A parameter-free approach is proposed to determine the Lagrange multiplier for the constraint of material volume in the level set method. It is inspired by the procedure of determining the threshold of sensitivity number in the BESO method. It first computes the difference between the volume of current design and the upper bound of volume. Then, the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material. Numerical examples proved that this approach is effective to constrain the volume. More importantly, there is no parameter in the proposed approach, which makes it convenient to use. In addition, the convergence is stable, and there is no big oscillation.

Among many methods for structure topology optimization [

The approaches reviewed above are successful to determine the Lagrange multiplier. In our present study, a new approach is proposed, i.e., a parameter-free approach using the bi-directional evolutionary structural optimization (BESO) method [

The boundary

Propagation of

where

The compliance minimization problem given by

where

where

Now, an important task is to determine

Although this approach can accurately enforce the volume constraint, the integration along boundary

where

When the volume of current design is larger than

Such an observation offers us the theoretical background to determine the Lagrange multiplier for the constraint of material volume in the level set method by using the BESO method.

BESO [

where _{e}

Inspired by the process in the BESO method for determining the threshold

In every optimization iteration, we first compute the difference between the volume of current design (denoted as ^{k}) and

If

In order to delete the redundant material

In other words, the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material.

Recall that the sensitivity number

As one can see in

Note that besides using the BESO method to determine the Lagrange multiplier, we can also use it to nucleate holes during the level set based topology optimization. The details of the hole nucleation are referred to our previous papers [

In the following examples, the properties of solid material are:

The criteria of convergence include [

and

In addition, we will terminate the optimization when

First, we do the optimization with the Lagrange multiplier being determined by using the BESO.

Second, the optimization is done with the Lagrange multiplier being determined by using the augmented Lagrange multiplier method.

As can be seen in

From

The design optimization problem is shown in

First, the optimization is done with the Lagrange multiplier being determined by using the BESO.

Second, the optimization is done with the Lagrange multiplier being determined by using the augmented Lagrange multiplier method.

Third, we put more holes into the initial design; we use a mesh with

Because the finite element mesh is finer, the threshold of sensitivity number becomes more accurate, and the Lagrange multiplier obtained by

The design optimization problem is shown in

First, we do the optimization with the Lagrange multiplier being determined by using the BESO.

Second, besides using the BESO to determine the Lagrange multiplier, we also use the BESO to nucleate hole. The details of the hole nucleation are referred to our previous papers [

Determining the Lagrange multiplier for the constraint of material volume is an indispensable task in the level set method. Although the approaches that can be found in the literature are successful, they also suffer from some drawbacks. In this paper, a new approach is proposed to determine the Lagrange multiplier. First, it computes the difference between the volume of current design and the upper bound of volume. Then, the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material. Several numerical examples demonstrated that this approach is effective. More importantly, there is no parameter in the proposed approach, which makes it convenient to use. In addition, the convergence is stable, and there is no big oscillation. In the future, the proposed approach will be further extended to deal with other optimization problems or other constraints, for instance, the microstructure optimization [