Hyperreduction framework for model calibration in plasticityinduced fatigue
 David Ryckelynck^{1}Email author and
 Djamel Missoum Benziane^{1}
DOI: 10.1186/s4032301600686
© The Author(s) 2016
Received: 15 October 2015
Accepted: 9 April 2016
Published: 4 May 2016
Abstract
Background:
Many mechanical experiments in plasticityinduced fatigue are prepared by the recourse to finite element simulations. Usual simulation outputs, like local stress estimations or lifetime predictions, are useful to choose boundary conditions and the shape of a specimen. In practice, many other numerical data are also generated by these simulations. But unfortunately, these data are ignored, although they can facilitate the calibration procedure. The focus of this paper is to illustrate a new simulation protocol for finiteelement model calibration. By the recourse to hyperreduction of mechanical models, more data science is involved in the proposed protocol, in order to solve less nonlinear mechanical equations during the calibration of mechanical parameters. Usually, the location of the crack initiation is very sensitive to the heterogeneities in the material. The proposed protocol is versatile enough in order to focus the hyperreduced predictions where the first crack is initiated during the fatigue test.
Methods:
In this paper, we restrict our attention to elastoplasticity or elastoviscoplasticity without damage nor crack propagation. We propose to take advantage of the duration of both the experiment design and the experimental protocol, to collect numerical data aiming to reduce the computational complexity of the calibration procedure. Until experimental data are available, we have time to prepare the calibration by substituting numerical data to nonlinear equations. This substitution is performed by the recourse to the hyperreduction method (Ryckelynck in J Comput Phys 202(1):346–366, 2005, Int J Numer Method Eng 77(1):75–89, 2009). An hyperreduced order model involves a reduced basis for the displacement approximation, a reduced basis for stress predictions and a reduced integration domain for the setting of reduced governing equations. The reduced integration domain incorporates a zone of interest that covers the location of the crack initiation. This zone of interest is updated according to experimental observations performed during the fatigue test.
Results:
Bending experiments have been performed to study the influence of a grain boundary on AM1 superalloy oligocyclic fatigue at high temperature. The proposed hyperreduction framework is shown to be relevant for the modeling of these experiments. To account for the microstructure generated by a real industrial casting process, the specimen has been machined in a turbine blade. The model calibration aims to identify the loading condition applied on the specimen in order to estimate the stress at the point where the first crack is initiated, before the crack propagation. The model parameters are related to the load distribution on the specimen. The calibration speedup obtained by hyperreduction is almost 1000, including the update of the reduced integration domain focused on the experimental location of the crack initiation. The related electricenergy saving is 99.9 %.
Keywords
Materials informatics Data science Model inversion Hyperreduction POD Calibration protocol Energy consumptionBackground
In the calibration framework, while the design of the experimental setup is performed, the parameter space is sampled according to few points in the parameter space. These sampling points are denoted by \(({\varvec{\mu }}_j)_{j=1}^m\). This is an ideal framework to practice empirical approaches to model reduction such as the proper orthogonal decomposition (POD) [24, 25], the reduced basis method [26], the APHR method [27]. These methods are qualified as empirical, because the reduced vectors are extracted from simulations results by considering these results as numerical data. The physics is in the simulation results, not in the extraction procedure conversely to reduced basis given by normal modes. But empirical approaches have proven their computational efficiency. For given simulation data, the POD method is generated by using a singular value decomposition of known FE solutions stored in a matrix [28]. In case of FE models having a large number of degrees of freedom (more than 100,000) and a large number of time steps (more than 50), we perform the computation of \(\mathbf {V}\) by the incremental algorithm proposed in [29].
In nonlinear mechanics of materials, the Galerkin projection does not provide sufficient simulation speedup during the online step, except when using the PGD method as proposed in [38]. As shown in [27, 39–43], the repeated evaluations of \(\mathbf {V}^T \, \mathbf {R}^n\) involved in the projection of the FE equations into the reduced space scale with \(\mathcal {N}\). It is often too much time consuming. In this paper, we reduce this complexity by using the hyperreduction method [9, 27]. With this method the reduced equations are setup on a reduced integration domain (RID) which is a subdomain of \(\Omega \). Then, the constitutive equations are evaluated only over the RID. Therefore, the stresses are not predicted outside of the RID. If the RID does not contain the point where the crack is initiated during the fatigue test, then we will have missing simulation outputs for the calibration of the life duration criterion. This paper aims to propose a convenient solution to this issue.
Methods
The RID receives the contribution of empirical modes and it is supplemented by a zone of interest. The former is a subdomain denoted by \(\Omega ^\psi \), the latter is a subdomain denoted by \(\Omega ^I\). In the proposed versatile hyperreduction approach, \(\Omega ^\psi \) is generated during the offline step of the calibration protocol, but \(\Omega ^I\) is chosen during the online step according to the experimental location of the crack initiation, as shown in the flowchart in Fig. 1. It enables a versatile approach to calibration by hyperreduction in the framework of plasticityinduced fatigue. The hyperreduction method aims at preserving the usual assembly loop on elements when computing the FE residuals. Such an approach facilitates the hyperreduction of various kind of nonlinear constitutive equations in mechanics. The RID is denoted by \(\Omega ^Z\). It is a collection of few elements of the original FE mesh, termed “reduced mesh”. The mesh downloaded in computer memory for the hyperreduced predictions is the reduced mesh, not the full original FE mesh.
Results and discussion on a bending specimen in cristal plasticity
In [6], bending experiments have been performed by Mélanie Leroy to study the influence of a grain boundary on AM1 superalloy oligocyclic fatigue at high temperature. Here the temperature field is not uniform over the specimen, but it does not vary during the load cycles. We matter about the grainboundary strength, and not about the weakest part of the turbine blade. Then, to account for the microstructure generated by a real industrial casting process, the specimen has been machined in a turbine blade. The model calibration aims to identify the loading condition applied on the specimen in order to estimate the stress at the point where the first crack was initiated, before the crack propagation. The model parameters are related to the load distribution on the specimen. The numerical method has been implemented in the research software named Zset (http://www.zsetsoftware.com).
We have selected a blade involving two grains. The Euler angles of each grains are \((13.9, 5.6, 0.9^\circ )\) and \((64.2, 17.8, 84.0^\circ )\). As shown in Fig. 2, slots have been designed and machined on the turbine blade in order to amplify the stress concentration factor around the grain boundary. The position of the slots, the boundary conditions and the magnitude of the mechanical loading have been chosen by recourse to FE simulations. This preliminary work, before doing experiments, including the machining of the slots, took almost 3 months. This gave us time to conduct the numerical simulation S2 and the data mining. Both exact locations of the crack initiation and the grain boundary around the crack are revealed at the end of the fatigue test. Here the term exact must be understood as “at the scale of the local element size in the mesh”.
The constitutive equation of AM1 follows the crystal plasticity theory proposed in [48]. It accounts for the thermal expansion and thermal sensitivity of plasticity. The expected life duration of the specimen in the framework of oligocyclic fatigue should be around 10,000 cycles. This is a constrain to account for, when choosing the magnitude of the load. Before the experiment, the life duration of the specimen was estimated by neglecting the effect of the grain boundary on the life duration criterion.
Once the shape of the specimen has been fixed and validated by a linear elastic simulation, we have access to the prediction of the elastic stress \(a \,{\varvec{\sigma }}^n_e\), for all \(a \in \mathbb {R}\), at one sampling point \({\varvec{\mu }}_1 = [0.5, \, 0.5]\) in the parameter space. This simulation is named S0. Here, a is a variable determined in order to have an estimated life duration of the specimen about 10,000 loading cycles. For the validation of the value of a, an elastoplastic simulation has to be performed. This elastoplastic simulation is the simulation S1. A sufficient number of loading cycles should be considered in order to forecast a stabilized strainstress cycle at the weakest point of the specimen. Here, we have considered five loading cycles only. It generates intermediate numerical data, constituting more than 62 Go in the computer memory.
The simulation outputs, related to displacement at points A, B and C, occupy only 72 ko in computer memory. Hence, the usual FE calibration procedure creates and then deletes 99.9998 % of the numerical data generated by the FE simulations, without any data mining.
As shown in Fig. 8, the error committed during the last cycle is much smaller than the error with respect to the full time interval. For the last cycle, the infinite norm of the discrepancy on the local stresses in \(\widehat{\Omega }\) is 1 %. It is 0.5 % for the displacements on points A, B and C. Approximation errors are much higher during the heating of the specimen, more than 50 %.
The online construction of the RID takes only 30 s. The RID and \(\Omega ^\psi \) are shown in Fig. 9. In this figure, \(\Omega ^\psi \) does not provide stress prediction close to the location of the crack initiation, contrary to \(\Omega ^Z\). The zone of interest \(\Omega ^I\) contains only few elements around the location of the first crack, as revealed by the fatigue test. Far from the crack initiation, \(\Omega ^Z\) and \(\Omega ^\psi \) are identical. Hence the proposed protocol is really relevant for the prediction of the stresses that contribute to the crack initiation. The RID involves 2569 nodes and 1000 elements. \(\Omega ^\psi \) includes the elements below the loading forces \(F_a\), \(F_b\), \(F_c\), \(F_d\), and the points A, B and C related to simulation outputs. When downloading the reducedmesh in the computer memory, the hyperreduced predictions are very fast: 263 s. The simulation speedup is 931, compared to the FE predictions. Regarding the numerical data, each HR prediction generates 241 times less numerical data than the FE simulation. The reduced basis \(\widehat{\mathbf {V}}\) is 180 times less memory demanding than \(\mathbf {V}\). Hence, the HR simulations could have been done on a processor having less main memory. Furthermore, the computational time being 931 times shorter, the electric energy saving by HR predictions is almost 99.9 %.
The calibration process was performed by the recourse to 20 HR parametric simulations. Then, the optimal parameters have been validated by using an usual FE simulation.
Conclusions
Accessing to the data being faster than accessing to the solutions to nonlinear mechanical equations, we obtain very fast calibration of finite element models in heterogeneous plasticity.
Compared to parallel computing, hyperreduction is less accurate, but it provides large speedup for numerical simulations. Furthermore, it provides energy power saving that does not occur in parallel computing. Here, we save up to 99.9 % of energy thanks to the simulation speedup.
A high speedup of almost 1000 can be obtained by downloading a reduced mesh in the computer memory. If not, the speedup factor of hyperreduced simulations is about 30. This is mostly explained by the time needed to read and write data for elements that are not in the reduced integration domain, although no mechanical computation is performed on these elements.
A versatile approach to hyperreduction is proposed. Hence the location of the reduced integration domain accounts for both numerical data and experimental data related to the location of the crack initiation. In the proposed example, the online construction of the reduced integration domain takes only 11 % of the duration of one hyperreduced simulation.
In the usual calibration procedure, 99.9998 % of numerical data generated by the design of the experimental setup are wasted, although these data enable huge computational time savings and electric energy savings when using the hyperreduction method. Moreover, hyperreduced simulations are less demanding in computational ressources. In the proposed example, each HR prediction generates 241 times less numerical data than the FE simulation. And the reduced bases restricted to a reduced mesh are 180 times less memory demanding than the full reduced bases.
In future work, we must improve de hyperreduced prediction of specimen heating. Fortunately, in the proposed example, the approximation error committed during the heating did not have a significant effect on the accuracy of the mechanical response of interest. This situation was very convenient for the calibration of the load position applied to the specimen.
Abbreviations
 PDE:

partial differential equation
 RID:

reduced integration domain
 HR:

hyperreduced
 FE:

finite element
Declarations
Authors' contributions
The theory represents the work by DR. Numerical results represents joint work by all authors. Both authors read and approved the final manuscript.
Acknowledgements
This study was supported by the PRC Structures Chaudes and FUI MECASIF funded by the French government.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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
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