# Parameter identification of two-time-scale nonlinear transient models

- Guillaume Puel
^{1}Email author and - Denis Aubry
^{1}

**2**:8

**DOI: **10.1186/s40323-015-0030-z

© Puel and Aubry 2015

**Received: **16 January 2015

**Accepted: **15 May 2015

**Published: **9 June 2015

## Abstract

The aim of this paper is to study two-time-scale nonlinear transient models and their associated parameter identification. When it is possible to consider two well-separated time scales, and when the fast component of the applied loading is periodic, a periodic time homogenization scheme, similar to what exists in space homogenization, can be used to derive a homogenized model. A parameter identification process for this latter is then proposed, and applied to an academic example, which allows to show the benefits of such a strategy.

### Keywords

Inverse problem Identification Adjoint state Time multiscale Periodic homogenization## Background

### Context

Material fatigue is one of the key phenomena whose study is essential to design and develop modern industrial parts: many of the classical methods used in the industry, such as those described in Ref. [1], allow to estimate fatigue life through cumulative formulas. However, these latter do not take into account the history associated with the loading, and tend to give poor results as soon as the loading frequencies are high and the inertia effects should be considered. One particularly difficult case of study corresponds to combined cycle fatigue (CCF), where two different periodic loads (generally, one being ‘slow’, the other ‘fast’) are involved. In the meantime, more and more complex material laws have been proposed to describe cyclic behaviors, and the ideal way to use these models consists in solving them in the time domain.

As far as the numerical calculation of a time-dependent model is concerned, the question of the computational cost can be of utmost relevance, especially when the considered model deals with fast phenomena, which require the use of very small time steps, when compared with the length of the time interval of study. In order to drastically reduce the computational cost, a periodic time homogenization method can be used when two well-separated time scales exist, and when the components of the applied loading are periodic with respect to the fast time scale [2]. The resulting homogenized model can then be solved with a drastically reduced computational cost, using time steps related to the slow time scale only, whereas the fast time scale is taken into account in an average way in the homogenization scheme.

In order to give accurate predictions, such time-homogenized models have to be compared with experimental data. The key point is to define an identification strategy able to deal with such models and using a process that remains cheap and efficient. The aim of this paper is to analyze on a specific academic example how a relevant identification strategy can be proposed, and to comment on the different choices that can be made throughout the whole identification process. First, the basic elements of the periodic time homogenization method are described, as well as one particular case of study showing the efficiency of the method; then, the general parameter identification strategy that will be adapted to the case of time-homogenized models is introduced. After this introduction, the main part of the paper focuses on a simple academic example allowing to give the theoretical developments associated with a truly two-time-scale identification strategy, and to analyze the associated numerical results upon which conclusions can be drawn.

### Periodic time homogenization method

#### Main ingredients

#### Cases of study

References on the time homogenization method still tend to be quite scarce, as showed a recent tentative review in Ref. [5]. Nevertheless, the possible applications are numerous, as we can see with a significant number of references dealing with topics as diverse as the behavior of viscoelastic materials in Ref. [6], the vibration of preloaded beams in Ref. [7], the ultrasonic welding of composites in Ref. [8], the ultrasonic imaging of tissues in Ref. [9], the cyclic loading of normally consolidated clay in Ref. [10] or the viscoelastic-viscoplastic behavior of polymers in Ref. [11].

### Gradient-based parameter identification using adjoint state formulations

To improve the quality of the model’s predictions, especially the fatigue life estimate, it is mandatory to compare computed quantities with experimental data and to define an identification scheme allowing to update the model’s parameters judiciously. In this section, a general description of the proposed identification strategy is given, which, as we will see, is relevant for both reference and time-homogenized problems: it will thus be illustrated in this latter case in the “Methods”.

#### Formulation of the identification problem

The identification problem consists in finding the parameter vector \({\mathbf p}_{opt}\) such that the solution \({\mathbf u}(t;{\mathbf p}_{opt})\) of (6) obtained with the parameters \({\mathbf p}_{opt}\) is as close to the available experimental data as possible. These latter are indeed compared with the corresponding quantities \({\mathbf A}{\mathbf u}(t;{\mathbf p})\), where \({\mathbf A}\) is a projection operator allowing to select, for each quantity, the closest DOF to the experimental measurement point. In order to use consistent notations, the corresponding experimental quantity is denoted \({\mathbf A}{{\mathbf u}}_{exp}(t)\); however, it does not mean that such a vector \({{\mathbf u}}_{exp}(t)\) actually exists.

#### Adjoint state formulation

The determination of this minimum is achieved using gradient-based minimization methods, therefore the question of avoiding local minima by means of an appropriate regularization process should be carefully addressed. In some cases, rather than using the classical Tikhonov regularization term, the a priori experience can be introduced in some specific ways, as in Ref. [16]. Similarly, the fact of using a homogenized model in the parameter identification process can introduce a regularizing effect, just as explained in Ref. [17]. However, we will not address here this specific question, but rather focus on the identification process itself.

## Methods

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 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

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

#### Evolution law

#### Zeroth-order time-homogenized problem

### 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.

#### Adjoint state problem

#### Misfit function’s gradient

## Results and discussion

### Validation of the time homogenization method

The studied bar, which is \(1\ {\text {m}}\) long, is discretized in ten quadratic finite elements. The surface force at \(x=L\) is a combined cycle load, defined as the sum of two sines of frequencies \(F\) and \(F/\xi\). For this case of study, the assumption (26) is verified, since the two loading frequencies are \(F=0.05\ {\text {Hz}}\) and \(F/\xi =500\ {\text {Hz}}\) (cyclic loads with respective amplitudes \(1\ {\text {kN}}\) and \(0.25\ {\text {kN}}\)) and the material considered is a typical steel (\(E=200\ {\text {GPa}}\) and \(\rho =7,800\ {\text {kg}}\, {\text {m}}^{-3}\)). Moreover, the damping ratio \(c_K\) is equal to \(10^{-5}\), hence verifying assumption (27).

The reference calculations consist in solving the Eqs. (11)–(15) of the reference problem for a time interval corresponding to the first slow loading period \([0,20]\mathrm {s}\): the time step is chosen according to the classical rule of thumb of 20 time steps per fast loading period, i.e. \(10^{-4}\ {\text {s}}\). MATLAB’s ‘ode45’ procedure is used.

The zeroth-order time-homogenized equations (37)−(41) are solved with the same algorithm, but the chosen time step is \(0.01\ {\mathrm {s}}\), which reduces by 100 the number of iterations required for the calculation. Since the loading is the sum of two periodic components (one slow and one fast), the residual problem, consisting in Eqs. (43)–(46), depends on the fast time scale only, and can be solved in the frequency domain once and for all.

Validation of the periodic time homogenization method

\(x=0\) | \(x=L\) | |
---|---|---|

Reference | 1.874891 × 10 | 1.652784 × 10 |

Time-homogenized | 1.874898 × 10 | 1.652791 × 10 |

### Parameter identification

Parameter identification process

Noise level | Number of iterations | \(K_{id}\) (\(\text {SI Units}\)) | \(n_{id}\) |
---|---|---|---|

0 | 24 | 100.89 × 10 | 9.94 |

10% | 23 | 100.96 × 10 | 9.95 |

20% | 22 | 100.87 × 10 | 9.94 |

50% | 37 | 99.40 × 10 | 9.78 |

100% | 35 | 97.36 × 10 | 9.56 |

In addition, to demonstrate the robustness of the process, centered Gaussian noise is added to the synthetic data used in the identification strategy: different levels (corresponding to a standard deviation proportional to the mean of the measured displacement) are proposed, and the associated identification results are listed in Table 2. They show that the identification works, even for high levels of noise: the choice of the misfit function (48) with the application of the fast-time average to the noisy data has allowed to filter significantly the introduced noise, hence strongly limiting its impact on the identification results. This is a particularly interesting property, whereas identifying using the reference model and the whole set of noisy data undoubtedly would have led to poor results: indeed, as said above, the fact of using a homogenized model in a parameter identification process generally introduces a regularizing effect, as explained in Ref. [17]; unfortunately, even in this academic case of study, the numerical cost for solving the reference identification problem is too prohibitive to rigorously check this property.

Eventually, the computational cost associated with the identification process is significantly reduced when compared with what is obtained when the inverse problem related to the reference problem (11)–(15) is considered, which allows to address the case of slow-evolving phenomena, requiring the simulation over a large time interval.

## Conclusions

Here we have proposed a first preliminary study of a two-time-scale parameter identification process, using time-homogenized models: the adaptation of a classical identification strategy based on an adjoint state formulation to estimate the misfit function’s gradient can be used in this specific framework: this leads, on the proposed example, to the determination of the time-homogenized counterpart of the adjoint solution associated with the reference identification problem. Despite its simplicity, the academical example studied here showed the relevance of the strategy and its reduced computational cost: these results can be viewed as a first step before dealing with more complex cases of study.

To come back more specifically to CCF life prediction, a possible prospect could be to use the experimental information available (with classical high cycle fatigue tests for instance) to tune correctly the parameters of the considered material laws. In addition, to further improve the reduction in the computational cost, the extension to models with three different time scales, such as those described in Ref. [5], should be straightforward as well.

## Declarations

### Authors’ contributions

DA started the project and the initial coding. GP developed the current version of the code, analysed the data and wrote the manuscript. Both collaborated to develop the analyses and expertise on the different aspects of the model. Both authors read and approved the final manuscript.

### Acknowledgements

The authors would like to acknowledge the support of the European Commission through the project PREMECCY, described in [15], which allowed to initiate this study.

### Compliance with ethical guidelines

**Competing interests** The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Fatemi A, Yang L (1998) Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. Int J Fatigue 20(1):9–34View ArticleGoogle Scholar
- Guennouni T, Aubry D (1986) Réponse homogénéisée en temps de structures sous chargements cycliques. Comptes rendus de l’Académie des sciences. Série II 303(20):1765–1768 (in French)MATHMathSciNetGoogle Scholar
- Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. Elsevier, BurlingtonMATHGoogle Scholar
- Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, BerlinMATHGoogle Scholar
- Puel G, Aubry D (2014) Efficient fatigue simulation using periodic homogenization with multiple time scales. Int J Multiscale Comput Eng 12(4):291–318View ArticleGoogle Scholar
- Yu Q, Fish J (2002) Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading. Comput Mech 29(3):199–211MathSciNetView ArticleGoogle Scholar
- Shishkina EV, Blekhman II, Cartmell MP, Gavrilov SN (2008) Application of the method of direct separation of motions to the parametric stabilization of an elastic wire. Nonlinear Dyn 54(4):313–331MATHMathSciNetView ArticleGoogle Scholar
- Lévy A, Le Corre S, Poitou A, Soccard E (2011) Ultrasonic welding of thermoplastic composites: modeling of the process using time homogenization. Int J Multiscale Comput Eng 9(1):53–72View ArticleGoogle Scholar
- Dontsov EV, Guzina BB (2012) Acoustic radiation force in tissue-like solids due to modulated sound field. J Mech Phys Solids 60(10):1791–1813MATHMathSciNetView ArticleGoogle Scholar
- Papon A, Yin Z-Y, Riou Y, Hicher P-Y (2013) Time homogenization for clays subjected to large numbers of cycles. Int J Numer Anal Methods Geomech 37(11):1470–1491View ArticleGoogle Scholar
- Haouala S, Doghri I (2015) Modeling and algorithms for two-scale time homogenization of viscoelastic–viscoplastic solids under large numbers of cycles. Int J Plast 70:98–125View ArticleGoogle Scholar
- Aubry D, Puel G (2010) CCF modelling with use of a two-timescale homogenization model. In: Proceedings of the international fatigue conference (Fatigue 2010), vol 2. Procedia Engineering, pp 787–796
- Devulder A, Aubry D, Puel G (2010) Two-time scale fatigue modelling: application to damage. Comput Mech 45(6):637–646MATHView ArticleGoogle Scholar
- Puel G, Aubry D (2012) Material fatigue simulation using a periodic time homogenization method. Eur J Comput Mech 21(3–6):312–324Google Scholar
- PREMECCY (2012) Predictive methods for combined cycle fatigue in gas turbine blades. Final Project Report AST5-CT-2006-030889, European Commission—6th RTD Framework Program
- Puel G, Bourgeteau B, Aubry D (2013) Parameter identification of transient nonlinear models for the multibody simulation of a vehicle chassis. Mech Syst Signal Process 36(2):354–369View ArticleGoogle Scholar
- Chung C-B, Kravaris C (1988) Identification of spatially discontinuous parameters in second-order parabolic systems by piecewise regularisation. Inverse Probl 4(4):973–994MATHMathSciNetView ArticleGoogle Scholar
- Constantinescu A, Tardieu N (2001) On the identification of elastoviscoplastic constitutive laws from indentation tests. Inverse Probl Eng 9(1):19–44View ArticleGoogle Scholar
- Plessix R-E (2006) A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys J Int 167(2):495–503View ArticleGoogle Scholar
- Papon A (2010) Numerical modeling of soil behavior under very large numbers of cycles: time homogenization and parameter identification. Ph.D. thesis, Ecole Centrale de Nantes
**(in French)** - Dormand JR, Prince PJ (1980) A family of embedded Runge–Kutta formulae. J Comput Appl Math 6(1):19–26MATHMathSciNetView ArticleGoogle Scholar
- Weideman JAC (2002) Numerical integration of periodic functions: a few examples. Am Math Mon 109(1):21–36MATHMathSciNetView ArticleGoogle Scholar
- Kesavan S, Saint Jean Paulin J (1997) Homogenization of an optimal control problem. SIAM J Control Optim 35(5):1557–1573MATHMathSciNetView ArticleGoogle Scholar
- Coleman TF, Li Y (1994) On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds. Math Progr 67(1–3):189–224MATHMathSciNetView ArticleGoogle Scholar