This section is organized as follows. The generalized standard constitutive equations are presented first. They are based on an incremental variational principle proposed in [33]. Then, we introduce the hyperreduced incremental problem to be solved. The following section is devoted to the construction of the FEequilibrated stress within the framework of the hyperreduced modeling. The next sections introduce the error estimator, the tailored norm related to the CRE and the partition of the CRE. We finish by remarks on the numerical implementation of the proposed approach.
Incremental potential for the constitutive equations
The constitutive laws are described by using an incremental potential in the framework of the irreversible thermodynamic processes. A priori error estimators and incremental variational formulations were introduced in [33] for mechanical problems of bodies undergoing large dynamic deformations. Extensions of this approach were proposed in [20,21] for effective response predictions of heterogeneous materials. The strain history is taken into account by using internal variables denoted by α. These variables are the lump sum of the history of material changes. This approach has its roots in the works by Biot [34], Ziegler [35], Germain [36] or Halpen and Nguyen [37] and has proven its ability to cover a broad spectrum of models in viscoelasticity, viscoplasticity, plasticity and also continuum damage mechanics. The FE solution is approximated by an hyperreduced predictions denoted by \(\textbf {u}_{\textit {HR}}^{n}\), \(\boldsymbol {\alpha }_{\textit {HR}}^{n}\), \(\boldsymbol {\sigma }_{\textit {HR}}^{n}\) respectively for the displacements, the internal variables and the Cauchy stress, at discrete time t
^{n}. For the sake of simplicity, we denote by \(\boldsymbol {\varepsilon }_{\textit {HR}}^{n}\) the strain tensor \(\boldsymbol {\varepsilon }(\textbf {u}_{\textit {HR}}^{n})\), in the framework of the infinitesimal strain theory.
According to the framework of the incremental variational formulations, the constitutive law is defined by a condensed incremental potential, denoted by \(w_{\Delta }(\boldsymbol {\varepsilon }_{\textit {HR}}^{n})\), such that:
$$ \boldsymbol{\sigma}^{n}_{HR} = \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}_{HR}^{n}), \quad n=1,\ldots,N_{t} \: $$
((10))
In practice, we do not implement the computation of the partial derivative of w
_{
Δ
} with respect to the component of the strain tensor. Nevertheless, Equation (10) is fulfilled up to the computer precision. The internal variables are solutions of the following equation [21]:
$$\begin{array}{*{20}l} \boldsymbol{\alpha}^{n}_{HR} = \text{arg} \: \text{Inf}_{\boldsymbol{\alpha}^{\star}} J(\boldsymbol{\varepsilon}^{n}_{HR},\boldsymbol{\alpha}^{\star}), \end{array} $$
((11))
$$\begin{array}{*{20}l} J(\boldsymbol{\varepsilon},\boldsymbol{\alpha}) = w (\boldsymbol{\varepsilon},\boldsymbol{\alpha}) + \left(t^{n}  t^{n1}\right) \: \varphi \left(\frac{\boldsymbol{\alpha}\boldsymbol{\alpha}^{n1}_{HR}(\textbf{x})}{t^{n}t^{n1}} \right), \end{array} $$
((12))
$$\begin{array}{*{20}l} w_{\Delta}(\boldsymbol{\varepsilon}) = \text{Inf}_{\boldsymbol{\alpha}^{\star}} J(\boldsymbol{\varepsilon},\boldsymbol{\alpha}^{\star}), \end{array} $$
((13))
where w(ε,α) is the free energy of the material, and \(\varphi (\dot {\boldsymbol {\alpha }})\) is its dissipation potential [36]. The two potentials w and φ are convex functions of their arguments (ε,α) and \(\dot {\boldsymbol {\alpha }}\) respectively, according to the theory of generalized standard materials [36,37]. Examples of constitutive laws can be found in [38]. A detailed example is given in the last section of this paper.
The convexity of w
_{
Δ
} is proved in [21] under the assumption that w and φ are convex functions. In the sequel, the explicit knowledge of the condensed incremental potential is not required for the mathematical formulation of the error estimator.
By recourse to the Legendre transformation, the dual potential of w
_{
Δ
}, denoted by \(w^{\star }_{\Delta }\), reads:
$$ w^{\star}_{\Delta}\left(\widetilde{\boldsymbol{\sigma}}\right) = \text{Sup}_{\boldsymbol{\varepsilon}^{\star}} \left(\boldsymbol{\varepsilon}^{\star} : \widetilde{\boldsymbol{\sigma}}  w_{\Delta}\left(\boldsymbol{\varepsilon}^{\star}\right)\right) $$
((14))
We restrict our attention to convex functions w
_{
Δ
}, hence:
$$ w_{\Delta}(\widetilde{\boldsymbol{\varepsilon}}) = \text{Sup}_{\boldsymbol{\sigma}^{\star}} \left(\widetilde{\boldsymbol{\varepsilon}} : \boldsymbol{\sigma}^{\star}  w^{\star}_{\Delta}(\boldsymbol{\sigma}^{\star})\right) $$
((15))
Therefore:
$$ w_{\Delta}(\widetilde{\boldsymbol{\varepsilon}}) + w^{\star}_{\Delta}(\widetilde{\boldsymbol{\sigma}})  \widetilde{\boldsymbol{\varepsilon}} : \widetilde{\boldsymbol{\sigma}} \ge 0 \quad \forall \: \widetilde{\boldsymbol{\varepsilon}}, \: \widetilde{\boldsymbol{\sigma}} $$
((16))
The definition of the partial derivatives \(\frac {\partial w_{\Delta }}{\partial \boldsymbol {\varepsilon }}\) and \(\frac {\partial w^{\star }_{\Delta }}{\partial \boldsymbol {\sigma }}\) is extended to fulfill the following equations:
$$\begin{array}{*{20}l} \widetilde{\boldsymbol{\sigma}}_{\varepsilon} = \text{arg} \: \text{Sup}_{\boldsymbol{\sigma}^{\star}} \left(\widetilde{\boldsymbol{\varepsilon}} : \boldsymbol{\sigma}^{\star}  w^{\star}_{\Delta}(\boldsymbol{\sigma}^{\star})\right)\Leftrightarrow \: \widetilde{\boldsymbol{\varepsilon}} = \frac{\partial w^{\star}_{\Delta}}{\partial \boldsymbol{\sigma}}(\widetilde{\boldsymbol{\sigma}}_{\varepsilon}) \end{array} $$
((17))
$$\begin{array}{*{20}l} \widetilde{\boldsymbol{\varepsilon}}_{\sigma} = \text{arg} \: \text{Sup}_{\boldsymbol{\varepsilon}^{\star}} \left(\boldsymbol{\varepsilon}^{\star} : \widetilde{\boldsymbol{\sigma}}  w_{\Delta}(\boldsymbol{\varepsilon}^{\star})\right)\Leftrightarrow \: \widetilde{\boldsymbol{\sigma}} = \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\widetilde{\boldsymbol{\varepsilon}}_{\sigma}) \end{array} $$
((18))
Hence, by construction, \(\boldsymbol {\sigma }^{n}_{\textit {HR}}\) and \(\boldsymbol {\varepsilon }_{\textit {HR}}^{n}\) fulfill the following equation:
$$ w_{\Delta}(\boldsymbol{\varepsilon}_{HR}^{n}) + w^{\star}_{\Delta}(\boldsymbol{\sigma}^{n}_{HR})  \boldsymbol{\varepsilon}_{HR}^{n} : \boldsymbol{\sigma}^{n}_{HR} = 0 $$
((19))
Hyperreduced setting
The continuous medium is occupying a domain Ω. The boundary ∂
Ω of Ω is denoted by ∂
_{
U
}
Ω∪∂
_{
F
}
Ω. On ∂
_{
U
}
Ω, there is the Dirichlet condition u=u
_{
c
}. On ∂
_{
F
}
Ω, there is a given force field F
^{n}. The FE displacement field belongs to a function space \(\textbf {u}_{c} +\mathcal {V}_{h}\). The reduced subspace is denoted by \(\mathcal {V}_{\textit {ROM}}\):
$$ \mathcal{V}_{ROM} = \text{span}(\boldsymbol{\psi}_{1}, \ldots, \boldsymbol{\psi}_{N_{\psi}}) \subset \mathcal{V}_{h} $$
((20))
We denote by V the matrix form of empirical modes defined on the FE basis:
$$ \boldsymbol{\psi}_{k}(\textbf{x}) = \sum_{i=1}^{N_{\xi}} \boldsymbol{\xi}_{i}(\textbf{x}) \: V_{ik}, \quad k=1, \ldots,N_{\psi} $$
((21))
By following the hyperreduction method [39], we generate the RID Ω
_{
Z
} by using the empirical modes \(({\boldsymbol \psi }_{k})_{k=1}^{N_{\psi }}\) and the strain modes \((\boldsymbol {\varepsilon }(\boldsymbol {\psi }_{k}))_{k=1}^{N_{\psi }}\). When the RID is known, we determine the set of available FE residual entries when the stress estimation is restricted to Ω
_{
Z
}. This set is denoted by such that:
$$ \mathcal{Z} = \{ i \in \{1, \ldots,N_{\xi} \}, \: \int_{\Omega \backslash \Omega_{Z}} {\boldsymbol{\xi}_{i}^{2}} \: d\Omega = 0\} $$
((22))
Therefore, we can define truncated test functions insuring a weak form of the balance of momentum restricted to the domain Ω
_{
Z
}, since these test functions are set to zero over Ω∖Ω
^{Z}. These truncated test functions are denoted by \(({\boldsymbol {\psi }_{k}^{Z}})_{k=1}^{N_{\psi }}\) such that:
$$ {\boldsymbol{\psi}_{k}^{Z}}(\textbf{x}) = \sum_{i \in \mathcal{Z}} \boldsymbol{\xi}_{i}(\textbf{x}) \: V_{ik}, \quad k=1, \ldots,N_{\psi} $$
((23))
The statement of the hyperreduced incremental problem is the following. Given a parameter vector μ, find an estimation of the reduced coordinates γ
^{n} such that \(\textbf {u}_{\textit {HR}}^{n} \in \textbf {u}_{c} + \mathcal {V}_{\textit {ROM}}\) fulfills the constitutive equations and the principle of virtual work:
$$\begin{array}{*{20}l} \textbf{u}^{n}_{HR}(\textbf{x}) = \textbf{u}_{c} + \sum_{k=1}^{N_{\psi}} \boldsymbol{\psi}_{k}(\textbf{x}) \: {\gamma^{n}_{k}} \quad \textbf{x} \in \Omega \end{array} $$
((24))
$$\begin{array}{*{20}l} \int_{\Omega_{Z}} \boldsymbol{\varepsilon}({\boldsymbol{\psi}_{k}^{Z}}) \: : \: \boldsymbol{\sigma}^{n}_{HRZ} \: d\Omega = \int_{\partial_{F} \Omega \cap \partial \Omega_{Z}} {\boldsymbol{\psi}_{k}^{Z}} \:.\: \textbf{F}^{n} \: d\Gamma \quad \forall \: k \in \{ 1,\ldots,N_{\psi} \} \end{array} $$
((25))
$$\begin{array}{*{20}l} \boldsymbol{\alpha}^{n}_{HR} = \text{arg} \: \text{Inf}_{\boldsymbol{\alpha}^{\star}} J(\boldsymbol{\varepsilon}(\textbf{u}^{n}_{HR}),\boldsymbol{\alpha}^{\star}) \quad \forall \textbf{x} \in \Omega_{Z} \end{array} $$
((26))
$$\begin{array}{*{20}l} \boldsymbol{\sigma}^{n}_{HRZ} = \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}(\textbf{u}^{n}_{HR})) \quad \forall \textbf{x} \in \Omega_{Z} \end{array} $$
((27))
where : is the contracted product for secondorder tensors. We must emphasize the fact that Equation (24) gives access to the displacement in all the domain Ω although the equilibrium is set only on Ω
_{
Z
}. Hence, when the hyperreduced solution is known, the stress \(\boldsymbol {\sigma }^{n}_{\textit {HR}}\) can be estimated on the full domain Ω by using Equation (10).
The FE equations are obtained by substituting in equations (24) to (26), the following items:

the shape functions \((\boldsymbol {\xi }_{i})_{i=1}^{N_{\xi }}\) for \((\boldsymbol {\psi }_{k})_{k=1}^{N_{\psi }}\),

the test functions \((\boldsymbol {\xi }_{i})_{i=1}^{N_{\xi }}\) for \(({\boldsymbol {\psi }_{k}^{Z}})_{k=1}^{N_{\psi }}\),

the full domain Ω for Ω
_{
Z
},

the variables \(((q_{i})_{i=1}^{N_{\xi }},\textbf {u}^{n}_{\textit {FE}}, \boldsymbol {\sigma }^{n}_{\textit {FE}}, \boldsymbol {\alpha }^{n}_{\textit {FE}})\) for \(\left ((\gamma _{k})_{k=1}^{N_{\psi }},\textbf {u}^{n}_{\textit {HR}}, \boldsymbol {\sigma }^{n}_{\textit {HR}}, \boldsymbol {\alpha }^{n}_{\textit {HR}}\right)\).
However, in Equation (27), the incremental potential related to the hyperreduced prediction is not the incremental potential of the FE prediction, because \(\boldsymbol {\alpha }^{n1}_{\textit {FE}}\) and \(\boldsymbol {\alpha }^{n1}_{\textit {HR}}\) may differ. So, \(\boldsymbol {\sigma }^{n}_{\textit {FE}}\) fulfills the constitutive equation, but it may not be a derivative of w
_{
Δ
}. Therefore, we introduce a correction term in stress, denoted by δ
σ
^{n}, such that:
$$ \boldsymbol{\sigma}_{FE}^{n} = \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}_{FE}^{n}) + \delta \boldsymbol{\sigma}^{n} $$
((28))
where \(\boldsymbol {\varepsilon }_{\textit {FE}}^{n} = \boldsymbol {\varepsilon }(\textbf {u}_{\textit {FE}}^{n})\) and δ
σ
^{n} account for the variation of the internal variables due to the difference between \(\left (\boldsymbol {\varepsilon }_{\textit {FE}}^{i}\right)_{i=1}^{n1}\) and \(\left (\boldsymbol {\varepsilon }_{\textit {HR}}^{i}\right)_{i=1}^{n1}\) at time instants before t
^{n}. The convexity of w and ϕ insures that the FE solution is unique, if no rigid mode is available.
Dual reducedsubspace
In this section, we introduce the technique of construction of the stress fields \(\widehat {\boldsymbol {\sigma }}^{n}\) fulfilling the finite element equilibrium equation at each discrete time t
^{n}. These stress fields are used in the next sections to build a CRE. The FE equilibrium equation is a linear equation upon stresses denoted by σ
^{n}. It defines an affine space such that \(\boldsymbol {\sigma }^{n} \in \boldsymbol {\sigma }^{n}_{N} + \mathcal {S}_{h}\), where \(\boldsymbol {\sigma }^{n}_{N}\) is a particular solution of Neumann boundary conditions (i.e. in the linear elastic case for instance), and \(\mathcal {S}_{h}\) is the following vector space:
$$ \mathcal{S}_{h} = \left\{\boldsymbol{\sigma} \in L^{2}(\Omega) \:  \: \int_{\Omega} \boldsymbol{\varepsilon}(\textbf{u}^{\star}) \: : \: \boldsymbol{\sigma} \: d\Omega = 0 \quad \forall \textbf{u}^{\star} \in \mathcal{V}_{h}\right\} $$
((29))
The following linear elastic problem gives us \(\boldsymbol {\sigma }^{n}_{N}\):
$$\begin{array}{*{20}l} \textbf{u}^{n}_{N} \in \mathcal{V}_{h} \end{array} $$
((30))
$$\begin{array}{*{20}l} \int_{\Omega} \boldsymbol{\varepsilon}(\textbf{u}^{\star}) \: : \: \boldsymbol{\sigma}^{n}_{N} \: d\Omega = \int_{\partial_{F} \Omega} \textbf{u}^{\star} \:.\: \textbf{F}^{n} \: d\Gamma \quad \forall \: \textbf{u}^{\star} \in \mathcal{V}_{h} \end{array} $$
((31))
$$\begin{array}{*{20}l} \boldsymbol{\sigma}^{n}_{N} = E_{o} \: \boldsymbol{\varepsilon}(\textbf{u}^{n}_{N}) \quad \forall \textbf{x} \in \Omega \end{array} $$
((32))
where \(E_{o} \in \mathbb {R}^{+}\) is an abritary constant.
By the recourse to the stress predicted by the FE simulation for μ=μ
^{1}, we obtain the snapshots of stress fields \(\left (\boldsymbol {\sigma }^{n}_{\textit {FE}}(\cdot ;\boldsymbol {\mu }^{1})  \boldsymbol {\sigma }^{n}_{N}\right)_{n=1}^{N_{t}}\) that span a subspace of \(\mathcal {S}_{h}\):
$$ \boldsymbol{\sigma}^{n}_{FE}\left(\cdot;\boldsymbol{\mu}^{1}\right)  \boldsymbol{\sigma}^{n}_{N} \in \mathcal{S}_{h}, \quad n=1,\ldots,N_{t} $$
((33))
Similarly to the approach proposed in [10] for elasticity, we introduce a dual reducedsubspace denoted by \(\mathcal {S}_{\textit {ROM}} \subset \mathcal {S}_{h}\). \(\mathcal {S}_{\textit {ROM}}\) is generated by the usual POD applied to \(\left (\boldsymbol {\sigma }^{n}_{\textit {FE}}(\cdot ;{\boldsymbol \mu }^{1})  \boldsymbol {\sigma }^{n}_{N}\right)_{n=1}^{N_{t}}\). Its dimension is denoted by \(N^{\sigma }_{\psi }\), and it is such that \(N^{\sigma }_{\psi } \le N_{t}\).
Hence, the hyperreduced predictions of the stress tensor can be projected into the space of admissible stresses fulfilling the FE equilibrium equation. This projection reads:
$$ \widehat{\boldsymbol{\sigma}}^{n} = \boldsymbol{\sigma}^{n}_{N} + \text{arg} \: \min_{\boldsymbol{\sigma}^{\star} \in \mathcal{S}_{ROM}} \ \boldsymbol{\sigma}^{n}_{HR}  \boldsymbol{\sigma}^{n}_{N}  \boldsymbol{\sigma}^{\star} \_{L^{2}(\Omega)} $$
((34))
The above minimization problem is a global L
^{2} projection of the stress \(\boldsymbol {\sigma }^{n}_{\textit {HR}}\) onto the reduced basis that span \(\mathcal {S}_{\textit {ROM}}\). The dual basis being a POD basis, it is orthonormal with respect to the L
^{2} scalar product. So the computational complexity of the stresses projection scales linearly with the number of Gauss points of the mesh and \(N^{\sigma }_{\psi }\). In practice, this computational complexity is negligible compared to the complexity of the evaluation of \(\boldsymbol {\sigma }^{n}_{\textit {HR}}\), by using the constitutive equations.
The constitutive relation error and its partition
The reference solution being incremental in time, no continuous formulation in time is considered for the error estimator. The error estimation proposed in [33] for incremental variational formulation, is related to an asymptotic convergence assumption in order to get an upper bound of the approximation error related to the FE discretization. This upper bound depends on a constant related of the weak form of the partial differential equations. Here, we propose to apply the Constitutive Relation Error method proposed in [22] to estimate the constant c
_{
η
} without assuming an asymptotic convergence of the HRAE.
Following the method proposed in [22], a constitutive relation error, denoted by η, can be introduced as follows:
$$\begin{array}{@{}rcl@{}} \eta(\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}}) = \sum_{n=1}^{N_{t}} \int_{\Omega} w_{\Delta}(\boldsymbol{\varepsilon}(\textbf{u}^{n}_{HR})) + w^{\star}_{\Delta}(\widehat{\boldsymbol{\sigma}}^{n})  \boldsymbol{\varepsilon}(\textbf{u}^{n}_{HR}) : \widehat{\boldsymbol{\sigma}}^{n} \: d\Omega \end{array} $$
((35))
where \(\widehat {\boldsymbol {\sigma }}^{n}\) is defined by Equation (34). The following properties hold:
$$\begin{array}{*{20}l} \eta(\widetilde{\textbf{u}}, \: \widetilde{\boldsymbol{\sigma}}) \ge 0 \quad \forall \: \widetilde{\textbf{u}} \in \textbf{u}_{c} + \mathcal{V}_{h}, \: \forall \: \widetilde{\boldsymbol{\sigma}} \end{array} $$
((36))
$$\begin{array}{*{20}l} \eta(\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}}) = 0 \quad \Leftrightarrow \quad \widehat{\boldsymbol{\sigma}}^{n} = \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}(\textbf{u}^{n}_{HR})) \quad \forall \textbf{x} \in \Omega\: \forall \: n \in \{1,\ldots,N_{t}\} \end{array} $$
((37))
$$\begin{array}{*{20}l} \eta(\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}}) = 0 \quad \Rightarrow \quad \textbf{e}^{n} = 0 \quad \forall \textbf{x} \in \Omega \:\forall n \in \{1,\ldots,N_{t}\} \end{array} $$
((38))
The first property (36) comes from the Legendre transformation (14). The property (37) comes from Equation (19). The proof of Property (38) is the following. If \(\eta (\textbf {u}^{n}_{\textit {HR}}, \: \widehat {\boldsymbol {\sigma }}^{n}) = 0\forall \: n \in \{1,\ldots,N_{t}\}\) then \((\textbf {u}^{n}_{\textit {HR}}, \widehat {\boldsymbol {\sigma }}^{n}, \boldsymbol {\alpha }^{n}_{\textit {HR}})\) fulfills the constitutive equations, the Dirichlet conditions and the FE equilibrium equation. As \((\textbf {u}^{n}_{\textit {HR}}, \widehat {\boldsymbol {\sigma }}^{n}, \boldsymbol {\alpha }^{n}_{\textit {HR}})\) fulfills all the equations of the FE problem, it is a solution of the FE problem and e
^{n}=0.
We propose for ∥e
^{n}∥_{
u
} an Hilbert norm parametrized by the approximate solution \(\textbf {u}^{n}_{\textit {HR}}\) and the exact solution \(\textbf {u}^{n}_{\textit {FE}}\). This parametrized norm is defined by:
$$\begin{array}{@{}rcl@{}} \ \textbf{e}^{n} \_{u}^{2} = \int_{\Omega} \boldsymbol{\varepsilon}(\textbf{e}^{n}) : \textbf{G}(\boldsymbol{\varepsilon}(\textbf{u}^{n}_{HR}),\boldsymbol{\varepsilon}(\textbf{u}^{n}_{FE})) : \boldsymbol{\varepsilon}(\textbf{e}^{n}) \: d\Omega \end{array} $$
((39))
where ε(·) is the symmetric part of the gradient and G a symmetric positivedefinite fourthorder tensor. If the identity tensor is substituted for G, we obtain a usual H
^{1}(Ω) norm. We assume that there is no rigid mode, neither in the FE solution nor in the reduced basis.
The following mathematical developments aim to establish a relation between \(\eta (\textbf {u}_{\textit {HR}}, \: \frac {\partial w_{\Delta }}{\partial \boldsymbol {\varepsilon }}(\boldsymbol {\varepsilon }_{\textit {FE}}))\) and the tailored norm ∥e
^{n}∥_{
u
}. We choose G by the recourse to the convexity of the incremental potential w
_{
Δ
}. According to the Legendre transformation, we have:
$$ w^{\star}_{\Delta}\left(\frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}\left(\boldsymbol{\varepsilon}^{n}_{FE}\right)\right) = \boldsymbol{\varepsilon}^{n}_{FE}: \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}\left(\boldsymbol{\varepsilon}^{n}_{FE}\right)  w_{\Delta}\left(\boldsymbol{\varepsilon}^{n}_{FE}\right), \: n=1,\ldots,N_{t}, $$
((40))
Hence, we can remove the contribution of the dual potential \(w_{\Delta }^{*}\) in \(\eta \left (\textbf {u}_{\textit {HR}}, \: \frac {\partial w_{\Delta }}{\partial \boldsymbol {\varepsilon }}(\boldsymbol {\varepsilon }_{\textit {FE}}) \right)\):
$$ \eta\left(\textbf{u}_{HR}, \: \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}_{FE})\right) = \sum_{n=1}^{N_{t}} \int_{\Omega} w_{\Delta}(\boldsymbol{\varepsilon}^{n}_{HR})  w_{\Delta}(\boldsymbol{\varepsilon}^{n}_{FE})  \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}^{n}_{FE}) :(\boldsymbol{\varepsilon}^{n}_{HR} \boldsymbol{\varepsilon}^{n}_{FE}) \: d\Omega $$
((41))
Let us define the scalar function f(λ):
$$\begin{array}{*{20}l} \lambda \in [0,1], \: f(\lambda) = \eta\left(\boldsymbol{\beta}(\lambda), \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}_{FE})\right), \end{array} $$
((42))
$$\begin{array}{*{20}l} \hbox{w.r.t} \quad \boldsymbol{\beta}^{n}(\lambda) = \textbf{u}_{FE}^{n} + \lambda \: \left(\textbf{u}_{HR}^{n}  \textbf{u}_{FE}^{n}\right), \: n=1,\ldots,N_{t} \end{array} $$
((43))
We assume that f is C
^{2} on [0,1]. The application of the Taylor Lagrange formula at order 2 leads to:
$$ \exists \: \lambda_{c} \in ]0, 1[ \: \hbox{s.t.} \: f(1) = f(0) + f^{\prime}(0) + \frac{1}{2} \: f^{\prime\prime}(\lambda_{c}) $$
((44))
The function evaluation and its derivatives read:
$$\begin{array}{*{20}l} f(0) = 0 \end{array} $$
((45))
$$\begin{array}{*{20}l} f(1) = \eta \left(\textbf{u}_{HR}, \: \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}\left(\boldsymbol{\varepsilon}_{FE}\right) \right) \end{array} $$
((46))
$$\begin{array}{*{20}l} f^{\prime}(0) = 0 \end{array} $$
((47))
$$\begin{array}{*{20}l} f^{\prime \prime}(\lambda) = \sum_{n=1}^{N_{t}} \int_{\Omega} \left(\boldsymbol{\varepsilon}_{HR}^{n}  \boldsymbol{\varepsilon}_{FE}^{n}\right) : \frac{\partial^{2} w_{\Delta}}{\partial \boldsymbol{\varepsilon}^{2}} \left(\boldsymbol{\varepsilon}\left(\boldsymbol{\beta}^{n}(\lambda)\right)\right) : \left(\boldsymbol{\varepsilon}_{HR}^{n}  \boldsymbol{\varepsilon}_{FE}^{n}\right) \: d\Omega \end{array} $$
((48))
Therefore, by choosing:
$$ \textbf{G}\left(\boldsymbol{\varepsilon}_{HR}^{n},\boldsymbol{\varepsilon}_{FE}^{n}\right) = \frac{1}{2} \: \frac{\partial^{2} w_{\Delta}}{\partial \boldsymbol{\varepsilon}^{2}}\left(\boldsymbol{\varepsilon}_{FE}^{n} + \lambda_{c} \: \left(\boldsymbol{\varepsilon}_{HR}^{n}  \boldsymbol{\varepsilon}_{FE}^{n} \right)\right) $$
((49))
we obtain the following property:
$$ \sum_{n=1}^{N_{t}} \left\ \textbf{e}^{n} \right\_{u}^{2} = \eta \left(\textbf{u}_{HR}, \: \frac{\partial w_{\Delta}}{\partial \boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}_{FE})\right) $$
((50))
This property is an intermediate result before establishing the relationship between \(\ \textbf {e}^{n} \_{u}^{2}\) and the error estimator. The incremental potential being strictly convex, G is definite positive. A schematic view of \(\sum _{n=1}^{N_{t}} \ \textbf {e}^{n} \_{u}^{2}\) is shown on Figure 1.
Similarly, a norm on stresses can be defined to measure the distance between the admissible stress \(\widehat {\boldsymbol {\sigma }}^{n}\) and \(\boldsymbol {\sigma }_{\textit {FE}}^{n}  \delta \boldsymbol {\sigma }^{n}\), by the recourse to the dual potential \(w_{\Delta }^{\star }\) such that:
$$\begin{array}{@{}rcl@{}} \left\ \widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n} \right\_{\sigma}^{2} = \int_{\Omega} \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n}\right) : \textbf{H}\left(\widehat{\boldsymbol{\sigma}}^{n}, \boldsymbol{\sigma}_{FE}^{n}  \delta \boldsymbol{\sigma}^{n}\right) : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n}\right) \: d \Omega\!\!\!\!\\ \end{array} $$
((51))
where H is a positive symmetric fourth order tensor. To establish the relation between H and the Hessian matrix of the potential \(w^{\star }_{\Delta }\), we consider \(\eta (\textbf {u}_{\textit {FE}}, \widehat {\boldsymbol {\sigma }})\). Intermediate stresses \((\textbf {S}^{n}(\lambda))_{n=1}^{N_{t}}\) are introduced such that:
$$ \textbf{S}^{n}(\lambda) = \boldsymbol{\sigma}_{FE}^{n}  \delta \boldsymbol{\sigma}^{n} + \lambda \: (\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n}), \: n=1,\ldots,N_{t} $$
((52))
By the recourse to the Taylor Lagrange formula, we obtain the following property:
$$\begin{array}{@{}rcl@{}} \lambda_{c} \in ]0,1[, \hbox{s.t} \: \sum_{n=1}^{N_{t}} \ \widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n} \_{\sigma}^{2} &=& \eta(\textbf{u}_{FE}, \widehat{\boldsymbol{\sigma}}) \\ \text{w.r.t} \quad \textbf{H}\left(\widehat{\boldsymbol{\sigma}}^{n}, \boldsymbol{\sigma}_{FE}^{n}  \delta \boldsymbol{\sigma}^{n}\right) &=& \frac{1}{2} \: \frac{\partial^{2} w^{\star}_{\Delta}}{\partial \boldsymbol{\sigma}^{2}}(\textbf{S}^{n}(\lambda_{c})) \end{array} $$
((53))
Property.
The proposed constitutive relation error fulfills the following partition:
$$ \eta(\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}}) = \sum_{n=1}^{N_{t}} \ \textbf{e}^{n} \_{u}^{2} + \sum_{n=1}^{N_{t}} \ \widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n} \_{\sigma}^{2} + \sum_{n=1}^{N_{t}} \int_{\Omega} \boldsymbol{\varepsilon}(\textbf{e}^{n}) : \delta \boldsymbol{\sigma}^{n} \: d\Omega $$
((54))
The last term of the sum is a coupling term between the error e
^{n} and the HRAE committed before the discrete time t
^{n}. Since we can’t certify that this term is positive, η is not an upper bound of the HRAE.
Thanks to the definition of δ
σ by Equation (28), we have the following properties:
$$\begin{array}{*{20}l} \eta(\textbf{u}_{FE}, \boldsymbol{\sigma}_{FE} \delta \boldsymbol{\sigma}) = 0 \end{array} $$
((55))
$$\begin{array}{*{20}l} \eta(\textbf{u}_{HR}, \: \boldsymbol{\sigma}_{FE} \delta \boldsymbol{\sigma}) = \sum_{n=1}^{N_{t}} \ \textbf{e}^{n} \_{u}^{2} \end{array} $$
((56))
We start proving the partition property by considering the following sum of terms:
$$\mathcal{A} = \eta (\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}})  \eta (\textbf{u}_{HR}, \: \boldsymbol{\sigma}_{FE} \delta \boldsymbol{\sigma})  \eta(\textbf{u}_{FE}, \widehat{\boldsymbol{\sigma}}) + \eta(\textbf{u}_{FE}, \boldsymbol{\sigma}_{FE} \delta \boldsymbol{\sigma}) $$
In , the contributions of w
_{
Δ
} and \(w^{\star }_{\Delta }\) are canceled when considering the definition of η, and we obtain:
$$\mathcal{A} = \sum_{n=1}^{N_{t}} \int_{\Omega} \boldsymbol{\varepsilon}_{HR}^{n} : \widehat{\boldsymbol{\sigma}}^{n} + \boldsymbol{\varepsilon}_{HR}^{n} : \left(\boldsymbol{\sigma}_{FE}^{n} \delta \boldsymbol{\sigma}^{n} \right) + \boldsymbol{\varepsilon}_{FE}^{n} : \widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\varepsilon}_{FE}^{n} : \left(\boldsymbol{\sigma}_{FE}^{n} \delta \boldsymbol{\sigma}^{n}\right) \: d\Omega $$
Then:
$$\begin{array}{@{}rcl@{}} \mathcal{A} &=& \sum_{n=1}^{N_{t}} \int_{\Omega}  \boldsymbol{\varepsilon}_{HR}^{n} : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n} \right)+ \boldsymbol{\varepsilon}_{FE}^{n} : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n}\right) \: d\Omega \\ & = & \sum_{n=1}^{N_{t}} \int_{\Omega} \left(\boldsymbol{\varepsilon}_{FE}^{n}  \boldsymbol{\varepsilon}_{HR}^{n}\right) : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} \right) \: d\Omega + \sum_{n=1}^{N_{t}} \int_{\Omega} \left(\boldsymbol{\varepsilon}_{FE}^{n} \boldsymbol{\varepsilon}_{HR}^{n}\right) : \delta \boldsymbol{\sigma} \: d\Omega, \end{array} $$
where \(\boldsymbol {\varepsilon }_{\textit {FE}}^{n} \boldsymbol {\varepsilon }_{\textit {HR}}^{n} = \boldsymbol {\varepsilon }(\textbf {e}^{n})\). Therefore, according to properties (53), (55) and (56), the following partition is fulfilled:
$$\begin{array}{@{}rcl@{}} \eta \left(\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}} \right) &= & \sum_{n=1}^{N_{t}} \ \textbf{e}^{n} \_{u}^{2} + \sum_{n=1}^{N_{t}} \ \widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} + \delta \boldsymbol{\sigma}^{n} \_{\sigma}^{2} \\ && + \sum_{n=1}^{N_{t}} \int_{\Omega} \boldsymbol{\varepsilon}(\textbf{e}^{n}) : \delta \boldsymbol{\sigma}^{n} \: d \Omega \\ && + \sum_{n=1}^{N_{t}} \int_{\Omega} \boldsymbol{\varepsilon}(\textbf{e}^{n}) : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n} \right) \: d\Omega \end{array} $$
Since \(\textbf {e}^{n} \in \mathcal {V}_{h}\) and \(\boldsymbol {\sigma }_{\textit {FE}}^{n}  \widehat {\boldsymbol {\sigma }}^{n} \in \mathcal {S}_{h}\), then:
$$\int_{\Omega} \boldsymbol{\varepsilon}(\textbf{e}^{n}) : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{FE}^{n}\right) \: d\Omega = 0 \quad \forall n $$
This ends the proof.
Property.
If e
^{n}=0, for n∈{1,…,N
_{
t
}} then \(\textbf {u}^{n}_{\textit {HR}} = \textbf {u}^{n}_{\textit {FE}}\), \(\boldsymbol {\alpha }^{n}_{\textit {HR}} = \boldsymbol {\alpha }^{n}_{\textit {FE}}\), δ
σ
^{n}=0 and:
$$\begin{array}{*{20}l} \eta(\textbf{u}_{FE}, \: \widehat{\boldsymbol{\sigma}}_{FE}) = \sum_{n=1}^{N_{t}} \ \widehat{\boldsymbol{\sigma}}^{n}_{FE}  \boldsymbol{\sigma}_{FE}^{n} \_{\sigma}^{2} \end{array} $$
((57))
$$\begin{array}{*{20}l} \widehat{\boldsymbol{\sigma}}^{n}_{FE} = \boldsymbol{\sigma}^{n}_{N} + \text{arg} \: \min_{\boldsymbol{\sigma}^{\star} \in \mathcal{S}_{ROM}} \ \boldsymbol{\sigma}^{n}_{FE}  \boldsymbol{\sigma}^{n}_{N}  \boldsymbol{\sigma}^{\star} \_{L^{2}(\Omega)} \end{array} $$
((58))
In the general case, the closer \(\widehat {\boldsymbol {\sigma }}^{n}\) to \(\boldsymbol {\sigma }_{\textit {FE}}^{n}  \delta \boldsymbol {\sigma }^{n}\) the better the error estimation. The proposed error estimator incorporates errors related to the projection of the stress onto the dual reduced basis.
Numerical implementation of the CRE
The numerical implementation of the incremental potential and its dual is not required for the error estimation. Let’s consider the following function of a scalar coordinate λ:
$$\begin{array}{*{20}l} \lambda \in [0,1] \:, \: g(\lambda) = w_{\Delta}\left(\boldsymbol{\varepsilon}_{HR}^{n}\right) + w^{\star}_{\Delta}(\boldsymbol{\tau}(\lambda))  \boldsymbol{\varepsilon}_{HR}^{n} : \boldsymbol{\tau}(\lambda), \end{array} $$
((59))
$$\begin{array}{*{20}l} \boldsymbol{\tau}(\lambda) = \boldsymbol{\sigma}_{HR}^{n} + \lambda \: \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{HR}^{n}\right) \end{array} $$
((60))
Then:
$$\begin{array}{*{20}l} g(0) = 0 \end{array} $$
((61))
$$\begin{array}{*{20}l} \eta(\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}}) = \sum_{n=1}^{N_{t}} \int_{\Omega} g(1) \: d \Omega \end{array} $$
((62))
$$\begin{array}{*{20}l} \frac{d \: g}{d \: \lambda}(\lambda) = \left(\frac{\partial w^{\star}_{\Delta}}{\partial \boldsymbol{\sigma}} (\boldsymbol{\tau}(\lambda))  \boldsymbol{\varepsilon}_{HR}^{n} \right) : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{HR}^{n}\right) \end{array} $$
((63))
Therefore, the following property holds:
$$\begin{array}{*{20}l} \eta(\textbf{u}_{HR}, \: \widehat{\boldsymbol{\sigma}}) = \sum_{n=1}^{N_{t}} \int_{\Omega} {\int_{0}^{1}} (\widehat{\boldsymbol{\varepsilon}}^{n}(\lambda)  \boldsymbol{\varepsilon}_{HR}^{n}) : \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{HR}^{n}\right) \: d \lambda \: d \Omega \end{array} $$
((64))
$$\begin{array}{*{20}l} \text{w.r.t.} \: \: \widehat{\boldsymbol{\varepsilon}}^{n}(\lambda) = \frac{\partial w^{\star}_{\Delta}}{\partial \boldsymbol{\sigma}} (\boldsymbol{\tau}(\lambda)), \quad \boldsymbol{\tau}(\lambda) = \boldsymbol{\sigma}_{HR}^{n} + \lambda \: \left(\widehat{\boldsymbol{\sigma}}^{n}  \boldsymbol{\sigma}_{HR}^{n}\right) \end{array} $$
((65))
Here, \(\widehat {\boldsymbol {\varepsilon }}^{n}(\lambda)\) is the strain tensor given by the constitutive equation upon the stress τ(λ) at time t
^{n} and the internal variables \(\boldsymbol {\alpha }^{n1}_{\textit {HR}}\) at time t
^{n−1}. Hence, the numerical estimation of \(\widehat {\boldsymbol {\varepsilon }}^{n}(\lambda)\) can be performed by the usual implementation of the constitutive equations of generalized standard materials [4042].
In the following numerical simulations, we have estimated the integral on λ by the value of first derivative of g for λ=1.
Numerical estimation of the constant c
_{
η
}
In practice, we have more modes than necessary to achieve accurate predictions for μ=μ
^{1}, because the POD of \((\textbf {u}^{n}_{\textit {FE}}(\cdot, \: \boldsymbol {\mu }^{1}))_{n=1}^{N_{t}}\) is accurate up to the numerical precision. The total number of available modes is denoted by \(\overline {N}_{\psi }\) such that \(\overline {N}_{\psi } \le \min (N_{\xi }, N_{t})\) and \(N_{\psi } \le \overline {N}_{\psi }\). This gives access to several hyperreduced models by restricting the number of modes involved in the reduced basis related to the displacements. The constant c
_{
η
} is setup by using these approximate hyperreduced solutions and the known stresses predicted by the FE model:
$$ c_{\eta} = \max_{N_{\psi} \in \left\{1, \ldots,\overline{N}_{\psi}\right\}} \frac{\sum_{n=1}^{N_{t}} \left\ s\left(\textbf{u}^{n}_{FE}\left(\cdot;\boldsymbol{\mu}^{1}\right)  s\left(\textbf{u}^{n}_{HR}\left(\cdot;\boldsymbol{\mu}^{1}\right)\right)\right. \right\_{F}^{2}}{\min \left\{\eta\left(\textbf{u}_{HR}\left(\cdot;\boldsymbol{\mu}^{1}\right), \: \boldsymbol{\sigma}_{FE}\left(\cdot;\boldsymbol{\mu}^{1}\right)\right) \:, \: \eta\left(\textbf{u}_{HR}\left(\cdot;\boldsymbol{\mu}^{1}\right), \: \widehat{\boldsymbol{\sigma}}\left(\cdot;\boldsymbol{\mu}^{1}\right) \right)\right\}} $$
((66))
Hence the constraint (9) is fulfilled. Let us denote by \(N^{c}_{\psi }\) the number of displacement modes for which the maximum in Equation (66) is reached. In our opinion, if we expect that the error estimator behaves like an upper bound, we should not take \(N^{c}_{\psi }\) modes to generate the final HR model. In the following example, we are setting \(N_{\psi } = N^{c}_{\psi }+1\).