Based on the bidirectional evolutionary structural optimization (BESO) method, the present article proposes an optimization
method for a thermal structure involving design-dependent convective
boundaries. Because the BESO method is incapable of keeping track of
convection boundaries, virtual elements are introduced to assist in
identifying the convection boundaries of the structure. In order to solve the
difficult issue of element assignment under a design-dependent convection boundary,
label matrixes are employed to modify the heat transfer matrix and the
equivalent temperature load vector of elements over topology iterations.
Additionally, the optimization objective is set to minimize the maximum
temperature of the structure in order to deal with the objective reasonableness, and
the

Structural topology optimization, as an effective method of structural weight reduction, has been paid more and more attention in recent years. However, a lightweight design for a thermal structure under various boundary conditions still faces great challenges (Deaton and Grandhi, 2015; Zhu et al., 2015), namely because studies under first-type (Dirichlet boundary) and second-type (Neumann boundary) boundary conditions are more common than those under convection (Robin boundary) boundary conditions (Zhou et al., 2016).

Previous topology optimization of thermal structure considering the Robin (third-type) boundary (i.e., the convection boundary) has mostly been founded on density-based and level set methods. In density-based optimizations, it is necessary to create explicit boundaries by setting a cutoff value for elements' density; some classical manners to do this have been proposed, including the following: the hat function (Iga et al., 2009; Dede et al., 2015), the peak interpolation method (Yin and Ananthasuresh, 2001), the element connection parameter method (Ho Yoon and Young Kim, 2005) and convection interpolation (Bruns, 2007; Alexandersen, 2011). However, the choice of the truncation value of the above methods depends heavily on the experience of the designer, and an inappropriate cutoff value may result in ill-defined boundaries, such as “islands” (i.e., isolated solid materials). Some studies have been based on the explicit bounds of the level set method in order to realize the implementation of convection boundaries (Ahn and Cho, 2010; Coffin and Maute, 2015; Li et al., 2022). However, the level set method has the disadvantage of initialization dependence; thus, the location of convection boundaries will greatly affect the optimization results. More importantly, the above studies did not consider the fact that a structure boundary might be partly convective (Hu et al., 2008; Wang and Qian, 2020; Yan et al., 2021). Hence, it is necessary to identify whether a boundary is convective, adiabatic or partly convective. However, the difficulty here is that the solid or fluid region is updated during the iterations. As a result, the convection boundary changes with the iteratively altered structural boundary, as shown in Fig. 1.

Change in the design-dependent convection boundary with iteration.

Bidirectional evolutionary structural optimization (BESO) methods are also one of the main methods for topology optimization. They have the advantage of good convergence and initialization independence (Radman, 2021) and have been widely used in structural heat transfer (Xia et al., 2018), frequency (Gan and Wang, 2021), acoustics (Pereira et al., 2022) and material microstructure optimization design (Qiao et al., 2019), among others. In addition, BESO-type methods can provide a clear, explicit boundary (Da et al., 2018), which has been employed to solve the transmissible load problem (Fuchs and Moses, 2000; Yang et al., 2005) and the pressure load problem (Picelli et al., 2015, 2017). However, according to the heat transfer equation, the convective boundary is more complex than the transmissible or pressure load, as the convective boundary affects not only the equivalent temperature load but also the heat transfer capacity of the structure itself. Therefore, the current BESO method needs to be further investigated with respect to the design-dependent convection boundary in order to improve its applications to more complicated boundary conditions.

This article is devoted to the topological optimization of thermal structures considering the design-dependent convection boundary based on the BESO method. The remainder of the paper is structured as follows: in Sect. 2, the difficulties involved with thermal topology optimization considering the convection boundary are outlined; in Sect. 3, the corresponding solution method is proposed, including the identification of the convection boundary and the assignment of the convection matrix; in Sect. 4, the selection of the topology optimization objective is discussed, and the corresponding sensitivity is analyzed; and, finally, in Sect. 5, numerical cases with various boundary conditions are analyzed to verify the effectiveness and good convergence of the proposed method.

For the heat transfer problem, there are three types of thermal boundary
conditions (Palani and Ganesan, 2007): the first-type boundary (

When the boundary conditions are satisfied, the steady-state heat equation of discrete elements is obtained using the finite element method
(FEM):

According to Eq. (3), the first term on the right of the equation represents
the contribution of conductivity, and the second term represents the
contribution of convection. According to Eq. (4), the first and the second
terms on the right of the equation reflect the equivalent temperature load
generated by given heat flow and given convection, respectively. Further, Eqs. (3) and (4) can be expressed as follows:

Moreover, the element heat transfer matrix and the equivalent temperature
load vector of element nodes are assembled using the FEM to form the heat
transfer matrix and the temperature load vector of structure nodes:

In the BESO method, when the value of topological variables is 1, the corresponding elements are solid elements (SEs); when the value is a small quantity, such as 0.001 (not 0 to avoid singularity), the corresponding elements are nonsolid elements with very poor properties (Huang and Xie, 2010). If the design-dependent convection boundary is considered, the boundaries can be classified into two types: convective boundaries and adiabatic boundaries. Therefore, in this article, the nonsolid elements should be further subclassified into fluid elements (FEs) and void elements (VEs). As a result, the convection boundary can be identified from the interfaces between the SEs and the connected FEs (Deshmukh and Warkhedkar, 2013; Chamkha et al., 2017).

However, due to the variation in the FEs, the VEs and the SEs with
optimization iteration, it is necessary to take measures to keep track of the
iteratively changed convection boundary. Additionally, according to Eq. (10), the assignment of

This section is dedicated to realizing the identification of the convection boundary in the design domain during the iteration processes (in Sect. 3.1) and to assigning the corresponding convection contribution matrix (in Sect. 3.2).

In 2D structures, convection can be classified into two categories: (1) top and bottom convection (TBC), in which the direction is vertical to the structure surface, and (2) side convection (SC), in which the direction is parallel to the structure surface (Alexandersen, 2011). In the TBC case, all sides of an element will be influenced by convection, and the structural boundaries in such problems are all convective. In the SC case, convection affects partial sides of an element, and it is necessary to further confirm whether a boundary of the element is convective. Compared with the optimal design under TBC, the optimal design under BC requires an extra scheme to identify convective boundaries (rather than merely based on the geometry boundary); therefore, the latter design it is more complex and will be investigated in this article. In 3D structures, there is only one type of convection; it is similar to SC in 2D structures, but the difference is that the convection affects partial planes of an element. Hence, the convection boundary identification method for SC proposed later in this article can be applied to both 2D and 3D structures.

As the BESO method cannot keep track of convective boundaries (Qiao et al., 2019), virtual elements are introduced in this article in order to distinguish FEs from nonsolid elements. Using convection acting on
the surface of a 2D structure as an example, a schematic diagram is shown in
Fig. 2 that outlines the following specific steps:

In the design domain, the identifier of the solid elements is labeled “1” and that of the nonsolid elements is labeled “2”.

A layer of virtual elements around the design domain are added, and their identifier is marked as “0”. It is noted that these virtual elements do not participate in FEM calculations.

All elements are then “traversed”. If the identifier of an element is 2 and the identifier of any adjacent element is simultaneously 0, the identifier is changed to 0; if this is not the case, the identifier remains at its original value. To ensure identification, it is necessary to carry out this process enough times. The number of times that this process is carried out is related to the number of elements with an initial identifier of 2. As there is the possibility that only the last element of the traversal is marked as 0 and that the identifiers of all the preceding elements are 2, we can only guarantee the validity of the element identifier conversion by ensuring that the minimum number of iterations is the same as the number of initial elements marked as 2.

All of the virtual elements are deleted. The remaining elements with an identifier value of 2, 1 and 0 are void elements, fluid elements and solid elements, respectively.

Identification of the convection boundaries.

After the identification of the convection boundary of solid elements, it is necessary to further modify the element heat transfer matrix and temperature load vector influenced by convection.

Because convection only affects part of the boundary of the element, this
paper introduces label matrixes to achieve the assignment of the matrixes
related to convection. There are two parameters that affect convection:
the convection coefficient and the ambient temperature. The label
matrix of the convection coefficient is denoted as the

If a component (such as

For the reader's understanding, using the convection acting on the
surface of a 2D structure as an example, the steps related to the assignment of the
convection matrix to elements by label matrixes are summarized as follows:

If an element is nonsolid element (its identifier is 0 or 2), the

Check the identifier of its neighboring elements counterclockwise from the left edge, and then fill the

The element heat transfer matrix and the element temperature load vector considering convection are obtained by Eqs. (12) and (13).

In the BESO method, the objective function and constraints of optimization are
expressed as follows:

The optimization objective in this article is exclusively involved with the
thermal field. It is initially assumed that the temperature field and load
are related to topological variables; thus, the objective function depends on
the temperature field, temperature load and topological variables, namely

In the present article, the optimization objective is set to minimize the
maximum temperature in the structural temperature field, which can directly
reflect the thermal conduction capability of structures. However, the
maximum temperature is a scalar with no gradient. Thus, the

For the BESO method, sensitivity determines the “birth and death” of elements (Xu et al., 2020), which can be solved by the gradient of the
objective function (Huang and Xie, 2009). From Eq. (17), the corresponding Lagrangian adjoint matrix of the objective function is

Based on the work in Sects. 3 and 4, some typical cases of thermal structures with convective boundary conditions are analyzed in this section, and the topological optimization flowchart is shown in Fig. 3.

The optimal flowchart for thermal structures with a convective boundary.

The convection boundary condition, also known as the mixed boundary,
can be combined with both the first-type boundary and the second-type boundary to
solve the temperature field. In order to illustrate all of the types of structural
optimization designs under convection conditions, topological optimization
problems considering combination boundary conditions including
design-dependent convection boundaries are classified into three categories:

structural optimization under a combination of convection boundaries and the first-type boundary conditions, i.e., the

structural optimization under a combination of convection boundaries and the second-type boundary conditions, i.e., the

structural optimization under a combination of convection boundary, first-type boundary and second-type boundary conditions, i.e., the

Parameters for the numerical calculations.

Note that “

In this subsection, the thermal structure optimization results under
a combination boundary conditions including convection and the temperature constraint
are discussed. The design domain is a 0.5 m

Initial conditions for the

In this case, as the ambient temperature (100

Optimal process for the

From Fig. 5a and b, with the iteration proceeding, it can be seen that the value of the objective function increases, which is a typical characteristic of BESO (Huang and Xie, 2008). The final maximum temperature and thermal compliance reached based on the proposed method with the maximum temperature as the objective function are 77.44

Additional numerical instability appeared in iteration 5–7 in Fig. 5a, which was caused by the increase in the convection area. As seen in Fig. 6, in iteration 5, the generated topological cavities are disconnected with the outer surface of the structure, and convection only acts on the same part on the surface as that under the initial conditions. However, in iteration 6, the inner cavities propagate to connect with the outer surface, resulting in the convection areas extending to the boundary of the cavities. The increase in the convection area means that convection provides more heat to the structure, leading to a rise in the temperature of the structure. For this problem, the proposed optimization method in the present article, based on the BESO strategy, can correct the inappropriate element elimination according to the sensitivity information. In the following iteration 7, the internal cavity boundaries are modified to disconnect from the external surface, and the optimization iterations appear stable. The resulting configuration and corresponding temperature field are shown in Fig. 7.

Connectivity between the internal cavities and the outer surface of
the structure in iterations 5 to 7:

In Fig. 7a, the outer surface (convection areas) of the structure remains
unchanged, which is reasonable to reduce the convection areas (as mentioned
earlier). A similar phenomenon was also pointed out by Wang and Qian (2020). Furthermore, the boundaries of the internal cavities are adiabatic, and the temperature in the cavities' body is 0

Initial conditions and optimization results for the

In this subsection, the optimization of the thermal structure under a combination of
boundary conditions including the convection boundary and the heat flux is
discussed. For the initial structure, there is an equivalent temperature
load

Initial conditions for the

In this case, as the ambient temperature (0

Optimal process for the

From Fig. 10a, when taking the maximum temperature as the objective function,
the distance between the convective boundary and the heat source becomes
closer and closer with proceeding iterations (as shown in Fig. 11),
which results in a reduction in the initial structural maximum temperature
(92.62

The change in distance between the convection boundary and the heat source (getting closer and closer) as well as the change in the corresponding structural temperature field (based on the average temperature of element nodes) with proceeding iterations.

However, as shown in Fig. 10b, when taking thermal compliance as the
optimization objective, the iterative curve continues to oscillate during
the volume fraction reduction phase (iterations 1–23). When the volume
fraction reaches the target value (iterations 23–68), the overall variation
trends in the maximum temperature and thermal compliance of the
configuration both increase with proceeding iterations. Finally,
the final maximum temperature converges to 98.35

As the thermal conductivity of the structure is isotropic (according to
Table 1 with

Comparison of configurations for isotropic and orthotropic thermal conductivity:

As mentioned in Fig. 10, the design dependence of convection boundaries has an
influence on the final optimal configuration. If the design-dependent convection
boundary becomes a fixed boundary, with the

Comparison of the results for the fixed

In this subsection, the optimization of the thermal structure under a combination of
conditions including the convection boundary, temperature constraint and heat flux
are discussed, i.e., the

In Fig. 14, when

Initial conditions and optimization results for the

The present study emphasizes a topology optimization method under
design-dependent convection conditions. The main conclusions are as follows:

A topology optimization method for a thermal structure with design-dependent convection boundaries is proposed by introducing auxiliary elements to identify the design-dependent convection boundary, employing label matrixes to assign convection-concerned matrixes and utilizing a more appropriate optimal objective to address the design-dependent loads and heat transfer properties induced by convection changes.

The effectiveness of the proposed topological method is illustrated using cases with complex thermal boundary conditions (the

In the

In the

Without convection, the equivalent temperature load is independent of
topological variables, and Eq. (21) can be simplified as follows:

The original code and data used and/or analyzed in the current study are available upon reasonable request from the corresponding author.

YG: conceptualization, methodology and writing – original draft preparation; DW: project administration; TG: data curation; XL: investigation; XY: validation; GX: resources; LC: supervision, conceptualization and writing – review and editing.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors gratefully acknowledge financial support from the Independent Innovation Foundation of the Aero Engine Corporation of China (AECC).

This research has been supported by the Independent Innovation Foundation of the AECC (grant no. ZZCX-2018-017).

This paper was edited by Haiyang Li and reviewed by Guangrong Chen and three anonymous referees.