Error estimation and adaptivity for incompressible, non–linear (hyper–)elasticity

A Galerkin finite element method is developed for non–linear, incompressible (hyper) elasticity, and a posteriori error estimates are derived for both linear functionals of the solution and linear functionals of the stress on a boundary where Dirichlet boundary conditions are applied. A second, higher order method for calculating a linear functional of the stress on a Dirichlet boundary is also presented together with an a posteriori error estimator for this approach. An implementation for a 2D model problem with known solution demonstrates the accuracy of the error estimators. Finally the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity

A Comparison of Numerical Methods used for\ud Finite Element Modelling of Soft Tissue\ud Deformation

Soft tissue deformation is often modelled using incompressible nonlinear elasticity, with solutions computed using the finite element method. There are a range of options available when using the finite element method, in particular, the polynomial degree of the basis functions used for interpolating position and pressure, and the type of element making up the mesh. We investigate the effect of these choices on the accuracy of the computed solution, using a selection of model problems motivated by typical deformations seen in soft tissue modelling. We set up model problems with discontinuous material properties (as is the case for the breast), steeply changing gradients in the body force (as found in contracting cardiac tissue), and discontinuous first derivatives in the solution at the boundary, caused by a discontinuous applied force (as in the breast during mammography). We find that the choice of pressure basis functions are vital in the presence of a material interface, higher-order schemes do not perform as well as may be expected when there are sharp gradients, and in general that it is important to take the expected regularity of the solution into account when choosing a numerical scheme

Cardiac Electromechanics: The effect of contraction model on the mathematical problem and accuracy of the numerical scheme

Models of cardiac electromechanics usually contain a contraction model determining the active tension induced at the cellular level, and the equations of nonlinear elasticity to determine tissue deformation in response to this active tension. All contraction models are dependent on cardiac electro-physiology, but can also be dependent on\ud the stretch and stretch-rate in the fibre direction. This fundamentally affects the mathematical problem being solved, through classification of the governing PDEs, which affects numerical schemes that can be used to solve the governing equations. We categorise contraction models into three types, and for each consider questions such as classification and the most appropriate choice from two numerical methods (the explicit and implicit schemes). In terms of mathematical classification, we consider the question of strong ellipticity of the total strain energy (important for precluding ‘unnatural’ material behaviour) for stretch-rate-independent contraction models; whereas for stretch-rate-dependent contraction models we introduce a corresponding third-order problem and explain how certain choices of boundary condition could lead to constraints on allowable initial condition. In terms of suitable numerical methods, we show that an explicit approach (where the contraction model is integrated in the timestep prior to the bulk deformation being computed) is: (i) appropriate for stretch-independent contraction models; (ii) only conditionally-stable, with the stability criterion independent of timestep, for contractions models which just depend on stretch (but not stretch-rate), and (iii) inappropriate for stretch-rate-dependent models

Party membership and campaign activity in Britain: The impact of electoral performance

The article examines the impact of electoral results on party membership and activity. Previous studies have focused on the long-term effects of electoral success or failure, suggesting that they may produce a spiral of demobilization or mobilization. The article shows that the dramatic change of electoral fortunes experienced by British parties at the 1997 general election broke this spiral, with the outcome leading to significant changes in the health and activity of local parties. It is concluded that dramatic election results can have significant implications for party organization

Error bounds on block Gauss Seidel solutions of coupled\ud multiphysics problems

Mathematical models in many fields often consist of coupled sub–models, each of which describe a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub–models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non–linear coupled fluid–temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration.\ud \ud Copyright c 2000 John Wiley & Sons, Ltd

The rigidity of periodic body-bar frameworks on the three-dimensional fixed torus

We present necessary and sufficient conditions for the generic rigidity of body-bar frameworks on the three-dimensional fixed torus. These frameworks correspond to infinite periodic body-bar frameworks in $\mathbb{R}^3$ with a fixed periodic lattice.Comment: 31 pages, 12 figure

Identification of QuiP, the Product of Gene PA1032, as the Second Acyl-Homoserine Lactone Acylase of Pseudomonas aeruginosa PAO1

The relevance of the acyl homoserine lactone (acyl-HSL) quorum signals N-3-oxododecanoyl-homoserine lactone (3OC12HSL) and N-butanoyl-homoserine lactone to the biology and virulence of Pseudomonas aeruginosa is well investigated. Previously, P. aeruginosa was shown to degrade long-chain, but not short-chain, acyl-HSLs as sole carbon and energy sources (J. J. Huang, J.-I. Han, L.-H. Zhang, and J. R. Leadbetter, Appl. Environ. Microbiol. 69:5941-5949, 2003). A gene encoding an enzyme with acyl-HSL acylase activity, pvdQ (PA2385), was identified, but it was not required for acyl-HSL utilization. This indicated that P. aeruginosa encodes another acyl-HSL acylase, which we identify here. A comparison of total cell proteins of cultures grown with long-acyl acyl-HSLs versus other substrates implicated the involvement of a homolog of PvdQ, the product of gene PA1032, for which we propose the name QuiP. Transposon mutants of quiP were defective for growth when P. aeruginosa was cultured in medium containing decanoyl-HSL as a sole carbon and energy source. Complementation with a functional copy of quiP rescued this growth defect. When P. aeruginosa was grown in buffered lysogeny broth, constitutive expression of QuiP in P. aeruginosa led to decreased accumulations of the quorum signal 3OC12HSL, relative to the wild type. Heterologous expression of QuiP was sufficient to confer long-chain acyl-HSL acylase activity upon Escherichia coli. Examination of gene expression patterns during acyl-HSL-dependent growth of P. aeruginosa further supported the involvement of quiP in signal decay and revealed other genes also possibly involved. It is not yet known under which "natural" conditions quiP is expressed or how P. aeruginosa balances the expression of its quorum-sensing systems with the expression of its acyl-HSL acylase activities

Knowledge and the artefact

This paper discusses ways that knowledge may be found in or through artefacts. One purpose is to suggest situations where artefacts might be central to a narrative, rather than secondary to a text. A second purpose is to suggest ways that design and production of artefacts might be instrumental in eliciting knowledge. Four general situations are proposed: (1) Simple Forms - an artefact demonstrates or describes a principle or technique. (2) Communication of Process - artefacts arising from a process make the process explicit. (3) Artefacts Within the Research - artefacts are instrumental in advancing the research by communicating ideas or information. (4) Knowledge Elicited by Artefacts - artefacts provide a stimulus or context which enables information to be uncovered. .</p

Global consensus Monte Carlo

To conduct Bayesian inference with large data sets, it is often convenient or necessary to distribute the data across multiple machines. We consider a likelihood function expressed as a product of terms, each associated with a subset of the data. Inspired by global variable consensus optimisation, we introduce an instrumental hierarchical model associating auxiliary statistical parameters with each term, which are conditionally independent given the top-level parameters. One of these top-level parameters controls the unconditional strength of association between the auxiliary parameters. This model leads to a distributed MCMC algorithm on an extended state space yielding approximations of posterior expectations. A trade-off between computational tractability and fidelity to the original model can be controlled by changing the association strength in the instrumental model. We further propose the use of a SMC sampler with a sequence of association strengths, allowing both the automatic determination of appropriate strengths and for a bias correction technique to be applied. In contrast to similar distributed Monte Carlo algorithms, this approach requires few distributional assumptions. The performance of the algorithms is illustrated with a number of simulated examples