The Level Set Topological Optimization Method based on the Constraints of Volume and Stress

. This paper research the topological optimization based on level set method which based on the reaction diffusion equation. The paper propose a Level set topological optimization method which put the compliance as the objective function and take the volume as the main constraint, the element stress as the auxiliary constraint to regulate the volume constraint. This method not only can significantly speed up the optimization process, but can improve the precision of calculation and ensure the stability of the results when meet the stress constraint. Eventually, the proposed method can convenient the level set topological optimization to apply in engineering practice.


Introduction
At present, continuous structure topological optimization has become an important research field of engineering application. The principle of level set method is simple and clear, which won't appear the phenomenon of checkerboard, mesh dependence. So that, this method has a good development prospect.
Level set method was proposed by Osher and Sethian [1] which used to study the dynamic interface tracking as a numerical method. The reaction diffusion equation was introduced to solve the level set function method for structural topological optimization by Masaki Otomori [2]. What's more, the design of structure topological optimization which based on Flexible objective function by Gradient method was proposed by C. G. Zhang and Z. H. Xiong [3]. Moreover, X.Gu introduces some methods of stress to solve the Structural numerical optimization problem [4]. J.H. Rong and S. Ge, G etc proposed a topological optimization method of continuous structure, which combine the displacement method with the analysis of stress sensitivity [5]. ICM method is proposed to solve the problem of topological optimization that Plate and shell with stress constrains by X. D. Liu, Y. K. Sui etc [6].

Math model of topology optimization
The idea of level set topological optimization is that we establish higher dimension curve or surface of level set function, and then the forms of structural topology was described in the (1)

The derivation formula of topological optimization
The objective function of the design is to minimize structural compliance, and get a reasonable structure. : where * V is the upper limit of volume constraint which is a shifty parameter, ijkl > @ V represents the allowable stress. Using Lagrange multiplier algorithm transform above formulas into a unconstrained Lagrange formula is shown as follows: where represents the new objective function, and O represents the Lagrange multipliers [2].

Volume constraints formula
Volume constraint is applied to controls the changed volume of material. The details of solve are shown as follows: where, the solve of D as follows: As shown in Eq. 5. We can see that D represent the value of D in times. With the change of unit stress, the value of is also changed which is the main constrain that can control the decrease of the material.
Besides, the value of is decided by D . What's more, the drop speed of volume constrain will increase when the maximum stress value fully meet the located conditions. On the other hand, the drop speed of volume constrain will decrease when the maximum stress value will meet the located conditions. From that, we can improve the accuracy of calculation.

The computational formula of stress
In this paper, the stress calculation formula as follows: , and its Michelle unit stress is shown in Eq. 7. In the Fig. 2, H and H are the error parameters, H is equal to 0.01, H is equal to 0.00004.

Numerical examples
Now, Here will prove the effectiveness and accuracy of level set topological optimization which based on the stress and volume constrains. The constrain and force is applied in a cantilever beam as shown in Fig. 3, besides the thickness value of the plate is one millimeter, and Young's modulus value of the material is , what's more, The value of poisson's ratio is equal to 0.3.

*
Fixed design domain D Figure 3. Structure diagram of the cantilever beam

Example 1
The wide L of the structure is equal to 80mm, high H is 64mm, and concentrated load is equal to 3500N. The allowable value of stress is equal to 218MPa; the result of example 1 is shown in Fig. 4 and Fig. 5. As shown in Fig. 4, we can see that blue line represents the maximum stress value in each iteration of the structure, red one represents the allowable stress value and the green line represents the rest of the volume fraction. Along with the iteration, the volume fraction decreased continuously the maximum stress value of structure increased continuously and it will approach the allowable stress values. After calculating, we can know that the volume fraction of the structure is 0.536, and the stress value is shown as > @ V V , which meet the condition that stress is not more than allowable stress.

Example 2
The wide L of the structure is equal to 40mm, high H is 80mm, and concentrated load is equal to 4000N. The allowable value of stress is equal to 249MPa; the result of example 2 is shown in Fig. 6 and Fig. 7. As shown in Fig. 6, we can see that blue line represents the maximum stress value in each iteration of the structure, red one represents the allowable stress value and the green line represents the rest of the volume fraction. Along with the iteration, the volume fraction decreased continuously the maximum stress value of structure increased continuously and it will approach the allowable stress values.
After calculating, we can know that the volume fraction of the structure is 0.358, and the stress value is equal to > @ V V , which meet the condition that stress is not more than allowable stress.

Summary
This paper introduce the stress constraint to the level set method for topological optimization, and set a changed parameters as the threshold value to regulate the volume of structure, which based on the relationship between the located allowable stress and the of unit stress at every iteration. What's more, the parameters not only speed up the optimization process when the maximum stress value fully meet the located conditions, but also can slow down the optimize process, improve the accuracy of calculation, ensure stability of the results when the maximum stress value will meet the located conditions. Eventually, the proposed method can convenient the level set topological optimization to apply in engineering practice.