Appendix. Equations for the calculation of R(x), W(y) and S(t)
The developments used to calculate Eqs. (18), (19) and (20) are presented next.
A.1 Computing R(x) from W(y) and S(t)
When W(y) and S(t) are known, the test function (17) reduces to:
$$ {T}^{*}\left(x,y,t\right)={R}^{*}(x)\cdot W(y)\cdot S(t) $$
(24)
and the weak form (13) writes:
$$ \begin{array}{l}{\displaystyle \underset{\varXi }{\int }{R}^{*}WS\rho {C}_P\left(RW\frac{dS}{dt}\right)d\varXi }+{\displaystyle \underset{\left({\varPhi}_4-{\varPhi}_4^s\right)\times \varGamma }{\int }{R}^{*}W(H)Sh\left(RW(H)S-{T}_{amb}\right) dxdt}+\\ {}+{\displaystyle \underset{\varPhi_4^s\times \varGamma }{\int }{R}^{*}W(H)S{h}_s\left(RW(H)S-{T}_{amb}\right) dxdt}+{\displaystyle \underset{\varXi }{\int }{k}_x\frac{d{R}^{*}}{dx}WS\frac{dR}{dx} WSd\varXi}+\\ {}+{\displaystyle \underset{\varXi }{\int }{k}_y{R}^{*}\frac{dW}{dy}SR\frac{dW}{dy}Sd\varXi }=-{\displaystyle \underset{\varXi }{\int }{R}^{*}WS{\zeta}^md\varXi }-\\ {}-{\displaystyle \underset{\varXi }{\int}\left(\begin{array}{cc}\hfill \frac{d{R}^{*}}{dx}WS\hfill & \hfill {R}^{*}\frac{dW}{dy}S\hfill \end{array}\right){\xi}^md\varXi }-{\displaystyle \underset{\left({\varPhi}_4-{\varPhi}_4^s\right)\times \varGamma }{\int }{R}^{*}W(H)Sh{\psi}^m dxdt}-\\ {}-{\displaystyle \underset{\varPhi_4^s\times \varGamma }{\int }{R}^{*}W(H)S{h}_s{\psi}^m dxdt}\end{array} $$
(25)
where ζ
m, ξ
m and ψ
m are the residuals at enrichment step m:
$$ \begin{array}{l}{\zeta}^m={\displaystyle \sum_{i=1}^{l=m}\rho {C}_P{X}_l(x){Y}_l(y)\frac{d{\varTheta}_l(t)}{dt}}-{\mathbf{Q}}^{visc}\\ {}{\xi}^m=\mathbf{K}{\displaystyle \sum_{l=1}^{l=m}\left(\begin{array}{c}\hfill \frac{d{X}_l(x)}{dx}{Y}_l(y){\varTheta}_l(t)\hfill \\ {}\hfill {X}_l(x)\frac{d{Y}_l(y)}{dy}{\varTheta}_l(t)\hfill \end{array}\right)}\\ {}{\psi}^m={\displaystyle \sum_{l=1}^{l=m}{X}_l(x){Y}_l(H){\varTheta}_l(t)}\end{array} $$
(26)
As all the functions involving coordinates y and t are known, they can be integrated in Ω
y
= [0, H] and Γ = [0, t
max]. These integrations are presented next. Note that, in order to simplify the number of Eq., (27) includes integrals in R that cannot be integrated at this moment, but will be used later.
$$ \begin{array}{ccc}\hfill {w}_1={\displaystyle \underset{\varOmega_y}{\int }{W}^2 dy\kern2.04em }\hfill & \hfill {s}_1={\displaystyle \underset{\varGamma }{\int }S\frac{dS}{dt}dt\;}\kern0.36em \hfill & \hfill {r}_1={\displaystyle \underset{\varOmega_x}{\int }{R}^2dx}\hfill \\ {}\hfill {w}_2^l={\displaystyle \underset{\varOmega_y}{\int }W{Y}_l dy\kern1.8em }\hfill & \hfill \kern0.36em {s}_{{}_2}^l={\displaystyle \underset{\varGamma }{\int }S\frac{d{\varTheta}_l}{dt}dt}\;\hfill & \hfill \kern0.84em {r}_2^l={\displaystyle \underset{\varOmega_x}{\int }R{X}_ldx}\hfill \\ {}\hfill {w}_3={\displaystyle \underset{\varOmega_y}{\int }Wdy\kern2.4em }\hfill & \hfill \begin{array}{l}{s}_3={\displaystyle \underset{\varGamma }{\int }Sdt\kern1.08em }\\ {}{s}_4={\displaystyle \underset{\varGamma }{\int }{S}^2dt\kern1.08em }\end{array}\hfill & \hfill\;{r}_3={\displaystyle \underset{\varOmega_x}{\int }Rdx\kern0.36em }\hfill \end{array} $$
(27)
then we can define:
$$ {\mathbf{K}}_x=\left(\begin{array}{cc}\hfill {\displaystyle \underset{\varOmega_y}{\int }{k}_x{W}^2 dy}{\displaystyle \underset{\varGamma }{\int }{S}^2dt}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\displaystyle \underset{\varOmega_y}{\int }{k}_z{\left(\frac{dW}{dy}\right)}^2 dy}{\displaystyle \underset{\varGamma }{\int }{S}^2dt}\hfill \end{array}\right) $$
(28)
and
$$ \begin{array}{l}{\xi}_x^m={\displaystyle \sum_{l=1}^{l=m}\left(\left(\begin{array}{cc}\hfill {\displaystyle \underset{\varOmega_y}{\int }{k}_xW{Y}_l dy}{\displaystyle \underset{\varGamma }{\int }S{\varTheta}_ldt}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\displaystyle \underset{\varOmega_y}{\int }{k}_z\frac{dW}{dy}\frac{d{Y}_l}{dy} dy}{\displaystyle \underset{\varGamma }{\int }S{\varTheta}_ldt}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \frac{d{X}_l(x)}{dx}\hfill \\ {}\hfill {X}_l(x)\hfill \end{array}\right)\right)}\\ {}{\psi}_x^m={\displaystyle \sum_{l=1}^{l=m}W(H){Y}_l(H){\displaystyle \underset{\varGamma }{\int }S}{\varTheta}_l(t)dt}\left({X}_l(x)\right)\end{array} $$
(29)
Then, Eq. (25) reduces to
$$ \begin{array}{l}{\displaystyle \underset{\varOmega_x}{\int }{R}^{*}{w}_1{s}_1\rho {C}_PRdx}+{\displaystyle \underset{\varOmega_x-{\varOmega}_s}{\int }{R}^{*}{\left(W(H)\right)}^2{s}_4 hRdx}+{\displaystyle \underset{\varOmega_s}{\int }{R}^{*}{\left(W(H)\right)}^2{s}_4{h}_sRdx}+\\ {}{\displaystyle \underset{\varOmega_x-{\varOmega}_s}{\int }{R}^{*}W(H){s}_3h\left(-{T}_{amb}\right)dx}+{\displaystyle \underset{\varOmega_s}{\int }{R}^{*}W(H){s}_3{h}_s\left(-{T}_{amb}\right)dx}+\\ {}{\displaystyle \underset{\varOmega_x}{\int}\left(\begin{array}{cc}\hfill \frac{d{R}^{*}}{dx}\hfill & \hfill {R}^{*}\hfill \end{array}\right){\mathbf{K}}_x\left(\begin{array}{c}\hfill \frac{dR}{dx}\hfill \\ {}\hfill R\hfill \end{array}\right)d\varXi }=-{\displaystyle \underset{\varOmega_x}{\int }{R}^{*}\left({\displaystyle \sum_{l=1}^{l=m}{w}_2^l{s}_2^l\rho {C}_P{X}_l-{w}_3{s}_3{\mathbf{Q}}^{visc}}\right)dx}-\\ {}{\displaystyle \underset{\varOmega_x}{\int}\left(\begin{array}{cc}\hfill \frac{d{R}^{*}}{dx}\hfill & \hfill {R}^{*}\hfill \end{array}\right){\xi}_x^mdx}-{\displaystyle \underset{\varOmega_x-{\varOmega}_s}{\int }{R}^{*}h{\psi}_x^mdx}-{\displaystyle \underset{\varOmega_s}{\int }{R}^{*}{h}_s{\psi}_x^mdx}\end{array} $$
(30)
A.2 Computing W(y) from R(x) and S(t)
When R(x) and S(t) are known, the test function (17) reduces to:
$$ {T}^{*}\left(x,y,t\right)=R(x)\cdot {W}^{*}(y)\cdot S(t) $$
(31)
and the weak form (13) reads:
$$ \begin{array}{l}{\displaystyle \underset{\varXi }{\int }R{W}^{*}S\rho {C}_P\left(RW\frac{dS}{dt}\right)d\varXi }+{\displaystyle \underset{\left({\varPhi}_4-{\varPhi}_4^s\right)\times \varGamma }{\int }R{W}^{*}(H)Sh\left(RW(H)S-{T}_{amb}\right) dxdt}+\\ {}+{\displaystyle \underset{\varPhi_4^s\times \varGamma }{\int }R{W}^{*}(H)S{h}_s\left(RW(H)S-{T}_{amb}\right) dxdt}+{\displaystyle \underset{\varXi }{\int }{k}_x\frac{dR}{dx}{W}^{*}S\frac{dR}{dx} WSd\varXi}+\\ {}+{\displaystyle \underset{\varXi }{\int }{k}_zR\frac{d{W}^{*}}{dy}SR\frac{dW}{dy}Sd\varXi }=-{\displaystyle \underset{\varXi }{\int }R{W}^{*}S{\zeta}^md\varXi }-\\ {}-{\displaystyle \underset{\varXi }{\int}\left(\begin{array}{cc}\hfill \frac{dR}{dx}{W}^{*}S\hfill & \hfill R\frac{d{W}^{*}}{dy}S\hfill \end{array}\right){\xi}^md\varXi }-{\displaystyle \underset{\left({\varPhi}_4-{\varPhi}_4^s\right)\times \varGamma }{\int }R{W}^{*}(H)Sh{\psi}^m dxdt}-\\ {}-{\displaystyle \underset{\varPhi_4^s\times \varGamma }{\int }R{W}^{*}(H)S{h}_s{\psi}^m dxdt}\end{array} $$
(32)
As all the functions involving the in-plane coordinate x and the time coordinate t are known, they can be integrated in \( {\varOmega}_x=\left[-{\scriptscriptstyle \frac{1}{2}}L,{\scriptscriptstyle \frac{1}{2}}L\right] \) and Γ = [0, t
max]. Thus, using the previous notation, we can define:
$$ {\mathbf{K}}_y=\left(\begin{array}{cc}\hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_x{\left(\frac{dR}{dx}\right)}^2dx}{\displaystyle \underset{\varGamma }{\int }{S}^2dt}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_y{R}^2dx}{\displaystyle \underset{\varGamma }{\int }{S}^2dt}\hfill \end{array}\right) $$
(33)
and
$$ \begin{array}{l}{\xi}_y^m={\displaystyle \sum_{l=1}^{l=m}\left(\left(\begin{array}{cc}\hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_x\frac{dR}{dx}\frac{d{X}_l}{dx}dx}{\displaystyle \underset{\varGamma }{\int }S{\varTheta}_ldt}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_yR{X}_ldx}{\displaystyle \underset{\varGamma }{\int }S{\varTheta}_ldt}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {Y}_l(y)\hfill \\ {}\hfill \frac{d{Y}_l(y)}{dy}\hfill \end{array}\right)\right)}\\ {}{\psi}_y^m=h{\displaystyle \sum_{l=1}^{l=m}{\displaystyle \underset{\varOmega_x-{\varOmega}_s}{\int }R}{X}_l(x)dx}{\displaystyle \underset{\varGamma }{\int }S}{\varTheta}_l(t)dt\left({Y}_l(H)\right)\\ {}{\psi}_{ys}^m={h}_s{\displaystyle \sum_{l=1}^{l=m}{\displaystyle \underset{\varOmega_s}{\int }R}{X}_l(x)dx}{\displaystyle \underset{\varGamma }{\int }S}{\varTheta}_l(t)dt\left({Y}_l(H)\right)\end{array} $$
(34)
Then, Eq. (32) reduces to
$$ \begin{array}{l}{\displaystyle \underset{\varOmega_y}{\int }{W}^{*}{r}_1{s}_1\rho {C}_PWdy}+{W}^{*}(H){\left.{r}_1\right|}_{\varOmega_x-{\varOmega}_s}{s}_4hW(H)+{W}^{*}(H){\left.{r}_1\right|}_{\varOmega_s}{s}_4{h}_sW(H)+\\ {}{W}^{*}(H){\left.{r}_3\right|}_{\varOmega_x-{\varOmega}_s}{s}_3h\left(-{T}_{amb}\right)+{W}^{*}(H){\left.{r}_3\right|}_{\varOmega_s}{s}_3{h}_s\left(-{T}_{amb}\right)+\\ {}{\displaystyle \underset{\varOmega_y}{\int}\left(\begin{array}{cc}\hfill {W}^{*}\hfill & \hfill \frac{d{W}^{*}}{dy}\hfill \end{array}\right){\mathbf{K}}_y\left(\begin{array}{c}\hfill W\hfill \\ {}\hfill \frac{dW}{dy}\hfill \end{array}\right) dy}=-{\displaystyle \underset{I}{\int }{W}^{*}\left({\displaystyle \sum_{l=1}^{l=m}{r}_2^l{s}_2^l\rho {C}_P{Y}_l-{r}_3{s}_3{\mathbf{Q}}^{visc}}\right) dy}-\\ {}{\displaystyle \underset{\varOmega_y}{\int}\left(\begin{array}{cc}\hfill {W}^{*}\hfill & \hfill \frac{d{W}^{*}}{dy}\hfill \end{array}\right){\xi}_y^m dy}-{W}^{*}(H)\left({\psi}_y^m+{\psi}_{ys}^m\right)\end{array} $$
(35)
A.3 Computing S(t) from R(x) and W(y)
When R(x) and W(y) are known, those obtained in 4.1 and 4.2, the test function (17) writes:
$$ {T}^{*}\left(x,y,t\right)=R(x)\cdot W(y)\cdot {S}^{*}(t) $$
(36)
and the weak form (13) reduces to:
$$ \begin{array}{l}{\displaystyle \underset{\varXi }{\int }RW{S}^{*}\rho {C}_P\left(RW\frac{dS}{dt}\right)d\varXi }+{\displaystyle \underset{\left({\varPhi}_4-{\varPhi}_4^s\right)\times \varGamma }{\int }RW(H){S}^{*}h\left(RW(H)S-{T}_{amb}\right) dxdt}+\\ {}+{\displaystyle \underset{\varPhi_4^s\times \varGamma }{\int }RW(H){S}^{*}{h}_s\left(RW(H)S-{T}_{amb}\right) dxdt}+{\displaystyle \underset{\varXi }{\int }{k}_x\frac{dR}{dx}W{S}^{*}\frac{dR}{dx} WSd\varXi}+\\ {}+{\displaystyle \underset{\varXi }{\int }{k}_yR\frac{dW}{dy}{S}^{*}R\frac{dW}{dy}Sd\varXi }=-{\displaystyle \underset{\varXi }{\int }RW{S}^{*}{\zeta}^md\varXi }-\\ {}-{\displaystyle \underset{\varXi }{\int}\left(\begin{array}{cc}\hfill \frac{dR}{dx}W{S}^{*}\hfill & \hfill R\frac{dW}{dy}{S}^{*}\hfill \end{array}\right){\xi}^md\varXi }-{\displaystyle \underset{\left({\varPhi}_4-{\varPhi}_4^s\right)\times \varGamma }{\int }RW(H){S}^{*}h{\psi}^m dxdt}-\\ {}-{\displaystyle \underset{\varPhi_4^s\times \varGamma }{\int }RW(H){S}^{*}{h}_s{\psi}^m dxdt}\end{array} $$
(37)
As all the functions involving the in-plane coordinate x and the thickness coordinate y are known, they can be integrated in \( {\varOmega}_x=\left[-{\scriptscriptstyle \frac{1}{2}}L,{\scriptscriptstyle \frac{1}{2}}L\right] \) and Ω
y
= [0, H]. Thus, using the previous notation, we can define:
$$ {\mathbf{K}}_t=\left(\begin{array}{cc}\hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_x{\left(\frac{dR}{dx}\right)}^2dx}{\displaystyle \underset{\varOmega_y}{\int }{W}^2 dy}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_y{R}^2dx}{\displaystyle \underset{\varOmega_y}{\int }{\left(\frac{dW}{dy}\right)}^2 dy}\hfill \end{array}\right) $$
(38)
and
$$ \begin{array}{l}{\xi}_t^m={\displaystyle \sum_{l=1}^{l=m}\left(\left(\begin{array}{cc}\hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_x\frac{dR}{dx}\frac{d{X}_l}{dx}dx}{\displaystyle \underset{\varOmega_y}{\int }W{Y}_l dy}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\displaystyle \underset{\varOmega_x}{\int }{k}_zR{X}_ldx}{\displaystyle \underset{\varOmega_y}{\int}\frac{dW}{dy}\frac{d{Y}_l}{dy} dy}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\varTheta}_l(t)\hfill \\ {}\hfill {\varTheta}_l(t)\hfill \end{array}\right)\right)}\\ {}{\psi}_t^m=h{\displaystyle \sum_{l=1}^{l=m}W(H){Y}_l(H){\displaystyle \underset{\varOmega_x-{\varOmega}_s}{\int }R}{X}_l(x)dx}\left({\varTheta}_l(t)\right)\\ {}{\psi}_{ts}^m={h}_s{\displaystyle \sum_{l=1}^{l=m}W(H){Y}_l(H){\displaystyle \underset{\varOmega_s}{\int }R}{X}_l(x)dx}\left({\varTheta}_l(t)\right)\end{array} $$
(39)
Then, Eq. (37) reduces to
$$ \begin{array}{l}{\displaystyle \underset{\varGamma }{\int }{S}^{*}{r}_1{w}_1\rho {C}_P\frac{dS}{dt}dt}+{\displaystyle \underset{\varGamma }{\int }{S}^{*}{\left.{r}_1\right|}_{\varOmega_x-{\varOmega}_s}{\left(W(H)\right)}^2 hSdt}+{\displaystyle \underset{\varGamma }{\int }{S}^{*}{\left.{r}_1\right|}_{\varOmega_s}{\left(W(H)\right)}^2{h}_sSdt}+\\ {}{\displaystyle \underset{\varGamma }{\int }{S}^{*}{\left.{r}_3\right|}_{\varOmega_x-{\varOmega}_s}W(H)h\left(-{T}_{amb}\right)dt}+{\displaystyle \underset{\varGamma }{\int }{S}^{*}{\left.{r}_3\right|}_{\varOmega_s}W(H){h}_s\left(-{T}_{amb}\right)dt}+\\ {}{\displaystyle \underset{\varGamma }{\int}\left(\begin{array}{cc}\hfill {S}^{*}\hfill & \hfill {S}^{*}\hfill \end{array}\right)\;{\mathbf{K}}_t\left(\begin{array}{c}\hfill S\hfill \\ {}\hfill S\hfill \end{array}\right)dt}=-{\displaystyle \underset{\varGamma }{\int }{S}^{*}\left({\displaystyle \sum_{l=1}^{l=m}{r}_2^l{w}_2^l\rho {C}_P\frac{d{\varTheta}_l}{dt}-{r}_3{w}_3{\mathbf{Q}}^{visc}}\right)dt}-\\ {}{\displaystyle \underset{\varGamma }{\int}\left(\begin{array}{cc}\hfill {S}^{*}\hfill & \hfill {S}^{*}\hfill \end{array}\right){\xi}_t^mdt}-{\displaystyle \underset{\varGamma }{\int }{S}^{*}\left({\psi}_t^m+{\psi}_{ts}^m\right)dt}\end{array} $$
(40)
