FEniCS in Linux Containers

FEniCS'15 Workshop at Imperial College London, 29th June - 1st July 2015

A volume-averaged nodal projection method for the Reissner-Mindlin plate model

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

A Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

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 $\boldsymbol{M} \in \mathcal{HZ} \subset H^{\mathrm{sym}}(\mathrm{Div})$. The latter results in highly accurate approximations of the bending moments $\boldsymbol{M}$ and in the rotation field being in the discontinuous Lebesgue space $\boldsymbol{\phi} \in [L]^2$, such that the Kirchhoff-Love constraint can be satisfied for $t \to 0$. In order to preserve optimal convergence rates across all variables for the case $t \to 0$, we present an extension of the formulation using Raviart-Thomas elements for the shear stress $\mathbf{q} \in \mathcal{RT} \subset H(\mathrm{div})$. 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 $\boldsymbol{M}$. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods.Comment: Additional implementation material in: https://github.com/Askys/NGSolve_HuZhang_Elemen

Containers for Portable, Productive, and Performant Scientific Computing

Containers are an emerging technology that holds promise for improving productivity and code portability in scientific computing. The authors examine Linux container technology for the distribution of a nontrivial scientific computing software stack and its execution on a spectrum of platforms from laptop computers through high-performance computing systems. For Python code run on large parallel computers, the runtime is reduced inside a container due to faster library imports. The software distribution approach and data that the authors present will help developers and users decide on whether container technology is appropriate for them. The article also provides guidance for vendors of HPC systems that rely on proprietary libraries for performance on what they can do to make containers work seamlessly and without performance penalty

Strain smoothing for compressible and nearly-incompressible finite elasticity

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

Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool

Time delay has been incorporated in models to reflect certain physical or biological meaning. The theory of delay differential equations (DDEs), which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behavior of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDE-Biftool, which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short discussion of an example application, which is a harvested predator-prey model with a single discrete time delay, will be presented first. We then implement this example model in DDE-Biftool, pointing out features where beginners need to be cautious. We end with a comparison of our theoretical and numerical results