Topology optimization of thermal problems in a nonsmooth variational setting: closed-form optimality criteria

Abstract

This paper extends the nonsmooth Relaxed Variational Approach (RVA) to topology optimization, proposed by the authors in a preceding work, to the solution of thermal optimization problems. First, the RVA topology optimization method is briefly discussed and, then, it is applied to a set of representative problems in which the thermal compliance, the deviation of the heat flux from a given field and the average temperature are minimized. For each optimization problem, the relaxed topological derivative and the corresponding adjoint equations are presented. This set of expressions are then discretized in the context of the finite element method and used in the optimization algorithm to update the characteristic function. Finally, some representative (3D) thermal topology optimization examples are presented to asses the performance of the proposed method and the Relaxed Variational Approach solutions are compared with the ones obtained with the level set method in terms of the cost function, the topology design and the computational cost.

Appendices

Appendix A: Finite element discretization

The finite element method (FEM) is used to discretize and solve the state-equation (18) and the required adjoint problems. The temperature field in $\varOmega$ is approximated via $C_0$ shape functions as follows²¹:

$$\begin{aligned} \varvec{\theta }_{{\chi }}(\mathbf{x })\equiv {\mathbf {N}}_{\theta }(\mathbf{x }){\hat{\varvec{\theta }}}_{\chi } \end{aligned}$$

(A.1)

where ${\mathbf {N}}_{\theta }(\mathbf{x })$ is the, temperature, shape-function matrix and ${{\hat{\varvec{\theta }}}}_{\chi }$ corresponds to the nodal temperature vector. Equivalently, the gradient of $\varvec{\theta }_{{\chi }}(\mathbf{x })$ is expressed as

$$\begin{aligned} {\varvec{\nabla } \theta }_{{\chi }}(\mathbf{x })\equiv {\mathbf {B}}(\mathbf{x }){{\hat{\varvec{\theta }}}}_{\chi } \end{aligned}$$

(A.2)

where ${\mathbf {B}}(\mathbf{x })$ denotes the gradient matrix. Then, introducing expressions (A.1) and (A.2) into the Fourier’s law, the heat flux, $\mathbf{{q}}_{\chi }(\mathbf{x })$, can be written as

$$\begin{aligned} \varvec{q}_{{\chi }}(\mathbf{x })\equiv -{{{\varvec{\kappa }}}}_{{\chi }}(\mathbf{x })\ {\mathbf {B}}(\mathbf{x }){{\hat{\varvec{\theta }}}}_{\chi } { \, . }\end{aligned}$$

(A.3)

Finally, the state Eq. (18), once the previous expressions are replaced, yields to

$$\begin{aligned} {\mathbb {K}}_{{\chi }}{{\hat{\varvec{\theta }}}}_{{\chi }}=\mathbf {f} \end{aligned}$$

(A.4)

with

$$\begin{aligned} \left\{ \begin{aligned}&\begin{aligned} {\mathbb {K}}_{{\chi }}=&\int _{\varOmega }{\mathbf {B}}^T(\mathbf{x })\ {{{\varvec{\kappa }}}}_{{\chi }}(\mathbf{x })\ {\mathbf {B}}(\mathbf{x })\,d{\varOmega }\\&-\int _{\partial _h\varOmega }\mathbf {N}_{\theta }^T(\mathbf{x })h\mathbf {N}_{\theta }(\mathbf{x }){ \, d\varGamma }\end{aligned} \\&\begin{aligned} \mathbf {f}=&\int _{\varOmega } \mathbf {N}_{\theta }^T(\mathbf{x }){r}_{{\chi }}(\mathbf{x }){ \, d\varOmega }\\&-\int _{\partial _{q}\varOmega }\mathbf {N}_{\theta } ^T(\mathbf{x }){\overline{q}}(\mathbf{x }){ \, d\varGamma }\\&-\int _{\partial _h\varOmega }\mathbf {N}_{\theta }^T(\mathbf{x })h{\theta _{amb}(\mathbf{x })}{ \, d\varGamma }\end{aligned} \end{aligned} \right. { \, , }\end{aligned}$$

(A.5)

where $\mathbb {K}_{{\chi }}$ and $\mathbf {f}$ stand for the stiffness matrix and the external forces vector, respectively.²²

A Laplacian smoothing is used to smooth the topology, control the filament size and avoid checkerboard patterns. The smooth discrimination function, ${\psi }_\tau$, corresponds to the solution of

$$\begin{aligned} \left\{ \begin{aligned}&{\psi }_{\tau }(\mathbf{x })-\epsilon ^2\varDelta _{{\mathbf{x}}}{\psi }_{\tau }(\mathbf{x })={\psi }(\mathbf{x })&\quad in \; \varOmega \\&\nabla _{{\mathbf{x}}}{\psi }_{\tau }(\mathbf{x })\cdot {\mathbf {n}}={0}&\quad on \; \partial \varOmega \end{aligned} \right. { \, , }\end{aligned}$$

(A.6)

where, $\varDelta _{{\mathbf{x}}}({{\mathbf{x}}},n)$ and $\nabla _{{\mathbf{x}}}({{\mathbf{x}}},n)$ stand for the Laplacian and gradient operators, respectively, and ${\mathbf {n}}$ is the outwards normal to the boundary of the analysis domain, $\partial \varOmega$. The FE discretization of Eq. (A.6), considering ${\psi }_\tau (\mathbf{x })={\mathbf {N}}(\mathbf{x }){\hat{\varvec{\psi }}_\tau }$, leads to the following system

$$\begin{aligned} \hat{\varvec{\psi }}_\tau ={\tilde{\mathbb {G}}}^{-1}{\mathbf{f}}({\psi }) \end{aligned}$$

(A.7)

with

$$\begin{aligned} \left\{ \begin{aligned}&\tilde{\mathbb {G}}=\tilde{\mathbb {M}}+{\epsilon ^{2}}\tilde{\mathbb {K}} \quad \rightarrow \\&\quad \rightarrow \quad \left\{ \begin{aligned}&\tilde{\mathbb {M}}=\int _{\varOmega }{\mathbf{N}}^T(\mathbf{x }){\mathbf {N}}(\mathbf{x }){ \, d\varOmega }\;\\&\tilde{\mathbb {K}}=\int _{\varOmega }\nabla {\mathbf{N}}^T(\mathbf{x }){\nabla {\mathbf{N}}(\mathbf{x }){ \, d\varOmega }}\;\\ \end{aligned} \right.&(a) \\&{\mathbf{f}}({\psi })=\int _\varOmega {{\mathbf {N}}^T(\mathbf{x }){\psi }(\mathbf{x }){ \, d\varOmega }}&(b) \end{aligned} \right. \end{aligned}$$

(A.8)

where ${\mathbf {N}}(\mathbf {x})$ stands for the standard interpolation matrix and ${\varvec{\hat{{\psi }}}}_\tau$ is the vector of nodal values of the field ${\psi }_\tau (\mathbf{x })$.

Appendix B: Thermal compliance minimization: cost function derivative

The topological sensitivity of the thermal compliance optimization problem (Eq. 24) is computed in detail in this section via the adjoint method and the Relaxed Topological Derivative (RTD). Let first rephrase the objective function, ${\mathcal {J}}^{(h_e)}({\chi })$, to incorporate the state Eq. (A.4)

$$\begin{aligned} \overline{\mathcal {J}}^{(h_e)}({\chi })=\frac{1}{2} \mathbf {f}^{T}{{\hat{\varvec{\theta }}}}_{{\chi }}- {\hat{\mathbf {w}}}^T \underbrace{\left( {\mathbb {K}}_{{\chi }}{{\hat{\varvec{\theta }}}}_{{\chi }}-\mathbf {f}\right) } _{{=\mathbf {0}}} { \, , }\end{aligned}$$

(B.1)

where ${\hat{\mathbf {w}}}$ corresponds to the solution of the adjoint state problem, as aforementioned. Computing the RTD of Eq. (B.1) and reordering terms, one arrives to

$$\begin{aligned} \begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}= \left( \frac{1}{2}\mathbf {f}^{T}-{\hat{\mathbf {w}}}^T{\mathbb {K}}_{{\chi }}\right) \dfrac{\delta {{\hat{\varvec{\theta }}}}_{{\chi }}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\\&\quad + \left( \frac{1}{2}\dfrac{\delta \mathbf {f}^T _{{\chi }}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }}-{\hat{\mathbf {w}}}^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }} + {\hat{\mathbf {w}}}^T\dfrac{\delta \mathbf {f}_{{\chi }}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\right) { \, . }\end{aligned} \end{aligned}$$

(B.2)

Substituting ${\hat{\mathbf {w}}}\equiv {\dfrac{1}{2}}{{\hat{\varvec{\theta }}}}_{{\chi }}$ in Eq. (B.2), and considering the state Eq. (A.4), the expression can be simplified to

$$\begin{aligned} \begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}&=\frac{1}{2}\underbrace{(\mathbf {f}^{T}-{{\hat{\varvec{\theta }}}}_{{\chi }}^T{\mathbb {K}}_{{\chi }})} _{{=\mathbf {0}}}\dfrac{\delta {{\hat{\varvec{\theta }}}}_{{\chi }}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\\&\quad +\left( \dfrac{\delta \mathbf {f}^T _{{\chi }}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }}-{{\hat{\varvec{\theta }}}}_{{\chi }}^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }}\right) \\&=\left[ \dfrac{\delta \mathbf {f}^T _{{\chi }}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }}- {\hat{\varvec{\theta }}}_{{\chi }}^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}(\mathbf{x }){\hat{\varvec{\theta }}}_{{\chi }}\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}} { \, . }\end{aligned} \end{aligned}$$

(B.3)

Then, considering Eqs. (14)–(17) and replacing the corresponding terms into Eq. (B.3), the Relaxed Topological Derivative of Eq. (B.1) can be expressed as

$$\begin{aligned} \begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}=\dfrac{\partial {{r}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}{\mathbf {N}}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }} \varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\\&\qquad -{{\hat{\varvec{\theta }}}}^T_{{\chi }}{\mathbf {B}}^T{(\hat{\mathbf{x }})}\dfrac{\partial {{{\varvec{\kappa }}}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}{\mathbf {B}}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }}\varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}\\&\quad =\left[ \dfrac{\partial {{r}_{{\chi }}}}{\partial {\chi }}{\mathbf {N}}(\mathbf{x }){{\hat{\varvec{\theta }}}}_{{\chi }} \right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\\&\qquad -\left[ \varvec{\nabla }\theta ^T_{{\chi }}(\mathbf{x })\dfrac{\partial {{\varvec{\kappa }}}_{{\chi }}}{\partial {{\chi }}}\varvec{\nabla }\theta _{{\chi }}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}\\&\quad =\left[ m_{{r}}{{\chi }}^{m_{{r}}-1}(\mathbf{x }){r}(\mathbf{x }){\mathbf {N}}(\mathbf{x }){{\hat{\varvec{\theta }}}}_{{\chi }} \right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\\&\qquad -\left[ m_{\kappa }{{\chi }}^{m_{\kappa }-1}(\mathbf{x })\varvec{\nabla }\theta ^T_{{\chi }}(\mathbf{x }){{\varvec{\kappa }}}(\mathbf{x })\varvec{\nabla }\theta _{{\chi }}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}{{\varDelta {\chi }_{\kappa }}({\hat{{\mathbf{x}}}}) } { \, , }\end{aligned} \end{aligned}$$

(B.4)

which is then written in terms of energy densities, to recover Eq. (28), as

$$\begin{aligned} \begin{aligned} \dfrac{\delta {\overline{\mathcal {J}}^{(h_e)}(\theta _{{\chi }})}}{\delta {\chi }}{(\hat{\mathbf{x }})}&=m_{{r}}\left( {{\chi }}_{{r}}{(\hat{\mathbf{x }})}\right) ^{m_{{r}}-1}{\overline{{\mathcal {U}}}_{{r}}}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\\&\quad -2m_{\kappa } \left( {\chi }_{\kappa }{(\hat{\mathbf{x }})}\right) ^{m_{\kappa }-1}{\overline{{\mathcal {U}}}}{(\hat{\mathbf{x }})}{{\varDelta {\chi }}_{\kappa }({\hat{{\mathbf{x}}}}) } { \, , }\end{aligned} \end{aligned}$$

(B.5)

where $\overline{{\mathcal {U}}}{(\hat{\mathbf{x }})}$ is the nominal heat conduction energy density and ${\overline{{\mathcal {U}}}_{{r}}}{(\hat{\mathbf{x }})}$ is the nominal heat source energy density, as described in Eq. (29).

Appendix C: Thermal cloaking via heat flux manipulation: cost function derivative

This section describes step-by-step the topological sensitivity computation of the thermal cloaking optimization problem (34), mimicking the procedure explained in “Appendix B”. Let us then define the extended cost function, $\overline{{\mathcal {J}}}^{(h_e)}({\chi })$, i.e.

$$\begin{aligned} \begin{aligned} \overline{{\mathcal {J}}}^{(h_e)}({\chi })&=\Bigg (\underbrace{\int _{\varOmega } {1_{\varOmega _c}(\mathbf{x })}\left| \mathbf{q}_{{\chi }}\left( {{\mathbf{x}}},{\theta }_{\chi }^{(1)}\right) -{\overline{\mathbf{q}}}(\mathbf{x })\right| ^2 { \, d\varOmega }}_{E\left( {\chi },{\theta }_{\chi }^{(1)}\right) }\Bigg )^\frac{1}{2} \\&\quad -\hat{\mathbf{w}}^T\underbrace{\left( {\mathbb {K}}_{{\chi }}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)}-\mathbf {f}^{(1)}\right) } _{{=\mathbf {0}}} { \, , }\end{aligned} \end{aligned}$$

(C.1)

which is subsequently derived through the RTD, yielding to

$$\begin{aligned} \begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}=&\frac{1}{2}\dfrac{1}{{\mathcal {J}}^{(h_e)}({\chi })}\dfrac{\delta E({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}-{\hat{\mathbf {w}}}^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} \\&-{\hat{\mathbf {w}}}^T{\mathbb {K}}_{{\chi }}\dfrac{\delta {\hat{\varvec{\theta }}}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}+ {\hat{\mathbf {w}}}^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\end{aligned} \end{aligned}$$

(C.2)

where

$$\begin{aligned} \left\{ \begin{aligned}&\dfrac{\delta E({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}= \left[ 2 \ { {1_{\varOmega _c}(\mathbf{x })}\left( \mathbf{q}_{{\chi }}\left( {{\mathbf{x}}},{\theta }_{\chi }^{(1)}\right) -{\overline{\mathbf{q}}}(\mathbf{x })\right) \dfrac{\delta \mathbf{q}_{\chi }({\chi })}{\delta {\chi }}(\mathbf{x })} \right] _{\mathbf{x }={\hat{\mathbf{x }}}} { \, , }\\&\dfrac{\delta \mathbf{q}_{\chi }({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}= -\dfrac{\delta {{{\varvec{\kappa }}}}_{{\chi }}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}{\varvec{\nabla }\theta }_{{\chi }}^{(1)}{(\hat{\mathbf{x }})}-{{{\varvec{\kappa }}}}_{{\chi }}\nabla \dfrac{\delta {\varvec{\theta }}_{{\chi }} ^{(1)}}{\delta {\chi }}{(\hat{\mathbf{x }})}{ \, . }\end{aligned} \right. \end{aligned}$$

(C.3)

Introducing expressions (C.3) into Eq. (C.2), and manipulating the terms, we obtain

$$\begin{aligned}&\frac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}\nonumber \\&\quad = { \underbrace{\scriptstyle {\left( -{\hat{\mathbf {w}}}^T{\mathbb {K}}_{{\chi }}-{\mathbf{C_1}}\left( {\chi },{\hat{\mathbf{x }}},{\theta }_{\chi }^{(1)}\right) {{{\varvec{\kappa }}}}_{{\chi }}\nabla \right) }}_{\mathbf{=0}}} \frac{\delta {{\hat{\varvec{\theta }}}}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\nonumber \\&\qquad -{\mathbf{C_1}}\left( {\chi },{\hat{\mathbf{x }}},{\theta }_{\chi }^{(1)}\right) \dfrac{\delta {{{\varvec{\kappa }}}}_{{\chi }}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}{\varvec{\nabla }\theta }^{(1)}_{{\chi }}{(\hat{\mathbf{x }})}\nonumber \\&\qquad -{\hat{\mathbf {w}}}^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} +{\hat{\mathbf {w}}}^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{ \, , }\end{aligned}$$

(C.4)

with

$$\begin{aligned} {\mathbf{C_1}}\left( {\chi },{\hat{\mathbf{x }}},{\theta }_{\chi }^{(1)}\right) = \dfrac{{1_{\varOmega _c}{(\hat{\mathbf{x }})}} \left( \mathbf{q}_{{\chi }}\left( {\hat{\mathbf{x }}},{\theta }_{\chi }^{(1)}\right) -{\overline{\mathbf{q}}}{(\hat{\mathbf{x }})}\right) }{{\mathcal {J}}^{(h_e)}({\chi })} { \, . }\end{aligned}$$

(C.5)

Now, the adjoint problem of Eq.  (C.4) is solved for ${{\hat{\mathbf{w}}}}\equiv {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}$, leading to

$$\begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}= -{\mathbf{C_1}}\left( {\chi },{\hat{\mathbf{x }}},{\theta }_{\chi }^{(1)}\right) \dfrac{\delta {{{\varvec{\kappa }}}}_{{\chi }}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}{\varvec{\nabla }\theta }_{{\chi }}^{(1)}{(\hat{\mathbf{x }})}\nonumber \\&\quad -\left( {{\hat{\varvec{\theta }}}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{{\hat{\varvec{\theta }}}}_{{\chi }}^{(1)} + \left( {{\hat{\varvec{\theta }}}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{ \, . }\end{aligned}$$

(C.6)

After applying the RTD to the corresponding terms, Eq. (C.6) reads as

$$\begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}=\left[ \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {N}}^T(\mathbf{x })\dfrac{\partial {{r}_{{\chi }}}}{\partial {\chi }}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}} \varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\nonumber \\&\quad -\left[ \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\right) ^T{\mathbf {B}}^T(\mathbf{x })\dfrac{\partial {{{\varvec{\kappa }}}_{{\chi }}}}{\partial {\chi }}(\mathbf{x }){\mathbf {B}}(\mathbf{x }){\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}\nonumber \\&\quad -\left[ {\mathbf{C_1}}\left( {\chi },{{\mathbf{x}}},{\theta }_{\chi }^{(1)}\right) \dfrac{\partial {{{\varvec{\kappa }}}}_{{\chi }}}{\partial {\chi }}(\mathbf{x }){\mathbf {B}}(\mathbf{x }){\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}{ \, . }\end{aligned}$$

(C.7)

Subsequently, relations (14) and (15) are considered in Eq. (C.7), which yields to

$$\begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}=\left[ m_{{r}}{{\chi }}^{m_{{r}}-1}\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {N}}^T(\mathbf{x }){r}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\nonumber \\&\quad -\left[ m_{\kappa }{{\chi }}^{m_{\kappa }-1}\left( \varvec{\nabla }\theta ^{(1)}_{{\chi }}\right) ^T(\mathbf{x }){{\varvec{\kappa }}}(\mathbf{x })\varvec{\nabla }\theta _{{\chi }}^{(2)}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}{{\varDelta {\chi }_{\kappa }}({\hat{{\mathbf{x}}}}) } \nonumber \\&\quad -\left[ m_{\kappa }{{\chi }}^{m_{\kappa }-1}{\mathbf{C_1}}\left( {\chi },{{\mathbf{x}}},{\theta }_{\chi }^{(1)}\right) {{{\varvec{\kappa }}}}(\mathbf{x }){\mathbf {B}}(\mathbf{x }){\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}} \varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}{ \, . }\end{aligned}$$

(C.8)

Finally, Eq. (C.8) can be reformulated, in terms of pseudo-energies, as

$$\begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}&= m_{{r}}\left( {{\chi }}_{{r}}{(\hat{\mathbf{x }})}\right) ^{m_{{r}}-1}{\overline{{\mathcal {U}}}_{{r}}}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\nonumber \\&-2m_{\kappa } \left( {\chi }_{\kappa }{(\hat{\mathbf{x }})}\right) ^{m_{\kappa }-1}{\overline{{\mathcal {U}}}}_{1-2}{(\hat{\mathbf{x }})}{{\varDelta {\chi }}_{\kappa }({\hat{{\mathbf{x}}}}) } \nonumber \\&-m_{\kappa } \left( {\chi }_{\kappa }{(\hat{\mathbf{x }})}\right) ^{m_{\kappa }-1}{\overline{{\mathcal {U}}}}_\mathbf{q}{(\hat{\mathbf{x }})}{{\varDelta {\chi }}_{\kappa }({\hat{{\mathbf{x}}}}) } { \, , }\end{aligned}$$

(C.9)

where $\overline{{\mathcal {U}}}_{1-2}{(\hat{\mathbf{x }})}$ is the nominal heat conduction energy density, ${\overline{{\mathcal {U}}}_{{r}}}{(\hat{\mathbf{x }})}$ is the nominal heat source energy density and ${\overline{{\mathcal {U}}}}_\mathbf{q}{(\hat{\mathbf{x }})}$ corresponds to the nominal heat flux energy density, as defined in Eq. (43).

Appendix D: Average temperature minimization: cost function derivative

Let us now proceed with the computation of the topological sensitivity of the average temperature minimization problem (50). As before, let $\overline{\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })$ be the extended cost function, considering the state equation through the Lagrange multiplier vector, ${\hat{\mathbf {w}}}$, defined as

$$\begin{aligned} \overline{\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })=C_2 \mathbf{1}_{\partial _c\varOmega }^T{\hat{\varvec{\theta }}}_{\chi }^{(1)}-\hat{\mathbf{w}}^T\underbrace{\left( {\mathbb {K}}_{{\chi }}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)}-\mathbf {f}^{(1)}\right) } _{{=\mathbf {0}}} { \, , }\end{aligned}$$

(D.1)

where $C_2=\left( \int _{\partial _c\varOmega } { \, d\varGamma }\right) ^{-1}$.

Applying the RTD to Eq. (D.1) and reordering its terms, one obtains

$$\begin{aligned} \begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}&= \left( -{\hat{\mathbf {w}}}^T{\mathbb {K}}_{{\chi }}+C_2 \mathbf{1}_{\partial _c\varOmega }^T \right) \dfrac{\delta {{\hat{\varvec{\theta }}}}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\\ -&{\hat{\mathbf {w}}}^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} +{\hat{\mathbf {w}}}^T\dfrac{\delta \mathbf {f}_{{\chi }} ^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{ \, , }\end{aligned} \end{aligned}$$

(D.2)

which is then simplified by choosing $\hat{\mathbf{w}}\equiv -C_2 {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}$, yielding to

$$\begin{aligned} \begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}&= C_2 \underbrace{\left( \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbb {K}}_{{\chi }}+\mathbf{1}_{\partial _c\varOmega }^T \right) }_{\displaystyle \mathbf{=0}}\dfrac{\delta {{\hat{\varvec{\theta }}}}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\\&\quad \ +C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} \\&\quad \ -C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\\&=C_2 \Bigg (\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} \\&\quad \ - \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\Bigg ) { \, . }\end{aligned} \end{aligned}$$

(D.3)

Equation (D.3) is finally discretized using the expressions in Sect. 1, which then reads as

$$\begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}={ C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {B}}^T{(\hat{\mathbf{x }})}\dfrac{\partial {{{\varvec{\kappa }}}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}{\mathbf {B}}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}}\nonumber \\&\qquad -C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {N}}^T{(\hat{\mathbf{x }})}\dfrac{\partial {{r}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\nonumber \\&\quad = {C_2 \left[ m_{\kappa }{{\chi }}^{m_{\kappa }-1}(\mathbf{x })\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {B}}^T(\mathbf{x }){{\varvec{\kappa }}}(\mathbf{x }){\mathbf {B}}(\mathbf{x }){\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}{{\varDelta {\chi }_{\kappa }}({\hat{{\mathbf{x}}}}) }} \nonumber \\&\qquad -C_2 \left[ m_{{r}}{{\chi }}^{m_{{r}}-1}(\mathbf{x })\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {N}}^T(\mathbf{x }){r}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}} \varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}{ \, . }\end{aligned}$$

(D.4)

The Relaxed Topological Derivative of the cost function (50) can be finally expressed in terms of energy densities as

$$\begin{aligned} \begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}&=2 C_2 m_{\kappa } \left( {\chi }_{\kappa }{(\hat{\mathbf{x }})}\right) ^{m_{\kappa }-1}{\overline{{\mathcal {U}}}}_{1-2}{(\hat{\mathbf{x }})}{{\varDelta {\chi }}_{\kappa }({\hat{{\mathbf{x}}}}) } \\&\quad -C_2 m_{{r}}\left( {{\chi }}_{{r}}{(\hat{\mathbf{x }})}\right) ^{m_{{r}}-1}{\overline{{\mathcal {U}}}_{{r}-2}}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}{ \, , }\end{aligned} \end{aligned}$$

(D.5)

where $\overline{{\mathcal {U}}}_{1-2}{(\hat{\mathbf{x }})}$ and ${\overline{{\mathcal {U}}}_{{r}-2}}{(\hat{\mathbf{x }})}$ are, respectively, the nominal heat conduction energy density and the nominal heat source energy density, both defined in Eq. (54).

Appendix E: Temperature variance minimization: cost function derivation

Let us now address the corresponding RTD computation of the cost function for the minimization of the temperature variance (Eq. 56), starting by defining the extended cost function as

$$\begin{aligned} \begin{aligned} \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })&=C_3 \left( {\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) \right) ^T \mathbb {M}_{\partial _c\varOmega } {\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) \\&\quad -\hat{\mathbf{w}}^T\underbrace{\left( {\mathbb {K}}_{{\chi }}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)}-\mathbf {f}^{(1)}\right) }_{{=\mathbf {0}}} { \, , }\end{aligned} \end{aligned}$$

(E.1)

where ${\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right)$ and $\mathbb {M}_{\partial _c\varOmega }$ are respectively defined as

$$\begin{aligned}&{\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) = {\hat{\varvec{\theta }}}_{\chi }^{(1)}-{\mathbb {I}}\,{\mathcal {J}}^{(h_e)}_{\text {av}}\left( \theta _{\chi }^{(1)}\right) { \, , }\\&\mathbb {M}_{\partial _c\varOmega } = \int _{\partial \varOmega } \mathbf{N}^T(\mathbf{x }){1}_{\partial _c\varOmega } (\mathbf{x })\mathbf{N}(\mathbf{x }){ \, d\varGamma }{ \, . }\end{aligned}$$

Applying the RTD to Eq. (E.1) and rearranging the expression, one arrives to

$$\begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}\nonumber \\&\quad = {\overbrace{\left( -{\hat{\mathbf {w}}}^T{\mathbb {K}}_{{\chi }}+2 C_{3} \left( {\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) \right) ^T \mathbb {M}_{\partial _c\varOmega } \right) }^{\displaystyle \mathbf{=0}}}\dfrac{\delta {{\hat{\varvec{\theta }}}}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\nonumber \\&\qquad -2 C_3 \left( {\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) \right) ^T \mathbb {M}_{\partial _c\varOmega } {\mathbb {I}} \dfrac{\delta {\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}\nonumber \\&\qquad -{\hat{\mathbf {w}}}^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} +{\hat{\mathbf {w}}}^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{ \, . }\end{aligned}$$

(E.2)

Then, the adjoint state equation can be readily identified from Eq. (E.2) and solved for ${{\hat{\mathbf{w}}}} \equiv -{C_{3}}{\hat{\varvec{\theta }}}_{{\chi }}^{(3)}$, resulting in

$$\begin{aligned} \begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}&= -2 C_3 \left( {\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) \right) ^T \mathbb {M}_{\partial _c\varOmega } {\mathbb {I}} \dfrac{\delta {\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}\\&\quad +C_3 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} \\&\quad -C_3\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{ \, , }\end{aligned} \end{aligned}$$

(E.3)

which can be, after inserting the RTD of ${\mathcal {J}}^{(h_e)}_{\text {av}}({\chi })$ (D.3), expressed as

$$\begin{aligned} \begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}=-2 C_3 \left( {\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) \right) ^T \mathbb {M}_{\partial _c\varOmega } {\mathbb {I}} \\&\quad \,\Bigg (C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} -C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}\Bigg ) \\&\quad +\,C_3 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T\dfrac{\delta {\mathbb {K}}_{{\chi }}}{\delta {\chi }}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)} -C_3 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T\dfrac{\delta \mathbf {f}_{{\chi }}^{(1)}}{\delta {{\chi }}}{(\hat{\mathbf{x }})}{ \, . }\end{aligned} \end{aligned}$$

(E.4)

Replacing the RTD of the stiffness matrix and the force vector into Eq. (E.4), one arrives to

$$\begin{aligned} \begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}=-2 C_3 {\mathcal A}\left( \theta _{\chi }^{(1)}\right) \\&\Bigg ( C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {B}}^T{(\hat{\mathbf{x }})}\dfrac{\partial {{{\varvec{\kappa }}}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}{\mathbf {B}}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}\\&\quad -C_2 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {N}}^T{(\hat{\mathbf{x }})}\dfrac{\partial {{r}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\Bigg )\\&\quad +C_3 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T{\mathbf {B}}^T{(\hat{\mathbf{x }})}\dfrac{\partial {{{\varvec{\kappa }}}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}{\mathbf {B}}{(\hat{\mathbf{x }})}{\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\varDelta {\chi }_{\kappa }{(\hat{\mathbf{x }})}\\&\quad -C_3 \left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T{\mathbf {N}}^T{(\hat{\mathbf{x }})}\dfrac{\partial {{r}_{{\chi }}}}{\partial {\chi }}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}{ \, , }\end{aligned} \end{aligned}$$

(E.5)

where ${\mathcal A}\left( \theta _{\chi }^{(1)}\right)$ is equal to $\left( {\mathcal T}_{\chi }\left( \theta _{\chi }^{(1)}\right) \right) ^T \mathbb {M}_{\partial _c\varOmega } {\mathbb {I}}$. Now we introduce the definition of the conductivity and the heat source with respect to the topology (Eqs. (14) and (15)) into expression (E.5), yielding to

$$\begin{aligned}&\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}=-2 C_3 {\mathcal A}\left( \theta _{\chi }^{(1)}\right) \nonumber \\&\quad \, \Bigg (C_2 \left[ m_{\kappa }{{\chi }}^{m_{\kappa }-1}\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {B}}^T(\mathbf{x }){{\varvec{\kappa }}}(\mathbf{x }){\mathbf {B}}(\mathbf{x }){\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}{{\varDelta {\chi }_{\kappa }}({\hat{{\mathbf{x}}}}) }\nonumber \\&\quad -C_2 \left[ m_{{r}}{{\chi }}^{m_{{r}}-1}\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(2)}\right) ^T{\mathbf {N}}^T(\mathbf{x }){r}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\Bigg )\nonumber \\&\quad +C_3 \left[ m_{\kappa }{{\chi }}^{m_{\kappa }-1}\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T{\mathbf {B}}^T(\mathbf{x }){{\varvec{\kappa }}}(\mathbf{x }){\mathbf {B}}(\mathbf{x }){\hat{\varvec{\theta }}}_{{\chi }}^{(1)}\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}{{\varDelta {\chi }_{\kappa }}({\hat{{\mathbf{x}}}}) } \nonumber \\&\quad -C_3 \left[ m_{{r}}{{\chi }}^{m_{{r}}-1}\left( {\hat{\varvec{\theta }}}_{{\chi }}^{(3)}\right) ^T{\mathbf {N}}^T(\mathbf{x }){r}(\mathbf{x })\right] _{{\mathbf {x}}=\hat{{\mathbf {x}}}}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}{ \, . }\end{aligned}$$

(E.6)

Finally, the sensitivity $\dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })}{\delta {\chi }}$ at point $\hat{{\mathbf{x}}}$ can be written as a sum of actual energies, which yields to

$$\begin{aligned} \dfrac{\delta \overline{\mathcal {J}}^{(h_e)}_{\text {vr}}({\chi })}{\delta {\chi }}{(\hat{\mathbf{x }})}&= -4C_3C_2m_{\kappa } \left( {\chi }_{\kappa }{(\hat{\mathbf{x }})}\right) ^{m_{\kappa }-1}{\overline{{\mathcal {U}}}}_{1-2}{(\hat{\mathbf{x }})}{{\varDelta {\chi }}_{\kappa }({\hat{{\mathbf{x}}}}) }\nonumber \\&\quad +C_3C_2m_{{r}}\left( {{\chi }}_{{r}}{(\hat{\mathbf{x }})}\right) ^{m_{{r}}-1}{\overline{{\mathcal {U}}}_{{r}-2}}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}\nonumber \\&\quad +2C_3m_{\kappa } \left( {\chi }_{\kappa }{(\hat{\mathbf{x }})}\right) ^{m_{\kappa }-1}{\overline{{\mathcal {U}}}}_{1-3}{(\hat{\mathbf{x }})}{{\varDelta {\chi }}_{\kappa }({\hat{{\mathbf{x}}}}) }\nonumber \\&\quad -C_3m_{{r}}\left( {{\chi }}_{{r}}{(\hat{\mathbf{x }})}\right) ^{m_{{r}}-1}{\overline{{\mathcal {U}}}_{{r}-3}}{(\hat{\mathbf{x }})}\varDelta {\chi }_{_{{r}}}{(\hat{\mathbf{x }})}{ \, , }\end{aligned}$$

(E.7)

where $\overline{{\mathcal {U}}}_{i-j}{(\hat{\mathbf{x }})}$ is the nominal heat conduction energy density for i-th and j-th temperature fields ($i,j=\{1,2,3\}$) and ${\overline{{\mathcal {U}}}_{{r}-k}}{(\hat{\mathbf{x }})}$ corresponds to the nominal heat source energy density for the k-th temperature field $(k=\{1,2,3\})$.