Conference Agenda

Many data-driven approaches in computational material engineering and mechanics rely on realistic volume elements for conducting numerical simulations. Examples include multiscale simulations based on neural networks or reduced-order models as well as the exploration and optimization of structure-property linkages. This motivates microstructure characterization and reconstruction (MCR). In previous contributions, MCRpy [1] has been introduced as a modular open-source tool for descriptor-based MCR, where any descriptors can be used for characterization and any loss function combining any descriptors can be minimized using any optimizer for reconstruction. A key feature of MCRpy is that differentiable descriptors are available and can be used in conjunction with gradient-based optimizers. This allows the underlying optimization problem to converge several orders of magnitude faster than with the previously used stochastic optimizers [2, 3]. While MCRpy and the gradient-based reconstruction have been presented in previous contributions for microstructures with multiple phases, the present contribution extends these concepts towards orientation maps.

After a brief introduction to MCRpy, the main difficulties of extending gradient- and descriptor-based microstructure reconstruction to orientation maps are discussed. Besides the symmetry of orientation itself after 360°, additional crystal symmetries need to be incorporated and singularities need to be avoided. For this reason, differentiable statistical descriptors are defined in terms of symmetrized harmonic basis functions defined on the 4D unit quaternion hypersphere [4]. Based on a generic combination of descriptors comprising two-point statistics of orientation information and the orientation variation, the optimization problem is defined in the fundamental region of a neo-Eulerian orientation space. These and other measures are motivated and discussed in detail. The capabilities of the method are demonstrated by exemplarily applying it to various microstructures. In this context, it is mentioned that all algorithms are made publicly available in MCRpy and it is demonstrated how to use them. Furthermore, it is shown how to extend MCRpy by defining a new microstructure descriptor in terms of any desired orientation representation or basis function and readily using it for reconstruction without additional implementation effort.

[1] Seibert, Raßloff, Kalina, Ambati, Kästner, Microstructure Characterization and Reconstruction in Python: MCRpy, IMMJ, 2022

[2] Seibert, Ambati, Raßloff, Kästner, Reconstructing random heterogeneous media through differentiable optimization, COMMAT, 2021

[3] Seibert, Raßloff, Ambati, Kästner, Descriptor-based reconstruction of three-dimensional microstructures through gradient-based optimization, Acta Materialia, 2022

[4] Mason, Analysis of Crystallographic Texture Information by the Hyperspherical Harmonic Expansion, PhD Thesis, 2009

In the context of homogenization approaches, data-driven methods entail advantages due to the ability to capture complex behaviour without the assumption of a specific material model. Constitutive model based data-driven methods approximating the constitutive relations by training artificial neural networks and the method of constitutive model free data-driven computational mechanics, directly incorporating stress-strain data in the analysis, are distinguished. Neural network based constitutive descriptions are one of the most widely used data-driven approaches in computational mechanics. In contrast to this, the method of distance minimizing data-driven computational mechanics enables to bypass the material modelling step entirely by iteratively obtaining a physically consistent solution, which is close to the material behaviour represented by the data. A generalization of this method providing increased robustness with respect to outliers in the underlying data set is the maximum entropy data-driven solver. Additionally, a tensor voting enhancement based on incorporating locally linear tangent spaces enables to interpolate in regions of sparse sampling.

In this contribution, a comparison of artificial neural networks and data-driven computational mechanics is carried out based on nonlinear elasticity. General differences between machine learning, distance minimizing as well as entropy maximizing based data-driven methods concerning pre-processing, required computational effort and solution procedure are pointed out. In order to demonstrate the capabilities of the proposed methods, numerical examples with synthetically created datasets obtained by numerical material tests are executed.

The accurate bending of sheet metal structures is critical in a variety of industrial and scientific contexts, whether it is to modify existing components or achieve specific shapes. Laser peen forming (LPF) is an advanced process for sheet metal applications that involves using mechanical shock waves to deform a specific area to a desired radius of curvature. The degree of deformation achieved through LPF is affected by various experimental factors such as laser energy, the number of peening sequences, and specimen thickness. Therefore, it is important to understand the complex dependencies and select the appropriate LPF process parameters for forming or correction purposes. This study aims to develop a data-driven approach to predict the deformation obtained from LPF for different process parameters. The experimental data is used to train, validate, and test an artificial neural network (ANN). The trained ANN successfully predicted the deformation obtained from LPF. An innovative process planning approach is developed to demonstrate the usability of ANN predictions in achieving the desired deformation in a treated area. The effectiveness of this approach is demonstrated on three benchmark cases involving thin Ti-6Al-4V sheets: deformation in one direction, bi-directional deformation, and modification of an existing deformation in pre-bent specimens via LPF.

Crystal plasticity is the result of the motion of line like crystal defects, the dislocations. While many traits of crystal plasticity may be described by phenomenological models, the description of the well-known patterning of dislocations as well as the phenomenon of single crystal work-hardening caused by dislocation multiplication during plastic deformation, ask for continuum models rooted more directly in the collective behavior of dislocations. A promising homogenization approach in this realm is the so-called Continuum Dislocation Dynamics (CDD) framework, which is based on conservation laws for tensorial dislocation density measures. In other words, the CDD theory can be considered as a continuum representation of dislocation networks through a hierarchy of tensorial dislocation variables. [1]

In this work, we derive nonlinear expressions for source terms as required in CDD for modeling work-hardening, which is arguably the most salient feature of metal-plasticity. [2] For that purpose we use modern data-driven discovery methods, like the Sparse Identification of Nonlinear Dynamics (SINDy), to describe the highly nonlinear dynamics of dislocation multiplication. The SINDy algorithm is capable of identifying the few predominant terms in the corresponding governing equations based on a model library of predefined, possibly high-dimensional spaces of nonlinear functions using sparse regression techniques. [3]

The SINDy algorithm is applied on a large database of Discrete Dislocation Dynamics (DDD) simulations of the plastic deformation of FCC single crystalline copper under constant strain rate in 120 different loading directions with neglected cross-slip. The extraction of the underlying data of dynamic CDD tensor variables, consisting of density, curvature and velocity tensors of n-th order, from the DDD data is performed by a recently developed algorithm.

References

[1] Thomas Hochrainer, S. Sandfeld, M. Zaiser, and P. Gumbsch. Continuum dislocation dynamics: towards a physically theory of plasticity. Journal of the Mechanics and Physics of Solids, 63(1):167–178, 2014.

[2] Markus Sudmanns, Markus Stricker, Daniel Weygand, Thomas Hochrainer and Katrin Schulz. Dislocation multiplication by cross-slip and glissile reaction in a dislocation based continuum formulation of crystal plasticity. Journal of the Mechanics and Physics of Solids, 132:103695, 2019.

[3] Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 113(15):3932–3937, 2016.

The crude Monte Carlo method delivers an unbiased estimate of the probability of failure. However, the accuracy of the approach, i.e., the variance of the estimate, depends on the number of evaluated samples. This number must be very large for estimations of a low probability of failure. If the evaluation of each sample is computationally expensive, the crude Monte Carlo simulation strategy is impracticable. To this end, subset simulations are used to reduce the required number of evaluations. Subset simulations require a Markov Chain Monte Carlo sampler, e.g., the random walk Metropolis-Hastings algorithm [1]. The algorithm, however, struggles with sampling in low-probability regions, especially if they are narrow. Therefore, advanced Markov Chain Monte Carlo simulations are preferred. In particular, the Hamiltonian Monte Carlo method explores the target distribution space rapidly. Driven by Hamiltonian dynamics, this sampler provides a non-random walk through the target distribution [2]. The incorporation of subset simulation and Hamiltonian Monte Carlo methods has shown promising results for reliability analysis [3]. However, gradient evaluations in the Hamiltonian Monte Carlo method are computationally expensive, especially when dealing with high-dimensional problems and evaluating long trajectories. Integrating Hamiltonian Neural Networks in Hamiltonian Monte Carlo simulations significantly speeds up the sampling [4]. The extension to latent Hamiltonian neural networks improves the expressivity by adding neurons to the last layer. Furthermore, the enhancement of the No U-Turns Sampler to the Hamiltonian Monte Carlo results in the efficient proposal of the following states [5]. During the exploration of low-probability regions, an online error monitoring calls the standard NUTS sampler if the latent Hamiltonian Neural Network estimates are inaccurate. Based on this recent enhancement, we provide an efficient sampling strategy for subset simulations using latent Hamiltonian neural networks to replace the gradient calculation and speed up the Hamiltonian Monte Carlo simulation.

[1] W.K. Hastings. Biometrika 57 (1970) 97-109.

[2] M. Betancourt. arXiv preprint, arXiv:1701.02434 (2017).

[3] Z. Wang, M. Broccardo, J. Song. Struct. Saf. 76 (2019) 51-67.

[4] D. Thaler, S.L.N. Dhulipala, F. Bamer, B. Markert, M.D. Shields. Proc. Appl. Math. Mech. (2023);22:e202200188.

[5] S.L.N. Dhulipapla, Y. Che, M.D. Shields. arXiv preprint, arXiv:2208.06120v1 (2022).

Artificial intelligence (AI) applications have recently gained widespread attention due to their capabilities in the domains of speech and image recognition as well as natural language processing. This has drawn research attention towards AI and artificial neural networks (ANNs) in particular within numerous branches of applied mathematics and computational mechanics. The challenge of generating extensive training data for supervised learning of ANNs can be addressed by incorporating laws of physics into ANNs. Most of so-called physics informed neural network (PINN) [1] frameworks for structural mechanics applications incorporate the partial differential equations (PDEs) governing a specific problem within the loss function in the form of energy methods [2] or collocation methods [3].

Many structural mechanics problems are governed by stiff PDEs resulting in locking effects which have already been recognized in the early days of finite element analysis. Locking effects are present for all known discretization schemes, not only for finite elements, independent of the polynomial order or smoothness of the shape functions. This applies to both Galerkin-type solution methods and also collocation methods based on the Euler-Lagrange equations of the specific boundary value problem [4].

In this contribution, we examine the impact of stiff PDEs or locking effects on the accuracy and efficiency of PINN-based numerical solutions of problems in structural mechanics. First investigations on the use of PINNs for solving shear deformable beam and plate problems are presented. Different types of beam and plate formulations, as well as different types of collocation-based loss functions are evaluated and compared with respect to accuracy and efficiency.

REFERENCES

[1] M. Raissi, P. Perdikaris, G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, Vol. 378, pp. 686-707. 2019

[2] E. Samaniego, C. Anitescu, S. Goswami, V.M. Nguyen-Thanh, H. Guo, K. Hamdia, X. Zhuang, T. Rabczuk. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, Vol. 362, 112790. 2020

[3] H. Guo, X. Zhuang, T. Rabczuk. A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate. Computers Materials & Continua. Vol. 59(2), pp. 433-456. 2019

[4] B. Oesterle, S. Bieber, R. Sachse, E. Ramm, M. Bischoff. Intrinsically locking-free formulations for isogeometric beam, plate and shell analysis. Proc. Appl. Math. Mech. 2018, 18:e20180039. 2018