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

in Journal of Applied Mechanics (in press)

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 ... [more ▼]

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 fi elds (which vary from one realization of the fi eld 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 fi elds 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 fi elds are therefore described using the copula theorem and Gaussian fi elds, which allow di fferent 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 fi elds and transform them into the new random fields. The bene fit 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 fi eld). Bayesian inference regards the parameters that are to be identi fied as random variables and requires a user-defi ned 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 su ciffient to accurately recover the distribution parameter values. [less ▲]

An adaptive multiscale quasicontinuum approach for mechanical simulations of elastoplastic periodic lattices

in Mechanics Research Communications (2022), 126

The quasicontinuum method is a multiscale method that combines locally supported coarse-grained domains, with small regions in which the microstructural model is fully resolved. This contribution proposes ... [more ▼]

The quasicontinuum method is a multiscale method that combines locally supported coarse-grained domains, with small regions in which the microstructural model is fully resolved. This contribution proposes the first adaptive formulation of the method for microstructural elastoplasticity. The microstructural model uses an elastoplastic beam description. The indicator for refinement is the occurrence of plastic deformation, such that plasticity can only occur in fully resolved regions. An illustrative numerical example of a scratch test of an elastoplastic Kelvin lattice demonstrates the capabilities of the resulting framework. [less ▲]

A hyper-reduction method using adaptivity to cut the assembly costs of reduced order models

in Computer Methods in Applied Mechanics and Engineering (2021), 380

At every iteration or timestep of the online phase of some reduced-order modelling schemes, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The ... [more ▼]

At every iteration or timestep of the online phase of some reduced-order modelling schemes, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The projected small linear systems are cheap to solve, but assembly and projection are now the dominant computational cost. In this paper we introduce a new hyper-reduction strategy called reduced assembly (RA) that drastically cuts these costs. RA consists of a triangulation adaptation algorithm that uses a local error indicator to con- struct a reduced assembly triangulation specially suited to the reduced order basis. Crucially, this reduced assembly triangulation has fewer cells than the original one, resulting in lower assembly and projection costs. We demonstrate the efficacy of RA on a Galerkin-POD type reduced order model (RAPOD). We show performance increases of up to five times over the baseline Galerkin-POD method on a non-linear reaction-diffusion problem solved with a semi-implicit time-stepping scheme and up to seven times for a 3D hyperelasticity problem solved with a continuation Newton-Raphson algorithm. The examples are implemented in the DOLFIN finite element solver using PETSc and SLEPc for linear algebra. Full code and data files to produce the results in this paper are provided as supplementary material. [less ▲]

UNSUPERVISED LEARNING BASED MODEL ORDER REDUCTION FOR HYPERELASTOPLASTICITY

Presentation (2021, January)

Non-localised contact between beams with circular and elliptical cross-sections

in Computational Mechanics (2020), 65

The key novelty of this contribution is a dedicated technique to e fficiently determine the distance (gap) function between parallel or almost parallel beams with circular and elliptical cross-sections ... [more ▼]

The key novelty of this contribution is a dedicated technique to e fficiently determine the distance (gap) function between parallel or almost parallel beams with circular and elliptical cross-sections. The technique consists of parametrizing the surfaces of the two beams in contact, fixing a point on the centroid line of one of the beams and searching for a constrained minimum distance between the surfaces (two variants are investigated). The resulting unilateral (frictionless) contact condition is then enforced with the Penalty method, which introduces compliance to the, otherwise rigid, beams' cross-sections. Two contact integration schemes are considered: the conventional slave-master approach (which is biased as the contact virtual work is only integrated over the slave surface) and the so-called two-half-pass approach (which is unbiased as the contact virtual work is integrated over the two contacting surfaces). Details of the finite element formulation which is suitably implemented using Automatic Di fferentiation techniques are presented. A set of numerical experiments shows the overall performance of the framework and allows a quantitative comparison of the investigated variants. [less ▲]

Fusing the Seth-Hill strain tensors to fit compressible elastic material responses in the nonlinear regime

in International Journal of Mechanical Sciences (2019), 163

Strain energy densities based on the Seth-Hill strain tensors are often used to describe the hyperelastic mechanical behaviours of isotropic, transversely isotropic and orthotropic materials for ... [more ▼]

Strain energy densities based on the Seth-Hill strain tensors are often used to describe the hyperelastic mechanical behaviours of isotropic, transversely isotropic and orthotropic materials for relatively large deformations. Since one parameter distinguishes which strain tensor of the Seth-Hill family is used, one has in theory the possibility to t the material response in the nonlinear regime. Most often for compressible deformations however, this parameter is selected such that the Hencky strain tensor is recovered, because it yields rather physical stress-strain responses. Hence, the response in the nonlinear regime is in practise not often tailored to match experimental data. To ensure that elastic responses in the nonlinear regime can more accurately be controlled, this contribution proposes three generalisations that combine several Seth-Hill strain tensors. The generalisations are formulated such that the stress-strain responses for in finitesimal deformations remain unchanged. Consequently, the identifi cation of the Young's moduli, Poisson's ratios and shear moduli is not a ffected. 3D fi nite element simulations are performed for isotropy and orthotropy, with an emphasis on the identifi cation of the new material parameters. [less ▲]

A stochastic Galerkin cell-based smoothed finite element method (SGCS-FEM)

in International Journal of Computational Methods (2019), 17(8),

In this paper, the cell based smoothed finite element method is extended to solve stochastic partial diff erential equations with uncertain input parameters. The spatial field of Young's moduli and the ... [more ▼]

In this paper, the cell based smoothed finite element method is extended to solve stochastic partial diff erential equations with uncertain input parameters. The spatial field of Young's moduli and the corresponding stochastic results are represented by Karhunen-Lo eve expansion and polynomial chaos expansion, respectively. The Young's Modulus of structure is considered to be random for stochastic static as well as free vibration problems. Mathematical expressions and the solution procedure are articulated in detail to evaluate the statistical characteristics of responses in terms of static displacements and free vibration frequencies. The feasibility and eff ectiveness of the proposed SGCS-FEM method in terms of accuracy and lower requirement on the mesh size in the solution domain over that of conventional FEM for stochastic problems are demonstrated by carefully chosen numerical examples. From the numerical study, it is inferred that the proposed framework is computationally less demanding without compromising accuracy. [less ▲]

A Tutorial on Bayesian Inference to Identify Material Parameters in Solid Mechanics

in Archives of Computational Methods in Engineering (2019)

The 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 ... [more ▼]

The 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. [less ▲]

Adaptive smoothed stable extended finite element method for weak discontinuities for finite elasticity

in European Journal of Mechanics. A, Solids (2019), 78

In this paper, we propose a smoothed stable extended finite element method (S2XFEM) by combining the strain smoothing with the stable extended finite element method (SXFEM) to efficiently treat inclusions ... [more ▼]

In this paper, we propose a smoothed stable extended finite element method (S2XFEM) by combining the strain smoothing with the stable extended finite element method (SXFEM) to efficiently treat inclusions and/or voids in hyperelastic matrix materials. The interface geometries are implicitly represented through level sets and a geometry based error indicator is used to resolve the geometry. For the unknown fields, the mesh is refined based on a recovery based error indicator combined with a quadtree decomposition guarantee the method’s accuracy with respect to the computational costs. Elements with hanging nodes (due to the quadtree meshes) are treated as polygonal elements with mean value coordinates as the basis functions. The accuracy and the convergence properties are compared to similar approaches for several numerical examples. The examples indicate that S2XFEM is computationally the most efficient without compromising the accuracy. [less ▲]

An equation-free, nested, concurrent multiscale approach without scale-separation

Scientific Conference (2018, September)