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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. [ABSTRACT FROM AUTHOR]

Highlights • 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. Abstract 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 C 1 -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. [ABSTRACT FROM AUTHOR]

At every iteration or timestep of the online phase of some reduced-order modelling schemes for non-linear or time-dependent systems, 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 become 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 construct 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 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. • Reduced Assembly method to cut assembly and projection costs in reduced order models. • Uses local error indicator and triangulation adaptivity. • Five to seven times speedup over baseline method. [ABSTRACT FROM AUTHOR]

In this work we develop new finite element discretisations of the shear-deformable Reissner–Mindlin plate problem based on the Hellinger–Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M ∈ HZ ⊂ H sym (Div). The latter results in highly accurate approximations of the bending moments M and in the rotation field being in the Lebesgue space ϕ ∈ [ L 2 ] 2 , such that the Kirchhoff–Love constraint can be satisfied for t → 0. In order to preserve optimal convergence rates across all variables for the case t → 0 , we present an extension of the formulation using Raviart–Thomas elements for the shear stress q ∈ RT ⊂ H (div). We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting. This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries. The robustness of the formulations and the construction of the Hu-Zhang element are tested for shear-locking, curved geometries and an L-shaped domain with a singularity in the bending moments M. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods. • Novel variational formulations of the Reissner-Mindlin plate via Hu-Zhang elements. • Well-posedness proofs, robustness proofs, and a priori error estimates. • New formulas for the Hu-Zhang basis via polytopal templates and Legendre polynomials. • Novel mappings of the base functions from the reference- to the physical element. • Techniques for the embedding of Dirichlet boundary conditions in the Hu-Zhang space. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present a robust and efficient form of the smoothed finite element method (S-FEM) to simulate hyperelastic bodies with compressible and nearly-incompressible neo-Hookean behaviour. The resulting method is stable, free from volumetric locking and robust on highly distorted meshes. To ensure inf-sup stability of our method we add a cubic bubble function to each element. The weak form for the smoothed hyperelastic problem is derived analogously to that of smoothed linear elastic problem. Smoothed strains and smoothed deformation gradients are evaluated on sub-domains selected by either edge information (edge-based S-FEM, ES-FEM) or nodal information (node-based S-FEM, NS-FEM). Numerical examples are shown that demonstrate the efficiency and reliability of the proposed approach in the nearly-incompressible limit and on highly distorted meshes. We conclude that, strain smoothing is at least as accurate and stable, as the MINI element, for an equivalent problem size. [ABSTRACT FROM AUTHOR]

The flow curves of polymers often reveal the onset of melt instabilities such as sharkskin, stick–slip, or gross melt fracture, in order of increasing shear rates. The focus of this work lies in the application of the Göttfert sharkskin option to the investigation of flow curves of styrene-butadiene rubber (SBR) compounds. The sharkskin option consists of highly sensitive pressure transducers located inside a slit die of a capillary rheometer. This tool allows the detection of in-situ pressure fluctuation characteristics of different melt instabilities. It is shown that the change of slope of the transition region in the flow curves is only linked to slip. Dynamic Mechanical Analysis (DMA) measurements furthermore show that the shear rate at which the change of slope can be observed shows the same temperature dependency as the viscous and elastic properties of the compounds. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We develop a novel a posteriori error estimator for the L 2 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. [ABSTRACT FROM AUTHOR]

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 behaviour 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. • We derive a viscoelastic constitutive model for uncured rubber compounds. • Model parameters are optimised using a two-stage global/local algorithm. • We demonstrate the predictive performance of the new model. [ABSTRACT FROM AUTHOR]