S. L. SUN ET AL.

Open Access IIM

194

In order to solve this kind of constrained optimization

model efficiently, we can convert it to an unconstrained

one with differentiable objective function according to

[4]. First, the volume constraint |Tj| > 0 can be converted

to the following maximum constraint:

2

1

max 0

4kj

ki

jii k

xT

xx

Txx x

(9)

Then construct the L penalty function

2

1

,max0,

4

i

kj

ki

ii jiik

xT

xx

xExTx xx

(10)

and replace the non-differentiable term in Equation (10)

by its aggregate function. Thus the differentiable objec-

tive function is obtained as:

2

,

ln 1exp4

i

kj

ki

ii

jii k

jxT

xx

xEx

qTxx x

q

(11)

When the penalty factor α is greater than some threshold

value α0 and the parameter q approaches to infinity, the

solution of the unconstrained model for minimizing the dif-

ferentiable objective function in Equation (11) approaches

to the solution of the original constrained model (8).

4.2. Solution Algorithm

For consideration of efficiency and precision, the scheme

that integrates chaos search and BFGS is employed to

solve the optimization model. Chaos search has good

global search ability, but its convergence rate is low.

BFGS method is one of the most representative algo-

rithms in solving unconstrained optimization problem

with good numerical stability. The combination of chaos

search and BFGS can make full use of both global search

ability of chaos search and fast convergence rate near the

optimal solution of BFGS.

Based on the above consideration, the solution of op-

timization problem in Step 3) can be divided into two

steps: first obtain the approximated solution by chaos

search algorithm as the initial input values; then do opti-

mization by BFGS algorithm.

The procedure of chaos search algorithm is simply de-

scribed as follows: first map the design variable xi to

chaotic variable, then to the objective function i

E

in

Equation (8) directly search the best solution that satis-

fies the constraints. The volume of each tetrahedron is

calculated before evaluating the objective function; there-

fore, if the current position does not satisfy the con-

straints, directly go to the next searching.

Then using the approximated solution by chaos search

algorithm as the initial input values, do optimization to

the differentiable objective function in Equation (11) by

the BFGS algorithm for optimal solution.

Due to stochastic property of chaos search algorithm,

the approximated solutions may be different. In practice

calculation, four or five approximated solutions obtained

by chaos search are taken as the initial inputs for BFGS

algorithm. Then the best solution by BFGS algorithm is

chosen as the final optimal solution.

5. Mesh Quality Improvement

To improve mesh quality, smoothing or topological op-

timization alone may not achieve ideal results. In general,

alternately applying smoothing and topological optimiza-

tion technique can improve mesh quality significantly [5].

Smoothing process changes the location and distribution

of nodes, and may provide further space of improving

mesh quality for topological process. Therefore, this pa-

per combines these two approaches, alternately applying

smoothing and topological optimization to achieve better

results. The most frequently used operations of topologi-

cal optimization are so-called basic or elementary flips.

Recently, Liu et al. [6] proposed a new topological opti-

mization operation named small polyhedron reconnec-

tion (SPR), to search the optimal topological configura-

tion in an enlarged region which usually includes 30 ~ 40

tetrahedrons. Delaunay triangulation can also be proved

to be a topological optimization tool to improve the qual-

ity of mesh. This paper adopts re-triangulation as topo-

logical approach, makes a new Delaunay triangulation

after nodes updating by the proposed smoothing scheme.

Such an approach can reduce the value of quality metric

based on interpolation error further [3] (Among all the

triangulations, Delaunay triangulation makes the value

take minimum [2]). Meanwhile, reconnection of nodes

also provides optimization space for next smoothing

step.

6. Examples and Conclusions

The proposed approach is tested by a number of exam-

ples to illustrate its effectiveness. Quality improvement is

performed by applying the presented smoothing scheme

and re-triangulation for topological optimization alterna-

tively. Only interior nodes are repositioned during

smoothing, while the nodes on boundaries keep fixed.

Three testing meshes are shown in Figure 2. The first

testing mesh consists of 10,926 nodes and 58,615 tetra-

hedrons initially. The second testing mesh consists of

19,149 nodes and 76,188 tetrahedrons. The third mesh in-

cludes 6534 nodes and 25,177 tetrahedrons initially. Table

1 shows the statistics of initial quality and quality after 10