We propose to study an academic example in order to discuss the different steps of the parameter identification problem associated with a time-homogenized model. To the best of our knowledge, it is the first time that such a two-time-scale identification strategy using an adjoint state formulation is proposed: only one alternative has been briefly studied in Ref. [20] with the use of a genetic method. First, we will recall how the identification process is formulated on a classical reference problem, i.e. with no time homogenization. Then, we will apply the identification strategy on the time-homogenized version of the problem, and see what connections exist with the identification process associated with the reference problem.

The case of study detailed here consists of a straight bar of length \(L\), clamped at one end, and withstanding at the other end a normal force with two periodic components. The measured displacement at this end is then used to determine the parameter values of the material elastic viscoplastic law.

### Reference problem

#### Forward problem

The dynamic equilibrium of the bar is a scalar equation, with the normal (nonviscous) stress \(\sigma (x,t)\) and the longitudinal displacement \(u(x,t)\) defined at each point \(x\in [0,L]\) and for any \(t\in [0,T]\):

$$\begin{aligned} {\partial _x}\sigma + c_K {\mathrm {d}}_t\partial _x\sigma = \rho {{\mathrm {d}}^2_t}u \end{aligned}$$

(11)

where \(\partial _x\) is the partial space derivative, \(\rho\) is the mass density, and \(c_K\) is the damping ratio where damping is assumed as proportional to stiffness. Homogeneous initial conditions for the displacement and the velocity are assumed. No load is applied along the bar, and, whereas this latter is clamped at \(x=0\), a surface force \(f_{\mathrm {s}}(t)\) is applied at \(x=L\) for any \(t\in [0,T]\):

$$\begin{aligned} u_{|x=0} = 0 \end{aligned}$$

(12)

$$\begin{aligned} \left( \sigma + c_K {\mathrm {d}}_t\sigma \right) _{|x=L} = f_{\mathrm {s}} \end{aligned}$$

(13)

This surface force is biperiodic, with the ratio of the associated low frequency over the fast one equal to \(10^{-4}\).

The constitutive relation links the normal (nonviscous) stress to the longitudinal strain along the bar for any \((x,t)\in [0,L]\times [0,T]\):

$$\begin{aligned} \sigma = E\left( {\partial _x}u-\varepsilon ^{\mathrm {p}}\right) \end{aligned}$$

(14)

where \(E\) is the Young’s modulus, and \(\varepsilon ^{\mathrm {p}}(x,t)\) stands for the longitudinal plastic strain. One assumes that this latter verifies a Norton’s viscoplastic evolution law for any \((x,t)\in [0,L]\times [0,T]\), which depends on the nonviscous stress only:

$$\begin{aligned} {{\mathrm {d}}_t}\varepsilon ^{\mathrm {p}}=\left( \frac{|\sigma |}{K}\right) ^n \mathop {\mathrm {sign}}\sigma \end{aligned}$$

(15)

with \(\varepsilon ^{\mathrm {p}}\,_{|t=0}=0\ \forall x\in [0,L]\). \(K\) and \(n\) are the two constant parameters to be identified.

The numerical implementation consists of quadratic 1D finite elements under MATLAB for the spatial discretization of the displacement, whereas the longitudinal plastic strain is linearly interpolated using the ‘external’ nodes of the different elements. The time integration uses the ‘ode45’ procedure, based on a fourth-order embedded Runge–Kutta formula according to Ref. [21]: depending on the number of finite elements used, the time step to be chosen can verify the classical rule of thumb of 20 time steps per fast cycle, or must be smaller than the maximal time step associated with the Courant–Friedrichs–Lewy condition, making in both cases the computational cost become unaffordable as soon as a high number of cycles is calculated.

#### Parameter identification process

It is assumed that experimental data consist of the knowledge of the displacement \(u_{exp}(t)\) measured at \(x=L\). The proposed misfit function is then:

$$\begin{aligned} {\mathcal {J}}(K,n)&= \frac{1}{2}\int _0^T (u(L,t;K,n)-u_{exp}(t) )^2{\mathrm {d}}t\nonumber \\& \quad + \frac{\alpha _K}{2}(K-K_0)^2 + \frac{\alpha _n}{2}(n-n_0)^2 \end{aligned}$$

(16)

where \(u(x,t;K,n)\) is the solution of the forward problem (11)–(15) with parameter values \(K\) and \(n\).

In order to derive the adjoint state problem, the following Lagrangian is introduced:

$$\begin{aligned} {\mathcal {L}}(u,\varepsilon ^{\mathrm {p}},K,n,z,\lambda )&= \frac{1}{2}\int _0^T (u(L,t)-u_{exp}(t) )^2{\mathrm {d}}t\nonumber \\& \quad + \frac{\alpha _K}{2}(K-K_0)^2 + \frac{\alpha _n}{2}(n-n_0)^2\nonumber \\& \quad - \int _0^T\int _0^L\rho ({\mathrm {d}}_t^2u)z\,{\mathrm {d}}x\,{\mathrm {d}}t + \int _0^Tf_{\mathrm {s}}z_{|x=L}\,{\mathrm {d}}t\nonumber \\& \quad - \int _0^T\int _0^L\left [E\left( {\partial _x}u-\varepsilon ^{\mathrm {p}}\right) +c_KE{\mathrm {d}}_t\left( {\partial _x}u-\varepsilon ^{\mathrm {p}}\right) \right ]{\partial _x}z\,{\mathrm {d}}x\,{\mathrm {d}}t \nonumber \\& \quad + \int _0^T\int _0^L\left[ {{\mathrm {d}}_t}\varepsilon ^{\mathrm {p}}-\left( \frac{E\left| {\partial _x}u-\varepsilon ^{\mathrm {p}}\right| }{K}\right) ^n\mathop {\mathrm {sign}}\left( {\partial _x}u-\varepsilon ^{\mathrm {p}}\right) \right] \lambda \,{\mathrm {d}}x\,{\mathrm {d}}t \end{aligned}$$

(17)

where \((u,\varepsilon ^{\mathrm {p}},K,n,z,\lambda )\) are considered as independent quantities. The minimization of the misfit function (16) under the constraints (11)–(15) is then equivalent to express the stationarity of the Lagrangian (17) with no constraint [19].

The adjoint states \(z(x,t)\) and \(\lambda (x,t)\) are Lagrange multipliers verifying, for any \((x,t)\in (0,L)\times [0,T]\), equations coming from the stationarity of the Lagrangian with respect to \(u\):

$$\begin{aligned} {\partial _x}\left[ E{\partial _x}z-c_KE{{\mathrm {d}}_t}{\partial _x}z+n\frac{E}{K}\left( \frac{E\left| {\partial _x}u-\varepsilon ^{\mathrm {p}}\right| }{K}\right) ^{n-1}\lambda \right] = \rho {{\mathrm {d}}_t^2}z \end{aligned}$$

(18)

$$\begin{aligned} \left( E{\partial _x}z-c_KE{{\mathrm {d}}_t}{\partial _x}z+n\frac{E}{K}\left( \frac{E\left| {\partial _x}u-\varepsilon ^{\mathrm {p}}\right| }{K}\right) ^{n-1}\lambda \right) _{|x=L} = u_{|x=L}-u_{exp} \end{aligned}$$

(19)

$$\begin{aligned} z_{|t=T} = 0, \quad {\mathrm {d}}_tz_{|t=T} = 0 \end{aligned}$$

(20)

and \(\varepsilon ^{\mathrm {p}}\):

$$\begin{aligned} {\mathrm {d}}_t\lambda = n\frac{E}{K}\left( \frac{E\left| {\partial _x}u-\varepsilon ^{\mathrm {p}}\right| }{K}\right) ^{n-1}\lambda + E{\partial _x}z-c_KE{{\mathrm {d}}_t}{\partial _x}z \end{aligned}$$

(21)

$$\begin{aligned} \lambda _{|t=T} =0 \end{aligned}$$

(22)

Formally, the latter equations correspond to a dynamic elasto–viscoplastic problem (with negative damping), when one considers the first Lagrange multiplier \(z\) as a kind of displacement (even if it is not homogeneous to a length) and the second Lagrange multiplier \(\lambda\) as a kind of plastic strain (even it is not dimensionless). The discrepancy between the forward model’s predictions and experimental data is directly imposed as a Neumann boundary condition at \(x=L\), i.e. where the measurements are available. One should notice that this problem is time-backward with final conditions (at \(t=T\)): once the adjoint state problem has been properly discretized with respect to space, the obtained time differential equations can be solved with classical time integration methods by applying a change in variables \(\theta =T-t\) in order to deal with initial conditions rather than final ones.

Once solved, the adjoint state solution is used to evaluate the misfit function’s gradient, whose components are equal to the partial derivatives of the Lagrangian (17) with respect to the parameters:

$$\begin{aligned} \frac{\partial {\mathcal {J}}}{\partial K}(K,n)&= \frac{\partial {\mathcal {L}}}{\partial K}(u(K,n),\varepsilon ^{\mathrm {p}}(K,n),K,n,z(u,\varepsilon ^{\mathrm {p}}),\lambda (u,\varepsilon ^{\mathrm {p}}))\nonumber \\&= \int _0^T\int _0^L\frac{n}{K}\left( \frac{E\left| \partial _xu-\varepsilon ^{\mathrm {p}}\right| }{K}\right) ^n\mathop {\mathrm {sign}}\left( \partial _xu-\varepsilon ^{\mathrm {p}}\right) \lambda \,{\mathrm {d}}x\,{\mathrm {d}}t \nonumber \\ & \quad + \alpha _K(K-K_0) \end{aligned}$$

(23)

$$\begin{aligned} \frac{\partial {\mathcal {J}}}{\partial n}(K,n)&= \frac{\partial {\mathcal {L}}}{\partial n} (u(K,n),\varepsilon ^{\mathrm {p}}(K,n),K,n,z(u,\varepsilon ^{\mathrm {p}}),\lambda (u,\varepsilon ^{\mathrm {p}})) \nonumber \\& = -\int _0^T\int _0^L \left( \frac{E\left| \partial _xu-\varepsilon ^{\mathrm {p}}\right| }{K}\right) ^n\mathop {\mathrm {sign}}\left( \partial _xu-\varepsilon ^{\mathrm {p}}\right) \log \left( \frac{E\left| \partial _xu-\varepsilon ^{\mathrm {p}}\right| }{K}\right) \lambda \,\mathrm {d}x\,{\mathrm {d}}t\nonumber \\& \quad + \alpha _n(n-n_0) \end{aligned}$$

(24)

One should notice that, beside the forward state solutions, only the second Lagrange multiplier \(\lambda\) appears in the previous expressions: this is logical here, because one is interested in the parameters \((K,n)\) of the evolution law (15) only. The first Lagrange multiplier \(z\) would have appeared in these expressions if one had tried to identify parameters associated with the dynamic equilibrium equation, such as the Young’s modulus \(E\) or the damping ratio \(c_K\).

These estimates of the misfit function’s gradient components are used by the minimization algorithm to determine the best search direction and step size. However, even if the strategy using the adjoint state problem is the cheapest one in terms of computational cost, this latter can be awfully prohibitive if a high number of fast cycles has to calculated, as it is the case here, since viscoplasticity is a slow-evolving phenomenon, implying that the change in the longitudinal viscoplastic strain is very small for each individual fast cycle.

### Time-homogenized problem

In order to obtain a drastic reduction in the computational cost, the periodic time homogenization method is applied. Following the different steps briefly described in the “Background”, the time-homogenized equations to be solved are obtained.

#### Dynamic equilibrium equation

The first equation to be dealt with is the dynamic equilibrium equation, which can be rewritten in a normalized way in order to compare the orders of magnitude of the different terms:

$$\begin{aligned} \partial _{\hat{x}}\hat{\sigma } + Fc_K{\mathrm {d}}_{\hat{t}}\partial _{\hat{x}}\hat{\sigma } = \frac{\rho L^2 F^2}{E}{\mathrm {d}}_{\hat{t}}^2\hat{u} \end{aligned}$$

(25)

where \(F\) is the frequency of the slow component of the loading, and the different normalized quantities are denoted as \(\hat{\bullet }\). Different cases arise, depending on the orders of magnitude of \(\rho L^2 F^2/E\) and \(Fc_K\). Here, we assume that:

$$\begin{aligned} \frac{\rho L^2 F^2}{E} = \beta \xi ^2 \end{aligned}$$

(26)

$$\begin{aligned} Fc_K = \gamma \xi \end{aligned}$$

(27)

where \(\beta \le O(1)\) and \(\gamma \le O(1)\). Physically, the first assumption (26) is equivalent to having \(L/\lambda =\sqrt{\beta }\), where \(\lambda\) is the wavelength associated with the fast loading frequency: this means that the order of magnitude of \(\lambda\) should be at least equal to the length of the bar.

Using these two assumptions and introducing an asymptotic expansion of the displacement \(u=u_0+\xi u_1+\xi ^2u_2+O(\xi ^3)\) allows to express the acceleration as follows:

$$\begin{aligned} \rho {\mathrm {d}}_t^2u = \frac{\beta E}{L^2F^2}u_0'' + \xi \frac{\beta E}{L^2F^2}\left( 2\dot{u}_0'+u_1''\right) + \xi ^2\frac{\beta E}{L^2F^2}\left( \ddot{u}_0+2\dot{u}_1'+u_2''\right) + O(\xi ^3) \end{aligned}$$

(28)

It can then be shown [5] that the two time scales are separable and that the homogenization method can be applied: the zeroth-order dynamic equilibrium equation becomes:

$$\begin{aligned} {\partial _x}\sigma _0 + \frac{\gamma }{F}\partial _x\sigma _0' = \frac{\beta E}{L^2F^2}u_0'' \end{aligned}$$

(29)

and the associated zeroth-order boundary condition at \(x=L\) reads:

$$\begin{aligned} \left( \sigma _0 + \frac{\gamma }{F}\sigma _0'\right) _{|x=L} = f_{\mathrm {s}} \end{aligned}$$

(30)

Eventually, by taking the fast-time average (4) of the two previous equations, and by using the quasiperiodicity assumption (5), one can obtain the time-homogenized zeroth-order dynamic equilibrium equation:

$$\begin{aligned} {\partial _x}\langle \sigma _0\rangle \ = 0 \end{aligned}$$

(31)

for any \((x,t)\in (0,L)\times [0,T]\), and the time-homogenized zeroth-order boundary condition:

$$\begin{aligned} \langle \sigma _0\rangle _{|x=L} = \langle f_{\mathrm {s}}\rangle \end{aligned}$$

(32)

which corresponds to the solution of a quasistatic problem.

#### Evolution law

In the same way, the evolution law (15) gives, up to order zero:

$$\begin{aligned} \frac{1}{\xi }{\varepsilon }^{\mathrm {p}}_0\,' + \left( \dot{{\varepsilon }}_0^{\mathrm {p}}+{\varepsilon }^{\mathrm {p}}_1\,' \right) + O (\xi ) = \left( \frac{|\sigma _0|}{K}\right) ^n \mathop {\mathrm {sign}}\sigma _0 + O(\xi ) \end{aligned}$$

(33)

First, the \(1/\xi\)-order has to be considered, giving: \(\varepsilon ^{\mathrm {p}}_0\,'= 0\). This is equivalent to say that the zeroth-order plastic strain only depends on the slow time variable:

$$\begin{aligned} \varepsilon _0^{\mathrm {p}} (x,t,\tau ) = \varepsilon _0^{\mathrm {p}} (x,t) \end{aligned}$$

(34)

The physical interpretation of this latter result is that viscoplasticity is a slow-evolving phenomenon, even if the material withstands a high-frequency loading. This slow variation is described by the zeroth-order evolution law, which can be expressed as:

$$\begin{aligned} \dot{\varepsilon }_0^{\mathrm {p}}+\varepsilon ^{\mathrm {p}}_1\,' = \left( \frac{|\sigma _0|}{K}\right) ^n \mathop {\mathrm {sign}}\sigma _0 \end{aligned}$$

(35)

In order to make the contribution of the first-order plastic strain disappear from this relation, the fast-time average (4) of this latter is evaluated. Since, on the one hand, Eq. (34) is equivalent to say that \(\langle \dot{\varepsilon }_0^{\mathrm {p}} \rangle \, =\dot{\varepsilon }_0^{\mathrm {p}}\), and, on the other hand, the quasiperiodicity assumption (5) implies that \(\langle \varepsilon ^{\mathrm {p}}_1\,'\rangle \,=0\), this average eventually gives that, for any \((x,t)\in [0,L]\times [0,T]\):

$$\begin{aligned} \dot{\varepsilon }_0^{\mathrm {p}} = \,\left\langle \left( \frac{|\sigma _0|}{K}\right) ^n \mathop {\mathrm {sign}}\sigma _0\right\rangle \end{aligned}$$

(36)

with a zero initial condition.

#### Zeroth-order time-homogenized problem

The final system consists of the following equations to be solved at the slow time scale only, for any \((x,t)\in (0,L)\times [0,T]\):

$$\begin{aligned} {\partial _x}\langle \sigma _0\rangle \ = 0 \end{aligned}$$

(37)

$$\begin{aligned} \langle \sigma _0\rangle \, = E\left( {\partial _x}\langle u_0\rangle -\varepsilon _0^{\mathrm {p}}\right) \end{aligned}$$

(38)

$$\begin{aligned} \langle u_0\rangle _{|x=0} = 0 \end{aligned}$$

(39)

$$\begin{aligned} \langle \sigma _0\rangle _{|x=L} = \langle f_{\mathrm {s}}\rangle \end{aligned}$$

(40)

$$\begin{aligned} \dot{\varepsilon }_0^{\mathrm {p}} = \,\left\langle \left( \frac{|\sigma _0|}{K}\right) ^n \mathop {\mathrm {sign}}\sigma _0\right\rangle \end{aligned}$$

(41)

The resulting system then corresponds to a quasistatic, elastic viscoplastic problem, where the loading consists of the slow component of the surface force.

Of course, the fast component does have an effect on these homogenized equations, by means of the evolution law (41): since this latter is nonlinear, it is required to calculate the zeroth-order ‘instantaneous’ stress field \(\sigma _0\) instead of directly using the corresponding homogenized quantity \(\langle \sigma _0\rangle\). To do this, it is mandatory to estimate the residual quantity \(\sigma _0^*(x,t,\tau )\), defined according to the following relation:

$$\begin{aligned} \sigma _0(x,t,\tau )=\,\langle \sigma _0\rangle (x,t)+\sigma _0^*(x,t,\tau ) \end{aligned}$$

(42)

This residual is then the solution of the following system of residual equations:

$$\begin{aligned} {\partial _x\sigma }_0^* + \frac{\gamma }{F}{\partial _x\sigma }_0^*\,' = \frac{\beta E}{L^2F^2} u_0^*\,'' \end{aligned}$$

(43)

$$\begin{aligned} \sigma _0^* = E\partial _xu_0^* \end{aligned}$$

(44)

$$\begin{aligned} u^*_{0\,|x=0} = 0 \end{aligned}$$

(45)

$$\begin{aligned} \left( \sigma ^*_0 + \frac{\gamma }{F}\sigma _0^*\,'\right) _{|x=L} = f_{\mathrm {s}}^* \end{aligned}$$

(46)

obtained by subtracting the equations of the time-homogenized problem (37)–(40) to the zeroth-order equations of the reference problem. This residual system consists in solving, for the fast time scale \(\tau\), and at each slow time step \(t_k\), a dynamic problem for a purely viscoelastic material, with a surface force \(f_{\mathrm {s}}^*=f_{\mathrm {s}}\ -\langle f_{\mathrm {s}}\rangle\). Since viscoplasticity (which is the only source of nonlinearity here) appears in the zeroth-order time-homogenized equations only, this residual problem is linear, and can be conveniently solved in the frequency domain. If needed, additional orders could be addressed, as detailed in Ref. [5].

The evaluation of the fast-time average appearing in (41) can be achieved by means of a numerical integration formula, such as the trapezoidal rule with \(N+1\) points, defined as:

$$\begin{aligned} \langle \alpha \rangle (t) \approx \frac{1}{N} \left( \frac{\alpha \left( t,0\right) }{2} + \sum _{j=1}^{N-1}\alpha \left( t,\frac{j}{N}\frac{1}{F}\right) +\frac{\alpha \left( t,\frac{1}{F}\right) }{2}\right) \end{aligned}$$

(47)

This choice is not only related to its formal simplicity, but can actually be justified using the conclusions presented in Ref. [22]: the author shows that the trapezoidal rule converges extremely fast when integrating smooth periodic functions, as it is the case here, and that no substantial additional gain would be obtained with the use of more elaborated formulas, such as Simpson’s for example.

### Associated parameter identification strategy

The identification strategy using adjoint state formulations described in the “Background” is applied now on the time-homogenized problem. First, a consistent misfit function has to be selected, then the adjoint state problem required to estimate the misfit function’s gradient is introduced.

#### Choice of the misfit function

The first step consists in describing, through the misfit function, the discrepancy between the time-homogenized model’s predictions and experimental data: indeed, on the one hand, the model has been solved on slow time steps only, whereas, on the other hand, the experimental data can be available on a much finer scale.

The most efficient choice in terms of computation cost is to use time-homogenized quantities in the misfit function (16), since it allows to address the time integral on slow time steps only:

$$\begin{aligned} {\mathcal J}_0(K,n)&= \frac{1}{2}\int _0^T|\langle u_0\rangle (L,t;K,n)\ -\langle u_{exp}\rangle (t)|^2\,{\text {d}}t \nonumber \\&\quad + \frac{\alpha _K}{2}(K-K_0)^2 +\frac{\alpha _n}{2}(n-n_0)^2 \end{aligned}$$

(48)

Whereas \(\langle u_0\rangle (L,t;E,K,n)\) is the solution of the zeroth-order time-homogenized forward problem (37)–(41), \(\langle u_{exp}\rangle (t)\) stands for the corresponding experimental quantity, which is obtained by fast-averaging the experimental data for each slow time step \(t_k\) of the time-homogenized displacement:

$$\begin{aligned} \langle u_{exp}\rangle (t_k) \,= \frac{F}{\xi }\int _{t_k}^{t_k+\frac{\xi }{F}}u_{exp}(t)\,{\mathrm {d}}t \end{aligned}$$

(49)

#### Adjoint state problem

The misfit function’s gradient is evaluated in the same way as in the section associated with the reference problem, using the solution of an adjoint state problem. Starting from the Lagrangian associated with the misfit function (48) and the time-homogenized forward problem (37)–(41), it is first obtained that the second Lagrange multiplier depends on the slow time scale only:

$$\begin{aligned} \lambda _0(x,t,\tau )=\lambda _0(x,t) \end{aligned}$$

(50)

because it is associated with the zeroth-order time-homogenized evolution law (41), which depends on \(t\) only. On the contrary, the first Lagrange multiplier is decomposed as \(\langle z_0\rangle (x,t)+z_0^*(x,t,\tau )\) to deal with the two systems (37)–(40) and (43)–(46) respectively.

Then, the two following PDEs corresponding to the adjoint state problem are obtained, for any \((x,t)\in (0,L)\times [0,T]\):

$$\begin{aligned} {\partial _x}\left[ E{\partial _x}\langle z_0\rangle +n\frac{E}{K}\left\langle \left( \frac{E\left| {\partial _x}u_0-\varepsilon _0^{\mathrm {p}}\right| }{K}\right) ^{n-1}\right\rangle \lambda _0\right] = 0 \end{aligned}$$

(51)

$$\begin{aligned} \left( E{\partial _x}\langle z_0\rangle +n\frac{E}{K}\left\langle \left( \frac{E\left| {\partial _x}u_0-\varepsilon _0^{\mathrm {p}}\right| }{K}\right) ^{n-1}\right\rangle \lambda _0\right) _{|x=L} = \langle u_0\rangle _{|x=L}-\langle u_{exp}\rangle \end{aligned}$$

(52)

$$\begin{aligned} \dot{\lambda }_0 = n\frac{E}{K}\left\langle \left( \frac{E\left| {\partial _x}u_0-\varepsilon _0^{\mathrm {p}}\right| }{K}\right) ^{n-1}\right\rangle \lambda _0 + E{\partial _x}\langle z_0\rangle \end{aligned}$$

(53)

$$\begin{aligned} \lambda _{0\,|t=T} =0 \end{aligned}$$

(54)

and they actually correspond to the zeroth-order time-homogenized versions of Eqs. (18)–(22) giving the adjoint state solutions associated with the identification process for the reference problem. Figures 4 and 5 show the comparison between the two corresponding adjoint state solutions \(\lambda\) and \(\lambda _0\) for the identification process detailed in the “Results and discussion”: the adjoint state solution \(\lambda _0\) corresponding to the time-homogenized problem is homogeneous along the bar, and is very close to the (homogeneous) fast-time average of the (heterogeneous) adjoint state solution \(\lambda\) associated with the reference problem.

Indeed, in this case, the previous equations can be decoupled in order to get the PDE verified by the second Lagrange multiplier \(\lambda _0\) for any \((x,t)\in [0,L]\times [0,T]\):

$$\begin{aligned} \partial _x\dot{\lambda }_0 = 0 \end{aligned}$$

(55)

$$\begin{aligned} \dot{\lambda }_{0\,|x=L} = \ \langle u_0\rangle _{|x=L}-\langle u_{exp}\rangle \end{aligned}$$

(56)

$$\begin{aligned} \lambda _{0\,|t=T} =0 \end{aligned}$$

(57)

finally implying that, for any \((x,t)\in [0,L]\times [0,T]\):

$$\begin{aligned} \dot{\lambda }_0 = \ \langle u_0\rangle _{|x=L}-\langle u_{exp}\rangle \end{aligned}$$

(58)

$$\begin{aligned} \lambda _{0\,|t=T} =0 \end{aligned}$$

(59)

which is a time-backward ODE with a final condition equal to zero. This equation can be solved using the slow time steps \(t_k\) only, which allows to derive the solution in a way as efficient as for the time-homogenized forward solution.

#### Misfit function’s gradient

The misfit function’s gradient then consists of the two following partial derivatives:

$$\begin{aligned} \frac{\partial \mathcal {J}_0}{\partial K}&= \int _0^T\int _0^L\left\langle \frac{n}{K}\left( \frac{E\left| \partial _xu_0-\varepsilon _0^{\mathrm {p}}\right| }{K}\right) ^n\mathop {\mathrm {sign}}\left( \partial _xu_0-\varepsilon _0^{\mathrm {p}}\right) \right\rangle \lambda _0\,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\&\quad +\alpha _K(K-K_0) \end{aligned}$$

(60)

$$\begin{aligned} \frac{\partial {\mathcal {J}}_0}{\partial n}&= -\int _0^T\int _0^L\left\langle \left( \frac{E\left| \partial _xu_0-\varepsilon _0^{\mathrm {p}}\right| }{K}\right) ^n\mathop {\mathrm {sign}}\left( \partial _xu_0-\varepsilon _0^{\mathrm {p}}\right) \log \left( \frac{E\left| \partial _xu_0-\varepsilon _0^{\mathrm {p}}\right| }{K}\right) \right\rangle \lambda _0\,{\mathrm {d}}x\,{\mathrm {d}}t\nonumber \\& \quad +\alpha _n(n-n_0) \end{aligned}$$

(61)

Once again, these relations correspond to the zeroth-order time-homogenized estimates of the two misfit function’s gradient components (23)–(24) obtained for the identification problem associated with the reference problem. Actually, this is a result that already occurs in periodic space homogenization, as shown for example in Ref. [23].