180 research outputs found

    Continuous boundary elements for potential problems

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    Imperial Users onl

    Master index to volumes 1–10

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    Fundamental solution based numerical methods for three dimensional problems: efficient treatments of inhomogeneous terms and hypersingular integrals

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    In recent years, fundamental solution based numerical methods including the meshless method of fundamental solutions (MFS), the boundary element method (BEM) and the hybrid fundamental solution based finite element method (HFS-FEM) have become popular for solving complex engineering problems. The application of such fundamental solutions is capable of reducing computation requirements by simplifying the domain integral to the boundary integral for the homogeneous partial differential equations. The resulting weak formulations, which are of lower dimensions, are often more computationally competitive than conventional domain-type numerical methods such as the finite element method (FEM) and the finite difference method (FDM). In the case of inhomogeneous partial differential equations arising from transient problems or problems involving body forces, the domain integral related to the inhomogeneous solutions term will need to be integrated over the interior domain, which risks losing the competitive edge over the FEM or FDM. To overcome this, a particular treatment to the inhomogeneous term is needed in the solution procedure so that the integral equation can be defined for the boundary. In practice, particular solutions in approximated form are usually applied rather than the closed form solutions, due to their robustness and readiness. Moreover, special numerical treatment may be required when evaluating stress directly on the domain surface which may give rise to hypersingular integral formulation. This thesis will discuss how the MFS and the BEM can be applied to the three-dimensional elastic problems subjected to body forces by introducing the compactly supported radial basis functions in addition to the efficient treatment of hypersingular surface integrals. The present meshless approach with the MFS and the compactly supported radial basis functions is later extended to solve transient and coupled problems for three-dimensional porous media simulation

    An immersed computational framework for multiphase fluid-structure interaction.

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    "The objective of this thesis is to further extend the application range of immersed computational approaches in the context of hydrodynamics and present a novel general framework for the simulation of fluid-structure interaction problems involving rigid bodies, flexible solids and multiphase flows. The proposed method aims to overcome shortcomings such as the restriction of having to deal with similar density ratios among different phases or the restriction to solve single-phase flows. The new framework will be capable of coping with large density ratios, multiphase flows and will be focussed on hydrodynamic problems. The two main challenges to be addressed are: - the representation, evolution and compatibility of the multiple fluid-solid interface - the proposition of unified framework containing multiphase flows, flexible structures and rigid bodies with possibly large density ratios First, a new variation of the original IBM is presented by rearranging the governing equations which define the behaviour of the multiple physics involved. The formulation is compatibile with the "one-fluid" equation for two phase flows and can deal with large density ratios with the help of an anisotropic Poisson solver. Second, deformable structures and fluid are modelled in a identical manner except for the deviatoric part of the Cauchy stress tensor. The challenging part is the calculation of the deviatoric part the Cauchy stress in the structure, which is expressed as a function of the deformation gradient tensor. The technique followed In this thesis is that original ISP, but re-expressed in terms of the Cauchy stress tensor. Any immersed rigid body is considered as an incompressible non-viscous continuum body with an equivalent internal force field which constrains the velocity field to satisfy the rigid body motion condition. The "rigid body" spatial velocity is evaluated by means of a linear least squares projection of the background fluid velocity, whilst the immersed force field emerges as a result of the linear momentum conversation equation. This formulation is convenient for arbitrary rigid shapes around a fixed point and the most general translation- rotation. A characteristic or indicator function, defined for each interacting continuum phase, evolves passively with the velocity field. Generally, there are two families of algorithms for the description of the interfaces, namely, Eulerian grid based methods (interface tracking). In this thesis, the interface capturing Level Set method is used to capture the fluid-fluid interface, due to its advantages to deal with possible topological changes. In addiction, an interface tracking Lagrangian based meshless technique is used for the fluid-structure interface due to its benefits at the ensuring mass preservation. From the fluid discretisation point of view, the discretisation is based on the standard Marker-and-Cell method in conjunction with a fractional step approach for the pressure/velocity decoupling. The thesis presents a wide range of applications for multiphase flows interacting with a variety of structures (i.e. rigid and deformable) Several numerical examples are presented in order to demonstrate the robustness and applicability of the new methodology. (Abstract shortened by ProQuest.).

    Splines and local approximation of the earth's gravity field

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    Bibliography: pages 214-220.The Hilbert space spline theory of Delvos and Schempp, and the reproducing kernel theory of L. Schwartz, provide the conceptual foundation and the construction procedure for rotation-invariant splines on Euclidean spaces, splines on the circle, and splines on the sphere and harmonic outside the sphere. Spherical splines and surface splines such as multi-conic functions, Hardy's multiquadric functions, pseudo-cubic splines, and thin-plate splines, are shown to be largely as effective as least squares collocation in representing geoid heights or gravity anomalies. A pseudo-cubic spline geoid for southern Africa is given, interpolating Doppler-derived geoid heights and astro-geodetic deflections of the vertical. Quadrature rules are derived for the thin-plate spline approximation (over a circular disk, and to a planar approximation) of Stokes's formula, the formulae of Vening Meinesz, and the L₁ vertical gradient operator in the analytical continuation series solution of Molodensky's problem

    Enriched and Isogeometric Boundary Element Methods for Acoustic Wave Scattering

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    This thesis concerns numerical acoustic wave scattering analysis. Such problems have been solved with computational procedures for decades, with the boundary element method being established as a popular choice of approach. However, such problems become more computationally expensive as the wavelength of an incident wave decreases; this is because capturing the oscillatory nature of the incident wave and its scattered field requires increasing numbers of nodal variables. Authors from mathematical and engineering backgrounds have attempted to overcome this problem using a wide variety of procedures. One such approach, and the approach which is further developed in this thesis, is to include the fundamental character of wave propagation in the element formulation. This concept, known as the Partition of Unity Boundary Element Method (PU-BEM), has been shown to significantly reduce the computational burden of wave scattering problems. This thesis furthers this work by considering the different interpolation functions that are used in boundary elements. Initially, shape functions based on trigonomet- ric functions are developed to increase continuity between elements. Following that, non-uniform rational B-splines, ubiquitous in Computer Aided Design (CAD) soft- ware, are used in developing an isogeometric approach to wave scattering analysis of medium-wave problems. The enriched isogeometric approach is named the eXtended Isogeometric Boundary Element Method (XIBEM). In addition to the work above, a novel algorithm for finding a uniform placement of points on a unit sphere is presented. The algorithm allows an arbitrary number of points to be chosen; it also allows a fixed point or a bias towards a fixed point to be used. This algorithm is used for the three-dimensional acoustic analyses in this thesis. The new techniques developed within this thesis significantly reduce the number of degrees of freedom required to solve a problem to a certain accuracy—this reduc- tion is more than 70% in some cases. This reduces the number of equations that have to be solved and reduces the amount of integration required to evaluate these equations

    Triangular finite element solution to boundary value problems

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    This Thesis discusses the triangular finite element solution to second order elliptic boundary value problems. The Barycentric Coordinate system, which some engineers call the areal coordinate system, is used throughout in this Thesis. Some fundamental parts of vector calculus are developed in this coordinate system, and are applied to the triangular finite element method

    Colloquium numerical treatment of integral equations

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