = 1.0 − J(E i ), J(E i ) is the scaled Jacobian value of given element E i .

Experimental Results
We have applied our hexahedral mesh improvement algorithm to several hexahedral meshes generated by the most widely used sweeping method, tetrahedron decomposition method, and grid-based method, corresponding to the Torque arm, Eight-shape, Rod and Base model respectively.To evaluate the mesh quality, the minimal and mean mesh quality value are defined as: where J(E) is the scaled Jacobian value of element E i , and n H is the element number of hexahedral meshes.In the following test examples, the mesh quality is visually illustrated by color map according to the scaled Jacobian value-the color of a hexahedron is blue if its scaled Jacobian value is 1.0 and red if its scaled Jacobian value is 0.0.Additionally, the mesh quality is quantitatively measured by histograms (in percentage) with respect to scaled Jacobian values.We also compare the method proposed in this paper with smart Laplace [9], GETMe [11] and local-to-global smoothing [20] methods to verify its effectiveness.
Figure 3 is a hexahedral mesh quality improvement of a torque arm model.The torque arm is a typical part in the automobile structure, which usually bears torque and compressive stress.The prerequisite for strength calculation of ( 14) torque arm is to generate high-quality hexahedral mesh.The given original mesh is generated by sweep method [26,27], displayed in Fig. 3a. Figure 3b-e are improved results obtained by smart Laplace, GETMe, local-to-global and our method.Distributions of scaled Jacobian (in percentage) are shown in Fig. 4 to compare the quality of different hexahedral meshes in Fig. 3.The minimal mesh quality value of the smoothed mesh achieved by smart Laplace, GETMe, local-to-global and our method is elevated to 0.24, 0.24, 0.5 and 0.51 respectively.As can be seen from the mesh details in Fig. 3, the initially generated mesh quality is low, which is easy to cause calculation errors.Smart Laplace and GETMe are failed to improve the minimal mesh quality because they did not consider the influence of the surface node on mesh quality.The proposed method achieves the same mesh quality as the local-to-global method, which meets the requirements of finite element analysis.Automation of hexahedral mesh generation has always been a major problem in the field of mesh generation.At present, the two most automated methods are tetrahedral splitting method and grid-based method, which often lead to a number of horrible quality hexahedral elements.Figure 5 is a hexahedral mesh optimization of an eight-shape model obtained by tetrahedral splitting method.The corresponding quality distributions of Fig. 5a-e is displayed in Fig. 6, where the detail plot shows that the minimal mesh quality is 0.02, 0.04, 0.07, 0.17 and 0.16 respectively.
The grid-based method [28,29] is one of the most widely used methods, which is suitable for any shape of solid model, and can obtain high quality internal mesh and low-quality surface mesh.The following two examples of mesh smoothing are performed for hexahedral meshes generated by grid-based method.Figure 7a shows the original unstructured hexahedral mesh of a rod model.As one of the key components of diesel engines, the rod requires finite element simulation of its structural strength and forming process.However, the quality of the original hexahedral mesh Through the smart Laplace, GETMe, local-to-global and our method, the improved hexahedral meshes show in Fig. 7b-e are obtained.From the mesh details shown in the Fig. 7, our method has significant optimization for low-quality surface hexahedral meshes.Figure 8 is a comparison of mesh quality of different meshes, in which the minimal mesh quality is 0.08, 0.09, 0.09, 0.14 and 0.19 respectively and our method has produced the largest proportion of regular elements.Figure 9 is a mesh quality improvement example of a base model with 129,148 hexahedrons and 146,546 nodes.Due to the number of nodes in this model is too large, the sparse matrix established by local-to-global method is too huge, resulting in errors in mesh quality improvement.However, the method in this paper only performs global optimization in local region, which avoids calculation errors.The details shown in the Fig. 9a indicate that the surface hexahedral mesh quality of the original model is poor, with a minimum quality of 0.01, and a reasonable threshold for local region recognition can be set to 0.25 (i.e.1/4 to 1/3 of the maximum and minimal Jacobian interval).With this specified threshold, 187 local regions are clustered in Fig. 9b.Our method can obtain hexahedral grids with acceptable quality, while smart Laplace and GETMe have little effect on improving the quality of surface hexahedral grids.The mesh quality comparison of different improved results is shown in Fig. 10, and the minimal quality of our improved mesh is 0.12.
Table 1 gives the mesh quality statistics of improved hexahedral mesh on Torque arm, Eight-shape, Rod and Base models and the computational costs are given in Table 2. Above all, the Smart Laplace method and GETMe method do not consider the nodes on the surface of the Hexahedral models, so they have little effect on improving the surface hexahedral mesh quality.Therefore, it is difficult to improve the mesh quality of the Hexahedral mesh models.The Local-to-global method takes into account the geometric features of the model and can greatly improve the quality of the mesh.However, this method only divides surface nodes into fixed points and movable points, making it difficult to adjust the position of nodes on sharp edges.In addition, the Local-to-global method requires global matrix calculations, which can be timeconsuming and may not even yield optimization results for large models.Therefore, this article proposes a strategy of region aggregation, which decomposes the global region into multiple local regions for optimization, so the calculation is obviously speeded up; Then, the feature of crease node was added, and a mesh regularization method based on geometric features was proposed to improve the  position of nodes on sharp edges.For models with many creased edges and a quantity of nodes, such as Rod and Base model, our method has advantages in mesh quality and computational efficiency compared to the Local-toglobal method.Analysis from the optimization results, the mesh size has a small influence on the optimization results.However, from the test results of Rod and Base model, it can be seen that the transition between different mesh sizes will affect the optimization results.The higher the quality of the original mesh, the more reasonable the topology structure of the mesh.The mesh quality optimization method that only adjusts the node position can achieve better results, such as the method proposed in this study.Geometric features have the greatest impact on mesh quality optimization.The smoother the area, the more the mesh quality is improved.Areas with higher curvature and more feature constraints are more difficult to obtain good mesh quality.Geometric features have the greatest impact on mesh quality optimization, and the smoother the area, the more the mesh quality is improved.Regions with higher curvature and more feature constraints are more difficult to obtain good mesh quality.

Conclusions
In this paper, we present an effective geometric constraintbased hexahedral mesh quality improvement method, whose innovation is mainly reflected in: (1) A mesh quality improvement strategy from single element to local region and then to global model is adopted, which avoids the large-scale energy Equation solving required for the global optimization of the hexahedral mesh.(2) The tensor voting theory is introduced to classify the nodes to four categories, which can be constrained in different forms.For a creased node, it is allowed to move only along its two connected crease edges, and its geometric constraint form is a combination of two planar constraints.A smooth surface node is a node that can only move on its tangent plane, and its corresponding geometric constraint is a single planar constraint.A surface corner is confined to its initial position, with a fixed geometric constraint.(3) A geometric constraints-based regularization algorithm for single hexahedron is proposed.Different geometric constraints are added to obtain a regular form for each hexahedral element by minimizing a quadratic functional in a least squares sense.(4) A global hexahedral mesh operation is performed locally for a number of regions with element whose quality is lower than the user-specified threshold.The mesh nodes outside these local regions are simultaneously relocated by weighting the contribution of new temporary node in each attached element.
As compared to other two geometry-based hexahedral mesh-smoothing methods, our method can achieve much better improved results in almost the same amount of time-consuming.Compared with local-to-global method, our method is faster and achieves almost the same effect of mesh quality improvement.This is useful and critical in the application of numerical simulations such as bulk forming simulation and structural optimization.
There are still many shortcomings in this method.Firstly, the parameters ε and η in node classification process are sensitive to the noise of meshes, which may lead to errors in judgment of node type.Secondly, for some hexahedral meshes with complex topology and large changes of element size, the improvement of mesh quality by this method is not obvious enough.The guarantee on mesh quality improvement by the combination of mesh topology modification and node position smoothing is an interesting problem we would like to tackle in the future.

Fig. 7 Fig. 8
Fig. 7 Mesh quality improvement of a rod model

Fig. 10
Fig. 10 Comparison of the mesh quality of a base model