Finite element aided design evolution of composite leaf spring

This paper shows the process of the virtual production development of the mechanical connection between the top leaf of a dual composite leaf spring system to a shackle using finite element methods. The commercial FEA package MSC/MARC has been used for the analysis. In the original design the joint was based on a closed eye-end. Full scale testing results showed that this configuration achieved the vertical proof load of 150 kN and 1 million cycles of fatigue load. However, a problem with delamination occurred at the interface between the fibres going around the eye and the main leaf body. To overcome this problem, a second design was tried using transverse bandages of woven glass fibre reinforced tape to wrap the section that is prone to delaminate. In this case, the maximum interlaminar shear stress was reduced by a certain amount but it was still higher than the material’s shear strength. Based on the fact that, even with delamination, the top leaf spring still sustained the maximum static and fatigue loads required, the third design was proposed with an open eye-end, eliminating altogether the interface where the maximum shear stress occurs. The maximum shear stress predicted by FEA is reduced significantly and a safety factor of around 2 has been obtained. Thus, a successful and safe design has been achieved.

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

A moving-mesh finite element method and its application to the numerical solution of phase-change problems

A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase. The algorithm uses a distributed conservation principle to determine nodal mesh velocities, which are then used to move the nodes. The nodal values are obtained from an ALE (Arbitrary Lagrangian-Eulerian) equation, which represents a generalization of the original algorithm presented in Applied Numerical Mathematics, 54:450--469 (2005). Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and, for the first time, two-phase Stefan problems in one and two space dimensions, paying particular attention to the implementation of the interface boundary conditions. Results are presented to demonstrate the accuracy and the effectiveness of the method, including comparisons against analytical solutions where available.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

Gradient recovery in adaptive finite-element methods for parabolic problems

We derive energy-norm a posteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the �rst completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique.Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA in the appendix.

A finite element method for nonlinear elliptic problems

We present a Galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations. A key tool is the discretization proposed in Lakkis and Pryer, 2011, allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretization method is that a recovered (finite element) Hessian is a byproduct of the solution process. We build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge–Amp`ere equation and the Pucci equation.

A finite element method for second order nonvariational elliptic problems

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of “finite element Hessian” and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasi-linear PDE, all in nonvariational form.

MetUM-GOML: a near-globally coupled atmosphere–ocean-mixed-layer model

Well-resolved air–sea interactions are simulated in a new ocean mixed-layer, coupled configuration of the Met Office Unified Model (MetUM-GOML), comprising the MetUM coupled to the Multi-Column K Profile Parameterization ocean (MC-KPP). This is the first globally coupled system which provides a vertically resolved, high near-surface resolution ocean at comparable computational cost to running in atmosphere-only mode. As well as being computationally inexpensive, this modelling framework is adaptable– the independent MC-KPP columns can be applied selectively in space and time – and controllable – by using temperature and salinity corrections the model can be constrained to any ocean state. The framework provides a powerful research tool for process-based studies of the impact of air–sea interactions in the global climate system. MetUM simulations have been performed which separate the impact of introducing inter- annual variability in sea surface temperatures (SSTs) from the impact of having atmosphere–ocean feedbacks. The representation of key aspects of tropical and extratropical variability are used to assess the performance of these simulations. Coupling the MetUM to MC-KPP is shown, for example, to reduce tropical precipitation biases, improve the propagation of, and spectral power associated with, the Madden–Julian Oscillation and produce closer-to-observed patterns of springtime blocking activity over the Euro-Atlantic region.

The role of local atmospheric forcing on the modulation of the ocean mixed layer depth in reanalysis and a coupled single column ocean model

The role of the local atmospheric forcing on the ocean mixed layer depth (MLD) over the global oceans is studied using ocean reanalysis data products and a single-column ocean model coupled to an atmospheric general circulation model. The focus of this study is on how the annual mean and the seasonal cycle of the MLD relate to various forcing characteristics in different parts of the world's ocean, and how anomalous variations in the monthly mean MLD relate to anomalous atmospheric forcings. By analysing both ocean reanalysis data and the single-column ocean model, regions with different dominant forcings and different mean and variability characteristics of the MLD can be identified. Many of the global oceans' MLD characteristics appear to be directly linked to different atmospheric forcing characteristics at different locations. Here, heating and wind-stress are identified as the main drivers; in some, mostly coastal, regions the atmospheric salinity forcing also contributes. The annual mean MLD is more closely related to the annual mean wind-stress and the MLD seasonality is more closely to the seasonality in heating. The single-column ocean model, however, also points out that the MLD characteristics over most global ocean regions, and in particular the tropics and subtropics, cannot be maintained by local atmospheric forcings only, but are also a result of ocean dynamics that are not simulated in a single-column ocean model. Thus, lateral ocean dynamics are essentially in correctly simulating observed MLD.

Modelling the lava dome extruded at Soufrière Hills Volcano, Montserrat, August 2005–May 2006. Part I: Dome shape and internal structure

Lava domes comprise core, carapace, and clastic talus components. They can grow endogenously by inflation of a core and/or exogenously with the extrusion of shear bounded lobes and whaleback lobes at the surface. Internal structure is paramount in determining the extent to which lava dome growth evolves stably, or conversely the propensity for collapse. The more core lava that exists within a dome, in both relative and absolute terms, the more explosive energy is available, both for large pyroclastic flows following collapse and in particular for lateral blast events following very rapid removal of lateral support to the dome. Knowledge of the location of the core lava within the dome is also relevant for hazard assessment purposes. A spreading toe, or lobe of core lava, over a talus substrate may be both relatively unstable and likely to accelerate to more violent activity during the early phases of a retrogressive collapse. Soufrière Hills Volcano, Montserrat has been erupting since 1995 and has produced numerous lava domes that have undergone repeated collapse events. We consider one continuous dome growth period, from August 2005 to May 2006 that resulted in a dome collapse event on 20th May 2006. The collapse event lasted 3 h, removing the whole dome plus dome remnants from a previous growth period in an unusually violent and rapid collapse event. We use an axisymmetrical computational Finite Element Method model for the growth and evolution of a lava dome. Our model comprises evolving core, carapace and talus components based on axisymmetrical endogenous dome growth, which permits us to model the interface between talus and core. Despite explicitly only modelling axisymmetrical endogenous dome growth our core–talus model simulates many of the observed growth characteristics of the 2005–2006 SHV lava dome well. Further, it is possible for our simulations to replicate large-scale exogenous characteristics when a considerable volume of talus has accumulated around the lower flanks of the dome. Model results suggest that dome core can override talus within a growing dome, potentially generating a region of significant weakness and a potential locus for collapse initiation.