Generative AI and digital twinning for the simulation of ductile/fragile fracture in segregated steel

 

 

Résilience des aciers de forge ségrégés. Etudes expérimentales et numériques réalisées dans le cadre du projet French Fab piloté par Anne-Françoise Gourgues.

 

PhD thesis 2021-2024 @ Mines ParisTech / PSL University. Simulation numérique avancée pour la compréhension des points bas en résilience des aciers forgés

 

https://www.mat.minesparis.psl.eu/Recrutements/Theses/Details/Simulation-numA-copy-rique-avancA-copy-e-pour-la-comprA-copy-hension-des-points-bas-en-rA-copy-silience-des-aciers-forgA-copy-s/37111

PhD thesis 2021-2024 @ Mines ParisTech / PSL University. Outils de prédiction de la rupture basés sur le couplage mesure de champs, calculs EFs et machine-learning

https://www.mat.minesparis.psl.eu/Recrutements/Theses/Details/-CONTRAT-DOCTORAL-Outils-de-prA-copy-diction-de-la-rupture-basA-copy-s-sur-le-couplage-mesure-de-champs-calculs-EFs-et-machine-learning/36638

CutFEM method to simulate composite fracture

Phase-field in the bulk, zero-thickness cohesive elements with friction contact at the matrix/inclusions interface

Multiple crack propagation with XFEM

Work of Dr Sutula, P. Kerfriden and S. Bordas, in collaboration with Soitec. 

 

D Sutula, P Kerfriden, T van Dam, SPA Bordas, Minimum energy multiple crack propagation. Part I: Theory and state of the art review, Engineering Fracture Mechanics 191, 205-224, 2018

D Sutula, P Kerfriden, T van Dam, SPA Bordas, Minimum energy multiple crack propagation. Part-II: Discrete solution with XFEM, Engineering Fracture Mechanics 191, 225-256, 2018

Bayesian optimisation for the selection of representative load paths in computational material testing (best UK Ph.D. in Comp. Mech. for Dr Olivier Goury)

The aim of computational material testing is to obtain the relationship between forces and extensions of a complex material through simulations, given some information about its microstructure. This is conceptually similar to pulling on a specimen experimentally, and reporting the force that needs to be applied to obtain an overall deformation of the material. Of course, replacing this costly experimental setup by simulations allows practitioners to investigate the use of new materials before manufacturing them, to play with the microstructure (e.g. varying the fiber content in advanced fiber-reinforced concrete) in order to design a material that fits particular engineering needs, or to test and control the reaction of the material when used for the design of a complex, multiscale engineering system.

However, computational material testing (or computational homogenisation), is very demanding in terms of computational resources. Typically, the material model needs to be solved at every material point of the engineering system of interest, which can be arbitrarily large. 

The solution that we have been investigating to alleviate this issue is to develop efficient Model order Reduction technique. In an offline stage, the material is tested using pre-defined loading scenarios, which leads to a set of particular mechanical states (the snapshot). Then, when a response of the material is required by the engineer for design purposes (i.e. online), it is obtained by performing an optimal interpolation in the space spanned by these pre-computed states, which reduces the overall computational cost by orders of magnitude.

Figure 1: Model order reduction for computational material testing 

The choice of the pre-defined load scenarios is key to the success of this approach. All the states that will need to be predicted should be sufficiently well represented by the states contained in the snapshot. At the present stage of our research, we consider that all potential load cases are equally likely. Although very general, this framework leads to an immense space of mechanical states to explore and represent.

Figure 2: random load-path generation, in 2D.
In 3D the paths belong to an hypercube of dimension 6

One first approach to explore this space of likely states is to choose random load paths (see Figure 2). In the case of damage mechanics, we constrain the random generation to follows states of strictly increasing energy dissipation, thereby constraining the generator to explore non-trivial (i.e. damaged) mechanical states. The reduced order models obtained by applying this simple idea are surprisingly efficient and reliable in our numerical test cases (simple elastic damageable multiphase representative volume element).

Our second approach is more advanced and aims at providing a quasi-optimal family of load-cases. The idea is to locate the load case that is the most incorrectly predicted when using previously generated mechanical states, and iterate until the accuracy of the reduced order model is acceptable. This data-driven approach is very appealing but requires a measure of goodness of prediction over all potential load-cases, which is difficult to (i) define and (ii) obtain at reasonable numerical costs. We have tackled the affordability issue by adopting:

1. A hierarchical description of the load paths using adaptive shape functions in time: start with proportional loading, then introduce an increasing number of kinks at arbitrary location on the load path (see figure 3)



Figure 3: Automatic path-generation through probabilistic worst-case
scenario detection (left: proportional loading, right:
space of load-paths containing a unique kink).

2. A Bayesian optimisation algorithm to detect the case of worst prediction for a given approximate description of potential load-cases. Precisely, we compute the residual of the governing equations at points chosen quasi-optimally using Gaussian process interpolation and the associated notion of maximum probability of improvement. Subsequently, we establish a relationship between a measure of this residual and the goodness of prediction, through a probabilistic regression technique.

This new approach to load paths selection provides some confidence on the accuracy of the constructed reduced model (we do not leave unexplored regions in the space of potential mechanical states), but it is significantly more expensive than the random generation approach: we must pay for reliability, in particular in the context of complex, nonlinear problems. This overhead may decrease when considering more complex material behaviours (visco-plasticity, anisotropy), owing to the fact that the random generator may require more time to detect small outliers in the space of potential mechanical states. 

More details about this research can be found in the Ph.D. thesis of Dr Olivier Goury.

Goury, O., Amsallem, D., Bordas, S.P.A. et al. Comput Mech (2016) 58: 213. https://doi.org/10.1007/s00466-016-1290-2

Automatic generation of the geometry of composite laminates from CAD data within an Isogeometric Analysis context

The CAD geometry of composite panels is usually described by a surface. In order to analyse the stress concentrations due to material heterogeneities (for instance to predict delamination), a 3D geometrical model has to be created.

We developed a reliable algorithm to create 3D layers of materials, given the mid-surface description in the form of NURBS. We showed that this technology could be easily used within the Isogeometric Analysis context for the damage assessment of thin panels directly from CAD.

Simulations by V.P. Nguyen, P. Kerfriden and S.P.A. Bordas.

Funding: FP7 ITN INSIST

https://doi.org/10.1016/j.compositesb.2013.12.018

Hybrid domain decomposition / model order reduction approach

From the Ph.D. thesis of Olivier Goury

Extraction of low-dimensional subspaces for the representation of unknown fields in stochastic fracture mechanics.

Our previous research has shown that Model Order reduction could help alleviate the computational burden associated with the simulation of propagating damage in composite structures. However, the region where damage localises (i.e. crack tip) must be treated using a full order model (i.e. no reduction). This is not necessarily a huge limitation. Indeed, damaged regions are very localised in a complex engineering system (e.g. vicinity of joints, holes or other stress concentrators).

The key point of this research is to find the region that must be excluded from the reduction. While our previous investigations used empirically defined damaged region, basically spheres on the materials points exhibiting maximum energy dissipation), we propose here an algebraic interpretation of the process zone: this will be defined as a region of minimum measure that needs to be excluded from the reduction technique in order to obtain approximate numerical predictions up to a given level of accuracy. Typically, the process zone becomes the region where the mechanical state changes rapidly with respect to parameter variations (i.e. chaotic behaviour).

We developed an algorithm to extract the algebraic process zone, which we called progressive Proper orthogonal decomposition. In essence, this is a Greedy algorithm that automatically finds a domain of given size over which a Singular Value Decomposition converges optimally (i.e. maximum correlation in the data). The size of this domain, together with the order of truncation of the SVD, is then selected in order to maximise the computational speed-up.

The algorithm has been applied successfully to the prediction of propagating damage in random media.

Local/Global model order reduction techniques for fracture

The idea of local/global model order reduction is to apply reduced basis techniques in the region of the domain where they are valid. Elsewhere, direct numerical simulation is performed.

  

In the case of quasi-brittle fracture, reduced spaces are unable to represent the solution in the process zone. The successive damage states need to be represented accurately, and are unfortunately not correlated to one another.

The process zones are extracted by an empirical procedure (a zone of predefined size around the points of the domain undergoing maximum damage rate). A reduced model of the complementary domain is built by a projection of the initial model in a reduced space spanned by a few snapshot-POD basis vectors (figure above). In the process zone itself, the reduced space is not completely discarded, as it contains valuable information about the solution. Instead, it is used as a coarse-grid preconditioner for an iterative algorithm to the direct solution. The iterative algorithm coupling the two solvers is illustrated in the figure below

We are currently investigating an algebraic extraction of the process zone, such that problem-dependence is avoided.

"On-the-fly" Reduced Order Modelling approach for the simulation of damage propagation in heterogeneous media

This research establishes a bridge between POD-based model order reduction techniques and the classical Newton/Krylov solvers. This bridge is used to derive an efficient algorithm to correct, “on-the-fly”, the reduced order modelling of highly nonlinear problems undergoing strong topological changes. Damage initiation problems are addressed and tackle via a corrected hyperreduction method. It is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm, even when strong topological changes are involved. 

Error controlled time-stepping procedure for nonlinear fracture

 

A very simple procedure to adapt "ont-the-fly" the time-stepping schemes in the simulation of delamination [Allix et al. 2010]. A statically admissible solution is reconstructed by interpolation between two sucessive time solutions. A measure of the non-verification of the constitutive law of this reconstructed solution over each time interval provides a valid error estimate for the time discretisation scheme. It is used to adapt the load steps (value of the arc-length parameter in this particular case).

Simulation of delamination in a bolted joint

An application of the multiscale domain decomposition method to joints in laminate. The scientific bottleneck in this work was the cost of solving the coarse-grid problem of the domain decomposition approach. We used the ideas developed by Dr. Gosselet and Prof. Rey [Gosselet and Rey 2003] on the acceleration of Krylov solvers to fasten the solution process.

Section of the deformed bolted joint when loaded in traction.

One of the practical lessons of this work was that the whole design process needs to be scalable on a parallel architecture. An efficient parallel solver is virtually useless if the pre and post-processing steps cannot handle large distributed data.

A nice video for the simulation of a smaller joint.

Simulations by Pierre Kerfriden

Multiscale simulation of delamination in composite laminates

This work makes use of the cohesive zone models developed in LMT Cachan by Prof. Allix. [Allix et al. 1992] to simulate inter-laminate crack propagation. The LaTIn domain [Ladeveze et al. 2002] decomposition solver permits to handle the nonlinear interface constitutive law efficiently. In a nutshell, the cohesive behaviour is lumped into the interfaces of the domain decomposition method. The resulting problems for each substructure are linear and can be factorised once for all at the start of the simulation. However, this process raises difficulties. The convergence rate of the iterative algorithm of the LaTIn domain decomposition approach is affected by the damage state of the cohesive interfaces. One of the major contribution of the work was to adapt this algorithm to the local nonlinearities, which enabled to retrieve the expected numerical efficiency [Kerfriden et al. 2009].

The very rich coarse scale problem of the LaTin domain decomposition method limits the scalability of the method. A Schur-based domain decomposition solver is developed to overcome this limit [Kerfriden et al. 2009].