Generation of energy-minimizing point sets on spheres and their application in mesh-free interpolation and differentiation

Kunc, Oliver; Fritzen, Felix

doi:10.1007/s10444-019-09726-5

Cited by 11 publications

(16 citation statements)

References 36 publications

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“…Some point sets of various sizes are included in the example program [18]. More detailed investigations on this topic and an open-source code of a point generation program are part of another work, [34]. Alternatively, Equal Area Points [35] may be used as a rough but quickly computable approximation of such point sets.…”

Section: Algorithmmentioning

confidence: 99%

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Kunc

Fritzen

2019

MCA

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The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5-100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method.

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Section: Algorithmmentioning

confidence: 99%

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Kunc

Fritzen

2019

MCA

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“…Multiscale analysis of reinforced concrete is conducted in [51], where ANNs are used to approximate the stress vs. crack opening material response based on mesoscale simulations (see also [52]). Radial numerically explicit potentials have been developed in [53] for hyperelastic materials for the acceleration of two-scale problems, which can be improved by recent strategies for the generation of points on hyperspheres, see, e.g., [54]. In [55], ANNs are used in combination with FE simulations in order to capture the hydro-mechanical coupling of porous media.…”

Section: Machine Learning Methods As An Alternative To Materials Modelsmentioning

confidence: 99%

Application of artificial neural networks for the prediction of interface mechanics: a study on grain boundary constitutive behavior

Fernández

Rezaei

Mianroodi

et al. 2020

Adv. Model. and Simul. in Eng. Sci.

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The present work aims at the identification of the effective constitutive behavior of 5 aluminum grain boundaries (GB) for proportional loading by using machine learning (ML) techniques. The input for the ML approach is high accuracy data gathered in challenging molecular dynamics (MD) simulations at the atomic scale for varying temperatures and loading conditions. The effective traction-separation relation is recorded during the MD simulations. The raw MD data then serves for the training of an artificial neural network (ANN) as a surrogate model of the constitutive behavior at the grain boundary. Despite the extremely fluctuating nature of the MD data and its inhomogeneous distribution in the traction-separation space, the ANN surrogate trained on the raw MD data shows a very good agreement in the average behavior without any data-smoothing or pre-processing. Further, it is shown that the trained traction-separation ANN captures important physical properties and is able to predict traction values for given separations not contained in the training data. For example, MD simulations show a transition in traction-separation behaviour from pure sliding mode under shear load to combined GB sliding and decohesion with intermediate hardening regime at mixed load directions. These changes in GB behaviour are fully captured in the ANN predictions. Furthermore, by construction, the ANN surrogate is differentiable for arbitrary separation and also temperature, such that a thermo-mechanical tangent stiffness operator can always be evaluated. The trained ANN can then serve for large-scale FE simulation as an alternative to direct MD-FE coupling which is often infeasible in practical applications.

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“…By considering that the six components of a symmetric macrostrain tensor can be related to a point lying on the R 6 space, we choose 100 points uniformly distributed in the unitary hypersphere in R 6 [26], to define the same number of corresponding macrostrains. Macrostrains are imposed to the RVE multiplied by a time factor χ, monotonically increasing from 0 to χ end = 0.1 for model A, and χ end = 0.02 for model B.…”

Section: Samplingmentioning

confidence: 99%

High performance reduction technique for multiscale finite element modeling (HPR-FE2): Towards industrial multiscale FE software

Raschi

Lloberas-Valls

Huespe

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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The authors have shown in previous contributions that reduced order modeling with optimal cubature applied to finite element square (FE 2 ) techniques results in a reliable and affordable multiscale approach, the HPR-FE 2 technique. Such technique is assessed here for an industrial case study of a generic 3D reinforced composite whose microstructure is represented by two general microcells accounting for different deformation mechanisms, microstrucural phases and geometry arrangement. Specifically, in this approach the microstrain modes used for building the reduced order model (ROM) are obtained through standard proper orthogonal decomposition (POD) techniques applied over snapshots of a representative sampling strain space. Additionally, a reduced number of integration points is obtained by exactly integrating the main free energy modes resulting from the sampling energy snapshots. The outcome consists of a number of dominant strain modes integrated over a remarkably reduced number of integration points which provide the support to evaluate the constitutive behavior of the microstructural phases. It is emphasized that stresses are computed according to the selected constitutive law at the reduced integration points and, therefore, the strategy inherits advantageous properties such as model completeness and customization of material properties. Overall results are discussed in terms of the consistency of the multiscale analysis, customization of the microscopic material parameters and speedup ratios compared to high-fidelity finite element (HF) simulations.

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Generation of energy-minimizing point sets on spheres and their application in mesh-free interpolation and differentiation

Cited by 11 publications

References 36 publications

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Application of artificial neural networks for the prediction of interface mechanics: a study on grain boundary constitutive behavior

High performance reduction technique for multiscale finite element modeling (HPR-FE2): Towards industrial multiscale FE software

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