Induced seismicity (earthquakes caused by injection or extraction of fluids in Earth's subsurface) is a major, new hazard in the United States, the Netherlands, and other countries, with vast economic consequences if not properly managed. Addressing this problem requires development of predictive simulations of how fluid-saturated solids containing frictional faults respond to fluid injection/extraction. Here we present a numerical method for linear poroelasticity with rate-and-state friction faults. A numerical method for approximating the fully coupled linear poroelastic equations is derived using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) framework. Well-posedness is shown for a set of physical boundary conditions in 1D and in 2D. The SBP-SAT technique is used to discretize the governing equations and show semi-discrete stability and the correctness of the implementation is verified by rigorous convergence tests using the method of manufactured solutions, which shows that the expected convergence rates are obtained for a problem with spatially variable material parameters. Mandel's problem and a line source problem are studied, where simulation results and convergence studies show satisfactory numerical properties. Furthermore, two problem setups involving fault dynamics and slip on faults triggered by fluid injection are studied, where the simulation results show that fluid injection can trigger earthquakes, having implications for induced seismicity. In addition, the results show that the scheme used for solving the fully coupled problem, captures dynamics that would not be seen in an uncoupled model. Future improvements involve imposing Dirichlet boundary conditions using a different technique, extending the scheme to handle curvilinear coordinates and three spatial dimensions, as well as improving the high-performance code and extending the study of the fault dynamics.

Este trabalho apresenta um método numérico para simular escoamentos viscoelásticos tridimensionais com superfícies livres governados pela equação constitutiva de Maxwell. O método numérico é uma extensão da técnica newtoniana GENSMAC3D para escoamentos viscoelásticos. As equações governantes para escoamentos incompressíveis cartesianos isotérmicos são apresentadas em detalhes. O tratamento do tensor não-newtoniano em contornos rígidos tridimensionais é apresentado em detalhes, bem como o cálculo da condição de contorno na superfície livre. As equações governantes são resolvidas pelo método de diferenças finitas numa malha deslocada tridimensional. O fluido é modelado pela técnica das partículas marcadoras e tratado como uma superfície linear por partes. O método numérico desenvolvido foi implementado no sistema Freeflow3D, e resultados numéricos obtidos na simulação de escoamentos tridimensionais governados pela equação constitutiva de Maxwell são apresentados. Adicionalmente, apresentamos uma validação mostrando a convergência do método desenvolvido nesse trabalho / This work presents a numerical method for solving three-dimensional viscoelastic flows with free surfaces governed by the Maxwell constitutive equation. The numerical method is an extension of the Newtonian technique GENSMAC3D to viscoelastic flows. The governing equations for Cartesian incompressible isothermic flows are presented in details. The treatment of the non-Newtonian tensor on three-dimensional rigid boundaries is given in details as well as the calculation of the boundary conditions on the free surface. The governing equations are solved by a finite difference method using a three-dimensional staggered grid. The fluid is described by marker particles and is represented by a piecewise linear surface. The numerical method developed in this work was implemented into the Freeflow3D system, and numerical results obtained from the simulation of complex three-dimensional flows are presented. Additionally, we present validation results and demonstrate the convergence of the method by performing mesh refinement

Made available in DSpace on 2015-03-04T18:57:57Z (GMT). No. of bitstreams: 1 Tese Luciana Santos .pdf: 8395333 bytes, checksum: 6fc53db0b60c462c2bcba5dca268b385 (MD5) Previous issue date: 2013-07-05 / In this work we deal with the problem of shallow water ows using finite difference schemes first applied to simple models of hyperbolic problems, as the advection equation and the gravity wave equations. For the treatment of shallow water equations, besides of semi lagrangian schemes and staggered grids, we make use of a semi-implicit finite difference scheme. Finally we describe a model based on the finite volume technique, where the conservation equations of momentum are discretized according to a semi-implicit finite difference scheme, using a lagrangean aproximation to convective terms, applied to a staggered grid, while the equation of mass conservation is discretized by a semi-implicit scheme applied to a non structured ortogonal grid. In this model the ow is determined by the free surface elevation and by the component of velocity normal to each side of the grid. The reconstruction of the velocity field of the complete shallow water equations is made by the depth integrated method. Applications include ow in a retangular closed bay, with periodic boundary conditions, a geophysical ow applied to a small portion of the Amazonas river and to problems with discontinuous initial conditions, like those occurring in dam break problems. / Neste trabalho tratamos o problema de escoamentos em águas rasas através de esquemas em diferenças finitas aplicados inicialmente a modelos simples de problemas hiperbólicos como a equação de advecção e as equações de ondas de gravidade. No tratamento das equações de águas rasas, além de esquemas semi lagrangeanos e de malhas staggered, é utilizado um esquema semi-implícito de diferenças finitas. Por fim é descrito um modelo baseado no método de volumes finitos, onde as equações de conservação de momentum são discretizadas de acordo com um esquema semi-implícito de diferenças finitas, com uma aproximação lagrangeana para os termos convectivos, aplicado a uma malha staggered, enquanto que a equações de conservação de massa é discretizada por um esquema semi-implícito aplicado a uma malha não estruturada ortogonal staggered. Nesse modelo o escoamento é determinado pela elevação da superfície livre e pela componente da velocidade normal a cada um dos lados da malha. A reconstrução do campo de velocidades do conjunto completo de equações de águas rasas é dada através do método da profundidade integrada. As aplicações incluem um escoamento em uma bacia retangular fechada, com condições de contorno periódicas, um escoamento geofísico aplicado a um trecho do rio Amazonas e problemas com condição inicial descontínua, do tipo dam break.

Submitted by Maria Cristina (library@lncc.br) on 2017-04-12T20:03:02Z No. of bitstreams: 1 Dissertacao_Juliano_Abimael.pdf: 1562533 bytes, checksum: 72a2f22f7a5dd247b98bf5da9985fc3e (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-04-12T20:03:23Z (GMT) No. of bitstreams: 1 Dissertacao_Juliano_Abimael.pdf: 1562533 bytes, checksum: 72a2f22f7a5dd247b98bf5da9985fc3e (MD5) / Made available in DSpace on 2017-04-12T20:03:59Z (GMT). No. of bitstreams: 1 Dissertacao_Juliano_Abimael.pdf: 1562533 bytes, checksum: 72a2f22f7a5dd247b98bf5da9985fc3e (MD5) Previous issue date: 2016-08-24 / Agencia Nacional de Pesquisa (ANP) / The classical methods of finite differences and Galerkin finite element are unable to eliminate the error of pollution effect for high wave numbers. Methods such as Galerkin Least Square (GLS) and Quasi Stabilized Finite Element Method (QSFEM) are methods that minimize error pollution is feasible, however, only in uniform grids. An important step to be taken is the study and development of methodologies that minimize the error pollution effect on non-uniform grids. In this line, the formulation Quasi Optimal Finite Difference (QOFD) obtained by numerical minimization of the functional truncation error for plane waves in an arbitrary direction, and has minimal pollution to stencils for uniform grids is a reliable method in more general meshes. In this work, and describe the methods mentioned above, we propose an approach that generates the same QOFD coefficients through the use of a radial basis functions, composed of the Bessel functions of the first kind and order zero. Furthermore, for wave equation in the time domain, we propose finite difference approximations to the high-order wave equation. This methodology will use a polynomial base constructed from the characteristic functions of this equation. / As metodologias clássicas de diferenças finitas e elementos finitos de Galerkin se mostram incapazes de eliminar o efeito de poluição do erro para altos números de onda. Métodos como Galerkin Least Square (GLS) e Quasi Stabilized Finite Element Method (QSFEM) são métodos que minimizam a poluição do erro sendo factíveis, contudo, apenas em malhas uniformes. Um passo importante a ser dado é o estudo e desenvolvimento de metodologias que minimizem o efeito de poluição do erro em malhas não-uniformes. Nessa linha, a formulação Quasi Optimal Finite Difference (QOFD), obtida numericamente pela minimização do funcional do erro de truncamento para ondas planas em direção arbitrária, além de ter mínima poluição para stencils sobre malhas uniformes é um método factível em malhas mais gerais. Neste trabalho, além de descrevermos os métodos citados anteriormente, propomos uma aproximação que gera os mesmos coeficientes do QOFD por meio do emprego de uma base radial de funções, composta pelas funções de Bessel de primeiro tipo e ordem zero. Além disso, para a equação da onda no domínio do tempo, propomos aproximações por diferenças finitas de alta ordem para a equação da onda. Tal metodologia fará uso de uma base polinomial construída a partir das funções características desta equação.

O presente trabalho de pesquisa tem por objetivo estudar a paralelização de algoritmos voltados à solução de equações diferenciais parciais. Esses algoritmos são utilizados para gerar a solução numérica das equações de Navier-Stokes em um escoamento bidimensional incompressível de um fluido newtoniano. As derivadas espaciais são calculadas através de um método de diferenças finitas compactas com a utilização de aproximações de altas ordens de precisão. Uma vez que o cálculo de derivadas espaciais com alta ordem de precisão da forma compacta adotado no presente estudo requer a solução de sistemas lineares tridiagonais, é importante realizar estudos voltados a resolução desses sistemas, para se obter uma boa performance. Ressalta-se ainda que a solução de sistemas lineares também faz-se presente na solução numérica da equação de Poisson. Os resultados obtidos decorrentes da solução das equações diferenciais parciais são comparados com os resultados onde se conhece a solução analítica, de forma a verificar a precisão dos métodos implementados. Os resultados do código voltado à resolução das equações de Navier-Stokes paralelizado para simulação de escoamentos incompressíveis são comparados com resultados da teoria de estabilidade linear, para validação do código final. Verifica-se a performance e o speedup do código em questão, comparando-se o tempo total gasto em função do número de elementos de processamento utilizados / The objective of the present work is to study the parallelization of partial differential equations. The aim is to achieve an effective parallelization to generate numerical solution of Navier-Stokes equations in a two-dimensional incompressible and isothermal flow of a Newtonian fluid. The spatial derivatives are calculated using compact finite differences approximations of higher order accuracy. Since the calculation of spatial derivatives with high order adopted in the present work requires the solution of tridiagonal systems, it is important to conduct studies to solve these systems and achieve good performance. In addiction, linear systems solution is also present in the numerical solution of a Poisson equation. The results generated by the solution of partial differential equations are compared to analytical solution, in order to verify the accuracy of the implemented methods. The numerical parallel solution of a Navier-Stokes equations is compared with linear stability theory to validate the final code. The performance and the speedup of the code in question is also checked, comparing the execution time in function of the number of processing elements

When pricing American-style options on d assets by Monte Carlo methods, one usually stores the simulated asset prices at all time steps on all paths in order to determine when to exercise the options. If N time steps and M paths are used; then the storage requirement is d · M · N. In this thesis, we give two simulation methods to price multi-asset American-style options, where the storage requirement only grows like (d + 1)M + N. The only additional computational cost is that we have to generate each random number twice instead of once. For machines with limited memory, we can now use larger values of M and N to improve the accuracy in pricing the options. / by Wong Chi Yan. / Adviser: Raymond H. Chan. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1708. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 79-82). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Title: Development of effective code for earthquake dynamic source simulations Author: Bc. Jan Premus Department: Department of Geophysics Supervisor: doc. RNDr. František Gallovič, Ph.D, Department of Geophysics Abstract: Dynamic rupture modeling coupled with strong motion data fitting offers an insight into physical mechanisms behind earthquake sources [Gallovic et al., 2019]. Running a large number of dynamic model simulations is required due to the nonlinearity of the inverse problem. The goal of this Thesis is a development of an efficient forward solver for the dynamic inversions. The fi- nite difference staggered grid code FD3D by Madariaga and Olsen [1998] served as a basis for the development, offering sufficient speed, but rather low accu- racy. Traction at split node implementation of the fault boundary condition and perfectly matched layers as the absorbing boundary condition were required to obtain desirable accuracy. In addition to the slip weakening friction law, fast ve- locity weakening friction law has been implemented, increasing the applicability of the code. We test the new code FD3D TSN using USGS/SCEC benchmarks TPV5 (slip-weakening friction) and TPV104 (fast rate weakening friction) [Harris et al., 2018], showing very good agreement with results calculated by advanced numerical...

<p>Today’s bridge design work in many cases demands a trustworthy dynamic analysis instead of using the traditional dynamic amplification factors. In this thesis a reliable 3D Bernoulli-Euler beam finite element model of the New Årsta Railway Bridge was prepared for thorough dynamic analysis using in situ bridge monitoring for correlation. The bridge is of the concrete box girder type with a heavily reinforced and prestressed bridge deck. The monitoring system was designed for long term monitoring with strain transducers embedded in the concrete and accelerometers mounted inside the edge beams and at the lower edge of the track slab.</p><p>The global finite element model used the exact bridge geometry but was simplified regarding prestressing cables and the two railway tracks. The prestressing cables and the tracks were consequently not included and an equivalent pure concrete model was identified.</p><p>A static macadam train load was eccentrically placed on one of the bridge’s two tracks. By using Vlasov’s torsional theory and thereby including constrained warping a realistic modulus of elasticity for the concrete without prestressing cables and stiffness contribution from the railway tracks was found. This was allowed by comparing measured strain from strain transducers with the linear elastic finite element model’s axial stresses. Mainly three monitoring bridge sections were used, each of which was modelled with plane strain finite elements subjected to sectional forces/moments from a static macadam train load and a separately calculated torsional curvature.</p><p>From the identified modulus of elasticity the global finite element model was updated for Poisson’s ratio and material density (mass) to correspond with natural frequencies from the performed signal analysis of accelerometer signals. The influence of warping on the natural frequencies of the global finite element model was assumed small and the bridge’s torsional behaviour was modelled to follow Saint-Venant’s torsional theory.</p><p>A first preliminary estimation of modal damping ratios was included. The results indicated that natural frequencies were in accordance between modelling and signal analysis results, especially concerning high energy modes. Estimated damping ratios for the first vibration modes far exceeded the lower limit value specified in bridge design codes and railway bridge dynamic analysis recommendations.</p>

Density functional theory (DFT) is not only an accurate but also a widely used theory for describing the quantum-mechanical electronic structure of matter. In this approach, the intractable problem of interacting electrons is simplified to a tractable problem of non-interacting electrons moving in an effective potential. Even with this simplification, DFT remains extremely computationally expensive. In particular, DFT scales cubically with respect to the number of atoms, which restricts the size of systems that can be studied. Orbital free density functional theory (OF-DFT) represents a simplification of DFT applicable to metallic systems that behave like a free-electron gas. Current implementations of OF-DFT employ the plane-wave basis, the global nature of the basis prevents the efficient use of modern high-performance computer archi- tectures. We present a real-space formulation and higher-order finite-difference implementation of periodic Orbital-free Density Functional Theory (OF-DFT). Specifically, utilizing a local reformulation of the electrostatic and kernel terms, we develop a gener- alized framework suitable for performing OF-DFT simulations with different variants of the electronic kinetic energy. In particular, we develop a self-consistent field (SCF) type fixed-point method for calculations involving linear-response kinetic energy functionals. In doing so, we make the calculation of the electronic ground-state and forces on the nuclei amenable to computations that altogether scale linearly with the number of atoms. We develop a parallel implementation of our method using Portable, Extensible Toolkit for scientific computations (PETSc) suite of data structures and routines. The communication between processors is handled via the Message Passing Interface(MPI). We implement this formulation using the finite-difference discretization, us- ing which we demonstrate that higher-order finite-differences can achieve relatively large convergence rates with respect to mesh-size in both the energies and forces. Additionally, we establish that the fixed-point iteration converges rapidly, and that it can be further accelerated using extrapolation techniques like Anderson mixing. We verify the accuracy of our results by comparing the energies and forces with plane-wave methods for selected examples, one of which is the vacancy formation energy in Aluminum. Overall, we demonstrate that the proposed formulation and implementation is an attractive choice for performing OF-DFT calculations.

A new method to perform direct numerical simulations of wall-bounded flows has been developed and implemented. The method uses high-order compact finite differences in wall-normal (for channel flow) or radial direction (for pipe flow) on a collocated grid, which gives high-accuracy results without the effectfof filtering caused by frequent interpolation as required on a staggered grid. The use of compact finite differences means that extreme clustering near the wall leading to small time steps in high-Reynolds number simulations is avoided. The influence matrix method is used to ensure a completely divergence-freesolution and all systems of equations are solved in banded form, which ensures an effcient solution procedure with low requirements for data storage. The method is unique in the sense that exactly divergence-free solutions on collocated meshes are calculated using arbitrary dffierence matrices. The code is validated for two flow cases, i.e. turbulent channel and turbulent pipe flow at relatively low Reynolds number. All tests show excellent agreement with analytical and existing results, confirming the accuracy and robustness ofthe method. The next step is to eciently parallelise the code so that high-Reynolds number simulations at high resolution can be performed. We furthermore investigated rare events occurring in the near-wall region of turbulent wall-bounded flows. We find that negative streamwise velocities and extreme wall-normal velocity uctuations are found rarely (on the order of 0:01%), and that they occur more frequently at higher Reynolds number. These events are caused by strong vortices lying further away from the wall and it appears that these events are universal for wall-bounded flows. / <p>QC 20150303</p>