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

Нестаціонарні теплові процеси в анізотропних твердих тілах

Дисертацію присвячено дослідженню теплових процесів в анізотропних твердих тілах за допомогою безсіткового методу розв’язання тривимірних задач нестаціонарної теплопровідності. Серед великого розмаїття задач математичної фізики, які нині успішно вирішуються, особливу роль займають задачі теплопровідності в анізотропних матеріалах. Насамперед це пов’язано з активним використанням анізотропних матеріалів при виготовленні великої кількості сучасних приладів та пристроїв, деталей конструкцій та машин – наприклад, трансформаторів із сердечниками з текстурованої сталі (в електротехніці), лопаток газотурбінних двигунів із жароміцних нікелевих сплавів з монокристалічною структурою (в авіації), п’єзоперетворювачів, електрооптичних модуляторів та рідкокристалічних індикаторів (в електронному приладобудуванні). Сучасні анізотропні матеріали зі складною структурою (наприклад, композитні матеріали, багатошарові матеріали, покриття, нанесені на підкладки) все частіше використовуються в новітніх інженерних розробках, а також в якості конструкційних матеріалів.У різних технологічних процесах і пристроях дані матеріали піддаються тепловому впливу, внаслідок чого в них відбуваються фізико-хімічні явища, зокрема зміна геометричних розмірів. Неконтрольоване теплове розширення конструкційних матеріалів може призвести до погіршення експлуатаційних характеристик пристрою, а також до аварійних ситуацій. Тому при створенні та використанні таких матеріалів необхідно враховувати анізотропію їх теплофізичних властивостей, а також досліджувати теплові процеси, які в них протікають. The dissertation deals with the study of thermal processes in anisotropic solids by meshless method for solving three-dimensional non-stationary heat conduction problems. Heat conduction problems in anisotropic solids play a significant role among the wide variety of problems of mathematical physics which are currently being successfully solved. First of all, it is associated with the active use of anisotropic materials in the manufacture of a large number of modern instruments and devices, structural parts and machines. For example, transformers with textured steel cores (in electrical engineering), gas-turbine engine blades of heat-resistant nickel alloys with a single-crystal structure (in aviation), piezoelectric transducers, electro-optic modulators and liquid crystal indicators (in electronic instrument engineering). Modern anisotropic materials with a complex structure (composite materials, multilayered materials, coatings on substrates, etc.) are increasingly used in advanced engineering designs and as structural materials. In various technological processes and devices, these materials are exposed to thermal effects, resulting in physical and chemical phenomena, including changes in geometric parameters. Uncontrolled thermal expansion of structural materials may lead to device performance degradation, as well as to emergency situations. Therefore, when creating and using such materials, it is necessary to take into account the anisotropy of their thermophysical properties, as well as to study the thermal processes occurring in them

Heat and moisture diffusion in spruce and wood panels computed from 3-D morphologies using the Lattice Boltzmann method

International audienceIn this paper, the Lattice Boltzmann method is used to simulate heat and mass diffusion in bio-based building materials. The numerical method is presented and the methodology developed to reduce the calculation time is described. The 3-D morphologies of spruce and wood fibers are obtained using synchrotron X-ray micro-to-mography. Equivalent macroscopic properties (heat conductivity and mass diffusivity) are therefore determined from the real micro-structure of the materials. The results reveal the anisotropy of the studied materials. The computed equivalent heat conductivity varies from − − 0.036 W m K 1 1 to − − 0.52 W m K 1 1 and the computed di-mensionless mass diffusivity varies from 0.0088 to 0.78 depending on the materials and on the diffusion directions. Using these results, morphology families are identified and simple expressions are proposed to predict the equivalent properties as a function of phase properties and solid fraction

Finite Block Method and Applications in Engineering with Functional Graded Materials

PhDFracture mechanics plays an important role in understanding the performance of all types of materials including Functionally Graded Materials (FGMs). Recently, FGMs have attracted the attention of various scholars and engineers around the world since its specific material properties can smoothly vary along the geometries. In this thesis, the Finite Block Method (FBM), based on a 1D differential matrix derived from the Lagrangian Interpolation Method, has been presented for the evaluation of the mechanical properties of FGMs on both static and dynamic analysis. Additionally, the coefficient differential matrix can be determined by a normalized local domain, such as a square for 2D, a cubic for 3D. By introducing the mapping technique, a complex real domain can be divided into several blocks, and each block is possible to transform from Cartesian coordinate (xyz) to normalized coordinate ( ) with 8 seeds for two dimensions and 20 seeds for three dimensions. With the aid of coefficient differential matrix, the differential equation is possible to convert to a series of algebraic functions. The accuracy and convergence have been approved by comparison with other numerical methods or analytical results. Besides, the stress intensity factor and T-stresses are introduced to assess the fracture characteristics of FGMs. The Crack Opening displacement is applied for the calculation of the stress intensity factor with the FBM. In addition, a singular core is adopted to combine with the blocks for the simulation of T stresses. Numerical examples are introduced to verify the accuracy of the FBM, by comparing with Finite Element Methods or analytical results. Finally, the FBM is applied for wave propagation problems in two- and three-dimensional porous mediums considering their poroelasticities. To demonstrate the accuracy of the present method, a one-dimensional analytical solution has been derived for comparison

Meshless Methods for the Neutron Transport Equation

Mesh-based methods for the numerical solution of partial differential equations (PDEs) require the division of the problem domain into non-overlapping, contiguous subdomains that conform to the problem geometry. The mesh constrains the placement and connectivity of the solution nodes over which the PDE is solved. In meshless methods, the solution nodes are independent of the problem geometry and do not require a mesh to determine connectivity. This allows the solution of PDEs on geometries that would be difficult to represent using even unstructured meshes. The ability to represent difficult geometries and place solution nodes independent of a mesh motivates the use of meshless methods for the neutron transport equation, which often includes spatially-dependent PDE coefficients and strong localized gradients. The meshless local Petrov-Galerkin (MLPG) method is applied to the steady-state and k-eigenvalue neutron transport equations, which are discretized in energy using the multigroup approximation and in angle using the discrete ordinates approximation. The MLPG method uses weighted residuals of the transport equation to solve for basis function expansion coefficients of the neutron angular flux. Connectivity of the solution nodes is determined by the shared support domain of overlapping meshless functions, such as radial basis functions (RBFs) and moving least squares (MLS) functions. To prevent oscillations in the neutron flux, the MLPG transport equation is stabilized by the streamline upwind Petrov-Galerkin (SUPG) method, which adds numerical diffusion to the streaming term. Global neutron conservation is enforced by using MLS basis and weight functions and appropriate SUPG parameters. The cross sections in the transport equation are approximated in accordance with global particle balance and without constraint on their spatial dependence or the location of the basis and weight functions. The equations for the strong-form meshless collocation approach are derived for comparison to the MLPG equations. Two integration schemes for the basis and weight functions in the MLPG method are presented, including a background mesh integration and a fully meshless integration approach. The method of manufactured solutions (MMS) is used to verify the resulting MLPG method in one, two and three dimensions. Results for realistic problems, including two-dimensional pincells, a reflected ellipsoid and a three-dimensional problem with voids, are verified by comparison to Monte Carlo simulations. Finally, meshless heat transfer equations are derived using a similar MLPG approach and verified using the MMS. These heat equation are coupled to the MLPG neutron transport equations, and results for a pincell are compared to values from a commercial pressurized water reactor.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145796/1/brbass_1.pd

Numerical Computation, Data Analysis and Software in Mathematics and Engineering

The present book contains 14 articles that were accepted for publication in the Special Issue “Numerical Computation, Data Analysis and Software in Mathematics and Engineering” of the MDPI journal Mathematics. The topics of these articles include the aspects of the meshless method, numerical simulation, mathematical models, deep learning and data analysis. Meshless methods, such as the improved element-free Galerkin method, the dimension-splitting, interpolating, moving, least-squares method, the dimension-splitting, generalized, interpolating, element-free Galerkin method and the improved interpolating, complex variable, element-free Galerkin method, are presented. Some complicated problems, such as tge cold roll-forming process, ceramsite compound insulation block, crack propagation and heavy-haul railway tunnel with defects, are numerically analyzed. Mathematical models, such as the lattice hydrodynamic model, extended car-following model and smart helmet-based PLS-BPNN error compensation model, are proposed. The use of the deep learning approach to predict the mechanical properties of single-network hydrogel is presented, and data analysis for land leasing is discussed. This book will be interesting and useful for those working in the meshless method, numerical simulation, mathematical model, deep learning and data analysis fields