An effective, simple, robust and locking-free plate formulation is proposed to analyze the static bending, buckling, and free vibration of homogeneous and functionally graded plates. The simple ...first-order shear deformation theory (S-FSDT), which was recently presented in Thai and Choi (2013) 11, is naturally free from shear-locking and captures the physics of the shear-deformation effect present in the original FSDT, whilst also being less computationally expensive due to having fewer unknowns. The S-FSDT requires C1-continuity that is simple to satisfy with the inherent high-order continuity of the non-uniform rational B-spline (NURBS) basis functions, which we use in the framework of isogeometric analysis (IGA). Numerical examples are solved and the results are compared with reference solutions to confirm the accuracy of the proposed method. Furthermore, the effects of boundary conditions, gradient index, and geometric shape on the mechanical response of functionally graded plates are investigated.
In industrial extrusion processes, increasing shear rates can lead to higher production rates. However, at high shear rates, extruded polymers and polymer compounds often exhibit melt instabilities ...ranging from stick‐slip to sharkskin to gross melt fracture. These instabilities result in challenges to meet the specifications on the extrudate shape. Starting with an existing published data set on melt instabilities in polymer extrusion, we assess the suitability of clustering, unsupervised machine learning algorithms combined with feature selection, to extract and identify hidden and important features from this data set, and their possible relationship with melt instabilities. The data set consists of both intrinsic features of the polymer as well as extrinsic features controlled and measured during an extrusion experiment. Using a range of commonly available clustering algorithms, it is demonstrated that the features related to only the intrinsic properties of the data set can be reliably divided into two clusters, and that in turn, these two clusters may be associated with either the stick‐slip or sharkskin instability. Furthermore, using a feature ranking on both the intrinsic and extrinsic features of the data set, it is shown that the intrinsic properties of molecular weight and polydispersity are the strongest indicators of clustering.
Unsupervised machine learning algorithms are used to predict two types of melt instabilities occurring during polymer extrusion. The influence of different physical and chemical properties on the clustering is analyzed by applying feature ranking procedures.
In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of ...derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a one-dimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney–Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.
•Derivative-driven Monte Carlo for fast propagation of uncertainty through models.•Automatic computation of the sensitivity derivative with FEniCS.•Comparison of the approach with multi-level polynomial chaos.•Superior performance of estimation for first and second order quantities.
•A general approach for the stochastic finite element analysis of soft tissue deformation is presented.•Sensitivity derivative Monte Carlo method to propagate uncertainty through hyperelastic models ...with stochastic parameters.•The proposed method is up to two orders of magnitude faster than the standard Monte Carlo approach.•The method is able to provide the user with statistical results on quantities of practical interest.•We applied the approach to a simple academic example and to the stochastic deformation of a brain.
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.
•Introducing our open-source FEniCS-Shells library that is freely available to the community.•Providing a novel interpretation and implementation of the MITC discretisation for plate and ...shells.•Improving and extending to nonlinear shells a PSRI approach initially introduced for linear shells by Arnold and Brezzi.•Proposing new verification tests for weakly nonlinear shell models.
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 Kármán 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.
We develop a novel a posteriori error estimator for the L2 error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes ...advantage of the semi-discretization scheme using rational approximations which allow to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank–Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. In addition we propose an algorithm to adapt both the finite element mesh and the rational scheme in order to balance the discretization errors. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy.
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.
In the seminal paper of Bank and Weiser (1985) 17 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 algorithmic approach to constructing hierarchical estimators of the Bank–Weiser type that is designed for implementation 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 for diverse Poisson problems and for mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. Two open source implementations in the DOLFIN and DOLFINx solvers of the FEniCS Project are provided as supplementary material.
We provide a new argument proving the reliability of the Bank–Weiser estimator for Lagrange piecewise linear finite elements in both dimensions 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.
We describe the SOniCS (SOFA + FEniCS) plugin to help develop an intuitive understanding of complex biomechanics systems. This new approach allows the user to experiment with model choices easily and ...quickly without requiring in-depth expertise. Constitutive models can be modified by one line of code only. This ease in building new models makes SOniCS ideal to develop surrogate, reduced order models and to train machine-learning algorithms for enabling real-time patient-specific simulations. SOniCS is thus not only a tool that facilitates the development of surgical training simulations but also, and perhaps more importantly, paves the way to increase the intuition of users or otherwise non-intuitive behaviors of (bio)mechanical systems. The plugin uses new developments of the FEniCSx project enabling automatic generation with FFCx of finite-element tensors, such as the local residual vector and Jacobian matrix. We verify our approach with numerical simulations, such as manufactured solutions, cantilever beams, and benchmarks provided by FEBio. We reach machine precision accuracy and demonstrate the use of the plugin for a real-time haptic simulation involving a surgical tool controlled by the user in contact with a hyperelastic liver. We include complete examples showing the use of our plugin for simulations involving Saint Venant–Kirchhoff, Neo-Hookean, Mooney–Rivlin, and Holzapfel Ogden anisotropic models as supplementary material.
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