Conference Agenda

Multiscale methods are often employed to study the effect of microstructure on macroscopic behaviour. For non-linear problems, these usually result in a two-scale formulation, where macro- and microstructure are simultaneously solved and coupled. If the microstructural features are much smaller compared to the macrostructural size, its effective behavior can be sufficiently predicted with first-order computational homogenization. However, if scale separation cannot be assumed or non-local effects due to buckling, softening, etc., emerge, higher-order methods, such as second-order homogenization [1], need to be considered. This formulation contains the second gradient of the displacement field, giving rise to a length-scale associated with the length-scale of the underlying unit cell, thus making it possible to capture size and non-local effects. Solving such problems is currently computationally expensive and typically infeasible for realistic applications, which limits the applicability of this method.

In this work, we address this issue by developing a reduced order model for second-order computational homogenization scheme based on Proper Orthogonal Decomposition. We consider different numerical examples and discuss different training strategies, computational savings and accuracy of the surrogate model.

Acknowledgements: This result is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 818473).

[1] Kouznetsova, V., Geers, M.G.D. and Brekelmans, W.A.M. (2002), Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Meth. Engng., 54: 1235-1260. https://doi.org/10.1002/nme.541

Manifold learning techniques such as Laplacian Eigenmaps (LE) [1] are commonly applied in fields like image and speech processing, to extract nonlinear trends from large sets of high-dimensional data [2]. Such techniques are also intriguing as model order reduction methods in multiscale solid mechanics: LE can capture nonlinearities in the solution manifolds of discretised physical problems [3]. Compared to POD-based algorithms, this may result in reduced order models yielding more accurate results with fewer parameters and lower computational effort [3]. Consequently, manifold learning techniques have been applied successfully to problems in fluid mechanics [3] and elastodynamics [4].

In the framework of the FE² method – in which computations on microscale representative volume elements (RVEs) are performed at each Gauss point of a macroscopic problem [5] – the payoff of such computational cost reduction may also be significant.

This contribution discusses the application of LE to model order reduction for RVE computations. Nonlinear, hyperelastic and elastoplastic behaviour is considered. The area of application comes with unique challenges and opportunities: for example, the mapping between reduced and original spaces and the projection of residuals onto reduced bases is not trivial [3]. On the other hand, the underlying PDEs [4] and the parametrisation of the RVE problem via a macroscopic deformation gradient and history variables [5] imply a strong (nonlinear) correlation in the unknown displacement degrees of freedom to be reduced. This talk will explore some of these challenges and opportunities.

[1] Belkin, Mikhail, and Partha Niyogi. "Laplacian eigenmaps for dimensionality reduction and data representation." Neural computation 15, no. 6 (2003): 1373-1396.

[2] Lee, John A., and Michel Verleysen. Nonlinear dimensionality reduction. Vol. 1. New York: Springer, 2007.

[3] Pyta, Lorenz Matthias. "Modellreduktion und optimale Regelung nichtlinearer Strömungsprozesse." PhD diss., Dissertation, RWTH Aachen University, 2018.

[4] Millán, Daniel, and Marino Arroyo. "Nonlinear manifold learning for model reduction in finite elastodynamics." Computer Methods in Applied Mechanics and Engineering 261 (2013): 118-131.

[5] Schröder, Jörg. "A numerical two-scale homogenization scheme: the FE 2-method." Plasticity and beyond: microstructures, crystal-plasticity and phase transitions (2014): 1-64.

Plasticity is the result of the motion and interaction of discrete dislocations in a crystalline material. Modelling plasticity at the crystal level based on discrete dislocation dynamics (DDD) is challenging due to the complexities associated with the dislocation activities of different slip planes. A data-driven approach provides an alternative method for simulating the complex behavior associated with plasticity at a small scale. We use methods based on dynamic mode decomposition1 (DMD) to analyze the DDD data2. We built reduced-order models for describing system dynamics with a few dominant modes. The models are built upon datasets of different physical resolution, e.g. dislocation density information resolved on the slip system level based on total dislocation densities, dislocation density vectors, or second-order dislocation alignment tensors. Different levels of spatial resolution are used to evaluate the effectiveness of models in reconstruction of the analysed data.

The modelling approach is then extended to forecast material response beyond the training dataset, for which we adopt more general (non-linear) Koopman operator theory and advanced stabilized DMD schemes, like shift invariant (physically informed) DMD or optimized DMD3. The different schemes are compared in their ability to predict the nonlinear behaviour in crystal plasticity from the DDD data.

REFERENCES

[1] Schmid, Peter J. "Dynamic mode decomposition of numerical and experimental data." Journal of fluid mechanics 656 (2010): 5-28.

[2] Akhondzadeh, Sh, Ryan B. Sills, Nicolas Bertin, and Wei Cai. "Dislocation density-based plasticity model from massive discrete dislocation dynamics database." Journal of the Mechanics and Physics of Solids 145 (2020): 104152.

[3] Askham, Travis, and J. Nathan Kutz. "Variable projection methods for an optimized dynamic mode decomposition." SIAM Journal on Applied Dynamical Systems 17, no. 1 (2018): 380-416.

The availability of high quality µ-CT images of materials allows for detailed multiscale simulation workflows in digital material characterization. In this case, data driven hybrid machine learning approaches are used to speed up full field approaches. Efficient and performance implementations of such data driven methods are essential for them being used for industrial applications.

This work concerns DMN (Deep Material Network) whose potential applications were exploited recently [1,2,3]. They only need linear elastic training data to identify equivalent laminate microstructure, which can be used to predict nonlinear behavior.

The industrial applicability of the DMN for short fiber reinforced plastic is investigated by comparing its speed and accuracy against direct numerical simulation results[4,5] on RVEs[6] using different physically nonlinear material behavior.

[1]- Liu, Z., Wu, C., & Koishi, M. (2019). A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials. Computer Methods in Applied Mechanics and Engineering, 345, 1138–1168.

[2]- Liu, Z., & Wu, C. (2019). Exploring the 3D architectures of deep material network in data-driven multiscale mechanics. Journal of the Mechanics and Physics of Solids, 127, 20–46.

[3]- Gajek, S., Schneider, M., & Böhlke, T. (2020). On the micromechanics of deep material networks. Journal of the Mechanics and Physics of Solids, 142, 103984.

[4]- Matthias Kabel, Dennis Merkert, & Matti Schneider (2015). Use of composite voxels in FFT-based homogenization. Computer Methods in Applied Mechanics and Engineering, 294, 168-188.

[5]- Matthias Kabel, Andreas Fink, & Matti Schneider (2017). The composite voxel technique for inelastic problems. Computer Methods in Applied Mechanics and Engineering, 322, 396-418.

[6]- Schneider, M. (2022). An algorithm for generating microstructures of fiber-reinforced composites with long fibers. International Journal for Numerical Methods in Engineering, 123(24), 6197-6219.

Modeling for the description and prediction of processes in nature often leads to partial differential equations. Solving these field equations can only be done analytically in very few cases, so that in practice numerical approximation methods are often used. Variational methods like the Galerkin method have proven to be very effective and are widely used in industry and research. To set up the system of equations, integration over the area to be calculated is necessary. For more complex geometries or nonlinear equations, analytical integration becomes difficult or even infeasible, so that integration is also often performed numerically in the form of weighted evaluations of the integrand, the Gauss quadrature. In order to benefit from the quasi-optimal accuracy of the Galerkin method according to Cea’s lemma in the linear case, the quadrature scheme must also be of sufficient accuracy. On the contrary, for more complex constitutive laws, under-integration is often used in engineering to save computational time. Based on a split of geometric and material nonlinearities, the present talk introduces a one-point integration scheme that is able to integrate polynomial shape functions of arbitrary order geometrically accurate. The material nonlinearity can be captured with the desired accuracy via a Taylor series expansion from the nonlinear state. As a demonstration the integration scheme is applied to two-dimensional polygonal shaped second order virtual elements where the quadratic projection is integrated via a single integration point.