A Variational Formulation of Dissipative Quasicontinuum Methods

Lattice systems and discrete networks with dissipative interactions are successfully employed as meso-scale models of heterogeneous solids. As the application scale generally is much larger than that of the discrete links, physically relevant simulations are computationally expensive. The QuasiContinuum (QC) method is a multiscale approach that reduces the computational cost of direct numerical simulations by fully resolving complex phenomena only in regions of interest while coarsening elsewhere. In previous work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally conservative QC methodology was generalized to a virtual-power-based QC approach that includes local dissipative mechanisms. In this contribution, the virtual-power-based QC method is reformulated from a variational point of view, by employing the energy-based variational framework for rate-independent processes (Mielke and Roub\'i\v{c}ek, Rate-Independent Systems: Theory and Application, Springer-Verlag, 2015). By construction it is shown that the QC method with dissipative interactions can be expressed as a minimization problem of a properly built energy potential, providing solutions equivalent to those of the virtual-power-based QC formulation. The theoretical considerations are demonstrated on three simple examples. For them we verify energy consistency, quantify relative errors in energies, and discuss errors in internal variables obtained for different meshes and two summation rules.Comment: 38 pages, 21 figures, 4 tables; moderate revision after review, one example in Section 5.3 adde

Identifying elastoplastic parameters with Bayes’ theorem considering output error, input error and model uncertainty

We discuss Bayesian inference for the identification of elastoplastic material parameters. In addition to errors in the stress measurements, which are commonly considered, we furthermore consider errors in the strain measurements. Since a difference between the model and the experimental data may still be present if the data is not contaminated by noise, we also incorporate the possible error of the model itself. The three formulations to describe model uncertainty in this contribution are: (1) a random variable which is taken from a normal distribution with constant parameters, (2) a random variable which is taken from a normal distribution with an input-dependent mean, and (3) a Gaussian random process with a stationary covariance function. Our results show that incorporating model uncertainty often, but not always, improves the results. If the error in the strain is considered as well, the results improve even more.The authors would like to acknowledge the financial support from the University of Luxembourg and the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) entitled \Towards real time multiscale simulation of cutting in nonlinear materials with applications to surgical simulation and computer guided surgery

A Bayesian framework to identify random parameter fields based on the copula theorem and Gaussian fields: Application to polycrystalline materials

For many models of solids, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application approaches the length scale of the micro-structure however, spatially fluctuating parameter fields (which vary from one realization of the field to another) can be incorporated to make the model capture the stochasticity of the underlying micro-structure. Randomly fluctuating parameter fields are often described as Gaussian fields. Gaussian fields however assume that the probability density function of a material parameter at a given location is a univariate Gaussian distribution. This entails for instance that negative parameter values can be realized, whereas most material parameters have physical bounds (e.g. the Young’s modulus cannot be negative). In this contribution, randomly fluctuating parameter fields are therefore described using the copula theorem and Gaussian fields, which allow different types of univariate marginal distributions to be incorporated, but with the same correlation structure as Gaussian fields. It is convenient to keep the Gaussian correlation structure, as it allows us to draw samples from Gaussian fields and transform them into the new random fields. The benefit of this approach is that any type of univariate marginal distribution can be incorporated. If the selected univariate marginal distribution has bounds, unphysical material parameter values will never be realized. We then use Bayesian inference to identify the distribution parameters (which govern the random field). Bayesian inference regards the parameters that are to be identified as random variables and requires a user- defined prior distribution of the parameters to which the observations are inferred. For the homogenized Young’s modulus of a columnar polycrystalline material of interest in this study, the results show that with a relatively wide prior (i.e. a prior distribution without strong assumptions), a single specimen is sufficient to accurately recover the distribution parameter values.Hussein Rappel and Lars Beex gratefully acknowledge the financial support of the Fonds National de la Recherche Luxembourg under grant number INTER/DFG/16/11501927

A tutorial on Bayesian inference to identify material parameters in solid mechanics

peer reviewedThe aim of this contribution is to explain in a straightforward manner how Bayesian inference can be used to identify material parameters of material models for solids. Bayesian approaches have already been used for this purpose, but most of the literature is not necessarily easy to understand for those new to the field. The reason for this is that most literature focuses either on complex statistical and machine learning concepts and/or on relatively complex mechanical models. In order to introduce the approach as gently as possible, we only focus on stress-strain measurements coming from uniaxial tensile tests and we only treat elastic and elastoplastic material models. Furthermore, the stress-strain measurements are created artificially in order to allow a one-to-one comparison between the true parameter values and the identified parameter distributions.Hussein Rappel, Lars A.A. Beex and St ephane P.A. Bordas would like to acknowledge the nancial support from the University of Luxembourg. St ephane P.A. Bordas also thanks the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) entitled "Towards real time multiscale simulation of cutting in nonlinear materials with applications to surgical simulation and computer guided surgery". Jack S. Hale is supported by the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND Grant No. 6693582)

Generalized quasicontinuum modeling of metallic lattices with geometrical and material nonlinearity and variability

We propose a generalized quasicontinuum method to model the mechanical response of 3D lattice structures. The method relies on the spatial coupling of fully-resolved domains and coarse-grained domains. In the fully-resolved domain, the full micro-structure is taken into account. In the coarse-grained domain, the kinematics of the micro-structure are individually interpolated based on their connectivity. On top of that, the contributions of the microstructure to the governing equations in the coarse-grained domain are sampled using only a few unit cells. In both domains, geometrical and material variability along the strut can be naturally taken into account using a 3D co-rotational beam finite element with embedded plastic hinges. We verify the approach for BCC lattices, demonstrating that the new method can capture both material and geometrical non-linearities of single struts at a fraction of the cost of a direct numerical simulation

An implementation of the reduced assembly proper orthogonal decomposition method

Please see the Bitbucket link below for full information. We will archive the code here in due course