A high-performance boundary element method and its applications in engineering

As a semi-numerical and semi-analytical method, owing to the inherent advantage, of boundary-only discretisation, the boundary element method (BEM) has been widely applied to problems with complicated geometries, stress concentration problems, infinite domain problems, and many others. However, domain integrals and non-symmetrical and dense matrix systems are two obstacles for BEM which have hindered the its further development and application. This thesis is aimed at proposing a high-performance BEM to tackle the above two drawbacks and broaden the application scope of BEM. In this thesis, a detailed introduction to the traditional BEM is given and several popular algorithms are introduced or proposed to enhance the performance of BEM. Numerical examples in heat conduction analysis, thermoelastic analysis and thermoelastic fracture problems are performed to assess the efficiency and correction of the algorithms. In addition, necessary theoretical derivations are embraced for establishing novel boundary integral equations (BIEs) for specific engineering problems. The following three parts are the main content of this thesis. (1) The first part (Part II consisting of two chapters) is aimed at heat conduction analysis by BEM. The coefficient matrix of equations formed by BEM in solving problems is fully-populated which occupy large computer memory. To deal with that, the fast multipole method (FMM) is introduced to energize the line integration boundary element method (LIBEM) to performs better in efficiency. In addition, to compute domain integrals with known or unknown integrand functions which are caused by heat sources or heterogeneity, a novel BEM, the adaptive orthogonal interpolation moving least squares (AOIMLS) method enhanced LIBEM, which also inherits the advantage of boundary-only discretisation, is proposed. Unlike LIBEM, which is an accurate and stable method for computing domain integrals, but only works when the mathematical expression of integral function in domain integrals is known, the AOIMLS enhanced LIBEM can compute domain integrals with known or unknown integral functions, which ensures all the nonlinear and nonhomogeneous problems can be solved without domain discretisation. In addition, the AOIMLS can adaptively avoid singular or ill-conditioned moment matrices, thus ensuring the stability of the calculation results. (2) In the second part (Part III consisting of four chapters), the thermoelastic problems and fracture problems are the main objectives. Due to considering thermal loads, domain integrals appear in the BIEs of the thermoelastic problems, and the expression of integrand functions is known or not depending on the temperature distribution given or not, the AOIMLS enhanced LIBEM is introduced to conduct thermoelasticity analysis thereby. Besides, a series of novel unified boundary integral equations based on BEM and DDM are derived for solving fracture problems and thermoelastic fracture problems in finite and infinite domains. Two sets of unified BIEs are derived for fracture problems in finite and infinite domains based on the direct BEM and DDM respectively, which can provide accurate and stable results. Another two sets of BIEs are addressed by employing indirect BEM and DDM, which cannot ensure a stable result, thereby a modified indirect BEM is proposed which performs much more stable. Moreover, a set of novel BIEs based on the direct BEM and DDM for cracked domains under thermal stress is proposed. (3) In the third part (Part IV consisting of one chapter), a high-efficiency combined BEM and discrete element method (DEM) is proposed to compute the inner stress distribution and particle breakage of particle assemblies based on the solution mapping scheme. For the stress field computation of particles with similar geometry, a template particle is used as the representative particle, so that only the related coefficient matrices of one template particle in the local coordinate system are needed to be calculated, while the coefficient matrices of the other particles, can be obtained by mapping between the local and global coordinate systems. Thus, the combined BEM and DEM is much more effective when modelling a large-scale particle system with a small number of distinct possible particle shapes. Furthermore, with the help of the Hoek-Brown criterion, the possible cracks or breakage paths of a particle can be obtained

Meshfree Methods Using Localized Kernel Bases

Radial basis functions have been used to construct meshfree numerical methods for interpolation and for solving partial differential equations. Recently, a localized basis of radial basis functions has been developed on the sphere. In this dissertation, we investigate applying localized kernel bases for interpolation, approximation, and for novel discretization methods for numerically solving partial differential equations and integral equations. We investigate methods for partial differential equations on spheres using newly explored bases constructed from radial basis functions and associated quadrature methods. We explore applications of radial basis functions to anisotropic nonlocal diffusion problems and we develop theoretical frameworks for these methods

Enriched finite element methods : advances & applications

This thesis presents advances and applications of the eXtended Finite Element Method (XFEM). The novelty of the XFEM is the enrichment of the primary variables in the elements intersected by the discontinuity surface by appropriate functions. The enrichment scheme carries the local behaviour of the problem and the main advantage is that the method does not require themesh to conform to the internal boundaries. But this flexibility comes with associated difficulties: (1) Blending problem; (2) Numerical integration of enrichment functions and (3) sub-optimal rate of convergence. This thesis addresses the difficulty in the numerical integration of the enrichment functions in the XFEM by proposing two new numerical integration schemes. The first method relies on conformal mapping, where the regions intersected by the discontinuity surface are mapped onto a unit disk. The second method relies on strain smoothing applied to discontinuous finite element approximations. By writing the strain field as a non-local weighted average of the compatible strain field, integration on the interior of the finite elements is transformed into boundary integration, so that no sub-division into integration cells is required. The accuracy and the efficiency of both the methods are studied numerically with problems involving strong and weak discontinuities. The XFEM is applied to study the crack inclusion interaction in a particle reinforced composite material. The influence of the crack length, the number of inclusions and the geometry of the inclusions on the crack tip stress field is numerically studied. Linear natural frequencies of cracked functionally graded material plates are studied within the framework of the XFEM. The effect of the plate aspect ratio, the crack length, the crack orientation, the gradient index and the influence of cracks is numerically studied. LATEX-ed Friday, October 14, 2011; 10:55am © Sundararajan NatarajanEThOS - Electronic Theses Online ServiceGBUnited Kingdo

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Enriched finite element methods: advances & applications

This thesis presents advances and applications of the eXtended Finite Element Method (XFEM). The novelty of the XFEM is the enrichment of the primary variables in the elements intersected by the discontinuity surface by appropriate functions. The enrichment scheme carries the local behaviour of the problem and the main advantage is that the method does not require themesh to conform to the internal boundaries. But this flexibility comes with associated difficulties: (1) Blending problem; (2) Numerical integration of enrichment functions and (3) sub-optimal rate of convergence. This thesis addresses the difficulty in the numerical integration of the enrichment functions in the XFEM by proposing two new numerical integration schemes. The first method relies on conformal mapping, where the regions intersected by the discontinuity surface are mapped onto a unit disk. The second method relies on strain smoothing applied to discontinuous finite element approximations. By writing the strain field as a non-local weighted average of the compatible strain field, integration on the interior of the finite elements is transformed into boundary integration, so that no sub-division into integration cells is required. The accuracy and the efficiency of both the methods are studied numerically with problems involving strong and weak discontinuities. The XFEM is applied to study the crack inclusion interaction in a particle reinforced composite material. The influence of the crack length, the number of inclusions and the geometry of the inclusions on the crack tip stress field is numerically studied. Linear natural frequencies of cracked functionally graded material plates are studied within the framework of the XFEM. The effect of the plate aspect ratio, the crack length, the crack orientation, the gradient index and the influence of cracks is numerically studied. LATEX-ed Friday, October 14, 2011; 10:55am © Sundararajan Nataraja