Probabilistic fracture mechanics by boundary element method

In this work, a new boundary element method is presented for the Probabilistic Fracture Mechanics analysis. The method developed allows the probabilistic analysis of cracked structure accomplished by the dual boundary element method (DBEM), in which the traction integral equation is used on one of the crack faces as opposed to the usual displacement integral equation. The stress intensity factors and their first order derivatives are evaluated for mode-I and mixed-mode fracture problems. A new boundary element formulation is derived and implemented to evaluate the design variables sensitivities. This method involves the solution of matrix systems formed by the direct differentiation of the discretised dual boundary element equations with respect to the each random parameter. The derivatives of fracture parameters with respect to design variables are calculated using implicit differentiation method (IDM) in DBEM for mode-I and mixed-mode fracture problems. The gradient of performance function is determined analytically and the total derivative method (TDM) is used in probabilistic fatigue crack growth problems. The randomness in the geometry, material property and the applied stress are considered in 2-D fracture problems; while initial crack size, final crack size, material property and applied stress are considered in fatigue crack growth. Uncertainties in other aspects of the problem can be included. First-Order Reliability Method (FORM) is used for predicting the reliability of cracked structures. The Hasofer Lind Rackwitz Fiessler algorithm is used to find the most probable point, referred as reliability index. Finally, the validation and applications of the stochastic boundary element coupled with FORM are presented. Numerical calculations are shown to be in good agreement either with the analytical solution or Monte Carlo Simulation

Delamination effect on response of a composite beam by wavelet spectral finite element method

Transform methods are very useful to solve the ordinary and partial differential equations. Fourier and Laplace transforms are the most commonly used transforms. Wavelet transforms are most popular with electrical and communication engineers to analyse the signals. From last few years, Wavelet transforms are in use for structural engineering problems, like solution of ordinary and partial differential equations. Dynamical problems in structural engineering fall under two categories, one involving low frequencies (structural dynamics problems) and the other involving high frequencies (wave propagation problems). Spectral Finite Element (SFE) method is a transform method to solve the high frequency excitation problems which are encountered in structural engineering. SFE based on Fourier transforms has high limitations in handling finite structures and boundary conditions. SFE based with wavelet transforms is a very good tool to analyse the dynamical problems and eliminate many limitations. In this project, a model for embedded de-laminated composite beam is developed using the wavelet based spectral finite element (WSFE) method for the de-lamination effect on response using wave propagation analysis. The simulated responses are used as surrogate experimental results for the inverse problem of detection of damage using wavelet filtering. The technique used to model a structure that, through width de-lamination subdivides the beam into base-laminates and sub-laminates along the line of de-lamination. The base-laminates and sub-laminates are treated as structural waveguides and kinematics are enforced along the connecting line. These waveguides are modeled as Timoshenko beams with elastic and inertial coupling and the corresponding spectral elements have three degrees of freedom, namely axial, transverse and shear displacements at each node. The internal spectral elements in the region of de-lamination are assembled assuming constant cross sectional rotation and equilibrium at the interfaces between the base-laminates and sub-laminates. Finally, the redundant internal spectral element nodes are condensed out to form two-noded spectral elements with embedded de-lamination. The response is being obtained by coding programs in MATLAB

Engineering evaluations and studies. Volume 2: Exhibit B, part 1

Ku-band communication system analysis, S-band system investigations, payload communication investigations, shuttle/TDRSS and GSTDN compatibility analysis are discussed

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

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

A finite element analysis of crack propagation problems with applications to seismology

Imperial Users onl

Efficient Geomechanical Simulations of Large-Scale Naturally Fractured Reservoirs Using the Fast Multipole-Displacement Discontinuity Method (FM-DDM)

Geothermal and unconventional reservoirs play an important role in supplying fuel for a growing energy demand in the United States. The development of such reservoirs relies on creating a fracture network to provide flow and transport conduits during injection and production operations. The Displacement Discontinuity Method (DDM) is frequently used for modeling the behavior of fractures embedded in elastic and poroelastic rocks. However, DDM requires the calculation of the influence among all fractures being computationally inefficient for large systems of cracks. It demands quadratic and cubic complexity of memory and solution time by direct methods, respectively, limiting its application to only small-scale situations. Recent fast summation techniques such as the Fast Multipole Method (FMM) have been used to speed up the solution of several boundary element problems using modest computational resources. FMM relies in accelerating matrix-vector products in iterative methods by splitting the computation of the influences among elements into near and far-field interactions. While the former are calculated similarly to the conventional DDM, the latter, where most of the interactions are found, are efficiently approximated by the FMM using analytical multipole and local expansions. However, in spite of its immediately apparent application in the geomechanic context, FMM has been limited to only certain fracture problems because those analytical expansions are only available for selected fundamental solutions and the development for new ones requires complex mathematical derivations even for those kernels of simple form. This work presents a new method called Fast Multipole–Displacement Discontinuity Method (FM-DDM) for an efficient flow-geomechanical simulation of large-scale naturally fractured reservoirs undergoing fluid injection and extraction. The approach combines both DDM and FMM using for the latter a kernel-independent version where multipole and local expansions are not required opening a range of potential applications within the geothermal and oil industries. Several case studies involving fracture networks with up to one hundred thousands of boundary elements were presented to evaluate accuracy, computational efficiency and applications of the FMM approach. From the results, FM-DDM showed an excellent agreement with well-known benchmark solutions outperforming DDM with linear complexity in both memory and execution time. In addition, a variety of large-scale geomechanical applications were efficiently evaluated with FM-DDM involving interactions between transverse hydraulic fractures and a fracture network, fast visualization of high-resolution stress distribution, and the design of exploitation strategies in elastic and poroelastic fractured reservoirs

Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations

Effective computational models of multiscale problems have to account for the impact of unresolved physics on the resolved scales. This dissertation advances our fundamental understanding of multiscale models and develops a mathematically rigorous closure modeling framework by combining the Mori-Zwanzig (MZ) formalism of Statistical Mechanics with the variational multiscale (VMS) method. This approach leverages scale-separation projectors as well as phase-space projectors to provide a systematic modeling approach that is applicable to complex non-linear partial differential equations. %The MZ-VMS framework is investigated in the context of turbulent flows. Spectral as well as continuous and discontinuous finite element methods are considered. The MZ-VMS framework leads to a closure term that is non-local in time and appears as a convolution or memory integral. The resulting non-Markovian system is used as a starting point for model development. Several new insights are uncovered: It is shown that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and, in the case of finite elements, inter-element jumps. Connections between MZ-based methods, artificial viscosity, and VMS models are explored. The MZ-VMS framework is investigated in the context of turbulent flows. Large eddy simulations of Burgers' equation, turbulent flows, and magnetohydrodynamic turbulence using spectral and discontinuous Galerkin methods are explored. In the spectral method case, we show that MZ-VMS models lead to substantial improvements in the prediction of coarse-grained quantities of interest. Applications to discontinuous Galerkin methods show that modern flux schemes can inherently capture memory effects, and that it is possible to guarantee non-linear stability and conservation via the MZ-VMS approach. We conclude by demonstrating how ideas from MZ-VMS can be adapted for shock-capturing and filtering methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145847/1/parish_1.pd

Fatigue reliability of ship structures

Today we are sitting on a huge wealth of structural reliability theory but its application in ship design and construction is far behind. Researchers and practitioners face a daunting task of dove-tailing the theoretical achievements into the established processes in the industry. The research is aimed to create a computational framework to facilitate fatigue reliability of ship structures. Modeling, transformation and optimization, the three key elements underlying the success of computational mechanics are adopted as the basic methodology through the research. The whole work is presented in a way that is most suitable for software development. The foundation of the framework is constituted of reliability methods at component level. Looking at the second-moment reliability theory from a minimum distance point of view the author derives a generic set of formulations that incorporate all major first and second order reliability methods (FORM, SORM). Practical ways to treat correlation and non- Gaussian variables are discussed in detail. Monte Carlo simulation (MCS) also accounts for significant part of the research with emphasis on variance reduction techniques in a proposed Markov chain kernel method. Existing response surface methods (RSM) are reviewed and improved with much weight given to sampling techniques and determination of the quadratic form. Time-variant problem is touched upon and methods to convert it to nested reliability problems are discussed. In the upper layer of the framework common fatigue damage models are compared. Random process simulation and rain-flow counting are used to study effect of wide-banded non-Gaussian process. At the center of this layer is spectral fatigue analysis based on SN curve and first-principle stress and hydrodynamic analysis. Pseudo-excitation is introduced to get linear equivalent stress RAO in the non-linear ship-wave system. Finally response surface method is applied to this model to calculate probability of failure and design sensitivity in the case studies of a double hull oil tanker and a bulk carrier