In order to improve the boundary mesh quality while maintaining the essential characteristics of discrete surfaces, a new approach combining optimizationbased smoothing and topology optimization is developed. The smoothing objective function is modified, in which two functions denoting boundary and interior quality, respectively, and a weight coefficient controlling boundary quality are taken into account. In addition, the existing smoothing algorithm can improve the mesh quality only by repositioning vertices of the interior mesh. Without destroying boundary conformity, bad elements with all their vertices on the boundary cannot be eliminated. Then, topology optimization is employed, and those elements are converted into other types of elements whose quality can be improved by smoothing. The practical application shows that the worst elements can be eliminated and, with the increase of weight coefficient, the average quality of boundary mesh can also be improved. Results obtained with the combined approach are compared with some common approach. It is clearly shown that it performs better than the existing approach.
Numerical simulation is an important component in diverse activities such as medical imaging, engineering design, and cinematic special effects. These simulations routinely rely on tetrahedral meshes to model the physical domain of interest. The popularity of tetrahedral meshes stems from their ability to accurately model extremely complex geometries. High resolution tetrahedral meshes are often particularly desirable, as they can improve numerical accuracy greatly at previously infeasible scales. Fortunately, the power of modern computing and data acquisition technologies has enabled the production of tetrahedral models with enormous size. However, it is normally difficult to acquire mesh in which all the elements are suitable for numerical computation. Poorly shaped tetrahedra in a mesh can result in numerical errors and increase the time cost to find a solution [
Recently, there are two main techniques to improve the mesh quality. The first modifies topology by inserting or deleting nodes as well as changing connectivity of nodes [
As to improving boundary quality of a surface mesh, considerable research has been conducted. Garimella et al. [
In this study, we are more concerned with optimizing boundary vertices to obtain better quality tetrahedra, rather than surface meshes. However, little research has been done to improve boundary surface quality in solid meshes. Klingner [
The goal of this paper is to improve the quality of boundary tetrahedra. It should be noted that, unlike Klingner’s work, surface vertices will not be moved. Topology optimization technique is adopted for converting elements that have all their vertices on the boundary into other type of elements, and the optimizationbased smoothing algorithm is used for improving mesh quality. Section
Due to inherent defect, the optimizationbased smoothing algorithm cannot improve boundary mesh quality through moving boundary nodes without distorting discrete surface. The boundary conformity cannot be guaranteed when surface nodes are permitted to move. For example, the elements in Figures
Four kinds of boundary elements.
The overall scheme is presented. Algorithmic details for each of the major steps in this scheme will be presented later in the paper.
Input: initial mesh, poor quality threshold value
Output: high quality mesh.
Compute the number of nodes and elements in initial mesh. If the element is a boundary one, mark the element with the boundary;
calculate the mesh quality of initial mesh based on the quality measure
in
Apply
Apply
Repeat Steps (3) to (5) until there are no poor quality elements in
Through studying the quality measures and optimization algorithms of tetrahedral meshes, the error function is derived by means of transforming the measures. The element’ distortion is disposed as the error. And the larger the distortion is, the bigger the value of the error. The maximum value of the error function is infinite, and the minimum is zero. The total error function of the mesh is the sum of the elements’ errors, and it is adopted as the objective function of the optimizationbased smoothing. The minimum value of objective function is solved for improving the mesh quality.
Generally speaking, a reasonable quality measure for elements should possess the following attributes.
It is invariant under translation, rotation, and scaling.
Normalization by an optimal value within a range [0,1], where 1 is for an equilateral tetrahedron and 0 is for a degenerate tetrahedron.
Ability to detect all possible badly shaped elements.
So an optimal quality measure
The reciprocal of (
When the element is a regular tetrahedron, the value of the function is 1. As one element degenerates, the value tends to be infinite. If the element is reverse or the volume is negative, the value of error is also infinite. The total error function of the mesh is defined as
An efficient and robust solver for the large system of equations presented by the optimization problem is needed. Better smoothing algorithms are based on numerical optimization [
▸
▸
▸
▸
▸
▸
go to step (3)
It is well known that the line search methods play a pivotal role on optimization problems. Locating a local minimum in the optimization problem with no constrains are prepared. All methods have the basic structure in common. In each iteration, a direction
▸
(2) compute
▸
Set
(4) return
▸
(6) else
▸
▸
Topology optimization algorithm is employed, which converts boundary elements that have all their nodes on the boundary into other types of boundary elements. Furthermore, the topology optimization does not necessarily improve the quality of elements, but those boundary elements must be eliminated. Pseudopodia for topology optimization are presented in Algorithm
those elements.
▸
Proposed by George [
Edge removal.
Edge swapping is a more complicated procedure [
Edge swapping.
Face swapping changes the local connectivity of a simplified mesh which converts all boundary elements into other boundary elements. Each interior face in a tetrahedral mesh separates two tetrahedra consisting of a total of five points. A large number of nonoverlapping tetrahedral configurations can be formed with these five points, but only two of them can be reconnected satisfactorily. Figure
Face swapping configuration of five points in three dimensions.
Figure
Distribution of poor quality elements.
Before optimization
Figure
Time for different weight coefficients.
The conclusion can be drawn that the proposed algorithm can dramatically reduce the number of poor quality elements in boundary and interior meshes. With the increase of weight coefficient, the overall and boundary mesh quality rises, while the time consumed increases rapidly. So in consideration of optimizing effect and efficiency, it is recommended the weight coefficient lies in the range of 0.7~0.8.
Figure
Statistics of the examples before optimization.
Model  Mesh size  Overall mesh quality  Boundary mesh quality  

Vertex number  Elem#  Worst  Average  Worst  Average  
Impeller  77710  428049  0.0052  0.5105  0.0052  0.4823 
Volute  36994  229101  0.0039  0.5074  0.0039  0.4761 
Models for mesh generation.
Impller
Volute
In Table
Statistics of the examples after optimization.
Model  Method  Overall mesh quality  Boundary mesh quality  Time/s  

Worst  Aver  Worst  Aver  
Impeller  Smoothing  0.0339  0.5760  0.0339  0.5360  122.9 
Combined  0.0671  0.5937  0.0671  0.5494  149.8  
 
Volute  Smoothing  0.0359  0.5679  0.0359  0.5486  102.8 
Combined  0.0593  0.5861  0.0593  0.5603  128.7 
To evaluate our mesh improvement algorithm, comparison with Freitag and OllivierGooch's method is made. And two cases of mesh (TIRE and RAND2) come courtesy of Freitag and OllivierGooch [
Statistics of the examples before and after optimization.
Model  Method  Before optimization  After optimization  

Min  Max  Min  Max  
TIRE  Freitag 

178.88°  13.67°  159.82° 
Combined  22.57°  141.54°  
 
RAND2  Freitag  0.10°  179.84°  7.50°  170.09° 
Combined  25.69°  132.27° 
The two mesh optimization algorithms have been implemented and tested for RAND2 with the distribution of dihedral angles, as shown in Figure
Mesh quality improvement for RAND2 with two algorithms.
Initial mesh
Optimized after Freitag algorithm
Optimized after proposed algorithm
A new combined algorithm based on optimizationbased smoothing and topology optimization is proposed for boundary mesh quality improvement. Error functions of elements determined based on an inverse of measure
This work was supported by The National Natural Science Foundation of China (no. 50825902, 51079062, 51109095, and 51179075), a Project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, Natural science fund in Jiangsu Province (BK2010346, BK2009006), and Postgraduate Innovation Foundation of Jiangsu Province (CXLX11_0576).