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Bubble-enriched smoothed finite element methods for nearly-incompressible solids ; ; Hale, Jack et al in Computer Modeling in Engineering and Sciences (in press) This work presents a locking-free smoothed finite element method (S-FEM) for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity. The proposed method overcomes ... [more ▼] This work presents a locking-free smoothed finite element method (S-FEM) for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity. The proposed method overcomes well-known issues of standard finite element methods (FEM) in the incompressible limit: the over-estimation of stiffness and sensitivity to severely distorted meshes. The concepts of cell-based, edge-based and node-based S-FEMs are extended in this paper to three-dimensions. Additionally, a cubic bubble function is utilized to improve accuracy and stability. For the bubble function, an additional displacement degree of freedom is added at the centroid of the element. Several numerical studies are performed demonstrating the stability and validity of the proposed approach. The obtained results are compared with standard FEM and with analytical solutions to show the effectiveness of the method. [less ▲] Detailed reference viewed: 66 (2 UL)A hyper-reduction method using adaptivity to cut the assembly costs of reduced order models Hale, Jack ; ; Baroli, Davide et al 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 ▲] Detailed reference viewed: 291 (43 UL)A cut finite element method for spatially resolved energy metabolism models in complex neuro-cell morphologies with minimal remeshing Farina, Sofia ; ; Hale, Jack et al in Advanced Modeling and Simulation in Engineering Sciences (2021), 8 A thorough understanding of brain metabolism is essential to tackle neurodegenerative diseases. Astrocytes are glial cells which play an important metabolic role by supplying neurons with energy. In ... [more ▼] A thorough understanding of brain metabolism is essential to tackle neurodegenerative diseases. Astrocytes are glial cells which play an important metabolic role by supplying neurons with energy. In addition, astrocytes provide scaffolding and homeostatic functions to neighboring neurons and contribute to the blood–brain barrier. Recent investigations indicate that the complex morphology of astrocytes impacts upon their function and in particular the efficiency with which these cells metabolize nutrients and provide neurons with energy, but a systematic understanding is still elusive. Modelling and simulation represent an effective framework to address this challenge and to deepen our understanding of brain energy metabolism. This requires solving a set of metabolic partial differential equations on complex domains and remains a challenge. In this paper, we propose, test and verify a simple numerical method to solve a simplified model of metabolic pathways in astrocytes. The method can deal with arbitrarily complex cell morphologies and enables the rapid and simple modification of the model equations by users also without a deep knowledge in the numerical methods involved. The results obtained with the new method (CutFEM) are as accurate as the finite element method (FEM) whilst CutFEM disentangles the cell morphology from its discretisation, enabling us to deal with arbitrarily complex morphologies in two and three dimensions. [less ▲] Detailed reference viewed: 72 (3 UL)Dynamic composition of solvers for coupled problems in DOLFINx Rehor, Martin ; Hale, Jack Presentation (2021, March 22) Recent developments in DOLFINx allow for the block assembly of linear algebraic systems arising from discretisations of coupled partial differential equations. Each algebraic block represents a subproblem ... [more ▼] Recent developments in DOLFINx allow for the block assembly of linear algebraic systems arising from discretisations of coupled partial differential equations. Each algebraic block represents a subproblem associated with a coupling of the unknown fields. Designing and implementing robust and scalable solution and preconditioning strategies for block-structured linear systems is an active area of research. In this contribution we show how DOLFINx can now exploit one of the most significant features of PETSc; the dynamic composition of the hierarchical solver and preconditioner options at runtime, see Brown et al [1]. The idea is inspired by the work of Kirby and Mitchell [2] that was originally implemented in the Firedrake Project. One of the most significant benefits of the approach is the possibility to construct advanced preconditioners that require structure beyond a purely algebraic problem description, eg the pressure-convection-diffusion (PCD) approximation of the Schur complement for the Navier–Stokes equations, see Silvester et al [3]. We illustrate the capabilities of our implementation on examples ranging from incompressible flow of a viscous fluid, through temperature-driven convection, to flows described by rate-type viscoelastic fluid models. References [1] J. Brown, M. G. Knepley, D. A. May, L. C. McInnes, and B. Smith, "Composable Linear Solvers for Multiphysics," in 2012 11th International Symposium on Parallel and Distributed Computing, Munich, Germany, Jun. 2012, pp. 55–62, doi: 10.1109/ISPDC.2012.16. [2] R. C. Kirby and L. Mitchell, "Solver Composition Across the PDE/Linear Algebra Barrier," SIAM J. Sci. Comput., vol. 40, no. 1, pp. C76–C98, 2017, doi: 10.1137/17M1133208. [3] H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. 2014, doi: 10.1093/acprof:oso/9780199678792.001.0001. Acknowledgements The present work is supported by the National Research Fund, Luxembourg in the frame of the Industrial Fellowship project RIFLE (13754363). The experiments presented in this work were carried out using the HPC facilities of the University of Luxembourg. [less ▲] Detailed reference viewed: 52 (6 UL)Hierarchical a posteriori error estimation of Bank-Weiser type in the FEniCS Project Bulle, Raphaël ; Hale, Jack ; et al E-print/Working paper (2021) In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of ... [more ▼] In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material. [less ▲] Detailed reference viewed: 49 (3 UL)Inverse deformation analysis: an experimental and numerical assessment using the FEniCS Project Mazier, Arnaud ; ; et al E-print/Working paper (2021) In this paper, we develop a framework for solving inverse deformation problems using the FEniCS Project finite element software. We validate our approach with experimental imaging data acquired from a ... [more ▼] In this paper, we develop a framework for solving inverse deformation problems using the FEniCS Project finite element software. We validate our approach with experimental imaging data acquired from a soft silicone beam under gravity. In contrast with inverse iterative algorithms that require multiple solutions of a standard elasticity problem, the proposed method can compute the undeformed configuration by solving only one modified elasticity problem. This modified problem has a complexity comparable to the standard one. The framework is implemented within an open-source pipeline enabling the direct and inverse deformation simulation directly from imaging data. We use the high-level Unified Form Language (UFL) of the FEniCS Project to express the finite element model in variational form and to automatically derive the consistent Jacobian. Consequently, the design of the pipeline is flexible: for example, it allows the modification of the constitutive models by changing a single line of code. We include a complete working example showing the inverse deformation of a beam deformed by gravity as supplementary material. [less ▲] Detailed reference viewed: 99 (15 UL)A comparison of constitutive models for describing the flow of uncured styrene-butadiene rubber Rehor, Martin ; Gansen, Alex ; et al in Journal of Non-Newtonian Fluid Mechanics (2020), 286 Uncured styrene-butadiene rubber (SBR) can be modelled as a viscoelastic material with at least two different relaxation mechanisms. In this paper we compare multi-mode constitutive models combining two ... [more ▼] Uncured styrene-butadiene rubber (SBR) can be modelled as a viscoelastic material with at least two different relaxation mechanisms. In this paper we compare multi-mode constitutive models combining two viscoelastic modes (linear and/or nonlinear) in three possible ways. Our particular choice of the two modes was inspired by models originally developed to describe the response of asphalt binders. We select the model that best fits the experimental data obtained from a modified stress relaxation experiment in the torsional configuration of the plate-plate rheometer. The optimisation of the five model parameters for each model is achieved by minimising the weighted least-squares distance between experimental observations and the computer model output using a tree-structured Parzen estimator algorithm to find an initial guess, followed by further optimisation using the Nelder-Mead simplex algorithm. The results show that the model combining the linear mode and the nonlinear mode is the most suitable variant to describe the observed behavior of SBR in the given regime. The predictive capabilities of the three models are further examined in changed experimental and numerical configurations. Full data and code to produce the figures in this article are included as supplementary material. [less ▲] Detailed reference viewed: 208 (19 UL)Removing the saturation assumption in Bank-Weiser error estimator analysis in dimension three Bulle, Raphaël ; ; Hale, Jack et al in Applied Mathematics Letters (2020), 107 We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes ... [more ▼] We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes the main novelty of our study. In addition, we present a numerical comparison of the Bank-Weiser and residual estimators for a three-dimensional test case. [less ▲] Detailed reference viewed: 134 (38 UL)Inverse simulation for retrieving the undeformed position for hyperelastic materials : application to breast simulations Mazier, Arnaud ; ; et al Scientific Conference (2020, July) Detailed reference viewed: 94 (29 UL)Practical aspects of the Bank-Weiser estimator implementation and Biomechanics applications. Bulle, Raphaël ; Bordas, Stéphane ; et al Scientific Conference (2020, July) Detailed reference viewed: 69 (8 UL)Investigation of the Sharkskin melt instability using optical Fourier analysis Gansen, Alex ; Rehor, Martin ; et al in Journal of Applied Polymer Science (2019), 137(24), 48806 An optical method allowing the characterization of melt flow instabilities typically occurring during an extrusion process of polymers and polymer compounds is presented. It is based on a camera‐acquired ... [more ▼] An optical method allowing the characterization of melt flow instabilities typically occurring during an extrusion process of polymers and polymer compounds is presented. It is based on a camera‐acquired image of the extruded compound with a reference length scale. Application of image processing and transformation of the calibrated image to the frequency domain yields the magnitude spectrum of the instability. The effectiveness of the before mentioned approach is shown on Styrene‐butadiene rubber (SBR) compounds, covering a wide range of silica filler content, extruded through a Göttfert capillary rheometer. The results of the image‐based analysis are compared with the results from the sharkskin option, a series of highly sensitive pressure transducers installed inside the rheometer. A simplified version of the code used to produce the optical analysis results is included as supplementary material. [less ▲] Detailed reference viewed: 128 (30 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: 618 (103 UL)A volume-averaged nodal projection method for the Reissner-Mindlin plate model ; ; Hale, Jack et al in Computer Methods in Applied Mechanics & Engineering (2018), 341 We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and ... [more ▼] We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and is devoid of shear-locking. The proposed approach uses linear maximum-entropy approximations and is built variationally on a two-field potential energy functional wherein the shear strain, written in terms of the primitive variables, is computed via a volume-averaged nodal projection operator that is constructed from the Kirchhoff constraint of the three-field mixed weak form. The stability of the method is rendered by adding bubble-like enrichment to the rotation degrees of freedom. Some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses. [less ▲] Detailed reference viewed: 177 (20 UL)Simple and extensible plate and shell finite element models through automatic code generation tools Hale, Jack ; ; Bordas, Stéphane et al in Computers & Structures (2018), 209 A large number of advanced finite element shell formulations have been developed, but their adoption is hindered by complexities of transforming mathematical formulations into computer code. Furthermore ... [more ▼] A large number of advanced finite element shell formulations have been developed, but their adoption is hindered by complexities of transforming mathematical formulations into computer code. Furthermore, it is often not straightforward to adapt existing implementations to emerging frontier problems in thin structural mechanics including nonlinear material behaviour, complex microstructures, multi-physical couplings, or active materials. We show that by using a high-level mathematical modelling strategy and automatic code generation tools, a wide range of advanced plate and shell finite element models can be generated easily and efficiently, including: the linear and non-linear geometrically exact Naghdi shell models, the Marguerre-von K ́arm ́an shallow shell model, and the Reissner-Mindlin plate model. To solve shear and membrane-locking issues, we use: a novel re-interpretation of the Mixed Interpolation of Tensorial Component (MITC) procedure as a mixed-hybridisable finite element method, and a high polynomial order Partial Selective Reduced Integration (PSRI) method. The effectiveness of these approaches and the ease of writing solvers is illustrated through a large set of verification tests and demo codes, collected in an open-source library, FEniCS-Shells, that extends the FEniCS Project finite element problem solving environment. [less ▲] Detailed reference viewed: 503 (46 UL)Uncertainty Quantification in Finite Element Models:Application to SoftTissue Biomechanics Hauseux, Paul ; Hale, Jack ; Bulle, Raphaël et al Scientific Conference (2018, July 23) We present probabilistic approaches aiming at the selection of the best constitutive model and to identify their parameters from experimental data. These parameters are always associated with some degree ... [more ▼] We present probabilistic approaches aiming at the selection of the best constitutive model and to identify their parameters from experimental data. These parameters are always associated with some degree of uncertainty. It is therefore important to study how this statistical uncertainty in parameters propagates to a safety-critical quantity of interest in the output of a model. Efficient Monte Carlo methods based on variance reduction techniques (Sensitivity Derivatives Monte Carlo methods [Hauseux et al. 2017] and MultiLevel Monte Carlo [Giles 2015] methods) are employed to propagate this uncertainty for both random variables and random fields. Inverse and forward problems are strongly connected. In a bayesian setting [Matthies et al. 2017], developing methods that reduce the number of evaluations of the forward model to an absolute minimum to achieve convergence is crucial for tractable computations. Numerical results in the context of soft tissue biomechanics are presented and discussed. [less ▲] Detailed reference viewed: 214 (5 UL)XDMF and ParaView: checkpointing format Habera, Michal ; Zilian, Andreas ; Hale, Jack et al Scientific Conference (2018, March 21) Checkpointing, i.e. saving and reading results of finite element computation is crucial, especially for long-time running simulations where execution is interrupted and user would like to restart the ... [more ▼] Checkpointing, i.e. saving and reading results of finite element computation is crucial, especially for long-time running simulations where execution is interrupted and user would like to restart the process from last saved time step. On the other hand, visualization of results in thid-party software such as ParaView is inevitable. In the previous DOLFIN versions (2017.1.0 and older) these two functionalities were strictly separated. Results could have been saved via HDF5File interface for later computations and/or stored in a format understood by ParaView - VTK’s .pvd (File interface) or XDMF (XDMFFile interface). This led to data redundancy and error-prone workflow. The problem essentially originated from incompatibilities between both libraries, DOLFIN and ParaView (VTK). DOLFIN’s internal representation of finite element function is based on vector of values of degrees of freedom (dofs) and their ordering within cells (dofmap). VTK’s representation of a function is given by it’s values at some points in cell, while ordering and geometric position of these points is fixed and standardised within VTK specification. For nodal (iso- and super-parametric) Lagrange finite elements (Pk , dPk ) both representations coincide up to an ordering. This allows to extend XDMF specification and introduce intermediate way of storing finite element function - intrinsic to both, ParaView and DOLFIN. The necessary work was done as a part of Google Summer of Code 2017 project Develop XDMF for- mat for visualisation and checkpointing, see https://github.com/michalhabera/gsoc-summary. New checkpointing functionality is exposed via write checkpoint() and read checkpoint() methods. [less ▲] Detailed reference viewed: 280 (31 UL)Using higher-order adjoints to accelerate the solution of UQ problems with random fields Hale, Jack ; Hauseux, Paul ; Bordas, Stéphane Poster (2018, January 08) A powerful Monte Carlo variance reduction technique introduced in Cao and Zhang 2004 uses local derivatives to accelerate Monte Carlo estimation. This work aims to: develop a new derivative-driven ... [more ▼] A powerful Monte Carlo variance reduction technique introduced in Cao and Zhang 2004 uses local derivatives to accelerate Monte Carlo estimation. This work aims to: develop a new derivative-driven estimator that works for SPDEs with uncertain data modelled as Gaussian random fields with Matérn covariance functions (infinite/high-dimensional problems) (Lindgren, Rue, and Lindström, 2011), use second-order derivative (Hessian) information for improved variance reduction over our approach in (Hauseux, Hale, and Bordas, 2017), demonstrate a software framework using FEniCS (Logg and Wells, 2010), dolfin-adjoint (Farrell et al., 2013) and PETSc (Balay et al., 2016) for automatic acceleration of MC estimation for a wide variety of PDEs on HPC architectures. [less ▲] Detailed reference viewed: 186 (27 UL)ECCOMAS Newsletter - Computational and Data Sciences in Luxembourg Beex, Lars ; Bordas, Stéphane ; Hale, Jack et al Report (2018) Detailed reference viewed: 1500 (160 UL)Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters Hauseux, Paul ; Hale, Jack ; et al in Applied Mathematical Modelling (2018), 62 We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than ... [more ▼] We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than the standard Monte Carlo method. The material model of interest can be altered by adjusting few lines of (FEniCS) code. The method is able to (1) provide the user with statistical confidence intervals on quantities of practical interest, such as the displacement of a tumour or target site in an organ; (2) quantify the sensitivity of the response of the organ to the associated parameters of the material model. We exercise the approach on the determination of a confidence interval on the motion of a target in the brain. We also show that for the boundary conditions under consideration five parameters of the Ogden-Holzapfel-like model have negligible influence on the displacement of the target zone compared to the three most influential parameters. The benchmark problems and all associated data are made available as supplementary material. [less ▲] Detailed reference viewed: 1005 (137 UL)Calculating the Malliavin derivative of some stochastic mechanics problems Hauseux, Paul ; Hale, Jack ; Bordas, Stéphane in PLoS ONE (2017), 12(12), 0189994 The Malliavin calculus is an extension of the classical calculus of variations from deterministic functions to stochastic processes. In this paper we aim to show in a practical and didactic way how to ... [more ▼] The Malliavin calculus is an extension of the classical calculus of variations from deterministic functions to stochastic processes. In this paper we aim to show in a practical and didactic way how to calculate the Malliavin derivative, the derivative of the expectation of a quantity of interest of a model with respect to its underlying stochastic parameters, for four problems found in mechanics. The non-intrusive approach uses the Malliavin Weight Sampling (MWS) method in conjunction with a standard Monte Carlo method. The models are expressed as ODEs or PDEs and discretised using the finite difference or finite element methods. Specifically, we consider stochastic extensions of; a 1D Kelvin-Voigt viscoelastic model discretised with finite differences, a 1D linear elastic bar, a hyperelastic bar undergoing buckling, and incompressible Navier-Stokes flow around a cylinder, all discretised with finite elements. A further contribution of this paper is an extension of the MWS method to the more difficult case of non-Gaussian random variables and the calculation of second-order derivatives. We provide open-source code for the numerical examples in this paper. [less ▲] Detailed reference viewed: 260 (38 UL) |
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