Дисертації з теми "Numerically stiff"

Some real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The partial differential equations (PDEs) governing such situations are categorized as "stiff" PDEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of DEs. Our attention has been focused on the explicit Exponential Time Differencing (ETD) integrators that are designed to solve stiff semi-linear problems. Semi-linear PDEs can be split into a linear part, which contains the stiffest part of the dynamics of the problem, and a nonlinear part, which varies more slowly than the linear part. The ETD methods solve the linear part exactly, and then explicitly approximate the remaining part by polynomial approximations. The first aspect of this project involves an analytical examination of the methods' stability properties in order to present the advantage of these methods in overcoming the stability constraints. Furthermore, we discuss the numerical difficulties in approximating the ETD coefficients, which are functions of the linear term of the PDE. We address ourselves to describing various algorithms for approximating the coefficients, analyze their performance and their computational cost, and weigh their advantages for an efficient implementation of the ETD methods. The second aspect is to perform a variety of numerical experiments to evaluate the usefulness of the ETD methods, compared to other competing stiff integrators, for integrating real application problems. The problems considered include the Kuramoto-Sivashinsky equation, the nonlinear Schrödinger equation and the nonlinear Thin Film equation, all in one space dimension. The main properties tested are accuracy, start-up overhead cost and overall computation cost, since these parameters play key roles in the overall efficiency of the methods.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003.
Includes bibliographical references (leaves 202-209). Also available in electronic version. Access restricted to campus users.

The aim of this project is to investigate the stability of cut slopes in stiff clay. The findings are subsequently applied to model stabilisation with piles, used to remediate failure of existing slopes and stabilise potentially unstable slopes created by widening transport corridors. Stiff clay is a strain softening material, meaning that soil strength reduces as the material is strained, for example in the formation of a slip surface. In an excavated slope this can lead to a progressive, brittle slope failure. Simulation of strain softening behaviour is therefore an important aspect to model. The interaction of piles and stiff clay cut slopes is investigated using the Imperial College Geotechnics section's finite element program ICFEP. In designing a suitable layout of the finite element mesh, preliminary analyses found the two existing local strain softening models to be very dependent on the size and arrangement of elements. To mitigate this shortcoming, a nonlocal strain softening model was implemented in ICFEP. This model controls the development of strain by relating the surrounding strains to the calculation of strain at that point, using a weighting function. Three variations of the nonlocal formulation are evaluated in terms of their mesh dependence. A parametric study with simple shear and biaxial compression analyses evaluated the new parameters required by the nonlocal strain softening model. The nonlocal results demonstrated very low mesh dependence and a clear improvement on the local strain softening models. In order to examine the mesh dependence of the new model in a boundary value problem compared to the local strain softening approach, excavated slope analyses without piles were first performed. The slope was modelled in plane strain with coupled consolidation. These analyses also investigated other factors such as the impact of adopting a small strain stiffness material model on the development of the failure mechanism and the impact of the spatial variation of permeability on the time to failure. The final set of analyses constructed vertical stabilisation piles in the excavated slope, represented as either solid elements or one dimensional beam elements. The development of various failure mechanisms for stiff clay cuttings was found to be dependent on pile location, pile diameter and pile length. This project provides an insight into the constitutive model and boundary conditions required to study stabilisation piles in a stiff clay cutting. The nonlocal model performed very well to reduce mesh dependence, confirming the biaxial compression results. However, the use of coupled consolidation was found to cause further mesh dependence of the results.

In many applications, ordinary differential equations can be additively partitioned \[y'=f(y)=\sum_{m=1}^{N}\f{}{m}(y).] It can be advantageous to discriminate between the different parts of the right-hand side according to stiffness, nonlinearity, evaluation cost, etc. In 2015, Sandu and G\"{u}nther \cite{sandu2015gark} introduced Generalized Additive Runge-Kutta (GARK) methods which are given by \begin{eqnarray*} Y_{i}^{\{q\}} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}a_{i,j}^{\{q,m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\\ & & \text{for } i=1,\dots,s^{\{q\}},\,q=1,\dots,N\\ y_{n+1} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}b_{j}^{\{m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\end{eqnarray*} with the corresponding generalized Butcher tableau \[\begin{array}{c|ccc} \c{}{1} & \A{1,1} & \cdots & \A{1,N}\\\vdots & \vdots & \ddots & \vdots\\ \c{}{N} & \A{N,1} & \cdots & \A{N,N}\\\hline & \b{}{1} & \cdots & \b{}{N}\end{array}\] The diagonal blocks $\left(\A{q,q},\b{}{q},\c{}{q}\right)$ can be chosen for example from standard Runge-Kutta methods, and the off-diagonal blocks $\A{q,m},\:q\neq m,$ act as coupling coefficients between the underlying methods. The case when $N=2$ and both diagonal blocks are implicit methods (IMIM) is examined. This thesis presents order conditions and simplifying assumptions that can be used to choose the off-diagonal coupling blocks for IMIM methods. Error analysis is performed for stiff problems of the form \begin{eqnarray*}\dot{y} & = & f(y,z)\\ \epsilon\dot{z} & = & g(y,z)\end{eqnarray*} with small stiffness parameter $\epsilon.$ As $\epsilon\to 0,$ the problem reduces to an index 1 differential algebraic equation provided $g_{z}(y,z)$ is invertible in a neighborhood of the solution. A tree theory is developed for IMIM methods applied to the reduced problem. Numerical results will be presented for several IMIM methods applied to the Van der Pol equation.

Ce travail est consacré à l’étude théorique et numérique de systèmes hyperboliques d’équations aux dérivées partielles et aux équations de transport, avec des termes de relaxation et des conditions aux bords. Dans la première partie, on étudie la stabilité raide d’approximations numériques par différences finies du problème mixte donnée initiale-donnée au bord pour l’équation des ondes amorties dans le quart de plan. Dans le cadre du schéma discret en espace, nous proposons deux méthodes de discrétisation de la condition de Dirichlet. La première est la technique de sommation par partie et la seconde est basée sur le concept de condition au bord transparente. Nous proposons également une comparaison numérique des deux méthodes, en particulier de leur domaine de stabilité. La deuxième partie traite de schémas numériques d’ordre élevé pour l’équation de transport avec une donnée entrante sur domaine borné. Nous construisons, implémentons et analysons la procédure de Lax-Wendroff inverse au bord entrant. Nous obtenons des taux de convergence optimaux en combinant des estimations de stabilité précises pour l’extrapolation des conditions au bord avec des développements de couche limite numérique. Dans la dernière partie, nous étudions la stabilité de solutions stationnaires pour des systèmes non conservatifs avec des termes géométrique et de relaxation. Nous démontrons que les solutions stationnaires sont stables parmi les solutions entropique processus, qui généralisent le concept de solutions entropiques faibles. Nous supposons essentiellement que le système est complété par une entropie partiellement convexe et que, selon la dissipation du terme de relaxation, la stabilité ou la stabilité asymptotique des solutions stationnaires est obtenue
The dissertation focuses on the study of the theoretical and numerical analysis of hyperbolic systems of partial differential equations and transport equations, with relaxation terms and boundary conditions. In the first part, we consider the stiff stability for numerical approximations by finite differences of the initial boundary value problem for the linear damped wave equation in a quarter plane. Within the framework of the difference scheme in space, we propose two methods of discretization of Dirichlet boundary condition. The first is the technique of summation by part and the second is based on the concept of transparent boundary conditions. We also provide a numerical comparison of the two numerical methods, in particular in terms of stability domain. The second part is about high order numerical schemes for transport equations with nonzero incoming boundary data on bounded domains. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at incoming boundary. We obtain optimal convergence rates by combining sharp stability estimate for extrapolation boundary conditions with numerical boundary layer expansions. In the last part, we study the stability of stationary solutions for non-conservative systems with geometric and relaxation source term. We prove that stationary solutions are stable among entropy process solution, which is a generalisation of the concept of entropy weak solutions. We mainly assume that the system is endowed with a partially convex entropy and, according to the entropy dissipation provided by the relaxation term, stability or asymptotic stability of stationary solutions is obtained

This thesis presents new numerical methods for solving differential equations with periodicity. Spectral methods for solving linear and nonlinear ODEs, linear ODE eigenvalue problems and linear time-dependent PDEs on a periodic interval are reviewed, and a novel approach for computing multiplication matrices is presented. Choreographies, periodic solutions of the n-body problem that share a common orbit, are computed for the first time to high accuracy using an algorithm based on approximation by trigonometric polynomials and optimization techniques with exact gradient and exact Hessian matrix. New choreographies in spaces of constant curvature are found. Exponential integrators for solving periodic semilinear stiff PDEs in 1D, 2D and 3D periodic domains are reviewed, and 30 exponential integrators are compared on 11 PDEs. It is shown that the complicated fifth-, sixth- and seventh-order methods do not really outperform one of the simplest exponential integrators, the fourth-order ETDRK4 scheme of Cox and Matthews. Finally, algorithms for solving semilinear stiff PDEs on the sphere with spectral accuracy in space and fourth-order accuracy in time are proposed. These are based on a new variant of the double Fourier sphere method in coefficient space and standard implicit-explicit time-stepping schemes. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform better. The algorithms described in each chapter of this thesis have been implemented in MATLAB and made available as part of Chebfun.

Various subsurface engineering activities, including the stimulation of unconventional hydrocarbon reservoirs, the development of geothermal energy and the drilling and blasting operations, have been increasingly carried out in sedimentary rocks with layering structures. The success of these activities is reliant on the formation of fracture network created by hydraulic fracturing or the extension and interconnection of fractures to break the layered rock. This thesis dedicates to the fracture behaviours of layered rock with alternating stiff and soft layers. First, a new damage-plasticity constitutive model which takes account of the effect of confining pressure and strain rate on the strength and post-peak behaviour is proposed for layered rocks’ components (e.g., stiff layers, soft layers, and layer interfaces) subjected to various loading scenarios. The robustness and accuracy of the new model are demonstrated by validating against available experimental results and by benchmarking with the reported simulations. Then, the new constitutive model is used to numerically explore the fracture evolution behaviour of layered rock discs in the Brazilian test. The effects of inclination angle, Young's modulus of layer interface and mechanical contrast ratio on the fracture mechanism of layered rock disc are investigated. Finally, a versatile hydromechanical coupled finite-discrete element method is employed to simulate the non-planar three-dimensional simultaneous growth of multiple hydraulic fractures in layered tight reservoirs with various mechanical contrast ratios. The mechanism behind the simultaneous growth and methods to promote the simultaneous growth are also discussed. The numerical results obtained from this thesis provide some guidelines in designing the engineering projects conducted in layered rocks and thus help field operators to maximize the productivity.

The primary aim of this thesis is to calculate the numerical solution of a given stiff system of ordinary differential equations. We deal with the implementation of the implicit Runge-Kutta methods, in particular for Radau IIA order 5 which is now a competitive method for solving stiff initial value problems. New software based on Radau IIA, called IRKMR5 written in MATLAB has been developed for fixed order (order 5) with variable stepsizes, which is quite efficient when it is used to solve stiff problems. The code is organized in a modular form so that it facilitates both the understanding of it and its modification whenever needed. The new software is not only more functional than its Fortran 77 Radau IIA counterpart but also more robust and better documented. When implicit methods are used to solve nonlinear problems it is necessary to solve systems of nonlinear algebraic equations. New investigations for a modified Newton iteration are undertaken. This new strategy manages the iterative solutions of nonlinear equations in the ODEs solver. It also involves when to re-evaluate the Jacobian and the iteration matrix. The strategy also significantly reduces the number of function evaluations and linear solves. We subsequently consider the mathematical analysis of the nonlinear algebraic equations that arise from using s-stage fully implicit Runge-Kutta methods. Results for uniqueness of solutions and an error bound was established. The termination criterion in the iterative solution of the nonlinear equations is also studied as well as two types of termination criterion (displacement and the residual test). The residual test has been compared with the displacement test on some test examples and the results are tabulated.

The mechanisms that control the lateral response of stiff column-supported shallow foundations, resulting from the application of horizontal load on shallow foundations supported by stiff columns, are uncertain. Stiff columns constructed in soft clayey soil have been used to support retaining walls and in such cases, the lateral thrust applied behind these geotechnical structures is a source of horizontal loading. For seismic events, stiff columns constructed in soft clayey soil have been used to support shallow foundations subjected to horizontal load coming from the upper structure of buildings. Due to its practical applications, it has become important to understand the consequences of subjecting a shallow foundation supported by stiff columns to horizontal load by identifying the factors that control the lateral response of such systems. A series of centrifuge tests were carried out to examine the lateral response of stiff column-supported shallow foundations. The experimental trends suggested that the thickness of the coarse-granular mattress placed above the soil-column composite, called the Load Transfer Platform (LTP), controlled the lateral capacity and the overall lateral response of these systems. A numerical study using the finite element method confirmed the experimental trends. A parametric analysis was conducted with the purpose of investigating the influence of different geometry-based and material-based variables in the lateral response of these systems. The results of the parametric analysis further confirmed the importance of the thickness of the LTP in controlling the lateral response. The parametric results also emphasized the contribution of other variables to this lateral response, and these variables included the undrained shear strength of the soft clayey soil around the stiff columns, the stiff column diameter, and the spacing of the stiff columns after they are constructed in the soft clayey soil.
Doctor of Philosophy

Based on the laboratory, field and numerical results, the post-rupture strength defined firstly by Burland (1990) is verified and introduced in this thesis to describe the two-stage post-peak strain-softening characteristics of stiff clays. The first softening stage is induced by the loss of interbonding between particles, i.e. the cohesive component of strength at relatively small displacement and hence termed as cementation loss; while the second is due to the gradual realignment, i.e. reorientation of clay particles at large displacements and can be called as gradual frictional resistance loss. A generalised constitutive model is then established to simulate the two-stage softening characteristics of stiff clays by modifying the Mohr-Coulomb model. This model is formulated and then implemented into an explicit finite difference program FLAC. The new model is applied to simulate laboratory tests such as triaxial compression tests and direct shear box test to depict the new model. The numerical results demonstrate the capability and efficiency of the modified softening model to model the two-stage strain-softening behaviour of stiff clays. A series of analyses of delayed progressive failure of cut slopes in stiff clays have been performed using two-stage softening model incorporating post-rupture strength. The numerical results reproduce well the progressive failure process, position of failure surface and failure time, which proves further the validity of the new model. Meanwhile, parametric analyses are also carried out to demonstrate the general influence of post-rupture strength. The results demonstrate that the slope stability with the adoption of two-stage softening model is reduced compared with that using one-stage softening model due to the quicker cohesion reduction with deviatoric plastic strain in the first softening stage of two-stage softening model. Both post-rupture strength concept and two-stage softening model are applied to the modelling of a famous case¿ Aznalcóllar dam failure under both inhomogeneous and homogeneous hypotheses. The simulations reproduced well the failure of Aznalcóllar dam including the location and shape of the slip surface, the progressive failure course and the development of pore water pressure in terms of the developments of shear strain rate, shear strain increment, displacement, velocity and strength parameter softening. The mechanism of Aznalcóllar dam failure is deemed to be progressive failure mainly due to the softening of Guadalquivir blue clay. The developments of average stress ratio, average residual factor, average brittleness, average stress path, the distribution of shear stress and mobilised strength parameters along the slip surface confirm further the mechanism of progressive failure of Aznalcóllar dam with these values to be intermediate between peak and residual values during the failure course. The post-rupture state could be thought as the average one at initial failure. At final failure, most part of the slip surface is at residual state, especially along the horizontal part. The Aznalcóllar dam failure is sensitive to the softening rate. Larger rates will induce earlier failure and no failure will occur with slow softening rates. Only an appropriate setting of softening rates can cause failure at final phase under both inhomogeneous and homogeneous hypotheses. Finally, the post-rupture concept is introduced to derive analytical solutions to limit pressure, the stress, strain, and displacement fields for the cylindrical cavity expansion in stiff overconsolidated clay. The results of computational examples and the similarity between numerical solution and analytical one verify the reasonableness of the analytical solution to cavity expansion in stiff clays with two-stage softening characteristics.
Basado en los datos de laboratorio y de campo y los resultados numéricos, la resistencia post-ruptura definida por Burland (1990) se verifica y introduce para describir las dos etapas de ablandamiento post-ruptura de las arcillas rígidas. La primera etapa del ablandamiento está inducida por la pérdida de inter-conexión entre las partículas, o de la componente cohesiva de la resistencia a desplazamientos relativamente pequeños; la segunda etapa es debida al realineamiento gradual o reorientación de las partículas de arcilla a grandes desplazamientos y se puede llamar como pérdida gradual de la resistencia friccional. Se establece un modelo constitutivo general para simular las características de ablandamiento en dos etapas de las arcillas rígidas, modificando el modelo Mohr-Coulomb. La formulación de este modelo es ilustrada detalladamente y luego implementada en un programa explícito de diferencia finitas FLAC. Se aplica el nuevo modelo para simular los ensayos de laboratorio como ensayos de compresión triaxial y ensayos de corte directo. Los resultados numéricos demuestran la capacidad y eficiencia del modelo modificado de ablandamiento para reproducir el comportamiento de ablandamiento por deformación en dos etapas de las arcillas rígidas. Se han realizado una serie de análisis de la rotura progresiva diferida de taludes en arcillas rígidas, utilizando el nuevo modelo de ablandamiento en dos etapas con incorporación de la resistencia de post-ruptura. Los resultados numéricos reproducen bastante bien el proceso progresivo de la rotura, la posición de la superficie de rotura y el tiempo de rotura, lo cual brinda validez adicional al nuevo modelo. Adicionalmente, se ha llevado a cabo un análisis paramétrico para demostrar la influencia general de la resistencia de post-ruptura. Los resultados demuestran que se reduce la estabilidad de los taludes con la adopción del modelo de ablandamiento de dos etapas en comparación con lo de solo una etapa, debido a la reducción más rápida de la cohesión con la deformación plástica en primera fase de ablandamiento del modelo de ablandamiento de dos etapas. El concepto de la resistencia post-ruptura y el modelo de ablandamiento en dos etapas son aplicados en la simulación numérica del famoso caso de la rotura de la presa de Aznalcóllar bajo las dos hipótesis no homogéneos y homogénea. Las simulaciones reproducen bien la rotura de la presa de Aznalcóllar incluyendo el proceso progresivo de la rotura, la ubicación y la forma de la superficie de deslizamiento y el desarrollo de la presión del agua en los poros, según el desarrollo del estado de la plasticidad, la velocidad de la deformación de corte, el incremento de la deformación de corte, el desplazamiento, la velocidad y parámetros de resistencia del ablandamiento. El desarrollo de la relación media de las tensiones, el factor residual, la fragilidad media, la trayectoria de tensiones media y la distribución de las tensiones de corte movilizadas y los parámetros de resistencia movilizados a lo largo de la superficie de deslizamiento, confirman el mecanismo de rotura progresiva de la presa de Aznalcóllar debida principalmente al ablandamiento de la masa de arcilla azul del Guadalquivir, con estos valores en el intermedio entre los valores pico y residual durante el proceso de la rotura. La rotura de la presa de Aznalcóllar es sensible a la velocidad de ablandamiento de la deformación de corte plástica. Finalmente, se introduce el concepto de la resistencia post-ruptura de las arcillas rígidas para derivar las soluciones analíticas para la presión límite, las tensiones, las deformaciones y campos de desplazamientos de la expansión de cavidades cilíndricas en arcillas rígidas. Los resultados de los ejemplos de cálculo y la similitud entre la solución numérica y analítica verifican la razonabilidad de la solución analítica a la expansión de la cavidad en arcillas rígidas con las características de ablandamiento de dos etapas.

There is ample evidence for the existence of dark matter (DM) from cosmological observations. However, despite zealous efforts by the experimental particle physics community over the past decades, no confirmed detection of a DM particle has been reported to date, and its nature hence remains a mystery. A central theme of this work is a particular type of interaction between DM and the visible Universe, whose detection (or non-detection) can potentially shed light onto the particle nature of DM, namely DM annihilation into standard model particles. We develop numerical and deep learning methods for different problems relating to DM annihilation, and we present and discuss the results obtained with our methods. Another subject matter is the cosmic age between Recombination and Reionisation, i.e. the Dark Ages and Cosmic Dawn, with a particular emphasis on the 21cm line from neutral hydrogen, whose detection by next-generation radio telescopes - most notably the Square Kilometer Array (SKA) - promises major discoveries in the coming years. First, we devise a new method for incorporating DM annihilation feedback (DMAF) into cosmological simulations, and we study the effect on the intergalactic medium and the 21cm signal before the Epoch of Reionisation. Then, we show that Generative Adversarial Networks (GANs) are powerful tools for the fast generation of realistic astrophysical mock data. Specifically, we use them to emulate the imprint of DMAF on the gas density field and to simulate 21cm tomography images as a function of different astrophysical parameters. Finally, we analyse the gamma-ray Galactic Centre Excess in the Fermi data, for which DM annihilation has been proposed as a possible explanation, harnessing deep learning techniques.

Numerical schemes for the partial differential equations used to characterize stiffly forced conservation laws are constructed and analyzed. Partial differential equations of this form are found in many physical applications including modeling gas dynamics, fluid flow, and combustion. Many difficulties arise when trying to approximate solutions to stiffly forced conservation laws numerically. Some of these numerical difficulties are investigated. A new class of numerical schemes is developed to overcome some of these problems. The numerical schemes are constructed using an infinite sequence of conservation laws. Restrictions are given on the schemes that guarantee they maintain a uniform bound and satisfy an entropy condition. For schemes meeting these criteria, a proof is given of convergence to the correct physical solution of the conservation law. Numerical examples are presented to illustrate the theoretical results.

This dissertation is a contribution to the development of a unified model of continuum mechanics, describing both fluids and elastic solids as a general continua, with a simple material parameter choice being the distinction between inviscid or viscous fluid, or elastic solids or visco-elasto-plastic media. Additional physical effects such as surface tension, rate-dependent material failure and fatigue can be, and have been, included in the same formalism. The model extends a hyperelastic formulation of solid mechanics in Eulerian coordinates to fluid flows by means of stiff algebraic relaxation source terms. The governing equations are then solved by means of high order ADER Discontinuous Galerkin and Finite Volume schemes on fixed Cartesian meshes and on moving unstructured polygonal meshes with adaptive connectivity, the latter constructed and moved by means of a in- house Fortran library for the generation of high quality Delaunay and Voronoi meshes. Further, the thesis introduces a new family of exponential-type and semi- analytical time-integration methods for the stiff source terms governing friction and pressure relaxation in Baer-Nunziato compressible multiphase flows, as well as for relaxation in the unified model of continuum mechanics, associated with viscosity and plasticity, and heat conduction effects. Theoretical consideration about the model are also given, from the solution of weak hyperbolicity issues affecting some special cases of the governing equations, to the computation of accurate eigenvalue estimates, to the discussion of the geometrical structure of the equations and involution constraints of curl type, then enforced both via a GLM curl cleaning method, and by means of special involution-preserving discrete differential operators, implemented in a semi-implicit framework. Concerning applications to real-world problems, this thesis includes simulation ranging from low-Mach viscous two-phase flow, to shockwaves in compressible viscous flow on unstructured moving grids, to diffuse interface crack formation in solids.

Ce travail est consacré à l'analyse asymptotique des problèmes de contact d'un domaine homogène et d'un domaine perforé. Les modèles physiques considérés sont linéaires stationnaires ou non-stationnaires, scalaires ou systèmes. Ils comportent deux paramètres de comportement opposé. Sur le bord des trous, nous considérons les conditions de Dirichlet et de Neumann homogènes. Nous analysons la dépendance des solutions et conditions aux limites effectives par rapport à la discontinuité à l'interface, grâce aux couches limites et aux développements asymptotiques et à l'étude des couches limites qui décrivent l'interaction entre les deux sous-domaines assez différents. Nous obtenons explicitement les problèmes homogénéisés avec les conditions aux limites homogénéisées associées sur le bord et sur l'interface

This thesis reports on combined experimental and computational investigations conducted on problems of state and combined state and parameter estimation applied to vibrating engineering structures. The standard dynamic state space modeling framework is adopted for this purpose, and the analysis is carried out using the Kalman filter (and its variants), particle filters, and Markov chain Monte Carlo (MCMC) samplers. A review of the relevant literature has revealed that these tools have not been applied to situations where the system under study and its numerical representation via the discretized process equation displays numerical stiff behaviour. This numerical stiffness is characterized by the presence of response components with widely separated decay rates and (or) frequencies of oscillations. A computationally efficient treatment of such systems calls for the application of implicit discretization schemes to deduce the discrete process equations from the governing semi-discretized equations of motion resulting from the application of the finite element method. The implicit nature of the process equation, however, poses several challenges in the analysis of the resulting dynamic state space model since most existing methods for this purpose assume explicit process equation models. The present thesis investigates the modifications needed to some of the existing Bayesian filters, such as the Kalman filter, extended Kalman filter, unscented Kalman filter, bootstrap filter, and sequential importance sampling particle filters so that the methods can be employed to allow for implicit process equation models. Some of these tools are then combined with MCMC samplers to tackle problems of combined state and parameter estimation problems. The thesis covers linear and nonlinear dynamical systems and allows for the identification of not only the dynamical system parameters but also the parameters associated with models for the process and measurement noises. The tools developed are applied to a suite of laboratory experimental models, which include shear building frame models containing an inerter element and piecewise geometrical nonlinear features and one-storey and five-storey asymmetric, bending-torsion coupled building frames. These frames are tested on a multi-axes earthquake simulator. Also studied are typical nonlinear dynamical systems such as a limit cycle oscillator, a multi-degree of freedom degrading inelastic frame model, and an elastically mounted pendulum undergoing large amplitude oscillations. The thesis is organized into an introductory chapter, a chapter that provides a review of literature, four contributing chapters, and a chapter that summarizes contributions made and makes a few suggestions for future research. The contributing chapters are sequenced as follows: (a) Chapter 3 considers problems of state estimation in stiff linear state space model and discusses the application of an implicit Kalman filtering strategy, (b) Chapter 4 presents the details of the modifications made to the extended Kalman filter, unscented Kalman filter, bootstrap filter, and sequential importance sampling filter and discusses the application of resulting algorithms to tackle problems of state estimation in a set of nonlinear dynamical systems, (c) Chapter 5 considers the problem of combined state and parameter estimation in linear stiff systems by combining implicit Kalman filter with the general adaptive metropolis algorithm, an MCMC sampler, and (d) Chapter 6 presents the formulations for the combined state and parameter estimation in nonlinear stiff systems using implicit unscented Kalman filter along with general adaptive Metropolis algorithm. The thesis has around 230 references covering a time window of 1964-2023.

A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge-Kutta- Chebyshev (RKC) scheme is adjusted to integrate diffusion. Spatial operator is de- scretised by second-order finite differences on a uniform grid. The overall solution is advanced over S fractional stiff integrations, where S corresponds to the number of RKC stages. The behavior of the scheme is analyzed by applying it to three simple problems. The results show that it achieves second-order accuracy, thus, preserving the formal accuracy of the original RKC. The presented development sets the stage for future extensions, particularly, to multidimensional reacting flows with detailed chemistry.

With the current events concerning terrorist attacks, it is imperative to perform research and development on issues related to ballistic protection. The need to protect soldiers from high impact velocity threats has become increasingly important and challenging. Within the scope of this work the aim is to develop an optimised armour configuration for an advanced ballistic helmet design, which is able to defeat impacts from high velocity rifle bullets. This is done using finite element modelling supported by results from experimental tests. The design presented here is based on four different layers, where: (i) the first layer is designed to break and erode the projectile, (ii) the second layer absorbs the kinetic energy of the projectile, (iii) the third layer minimises the back face deflection and, finally, (iv) a fourth layer absorbs the shock wave of the initial impact and provides the necessary standoff (required by the back face deflection) for the first three layers, so that direct contact between these layers and the head does not occur. The results obtained by simulation with the finite element method (using LS-DynaTM) demonstrate that the models agree with the experimental results. A detailed numerical study of the diferent layers as well as the 7.62x39 M43 projectile was made. A good correlation between numerical and experimental results of the ammunition and armour materials was achieved, as well as between numerical and experimental results in terms of the depth of indentation as a function of impact velocity of the new ballistic helmet design. The last two sets of numerical analysis made for the helmet shell configuration was relative to the shock absorbing layer. The first set of simulations consisted of introducing rigid boundaries to the composite layer of the at panel. A second set of simulations considered the composite layer of the at panel to be attached to a rigid frame, without fixing this frame. From the simulation results, a shock-absorbing layer can be designed in such a way as to significantly reduce the risk on behind-helmet blunt trauma, and with acceptable force transfer to the head. An optimum standoff distance was determined for a ballistic helmet concept able to stop the M43 Kalashnikov projectile.
Face aos sucessivos eventos relativos a ataques terroristas, é imperativo realizar investigação científica e desenvolvimento em questões relacionadas com a proteção balística. O objetivo principal do trabalho que aqui se apresenta _e desenvolver um novo capacete balístico capaz de parar projéteis de alta velocidade, usando modelos de elementos finitos, validados com base em resultados de testes experimentais. O modelo de capacete aqui apresentado _e composto por quatro diferentes camadas, onde: (i) a primeira é capaz de deformar e fraturar o projétil, especialmente o núcleo de aço, ajudando a reduzir a sua velocidade; (ii) a segunda camada absorve energia cinética do projétil, (iii) a terceira limita a deflexão da face anterior e, finalmente, (iv) a quarta camada absorve a onda de choque do impacto inicial e garante a distância necessária para evitar o contato dessas camadas com a cabeça. Foi também realizado um estudo numérico detalhado das diferentes partes do projétil 7.62 x 39 M43. Obteve-se uma boa correlação entre os resultados numéricos (usando o software LS-DynaTM) e experimentais para os modelos do projétil quer do equipamento de proteção pessoal (capacete). Atingiu-se também uma boa correlação em termos de velocidade de impacto em função da profundidade de deformação do novo desenho de capacete balístico. Realizou-se uma análise numérica mais detalhada para a configuração do capacete relativa _a camada de absorção da onda de choque. Um primeiro conjunto de simulações consistiu em introduzir limites rígidos nas extremidades das três primeiras camadas. Um segundo conjunto de simulações considerou as três primeiras camadas anexadas a uma estrutura rígida, fixa no capacete. A partir dos resultados numéricos, conclui-se ser possível projetar uma camada de absorção da onda de choque de maneira a reduzir significativamente o risco de traumatismo craniano causado pelo impacto no capacete. Uma distância mínima entre a cabeça e o capacete pode, portanto, ser determinada para um novo modelo de capacete balístico capaz de parar o projétil M43 Kalashnikov.
Programa Doutoral em Engenharia Mecânica

We wish to apply the newly developed multigrid method [14] to the solution of ODEs resulting from the semi-discretization of time dependent PDEs by the method of lines. In particular, we consider the general form of two important PDE equations occuring in practice, viz. the nonlinear diffusion equation and the telegraph equation. Furthermore, we briefly examine a practical area, viz. atmospheric physics where we feel this method might be of significance. In order to offer the method to a wider range of PC users we present a computer program, called PDEMGS. The purpose of this program is to relieve the user of much of the expensive and time consuming effort involved in the solution of nonlinear PDEs. A wide variety of examples are given to demonstrate the usefulness of the multigrid method and the versatility of PDEMGS.
Thesis (M.Sc.)-University of Natal, Durban, 1992.

In this work, we present an amplitude-shape method for solving evolution problems described by partial differential equations. The method is capable of recognizing the special structure of many evolution problems. In particular, the stiff system of ordinary differential equations resulting from the semi-discretization of partial differential equations is considered. The method involves transforming the system so that only a few equations are stiff and the majority of the equations remain non-stiff. The system is treated with a mixed explicit-implicit scheme with a built-in error control mechanism. This approach proved to be very effective for the solution of stiff systems of equations describing spatially dependent chemical kinetics.
Thesis (Ph.D.)-University of Natal, 1997.

abstract: Ordered buckling of stiff films on elastomeric substrates has many applications in the field of stretchable electronics. Mechanics plays a very important role in such systems. A full three dimensional finite element analysis studying the pattern of wrinkles formed on a stiff film bonded to a compliant substrate under the action of a compressive force has been widely studied. For thin films, this wrinkling pattern is usually sinusoidal, and for wide films the pattern depends on loading conditions. The present study establishes a relationship between the effect of the load applied at an angle to the stiff film. A systematic experimental and analytical study of these systems has been presented in the present study. The study is performed for two different loading conditions, one with the compressive force applied parallel to the film and the other with an angle included between the application of the force and the alignment of the stiff film. A geometric model closely resembling the experimental specimen studied is created and a three dimensional finite element analysis is carried out using ABAQUS (Version 6.7). The objective of the finite element simulations is to validate the results of the experimental study to be corresponding to the minimum total energy of the system. It also helps to establish a relation between the parameters of the buckling profile and the parameters (elastic and dimensional parameters) of the system. Two methods of non-linear analysis namely, the Newton-Raphson method and Arc-Length method are used. It is found that the Arc-Length method is the most cost effective in terms of total simulation time for large models (higher number of elements).The convergence of the results is affected by a variety of factors like the dimensional parameters of the substrate, mesh density of the model, length of the substrate and the film, the angle included. For narrow silicon films the buckling profile is observed to be sinusoidal and perpendicular to the direction of the silicon film. As the angle increases in wider stiff films the buckling profile is seen to transit from being perpendicular to the direction of the film to being perpendicular to the direction of the application of the pre-stress. This study improves and expands the application of the stiff film buckling to an angled loading condition.
Dissertation/Thesis
M.S. Mechanical Engineering 2010

Stochastic differential equations(SDEs) play an important role in many branches of engineering and science including economics, finance, chemistry, biology, mechanics etc. SDEs (with m-dimensional Wiener process) arising in many applications do not have explicit solutions, which implies the development of effective numerical methods for such systems. For SDEs, one can classify the numerical methods into three classes: fully implicit methods, semi-implicit methods and explicit methods. In order to solve SDEs, the computation of Newton iteration is necessary for the implicit and semi-implicit methods whereas for the explicit methods we do not need such computation. In this thesis the common theme is to construct explicit numerical methods with strong order 1.0 and 1.5 for solving Itˆo SDEs. The five-stage Milstein(FSM)methods, split-step forward Milstein(SSFM)methods and M-stage split-step strong Taylor(M-SSST) methods are constructed for solving SDEs. The FSM, SSFM and M-SSST methods are fully explicit methods. It is proved that the FSM and SSFM methods are convergent with strong order 1.0, and M-SSST methods are convergent with strong order 1.5.Stiffness is a very important issue for the numerical treatment of SDEs, similar to the case of deterministic ordinary differential equations. Stochastic stiffness can lead someone to use smaller step-size for the numerical simulation of the SDEs. However, such issues can be handled using numerical methods with better stability properties. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the FSM and SSFM methods are larger than the Milstein and three-stage Milstein methods. The M-SSST methods possess large mean square stability region as compared to the order 1.5 strong Itˆo-Taylor method. SDE systems simulated with the FSM, SSFM and M-SSST methods show the computational efficiency of the methods. In this work, we also consider the problem of computing numerical solutions for stochastic delay differential equations(SDDEs) of Itˆo form with a constant lag in the argument. The fully explicit methods, the predictor-corrector Euler(PCE)methods, are constructed for solving SDDEs. It is proved that the PCE methods are convergent with strong order γ = ½ in the mean-square sense. The conditions under which the PCE methods are MS-stable and GMS-stable are less restrictive as compared to the conditions for the Euler method.

Stochastic differential equations(SDEs) play an important role in many branches of engineering and science including economics, finance, chemistry, biology, mechanics etc. SDEs (with m-dimensional Wiener process) arising in many applications do not have explicit solutions, which implies the development of effective numerical methods for such systems. For SDEs, one can classify the numerical methods into three classes: fully implicit methods, semi-implicit methods and explicit methods. In order to solve SDEs, the computation of Newton iteration is necessary for the implicit and semi-implicit methods whereas for the explicit methods we do not need such computation. In this thesis the common theme is to construct explicit numerical methods with strong order 1.0 and 1.5 for solving Itˆo SDEs. The five-stage Milstein(FSM)methods, split-step forward Milstein(SSFM)methods and M-stage split-step strong Taylor(M-SSST) methods are constructed for solving SDEs. The FSM, SSFM and M-SSST methods are fully explicit methods. It is proved that the FSM and SSFM methods are convergent with strong order 1.0, and M-SSST methods are convergent with strong order 1.5.Stiffness is a very important issue for the numerical treatment of SDEs, similar to the case of deterministic ordinary differential equations. Stochastic stiffness can lead someone to use smaller step-size for the numerical simulation of the SDEs. However, such issues can be handled using numerical methods with better stability properties. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the FSM and SSFM methods are larger than the Milstein and three-stage Milstein methods. The M-SSST methods possess large mean square stability region as compared to the order 1.5 strong Itˆo-Taylor method. SDE systems simulated with the FSM, SSFM and M-SSST methods show the computational efficiency of the methods. In this work, we also consider the problem of computing numerical solutions for stochastic delay differential equations(SDDEs) of Itˆo form with a constant lag in the argument. The fully explicit methods, the predictor-corrector Euler(PCE)methods, are constructed for solving SDDEs. It is proved that the PCE methods are convergent with strong order γ = ½ in the mean-square sense. The conditions under which the PCE methods are MS-stable and GMS-stable are less restrictive as compared to the conditions for the Euler method.

OBJECTIVE: To estimate the resource costs of providing wound care for the 488,000 catchment population of the Bradford and Airedale primary care trust (PCT). METHOD: A wound survey was carried out over a one-week period in March 2007 covering three hospitals in two acute trusts, district nurses, nursing homes and residential homes within the geographical area defined by the PCT. The survey included information on the frequency of dressing change, treatment time and district nurse travel time. The resource costs of wound care in the PCT were estimated by combining this information with representative costs for the UK National Health Service and information on dressing spend. RESULTS: Prevalence of patients with a wound was 3.55 per 1000 population. The majority of wounds were surgical/trauma (48%), leg/foot (28%) and pressure ulcers (21%). Prevalence of wounds among hospital inpatients was 30.7%. Of these, 11.6% were pressure ulcers, of which 66% were hospital-acquired. The attributable cost of wound care in 2006-2007 was pounds 9.89 million: pounds 2.03 million per 100,000 population and 1.44% of the local health-care budget. Costs included pounds 1.69 million spending on dressings, 45.4 full-time nurses (valued at pounds 3.076 million) and 60-61 acute hospital beds (valued at pounds 5.13 million). CONCLUSION: The cost of wound care is significant. The most important components are the costs of wound-related hospitalisation and the opportunity cost of nurse time. The 32% of patients treated in hospital accounted for 63% of total costs. Putting in place care pathways to avoid hospitalisation and avoiding the development of hospital-acquired pressure ulcers and other wound complications are important ways to reduce costs. DECLARATION OF INTEREST: John Posnett is an employee of Smith & Nephew.