In the following, the strong and weak forms as well as the finite element formulation of the thermomechanically coupled problem are derived in terms of the ALE description. Isotropic, homogeneous and thermoelastic materials are initially assumed. Plane strain and linear elasticity are later assumed in the derivation of the FE formulation.
The local balance of momentum equation with respect to a volume element in the initial configuration
\(\varOmega \) is
$$\begin{aligned} \rho \varvec{a} - \varvec{P}\cdot \varvec{\nabla }_X- \varvec{B} = 0 \,\,\, \text {in } \varOmega , \end{aligned}$$
(4)
where
\(\varvec{P}= \varvec{P}(\varvec{F}, \theta )\) is the first Piola–Kirchhoff stress tensor,
\(\rho \) is the density in the initial configuration,
\(\varvec{B}\) is the external body force per unit volume (in the initial configuration) and
\(\varvec{a}\) is the acceleration:
\(\varvec{a} = D_{tt}\varvec{x} = D_{tt}\varvec{u}\), where
\(\left. D_t(\cdot ) :=\partial (\cdot )/\partial t\right| _{\varvec{X}}\)
1,
\(\left. D_{tt}(\cdot ) :=\partial ^2(\cdot )/\partial t^2\right| _{\varvec{X}}\).
\(\varvec{\nabla }_X\) is the vector differential operator with respect to
\(\varOmega \). Boundary conditions are imposed as
\((\varGamma :=\partial \varOmega )\)
$$\begin{aligned} \left\{ \begin{array}{l} {\varvec{T}} = {\varvec{T}}_{\mathrm{P}} \quad {\text {on}}\; \varGamma _{\mathrm{N}_u},\\ {\varvec{u}} = {\varvec{u}}_{\mathrm{P}} \quad {\text {on}}\; \varGamma _{\mathrm{D}_u}, \end{array}\right. \end{aligned}$$
(5)
where
\(\varvec{T} :=\varvec{P}\cdot \varvec{N}\) (
\(\varvec{N}\) is the outward normal),
\(\varvec{T}_\mathrm{P}\) is the prescribed external traction per unit area with respect to
\(\varOmega \) and
\(\varvec{u}_\mathrm{P}\) represents prescribed displacements.
In terms of the ALE description, the momentum balance equation takes the form
2 [
12]
$$\begin{aligned} \hat{\rho }\Big [\varvec{\ddot{\bar{X}}}+ d_{tt}\varvec{\hat{u}}+ 2\left[ (d_t\varvec{\hat{u}})\otimes \varvec{\hat{\nabla }}\right] \cdot \varvec{\bar{v}}+ \varvec{\hat{F}}\cdot (D_t\varvec{\bar{v}}) \nonumber \\ +\varvec{\hat{G}}\mathbin {:}(\varvec{\bar{v}}\otimes \varvec{\bar{v}})\Big ] - \varvec{\hat{P}}\cdot \varvec{\hat{\nabla }}-\varvec{\hat{B}}= 0 \,\,\, \text {in } \hat{\varOmega }, \end{aligned}$$
(6)
where
\(d_t(\cdot ):=\partial (\cdot )/\partial t|_{\varvec{\hat{X}}}\),
\(d_{tt}(\cdot ):=\partial ^2(\cdot )/\partial t^2|_{\varvec{\hat{X}}}\) are referential time derivatives,
\(\varvec{\hat{F}}:=\varvec{\hat{x}}\otimes \varvec{\hat{\nabla }}\),
\(\varvec{\hat{G}}:=\varvec{\hat{x}}\otimes \varvec{\hat{\nabla }}\otimes \varvec{\hat{\nabla }}\), and
$$\begin{aligned} \varvec{\bar{v}}=D_t\varvec{\hat{X}}=\left\{ \begin{array}{ll} \varvec{\dot{R}}\cdot \varvec{R}^{\mathrm {T}}\cdot (\varvec{\hat{X}}-\varvec{X}_0) &{} \text {for } \varvec{\hat{X}}\in \hat{\varOmega }^{\mathrm{c}}\\ -\varvec{\dot{\bar{X}}}&{} \text {for } \varvec{\hat{X}}\in \hat{\varOmega }^{\mathrm{p}} \end{array} \right. \end{aligned}$$
(7)
is the convective velocity pertinent to the map
\(\varvec{\hat{\phi }}\) (see Eq. (
1)). Note that due to the nature of the rotation tensor,
\(\varvec{\hat{\nabla }}\cdot \varvec{\bar{v}}= 0\).
\(\hat{(\cdot )}\) denotes quantities related to
\(\hat{\varOmega }\). In particular,
\(\varvec{\hat{P}}=\varvec{P}\cdot \varvec{\hat{f}}^{\mathrm {T}}\) is the push-forward of the first Piola–Kirchhoff stress tensor to
\(\hat{\varOmega }\).
For a thermoelastic material, the constitutive relation
\(\varvec{P}(\varvec{F}, \theta )\) for the first Piola–Kirchhoff stress can be defined from a free energy function
\(\varPsi (\varvec{F}, \theta )\) such that
\(\varvec{P}(\varvec{F}, \theta )=\partial \varPsi (\varvec{F}, \theta )/\partial \varvec{F}\). For an isotropic material,
\(\varPsi \) should be independent of any rotation prior to the deformation, i.e.
$$\begin{aligned} \varPsi (\varvec{F}\cdot \varvec{R}, \theta ) = \varPsi (\varvec{F}, \theta ), \end{aligned}$$
(8)
for an arbitrary deformation gradient
\(\varvec{F}\), temperature
\(\theta \) and rotation tensor
\(\varvec{R}\). Consequently, the derivatives of
\(\varPsi \) with respect to
\(\varvec{F}\) (i.e. the stress) must satisfy the conditions
$$\begin{aligned} \frac{\partial {\varPsi (\varvec{\tilde{F}}\cdot \varvec{R},\theta )}}{\partial {\varvec{\tilde{F}}}}&= \frac{\partial {\varPsi (\varvec{\tilde{F}},\theta )}}{\partial {\varvec{\tilde{F}}}} \Rightarrow \varvec{P}(\varvec{\tilde{F}}\cdot \varvec{R}, \theta )\cdot \varvec{R}^{\mathrm {T}}\nonumber \\&= \varvec{P}(\varvec{\tilde{F}}, \theta ) \end{aligned}$$
(9)
for any
\(\varvec{\tilde{F}}\) and any
\(\varvec{R}\). As a consequence,
$$\begin{aligned} \varvec{P}(\varvec{\hat{F}}\cdot \varvec{\hat{f}}, \theta )\cdot \varvec{\hat{f}}^{\mathrm {T}}=\varvec{P}(\varvec{\hat{F}}, \theta ) \end{aligned}$$
(10)
for any
\(\varvec{\hat{f}}\) being either a rotation (as in the cylinder) or the identity (as in the plate). Hence, the original constitutive model can be used and
\(\varvec{\hat{P}}\)
\(( = \varvec{\hat{P}}(\varvec{\hat{F}}, \theta ) = \varvec{P}(\varvec{F}, \theta )\cdot \varvec{\hat{f}}^{\mathrm {T}}= \varvec{P}(\varvec{\hat{F}}\cdot \varvec{\hat{f}}, \theta )\cdot \varvec{\hat{f}}^{\mathrm {T}}= \varvec{P}(\varvec{\hat{F}}, \theta ))\) can be expressed independently of
\(\varvec{\hat{f}}\). For small deformation/temperature thermoelasticity (
\(\varvec{F}\approx \varvec{I}\) and
\(\theta \approx \theta ^{\mathrm{ref}}\)) it is commonly adopted that
\(\varvec{P}\approx \varvec{\mathsf {E}}\mathbin {:}[\varvec{H}- \alpha \bar{\theta }\varvec{I}]\), where
\(\varvec{H}=\varvec{F}-\varvec{I}\),
\(\varvec{\mathsf {E}}\) is the elasticity tensor,
\(\alpha \) is the thermal expansion coefficient and
\(\bar{\theta }= \theta - \theta ^{\mathrm{ref}}\) is the excess temperature with respect to the reference
\(\theta ^{\mathrm{ref}}\). Therefore, the linearization of
\(\varvec{\hat{P}}\) for small strains (
\(\varvec{\hat{F}}\approx \varvec{I}\)) and small temperature fluctuations (
\(\theta \approx \theta ^{\mathrm{ref}}\)), becomes
$$\begin{aligned} \varvec{\hat{P}}= \varvec{\mathsf {E}}\mathbin {:}[\varvec{\hat{H}}-\alpha \bar{\theta }\varvec{I}] = \varvec{\mathsf {E}}\mathbin {:}\varvec{\hat{H}}-3K\alpha \bar{\theta }\varvec{I}, \end{aligned}$$
(11)
where
\(\varvec{\hat{H}}=\varvec{\hat{F}}-\varvec{I}\) and
\(K\) is the bulk modulus. Hence, for an unconstrained specimen (
\(\varvec{\hat{P}}= 0\)), the temperature driven deformation is
\(\varvec{\hat{F}}= (1 + \alpha \bar{\theta })\varvec{I}\).
The boundary conditions can in the ALE framework be phrased as
$$\begin{aligned} \left\{ \begin{array}{l} \varvec{\hat{T}}^{\mathrm{TOT}}+ \hat{\rho }(\varvec{\hat{H}}\cdot \varvec{\bar{v}})(\varvec{\bar{v}}\cdot \varvec{\hat{N}}) = \varvec{\hat{T}}_\mathrm{P} \quad {\text {on}}\; \hat{\varGamma }_{\mathrm{R}_u}\\ \varvec{\hat{u}}= \varvec{\hat{u}}_\mathrm{P} \quad {\text {on}}\; \hat{\varGamma }_{\mathrm{D}_u}, \end{array}\right. \end{aligned}$$
(12)
where
\(\varvec{\hat{T}}^{\mathrm{TOT}}\) in the Robin-type boundary condition (
12a) is the natural boundary traction obtained via integration by parts in the weak form (see Sect.
3.2).
The strong form of the energy balance equation with respect to the initial configuration
\(\varOmega \) is [
13,
14]
$$\begin{aligned}&(\theta ^{\mathrm{ref}}+\bar{\theta }){\beta }\mathbin {:}(D_t\varvec{F}) + \rho c D_t\bar{\theta }\nonumber \\&\quad +\, \varvec{q}\cdot \varvec{\nabla }_X- r = 0 \,\,\, \text {in } \varOmega , \end{aligned}$$
(13)
where
$$\begin{aligned} {\beta }= -\frac{\partial ^2{\varPsi }}{\partial {\theta }\partial {\varvec{F}}} = -\frac{\partial {\varvec{P}}}{\partial {\theta }}, \end{aligned}$$
(14)
is the deformation-induced heat source,
\(c\) is the mass specific heat capacity,
\(\varvec{q}\) is the heat flux and
\(r\) is the external heat power per unit volume (in the initial configuration). Boundary conditions are imposed as
$$\begin{aligned} \left\{ \begin{array}{l} q_N = q_{N, {\mathrm {P}}} \quad {\text {on}}\; \varGamma _{\mathrm{N}_\theta }\\ \bar{\theta }= \bar{\theta }_{\mathrm{P}} \quad {\text {on}}\; \varGamma _{\mathrm{D}_\theta }, \end{array}\right. \end{aligned}$$
(15)
where
\(q_N :=\varvec{q}\cdot \varvec{N}\). In terms of the ALE description, the energy balance equation takes the form
$$\begin{aligned}&(\theta ^{\mathrm{ref}}+\bar{\theta }){\hat{\beta }}\mathbin {:}\left[ d_t\varvec{\hat{H}}+ (\varvec{\hat{F}}\cdot \varvec{\bar{v}})\otimes \varvec{\hat{\nabla }}\right] \nonumber \\&\quad +\hat{\rho }c\left( \varvec{\hat{\nabla }}\bar{\theta }\cdot \varvec{\bar{v}}+ d_t\bar{\theta }\right) + \varvec{\hat{q}}\cdot \varvec{\hat{\nabla }}- \hat{r}= 0 \,\,\, \text {in } \hat{\varOmega }, \end{aligned}$$
(16)
where
\(\hat{(\cdot )}\) denotes quantities related to
\(\hat{\varOmega }\). In particular,
\({\hat{\beta }}={\beta }\cdot \varvec{\hat{f}}^{\mathrm {T}}\) and
\(\varvec{\hat{q}}=\varvec{\hat{f}}\cdot \varvec{q}\) were introduced. Furthermore, for isotropic materials,
$$\begin{aligned} {\hat{\beta }}= {\beta }(\varvec{F}, \theta )\cdot \varvec{\hat{f}}^{\mathrm {T}}&= -\frac{\partial {\left[ \varvec{P}(\varvec{F},\theta )\cdot \varvec{\hat{f}}^{\mathrm {T}}\right] }}{\partial {\theta }}\nonumber \\&= -\frac{\partial {\varvec{P}(\varvec{\hat{F}},\theta )}}{\partial {\theta }} = {\beta }(\varvec{\hat{F}}, \theta ), \end{aligned}$$
(17)
where Eq. (
10) was used and a known constitutive relation
\({\beta }(\varvec{F}, \theta )\) in the initial configuration was assumed. Hence,
$$\begin{aligned} {\hat{\beta }}= -\frac{\partial {\varvec{P}(\varvec{\hat{F}}, \theta )}}{\partial {\theta }} = -\frac{\partial {\varvec{\hat{P}}}}{\partial {\theta }} = 3K\alpha \varvec{I}, \end{aligned}$$
(18)
where the last equality is valid for the linearized case (see Eq. (
11)).
Furthermore, in analogy with the result for the first Piola–Kirchhoff stress, it can be shown that for an isotropic material (and for
\(\varvec{\hat{f}}\) being a rotation or the identity tensor),
$$\begin{aligned} \varvec{\hat{q}}= \varvec{q}(\varvec{\nabla }_X\bar{\theta })\cdot \varvec{\hat{f}}^{\mathrm {T}}= \varvec{q}(\varvec{\hat{\nabla }}\bar{\theta }\cdot \varvec{\hat{f}})\cdot \varvec{\hat{f}}^{\mathrm {T}}= \varvec{q}(\varvec{\hat{\nabla }}\bar{\theta }), \end{aligned}$$
(19)
where it was tacitly assumed that the heat flux only depends on the gradient of the temperature with respect to the initial configuration. In particular, the linear Fourier’s law is henceforth adopted, whereby it is obtained that
$$\begin{aligned} \varvec{\hat{q}}=-k\varvec{\hat{\nabla }}\bar{\theta }, \end{aligned}$$
(20)
where
\(k\) is the constant heat conductivity.
The boundary conditions can in the ALE framework be phrased as
$$\begin{aligned} \left\{ \begin{array}{l} \hat{q}^{\mathrm{TOT}}_{\hat{N}} - \hat{\rho }c\bar{\theta }(\varvec{\bar{v}}\cdot \varvec{\hat{N}}) = \hat{q}_{\hat{N},{\mathrm {P}}} \quad {\text {on}}\; \hat{\varGamma }_{\mathrm{R}_\theta }\\ \bar{\theta }= \bar{\theta }_{\mathrm{P}} \quad {\text {on}}\; \hat{\varGamma }_{\mathrm{D}_\theta }, \end{array}\right. \end{aligned}$$
(21)
where
\(\hat{q}^{\mathrm{TOT}}_{\hat{N}}\) in the Robin-type boundary condition (
21a) is the natural boundary flux obtained via integration by parts in the weak form (see Sect.
3.2).
Linearizing the energy balance Eq. (
16) for small strains and small temperature fluctuations gives
$$\begin{aligned}&\theta ^{\mathrm{ref}}{\hat{\beta }}\mathbin {:}(d_t\varvec{\hat{H}}+ (\varvec{\hat{H}}\cdot \varvec{\bar{v}})\otimes \varvec{\hat{\nabla }}) + (\theta ^{\mathrm{ref}}+\bar{\theta }){\hat{\beta }}\mathbin {:}[\varvec{\bar{v}}\otimes \varvec{\hat{\nabla }}]\nonumber \\&\quad +\,\, \hat{\rho }c\left( \varvec{\hat{\nabla }}\bar{\theta }\cdot \varvec{\bar{v}}+ d_t\bar{\theta }\right) + \varvec{\hat{q}}\cdot \varvec{\hat{\nabla }}- \hat{r}= 0 \,\,\, \text {in } \hat{\varOmega }. \end{aligned}$$
(22)
For isotropic materials, (
18) is valid, and we obtain
$$\begin{aligned}&3K\alpha \theta ^{\mathrm{ref}}(\varvec{I}\mathbin {:}d_t\varvec{\hat{H}}+ (\varvec{\hat{H}}\cdot \varvec{\bar{v}})\cdot \varvec{\hat{\nabla }})\nonumber \\&\quad +\,\, \hat{\rho }c\left( \varvec{\hat{\nabla }}\bar{\theta }\cdot \varvec{\bar{v}}+ d_t\bar{\theta }\right) + \varvec{\hat{q}}\cdot \varvec{\hat{\nabla }}- \hat{r}= 0 \,\,\, \text {in } \hat{\varOmega }, \end{aligned}$$
(23)
where it was used that
\(\varvec{\bar{v}}\) is divergence free (see Eq. (
7)).
In summary: For homogeneous, isotropic materials, the residuals of the linearized strong form of the transient thermomechanically coupled problem are
$$\begin{aligned} \varvec{R}^\mathrm{s}_u(\varvec{\hat{u}},\bar{\theta })&= \hat{\rho }\Big [\varvec{\ddot{\bar{X}}}+ d_{tt}\varvec{\hat{u}}+ 2\left[ (d_t\varvec{\hat{u}})\otimes \varvec{\hat{\nabla }}\right] \cdot \varvec{\bar{v}}\nonumber \\&+\, \varvec{\hat{F}}\cdot (D_t\varvec{\bar{v}}) + \varvec{\hat{G}}\mathbin {:}(\varvec{\bar{v}}\otimes \varvec{\bar{v}})\Big ] - \varvec{\hat{P}}\cdot \varvec{\hat{\nabla }}-\varvec{\hat{B}}= 0,\nonumber \\ R^\mathrm{s}_{\theta }(\varvec{\hat{u}},\bar{\theta })&= 3K\alpha \theta ^{\mathrm{ref}}(\varvec{I}\mathbin {:}d_t\varvec{\hat{H}}+ (\varvec{\hat{H}}\cdot \varvec{\bar{v}})\cdot \varvec{\hat{\nabla }})\nonumber \\&+\, \hat{\rho }c\left( \varvec{\hat{\nabla }}\bar{\theta }\cdot \varvec{\bar{v}}+ d_t\bar{\theta }\right) + \varvec{\hat{q}}\cdot \varvec{\hat{\nabla }}- \hat{r}= 0, \end{aligned}$$
(24)
where the linearized form of
\(\varvec{\hat{P}}\) is given in (
11). It is clear from (
11) that
\(\varvec{\hat{P}}\) is temperature dependent, due to the influence of thermal expansion. This constitutes the influence of the temperature field on the momentum balance equation. Furthermore, the deformation-dependent terms in the second equation above represents the Gough–Joule effect: reversible heating/cooling of the material resulting from a nonzero strain rate [
15]. In the ALE context, this term is split into a referential derivative and a convective term (as seen above). In a stationary analysis, the former vanishes. It should be noted that the Gough–Joule effect is negligible for thermoelastic materials [
13]. Consequently, the thermomechanical coupling is one-sided in this case.
It can be seen that when the translational and rotational velocity of the system is constant in time, \(\varvec{\bar{v}}\) and \(\varvec{\ddot{\bar{X}}}\) are constant in time (specifically, \(\varvec{\ddot{\bar{X}}}= 0\)) and the time dependence in the above equations is confined to the solution fields (and the external loads).
If stationary rolling conditions are assumed, all referential time derivatives (
\(d_t\),
\(d_{tt}\)) as well as
\(\varvec{\ddot{\bar{X}}}\) are zero, resulting in a time-independent problem involving the residuals
$$\begin{aligned} \varvec{R}^\mathrm{s}_u(\varvec{\hat{u}},\bar{\theta })&= \hat{\rho }\left[ \varvec{\hat{F}}\cdot (D_t\varvec{\bar{v}}) +\varvec{\hat{G}}\mathbin {:}(\varvec{\bar{v}}\otimes \varvec{\bar{v}})\right] \nonumber \\&- \varvec{\hat{P}}\cdot \varvec{\hat{\nabla }}-\varvec{\hat{B}},\nonumber \\ R^\mathrm{s}_{\theta }(\varvec{\hat{u}},\bar{\theta })&= 3K\alpha \theta ^{\mathrm{ref}}(\varvec{\hat{H}}\cdot \varvec{\bar{v}})\cdot \varvec{\hat{\nabla }}+ \hat{\rho }c\varvec{\hat{\nabla }}\bar{\theta }\cdot \varvec{\bar{v}}\nonumber \\&+\, \varvec{\hat{q}}\cdot \varvec{\hat{\nabla }}- \hat{r}. \end{aligned}$$
(25)
The weak form is obtained by weighting the local expressions in Eq. (
24) with arbitrary (time-independent) test functions
\((\delta \varvec{\hat{u}}, \delta \bar{\theta })\in \varvec{\mathcal {V}}_u^0\times \mathcal {V}_\theta ^0\), integrating over the whole domain
\(\hat{\varOmega }\) and performing integration by parts. The weak residuals are thus defined as
$$\begin{aligned}&\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot \varvec{R}^\mathrm{s}_u(\varvec{\hat{u}},\bar{\theta })} \,d{V} \nonumber \\&\quad = \hat{\rho }\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot d_{tt}\varvec{\hat{u}}} \,d{V} \nonumber \\&\qquad +\, 2\hat{\rho }\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot \left[ (d_t\varvec{\hat{u}})\otimes \varvec{\hat{\nabla }}\right] \cdot \varvec{\bar{v}}} \,d{V}\nonumber \\&\qquad + \int _{\hat{\varOmega }} {(\delta \varvec{\hat{u}}\otimes \varvec{\hat{\nabla }})\mathbin {:}\varvec{\hat{P}}^{\mathrm{TOT}}} \,d{V} +\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot \varvec{\hat{r}}} \,d{V} \nonumber \\&\qquad -\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot \varvec{\hat{B}}^{\mathrm{TOT}}} \,d{V} -\int _{\hat{\varGamma }_{\mathrm{N}_u}} {\delta \varvec{\hat{u}}\cdot \varvec{\hat{T}}^{\mathrm{TOT}}} \,d{A}, \end{aligned}$$
(26)
where
$$\begin{aligned}&\varvec{\hat{P}}^{\mathrm{TOT}}:=\varvec{\mathsf {E}}\mathbin {:}\varvec{\hat{H}}- \hat{\rho }\varvec{\hat{H}}\cdot (\varvec{\bar{v}}\otimes \varvec{\bar{v}}) - 3K\alpha \bar{\theta }\varvec{I},\\&\varvec{\hat{r}}:=\hat{\rho }\varvec{\hat{H}}\cdot (D_t\varvec{\bar{v}}-(\varvec{\bar{v}}\otimes \varvec{\bar{v}})\cdot \varvec{\hat{\nabla }}),\\&\varvec{\hat{B}}^{\mathrm{TOT}}:=\varvec{\hat{B}}- \hat{\rho }D_t\varvec{\bar{v}}- \hat{\rho }\varvec{\ddot{\bar{X}}},\\&\varvec{\hat{T}}^{\mathrm{TOT}}:=\varvec{\hat{T}}- \hat{\rho }(\varvec{\hat{H}}\cdot \varvec{\bar{v}})(\varvec{\bar{v}}\cdot \varvec{\hat{N}}), \end{aligned}$$
$$\begin{aligned}&\int _{\hat{\varOmega }}{\delta \bar{\theta }R^\mathrm{s}_{\theta }(\varvec{\hat{u}},\bar{\theta })} \,d{V} \nonumber \\&\quad = 3K\alpha \theta ^{\mathrm{ref}}\int _{\hat{\varOmega }} {\delta \bar{\theta }\varvec{I}\mathbin {:}d_t\varvec{\hat{H}}} \,d{V} \nonumber \\&\qquad +\,3K\alpha \theta ^{\mathrm{ref}}\int _{\hat{\varGamma }} {\delta \bar{\theta }(\varvec{\hat{H}}\cdot \varvec{\bar{v}})\cdot \varvec{\hat{N}}} \,d{A} \nonumber \\&\qquad -\,\, 3K\alpha \theta ^{\mathrm{ref}}\int _{\hat{\varOmega }} {(\varvec{\hat{\nabla }}\delta \bar{\theta })\cdot (\varvec{\hat{H}}\cdot \varvec{\bar{v}})} \,d{V} \nonumber \\&\qquad +\,\int _{\hat{\varOmega }} {(\varvec{\hat{\nabla }}\delta \bar{\theta })\cdot [k\varvec{\hat{\nabla }}\bar{\theta }- \hat{\rho }c\varvec{\bar{v}}\bar{\theta }]} \,d{V} \nonumber \\&\qquad +\,\, \hat{\rho }c\int _{\hat{\varOmega }} {\delta \bar{\theta }d_t\bar{\theta }} \,d{V} -\,\int _{\hat{\varOmega }} {\delta \bar{\theta }\hat{r}} \,d{V} \nonumber \\&\qquad +\int _{\hat{\varGamma }_{\mathrm{N}_\theta }} {\delta \bar{\theta }\hat{q}^{\mathrm{TOT}}_{\hat{N}}} \,d{A}, \end{aligned}$$
(27)
where
$$\begin{aligned} \hat{q}^{\mathrm{TOT}}_{\hat{N}} :=\hat{q}_{\hat{N}} + \hat{\rho }c\bar{\theta }(\varvec{\bar{v}}\cdot \varvec{\hat{N}}), \end{aligned}$$
and it was used that
$$\begin{aligned} \varvec{\hat{q}}=-k\varvec{\hat{\nabla }}\bar{\theta }. \end{aligned}$$
The term
\(\varvec{\hat{T}}^{\mathrm{TOT}}\) emerges from the ALE formulation of the momentum balance equation. Thus, prescribing
\(\varvec{\hat{T}}^{\mathrm{TOT}}\) on
\(\hat{\varGamma }_{\mathrm{N}_u}\) constitutes a natural (Neumann) boundary condition:
$$\begin{aligned} \varvec{\hat{T}}^{\mathrm{TOT}}_\mathrm{P} = \varvec{\hat{T}}- \hat{\rho }(\varvec{\hat{H}}\cdot \varvec{\bar{v}})(\varvec{\bar{v}}\cdot \varvec{\hat{N}}) \,\,\, \text {on } \hat{\varGamma }_{\mathrm{N}_u}. \end{aligned}$$
If instead the intrinsic (physical) traction
\(\varvec{\hat{T}}:=\varvec{\hat{P}}\cdot \varvec{\hat{N}}\) is prescribed, a Robin-type boundary condition is obtained:
$$\begin{aligned} \varvec{\hat{T}}^{\mathrm{TOT}}+ \hat{\rho }(\varvec{\hat{H}}\cdot \varvec{\bar{v}})(\varvec{\bar{v}}\cdot \varvec{\hat{N}}) = \varvec{\hat{T}}_\mathrm{P} \,\,\, \text {on } \hat{\varGamma }_{\mathrm{R}_u}, \end{aligned}$$
where the boundary
\(\hat{\varGamma }_{\mathrm{N}_u}\) was simply renamed
\(\hat{\varGamma }_{\mathrm{R}_u}\) in order to reflect the type of boundary condition in effect. Similarly, prescribing the quantity
\(\hat{q}^{\mathrm{TOT}}_{\hat{N}}\) on
\(\hat{\varGamma }_{\mathrm{N}_\theta }\) constitutes a natural (Neumann) boundary condition:
$$\begin{aligned} \hat{q}^{\mathrm{TOT}}_{\hat{N}\mathrm {P}} = \hat{\rho }c\bar{\theta }(\varvec{\bar{v}}\cdot \varvec{\hat{N}}) + \hat{q}_{\hat{N}} \,\,\, \text {on } \hat{\varGamma }_{\mathrm{N}_\theta }. \end{aligned}$$
If instead the intrinsic (physical) heat flux
\(\varvec{\hat{q}}\) is prescribed, a Robin-type boundary condition is obtained:
$$\begin{aligned} \hat{q}^{\mathrm{TOT}}_{\hat{N}} - \hat{\rho }c\bar{\theta }(\varvec{\bar{v}}\cdot \varvec{\hat{N}}) = \hat{q}_{\hat{N},\mathrm {P}} \,\,\, \text {on } \hat{\varGamma }_{\mathrm{R}_\theta }, \end{aligned}$$
where the boundary
\(\hat{\varGamma }_{\mathrm{N}_\theta }\) was renamed
\(\hat{\varGamma }_{\mathrm{R}_\theta }\).
In order to state the final version of the weak form, trial and test spaces are introduced for the respective solution fields
\(\varvec{\hat{u}}(\varvec{\hat{X}},t)\) and
\(\bar{\theta }(\varvec{\hat{X}},t)\):
$$\begin{aligned} \varvec{\mathcal {V}}_u&= \{\varvec{v}: \varvec{v}=\varvec{\hat{u}}_\mathrm{P} \text { on } \hat{\varGamma }_{\mathrm{D}_u}, \varvec{v} \text { sufficiently regular}\},\nonumber \\ \varvec{\mathcal {V}}_u^0&= \{\varvec{v}: \varvec{v}=\varvec{0} \text { on } \hat{\varGamma }_{\mathrm{D}_u}, \varvec{v} \text { sufficiently regular}\},\nonumber \\ \mathcal {V}_\theta&= \{v: v=\bar{\theta }_\mathrm{P} \text { on } \hat{\varGamma }_{\mathrm{D}_\theta }, v \text { sufficiently regular}\},\nonumber \\ \mathcal {V}_\theta ^0&= \{v: v=0 \text { on } \hat{\varGamma }_{\mathrm{D}_\theta }, v \text { sufficiently regular}\}. \end{aligned}$$
(28)
The exact meaning of
sufficiently regular is not elaborated here (see eg. Brenner and Scott [
16]). The weak form of the ALE boundary value problem derived in the previous section can now be stated as: Find
\(\varvec{\hat{u}}\in \varvec{\mathcal {V}}_u\) and
\(\bar{\theta }\in \mathcal {V}_\theta \) such that
$$\begin{aligned} R^\mathrm{w}_u(\varvec{\hat{u}},\bar{\theta };\delta \varvec{\hat{u}})&= 0 \,\,\, \forall \delta \varvec{\hat{u}}\in \varvec{\mathcal {V}}_u^0,\nonumber \\ R^\mathrm{w}_{\theta }(\varvec{\hat{u}},\bar{\theta };\delta \bar{\theta })&= 0 \,\,\, \forall \delta \bar{\theta }\in \mathcal {V}_\theta ^0, \end{aligned}$$
(29)
where the residuals are obtained by inserting the aforementioned Robin-type boundary conditions into the integral expressions in Eqs. (
26), (
27):
$$\begin{aligned} R^\mathrm{w}_u(\varvec{\hat{u}},\bar{\theta };\delta \varvec{\hat{u}})&:= \hat{\rho }\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot d_{tt}\varvec{\hat{u}}} \,d{V} \nonumber \\&+\, 2\hat{\rho }\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot \left[ (d_t\varvec{\hat{u}})\otimes \varvec{\hat{\nabla }}\right] \cdot \varvec{\bar{v}}} \,d{V}\nonumber \\&+ \int _{\hat{\varOmega }} {(\delta \varvec{\hat{u}}\otimes \varvec{\hat{\nabla }})\mathbin {:}\varvec{\hat{P}}^{\mathrm{TOT}}} \,d{V} \!+\!\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot \varvec{\hat{r}}} \,d{V} \nonumber \\&-\,\int _{\hat{\varOmega }} {\delta \varvec{\hat{u}}\cdot \varvec{\hat{B}}^{\mathrm{TOT}}} \,d{V} -\int _{\hat{\varGamma }_{\mathrm{R}_u}} {\delta \varvec{\hat{u}}\cdot \varvec{\hat{T}}_\mathrm{P}} \,d{A}\nonumber \\&+\,\hat{\rho }\int _{\hat{\varGamma }_{\mathrm{R}_u}} {\delta \varvec{\hat{u}}\cdot (\varvec{\hat{H}}\cdot \varvec{\bar{v}}) (\varvec{\bar{v}}\cdot \varvec{\hat{N}})} \,d{A},\end{aligned}$$
(30)
$$\begin{aligned} R^\mathrm{w}_{\theta }(\varvec{\hat{u}},\bar{\theta };\delta \bar{\theta })&:= 3K\alpha \theta ^{\mathrm{ref}}\int _{\hat{\varOmega }} {\delta \bar{\theta }\varvec{I}\mathbin {:}d_t\varvec{\hat{H}}} \,d{V}\nonumber \\&+\,3K\alpha \theta ^{\mathrm{ref}}\int _{\hat{\varGamma }} {\delta \bar{\theta }(\varvec{\hat{H}}\cdot \varvec{\bar{v}})\cdot \varvec{\hat{N}}} \,d{A} \nonumber \\&-\,\, 3K\alpha \theta ^{\mathrm{ref}}\int _{\hat{\varOmega }} {(\varvec{\hat{\nabla }}\delta \bar{\theta })\cdot (\varvec{\hat{H}}\cdot \varvec{\bar{v}})} \,d{V} \nonumber \\&+\,\int _{\hat{\varOmega }} {(\varvec{\hat{\nabla }}\delta \bar{\theta })\cdot [k\varvec{\hat{\nabla }}\bar{\theta }- \hat{\rho }c\varvec{\bar{v}}\bar{\theta }]} \,d{V} \nonumber \\&+\, \hat{\rho }c\int _{\hat{\varOmega }} {\delta \bar{\theta }d_t\bar{\theta }} \,d{V} -\int _{\hat{\varOmega }} {\delta \bar{\theta }\hat{r}} \,d{V} \nonumber \\&+\,\int _{\hat{\varGamma }_{\mathrm{R}_\theta }} {\delta \bar{\theta }\hat{q}_{\hat{N},\mathrm {P}}} \,d{A} \nonumber \\&+\,\hat{\rho }c\int _{\hat{\varGamma }_{\mathrm{R}_\theta }} {\delta \bar{\theta }\bar{\theta }(\varvec{\bar{v}}\cdot \varvec{\hat{N}})} \,d{A}. \end{aligned}$$
(31)