Analysis of polyhedral domed structures composed of flat plates of sandwich material

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A software perspective on infinite elements for wave diffraction and wave forces on marine risers

This thesis describes work on the problem of the scattering of water waves by fixed objects. The method used to solve this problem is that of finite and infinite elements. In particular the development of a new wave infinite element is described. Various aspects of the wave scattering problem are considered, but always from the perspective of the numerical methods, the algorithms and the computer implementations used. These deal not only with the modelling of the wave equations, but also the pre and post processing of the finite element algorithms. This encompasses the generation of suitable finite element meshes, in an accurate and economical way, and the presentation of the results, particularly as accurate contour plots of the wave surface. The first two chapters gives a brief introduction to water waves, and a summary of the basic concepts of finite and infinite elements. In the third chapter the new infinite element for waves, which is a development of an earlier infinite element, is described in detail, including the new mapping, the necessary shape functions and the integration of the element matrix. The earlier infinite element was restricted to the exterior of circular problems. For scattering objects of large aspect ratio this led to meshes with many finite elements, which performed no useful function, and which were computationally wasteful. The mapping in the new infinite element allows the mesh of infinite elements to be tailored to the shape of the diffracting body, without any observed loss of accuracy. It is therefore much more flexible and computationally efficient, because the infinite elements no longer need to be placed radially. The next three chapters, concentrate on the computer science aspects of the implementation of the finite and infinite elements dealing with the linked list data structures for storage of the element information, the special purpose mesh generation programs, which make it possible to analyse a large range of practical scattering problems and the plotting programs for the display of the results. The chief work in chapter six is the implementation of the Akin and Grey accurate predictor-corrector contour plotting algorithm, with colour fill. The advantage of an accurate contour plotting algorithm is that any discontinuities in the contours represent discontinuities in the results, rather than plotting deficiencies. Chapter seven shows results which validate the new infinite element, particularly on the problem of waves diffracted by an ellipse. In the remaining chapters eight to eleven, the emphasis is on a practical problem of the wave forces on groups of risers, which are the tubes which carry hydrocarbons from the sea-bed to the working areas of offshore platforms. The aim was to see if the forces on a group of risers were different from the sum of the forces on the individual risers, calculated on the assumption that the risers do not modify the wave field. The conclusion is that more detailed studies may well bring financial benefits to the companies operating offshore installations

On wave boundary elements for radiation and scattering problems with piecewise constant impedance

Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this paper, a new type of interpolation for the wave field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation is found in this paper to be highly successful. Results are shown for a variety of scattering/radiating problems from convex and nonconvex obstacles on which are prescribed piecewise constant Robin conditions. Notable results include a conclusion that, using this new formulation, only approximately three degrees of freedom per wavelength are required

Ground Waves Generated by Pile Driving, and Structural Interaction

Pre-formed steel or concrete pile elements are installed by high energy impact or vibro-driver, which causes outgoing ground waves. In severe cases, adjacent buildings are at risk of damage. Assessment of risk is conventionally by reference to threshold limits of vibration. The global approach considers neither the interactive effects between ground and structure, nor frequency and duration. Here, firstly, the dynamics of a pile head impact and of the transmission of a portion of the energy into the ground were modelled by a combination of finite elements (FE), springs and dashpots. The boundary disturbances were then applied to a second model of the soil as an elastic half space. This outer model was constructed of axisymmetric finite and infinite elements for calibration against on-site measurements. The infinite elements (IE) represented a wider zone, and avoided spurious wave reflections at boundaries. Next, the verified ground disturbances adjacent to the pile were used as input to a three-dimensional FE/IE wedge-shaped model of a ‘slice’ of the axisymmetric system. Various structural forms, of steel frame structures and of brick walls, were added, giving a dynamic soil-structure analysis. Results show the responses of flexible and stiff structures to outgoing waves caused by impact pile driving and vibro-driving

A high-wavenumber boundary-element method for an acoustic scattering problem

In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval $[a,b]$, which only requires the discretization of $[a,b]$, we show theoretically and experimentally that the $L_2$ error in computing the acoustic field on $[a,b]$ is ${\cal O}(\log^{\nu+3/2}|k(b-a)| M^{-(\nu+1)})$, where $M$ is the number of degrees of freedom and $k$ is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems

Tomographic imaging and scanning thermal microscopy: thermal impedance tomography

The application of tomographic imaging techniques developed for medical applications to the data provided by the scanning thermal microscope will give access to true three-dimensional information on the thermal properties of materials on a mm length scale. In principle, the technique involves calculating and inverting a sensitivity matrix for a uniform isotropic material, collecting ordered data at several modulation frequencies, and multiplying the inverse of the matrix with the data vector. In practice, inversion of the matrix in impractical, and a novel iterative technique is used. Examples from both simulated and real data are given

Formation of a primitive ectoderm like cell population, EPL cells, from ES cells in response to biologically derived factors

The primitive ectoderm of the mouse embryo arises from the inner cell mass between 4.75 and 5.25 days post coitum, around the time of implantation. Positioned at a pivotal time in development, just prior to formation of the three germ layers of the embryo proper, the primitive ectoderm responds directly to the signals generated during gastrulation. We have identified a conditioned medium, MEDII, which caused the homogeneous conversion of ES cells to a morphologically distinct cell population, termed early primitive ectoderm-like (EPL) cells. EPL cells expressed the pluripotent cell markers Oct4, SSEA1 and alkaline phosphatase. However, the formation of EPL cells was accompanied by alterations in Fgf5, Gbx2 and Rex1 expression, a loss in chimaera forming ability, changes in factor responsiveness and modified differentiation capabilities, all consistent with the identification of EPL cells as equivalent to the primitive ectoderm population of the 5.5 to 6.0 days post coitum embryo. EPL cell formation could be reversed in the presence of LIF and withdrawal of MEDII, which suggested that EPL cell formation was not a terminal differentiation event but reflected the ability of pluripotent cells to adopt distinct cell states in response to specific factors. Partial purification of MEDII revealed the presence of two separable biological activities, both of which were required for the induction and maintenance of EPL cells. We show here the first demonstration of uniform differentiation of ES cells in response to biological factors. The formation of primitive ectoderm, both in vivo and in vitro, appears to be an obligatory step in the differentiation of the inner cell mass or ES cells into cell lineages of the embryonic germ layers. EPL cells potentially represent a model for the development of lineage specific differentiation protocols and analysis of gastrulation at a molecular level. An understanding of the active components of MEDII may provide a route for the identification of factors which induce primitive ectoderm formation in vivo.Joy Rathjen, Julie-Anne Lake, Michael D. Bettess, Jenny M. Washington, Gavin Chapman and Peter D. Rathje

A semi-analytical scheme for highly oscillatory integrals over tetrahedra

This is the peer reviewed version of the following article: [Hospital-Bravo, R., Sarrate, J., and Díez, P. (2017) A semi-analytical scheme for highly oscillatory integrals over tetrahedra. Int. J. Numer. Meth. Engng, 111: 703–723. doi: 10.1002/nme.5474], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5474/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.This paper details a semi-analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the Partition of Unity Method (PUM) enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non-oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared to standard Gauss-Legendre quadratures.Peer ReviewedPostprint (author's final draft