The structure optimization design under thermo-mechanical coupling is a difficult problem in the topology optimization field. An adaptive growth algorithm has become a more effective approach for structural topology optimization. This paper proposed a topology optimization method by an adaptive growth algorithm for the stiffener layout design of box type load-bearing components under thermo-mechanical coupling. Based on the stiffness diffusion theory, both the load stiffness matrix and the heat conduction stiffness matrix of the stiffener are spread at the same time to make sure the stiffener grows freely and obtain an optimal stiffener layout design. Meanwhile, the objectives of optimization are the minimization of strain energy and thermal compliance of the whole structure, and thermo-mechanical coupling is considered. Numerical studies for square shells clearly show the effectiveness of the proposed method for stiffener layout optimization under thermo-mechanical coupling. Finally, the method is applied to optimize the stiffener layout of box type load-bearing component of the machining center. The optimization results show that both the structural deformation and temperature of the load-bearing component with the growth stiffener layout, which are optimized by the adaptive growth algorithm, are less than the stiffener layout of shape ‘#’ stiffener layout. It provides a new solution approach for stiffener layout optimization design of box type load-bearing components under thermo-mechanical coupling.

Stiffened structures are applied widely in aeronautics, machine tool, automobile, and so on. These structures are composed of an unstiffened structure (plate or shell element) and stiffeners (Beam elements). The proper distribution of stiffeners on the structure can improve the structural mechanical performance effectively [

The internal structure optimization of box type support load-bearing component is the stiffener layout optimization design, essentially. In traditional, the stiffener layout design is usually referring to the regular shape, such as shape ‘#’, shape ‘*’, shape ‘X’, and so on. It is no doubt that the regular shape stiffener layout is simple to design and easy to manufacture, but actually, the stiffener layout design is still blind because it is difficult to improve both mechanics and thermotics of load-bearing components for requiring stiffness and material savings effectively. Therefore, a new optimization method for stiffener layout design should be developed, especially for the structure under thermo-mechanical coupling.

Actually, stiffener layout optimization is the problem for searching optimal material distribution [

The homogenization method and the density method as the traditional structure topology optimization methods are adopted frequently [

To overcome the disadvantage of traditional optimization methods, some new methods were developed. Bojczuk et al. [

In the structure optimization for heat conduction, it is still focus on cross-sectional area optimization [

To optimal stiffeners layout pattern of box type load-bearing component, this paper is organized as follows: Firstly, constructs the equivalent model of box type support of the machining center under thermo-mechanical coupling, then the stiffness diffusion theory is extended to the load and heat conduction model to develop structure adaptive growth algorithm. Secondly, extends the optimization objective to minimum strain energy and thermal compliance, then an adaptive growth algorithm is developed to optimize stiffener layout under thermo-mechanical coupling. Thirdly, to evaluate the method, the stiffener layout of square shell under thermo-mechanical coupling is optimized by the adaptive growth algorithm. Finally, the adaptive growth algorithm is used to optimize the stiffener of box type support.

The principle of adaptive growth algorithm under thermo-mechanical coupling is the calculation of two parallel levels of the objective function, load, and heat conduction. The two parallel levels of objective function and optimization are connected by stiffener.

In the adaptive growth algorithm of structure load optimization, the objective of the optimization is to find the stiffener which can be minimized the whole strain energy. In adaptive growth considering thermal loads, the objective of the optimization is to find a layout of the heat conduction rod to a minimum heat dissipation of structure.

The adaptive growth algorithm of this paper can construct the connection between the adaptive growth algorithm of structure and heat conduction structure, and the unit of connection is the growth stiffener. The growth stiffeners under thermal coupling are improved in terms of structural stiffness and heat dissipation. In terms of load, it is equivalent to adding a stiffener. At the heat level, it is equivalent to adding a high thermal conductivity rod.

After the stiffener growth, both the stiffness matrix and heat conduction stiffness matrix are updated based on the stiffness diffusion theory. The stiffness matrix updated under thermo-mechanical coupling is shown in

The interrelationship between the high heat conduction rod and stiffener is considered during the stiffener growth. The flow chart of growth stiffener layout under thermo-mechanical coupling is shown in

Considering the influence of thermo-mechanical coupling, the weighting factor is used to represent the influence of compliance and heat dissipation. The optimization goal is to optimize the comprehensive performance of the stiffener, and the density of the stiffener is the design variable. Under the condition of a given volume constraint, find the stiffener distribution of the best structure. In the multi-objective topology optimization problem, the topology optimization model is as follows:
_{i}_{T}

The flow of the adaptive growth algorithm under thermo-mechanical coupling is as follows:

Set the specific size of the box type load-bearing component, and the correlation coefficient of material properties, including elastic modulus, Poisson’s ratio, and thermal conductivity.

At the structural level and the heat level, the upper surface of the box type load-bearing component is divided into grids. The grid division at the structural level and heat level is completely consistent, this is, keeping the same grid size.

The overall stiffness matrix Struct_K of the base structure divided by free quadrilateral element grid and the overall stiffness matrix Heat_K of the base structure divided by free quadrilateral element grid are calculated.

Selected seed growth points at the selected structural level and at the heat level.

The structural seed growth point and the heat conduction seed growth point grow at the same time, and the mutual influence relationship is considered during the growth process.

Stiffness diffusion is carried out on the stiffener at the structural level. At the same time, the growth stiffener generated at the heat level is replaced by the stiffener at the same position. Then introduce the stiffener to the structural level, and stiffness diffusion for the stiffener. Finally, the overall stiffness matrix updated at the structural level is obtained.

Stiffness diffusion is carried out on the high heat conduction rod of the heat level. At the same time, the growth ribs generated at the structural level are replaced by high thermal conductivity rods in the same position and introduced to the heat level. Then stiffness diffusion for the high thermal conductivity rods. Finally, the overall stiffness matrix of the updated at the heat level is obtained.

When the growing rib grows out of the boundary or the total volume of the growing rib reaches the predetermined upper limit, the growth is stopped. Otherwise, the growth is continued.

Finally, the structural layout of the growth rib under the action of thermo-mechanical coupling is obtained.

The square shell is shown in

The temperature distribution and thermo-mechanical coupling with initial condition are shown in

The temperature distribution and structural deformation under thermo-mechanical coupling of shape “#” stiffener layout are shown in

Applying the proposed adaptive growth algorithm of this paper to stiffener layout design, the growth situations from Step 1 to Step 30 are shown in

The volume of growth stiffener exceed limit;

The stiffener grows to a point beyond the pre-defined boundary;

The growth steps exceed preset.

After growth stiffener layout optimization, the temperature distribution and structure of growth stiffener under thermo-mechanical coupling are analyzed by ANSYS, the results are shown in

Comparing the temperature distribution and the deformation of the structure without stiffener, the structure with shape ‘#’ stiffener layout and the structure with stiffener layout, which is grown by an adaptive growth algorithm. In the temperature distribution, the maximum temperature of the structure without stiffener is 118.592°C, the structure with shape ‘#’ stiffener layout is 102.488°C, and the structure with growth stiffener layout is 76.265°C. In the structural deformation, the maximum deformation of structure without stiffener is 0.84057 mm, the structure with shape ‘#’ stiffener layout is 0.83017 mm, and the structure with growth stiffener layout is 0.79382 mm.

From the results, we can conclude that using the adaptive growth algorithm to optimize stiffener layout of square shell, both the maximum temperature and structural deformation are decreased significantly. Compared with the analysis results of the structure with shape ‘#’ stiffener layout and structure with growth stiffener layout, the maximum temperature and structural deformation with growth stiffener layout are decreased 25.59% and 4.38%, respectively.

The results of square shell show the superiority of the adaptive growth algorithm for stiffener layout optimization under thermo-mechanical coupling:

Improving the structure stiffness and decreasing temperature simultaneously;

Saving the material compared with the shape ‘#’ stiffener layout to obtain the same performance;

The adaptive growth algorithm is more efficient.

Using adaptive growth algorithm of this paper proposed, the stiffener layout of the body (load-bearing box type component) of M800 machining center is optimized. Before stiffener optimization, the body is equivalent to a simple model. The simplified model of the body is shown in

The load conditions of the body are shown in

The optimization is from the top surface of body; the ground structure and thermal are built, and the seed point and stiffness spread radius are set up. Then the stiffener is grown by an adaptive growth algorithm. The processing of stiffener growth is shown in

After the stiffener growth, the stiffener layout should be round to obtain the growth rhythm, and simplified some repeat growth stiffener, the processing of simplified is shown in

The models of the body with shape ‘#’ stiffener layout and with growth stiffener layout are shown in

The temperature distribution and structural deformation of body with shape ‘#’ stiffener layout and growth stiffener layout under thermo-mechanical coupling are shown in

Comparing the analysis results of temperature distribution and structural deformation with two different stiffener layouts, shape “#” stiffener layout and growth stiffener layout, we can conclude that the maximum temperature of the body with a growth stiffener layout by using an adaptive growth algorithm decreased from 26.387°C to 25.965°C, dropping 1.6%. The structural deformation under thermo-mechanical coupling is also decreased obviously. The maximum structural deformation of the body with shape ‘#’ stiffener layout is 7.4159 μm, and the maximum structural deformation of the body with growth stiffener layout is 5.8492 μm, dropping 1.5667 μm, 21.1%.

This paper developed an adaptive growth algorithm for stiffener layout design under thermo-mechanical coupling based on the topology optimization theory of heat conduction and stiffening structurally. Compared with the regular shape stiffener layout, both the temperature and structural deformation of the structure with the growth stiffener layout is reduced obviously. The square shell case shows the adaptive growth algorithm for stiffener layout design is more effective, practicable, and applicable. Using this approach, the stiffener layout of box type load-bearing components is optimized, and the temperature and deformation are reduced obviously compared with the traditional stiffener layout.

From the numerical results, we can conclude as follows:

The adaptive growth algorithm based on thermo-mechanical coupling can improve the stiffness and heat dissipation performance of the structure through stiffeners. The maximum temperature and structural deformation of structure with growth stiffener layout are reduced obviously.

The stiffener layout optimization of square shell shows that the maximum temperature and the deformation under thermo-mechanical coupling of the structure with growth stiffener are 76.265°C and 0.79382 mm. Comparing the structure with shape ‘#’ stiffener, the maximum temperature and deformation decreased by 25.59% and 4.38%, respectively.

The stiffener layout optimization of the machining center body shows the maximum temperature and the deformation under thermo-mechanical coupling of the structure of the body with growth stiffener are 25.965°C and 5.8492 μm. Comparing the structure of body with shape ‘#’ stiffener, the maximum temperature and deformation decreased by 1.6% and 21.1%, respectively.

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.