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Abstract

In this article, a design sensitivity analysis method is developed for topology optimization of steady-state conductive thermal problems subject to design-dependent thermal loads using density gradients–based boundary detection. In a real physical heat conduction problem, a design-dependent thermal load through the convective boundary is an important factor; however, it is not easy to impose boundary conditions in heat conduction problem due to the difficulties of exact boundary representation. Here, the detection of convection boundaries is available by the density distinction between void and materials, which is similar to a gradient operator in image processing. Applying density gradients and Dirac delta function of densities, we impose a design-dependent load effect on thermal analysis and perform the design sensitivity analysis for topology optimization. At each optimization process, it is noted that the results of thermal analysis and the design sensitivities are influenced by the design-dependent thermal load. Through demonstrative numerical examples, the proposed approach using density gradients is proven to work yielding meaningful optimization results. The developed approach is applicable to the plant engineering and nuclear industries where thermo-mechanical coupling occurs inevitably by the thermal convection.

Keywords

Topology design optimization design-dependent load heat conduction design sensitivity analysis boundary detection

Introduction

A topology design optimization method is to search for a suitable material layout to maximize the system performances under design constraints. Since a homogenization-based topology optimization approach has been introduced by Bendsøe and Kikuchi,¹ many topology optimization methods of structural and thermal problems have been developed.² Among them, the density-based optimization method is known easy for implementation perspective than others. In this method, a bulk material density as a design variable varies from zero to unity representing the void state and solid state, respectively, to define the material in the continuum design domain.^3–5

Topology optimization in heat transfer problem has been an active topic during the last 20 years. The review paper by Dbouk⁶ summarizes well the researches for optimizing two-dimensional and three-dimensional thermal problems based on conductive,^7–10 convective,^11,12 and conjugate heat transfer.^13,14 In the heat conduction problem, there are three representative loads – internal thermal flux, heat generation, and heat transfer through conduction or convection boundaries. The internal thermal flux is a dead load about to design, and others are affected by the design change during the optimization process that is called a design-dependent load. Generally, there have been many trials to apply them optimization process. Gersborg-Hansen et al.¹⁵ applied design-dependent loads as internal heat generation–related domain, using finite volume method. Bruns¹⁶ considered the heat convection which arises to bulk material density over the domain, using finite volume method. Iga et al.¹⁷ introduced heat transfer coefficients concerning shape dependencies. Joo et al.¹⁸ adopted a surrogate model to consider natural convection in the topology optimization process. Recently, Makhija and Beran¹⁹ performed a concurrent shape and topology optimization for steady conjugate heat transfer problem. Convective heat transfer is design-dependent by nature and the accurate boundary consideration is a core part of this subject.

To consider the convection heat transfer accurately through the boundary, we adopt an implicit boundary concept similar to a level set function.^20,21 Detection of convection boundaries is available by the density distinction between void and materials, which is similar to a gradient operator in image processing. Applying density gradients and Dirac delta function of densities, we impose a design-dependent load effect to heat conduction analysis and perform a design sensitivity analysis (DSA) for topology optimization. At each optimization process, it is noted that results of thermal analysis and design sensitivities are influenced by the design-dependent thermal load due to the boundary change. The proposed boundary detection technique with density gradients provides a simple but accurate boundary representation applicable in the topology optimization with the design-dependent thermal loads.

Heat conduction problems and convective boundary detection

We consider a body occupying an open domain $Ω$ in a space bounded by a closed boundary Γ, as shown in Figure 1. Isotropic material properties are assumed in the domain $Ω$ . Three boundaries of a temperature boundary $Γ_{T}^{0}$ , a flux boundary $Γ_{T}^{1}$ , and a convective boundary $Γ_{T}^{2}$ are considered, where boundaries are mutually disjointed. The rate of internal heat generation Q is assumed. A prescribed boundary temperature $T_{0}$ on $Γ_{T}^{0}$ and a prescribed heat flux $q$ on $Γ_{T}^{1}$ are imposed and ambient temperature $T_{\infty}$ on the convection boundary $Γ_{T}^{2}$ is considered. $n$ represents an outward unit vector normal to the boundaries.

Figure 1.

Heat conduction body in space.

Here a steady-state heat conduction equation with the internal heat generation Q in the body is written as²²

- κ T_{, ii} = Q (i = 1, 2, 3)

(1)

where T is a temperature field and $κ$ is a positive thermal conductivity and assumed independent of temperature. Three kinds of boundary conditions, temperature, flux on the boundary, and convective heat transfer are applied on the surface of the body as²²

T = T_{0} on Γ_{T}^{0}

(2)

κ T_{, i} n_{i} = q on Γ_{T}^{1}

(3)

and

κ T_{, i} n_{i} + h (T - T_{\infty}) = 0 on Γ_{T}^{2}

(4)

where $h$ is a positive convection coefficient.

Using the virtual temperature field $\bar{T}$ satisfying homogeneous boundary conditions, the weak form of equation (1) is written by²²

\int_{Ω} (- κ T_{, ii} - Q) \bar{T} d Ω = 0 for all \bar{T} \in \bar{Y}

(5)

Equation (5) is regarded as a principle of virtual work. Applying the boundary conditions in equations (2)–(4), equation (5) can be rewritten as²²

\begin{matrix} \int_{Ω} κ δ_{ij} T_{, i} {\bar{T}}_{, j} d Ω - \int_{Ω} Q \bar{T} d Ω - \int_{Γ_{T}^{1}} q \bar{T} d Γ \\ + \int_{Γ_{T}^{2}} h (T - T_{\infty}) \bar{T} d Γ = 0, for all \bar{T} \in \bar{Y} \end{matrix}

(6)

A bilinear thermal energy form is defined as²²

A (T, \bar{T}) \equiv \int_{Ω} T_{, i} κ δ_{ij} {\bar{T}}_{, j} d Ω + \int_{Γ_{T}^{2}} hT \bar{T} d Γ

(7)

and a linear load form is given by²²

L (\bar{T}) \equiv \int_{Ω} Q \bar{T} d Ω + \int_{Γ_{T}^{1}} q \bar{T} d Γ + \int_{Γ_{T}^{2}} h T_{\infty} \bar{T} d Γ

(8)

Equation (6) can be rewritten as

Find T \in Y such that A (T, \bar{T}) = L (\bar{T}) for all \bar{T} \in \bar{Y}

(9)

During the optimization process, the heat convection boundary plays a critical role and needs to be detected to capture the topological change exactly. Here, we adopt Dirac delta function to apply the design-dependent effect. Using these functions, boundary integral terms in equations (7) and (8) can be converted²³

A (T, \bar{T}) \equiv \int_{Ω} κ \nabla T \cdot \nabla {\bar{T}}_{, j} d Ω + \int_{Ω} hT \bar{T} δ (ϕ) | \nabla ϕ | d Ω

(10)

L (\bar{T}) \equiv \int_{Ω} Q \bar{T} d Ω + \int_{Γ_{T}^{1}} q \bar{T} d Γ + \int_{Ω} h T_{\infty} \bar{T} δ (ϕ) | \nabla ϕ | d Ω

(11)

where $ϕ = ρ - ρ_{m}$ , $ρ_{m}$ is the density of solid material criterion. This process is the same used in the level set methods using the signed distance function.²³ It can be expected two advantageous effects. One is the indirect calculation, converting boundary into domain term, and the other is detecting convection boundary. The former is boundary integration is replaced for equivalent volume integration to implement easily. Approximate functions for $δ (x)$ are followed as²⁴

δ (x) = {\begin{matrix} \frac{3 (1 - α)}{4 Δ} (1 - \frac{x^{2}}{Δ^{2}}) if | x | \leq Δ \\ 0 if | x | > Δ \end{matrix}

(12)

where a small positive number $α$ is adopted to avoid a numerical singularity and the support length of Dirac delta function is defined by $Δ$ . The latter merit is, above properties of approximate function facilitate convection boundary detection that density of material criterion ranges from $ρ_{m} - Δ$ to $ρ_{m} + Δ$ . Density values within the support length $Δ$ are considered as boundaries during optimization. The larger the support length $Δ$ is, the more the detected boundary is blended. The value of $Δ$ is determined by the mesh size. The term $δ (ϕ) | \nabla ϕ |$ is similar to a gradient operator usually used in the edge detection of images. In image processing, once gradient values of pixels are computed, pixels having large gradient values become the possible boundary edge pixels, traced in the perpendicular direction to the gradient direction. In this article, the same concept is adopted to detect the solid boundary implicitly using the density gradients. Figure 2 shows graphically how the boundary detection works with the dependency of the supporting length. For a given density distribution, the contour of $δ (x)$ is represented by varying $Δ$ parameters. The element size is 0.1 and other parameters $α$ and $ρ_{m}$ are set as $5.0 \times 10^{- 4}$ and 0.5, respectively. The number of elements for the convection boundary varies based on what size of the support length of Dirac delta function is chosen.

Figure 2.

Boundary detection using density gradients: (a) given density distribution, (b) detected boundary at $Δ = 5.0 \times 10^{- 3}$ , (c) detected boundary at $Δ = 2.0 \times 10^{- 2}$ , and (d) detected boundary at $Δ = 5.0 \times 10^{- 2}$ .

Continuum-based design sensitivity analysis

Direct differentiation method

In this section, a shape sensitivity analysis considering a non-shape design variable vector $u$ representing the thermal conductivity of each element is considered using a direct differentiation method. For the given design $u$ , equation (9) can be written as

Find T \in Y such that A_{u} (T, \bar{T}) = L_{u} (\bar{T}) for all \bar{T} \in \bar{Y}

(13)

where the subscript u indicates the design dependence. The variational equation to the perturbed design $u + τ δ u$ is stated as²²

A_{u + τ δ u} (T_{τ}, \bar{T}) = L_{u + τ δ u} (\bar{T}) for all \bar{T} \in \bar{Y}

(14)

Using equation (14), we obtain the first-order variations in equation (13) with respect to the design variable $u$ as²²

A'_{δ u} (T, \bar{T}) \equiv {\frac{d}{d τ} A_{u + τ δ u} ({\tilde{T}}_{τ}, \bar{T}) |}_{τ = 0}

(15)

and

L'_{δ u} (\bar{T}) \equiv {\frac{d}{d τ} L_{u + τ δ u} (\bar{T}) |}_{τ = 0}

(16)

where the symbol “∼” indicates the suppressed dependence on design variation.

By the chain rule of differentiation and equation (15), the following equality holds²²

{\frac{d}{d τ} [A_{u + τ δ u} (T (u + τ δ u), \bar{T})] |}_{τ = 0} = A'_{δ u} (T, \bar{T}) + A_{u} (T', \bar{T})

(17)

The first-order variation of equation (15) using equations (16) and (17) gives²²

A_{u} (T', \bar{T}) = L'_{δ u} (\bar{T}) - A'_{δ u} (T, \bar{T}) for all \bar{T} \in \bar{Y}

(18)

Next, a general performance functional written in integral form is considered as²²

ψ = \int_{Ω} g_{1} (u, T, \nabla T) d Ω + \int_{Γ} g_{2} (u, T, \nabla T) d Γ

(19)

By taking the first-order variation of the performance functional gives the following expression²²

\begin{matrix} ψ' = {\frac{d}{d τ} ψ (u + τ δ u) |}_{τ = 0} = \frac{d ψ}{d u} δ u \\ = {\frac{d}{d τ} [\int_{Ω} g_{1} {u + τ δ u, T (u + τ δ u), \nabla T (u + τ δ u)} d Ω + \int_{Γ} g_{2} {u + τ δ u, T (u + τ δ u), \nabla T (u + τ δ u)} d Γ] |}_{τ = 0} \\ = \int_{Ω} (g_{1, u} δ u + g_{1, T} T' + g_{1, \nabla T} \nabla T') d Ω + \int_{Γ} (g_{2, u} δ u + g_{2, T} T' + g_{2, \nabla T} \nabla T') d Γ \end{matrix} (20)

Through this article, the prime symbol represents the first-order variation in the calculus of variations.

Adjoint variable method

Next, we define an adjoint equation for the given heat conduction problems. The adjoint equation is obtained by replacing the implicit dependence terms in equation (20) by a virtual temperature $\bar{λ}$ and equating the terms involving $\bar{λ}$ to the bilinear thermal energy form $A_{u} (λ, \bar{λ})$ as²²

\begin{matrix} A_{u} (λ, \bar{λ}) & = \int_{Ω} (g_{1, T} \bar{λ} + g_{1, \nabla T} \nabla \bar{λ}) d Ω \\ + \int_{Γ} (g_{2, T} \bar{λ} + g_{2, \nabla T} \nabla \bar{λ}) d Γ for all \bar{λ} \in \bar{Y} \end{matrix}

(21)

Since $\bar{T} and T' \in \bar{Y}$ and $λ and \bar{λ} \in \bar{Y}$ , equations (18) and (21) can be rewritten as²²

A_{u} (T', λ) = L'_{δ u} (λ) - A'_{δ u} (T, λ) for all λ \in \bar{Y}

(22)

and

\begin{matrix} A_{u} (λ, T') = \int_{Ω} (g_{1, T} T' + g_{1, \nabla T} \nabla T') d Ω \\ + \int_{Γ} (g_{2, T} T' + g_{2, \nabla T} \nabla T') d Γ for all T' \in \bar{Y} \end{matrix}

(23)

Utilizing the property of a symmetric operator, $A_{u} (•, •)$ , we obtain the following equation²²

\begin{matrix} \int_{Ω} (g_{1, T} T' + g_{1, \nabla T} \nabla T') d Ω + \int_{Γ} (g_{2, T} T' + g_{2, \nabla T} \nabla T') d Γ \\ = L'_{δ u} (λ) - A'_{δ u} (T, λ) \end{matrix}

(24)

Substituting equation (24) into equation (20), we have²²

ψ' = \int_{Ω} g_{1, u} δ u d Ω + \int_{Γ} g_{2, u} δ u d Γ + L'_{δ u} (λ) - A'_{δ u} (T, λ)

(25)

The computational efficiency and accuracy of the obtained adjoint equation are demonstrated in “Numerical examples” section.

Formulation of topology design optimization with convection boundary

In the topology optimization of thermal systems, the objective is to find the material distribution minimizing the thermal energy in the system under prescribed thermal loadings. A normalized bulk material density function u defines the material distribution. It has a continuous value from zero to one that implies void state and solid material, respectively. In the finite element method in which the material properties shape functions are defined inside each element, the bulk material densities are constant in each element and they are used as design variables associated with the thermal conductivity as

κ_{i} = u_{i}^{P} κ_{0}, (i = 1, 2, \dots, NE)

(26)

0 < u_{min} \leq u_{i} \leq 1

(27)

where $κ_{0}$ is the thermal conductivity and P is a penalty parameter for a concentrated distribution of material. To avoid the numerical singularity, the lower bound of material distribution, $ρ_{min}$ , is introduced. A standard form of topology design optimization heat conduction problem is stated as²²

Minimize Π = \int_{Ω} QTd Ω + \int_{Γ_{T}^{1}} qTd Γ

(28)

Subject to \int_{Ω} ud Ω \leq V_{allowable}

(29)

where $T$ , $Q$ , $q$ , $T_{\infty}$ , and $V_{allowable}$ are a temperature field, an internal heat generation, a prescribed heat flux, a given ambient temperature, and an allowable volume by the design, respectively.

For the structural or thermal optimization problems dealing with convex problems, gradient-based optimization methods are preferred. It requires the design sensitivity, that is, the change of the performance measure with respect to the design variables. For this purpose, the continuum-based adjoint variable method is known to be the most efficient and accurate.

For thermal compliance functional

\begin{matrix} ψ = \int_{Ω} g_{1} (u, T, \nabla T) d Ω + \int_{Γ} g_{2} (u, T, \nabla T) d Γ \\ = L_{u} (T) = Π \end{matrix}

(30)

The adjoint equation for thermal compliance is rewritten as

\begin{matrix} A_{u} (λ, \bar{λ}) = \int_{Ω} (g_{1, T} \bar{λ} + g_{1, \nabla T} \nabla \bar{λ}) d Ω \\ + \int_{Γ} (g_{2, T} \bar{λ} + g_{2, \nabla T} \nabla \bar{λ}) d Γ \\ = \int_{Ω} Q \bar{λ} d Ω + \int_{Γ_{T}^{1}} q \bar{λ} d Γ \end{matrix}

(31)

The first-order variation of the performance functional, that is, compliance sensitivity using equation (25) is obtained as

ψ' = \int_{Ω} g_{1, u} δ u d Ω + \int_{Γ} g_{2, u} δ u d Γ + L'_{δ u} (λ) - A'_{δ u} (T, λ)

(32)

Using equation (31), equation (32) is reduced to

\begin{matrix} ψ' = \int_{Ω} Q λ d Ω + \int_{Γ_{T}^{1}} q λ d Γ + \int_{Ω} h T_{\infty} λ δ (ϕ) | \nabla ϕ | d Ω \\ - \int_{Ω} h T_{\infty} T \nabla \cdot (δ (ϕ) \frac{\nabla ϕ}{| \nabla ϕ |}) λ d Ω \\ + \int_{Ω} h T_{\infty} T δ (ϕ) \frac{\nabla ϕ \cdot \nabla λ}{| \nabla ϕ |} d Ω + \\ - \int_{Ω} κ' \nabla T \cdot \nabla λ d Ω + \int_{Ω} hT λ \nabla \cdot (δ (ϕ) \frac{\nabla ϕ}{| \nabla ϕ |}) λ d Ω \\ - \int_{Ω} hT λ δ (ϕ) \frac{\nabla ϕ \cdot \nabla λ}{| \nabla ϕ |} d Ω \end{matrix}

(33)

Numerical examples

Example 1: design sensitivity verification with convection boundary

This example is to verify the proposed DSA method in the temperature field and the effects of applying the convection boundary to optimization. Consider a square plate composed of 2500 plane elements and 2601 nodes as shown in Figure 3. The ambient temperature is 0°C. Heat flux, $q = 1000 W / m^{2}$ , is applied at the left corner and right corner of the bottom side, and internal heat generator $Q = 100 W / m^{3}$ is distributed uniformly over the domain. The thermal conductivity coefficient of $κ = 1.4 W / m ° C$ and the convective heat transfer coefficient of $h = 0.5 W / m^{2} ° C$ are used in this problem. During the optimization step, one material distribution is selected in Figure 4(a). From this bulk density distribution, the heat convection boundary is easily obtained as shown in Figure 4(b). The thermal conductivity coefficient is selected as the design variable for the verification purpose of design sensitivity and performance measure is the thermal compliance function defined in the paper.

Figure 3.

Square plate subjected to thermal boundary conditions.

Figure 4.

(a) Bulk material density distribution and (b) heat convection boundary.

The thermal compliance sensitivity of each element with respect to the design variable is plotted in Figure 5. As expected, the sensitivity values of the thermal compliance are sensitive at a high material density or heat convection boundary.

Figure 5.

Thermal compliance sensitivity.

The obtained sensitivity values by the adjoint variable method are compared with the finite difference sensitivities during the optimization process in Table 1. $d Π / d τ$ stands for the analytical sensitivity using the adjoint variable method (AVM) and $Δ Π / Δ τ$ represents the central finite difference sensitivity. Noting that Ⓐ, Ⓓ, Ⓖ, Ⓙ are heat convection applied element at detected boundaries; excellent agreements are observed.

Table 1.

Design sensitivity comparison.

	$\frac{d Π}{d τ}$	$\frac{Δ Π}{Δ τ}$	$\frac{Δ Π}{Δ τ} / \frac{Δ Π}{Δ τ} \frac{d Π}{d τ} \frac{d Π}{d τ} (%)$		$\frac{d Π}{d τ}$	$\frac{Δ Π}{Δ τ}$	$\frac{Δ Π}{Δ τ} / \frac{Δ Π}{Δ τ} \frac{d Π}{d τ} \frac{d Π}{d τ} (%)$
Ⓐ	−1345.64	−1337.10	100.64	Ⓕ	−930.55	−928.13	100.26
Ⓑ	−535.38	−533.59	100.34	Ⓖ	−117.30	−117.08	100.18
Ⓒ	−20,503.45	−20,444.76	100.29	Ⓗ	−169.66	−169.19	100.28
Ⓓ	−746.16	−744.69	100.20	Ⓘ	−88,731.85	−87,798.51	101.06
Ⓔ	−738.73	−736.86	100.25	Ⓙ	−3099.53	−3079.31	100.66

Only 0.15 s is required for the AVM, which does only 0.05% of CPU time for the finite difference one in Table 2. This huge difference is caused by the heat convection term calculation related $δ (ϕ) | \nabla ϕ |$ in finite difference method (FDM).

Table 2.

CPU time comparison.

CPU time (s)
FDM	AVM
319.40	0.16

FDM: finite difference method; AVM: adjoint variable method.

Example 2: topology optimization with convection boundary

To understand the effect of convective heat flux, two topology optimization examples are considered for the coefficients of convective heat transfer, $h = 0.02 W / m^{2} ° C$ and $h = 0.6 W / m^{2} ° C$ . For two cases, the design domain, boundary and loading conditions are shown in Figures 6 and 8, 2500 plane square elements and 2601 nodes are used in each model and the ambient temperature is 0°C. The same material properties as the previous example’s are used and the bulk material densities are the design variable. The penalty parameter of value 2 for the topology optimization is used in the numerical analysis. The volume constraint is 30% of the original one. Results without convective heat transfer and with convective heat transfer are represented. Through topological optimization result and temperature distribution, we can notice differences whether convection boundary effect works or not.

Figure 6.

Topology design results by heat convection boundary ( $h = 0.02 W / m^{2}^{°} C$ ): (a) design domain and initial boundary conditions, (b) result without heat convection, (c) result with heat convection, (d) temperature without eat convection, and(e) temperature with heat convection.

The first case solves a square domain model subjected to heat flux $q = 1000 W / m^{2}$ , a uniformly distributed heat generation $Q = 100 W / m^{3}$ and heat convection boundary of the convection coefficient, $h = 0.02 W / m^{2} ° C$ . The parameter $ρ_{m}$ is chosen properly according to the changed bulk material density and criterion of considered boundary during the optimization. The final material distributions and temperature distribution at the final stage of the topology optimization are also shown in Figure 5. By the convection heat transfer through the boundary, different topology results are obtained. In heat conduction problems, the purpose of optimization is to find the optimal material layout yielding thermally stiff structure. In other words, we want to find the material layout difficult to increase the temperature in the system for the given thermal loading and boundary conditions. Figure 6(b) and (c) shows the optimized density distribution without and with heat convection, respectively.

Due to the small value of the convection coefficient, the difference between the two cases is not much. To investigate the effect of topology optimization, the temperature distributions in the two cases are compared. From Figures 6(d) and (e), the overall temperature with the convection boundary is lower that one without convection boundary, as is expected. It can be thought that thermal energy can flow in and out more easily, through the convection boundary. As shown, the effect of heat convection is clearer in the comparison of temperature distribution, showing a smooth transition of temperature profiles.

Figure 7 shows the optimization history without convection boundary and with convection boundary. As we can surmise from Figure 6(d) and (e), the thermal compliance with convection boundary has a smaller value than without one. Different to Example 1 with a simple boundary, the support length $Δ$ plays an important role to detect boundaries in Example 2. Qualitatively, the larger support length $Δ$ results in the blended boundary. The fluctuation of the objective function in Figure 7(b) is caused by the change of the convection boundary.

Figure 7.

Optimization history for case 1 ( $h = 0.02 W / m^{2}^{°} C$ ): (a) compliance history without heat convection and (b) compliance history with heat convection.

To assess the effect of the convective heat transfer, the same rectangular model is evaluated using the different value of the convection coefficient, $h = 0.6 W / m^{2} ° C$ . Same values of heat flux $q = 1000 W / m^{2}$ at each corner, a uniformly distributed heat generation $Q = 100 W / m^{3}$ is used. The two final material distributions and temperature field after the topology optimization are also shown in Figure 8. Figure 8(b) and (c) shows the optimized density distribution without and with heat convection, respectively. Compared to the previous example with the heat convection coefficient, $h = 0.02 W / m^{2} ° C$ , the convection heat transfer is dominant with the higher heat convection coefficient, $h = 0.6 W / m^{2} ° C$ , which results in the longer convection boundary development or perimeter as shown in Figure 8(c). Due to the effect of the convection heat transfer, the optimal model (Figure 8(c) and (e)) shows a low-temperature distribution throughout the whole domain. In Figure 8(e), as expected, the temperature distribution shows a smooth transition of temperatures with the convective heat transfer.

Figure 8.

Topology design results by heat convection boundary ( $h = 0.6 W / m^{2}^{°} C$ ): (a) design domain and initial boundary conditions, (b) result without heat convection, (c) result with heat convection, (d) temperature without heat convection, and(e) temperature with heat convection.

Figure 9 shows the optimization history without convection boundary and with convection boundary. Thermal compliance with convection boundary goes somewhat up and down, but it converges to a smaller value than without one.

Figure 9.

Optimization history for case 2 ( $h = 0.6 W / m^{2}^{°} C$ ): (a) thermal compliance history without heat convection and(b) thermal compliance history with heat convection.

Conclusion

To describe the boundary geometry in the topology optimization of conductive thermal problems with heat convection, we propose the density gradients–based boundary detection approach. Using the proposed technique, a weak formulation for conductive thermal problems including heat convection in steady state and the continuum-based adjoint DSA method are derived for topology optimization. In heat conduction problem subject to design-dependent load, the sensitivity term of convective heat flow plays a role significantly if the temperature difference between solid and fluid is much.

Through several numerical examples, the accuracy of the analytical adjoint DSA method is compared to the finite difference sensitivity showing good agreement. Through demonstrative numerical examples, the proposed approach using density gradients is proven to work yielding meaningful optimization results. In the consideration of design-dependent thermal loading, we obtain different topology designs depending on the heat flow through the convection boundary. The proposed boundary detection technique with density gradients provides a simple but accurate boundary representation applicable in the topology optimization problems subject to the thermally design-dependent loads and it is applicable to the plant engineering and nuclear industries where thermo-mechanical coupling occurs inevitably by the thermal convection.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science, ICT, and Future Planning) (no. NRF-2018R1D1A1B07050370), and the Human Resources Development Program (grant no. 20174010201350) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grants funded by the Korea government (Ministry of Trade, Industry, and Energy).

ORCID iD

Bonyong Koo

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