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 ▲]

Bayesian model uncertainty quantification for hyperelastic soft tissue models

in Data-Centric Engineering (2021)

Patient-specific surgical simulations require the patient-specific identification of the constitutive parameters. The sparsity of the experimental data and the substantial noise in the data (e.g ... [more ▼]

Patient-specific surgical simulations require the patient-specific identification of the constitutive parameters. The sparsity of the experimental data and the substantial noise in the data (e.g., recovered during surgery) cause considerable uncertainty in the identification. In this exploratory work, parameter uncertainty for incompressible hyperelasticity, often used for soft tissues, is addressed by a probabilistic identification approach based on Bayesian inference. Our study particularly focuses on the uncertainty of the model: we investigate how the identified uncertainties of the constitutive parameters behave when different forms of model uncertainty are considered. The model uncertainty formulations range from uninformative ones to more accurate ones that incorporate more detailed extensions of incompressible hyperelasticity. The study shows that incorporating model uncertainty may improve the results, but this is not guaranteed. [less ▲]

Frictional interactions for non-localised beam-to-beam and beam-inside-beam contact

in International Journal for Numerical Methods in Engineering (2021), 122(7), 1706-1731

This contribution presents the extensions of beam-to-beam and beam-inside-beam contact schemes of the same authors towards frictional interactions. Since the schemes are based on the beams’ true surfaces ... [more ▼]

This contribution presents the extensions of beam-to-beam and beam-inside-beam contact schemes of the same authors towards frictional interactions. Since the schemes are based on the beams’ true surfaces (instead of surfaces implicitly deduced from the beams’ centroid lines), the presented enhancements are not only able to account for frictional sliding in the beams’ axial directions, but also in the circumferential directions. Both the frictional beam-to-beam approach as well as the frictional beam-inside-beam approach are applicable to shear-deformable and shear-undeformable beams, as well as to beams with both circular and elliptical cross-sections (although the cross-sections must be rigid). A penalty formulation is used to treat unilateral and frictional contact constraints. FE implementation details are discussed, where automatic differentiation techniques are used to derive the implementations. Simulations involving large sliding displacements and large deformations are presented for both beam-to-beam and beam-inside-beam schemes. All simulation results are compared to those of the frictionless schemes. [less ▲]

Beam-inside-beam contact: Mechanical simulations of slender medical instruments inside the human body

in Computer Methods and Programs in Biomedicine (2020), 196

Background and Objective This contribution presents a rapid computational framework to mechanically simulate the insertion of a slender medical instrument in a tubular structure such as an artery, the ... [more ▼]

Background and Objective This contribution presents a rapid computational framework to mechanically simulate the insertion of a slender medical instrument in a tubular structure such as an artery, the cochlea or another slender instrument. Methods Beams are employed to rapidly simulate the mechanical behaviour of the medical instrument and the tubular structure. However, the framework’s novelty is its capability to handle the mechanical contact between an inner beam (representing the medical instrument) embedded in a hollow outer beam (representing the tubular structure). This “beam-inside-beam” contact framework, which forces two beams to remain embedded, is the first of its kind since existing contact frameworks for beams are “beam-to-beam” approaches, i.e. they repel beams from each other. Furthermore, we propose contact kinematics such that not only instruments and tubes with circular cross-sections can be considered, but also those with elliptical cross-sections. This provides flexibility for the optimization of patient-specific instruments. Results The results demonstrate that the framework’s robustness is substantial, because only a few increments per simulation and a few iterations per increment are required, even though large deformations, large rotations and large curvature changes of both the instrument and tubular structure occur. The stability of the framework remains high even if the modulus of the inner tube is thousand times larger than that of the outer tube. A mesh convergence study furthermore exposes that a relatively small number of elements is required to accurately approach the reference solution. Conclusions The framework’s high simulation speed originates from the exploitation of the rigidity of the beams’ cross-sections to quantify the exclusion between the inner and the hollow outer beam. This rigidity limits the accuracy of the framework at the same time, but this is unavoidable since simulation accuracy and simulation speed are two competing interests. Hence, the framework is particularly attractive if simulation speed is preferred over accuracy. [less ▲]

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

in Computer Methods in Applied Mechanics and Engineering (2020), 366(112878),

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 ... [more ▼]

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. [less ▲]

Contact between shear-deformable beams with elliptical cross-sections

in Acta Mechanica (2020), 231

Slender constituents are present in many structures and materials. In associated mechanical models, each slender constituent is often described with a beam. Contact between beams is essential to ... [more ▼]

Slender constituents are present in many structures and materials. In associated mechanical models, each slender constituent is often described with a beam. Contact between beams is essential to incorporate in mechanical models, but associated contact frameworks are only demonstrated to work for beams with circular cross-sections. Only two studies have shown the ability to treat contact between beams with elliptical cross-sections, but those frameworks are limited to point-wise contact, which narrows their applicability. This contribution presents initial results of a framework for shear-deformable beams with elliptical cross-sections if contact occurs along a line or at an area (instead of at a point). This is achieved by integrating a penalty potential over one of the beams’ surfaces. Simo-Reissner Geometrically Exact Beam (GEB) elements are employed to discretise each beam. As the surface of an assembly of such beam elements is discontinuous, a smoothed surface is introduced to formulate the contact kinematics. This enables the treatment of contact for large sliding displacements and substantial deformations. [less ▲]

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 ▲]

Equation-free multiscale modeling of metallic lattices with geometrical and material nonlinearity and variability

Scientific Conference (2019, September 05)

An nonlinear equation-free concurrent multiscale numerical framework, being the generalization of the quasicontinuum method [2] is proposed in this contribution to model 3D metallic lattice structures ... [more ▼]

An nonlinear equation-free concurrent multiscale numerical framework, being the generalization of the quasicontinuum method [2] is proposed in this contribution to model 3D metallic lattice structures. The proposed equation-free multiscale method (EFMM) relies on the use of fully-resolved domains (FRD) in which all of the details of the lattice micro-structure are captured, and of coarse-grained domains (CGD) in which a model reduction is performed by interpolation and summation steps. The particularity of the lattice geometry description is that cross section variations along the lattice struts (that are experimentally observed as a result of the manufacturing process) are explicitly represented by their discretization in several beam finite elements, both in the FRDs and CGDs. The interpolation step of the EFMM refers to a kinematic approximation of the lattice deformation within CGDs based on the movement of a reduced number of material points at the CGD corners. One of the originalities of this work is the consideration of a separate interpolation of each type of degrees of freedom within the CGDs, as a function of the connectivity of the lattice beam nodes (i.e. taking the location of different cross sections into account) and their kinematical pattern. This, together with accounting for plasticity, by the development and implementation of a 3D co-rotational beam finite element [1] with embedded plastic hinges [3], are unprecedented and original contributions. The EFMM is applied to metallic BCC lattices with various sizes and loading conditions. By comparing to direct numerical simulation (DNS), it is shown that both material and geometrical non-linearities can be captured at a fraction of the DNS cost (the computational time is reduced by 97.27% while introducing an error of only 3.76%). [less ▲]

Clustering Based Model Order Reduction For Hyper Elastoplastic Material Models

Presentation (2019, July 29)