A discontinuous Galerkin method for non-linear electro-thermo-mechanical problems: application to shape memory composite materials

Abstract

A coupled electro-thermo-mechanical discontinuous Galerkin (DG) method is developed considering the non-linear interactions of electrical, thermal, and mechanical fields. In order to develop a stable discontinuous Galerkin formulation the governing equations are expressed in terms of energetically conjugated fields gradients and fluxes. Moreover, the DG method is formulated in finite deformations and finite fields variations. The multi-physics DG formulation is shown to satisfy the consistency condition, and the uniqueness and optimal convergence rate properties are derived under the assumption of small deformation. First the numerical properties are verified on a simple numerical example, and then the framework is applied to simulate the response of smart composite materials in which the shape memory effect of the matrix is triggered by the Joule effect.

Appendices

Appendix 1: Constitutive behaviors

The objective of this section is to summarize large deformation constitutive theories in order to model the response of Shape Memory Polymers composites (SMPC) subjected to a variety of ETM histories. The composite material system is obtained by defining two separate models, one for carbon fiber and another one for SMPs. For carbon fibers, a transversely isotropic hyperelastic model is considered while an elasto-visco-plastic model is considered for the shape memory polymers.

1.1 Material model of carbon fiber

Carbon fiber is a transversely isotropic material and subsequently the number of mechanical constants is reduced to 5 because of the in-plane isotropy:

$$\begin{aligned} E^{{\text{T}}} & = E_{1} = E_{2} \ne E_{3} = E^{{\text{L}}} , \\ \nu ^{{{\text{TT}}}} & = \nu _{{12}} = \nu _{{21}} \ne \nu _{{13}} = \nu _{{23}} = \nu ^{{{\text{TL}}}} \\ G^{{{\text{LT}}}} & = G_{{13}} = G_{{23}} = G_{3} = G^{{\text{L}}} . \\ \end{aligned}$$

(113)

The missing in-plane shear modulus ${{G}}^{\text {TT}}$ is obtained from ${\nu}^{\text {TT}}$ and ${{{E}}^{\text {T}}}$, with

$$\begin{aligned} \begin{aligned} {{G}}^{\text {TT}}={{G}}_{\mathrm{12}}=\frac{{{E}}^{\text {T}}}{2(1+{\nu}^{\text {TT}})}. \end{aligned} \end{aligned}$$

(114)

In the previous relations, the subscript 3 or the superscript $\text {L}$ refers to the fiber direction and 1, 2, or $\text {T}$ is a direction transverse to the fiber direction. Along the longitudinal direction the Poisson ratios are not symmetric but instead satisfy $\frac{{\nu}_{ij}}{{{E}}_{i}}=\frac{{\nu}_{ji}}{{{E}}_{j}}$.

In order to model the carbon fiber, we have considered the equation proposed by Bonet et al. [10], with some modifications proposed by Wu et al. [70], since the original formulation considered that ${\nu}^{\text {TL}} ={\nu}^{\text {TT}}$, to describe the isotropic hyperelastic solids in the large strain regime. In addition, we have added the thermal contribution, characterized by the thermal expansion term $\alpha _{\mathrm{th}}$. In this formulation, the strain energy density $\psi$ consists of an isotropic component $\psi ^{\text {is}}$ and of an orthotropic transversely isotropic component $\psi ^{\text {tr}}$ such that $\psi =\psi ^{\text {is}}+\psi ^{\text {tr}}$. The Neo–Hookean equation is used to model the isotropic part, such that

$$\psi ^{{{\text{is}}}} = \frac{1}{2}G^{{{\text{TT}}}} ({\text{tr}}{\mathcal{\varvec{C}}} - 3) - G^{{{\text{TT}}}} {\text{ln}}J + \frac{1}{2}\lambda ({\text{ln}}J - 3\alpha ^{\prime}_{{{\text{th}}}} (T - T_{0} ))^{2} ,$$

(115)

where this energy density function has been defined by C. Miehe in [46], and where $\alpha '_{\mathrm{th}} = \alpha _{\mathrm{th}} \frac{\lambda +2/3G^{\text {TT}}}{\lambda}$ in order to recover the usual dilation coefficient definition of isotropic materials. The orthotropic transversely isotropic component is obtained from a generalization of the model proposed by Bonet et al. [10] and enhanced by Wu et al. [70]:

$$\begin{aligned} \psi ^{\text {tr}}=\left[ \alpha ^{\text {tr}}+2\beta ^{\text {tr}}(\text {ln} J-3\alpha _{\mathrm{th}}'(T- T_0))+\gamma ^{\text {tr}}({\text {I}}_4-1)\right] ({\text {I}}_4-1)-\frac{1}{2}\alpha ^{\text {tr}}({\text {I}}_5-1), \end{aligned}$$

(116)

where ${\text {I}}_4$ and ${\text {I}}_5$ denote the two new pseudo invariants of $\mathbf{C}$ expressed as [57, 58],

$$\begin{aligned} \begin{aligned} {\text {I}}_4=\varvec{A}\cdot \pmb {C}\cdot \varvec{A}\quad \text {and}\quad {\text {I}}_5=\varvec{A}\cdot \pmb {C}^2\cdot \varvec{A}\,, \end{aligned} \end{aligned}$$

(117)

with the unit vector $\varvec{A}$ defining the main direction of orthotropy (fiber direction) in the undeformed configuration.

The parameters of the model Eq. (116), $\lambda ,\,{{G}}^{\text {TT}},\, \alpha ^{\text {tr}},\,\beta ^{\text {tr}}$ and $\gamma ^{\text {tr}}$ are obtained from the measured properties Eqs. (113, 114) as

$$\begin{aligned} \begin{aligned}&\lambda =\frac{{{E}}^{\text {T}}({\nu}^{\text {TT}}+ n {{\nu}^{\text {TL}}}^2)}{ m(1+{\nu}^{\text {TT}})},\,\quad {{G}}^{\text {TT}}=\frac{{{E}}^{\text {T}}}{2(1+{\nu}^{\text {TT}})},\;\alpha ^{\text {tr}}={{G}}^{\text {TT}}-{{G}}^{\text {LT}}\,,\\& \beta ^{\text {tr}}=\frac{{{E}}^{\text {T}}\left[ n{\nu}^{\text {TL}}(1+{\nu}^{\text {TT}}-{\nu}^{\text {TL}})-{\nu}^{\text {TT}}\right]}{4 m(1+{\nu}^{\text {TT}})},\; \gamma ^{\text {tr}}=\frac{{{E}}^{\text {T}}(1-{\nu}^{\text {TT}})}{8m}-\frac{\lambda +2{{G}}^{\text {TT}}}{8}+\frac{\alpha ^{\text {tr}}}{2}-\beta ^{\text {tr}},\\& m=1-{\nu}^{\text {TT}}-2 n{{\nu}^{\text {TL}}}^2\,, \quad n=\frac{{{E}}^{\text {L}}}{{{E}}^{\text {T}}}. \end{aligned} \end{aligned}$$

(118)

The second Piola–Kirchhoff stress tensor can be obtained by differentiating the free energy in terms of the right Cauchy–Green strain tensor $\pmb {S}=2\frac{\partial \psi}{\partial \pmb {{C}}}$ leading to

$$\begin{aligned}&\begin{aligned} \pmb {S}= \pmb {S}^{\text {is}}+\pmb {S}^{\text {tr}}, \end{aligned} \text{ with} \end{aligned}$$

(119)

$$\begin{aligned}&\pmb {S}^{\text {is}}=\lambda {\text {ln}}{{J}}\pmb {{C}}^{-1}+{{G}}^{\text {TT}}(\pmb {I}-{\mathbf{C}}^{-1})-3\lambda \alpha '_{\mathrm{th}}(T- T_0)\pmb {{C}}^{-1}, \end{aligned}$$

(120)

where $\pmb {I}$ is the identity tensor, and with

$$\begin{aligned} \begin{aligned} \pmb {S}^{\text {tr}}&=2\beta ^{\text {tr}}({\text {I}}_4-1)\pmb {{C}}^{-1}+ 2\left[ \alpha ^{\text {tr}}+2\beta ^{\text {tr}}( {\text {ln}}{{J}}-3\alpha '_{\mathrm{th}}(T-T_0))+2\gamma ^{\text {tr}}({\text {I}}_4-1)\right] \varvec{A}\otimes \varvec{A}\\&\quad-\alpha ^{\text {tr}}(\pmb {{C}}\cdot \varvec{A}\otimes \varvec{A}+\varvec{A}\otimes \pmb {{C}}\cdot \varvec{A}). \end{aligned} \end{aligned}$$

(121)

Then the first Piola–Kirchhoff stress tensor is evaluated from the second Piola–Kirchhoff stress tensor as

$$\begin{aligned} \pmb {P}=\pmb {F}\cdot \pmb {S}. \end{aligned}$$

(122)

The stiffness is computed following [10, 70].

1.2 Elasto-visco-plastic formulation for SMP

In this section, we summarize the work of Srivastava et al. [59] to model the SMP behavior above and below glass transition.

1.2.1 Kinematics

To model the inelastic response of the amorphous polymeric materials, it is assumed that the deformation gradient $\pmb {{F}}$ may be multiplicatively decomposed into elastic and plastic parts

$$\begin{aligned} \pmb {{F}}=\pmb {{F}}^{\text {e}({\alpha})} \cdot \pmb {{F}}^{\text {p}({\alpha})} \; \text {with} \; \text {det}\pmb {{F}}^{\text {e}({\alpha})}= {J}^{\text {e}{(\alpha )}}= {J}>0 \; \text {and} \; \text {det}\pmb {{F}}^{\text {p}({\alpha})}=1\,, \end{aligned}$$

(123)

where $\pmb {{F}}^{\text {e}({\alpha})}$ is the elastic distortion and $\pmb {{F}}^{\text {p}({\alpha})}$ is the inelastic distortion with $\pmb {{F}}^{\text {p}{(\alpha )}}(\varvec{X},\,0)=\pmb { I}$. In these equations we have considered the possibility to account for several mechanisms ${\alpha}= 1,2,3$. Moreover, the elastic decomposition of the deformation gradient can be written as

$$\begin{aligned} \pmb {{F}}^{\text {e}{(\alpha )}}=\pmb {{R}}^{\text {e}{(\alpha )}} \cdot \pmb {{U}}^{\text {e}({\alpha})}\,, \end{aligned}$$

(124)

with the elastic right and left Cauchy–Green strain tensors respectively equal to

$$\begin{aligned} \pmb {{C}}^{\text {e}{(\alpha )}}=\pmb {{U}}^{\text {e}({\alpha})^2}=\pmb {{F}}^{\text {e}{(\alpha )}\text {T}}\cdot \pmb {{F}}^{\text {e}{(\alpha )}}\,\text{and} \quad \pmb {{B}}^{\text {e}{(\alpha )}}=\pmb {{F}}^{\text {e}{(\alpha )}}\cdot \pmb {{F}}^{\text {e}{(\alpha )}\text {T}}\,. \end{aligned}$$

(125)

1.2.2 Elasto-visco-plasticity

The material may be idealized to be isotropic. Accordingly, all constitutive functions are presumed to be isotropic in character. Let us assume that the free energy has the separable form

$$\begin{aligned} \psi _R= \sum _\alpha \ \psi ^{(\alpha )}(\varPhi _{\pmb {{C}}^{{\text {e}}(\alpha )}},\, T), \end{aligned}$$

(126)

where $\varPhi _{\pmb {c}^{{\text {e}}(\alpha )}}$ represents a list of the principle invariants of ${\pmb {{C}}^{{\text {e}}(\alpha )}}$ and T is the temperature. The Cauchy stress is decomposed in terms of the mechanisms $\pmb {\sigma}= \sum _{(\alpha )}\pmb {\sigma}^{(\alpha )}$ with

$$\begin{aligned} \pmb {\sigma}^{(\alpha )}= \dfrac{1}{{J}}\;\pmb { F}\;\pmb {{S}}^{(\alpha )}\; \pmb {{F}}^{\text {T}}=\frac{1}{{J}}\;\pmb { F}^{{\text {e}} (\alpha )}\;\pmb {{S}}^{e(\alpha )}\; \pmb {{F}}^{{\text {e}}(\alpha )\text {T}}, \end{aligned}$$

(127)

where $\pmb {{S}}^{e(\alpha )}$ is the symmetric elastic second Piola–Kirchhoff stress

$$\begin{aligned} \pmb {{S}}^{e(\alpha )}=2\frac{\partial \psi ^{(\alpha )}(\varPhi _{\pmb {{c}}^{{\text {e}}(\alpha )}},\,T)}{\partial \pmb { C}^{{\text {e}}(\alpha )}}. \end{aligned}$$

(128)

Moreover, the first Piola–Kirchhoff stress tensor can be computed following

$$\begin{aligned} \pmb {{P}}^{(\alpha )} ={ J}\; \pmb \sigma ^{(\alpha )} \pmb {{F}}^{-\text {T}}=\pmb {{F}}^{{\text {e}}(\alpha )}\;\pmb {{S}}^{{\text {e}}(\alpha )}\;\pmb {{F}}^{\text{p}(\alpha ){- \text {T}}}=\pmb {{F}}\;\pmb {{F}}^{\text{p}(\alpha )-1}\;\pmb {{S}}^{{\text {e}}(\alpha )} \;\pmb {{F}}^{\text{p}(\alpha ){- \text {T}}}\,. \end{aligned}$$

(129)

The driving stress of the plastic flow is the symmetric Mandel stress, which is defined as

$$\begin{aligned} \pmb {{M}}^{{\text {e}}(\alpha )}={J}\;\pmb {{R}}^{{\text {e}}(\alpha )\text {T}}\pmb \sigma ^{\text{(}\alpha )}\pmb {{R}}^{{\text {e}}(\alpha )}=\pmb {{U}}^{{\text {e}}(\alpha )}\pmb {{S}}^{{\text {e}}(\alpha )}\pmb {{U}}^{{\text {e}}(\alpha )}=\pmb {{C}}^{{\text {e}}(\alpha )}\pmb {{S}}^{{\text {e}}(\alpha )}, \end{aligned}$$

(130)

where $\pmb { {M}}^{{\text {e}}(\alpha )}$ is the elastic Mandel stress, $\pmb {{R}}^{{\text {e}}(\alpha )}$ is the rotation matrix, and where it has been assumed that $\pmb {{C}}^{{\text {e}}(\alpha )}$ and $\pmb {{S}}^{{\text {e}}(\alpha )}$ permute. The corresponding equivalent shear stress is given by

$$\begin{aligned} \bar{\tau}^{(\alpha )}=\frac{1}{\sqrt{2}}|\pmb {{M}}^{{\text {e}}(\alpha )}_0 |, \end{aligned}$$

(131)

where $\pmb {{M}}^{{\text {e}}(\alpha )}_0$ is the deviatoric part of the Mandel stress

$$\begin{aligned} \pmb {{M}}^{{\text {e}}(\alpha )}_0=\pmb {{M}}^{{\text {e}}(\alpha )}+\overline{{p}}\pmb { I}\,, \quad \text { with} \quad \overline{{p}}=-\frac{1}{3}\text {tr}\pmb {{M}}^{{\text {e}}(\alpha )}\,, \end{aligned}$$

(132)

and $|\pmb {{M}}^{{\text {e}}(\alpha )}_0 |=\sqrt{\pmb {{M}}^{{\text {e}}(\alpha )}_0 :\pmb {{M}}^{{\text {e}}(\alpha )}_0}$ is the norm of the deviatoric part of the Mandel stress.

In order to account for the major strain-hardening and softening characteristics of polymeric materials observed during visco-plastic deformation, Srivastava et al. [59] have introduced macroscopic internal variables $\pmb {\xi}^{(\alpha )}$ to represent important aspects of the microstructural resistance to plastic flow. The plastic flow follows

$$\begin{aligned} \dot{\pmb {{F}}}^{\text{p}(\alpha )}=\pmb {{D}}^{\text{p}(\alpha )} \pmb {{F}}^{\text{p}(\alpha )}, \end{aligned}$$

(133)

where each $\pmb { {F}}^{\text{p}(\alpha )}$ is to be regarded as an internal variable part of $\pmb {\xi}^{(\alpha )}$, and which is defined as a solution of the differential equation

$$\begin{aligned} \pmb { D}^{\text{p}(\alpha )}= \dot{\epsilon}^{\text{p}(\alpha )}\left( \frac{\pmb {{M}}^{{\text {e}}(\alpha )}_0}{2\bar{\tau}^\alpha}\right) , \end{aligned}$$

(134)

where $\pmb {D}^{\text{p}}$ is the plastic stretching tensor, and $\dot{\epsilon}^{\text{p}(\alpha )}=\sqrt{2}|\pmb {D}^{\text{p}(\alpha )}|$ is the equivalent plastic shear strain rate.

Therefore for given $\bar{\tau}^{(\alpha )}$ and $\pmb {\varLambda}^{(\alpha )}= (\pmb { {C}}^{{\text {e}}(\alpha )},\,\pmb {\xi}^{(\alpha )},\, T)$ a list of constitutive variables, the equivalent plastic shear strain rate $\dot{\epsilon}^{\text{p}(\alpha )}$ is obtained by solving a scalar strength relation such as

$$\begin{aligned} \bar{\tau}^{(\alpha )}=\varUpsilon ^{(\alpha )}(\pmb {\varLambda}^{(\alpha )},\dot{\epsilon}^{\text{p}(\alpha )}), \end{aligned}$$

(135)

where the strength function $\varUpsilon ^{(\alpha )}(\pmb {\varLambda}^{(\alpha )},\dot{\epsilon}^{\text{p}(\alpha )})$ is an isotropic function of its arguments.

1.2.3 Partial differential governing equations

In order to complete Eq. (7), y, the internal energy per unit mass, is defined as $y=c_{\text{v}} T$, where the volumetric heat capacity per unit mass is a function of the glass transition temperature, and is defined as follows

$$c_{{\text{v}}} = \left\{ {\begin{array}{*{20}l} {c_{0} - c_{1} (T - T_{{\text{g}}} )} \hfill & {{\text{if}}\;T \le T_{{\text{g}}}, } \hfill \\ {c_{0} } \hfill & {{\text{if}}\;T > T_{\text{g}}.} \hfill \\ \end{array} } \right.$$

(136)

Moreover, $\bar{F}$, the body source of heat, is expressed as

$$\begin{aligned} \bar{F}= {Q}_{{\text{r}}} +\sum _{\alpha} \bar{\tau}^{(\alpha )} \dot{\epsilon}^{\text{p}(\alpha )}+ T\frac{\partial ^2\psi ^{{\text {e}}(\alpha )}}{\partial {\pmb { C}^{{\text {e}}(\alpha )}} \partial {\text {T}}}:\dot{\pmb { C}}^{{\text {e}}(\alpha )}, \end{aligned}$$

(137)

where ${Q}_{{\text{r}}}$ is the scalar heat supply measured per unit reference volume and the last term of the right hand side is the thermo-elastic damping term which is neglected. Instead it is assumed that only a fraction $\text{v}$ of the rate of the plastic dissipation contributes to the temperature change

$$\begin{aligned} \bar{F}= {Q}_{{\text{r}}} + v\sum _{\alpha} \bar{\tau}^{(\alpha )} \dot{\epsilon}^{\text{p}(\alpha )}, \end{aligned}$$

(138)

where $0\le v\le 1$.

The glass transition in amorphous polymers depends on the equivalent shear strain rate $\dot{\epsilon}=\sqrt{2}|\pmb { D}_0|$ to which the material is subjected, where $\pmb {{D}}_0=\pmb {{D}}-\frac{1}{3}\text{tr} \pmb {{D}}\; \pmb {{I}}$ denotes the total deviatoric stretching tensor, with

$$\begin{aligned} \pmb {{D}}=\frac{1}{2} (\dot{\pmb {{F}}} \;\pmb { F}^{-1}+\pmb { F}^{-\text {T}}\dot{\pmb { F}}^{ \text {T}})\,. \end{aligned}$$

(139)

Eventually, the glass transition $T_{\mathrm{g}}$ is calculated from the following expression

$$T_{{\text{g}}} = \left\{ {\begin{array}{*{20}l} {T_{{\text{r}}}} \hfill & {{\text{if}}\;\dot{\epsilon} \le \varepsilon _{{\text{r}}} ,} \hfill \\ {T_{{\text{r}}} + n\log \left( {\frac{{\dot{\epsilon}}}{{\epsilon _{{\text{r}}}}}} \right)} \hfill & {{\text{if}}\;\dot{\epsilon} > \varepsilon _{{\text{r}}} ,} \hfill \\ \end{array}} \right.$$

(140)

where ${T}_{\text{r}}$ is the reference glass transition temperature at low strain rate, $\dot{\epsilon}$ is the shear strain rate, and $\epsilon _{\text{ r }}$ is the reference strain rate.

1.2.4 The first micromechanism ($\alpha =1$)

The first micromechanism ($\alpha = 1$) represents an elastic resistance due to intermolecular energetic bond-stretching and a dissipation due to the thermally-activated plastic flow following chain segment rotation and relative slippage of the polymer chains between neighboring cross-linkage points.

The following simple generalization of the classical strain energy function of infinitesimal isotropic elasticity is considered, and uses a logarithmic measure

$$\begin{aligned} \pmb {{E}}^{{\text {e}}(1)} =\frac{1}{2} \ln \pmb {{C}}^{{\text {e}}(1)}\,, \end{aligned}$$

(141)

of the finite elastic strain [4]. The form of the elastic free energy is thus defined as

$$\begin{aligned} {\psi}^{{\text {e}}(1)}={{G}|\pmb {{E}}^{{\text {e}}(1)}_0|^2}+{ \frac{1}{2} {K}\left( \text {tr}{\pmb {E}^{{\text {e}}(1)}}\right) ^2 -3 {K}\left( \text {tr}{\pmb { E}^{{\text {e}}(1)}}\right) { \alpha _{\text {th}}} (T- T_0)}+ \tilde{f}(T)\,, \end{aligned}$$

(142)

where the deviatoric part of the logarithmic strain is denoted by $\pmb {E}^{{\text {e}}}_0$, $\tilde{f}(T)$ is an entropy contribution to the free energy related to the temperature dependent specific heat of the material, and where the temperature dependent parameters ${G}(T), \;K (T),\; \alpha _{\text{th}} (T)$ are respectively the shear modulus, bulk modulus, and the coefficient of thermal expansion.

The Mandel stress is thus obtained from

$$\begin{aligned} \pmb {{M}}^{{\text {e}}(1)}=2\pmb { C}^{{\text {e}}{(1)}}\frac{\partial \psi ^{{\text {e}}(1)}(\pmb { E}^{{\text {e}}(1)},\text {T})}{\partial \pmb { C}^{{\text {e}}^{(1)}}}=\frac{\partial \psi ^{{\text {e}}(1)}(\pmb { E}^{{\text {e}}(1)},\,T)}{\partial \pmb { E}^{{\text {e}}{(1)}}}\,, \end{aligned}$$

(143)

if $\pmb { C}^{{\text {e}}{(1)}}$ and $\pmb {{M}}^{{\text {e}}(1)}$ permute. It should be noted that in this work $\pmb { E}^{{\text {e}}{(1)}}$ is computed by using a Taylor series approximation of Eq. (141), and not through the eigenvalue decomposition. Substituting Eq. (142) in Eq. (143), as $|\pmb { {E}}^{{\text {e}}(1)}_0|= \pmb { {E}}^{{\text {e}}(1)}_0: \pmb { {E}}^{{\text {e}}(1)}_0$ one can get directly $\pmb {{M}}^{{\text {e}}(1)}$ as

$$\begin{aligned} \pmb {{M}}^{{\text {e}}(1)}=2 {G} \pmb {E}^{{\text {e}}(1)}_0 + K \left( \text {tr}\pmb {E}^{{\text {e}}(1)}\right) \pmb { I}\;-3 K \alpha _{\text{th}} ( T- T_0)\pmb { I}\,. \end{aligned}$$

(144)

The thermal expansion is taken to have a bilinear temperature dependence, with the slope $\alpha _{{{\text{th}}}} = \alpha _{{\text{r}}} {\text{}}$ above the glass transition temperature and the slope $\alpha _{\mathrm{th}}=\alpha _{\text{gl}}$ below it.

Moreover, the evaluation equation for ${\pmb { {F}}}^{\text{p}(1)}$ follows Eqs. (133–134) with the thermally-activated relation for the equivalent plastic strain rate following

$$\dot{\epsilon }^{{{\text{p}}(1)}} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{if}}\;\;\tau^{{\text{e}}(1)} \le 0}, \hfill \\ {\dot{\epsilon }_{0}^{{(1)}} \exp \left( { - \frac{1}{\xi }} \right)\exp \left( { - \frac{{Q(T)}}{{K_{{\text{B}}} T}}} \right)\left[ {\sinh \left( {\frac{{\tau ^{{{\text{e}}(1)}} {\mkern 1mu} V^{\prime}}}{{2{\mkern 1mu} K_{{\text{B}}} T}}} \right)} \right]^{{1/m^{{(1)}} }} } \hfill & {{\text{if}}\;\;\tau^{{\text{e}}(1)} > 0}, \hfill \\ \end{array} } \right.$$

(145)

where $\dot{\epsilon}^{\text{p}(1)}$ is the plastic strain rate, the parameter $\epsilon _{0}^{\pmb .(1)}$ is a pre-exponential factor with units of 1/time, $K_{{\text{B}}}$ is Boltzmann’s constant, $V'$ is an activation volume, $m^{(1)}$ is the sensitive parameter for the strain rate, Q(T) is the temperature dependent activation energy, and $\tau ^{e(1)}$ denotes a net shear stress for the thermally activated flow

$$\begin{aligned} \tau ^{{\text {e}}(1)}=\bar{\tau}^{(1)}-( S_{{\text{a}}}+ S_{{\text{b}}}+\alpha _{{\text{p}}} \bar{p}), \end{aligned}$$

(146)

with $\alpha _{\text{p}}\geqslant 0$ a parameter introduced to account for the pressure sensitivity. The term $\exp {(-\dfrac{1}{\xi})}$ in Eq. (145) represents a concentration of flow defects, with

$$\xi = \left\{ {\begin{array}{*{20}l} {\xi _{{{\text{gl}}}} } \hfill & {{\text{if}}\;{T} \le T_{\text{g}}}, \hfill \\ {\xi _{{{\text{gl}}}} + d(T - T_{\text{g}})} \hfill & {{\text{if}}\;{T} > T_{\text{g}}}. \hfill \\ \end{array} } \right.$$

(147)

For the first micromechanism, besides the plastic strain gradient, the list $\varvec{\xi}^1=( \varphi ,{S}_{{\text{a}}},{S}_{{\text{b}}})$ of internal variables consists of three positive scalars, where the variable $\varphi \ge 0$ and ${S}_{{\text{a}}} \ge 0$ are introduced to model the yield peak which is observed in the stress-strain response of glassy polymers and ${S}_{{\text{b}}} \ge 0$ is introduced to model the isotropic hardening at high strain. The evolution equations of $\dot{S}_{{\text{a}}}$ and $\dot{\varphi}$ are governed by

$$\dot{S}_{{\text{a}}} = h_{{\text{a}}} \left[ {b(\varphi ^{*} - \varphi ) - S_{{\text{a}}} } \right]\dot{\epsilon }^{{{\text{p}}(1)}} \quad {\text{with}}\;{\text{initial}}\;{\text{value}}\;S_{{\text{a}}} = S_{{{\text{a}}0}} ,$$

(148)

$$\dot{\varphi } = g(\varphi ^{*} - \varphi )\dot{\epsilon }^{{{\text{p}}(1)}} \quad {\text{with}}\;{\text{initial}}\;{\text{value}}\;\varphi = \varphi _{0} ,$$

(149)

with $\varphi ^*$ as

$$\varphi ^{*} (\dot{\epsilon }^{{{\text{p}}(1)}} ,T) = \left\{ {\begin{array}{*{20}l} {z\left( {\left( {1 - \frac{T}{{T{\text{g}}}}} \right)^{r} + h_{{\text{g}}} } \right)\;\left( {\frac{{\dot{\epsilon }^{{{\text{p}}(1)}} }}{{\dot{\epsilon }_{{\text{r}}} }}} \right)^{s} } \hfill & {{\text{if}}\;T \le T_{{\text{g}}} \;{\text{and}}\;\dot{\epsilon }^{{{\text{p}}(1)}} > 0 ,} \hfill \\ z\;h_{\text{g}}\left( {\frac{{\dot{\varepsilon }^{{{\text{p(1)}}}} }}{{\varepsilon _{{\text{r}}} }}} \right)^{s} \hfill & {{\text{if}}\;T > T_{{\text{g}}} \;{\text{and}}\;\dot{\epsilon }^{{{\text{p}}(1)}} > 0,} \hfill \\ \end{array} } \right.$$

(150)

which represents the temperature and strain rate dependency of $\varphi$, where $z,\,r, \,{h}_{\mathrm{g}}$, and s are taken to be constants. In particular we have introduced $h_{\mathrm{g}}$ to get a small value for $\varphi ^*$ when $T>T_{{\text {g}}}$, instead of 0 in order to improve the convergence of the numerical model. Then the evolution of $S_{{\text{b}}}$ is governed by

$$\begin{aligned} S_{{\text{b}}}= S_{\text{b}0}+ H_{{\text{b}}}(\bar{\lambda}-1)^a , \,\,\,\bar{\lambda}=\sqrt{\text{tr} \pmb {C} /3}, \end{aligned}$$

(151)

where $\bar{\lambda}$ is an effective stretch which increases or decreases as the overall stretch increases or decreases, and where $H_{{\text{b}}}(T)$ is a temperature dependent hardening parameter.

The temperature evolutions of $H_{{\text{b}}}(T)$, Q(T) , G(T), and of the Poisson ratio ${\nu}({T})$ follow a law

$$\begin{aligned} \mathbf{\cdot} (T)=\frac{1}{2}\left( \mathbf{\cdot} _{\text{ gl }}+ \mathbf{\cdot} _{{\text{ r }}}\right) -\frac{1}{2}\left( \mathbf{\cdot} _{\text{ gl }}-\mathbf{\cdot} _{{\text{ r }}}\right) \tanh \left( \frac{1}{\Delta }({T}-{T}_{{\text {g}}})\right) - L_{\mathbf{\cdot} } (T-T_{{\text {g}}}), \end{aligned}$$

(152)

where $\mathbf{\cdot}_{\mathrm{gl}}$ and $\mathbf{\cdot}_{\text{r}}$ are the values in glassy and rubbery regions, and where $L_\mathbf{\cdot}$ represents the slope of the temperature variation of $\mathbf{\cdot}$, and takes the value of $L_\mathbf{\cdot} =L_{\mathbf{\cdot} _{\mathrm{gl}}}\,\text{if}\, T\le {T}_{\mathrm{g}}$ and $L_\mathbf{\cdot} =L_{\mathbf{\cdot} _{\mathrm{r}}}\,\text{if}\, T> {T}_{\mathrm{g}}$. The temperature dependence of the bulk modulus K(T) is then obtained by using the standard relation for isotropic materials ${K}(T)= {G}(T)\frac{2(1+\nu (T))}{3(1-2\;\nu (T))}$.

1.2.5 The second micromechanism ($\alpha =2$)

The second micromechanism ($\alpha = 2$) represents the molecular chains between mechanical crosslinks. At temperatures below $T_{\mathrm{g}}$ the polymer exhibits a significant amount of mechanical crosslinking which disintegrates when the temperature is increased above $T_{\mathrm{g}}$.

Only deviatoric contributions are considered in the free energy function

$$\begin{aligned} \psi ^{(2)}=\bar{\psi}^{(2)}(\bar{\pmb { C}}^{e(2)},\,T)=-\frac{1}{2}\; \mu ^{(2)} I^{(2)}_{{\text{m}}} \ln \left( 1-\frac{\text {tr} \bar{\pmb {C}}^{{\text {e}}(2)}-3}{I^{(2)}_{{\text{m}}}}\right) \,, \end{aligned}$$

(153)

where $\bar{\pmb { {C}}}^{{\text {e}}(2)}=\bar{\pmb { {F}}}^{{\text {e}}(2)\text {T}} \bar{\pmb { \text {F}}}^{{\text {e}}(2)}=\text {J}^{-\frac{2}{3}} \pmb {{C}}^{{\text {e}}(2)}$ denotes the distrotional (or volume preserving) right Cauchy strain tensor, the parameter $I^{(2)}_{{\text{m}}}$ is taken to be temperature constant, and where $\mu ^{(2)}$ is the rubbery shear modulus, which follows

$$\begin{aligned} \mu ^{(2)}=\mu _{\mathrm{g}} \text {exp} (- N( T- T_{\mathrm{g}}))\,, \end{aligned}$$

(154)

with $\mu _{\mathrm{g}}$ the value of $\mu ^{(2)}$ at the glass transition temperature, and N a parameter that represents the slope of temperature variation on a logarithmic scale.

The corresponding Mandel stress is evaluated from Eqs. (130) and (153) as

$$\begin{aligned} \pmb {{M}}^{{\text {e}}(2)}=\pmb {{C}}^{{\text {e}}(2)}\;2\;\frac{\partial {\bar{\psi}^{(2)}}}{\partial \pmb {C}^{{\text {e}}(2)}} =\mu ^{(2)}\left( 1-\frac{\text {tr} \bar{\pmb {{\text{C}}}}^{{\text {e}}(2)}-3}{ I^{(2)}_{{\text{m}}}}\right) ^{-1}\bar{\pmb {{C}}}^{{\text {e}}(2)}_0, \end{aligned}$$

(155)

where $\bar{\pmb { {C}}}^{{\text {e}}(2)}_0=\bar{\pmb { {C}}}^{{\text {e}}(2)}-\frac{1}{3}\text {tr} \bar{\pmb { {C}}}^{{\text {e}}(2)} \pmb { I}$ is the deviatoric part of $\pmb { {C}}^{{\text {e}}(2)}$ the right Cauchy Green tensor. Clearly, as $\bar{\pmb {{C}}}^{{\text {e}}(2)}$ and ${\pmb {{C}}}^{{\text {e}}(2)}$ permute, ${\pmb { {M}}^{{\text {e}}(2)}}$ and $\bar{\pmb { {C}}}^{{\text {e}}(2)}$ permute as well.

For the second mechanism, the equivalent plastic strain rate follows

$$\begin{aligned} \dot{\epsilon}^{\text{p}(2)}=\dot{\epsilon}^{(2)}_0\left( \frac{\bar{\tau}^{(2)}}{S^{(2)}}\right) ^{\frac{1}{m^{(2)}}}, \end{aligned}$$

(156)

where $\dot{\epsilon}^{(2)}_0$ is a reference plastic shear strain rate, $m^{(2)}$ is the positive valued strain rate sensitivity parameter, and $S^{(2)}$ is a temperature dependent parameter which follows (152).

1.2.6 The third micromechanism ($\alpha =3$)

The third micromechanism ($\alpha = 3$) introduces the molecular chains between chemical crosslinks and represents the resistance due to changes in the free energy upon stretching of the molecular chains between the crosslinks.

The free energy is a function of the deviatoric tensor $\bar{\pmb {{C}}}=\bar{\pmb { {F}}}^{\text {T}} \bar{\pmb {{F}}}= {J}^{-\frac{2}{3}}\pmb {{C}}$, and is given by a deviatoric form

$$\begin{aligned} \psi ^{(3)}=\bar{\psi}^{(3)}(\bar{\pmb { C}})=-\frac{1}{2}\; \mu ^{(3)} I^{(3)}_{{\text{m}}} \ln \left( 1-\frac{\text {tr} \bar{\pmb {C}}-3}{I^{(3)}_{{\text{m}}}}\right) , \end{aligned}$$

(157)

where the material constants $\mu ^{(3)}>0$ and $I^{(3)}_{{\text{m}}}>0$ are assumed to be temperature-independent.

The free energy (157) yields the corresponding second Piola stress $\pmb {{S}}^{(3)}$ as

$$\begin{aligned} \pmb { {S}}^{\text{(}3)}&=2\frac{\partial \bar{ \psi}^{(3)}}{\partial {\bar{\pmb {C}}}}:\frac{\partial \bar{\pmb { C}}}{\partial {\pmb { C}}} ={J}^{-\frac{2}{3}}\mu ^{(3)}\left( 1-\frac{\text {tr} {\bar{\pmb {{\text{C}}}}}-3}{\pmb { {I}}^{(3)}_{{\text{m}}}}\right) ^{-1} \left[ \pmb { I}-\frac{1}{3} \left( \text {tr} \bar{\pmb { C}}\right) {\bar{\pmb {{\text{C}}}}}^{-1} \right] \,. \end{aligned}$$

(158)

1.2.7 Finite increment form of the shape memory polymer constitutive law

The constitutive laws are formulated in a finite strain setting and solved following the predictor-corrector scheme during the time interval [$t_{n} ;\;t_{n+1}$], where we use the subscript $n$ for the previous time $t_{n}$ and $n+1$ for the current time $t_{n+1}$. The formulation can be summarized as follows:

• Prediction step The plastic deformation gradient is initialized to the value at the previous step $\pmb {{F}}^{\text {p}({\alpha})}_{(\text {pr})}=\pmb {{F}}^{\text {p} ({\alpha})}_{n}$, and the elastic deformation follows

  $$\begin{aligned} \pmb {{F}}^{\text {e}({\alpha})}_{n+1}=\pmb {{F}}_{n+1}\pmb { {F}}^{\text {p}(\alpha )^{-1}}_{n}\,. \end{aligned}$$

  (159)

• Correction step In this step we solve the system of equations that has been presented for each mechanism. To extract the plastic increment using the evaluation equation of the plastic deformation gradient during the time step between the configurations n and n+1, we consider

  $$\begin{aligned} \pmb {{F}}^{\text{p}({\alpha})}_{n+1}=\exp (\Delta {\pmb {\text {D}}}^{\text{p}({\alpha})})\pmb { {F}}^{\text{p}({\alpha})}_{n}\,, \end{aligned}$$

  (160)

  with

  $$\begin{aligned} {\Delta}\pmb {{D}}^{\text{p}({\alpha})}=(\epsilon ^{\text{p}({\alpha})}_{n+1}-\epsilon ^{\text{p}({\alpha})}_{n})\frac{\pmb {{M}}^{{\text {e}}({\alpha})}}{2\bar{\tau}^{({\alpha})}}=\Delta {\epsilon}^{\text{p}({\alpha})} \left( \dfrac{\pmb { {M}}^{{\text {e}}({\alpha})}}{2\bar{\tau}^{({\alpha})}} \right) \,. \end{aligned}$$

  (161)

More details about the predictor-corrector algorithm and the stiffness computation can be found in [26].

Appendix 2: The finite element formulation

Using the interpolations (33–34), the gradients are readily obtained by:

$$\begin{aligned} \nabla _0{\varvec{u}_{\mathrm{h}}}&\,= \, \mathbf{u}^{a}\otimes \nabla _0{{N}}^{a}_{{\mathbf{u}}} \, , \quad \nabla _0{{{{\mathbf{M}}}_{{\text {h}}}}}=\nabla _0{\mathbf{N}}^{a}_{{{\mathbf{M}}}}\;{{\mathbf{M}}}^{a} \,, \end{aligned}$$

(162)

$$\begin{aligned} \nabla _0{\delta}{\varvec{u}_{\mathrm{h}}}&\,= \, {\delta}{} \mathbf{u}^{a}\otimes \nabla _0{{N}}^{a}_{{\mathbf{u}}} \, , \quad \nabla _0{\delta}{{{{\mathbf{M}}}_{{\text {h}}}}}=\nabla _0{\mathbf{N}}^{a}_{{{\mathbf{M}}}}\;{\delta}{{\mathbf{M}}}^{a}\,, \end{aligned}$$

(163)

where $\nabla _0{{N}}^{a}_{{\mathbf{u}}}$ and $\nabla _0{\mathbf{N}}^{a}_{{\mathbf{M}}}= \left( \begin{array}{cc} \nabla _0{{N}}^{a}_{{\text {f}}_{{\text {V}}}} &{} \varvec{0}\\ \varvec{0} &{} \nabla _0{{N}}^{a}_{{\text {f}}_{\mathrm{T}}} \end{array} \right)$ are the gradients of the shape functions at node a.

1.1 Nodal forces

The expressions of the nodal forces (36) are obtained by substituting the interpolations (33–34) and (162–163) in the weak formulation (31).

First the mechanical contribution reads

$$\begin{aligned} {{\mathbf{F}}}_{\mathbf{u}{\text {ext}}}^{a}&= \sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{N}} \varOmega _0\right) ^{\text {s}}} {{N}}^{a}_{{\mathbf{u}}} \bar{\varvec{{T}}}{\text {dS}}_0 \\&\quad-\,\sum _{\mathrm{s}}\int _{\left( \partial _{\mathrm{D}} \varOmega _0\right) ^{\text {s}}} \left( \bar{\varvec{{u}}} \otimes \varvec{{N}} : \pmb {\mathcal {H}}\right) \cdot \nabla _0{{N}}^{a}_{{\mathbf{u}}} {\text {dS}}_0\nonumber \\&\quad+ \sum _{\mathrm{s}}\int _{\left( \partial _{\mathrm{D}} \varOmega _0\right) ^{\text {s}}} \left( \bar{\varvec{{u}}} \otimes \varvec{N} : \frac{ \pmb {\mathcal {H}}{\mathcal {B}}}{{h}_{\mathrm{s}}} \right) \cdot \varvec{{N}} {{N}}^{a}_{{\mathbf{u}}} {\text {dS}}_0\nonumber \nonumber \\&\quad-\,\int _{\partial _{\mathrm{D}}\varOmega _{0\mathrm{h}}} \left( \pmb {\mathcal {Y}}({\bar{{\mathbf{M}}}}){\bar{{\mathbf{M}}}}-\pmb {\mathcal {Y}}({\bar{{\mathbf{M}}}_0}){\bar{{\mathbf{M}}}_0} \right) \cdot \varvec{{N}}{{N}}^{a}_{{\mathbf{u}}} {\text {dS}}_0\;,\end{aligned}$$

(164)

$$\begin{aligned} {{\mathbf{F}}}_{\mathbf{u}{\text {int}}}^{a}&\,= \, \sum _{{\text {e}}} \int _{\varOmega ^\text {e}_0} \pmb {{P}}(\pmb {F}_{\mathrm{h}},\, {\mathbf{M}}_{\mathrm{h}})\cdot \nabla _0{{N}}^{a}_{{\mathbf{u}}} {\text {d}}\varOmega _0,\,\text {and}\end{aligned}$$

(165)

$$\begin{aligned} {\mathbf{F}}_{\mathbf{u}{\text {I}}}^{a\pm}=\, & {} {\mathbf{F}}_{{\mathbf{u}{\text {I1}}}}^{a\pm}+{\mathbf{F}}_{{\mathbf{u}{\text {I2}}}}^{a\pm}+{\mathbf{F}}_{{\mathbf{u}{\text {I3}}}}^{a\pm}\,, \end{aligned}$$

(166)

with the three mechanical contributions to the interface forces related to the degrees of freedom of the nodes $a\pm$ on each side of the interface elements reading¹

$$\begin{aligned} {{\mathbf{F}}}_{\mathbf{u}{\text {I1}}}^{a\pm}= & {} \sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{I}} \varOmega _0\right) ^{\text {s}}}(\pm {{N}}^{a\pm}_{{\mathbf{u}}}) \left\langle \pmb {{P}}(\pmb {F}_{\mathrm{h}},\, {\mathbf{M}}_{\mathrm{h}})\right\rangle \cdot \varvec{{N}}^{-} {\text {dS}}_0,\end{aligned}$$

(167)

$$\begin{aligned} {{\mathbf{F}}}_{\mathbf{u}{\text {I2}}}^{a\pm}= & {} \frac{1}{2}\sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{I}} \varOmega _0\right) ^{\text {s}}}\llbracket {\varvec{{u}}_{\mathrm{h}}}\rrbracket \otimes \varvec{{N}}^{-} :\pmb {\mathcal {H}}^\pm \cdot \nabla _0{{N}}^{a\pm}_{{\mathbf{u}}}{\text {dS}}_0,\end{aligned}$$

(168)

$$\begin{aligned} {{\mathbf{F}}}_{\mathbf{u}{\text {I3}}}^{a\pm}= & {} \sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{I}} \varOmega _0\right) ^{\text {s}}}(\llbracket {\varvec{{u}}_{\mathrm{h}}}\rrbracket \otimes \varvec{{N}}^{-}): \left\langle \frac{\pmb {\mathcal {H}}{\mathcal {B}}}{{h}_{\mathrm{s}}} \right\rangle \cdot \varvec{{N}}^{-}(\pm {{N}}^{a\pm}_{{\mathbf{u}}}) {\text {dS}}_0. \end{aligned}$$

(169)

Secondly, the electro-thermal contributions read

$$\begin{aligned} {{\mathbf{F}}_{{\mathbf{M}}_{{\text {ext}}}}^{a}}&= {} \sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{N}} \varOmega _0\right) ^{\text {s}}}{\mathbf{N}}^{a}_{{\mathbf{M}}} \bar{{\mathbf{J}}}{\text {dS}}_0-\sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{D}} \varOmega _0\right) ^{\text {s}}} \nabla _0 {{\mathbf{N}}}^{a^{\text {T}}}_{{\mathbf{M}}} {\pmb {{Z}}_0}(\pmb {F}_{\mathrm{h}},\bar{{\mathbf{M}}}) \bar{\mathbf{M}}_{{\mathbf{N}}}{\text {dS}}_0 \nonumber \\&\quad+\sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{D}} \varOmega _0\right) ^{\text {s}}} {\mathbf{N}}^{a}_{{\mathbf{M}}}{\bar{{\mathbf{N}}}_{{\mathbf{M}}}}^{\text {T}} {\pmb {{Z}}_0}(\pmb {F}_{\mathrm{h}},\bar{{\mathbf{M}}})\frac{\mathcal {B}}{{h}_{\mathrm{s}}} \bar{{\mathbf{M}}}_{{\mathbf{N}}} {\text {dS}}_0\,, \end{aligned}$$

(170)

$$\begin{aligned} {{\mathbf{F}}_{{\mathbf{M}}_{{\text {int}}}}^{a}}= & {} \sum _{{\text {e}}} \int _{\varOmega ^\text {e}_0}\nabla _0{{\mathbf{N}}}^{a^{\text {T}}}_{{\mathbf{M}}}{\mathbf{J}}(\pmb {F}_{\mathrm{h}},\,{\mathbf{M}}_{\mathrm{h}},\nabla {\mathbf{M}}_{{\text {h}}}){\text {d}}{{\varOmega}}_0+ \sum _{{\text {e}}} \int _{\varOmega ^\text {e}_0} {{\mathbf{N}}}^{a^{\text {T}}}_{{\mathbf{M}}}{\mathbf{I}}_{\mathrm{i}}{\text {d}}{{\varOmega}}_0 ,\,\text {and} \end{aligned}$$

(171)

$$\begin{aligned} {{\mathbf{F}}_{{\mathbf{M}}_{{\text {I}}}}^{a\pm}}= {} {\mathbf{F}}_{{\mathbf{M}}_{{\text {I1}}}}^{a\pm}+ {\mathbf{F}}_{{\mathbf{M}}_{{\text {I2}}}}^{a\pm}+{\mathbf{F}}_{{\mathbf{M}}_{{\text {I3}}}}^{a\pm}, \end{aligned}$$

(172)

with the three electric contributions to the interface forces$^{1}$

$$\begin{aligned} {{\mathbf{F}}_{{\mathbf{M}}_{{\text {I1}}}}^{a\pm}}= & {} \sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{I}} \varOmega _0\right) ^{\text {s}}} (\pm {\mathbf{N}}^{a\pm}_{{\mathbf{M}}})\left( \bar{{\mathbf{N}}}^{-}_{{\mathbf{M}}}\right) ^{\text {T}}\left\langle {{\mathbf{J}}}(\pmb {F}_{\mathrm{h}},\,{\mathbf{M}}_{{\text {h}}},\,\nabla {\mathbf{M}}_{{\text {h}}}) \right\rangle {\text {dS}}_0, \end{aligned}$$

(173)

$$\begin{aligned} {{\mathbf{F}}_{{\mathbf{M}}_{{\text {I2}}}}^{a\pm}}= & {} \frac{1}{2}\sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{I}} \varOmega _0\right) ^{\text {s}}} \left( {\nabla _0{ {{\mathbf{N}}}^{a\pm ^{\text {T}}}_{{\mathbf{M}}}}} \pmb {{Z}}^\pm _0({\pmb {F}}_{\mathrm{h}},\, {\mathbf{M}}_{{\text {h}}})\right) \llbracket {\mathbf{M}}_{{\text {h}}_{{\mathbf{N}}}}\rrbracket {\text {dS}}_0, \end{aligned}$$

(174)

$$\begin{aligned} {{\mathbf{F}}_{{\mathbf{M}}_{{\text {I3}}}}^{a\pm}}= & {} \sum _{\mathrm{s}} \int _{\left( \partial _{\mathrm{I}} \varOmega _0\right) ^{\text {s}}}(\pm {{\mathbf{N}}}^{a\pm}_{{\mathbf{M}}})\left( {\bar{\mathbf{{N}}}^{-}_{{\mathbf{M}}}}\right) ^{\text {T}} \left\langle \pmb {{Z}}_0({\pmb {F}}_{\mathrm{h}},\, {\mathbf{M}}_{{\text {h}}})\frac{ {\mathcal {B}}}{{h}_{\mathrm{s}}} \right\rangle \llbracket {\mathbf{M}}_{{\text {h}}_{{\mathbf{N}}}}\rrbracket {\text {dS}}_0\,. \end{aligned}$$

(175)

1.2 Tangent operators

In order to derive the tangent matrix $\pmb {\mathbb {K}}_{{\mathbf{G}}}^{{ab}}=\frac{ \partial {\mathbf{F}}_{\mathrm{ext}}^{a}}{\partial {{\mathbf{G}}}^{b}}-\frac{\partial {\mathbf{F}}_{\mathrm{int}}^{a}}{\partial {{\mathbf{G}}}^{b}}-\frac{\partial {\mathbf{F}}_{\mathrm{I}}^{a}}{ \partial {{\mathbf{G}}}^{b}}$, the system (38) is rewritten

$$\begin{aligned} \begin{aligned} \left( \begin{array}{cc} \pmb {\mathbb {K}}_{{\mathbf{u}}{\mathbf{u}}} &{}\pmb {\mathbb {K}}_{\mathbf{u} {\mathbf{M}}}\\ \pmb {\mathbb {K}}_{ {\mathbf{M}}{\mathbf{u}}} &{} \pmb {\mathbb {K}}_{{\mathbf{M}} {\mathbf{M}}} \end{array} \right) \left( \begin{array}{c}{ \Delta \mathbf{u}}\\ \Delta {\mathbf{M}}\end{array} \right) =-\left( \begin{array}{c}{\mathbf{R}}_{\mathbf{u}}(\mathbf{u}, {\mathbf{M}})\\ {\mathbf{R}}_{ {\mathbf{M}}}(\mathbf{u}, {\mathbf{M}})\end{array} \right) . \end{aligned} \end{aligned}$$

(176)

The stiffness matrix has been decomposed into four sub-matrices as shown in Eq. (176) with respect to the discretization of the five independent field variables (d for displacement ${\mathbf{u}}$, and 2 for ${\mathbf{M}}$ (for ${{\text {f}}_{\mathrm{V}}}$, and ${{\text {f}}_{\mathrm{T}}}$)), and can be obtained in a straighforward way from the internal forces, see details in [26].

Appendix 3: Derivation of the numerical properties

1.1 Taylor’s remainders

The remainder terms of Eqs. (60–60) are obtained by defining $\mathbf{V}^{{\text {t}}}={\mathbf{G}}+ t(\mathbf{V}-{\mathbf{G}})$, $\nabla \mathbf{Q}^{{\text {t}}}=\nabla {\mathbf{G}}+ t(\nabla \mathbf{Q}-\nabla {\mathbf{G}})$. They can thus be evaluated by

$$\begin{aligned} \bar{\mathbf{w}}_{{\mathbf{G}}}({\mathbf{G}},\nabla {\mathbf{G}})=\int _0^1\mathbf{w}_{{\mathbf{G}}}(\mathbf{V}^{{\text {t}}},\nabla \mathbf{Q}^{{\text {t}}}) {\text {d}} t, \;\;\;\bar{\mathbf{w}}_{\nabla {\mathbf{G}}}({\mathbf{G}})=\int _0^1\mathbf{w}_{\nabla {\mathbf{G}}}(\mathbf{V}^{{\text {t}}}) {\text {d}} t \,, \end{aligned}$$

(177)

and by

$$\begin{aligned} \bar{\mathbf{w}}_{{\mathbf{G}}{} {\mathbf{G}}}(\mathbf{V},\nabla \mathbf{Q})=\int _0^1(1-t){\mathbf{w}}_{{\mathbf{G}}{} {\mathbf{G}}}(\mathbf{V}^{{\text {t}}},\nabla \mathbf{Q}^{{\text {t}}}) {\text {d}} t\,,\; \bar{\mathbf{w}}_{{\mathbf{G}}\nabla {\mathbf{G}}}(\mathbf{V})=\int _0^1(1- t){\mathbf{w}}_{{\mathbf{G}}\nabla {\mathbf{G}}}(\mathbf{V}^{{\text {t}}}) {\text {d}} t\,, \end{aligned}$$

(178)

with the partial derivatives $\mathbf{w}_{{\mathbf{G}}{} {\mathbf{G}}}({\mathbf{G}},\nabla {\mathbf{G}})=\mathbf{v}_{{\mathbf{G}}{} {\mathbf{G}}}({\mathbf{G}}) \nabla {\mathbf{G}}$ and $\mathbf{w}_{{\mathbf{G}}\nabla {\mathbf{G}}}({\mathbf{G}})=\mathbf{v}_{{\mathbf{G}}}({\mathbf{G}})$ of ${\mathbf{w}}({\mathbf{G}},\nabla {\mathbf{G}})$, since $\mathbf{w}_{\nabla {\mathbf{G}}\nabla {\mathbf{G}}}({\mathbf{G}})=0$.

The remainder terms of Eqs. (62–63) read

$$\begin{aligned} \begin{aligned} \bar{\mathbf{d}}_{{\mathbf{G}}}({\mathbf{G}},\nabla {\mathbf{G}})=\int _0^1\mathbf{d}_{{\mathbf{G}}}(\mathbf{V}^{{\text {t}}},\nabla \mathbf{Q}^{{\text {t}}}) {\text {dt}} , \;\;\;\bar{\mathbf{d}}_{\nabla {\mathbf{G}}}({\mathbf{G}})=\int _0^1\mathbf{d}_{\nabla {\mathbf{G}}}(\mathbf{V}^{{\text {t}}}) {\text {dt}} \,, \end{aligned} \end{aligned}$$

(179)

and

$$\begin{aligned} \begin{aligned} \bar{\mathbf{d}}_{{\mathbf{G}}{} {\mathbf{G}}}(\mathbf{V},\nabla \mathbf{Q})=\int _0^1(1-t){\mathbf{d}}_{{\mathbf{G}}{} {\mathbf{G}}}(\mathbf{V}^{{\text {t}}},\nabla \mathbf{Q}^{{\text {t}}}) {\text {d}} t\,,\; \bar{\mathbf{d}}_{{\mathbf{G}}\nabla {\mathbf{G}}}(\mathbf{V})=\int _0^1(1-t){\mathbf{d}}_{{\mathbf{G}}\nabla {\mathbf{G}}}(\mathbf{V}^{{\text {t}}}) {\text {d}} t\,. \end{aligned} \end{aligned}$$

(180)

Finally, the remainder terms of Eqs. (64–65) read

$$\begin{aligned} \bar{\mathbf{p}}_{{\mathbf{G}}}({\mathbf{G}})=\int _0^1\mathbf{p}_{{\mathbf{G}}}(\mathbf{V}^{{\text {t}}}) {\text {dt}}, \quad \bar{\mathbf{p}}_{{\mathbf{G}}{} {\mathbf{G}}}(\mathbf{V})=\int _0^1(1-{\text {t}}){\mathbf{p}}_{{\mathbf{G}}{} {\mathbf{G}}}(\mathbf{V}^{{\text {t}}}) {\text {dt}}\,. \end{aligned}$$

(181)

1.2 Application of the Taylor’s expansion

The first term of Eq. (70) is rewritten, using the Taylor’s expansion defined in Eq. (60), as

$$\begin{aligned} &\int _{ \varOmega _{\mathrm{h}}} (\nabla \delta {\mathbf{G}}_{\mathrm{h}})^{\text {T}}( {\mathbf{w}}({\mathbf{G}}^{{\text {e}}},\nabla {\mathbf{G}}^{{\text {e}}})-{\mathbf{w}}({\mathbf{G}}_{\mathrm{h}},\nabla {\mathbf{G}}_{\mathrm{h}})) {\text {d}}{\varOmega} =\int _{ \varOmega _{\mathrm{h}}} (\nabla \delta {\mathbf{G}}_{\mathrm{h}})^{\text {T}}({\mathbf{w}}_{{\mathbf{G}}}({\mathbf{G}}^{{\text {e}}},\nabla {\mathbf{G}}^{{\text {e}}})({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})) {\text {d}}{\varOmega} \\&\quad + \int _{ \varOmega _{\mathrm{h}}} (\nabla \delta {\mathbf{G}}_{\mathrm{h}})^{\text {T}}({\mathbf{w}}_{\nabla {\mathbf{G}}}({\mathbf{G}}^{{\text {e}}})(\nabla {\mathbf{G}}^{{\text {e}}}-\nabla {\mathbf{G}}_{\mathrm{h}})) {\text {d}}{\varOmega} -\int _{ \varOmega _{\mathrm{h}}} (\nabla \delta {\mathbf{G}}_{\mathrm{h}})^{\text {T}}(\bar{\mathbf{R}}_{{\mathbf{w}}} ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}},\nabla {\mathbf{G}}^{{\text {e}}}-\nabla {\mathbf{G}}_{\mathrm{h}})) {\text {d}}{\varOmega}\,. \end{aligned}$$

(182)

Similarly, the second term of Eq. (70) is rewritten, using the Taylor’s expansion defined in Eq. (62), as

$$\begin{aligned} \begin{aligned}&\int _{ \varOmega _{\mathrm{h}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}} \left( \tilde{\mathbf{o}}({\mathbf{G}}^{{\text {e}}}) \nabla {\mathbf{G}}^{{\text {e}}}-\tilde{\mathbf{o}}({\mathbf{G}}_{\mathrm{h}}) \nabla {\mathbf{G}}_{\mathrm{h}}\right) {\text {d}}{\varOmega} =\int _{ \varOmega _{\mathrm{h}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}} \left( {\mathbf{d}}({\mathbf{G}}^{{\text {e}}}, \nabla {\mathbf{G}}^{{\text {e}}})-{\mathbf{d}}({\mathbf{G}}_{\mathrm{h}},\nabla {\mathbf{G}}_{\mathrm{h}})\right) {\text {d}}{\varOmega}\\&\quad =\int _{ \varOmega _{\mathrm{h}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}} {\mathbf{d}}_{{\mathbf{G}}}({\mathbf{G}}^{{\text {e}}},\nabla {\mathbf{G}}^{{\text {e}}}) ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})) {\text {d}}{\varOmega} +\int _{ \varOmega _{\mathrm{h}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}} {\mathbf{d}}_{\nabla {\mathbf{G}}}({\mathbf{G}}^{{\text {e}}}) (\nabla {\mathbf{G}}^{{\text {e}}}-\nabla {\mathbf{G}}_{\mathrm{h}})) {\text {d}}{\varOmega}\\&\quad \quad-\,\int _{ \varOmega _{\mathrm{h}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}}\bar{\mathbf{R}}_{{\mathbf{d}}} ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}},\nabla {\mathbf{G}}^{{\text {e}}}-\nabla {\mathbf{G}}_{\mathrm{h}}){\text {d}}{\varOmega}\,. \end{aligned} \end{aligned}$$

(183)

Likewise, the third term is rewritten, using the Taylor’s expansion defined in Eq. (60), as

$$\begin{aligned} \begin{aligned}&\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket \delta {\mathbf{G}}_{{\text {h}}_{{\mathbf{n}}}}^{\text {T}} \rrbracket \left\langle {\mathbf{w}}({\mathbf{G}}^{{\text {e}}},\nabla {\mathbf{G}}^{{\text {e}}})-{\mathbf{w}}({\mathbf{G}}_{\mathrm{h}},\nabla {\mathbf{G}}_{\mathrm{h}})\right\rangle {\text {dS}}\\&\quad =\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket \delta {\mathbf{G}}_{{\text {h}}_{{\mathbf{n}}}}^{\text {T}} \rrbracket \left\langle {\mathbf{w}}_{{\mathbf{G}}}({\mathbf{G}}^{{\text {e}}},\nabla {\mathbf{G}}^{{\text {e}}})({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})\right\rangle {\text {dS}} \\&\quad \quad+ \int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket \delta {\mathbf{G}}_{{\text {h}}_{{\mathbf{n}}}}^{\text {T}}\rrbracket \left\langle {\mathbf{w}}_{\nabla {\mathbf{G}}}({\mathbf{G}}^{{\text {e}}})(\nabla {\mathbf{G}}^{{\text {e}}}-\nabla {\mathbf{G}}_{\mathrm{h}})\right\rangle {\text {dS}}\\&\quad \quad-\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket \delta {\mathbf{G}}_{{\text {h}}_{{\mathbf{n}}}}^{\text {T}} \rrbracket \left\langle \bar{\mathbf{R}}_{{\mathbf{w}}} ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}},\nabla {\mathbf{G}}^{{\text {e}}}-\nabla {\mathbf{G}}_{\mathrm{h}})\right\rangle {\text {dS}}. \end{aligned} \end{aligned}$$

(184)

The fifth term of Eq. (70) is developed by using the definition of $\mathbf{p}^{\text {T}}({\mathbf{G}})={\mathbf{G}}^{\text {T}}{} \mathbf{o}^{\text {T}}({\mathbf{G}})$ and using the Taylor’s expansion defined in Eq. (64), leading to

$$\begin{aligned} \begin{aligned}&-\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket {\mathbf{G}}^{{\text {e}}^{\text {T}}} {{\mathbf{o}}}^{\text {T}}({\mathbf{G}}^{{\text {e}}}) -{\mathbf{G}}_{{\text {h}}}^{\text {T}}{{\mathbf{o}}}^{\text {T}}({\mathbf{G}}_{\mathrm{h}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}} = \\&\quad -\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket ({\mathbf{G}}^{{\text {e}}^{\text {T}}}-{\mathbf{G}}_{{\text {h}}}^{\text {T}})\mathbf{p}^{\text {T}}_{{\mathbf{G}}}({\mathbf{G}}^{{\text {e}}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}} +\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket \bar{\mathbf{R}}_{{\mathbf{p}}}^{\text {T}} ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})\rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}}. \end{aligned} \end{aligned}$$

(185)

However, as $\mathbf{p}^{\text {T}}_{{\mathbf{G}}}=-{{\mathbf{o}}}^{\text {T}}({\mathbf{G}})$, using Eq. (66), this last term is also rewritten as

$$\begin{aligned} \begin{aligned}&-\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket {\mathbf{G}}^{{\text {e}}^{\text {T}}} {{\mathbf{o}}}^{\text {T}}({\mathbf{G}}^{{\text {e}}}) -{\mathbf{G}}_{{\text {h}}}^{\text {T}}{{\mathbf{o}}}^{\text {T}}({\mathbf{G}}_{\mathrm{h}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}}\\= {}&\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket ({\mathbf{G}}^{{\text {e}}^{\text {T}}}-{\mathbf{G}}_{{\text {h}}}^{\text {T}})\mathbf{o}^{\text {T}}({\mathbf{G}}^{{\text {e}}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}} \\&\quad -\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({\mathbf{G}}_{\mathrm{h}}) ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}}. \end{aligned} \end{aligned}$$

(186)

Finally, using the definition of the ${\tilde{ \mathbf{\cdot} }}$ operator ${\mathbf{G}}^{\text {T}}{{\mathbf{o}}}^{\text {T}}({\mathbf{G}}^{\prime}) \delta {\mathbf{G}}_{{\mathbf{n}}}={\mathbf{G}}^{\text {T}}_{{\mathbf{n}}}\tilde{{\mathbf{o}}}^{\text {T}}({\mathbf{G}}^{\prime}) \delta {\mathbf{G}}$ ², and Eq. (186) is rewritten as

$$\begin{aligned} \begin{aligned}&-\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket {\mathbf{G}}^{{\text {e}}^{\text {T}}} {{\mathbf{o}}}^{\text {T}}({\mathbf{G}}^{{\text {e}}}) -{\mathbf{G}}_{{\text {h}}}^{\text {T}}{{\mathbf{o}}}^{\text {T}}({\mathbf{G}}_{\mathrm{h}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}}\\&\quad =\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket ({\mathbf{G}}^{{\text {e}}^{\text {T}}}_{{\mathbf{n}}}-{\mathbf{G}}_{{\text {h}}_{{\mathbf{n}}}}^{\text {T}}) \tilde{\mathbf{o}}^{\text {T}}({\mathbf{G}}^{{\text {e}}})\rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}}\right\rangle {\text {dS}} \\&\quad -\int _{\partial _{{\text {I}}} \varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}} \varOmega _{\mathrm{h}}}\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({\mathbf{G}}_{\mathrm{h}}) ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}}. \end{aligned} \end{aligned}$$

(187)

1.3 General properties of the finite element method and Hilbert spaces

The following properties will be used in the numerical properties derivation.

Lemma 5

(Interpolation inequality) For all ${\mathbf{G}}\in \left( \text {H}^{s}(\varOmega ^{{\text {e}}})\right) ^{n}$ there exists a sequence ${\mathbf{G}}^{\text {h}}\in \left( \mathbb {P}^{k}(\varOmega ^{{\text {e}}})\right) ^{n}$ and a positive constant ${\text{C}}_{{\mathcal {D}}}^{k}$ depending on s and k but independent of ${\mathbf{G}}$ and $h_{\mathrm{s}}$, such that

• 1. for any $0\le n\le s$

  $$\begin{aligned} \begin{aligned}&\parallel {\mathbf{G}}-{\mathbf{G}}^{\text {h}}\parallel _{\mathrm{H}^{n}(\varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {D}}}^{k} h_{\mathrm{s}}^{\mu -n} \parallel {\mathbf{G}}\parallel _{\mathrm{H}^{s}(\varOmega ^{{\text {e}}})}, \end{aligned} \end{aligned}$$

  (188)

• 2. for any $0\le n\le s-1+\frac{2}{r}$

  $$\begin{aligned} \begin{aligned}&\parallel {\mathbf{G}}-{\mathbf{G}}^{\text {h}}\parallel _{\text{W}^{n}_{r}(\varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {D}}}^{k} h_{\mathrm{s}}^{\mu -n-1+\frac{2}{r}} \parallel {\mathbf{G}}\parallel _{\mathrm{H}^{s}(\varOmega ^{{\text {e}}})},\text { if } d=2, \end{aligned} \end{aligned}$$

  (189)

• 3. for any $s>n+\frac{1}{2}$

  $$\begin{aligned} \begin{aligned}&\parallel {\mathbf{G}}-{\mathbf{G}}^{\text {h}}\parallel _{\mathrm{H}^{n}(\partial \varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {D}}}^{k} h_{\mathrm{s}}^{\mu -n-\frac{1}{2}} \parallel {\mathbf{G}}\parallel _{\mathrm{H}^{s}(\varOmega ^{{\text {e}}})}, \end{aligned} \end{aligned}$$

  (190)

where $\mu ={\text {min}}\left\{ s,\, k+1\right\}$.

The proof of the first and third properties can be found in [7], and the proof of the second property in the particular case of ${d}=2$ can be found in [1, 2], see also the discussion by [23].

Remarks

• (1) The approximation property in (2) is still valid for $r=\infty$, see [1].

• (2) For ${\mathbf{G}}\in \text{X}_{s}$, let us define the interpolant ${\text {I}}_{\mathrm{h}}{} {\mathbf{G}}\in \text{X}^{k}$ by ${\text {I}}_{\mathrm{h}}{} {\mathbf{G}}|_{\varOmega ^{{\text {e}}}}={\mathbf{G}}^{\text {h}}( {\mathbf{G}}|_{\varOmega ^{{\text {e}}}})$, which means ${\text {I}}_{\mathrm{h}}{} {\mathbf{G}}$ satisfies the interpolant inequality property provided in Lemma 5 on $\varOmega _{\mathrm{h}}$, see [28].

Lemma 6

(Trace inequality) For all ${\mathbf{G}}\in \left( \text {H}^{s+1}(\varOmega ^{{\text {e}}})\right) ^{n}$, there exists a positive constant ${\text{C}}_{{\mathcal {T}}}$, such that

$$\begin{aligned} \begin{aligned}&\parallel {\mathbf{G}}\parallel ^{r}_{{\text{W}}^{s}_{r}(\partial \varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {T}}} \left( \frac{1}{h_{\mathrm{s}}} \parallel {\mathbf{G}}\parallel ^{r}_{{\text{W}}^{s}_{r}(\varOmega ^{{\text {e}}})}+\parallel {\mathbf{G}}\parallel ^{r-1}_{{\text{W}}^{s}_{2r-2}(\varOmega ^{{\text {e}}})} \parallel \nabla ^{s+1}{} {\mathbf{G}}\parallel _{{\text {L}}^{2}(\varOmega ^{{\text {e}}})} \right) , \end{aligned} \end{aligned}$$

(191)

where $s=0, 1$ and $r=2, 4$, or in other words

$$\begin{aligned} \begin{aligned}&\parallel {\mathbf{G}}\parallel ^2_{\mathrm{L}^{2}(\partial \varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {T}}} \left( \frac{1}{h_{\mathrm{s}}} \parallel {\mathbf{G}}\parallel ^2_{{\text {L}}^{2}(\varOmega ^{{\text {e}}})}+\parallel {\mathbf{G}}\parallel _{{\text {L}}^{2}(\varOmega ^{{\text {e}}})} \parallel \nabla {\mathbf{G}}\parallel _{{\text {L}}^{2}(\varOmega ^{{\text {e}}})} \right) ,\\&\parallel {\mathbf{G}}\parallel ^4_{\mathrm{L}^{4}(\partial \varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {T}}} \left( \frac{1}{h_{\mathrm{s}}} \parallel {\mathbf{G}}\parallel ^4_{{\text {L}}^{4}(\varOmega ^{{\text {e}}})}+ \parallel {\mathbf{G}}\parallel ^{3}_{{\text {L}}^6(\varOmega ^{{\text {e}}})} \parallel \nabla {\mathbf{G}}\parallel _{{\text {L}}^{2}(\varOmega ^{{\text {e}}})} \right) . \end{aligned} \end{aligned}$$

(192)

The first equation, $s=0$ and $r=2$, is proved in [52], and the second one, $r=4$ and $s=0$, is proved in [22].

Lemma 7

(Trace inequality on the finite element space) For all ${\mathbf{G}}_{\mathrm{h}}\in \left( \mathbb {P}^{k}(\varOmega ^{{\text {e}}})\right) ^{n}$ there exists a constant ${\text{C}}_{{\mathcal {K}}}^{k}>0$ depending on k, such that

$$\begin{aligned} \parallel \nabla ^{\text{l}} {\mathbf{G}}_{\mathrm{h}} \parallel _{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}\le {{\text{C}}_{{\mathcal {K}}}^{k}}{{{h}}_{\mathrm{s}}^{-\frac{1}{2}}} \parallel \nabla ^{\text{l}} {\mathbf{G}}_{\mathrm{h}} \parallel _{\mathrm{L}^2(\varOmega ^{\text {e}})}\,\,\;\;\; l =0,\,1, \end{aligned}$$

(193)

where ${\text{C}}_{{\mathcal {K}}}^{k}=\text {sup}_{{\mathbf{G}}_{\mathrm{h}}\in {(\text{P}_{k}(\varOmega ^{\text {e}})})^n}\frac{{h}_{\text s}\parallel \nabla {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}}{\parallel \nabla {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{L}^2(\varOmega ^{\text {e}})}}$ is a constant which depends on the degree of the polynomial approximation only with ${h}_{\text s}=\frac{\vert \varOmega ^{\text {e}}\vert}{\vert \partial \varOmega ^{\text {e}}\vert}$, see [25] for more details.

Lemma 8

(Inverse inequality) For ${\mathbf{G}}_{\mathrm{h}}\in \left( \mathbb {P}^{k}(\varOmega ^{{\text {e}}})\right) ^{n}$ and $r\ge 2$, there exists ${\text{C}}_{{\mathcal {I}}}^{k}>0$, such that

$$\begin{aligned}&\begin{aligned}&\parallel {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^{r}(\varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {I}}}^{k} {h}_{\text s}^{\frac{d}{r}-\frac{d}{2}} \parallel {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^{2}(\varOmega ^{{\text {e}}})}, \end{aligned} \end{aligned}$$

(194)

$$\begin{aligned}&\begin{aligned}&\parallel {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^{r}(\partial \varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {I}}}^{k} h_{\mathrm{s}}^{\frac{d-1}{r}-\frac{d-1}{2}} \parallel {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^{2}(\partial \varOmega ^{{\text {e}}})} , \end{aligned} \end{aligned}$$

(195)

$$\begin{aligned}&\begin{aligned}&\parallel \nabla {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^{2}(\varOmega ^{{\text {e}}})} \le {\text{C}}_{{\mathcal {I}}}^{k} h_{\mathrm{s}}^{{-1}} \parallel {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^{2}(\varOmega ^{{\text {e}}})} . \end{aligned} \end{aligned}$$

(196)

The proof of these properties can be found in [12, Theorem 3.2.6]. Note that Eqs. (194–195) involve the space dimension ${d}$.

Lemma 9

(Relation between energy norms on the finite element space) From [69], for ${{\mathbf{G}}_{\mathrm{h}}}\in \text{X}^{k}$, there exists a positive constant ${\text{C}}^{k}$, depending on k, such that

$$\begin{aligned} \begin{aligned} \mid \parallel {{\mathbf{G}}_{\mathrm{h}}}\parallel \mid _1\le {\text{C}}^{k}\mid \parallel {{\mathbf{G}}_{\mathrm{h}}}\parallel \mid . \end{aligned} \end{aligned}$$

(197)

The demonstration directly follows by bounding the extra terms $\sum _{{\text {e}}} {h}_{\text s}\parallel {\mathbf{G}}\parallel ^2_{\mathrm{H}^1(\partial \varOmega ^{{\text {e}}})}$ of the norm defined by Eq. (77), in comparison to the norm defined by Eq. (76), using successively the trace inequality, Eq. (192), and the inverse inequality, Eq. (196), for the first term, and the trace inequality on the finite element space, Eq. (193), for the second term.

Lemma 10

(Energy bound of interpolant error) Let ${\mathbf{G}}^{{\text {e}}}\in \text{X}_{s}, s\ge 2$, and let ${\text {I}}_{\mathrm{h}}{} {\mathbf{G}}\in \text{X}^k$, be its interpolant. Therefore, there is a constant ${\text{C}}^{k}>0$ independent of $h_{\mathrm{s}}$, such that

$$\begin{aligned} \begin{aligned} \mid \parallel {\mathbf{G}}^{{\text {e}}}-{\text {I}}_{\mathrm{h}}{} {\mathbf{G}}\parallel \mid _1\le {\text{C}}^{k}{{h}_{\text{s}}^{\mu -1}}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}, \end{aligned} \end{aligned}$$

(198)

with $\mu =\text {min}\left\{ {s,\, k+1}\right\}$. The proof follows from Lemma 5, Eq. (188), and Eq. (190), applied on the mesh dependent norm (77).

1.4 The bound of the non-linear term ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$

1.4.1 Intermediate bounds

To bound the terms of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$, we have recourse to the following intermediate bounds, which are derived for the particular case of $d=2$.

Lemma 11

(Intermediate bounds) Let $\pmb \xi ={\text {I}}_{\mathrm{h}}{} {\mathbf{G}}-{\mathbf{y}},\,\delta {\mathbf{G}}_{\mathrm{h}}\in \text{X}^{k}$, $\pmb \eta ={\mathbf{G}}^{{\text {e}}} -{\text {I}}_{\mathrm{h}}{} {\mathbf{G}}\in \text{X}$ and $\pmb \zeta =\pmb \xi + \pmb \eta$. The terms in $\pmb \xi$ can be bounded by recourse to the trace inequality, see Lemma 6, and to the inverse inequality, see Lemma 8, while bounding the terms in $\pmb \eta$ makes use of the trace inequality, see Lemma 6 and of the interpolation inequality, see Lemma 5 for $d=2$. Using the definition of the ball (91–92) thus leads to the different contributions, see [23, 26] for details,

$$\begin{aligned}&\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^2_{\mathrm{L}^2(\varOmega ^{{\text {e}}})}\right) ^{\frac{1}{2}} \le C^{k} \sigma \le C^{k} h_{\mathrm{s}}^{\mu -1-\varepsilon}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}\,, \end{aligned}$$

(199)

$$\begin{aligned}&\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}}\le C^{k} h_{\mathrm{s}}^{-\frac{1}{2}}\sigma \le C^{k}{ h_{\mathrm{s}}^{\mu -\frac{3}{2}-\varepsilon}}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}\,, \end{aligned}$$

(200)

$$\begin{aligned}&\left( \sum _{{\text {e}}}\parallel \nabla \pmb \zeta \parallel ^2_{\mathrm{L}^2(\varOmega ^{{\text {e}}})}\right) ^{\frac{1}{2}}\le C^{k}\sigma \le C^{k}{{ h_{\mathrm{s}}^{\mu -1-\varepsilon}}}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}\,, \end{aligned}$$

(201)

$$\begin{aligned}&\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}}\le C^{k}{ h_{\mathrm{s}}^{-\frac{3}{4}}}\sigma \le C^{k}{ h_{\mathrm{s}}^{\mu -\frac{7}{4}-\varepsilon}}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}\,, \end{aligned}$$

(202)

$$\begin{aligned}&\left( \sum _{{\text {e}}}\parallel \llbracket \pmb \zeta \rrbracket \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}}\le C^{k} h_{\mathrm{s}}^{\frac{1}{4}}\sigma \le C^{k} h_{\mathrm{s}}^{\mu -\frac{3}{4}-\varepsilon}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}\,, \end{aligned}$$

(203)

$$\begin{aligned}&\left( \sum _{{\text {e}}}\parallel \nabla \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}}\le C^{k}{ h_{\mathrm{s}}^{-\frac{3}{4}}}\sigma \le C^{k}{ h_{\mathrm{s}}^{\mu -\frac{7}{4}-\varepsilon}}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} \end{aligned}$$

(204)

with $\mu =\text {min} \left\{ s,\, k+1\right\}$. Moreover, using the inverse inequality, see Lemma 8, one has

$$\begin{aligned} {\left\{ \begin{array}{ll} \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\text{W}^1_4(\varOmega ^{{\text {e}}})}&{}\le C_{{\mathcal {I}}}^{k} h_{\mathrm{s}}^{-\frac{1}{2}} \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\text{H}^1(\varOmega ^{{\text {e}}})}\,,\\ \mid \delta {\mathbf{G}}_{\mathrm{h}}\mid _{\text{W}^1_4(\varOmega ^{{\text {e}}})}&{}\le C_{{\mathcal {I}}}^{k} h_{\mathrm{s}}^{-\frac{1}{2}} \mid \delta {\mathbf{G}}_{\mathrm{h}}\mid _{\text{H}^1(\varOmega ^{{\text {e}}})}\,. \end{array}\right.} \end{aligned}$$

(205)

1.4.2 Bounds of the different contributions

The bound of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$ follows from the argumentation reported in [23] and is nominated by the term with the largest bound.

The first term of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$, defined in Eq. (74), can be expanded using Eq. (61) as

$$\begin{aligned} \mathcal {I}_{1}&=\int _{ \varOmega _{\mathrm{h}}}(\nabla \delta {\mathbf{G}})_{\mathrm{h}}^{\text {T}}\bar{\mathbf{R}}_{{\mathbf{w}}} (\pmb \zeta ,\nabla \pmb \zeta ) {\text {d}}{\varOmega}\\&=\sum _{{\text {e}}} \int _{\varOmega ^{\text {e}}}(\nabla \delta {\mathbf{G}}_{\mathrm{h}})^{\text {T}}(\pmb \zeta ^{\text {T}}\bar{{\mathbf{w}}}_{{{\mathbf{G}}}{{\mathbf{G}}}} (\mathbf{y},\nabla \mathbf{y})\pmb \zeta ) {\text {d}}{\varOmega}\\&\quad+\,2\sum _{{\text {e}}} \int _{\varOmega ^{\text {e}}}(\nabla \delta {\mathbf{G}})_{\mathrm{h}}^{\text {T}}\left( \pmb \zeta ^{\text {T}}\bar{{\mathbf{w}}}_{{\mathbf{G}}\nabla {\mathbf{G}}}(\mathbf{y}) \nabla \pmb \zeta \right) {\text {d}}{\varOmega}\\&=\mathcal {I}_{11}+2\mathcal {I}_{12}. \end{aligned}$$

(206)

The two term of the right hand side of Eq. (206) are bounded by using the generalized Hölder’s inequality, the generalized Cauchy–Schwartz’ inequality, the definition of $C_{{\text {y}}}$ in Eq. (67), and the bounds (199, 200, 201, and 202) as

$$\begin{aligned}\mid \mathcal {I}_{11}\mid&\le C_{{\text {y}}}\sum _{{\text {e}}}\parallel \pmb \zeta \parallel _{\text{L}^4(\varOmega ^{\text {e}})}\parallel \pmb \zeta \parallel _{\mathrm{L}^2(\varOmega ^{\text {e}})}\parallel \nabla \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^4(\varOmega ^{\text {e}})}\\&\le C_{{\text {y}}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\text{L}^4(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{4}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^2_{\mathrm{L}^2(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{2}}\left( \sum _{{\text {e}}} \parallel \nabla \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^4_{\mathrm{L}^4(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{4}}\\&\le C^{k} C_{{\text {y}}} h_{\mathrm{s}}^{{\mu}-2-\varepsilon}\sigma \mid \delta {\mathbf{G}}_{\mathrm{h}}\mid _{\text{H}^1(\varOmega _{\mathrm{h}})} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}, \end{aligned}$$

(207)

$$\begin{aligned} \mid \mathcal {I}_{12}\mid&\le C_{{\text {y}}}\sum _{{\text {e}}}\parallel \pmb \zeta \parallel _{\text{L}^4(\varOmega ^{\text {e}})}\parallel \nabla \pmb \zeta \parallel _{\mathrm{L}^2(\varOmega ^{\text {e}})}\parallel \nabla \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\mathrm{L}^4(\varOmega ^{{\text {e}}})}\\&\le C_{{\text {y}}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\text{L}^4(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{4}} \left( \sum _{{\text {e}}}\parallel \nabla \pmb \zeta \parallel ^2_{\mathrm{L}^2(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{2}}\left( \sum _{{\text {e}}} \parallel \nabla \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^4_{\mathrm{L}^4(\varOmega ^{{\text {e}}})}\right) ^{ \frac{1}{4}}\\&\le C^{k} C_{{\text {y}}} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \mid \delta {\mathbf{G}}_{\mathrm{h}}\mid _{\text{H}^1(\varOmega _{\mathrm{h}})}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}. \end{aligned}$$

(208)

Combining the above results leads to

$$\begin{aligned} \mid \mathcal {I}_{1}\mid \le C^{k} C_{{\text {y}}} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \mid \delta {\mathbf{G}}_{\mathrm{h}}\mid _{\text{H}^1(\varOmega _{\mathrm{h}})}\parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}. \end{aligned}$$

(209)

The second term of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$, defined in Eq. (74), becomes by using Eq. (61),

$$\begin{aligned} \begin{aligned} \mathcal {I}_{2}=&\int _{\partial _{{\text {I}}}\varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}}\varOmega _{\mathrm{h}}} \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}^{\text {T}} \rrbracket \left\langle \pmb \zeta ^{\text {T}}\bar{{\mathbf{w}}}_{{{\mathbf{G}}}{{\mathbf{G}}}} (\mathbf{y},\nabla \mathbf{y})\pmb \zeta \right\rangle {\text {dS}}\\&+2\int _{\partial _{{\text {I}}}\varOmega _{\mathrm{h}}\cup \partial _{\mathrm{D}}\varOmega _{\mathrm{h}}} \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}^{\text {T}} \rrbracket \left\langle \pmb \zeta ^{\text {T}}\bar{{\mathbf{w}}}_{{\mathbf{G}} \nabla {\mathbf{G}}}(\mathbf{y}) \nabla \pmb \zeta \right\rangle {\text {dS}} = \mathcal {I}_{21}+2\mathcal {I}_{22}\,. \end{aligned} \end{aligned}$$

(210)

The two terms of the right hand side of Eq. (210) are bounded by using the generalized Hölder's inequality, the generalized Cauchy–Schwartz’ inequality, the definition of $C_{{\text {y}}}$ in Eq. (67), and the bounds (202, 204), yielding

$$\begin{aligned}&\begin{aligned} \mid \mathcal {I}_{21}\mid&\le C_{{\text {y}}} h_{\mathrm{s}}^{\frac{1}{2}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})} \right) ^\frac{1}{2}\left( \sum _{{\text {e}}} h_{\mathrm{s}}^{-1}\parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}^{\text {T}} \rrbracket \parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{2}}\\&\le C^{k} C_{{\text {y}}} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{{-1}}\parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket \parallel _{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}^2 \right) ^{\frac{1}{2}}\,, \end{aligned} \end{aligned}$$

(211)

$$\begin{aligned}&\begin{aligned} \mid \mathcal {I}_{22}\mid&\le C_{{\text {y}}} h_{\mathrm{s}}^{\frac{1}{2}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})} \right) ^\frac{1}{4}\left( \sum _{{\text {e}}}\parallel \nabla \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})} \right) ^\frac{1}{4} \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{-1}\parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}^{\text {T}} \rrbracket \parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{2}}\\&\le C^{k} C_{{\text {y}}} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{{-1}} \parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket \parallel _{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}^2 \right) ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

(212)

We now substitute Eqs. (211, 212) in Eq. (210) to obtain the final bound of the second term of $\mathcal {N}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$ as

$$\begin{aligned} \mid \mathcal {I}_{2}\mid \le C^{k} C_{{\text {y}}} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{{-1}}\parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket \parallel _{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}^2 \right) ^{\frac{1}{2}}\,. \end{aligned}$$

(213)

Furthermore, for the third term of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$ as decomposed in Eq. (74), using Taylor’s expansion as in Eq. (60–60), the generalized Hölder’s inequality, the generalized Cauchy–Schwartz’ inequality, the definition of $C_{{\text {y}}}$ in Eq. (67), and the bounds (202, 203), leads to

$$\begin{aligned} \begin{aligned} \mid \mathcal {I}_{3}\mid&\le \sum _{{\text {e}}}\mid \int _{\partial_{\text{I}} \varOmega ^{{\text {e}}}}\llbracket \pmb \zeta _{\mathbf{n}}^{\text {T}} \rrbracket \left( \pmb \zeta ^{\text {T}}\bar{{\mathbf{w}}}_{\nabla {{\mathbf{G}}}{} {\mathbf{G}}}(\mathbf{y})\nabla \delta {\mathbf{G}}_{\mathrm{h}}\right) {\text {dS}} \mid \\&\le C_{{\text {y}}} h_{\mathrm{s}}^{-\frac{1}{2}}\left( \sum _{{\text {e}}}\parallel \llbracket \pmb \zeta \rrbracket \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}} \left( \sum _{{\text {e}}} h_{\mathrm{s}} \parallel \nabla \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\\&\le C_{{\text {y}}} C^{k} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left( \sum _{{\text {e}}} h_{\mathrm{s}} \mid \delta {\mathbf{G}}_{\mathrm{h}}\mid ^2_{\mathrm{H}^1(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\,. \end{aligned} \end{aligned}$$

(214)

Likewise, the fourth term of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$ defined in Eq. (74) is bounded using a Taylor’s expansion as in Eqs. (60–60), the generalized Hölder's inequality, the generalized Cauchy–Schwartz’ inequality, the definition of $C_{{\text {y}}}$ in Eq. (67), and the bounds (202, 203) leading to

$$\begin{aligned} \mid \mathcal {I}_{4}\mid&\le \sum _{{\text {e}}}\mid \int _{\partial_{\text{I}} \varOmega ^{{\text {e}}}}\llbracket \pmb \zeta _{\mathbf{n}}^{\text {T}} \rrbracket \left( \frac{{\mathcal {B}}}{{h}_{\mathrm{s}}}\pmb \zeta ^{\text {T}}\bar{{\mathbf{w}}}_{\nabla {{\mathbf{G}}}{} {\mathbf{G}}}(\mathbf{y})\right) \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket {\text {dS}}\mid \\&\le C_{{\text {y}}} h_{\mathrm{s}}^{-\frac{1}{2}} \left( \sum _{{\text {e}}}\parallel \llbracket \pmb \zeta \rrbracket \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})} \right) ^\frac{1}{4}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})} \right) ^\frac{1}{4} \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{-1}\parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket \parallel _{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}^2 \right) ^{\frac{1}{2}} \\&\le C^{k} C_{{\text {y}}} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{{-1}}\parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket \parallel _{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}^2 \right) ^{\frac{1}{2}}\,.\end{aligned}$$

(215)

Then the bound of the fifth term of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$ defined in Eq. (74) is derived using Eq. (231) developed in Appendix section “Declaration related to the fifth term of ${\mathcal{N}}({{\mathbf{G}}}^{{\text{e}}} ,{\mathbf{y}};\delta {{\mathbf{G}}}_{{\text{h}}} )$ ”, following

$$\begin{aligned} \begin{aligned} \mid \mathcal {I}_{5}\mid \le&2 C_{{\text {y}}} \sum _{{\text {e}}}\mid \int _{\partial _{{\text {I}}}\varOmega ^{{\text {e}}}\cup \partial _{\mathrm{D}}\varOmega ^{{\text {e}}}} \llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \rrbracket \,\mathbf{I}\,( {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}}) \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} {\text {dS}}\mid \\&+\frac{1}{8} C_{{\text {y}}}\sum _{{\text {e}}}\mid \int _{\partial _{{\text {I}}}\varOmega ^{{\text {e}}}} \llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \rrbracket \,\mathbf{I}\, \llbracket {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}}\rrbracket \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\rrbracket {\text {dS}}\mid = \mid \mathcal {I}_{51}\mid +\mid \mathcal {I}_{52}\mid \,, \end{aligned} \end{aligned}$$

(216)

where $\mathbf{I}$ is a matrix of unit norm and has the same size of $\bar{{\mathbf{o}}}_{{\mathbf{G}}}^{\text {T}}$. Using the generalized Hölder's inequality, the generalized Cauchy–Schwartz’ inequality, and the bounds (202, 203) one has

$$\begin{aligned}&\begin{aligned} \mid \mathcal {I}_{51}\mid&\le 2 C_{{\text {y}}}\sum _{{\text {e}}}\mid \int _{\partial \varOmega ^{{\text {e}}}}\llbracket \pmb \zeta ^{\text {T}} \rrbracket \mathbf{I}\left( \pmb \zeta \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right) {\text {dS}} \mid \\&\le 2 C_{{\text {y}}} h_{\mathrm{s}}^{-\frac{1}{2}}\left( \sum _{{\text {e}}}\parallel \llbracket \pmb \zeta \rrbracket \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}} \left( \sum _{{\text {e}}}\text {h}_{\mathrm{s}} \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\\&\le 2 C_{{\text {y}}} C^{k} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left( \sum _{{\text {e}}}\text {h}_{\mathrm{s}} \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\,, \end{aligned} \end{aligned}$$

(217)

$$\begin{aligned} \mid \mathcal {I}_{52}\mid&\le \frac{1}{8} C_{{\text {y}}}\sum _{{\text {e}}}\mid \int _{\partial \varOmega ^{{\text {e}}}}\llbracket \pmb \zeta ^{\text {T}} \rrbracket \mathbf{I}\llbracket \pmb \zeta \rrbracket \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket {\text {dS}} \mid \\&\le \frac{1}{8} C_{{\text {y}}} h_{\mathrm{s}}^{\frac{1}{2}}\left( \sum _{{\text {e}}}\parallel \llbracket \pmb \zeta \rrbracket \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}}\left( \sum _{{\text {e}}}\parallel \llbracket \pmb \zeta \rrbracket \parallel ^4_{\mathrm{L}^4(\partial \varOmega ^{{\text {e}}})}\right) ^{\frac{1}{4}} \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{-1} \parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}}\rrbracket \parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\\&\le \frac{1}{8} C_{{\text {y}}} C^{k} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -\varepsilon}\sigma \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{-1} \parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}}\rrbracket \parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\,. \end{aligned}$$

(218)

Combining Eqs. (217 and 218) leads to the final bound

$$\begin{aligned} \begin{aligned} \mid \mathcal {I}_{5}\mid&\le 2 C_{{\text {y}}} C^{k} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left( \sum _{{\text {e}}}\text {h}_{\mathrm{s}} \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\\&\quad+\frac{1}{8} C_{{\text {y}}} C^{k} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -\varepsilon}\sigma \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{-1} \parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}}\rrbracket \parallel ^2_{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}\,. \end{aligned} \end{aligned}$$

(219)

Finally to bound the last term of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$ defined in Eq. (74), we rewrite it using Eq. (63) as

$$\begin{aligned} \mathcal {I}_{6}&=\int _{ \varOmega _{\mathrm{h}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}}\bar{\mathbf{R}}_{{\mathbf{d}}} (\pmb \zeta ,\nabla \pmb \zeta ) {\text {d}}{\varOmega}\\&=\sum _{{\text {e}}} \int _{\varOmega ^{\text {e}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}}(\pmb \zeta ^{\text {T}}\bar{{\mathbf{d}}}_{{{\mathbf{G}}}{{\mathbf{G}}}} (\mathbf{y},\nabla \mathbf{y})\pmb \zeta ) {\text {d}}{\varOmega} +\,2\sum _{{\text {e}}} \int _{\varOmega ^{\text {e}}} \delta {\mathbf{G}}_{\mathrm{h}}^{\text {T}}(\pmb \zeta ^{\text {T}}\bar{{\mathbf{d}}}_{{{\mathbf{G}}}\nabla {{\mathbf{G}}}} (\mathbf{y})\nabla \pmb \zeta ) {\text {d}}{\varOmega}\\&=\mathcal {I}_{61}+2\mathcal {I}_{62}.\end{aligned}$$

(220)

The two contributions are bounded using the generalized Hölder’s inequality, the generalized Cauchy–Schwartz’ inequality, the definition of $C_{{\text {y}}}$ in Eq. (67), the bounds (200, 201), and the inverse inequality of Lemma 8, following

$$\begin{aligned} \mid \mathcal {I}_{61}\mid&\le C_{{\text {y}}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{4}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\mathrm{L}^4(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{4}}\left( \sum _{{\text {e}}} \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{L}^2(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{2}}\\&\le C^{k} C_{{\text {y}}} h_{\mathrm{s}}^{{\mu}-2-\varepsilon}\sigma \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\text{L}^2(\varOmega _{\mathrm{h}})} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}. \end{aligned}$$

(221)

$$\begin{aligned} \mid \mathcal {I}_{62}\mid&\le C_{{\text {y}}}\left( \sum _{{\text {e}}}\parallel \pmb \zeta \parallel ^4_{\text{L}^4(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{4}}\left( \sum _{{\text {e}}}\parallel \nabla \pmb \zeta \parallel ^2_{\mathrm{L}^2(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{2}}\left( \sum _{{\text {e}}} \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^4_{\mathrm{L}^4(\varOmega ^{\text {e}})}\right) ^{ \frac{1}{4}}\\&\le C^{k} C_{{\text {y}}} h_{\mathrm{s}}^{{\mu}-2-\varepsilon}\sigma \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\text{L}^2(\varOmega _{\mathrm{h}})} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}. \end{aligned}$$

(222)

Substituting Eqs. (221, 222) in Eq. (220) leads to

$$\begin{aligned} \begin{aligned} \mid \mathcal {I}_{6}\mid&\le C^{k} C_{{\text {y}}} h_{\mathrm{s}}^{{\mu}-2-\varepsilon}\sigma \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\text{L}^2(\varOmega _{\mathrm{h}})} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})}. \end{aligned} \end{aligned}$$

(223)

Combining Eqs. (209, 213, 214, 215, 216, and 223), yields

$$\begin{aligned} \begin{aligned}&\mid \mathcal {N}({\mathbf{G}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})\mid \le {\text{C}}^{k}\text{C}_{{\text {y}}} \parallel {\mathbf{G}}^{{\text {e}}}\parallel _{\mathrm{H}^{s}(\varOmega _{\mathrm{h}})} h_{\mathrm{s}}^{\mu -2-\varepsilon}\sigma \left[ \parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel _{\text{H}^1(\varOmega _{\mathrm{h}})}\right. \\&\quad +\left. \left( \sum _{{\text {e}}} h_{\mathrm{s}}\parallel \delta {\mathbf{G}}_{\mathrm{h}}\parallel ^2_{\mathrm{H}^1(\partial \varOmega ^{{\text {e}}})} \right) ^{\frac{1}{2}}+ \left( \sum _{{\text {e}}} h_{\mathrm{s}}^{{-1}}\parallel \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} \rrbracket \parallel _{\mathrm{L}^2(\partial \varOmega ^{{\text {e}}})}^2 \right) ^{\frac{1}{2}}\right] \,. \end{aligned} \end{aligned}$$

(224)

Finally, using the definition of the energy norm (77), this results yields the bound (95) of ${\mathcal {N}}({{\mathbf{G}}}^{{\text {e}}},\mathbf{y};\delta {\mathbf{G}}_{\mathrm{h}})$.

1.4.3 Declaration related to the fifth term of ${\mathcal{N}}({{\mathbf{G}}}^{{\text{e}}} ,{\mathbf{y}};\delta {{\mathbf{G}}}_{{\text{h}}} )$

Using the identities $\left[\kern-0.15em\left[ {{\text{ab}}} \right]\kern-0.15em\right] = \left[\kern-0.15em\left[ {\text{a}} \right]\kern-0.15em\right]\left\langle {\text{b}} \right\rangle + \left\langle {\text{a}} \right\rangle \left[\kern-0.15em\left[ {\text{b}} \right]\kern-0.15em\right]$ and $\left\langle {\text{a}} \right\rangle \left\langle {\text{b}} \right\rangle = \left\langle {{\text{ab}}} \right\rangle - \frac{1}{4}\left[\kern-0.15em\left[ {\text{a}} \right]\kern-0.15em\right]\left[\kern-0.15em\left[ {\text{b}} \right]\kern-0.15em\right]$ on ${\partial _{{\text {I}}}\varOmega _{\mathrm{h}}}$, the term

$\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}}) ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle$ can be rewritten with an abuse of notations on the product operator as

$$\begin{aligned} \begin{aligned}&\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}}) ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle = \left\langle ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}})\delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle \llbracket {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}} \rrbracket \\&\quad -\frac{1}{4}\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}})\rrbracket \llbracket {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}}\rrbracket \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\rrbracket + \llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}})\rrbracket \left\langle {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}} \right\rangle \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle \,. \end{aligned} \end{aligned}$$

(225)

Now, we need to solve explicitly the term $\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}})\rrbracket$, where $\bar{{\mathbf{o}}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}})$ corresponds to $-\bar{\mathbf{p}}_{{\mathbf{G}}{} {\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}})$ defined in Eq. (181) with ${\mathbf{p}}_{{\mathbf{G}}{} {\mathbf{G}}}(\mathbf{V}^{{\text {t}}})=-{\mathbf{o}}_{{\mathbf{G}}}(\mathbf{V}^{{\text {t}}})$, yielding

$$\begin{aligned} &\bar{\mathbf{o}}_{\mathbf{{G}}}({\mathbf{G}}_{\mathrm{h}})=\int _0^1(1-{\text {t}}){\mathbf{o}}_{\mathbf{{G}}}(\mathbf{V}^{{\text {t}}}) {\text{dt}}, \end{aligned}$$

(226)

with $\mathbf{V}^{{\text {t}}}={\mathbf{G}}^{{\text {e}}}+{\text {t}}({\mathbf{G}}_{\mathrm{h}}-{\mathbf{G}}^{{\text {e}}})$. As ${\mathbf{o}}_{\mathbf{{G}}}$ only involves terms in $\frac{2}{{\text {f}}_{\mathrm{T}}^3}$, we compute $\bar{\alpha}$ the nonzero component.

$$\begin{aligned} \begin{aligned} \bar{\alpha}=3\text{K} \alpha _{\mathrm{th}}\int _0^1(1-{\text {t}})(\frac{2}{[{\text {f}}_{\mathrm{T}}^{\text {e}}+{\text {t}}({\text {f}}_{\mathrm{T}}-{\text {f}}_{\mathrm{T}}^{\text {e}})]^3} )\text {dt}. \end{aligned} \end{aligned}$$

(227)

For simplicity, let us define $\lambda$ as

$$\begin{aligned} \begin{aligned} \lambda =\int _0^1(1-{\text {t}})\frac{2}{[{\text {f}}_{\mathrm{T}}^{\text {e}}+{\text {t}}({\text {f}}_{\mathrm{T}}-{\text {f}}_{\mathrm{T}}^{\text {e}})]^3} \text {dt}=\frac{1}{{\text {f}}_{\mathrm{T}}{\text {f}}_{\mathrm{T}}^{{\text {e}}^2}}\,. \end{aligned} \end{aligned}$$

(228)

It can be noticed that to evaluate $\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}})\rrbracket$, we need $\lambda ({\text {f}}_{\mathrm{T}}^{\text {e}}-{\text {f}}_{\mathrm{T}})$ which reads

$$\begin{aligned} \begin{aligned} \lambda ({\text {f}}_{\mathrm{T}}^{\text {e}}-{\text {f}}_{\mathrm{T}})&=\frac{1}{{\text {f}}_{\mathrm{T}}{\text {f}}_{\mathrm{T}}^{{\text {e}}^2}}\left( {\text {f}}_{\mathrm{T}}^{\text {e}}-{\text {f}}_{\mathrm{T}}\right) =\frac{1}{{\text {f}}_{\mathrm{T}}^{{\text {e}}}}\left( \frac{1}{{\text {f}}_{\mathrm{T}}}-\frac{1}{{\text {f}}_{\mathrm{T}}^{{\text {e}}}}\right) , \end{aligned} \end{aligned}$$

(229)

and the jump of the last result is

$$\begin{aligned} \begin{aligned} \llbracket \lambda ({\text {f}}_{\mathrm{T}}^{\text {e}}-{\text {f}}_{\mathrm{T}})\rrbracket&=\left\{ \begin{array}{ll}\frac{1}{{\text {f}}_{\mathrm{T}}^{{\text {e}}}}(\frac{1}{{\text {f}}_{\mathrm{T}}^+}-\frac{1}{{\text {f}}_{\mathrm{T}}^{{\text {e}}}}-\frac{1}{{\text {f}}_{\mathrm{T}}^-}+\frac{1}{{\text {f}}_{\mathrm{T}}^{{\text {e}}}})=-\frac{1}{{\text {f}}_{\mathrm{T}}^{{\text {e}}}{{\text {f}}_{\mathrm{T}}^+}{{\text {f}}_{\mathrm{T}}^-}} \llbracket {{\text {f}}_{\mathrm{T}}}-{\text {f}}_{\mathrm{T}}^{\text {e}}\rrbracket \quad \text {on}\,{\partial _{{\text {I}}}\varOmega _{\mathrm{h}}}\\ \frac{1}{{\text {f}}_{\mathrm{T}}{\text {f}}_{\mathrm{T}}^{{\text {e}}^2}}({\text {f}}_{\mathrm{T}}-{\text {f}}_{\mathrm{T}}^{\text {e}})=-\frac{1}{{\text {f}}_{\mathrm{T}}{\text {f}}_{\mathrm{T}}^{{\text {e}}^2}}\llbracket {{\text {f}}_{\mathrm{T}}-{\text {f}}_{\mathrm{T}}^{\text {e}}}\rrbracket \quad \text {on}\,{\partial _{\mathrm{D}}\varOmega _{\mathrm{h}}}. \end{array} \right. \end{aligned} \end{aligned}$$

(230)

Hence considering this equation in the matrix form, then substituting it in Eq. (225), and using the definition of $C_{{\text {y}}}$ in Eq. (67), lead to

$$\begin{aligned} \begin{aligned}&\mid \sum _{{\text {e}}}\int _{\partial _{{\text {I}}}\varOmega ^{{\text {e}}}\cup \partial _{\mathrm{D}}\varOmega ^{{\text {e}}}}\llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \bar{{\mathbf{o}}}^{\text {T}}_{{\mathbf{G}}}({{\mathbf{G}}}_{\mathrm{h}}) ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}}) \rrbracket \left\langle \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\right\rangle {\text {dS}}\mid \\&\quad \le C_{{\text {y}}} \sum _{{\text {e}}}\mid \int _{\partial _{{\text {I}}}\varOmega ^{{\text {e}}}}({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \mathbf{I} \llbracket {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}} \rrbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} {\text {dS}}\mid \\&\quad \quad+\frac{1}{8} C_{{\text {y}}}\sum _{{\text {e}}}\mid \int _{\partial _{{\text {I}}}\varOmega ^{{\text {e}}}} \llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \rrbracket \mathbf{I} \llbracket {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}}\rrbracket \llbracket \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}}\rrbracket {\text {dS}}\mid \\&\quad \quad+ C_{{\text {y}}}\sum _{{\text {e}}}\mid \int _{\partial _{{\text {I}}}\varOmega ^{{\text {e}}}} \llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \rrbracket \mathbf{I}( {\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}}) \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} {\text {dS}}\mid \\&\quad \quad+ C_{{\text {y}}}\sum _{{\text {e}}}\mid \int _{\partial _{\mathrm{D}}\varOmega ^{{\text {e}}}} \llbracket ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{\mathrm{h}})^{\text {T}} \rrbracket \mathbf{I} ({\mathbf{G}}^{{\text {e}}}-{\mathbf{G}}_{{\text {h}}}) \delta {\mathbf{G}}_{\mathrm{h}_{\mathbf{n}}} {\text {dS}}\mid \,, \end{aligned} \end{aligned}$$

(231)

where ${\mathbf{I}}$ is a matrix of unit norm and has the same size of $\overline{{\mathbf{o}}} _{{{\mathbf{G}}}}^{{\text{T}}}$.