Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions

A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provide

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

Exploring Phononic Properties of Two-Dimensional Materials using Machine Learning Interatomic Potentials

Phononic properties are commonly studied by calculating force constants using the density functional theory (DFT) simulations. Although DFT simulations offer accurate estimations of phonon dispersion relations or thermal properties, but for low-symmetry and nanoporous structures the computational cost quickly becomes very demanding. Moreover, the computational setups may yield nonphysical imaginary frequencies in the phonon dispersion curves, impeding the assessment of phononic properties and the dynamical stability of the considered system. Here, we compute phonon dispersion relations and examine the dynamical stability of a large ensemble of novel materials and compositions. We propose a fast and convenient alternative to DFT simulations which derived from machine-learning interatomic potentials passively trained over computationally efficient ab-initio molecular dynamics trajectories. Our results for diverse two-dimensional (2D) nanomaterials confirm that the proposed computational strategy can reproduce fundamental thermal properties in close agreement with those obtained via the DFT approach. The presented method offers a stable, efficient, and convenient solution for the examination of dynamical stability and exploring the phononic properties of low-symmetry and porous 2D materials

Two-dimensional models of hydrodynamical accretion flows into black holes

We present a systematic numerical study of two-dimensional axisymmetric accretion flows around black holes. The flows have no radiative cooling and are treated in the framework of the hydrodynamical approximation. The models calculated in this study cover the large range of the relevant parameter space. There are four types of flows, determined by the values of the viscosity parameter $\alpha$ and the adiabatic index $\gamma$: convective flows, large-scale circulations, pure inflows and bipolar outflows. Thermal conduction introduces significant changes to the solutions, but does not create a new flow type. Convective accretion flows and flows with large-scale circulations have significant outward-directed energy fluxes, which have important implications for the spectra and luminosities of accreting black holes.Comment: 43 pages, 23 figures, submitted to Ap

Numerical study of geometrical dispersion in self-affine rough fractures

We report a numerical study of passive tracer dispersion in fractures with rough walls modeled as the space between two complementary self-affine surfaces rigidly translated with respect to each other. Geometrical dispersion due to the disorder of the velocity distribution is computed using the lubrication approximation. Using a spectral perturbative scheme to solve the flow problem and a mapping coordinate method to compute dispersion, we perform extensive ensemble averaged simulations to test theoretical predictions on the dispersion dependence on simple geometrical parameters. We observe the expected quadratic dispersion coefficient dependence on both the mean aperture and the relative shift of the crack as of well as the anomalous dispersion dependence on tracer traveling distance. We also characterize the anisotropy of the dispersion front, which progressively wrinkles into a self-affine curve whose exponent is equal to that of the fracture surface