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A Bayesian framework to identify random parameter fields based on the copula theorem and Gaussian fields: Application to polycrystalline materials Rappel, Hussein ; ; et al 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 ▲] Detailed reference viewed: 82 (5 UL)A stochastic Galerkin cell-based smoothed finite element method (SGCS-FEM) ; Beex, Lars ; Bordas, Stéphane et al in International Journal of Computational Methods (in press) 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 ▲] Detailed reference viewed: 88 (0 UL)Non-localised contact between beams with circular and elliptical cross-sections Magliulo, Marco ; Lengiewicz, Jakub ; Zilian, Andreas et al in Computational Mechanics (in press) 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 ▲] Detailed reference viewed: 18 (3 UL)Contact between shear-deformable beams with elliptical cross-sections Magliulo, Marco ; Zilian, Andreas ; Beex, Lars in Acta Mechanica (in press) 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 ▲] Detailed reference viewed: 75 (6 UL)Fatigue phase-field damage modeling of rubber using viscous dissipation: Crack nucleation and propagation Loew, Pascal Juergen ; Peters, Bernhard ; Beex, Lars in Mechanics of Materials (2020), 142 By regularizing sharp cracks within a pure continuum setting, phase-damage models offer the ability to capture crack nucleation as well as crack propagation. Crack branching and coalescence can ... [more ▼] By regularizing sharp cracks within a pure continuum setting, phase-damage models offer the ability to capture crack nucleation as well as crack propagation. Crack branching and coalescence can furthermore be described without any additional efforts, as geometrical descriptions of the cracks are not required. In this contribution, we extend our previous phase-field model for rate-dependent fracture of rubbers in a finite strain setting (Loew et al., 2019) to describe damage under cyclic loading. The model is derived from the balance of mechanical energy and introduces a fatigue damage source as a function of the accumulated viscous dissipation under cyclic loading. We use uniaxial cyclic tension to present the influence of the fatigue material parameters and to confirm the model’s energy balance. The parameters are subsequently identified using monotonic and cyclic experiments of a plane stress nature. Finally, the model is validated by separate experiments, which demonstrate that the model accurately predicts (fatigue) crack nucleation as well as propagation. [less ▲] Detailed reference viewed: 22 (5 UL)Adaptive equation-free multiscale modeling of metallic lattices with geometrical nonlinearity and variability Chen, Li ; ; Beex, Lars et al Scientific Conference (2019, September 12) An equation-free concurrent multiscale framework is proposed to model 3D metallic lattice structures. The proposed equation-free multiscale method (EFMM) is effectively a generalization of the ... [more ▼] An equation-free concurrent multiscale framework is proposed to model 3D metallic lattice structures. The proposed equation-free multiscale method (EFMM) is effectively a generalization of the quasicontinuum method [2] and relies on the use of fully-resolved domains (FRD) in which all 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 geometrical description is that cross section variations along the lattice struts (caused by 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 displacement of a reduced number of material points. One of the originalities of this work is the consideration of a separate interpolation of each type of kinematic variables 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 geometric nonlinearity, by the development and implementation of a 3D co-rotational beam finite element [1], are innovative contributions. Choosing the appropriate sizes of the FRDs and the CGDs for a lattice to be simulated is a trade-off because larger FRDs prevail the accuracy but compromise the efficiency while larger CGDs do the opposite. Since the required sizes of the FRDs and CGDs are generally not known a priori for specific applications, an adaptive coarse-graining strategy is developed. To be specific, the whole lattice is initially considered as a CGD. Two kinds of error indicator are proposed (e.g. the Zienkiewicz-Zhu error indicator [4, 3] and the error indicator based on the discrepancy of strain energy). The error indicator guides on: 1) introducing more material points and rearranging the interpolation for the CGDs; 2) changing the localization-prone parts of the lattice into FRDs. The adaptive EFMM is applied to metallic BCC lattices with various sizes and loading conditions. By comparing to the results of those of the direct numerical simulation (DNS), it is shown that geometrical non-linearities can be captured at a fraction of the DNS cost. [less ▲] Detailed reference viewed: 13 (1 UL)Equation-free multiscale modeling of metallic lattices with geometrical and material nonlinearity and variability Chen, Li ; ; Beex, Lars et al 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 ▲] Detailed reference viewed: 13 (1 UL)Fusing the Seth-Hill strain tensors to fit compressible elastic material responses in the nonlinear regime Beex, Lars 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 ▲] Detailed reference viewed: 106 (4 UL)Bayesian Identification of Mean-Field Homogenization model parameters and uncertain matrix behavior in non-aligned short fiber composites ; ; et al in Composite Structures (2019), 220 We present a stochastic approach combining Bayesian Inference (BI) with homogenization theories in order to identify, on the one hand, the parameters inherent to the model assumptions and, on the other ... [more ▼] We present a stochastic approach combining Bayesian Inference (BI) with homogenization theories in order to identify, on the one hand, the parameters inherent to the model assumptions and, on the other hand, the composite material constituents behaviors, including their variability. In particular, we characterize the model parameters of a Mean-Field Homogenization (MFH) model and the elastic matrix behavior, including the inherent dispersion in its Young's modulus, of non-aligned Short Fibers Reinforced Polymer (SFRP) composites. The inference is achieved by considering as observations experimental tests conducted at the SFRP composite coupons level. The inferred model and material law parameters can in turn be used in Mean-Field Homogenization (MFH)-based multi-scale simulations and can predict the confidence range of the composite material responses. [less ▲] Detailed reference viewed: 121 (11 UL)Estimating fibres' material parameter distributions from limited data with the help of Bayesian inference Rappel, Hussein ; Beex, Lars in European Journal of Mechanics. A, Solids (2019), 75 Numerous materials are essentially structures of discrete fibres, yarns or struts. Considering these materials at their discrete scale, one may distinguish two types of intrinsic randomness that affect ... [more ▼] Numerous materials are essentially structures of discrete fibres, yarns or struts. Considering these materials at their discrete scale, one may distinguish two types of intrinsic randomness that affect the structural behaviours of these discrete structures: geometrical randomness and material randomness. Identifying the material randomness is an experimentally demanding task, because many small fibres, yarns or struts need to be tested, which are not easy to handle. To avoid the testing of hundreds of constituents, this contribution proposes an identification approach that only requires a few dozen of constituents to be tested (we use twenty to be exact). The identification approach is applied to articially generated measurements, so that the identified values can be compared to the true values. Another question this contribution aims to answer is how precise the material randomness needs to be identified, if the geometrical randomness will also influence the macroscale behaviour of these discrete networks. We therefore also study the effect of the identified material randomness to that of the actual material randomness for three types of structures; each with an increasing level of geometrical randomness. [less ▲] Detailed reference viewed: 164 (32 UL)Multiscale fracture: a natural connection between reduced order models and homogenisation Bordas, Stéphane ; Beex, Lars ; Chen, Li et al Scientific Conference (2019, May 13) Detailed reference viewed: 96 (9 UL)A Tutorial on Bayesian Inference to Identify Material Parameters in Solid Mechanics Rappel, Hussein ; Beex, Lars ; Hale, Jack et al 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 ▲] Detailed reference viewed: 501 (93 UL)Identifying elastoplastic parameters with Bayes' theorem considering double error sources and model uncertainty Rappel, Hussein ; Beex, Lars ; et al in Probabilistic Engineering Mechanics (2019), 55 We discuss Bayesian inference for the identi cation of elastoplastic material parameters. In addition to errors in the stress measurements, which are commonly considered, we furthermore consider errors in ... [more ▼] We discuss Bayesian inference for the identi cation 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. [less ▲] Detailed reference viewed: 279 (61 UL)Rate-dependent phase-field damage modeling of rubber and its experimental parameter identification Loew, Pascal Juergen ; Peters, Bernhard ; Beex, Lars in Journal of the Mechanics and Physics of Solids (2019) Phase-field models have the advantage in that no geometric descriptions of cracks are required, which means that crack coalescence and branching can be treated without additional effort. Miehe and ... [more ▼] Phase-field models have the advantage in that no geometric descriptions of cracks are required, which means that crack coalescence and branching can be treated without additional effort. Miehe and Schänzel (2014) introduced a rate-independent phase-field damage model for finite strains in which a viscous damage regularization was proposed. We extend the model to depend on the loading rate and time by incorporating rubber’s strain-rate dependency in the constitutive description of the bulk, as well as in the damage driving force. The parameters of the model are identified using experiments at different strain rates. Local strain fields near the crack tip, obtained with digital image correlation (DIC), are used to help identify the length scale parameter. Three different degradation functions are assessed for their accuracy to model the rubber’s rate-dependent fracture. An adaptive time-stepping approach with a corrector scheme is furthermore employed to increase the computational efficiency with a factor of six, whereas an active set method guarantees the irreversibility of damage. Results detailing the energy storage and dissipation of the different model constituents are included, as well as validation results that show promising capabilities of rate-dependent phase-field modeling. [less ▲] Detailed reference viewed: 124 (16 UL)Non-localized Contact Between Beams with Non-Circular Cross Sections Magliulo, Marco ; Zilian, Andreas ; Beex, Lars in Proceedings in Applied Mathematics and Mechanics (2019) In this contribution, we introduce a contact formulation between beams finite elements with (hyper)elliptical cross sections. The contact scheme allows to model scenarios in which the contact area is ... [more ▼] In this contribution, we introduce a contact formulation between beams finite elements with (hyper)elliptical cross sections. The contact scheme allows to model scenarios in which the contact area is finite or the contact area occurs along a line. Although some contact schemes are yet able to do this, they require one of the beams to have a circular cross section. Here however, we focus on non-circular cross-sections. Consequently, new projections are required, in which the beam surfaces are used explicitly to formulate contact kinematics. [less ▲] Detailed reference viewed: 88 (12 UL)Adaptive smoothed stable extended finite element method for weak discontinuities for finite elasticity ; ; Beex, Lars et al 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 ▲] Detailed reference viewed: 38 (0 UL)A hyper-reduction method using adaptivity to cut the assembly costs of reduced order models Hale, Jack ; ; Baroli, Davide et al E-print/Working paper (2019) 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 ▲] Detailed reference viewed: 201 (28 UL)An equation-free, nested, concurrent multiscale approach without scale-separation Beex, Lars ; Scientific Conference (2018, September) Detailed reference viewed: 47 (0 UL)Classification of states and model order reduction of large scale Chemical Vapor Deposition processes with solution multiplicity ; ; Beex, Lars et al in Computers and Chemical Engineering (2018), 121 This paper presents an equation-free, data-driven approach for reduced order modeling of a Chemical Vapor Deposition (CVD) process. The proposed approach is based on process information provided by ... [more ▼] This paper presents an equation-free, data-driven approach for reduced order modeling of a Chemical Vapor Deposition (CVD) process. The proposed approach is based on process information provided by detailed, high-fidelity models, but can also use spatio-temporal measurements. The Reduced Order Model (ROM) is built using the method-of-snapshots variant of the Proper Orthogonal Decomposition (POD) method and Artificial Neural Networks (ANN) for the identification of the time-dependent coefficients. The derivation of the model is completely equation-free as it circumvents the projection of the actual equations onto the POD basis. Prior to building the model, the Support Vector Machine (SVM) supervised classification algorithm is used in order to identify clusters of data corresponding to (physically) different states that may develop at the same operating conditions due to the inherent nonlinearity of the process. The different clusters are then used for ANN training and subsequent development of the ROM. The results indicate that the ROM is successful at predicting the dynamic behavior of the system in windows of operating parameters where steady states are not unique. [less ▲] Detailed reference viewed: 72 (6 UL)Identifying fibre material parameter distributions with little experimental efforts Rappel, Hussein ; Beex, Lars ; Bordas, Stéphane Scientific Conference (2018, July 23) Detailed reference viewed: 79 (15 UL) |
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