A simple yet effective technique is used to obtain a numerical stability criteria for explicit time-marching algorithms in elastic-viscoplasticity. The resulting stability criteria are capable of accounting for non-associative and work hardening viscoplasticity for a wide variety of constitutive laws of the Perzyna-type. Conservative estimates for maximum permissible time step are obtained. This thesis investigates the level of conservativeness by considering different problems exhibiting various levels of constraint. Using the proposed stability criterion, assuming a linear flow function, non-hardening and uniform material properties, it is shown that the initial strain algorithm for plasticity and the initial strain viscoplastic algorithms are numerically the same. The intuitive approach used to obtain an estimate of maximum permissible time step was also used to develop an unconditionally stable implicit time marching scheme which avoids expensive matrix inversions. / Thesis / Master of Engineering (ME)

Background. Fluid simulation is an area of ongoing research. In recent years, simulators have become more realistic and stable, partly by employing the condition of having divergence-free velocity fields. A divergence-free velocity field is a strict constraint that requires a high level of correctness in a simulation. Another recent development is in the subject of performance optimization, where asynchronous time integration is used. Asynchronous time integration means integrating different parts of a fluid with varying time step sizes. Doing so leads to overall larger time step sizes, which improves performance. This thesis combines the divergence-free velocity field condition with asynchronous time stepping in a particle-based simulator. Objectives. This thesis aims to achieve a performance speedup by implementing asynchronous time integration into an existing particle-based simulator that assures the velocity field is divergence-free. Methods. With an open source simulator employing a divergence-free velocity field as a starting point, asynchronous time integration is implemented. This is achieved by dividing the fluid into three regions, each with their own time step sizes. Introducing asynchronous time integration means significantly lowering the stability of a simulation. This is countered by implementing additional steps to increase stability. Results. Roughly a 40\% speedup is achieved in two out of three scenes, with similar visual results as the original synchronous simulation. In the third scene, there is no performance speedup as the performance is similar to that of the original simulation. The two first scenes could be sped up further with more aggressive settings for asynchronous time integration. This is however not possible due to stability issues, which are also the cause for the third scene not resulting in any speedup. Conclusions. Asynchronous simulation is shown to be a valid option even alongside a divergence solver. However, occasional unrealistic behavior resembling explosions among the particles do occur. Besides from being undesirable behavior, these explosions also decrease performance and prevent more aggressive performance settings from being used. Analysis of their cause, attempted solutions and potential future solutions are provided in the discussion chapter.

This study seeks to reduce the cost of numerically solving non-linear partial differential equations by reducing the number of computations without compromising accuracy. This was done by using accurate local time stepping. This algorithm uses local time stepping but compensates for the inconsistencies in the temporal dimension by interpolations and/or extrapolations. Reduction in computations are obtained by time-stepping only a particular region with small time steps. A shock tube problem and a detonation wave were the two test cases considered. The performance of the solution using this algorithm was compared with an algorithm that does not use accurate local time stepping.

Many fields in science and engineering require large-scale numerical simulations of complex systems described by differential equations. These systems are typically multi-physics (they are driven by multiple interacting physical processes) and multiscale (the dynamics takes place on vastly different spatial and temporal scales). Numerical solution of such systems is highly challenging due to the dimension of the resulting discrete problem, and to the complexity that comes from incorporating multiple interacting components with different characteristics. The main contributions of this dissertation are the creation of new families of time integration methods for multiscale and multiphysics simulations, and the development of industrial-strengh tools for sensitivity analysis. This work develops novel implicit-explicit (IMEX) general linear time integration methods for multiphysics and multiscale simulations typically involving both stiff and non-stiff components. In an IMEX approach, one uses an implicit scheme for the stiff components and an explicit scheme for the non-stiff components such that the combined method has the desired stability and accuracy properties. Practical schemes with favorable properties, such as maximized stability, high efficiency, and no order reduction, are constructed and applied in extensive numerical experiments to validate the theoretical findings and to demonstrate their advantages. Approximate matrix factorization (AMF) technique exploits the structure of the Jacobian of the implicit parts, which may lead to further efficiency improvement of IMEX schemes. We have explored the application of AMF within some high order IMEX Runge-Kutta schemes in order to achieve high efficiency. Sensitivity analysis gives quantitative information about the changes in a dynamical model outputs caused by caused by small changes in the model inputs. This information is crucial for data assimilation, model-constrained optimization, inverse problems, and uncertainty quantification. We develop a high performance software package for sensitivity analysis in the context of stiff and nonstiff ordinary differential equations. Efficiency is demonstrated by direct comparisons against existing state-of-art software on a variety of test problems. / Ph. D.

No description available.

A simulação de escoamentos com superfícies livres vem ganhando importância ao longo dos últimos anos devido às várias aplicações práticas em que esse tipo de escoamento está envolvido. Dentre os métodos numéricos existentes para a simulação de escoamentos, temos o GENSMAC, que é uma técnica numérica para simular escoamentos newtonianos com superfícies livres. A implementação de métodos semi-implícitos para a discretização temporal das equações de Navier-Stokes permitiu uma relaxação significativa na restrição no passo temporal, reduzindo consideravelmente o custo computacional na simulação de escoamentos com Re 1. Mas, mesmo no caso dos métodos semi-implícitos, o passo temporal não pode aumentar além de certos limites, bem aquém daquele da restrição CFL, sem provocar sérios problemas de precisão nos resultados numéricos e consequente aparecimento de resultados não físicos. Portanto, mesmo na formulação semi-implícita, uma restrição no passo temporal é aplicada. Neste trabalho, analisamos e implementamos no sistema FREEFLOW2D uma estratégia de adaptação do passo temporal de maneira a garantir a estabilidade e a precisão utilizando o maior passo temporal possível. A eficiência e robustez da técnica incorporada à formulação implícita do GENSMAC são demonstradas na solução de problemas bidimensionais complexos com superfícies livres e baixo número de Reynolds, incluindo os problemas do inchamento do extrudado e jet flow / The simulation of free surfaces flows has gained importance in the recent years due to the many practical applications of this type of flow. Among the many numerical methods available for the simulation of fluid flows, there is GENSMAC, which is a numerical technique to simulate Newtonian flows with free surfaces. The implementation of semi-implicitmethods for the temporal discretization of the Navier-Stokes equations allowed a significant loosening in the time step restriction, reducing considerably the computational cost of the simulation of flows with Re 1. But, even with the semi-implicit methods, the time step cannot increase beyond certain limits, well below the CFL restriction, without causing serious accuracy problems in the numerical results and the consequent appearance of non-physical results. Therefore, even in the semi-implicit formulation, a time step restriction is applied. In this work, we analyse and implement in the FREEFLOW2D system a strategy for adaptive time-stepping in order to ensure stability and precision while using the largest possible time step. The efficiency and robustness of the technique incorporated to the implicit formulation of GENSMAC are demonstrated in the solution of two-dimensional complex problems with free surfaces and low Reynolds numbers, including the swelling of the extrudate and jet flow problems

Conventional coupling paradigms used nowadays to couple various physics components in reactor analysis problems can be inconsistent in their treatment of the nonlinear terms. This leads to usage of smaller time steps to maintain stability and accuracy requirements thereby increasing the computational time. These inconsistencies can be overcome using better approximations to the nonlinear operator in a time stepping strategy to regain the lost accuracy. This research aims at finding remedies that provide consistent coupling and time stepping strategies with good stability properties and higher orders of accuracy. Consistent coupling strategies, namely predictive and accelerated methods, were introduced for several reactor transient accident problems and the performance was analyzed for a 0-D and 1-D model. The results indicate that consistent approximations can be made to enhance the overall accuracy in conventional codes with such simple nonintrusive techniques. A detailed analysis of a monoblock coupling strategy using time adaptation was also implemented for several higher order Implicit Runge-Kutta (IRK) schemes. The conclusion from the results indicate that adaptive time stepping provided better accuracy and reliability in the solution fields than constant stepping methods even during discontinuities in the transients. Also, the computational and the total memory requirements for such schemes make them attractive alternatives to be used for conventional coupling codes.

This thesis develops a generic framework based on the Fourier transform for pricing and hedging of various options in equity, commodity, currency, and insurance markets. The pricing problem can be reduced to solving a partial integro-differential equation (PIDE). The Fourier Space Time-stepping (FST) framework developed in this thesis circumvents the problems associated with the existing finite difference methods by utilizing the Fourier transform to solve the PIDE. The FST framework-based methods are generic, highly efficient and rapidly convergent. The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For path-independent options, prices can be obtained for a range of stock prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle barriers, free boundaries, and exercise policies. The thesis includes applications of the FST framework-based methods to a wide range of option pricing problems. Pricing of single- and multi-asset, European and path-dependent options under independent-increment exponential Levy stock price models, common in equity and insurance markets, can be done efficiently via the cornerstone FST method. Mean-reverting Levy spot price models, common in commodity markets, are handled by introducing a frequency transformation, which can be readily computed via scaling of the option value function. Generating stochastic volatility, to match the long-term equity options market data, and stochastic skew, observed in currency markets, is addressed by introducing a non-stationary extension of multi-dimensional Levy processes using regime-switching. Finally, codependent jumps in multi-asset models are introduced through copulas. The FST methods are computationally efficient, running in O(MN^d log_2 N) time with M time steps and N space points in each dimension on a d-dimensional grid. The methods achieve second-order convergence in space; for American options, a penalty method is used to attain second-order convergence in time. Furthermore, graphics processing units are utilized to further reduce the computational time of FST methods.

This thesis develops a generic framework based on the Fourier transform for pricing and hedging of various options in equity, commodity, currency, and insurance markets. The pricing problem can be reduced to solving a partial integro-differential equation (PIDE). The Fourier Space Time-stepping (FST) framework developed in this thesis circumvents the problems associated with the existing finite difference methods by utilizing the Fourier transform to solve the PIDE. The FST framework-based methods are generic, highly efficient and rapidly convergent. The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For path-independent options, prices can be obtained for a range of stock prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle barriers, free boundaries, and exercise policies. The thesis includes applications of the FST framework-based methods to a wide range of option pricing problems. Pricing of single- and multi-asset, European and path-dependent options under independent-increment exponential Levy stock price models, common in equity and insurance markets, can be done efficiently via the cornerstone FST method. Mean-reverting Levy spot price models, common in commodity markets, are handled by introducing a frequency transformation, which can be readily computed via scaling of the option value function. Generating stochastic volatility, to match the long-term equity options market data, and stochastic skew, observed in currency markets, is addressed by introducing a non-stationary extension of multi-dimensional Levy processes using regime-switching. Finally, codependent jumps in multi-asset models are introduced through copulas. The FST methods are computationally efficient, running in O(MN^d log_2 N) time with M time steps and N space points in each dimension on a d-dimensional grid. The methods achieve second-order convergence in space; for American options, a penalty method is used to attain second-order convergence in time. Furthermore, graphics processing units are utilized to further reduce the computational time of FST methods.

A simulação de escoamentos com superfícies livres vem ganhando importância ao longo dos últimos anos devido às várias aplicações práticas em que esse tipo de escoamento está envolvido. Dentre os métodos numéricos existentes para a simulação de escoamentos, temos o GENSMAC, que é uma técnica numérica para simular escoamentos newtonianos com superfícies livres. A implementação de métodos semi-implícitos para a discretização temporal das equações de Navier-Stokes permitiu uma relaxação significativa na restrição no passo temporal, reduzindo consideravelmente o custo computacional na simulação de escoamentos com Re 1. Mas, mesmo no caso dos métodos semi-implícitos, o passo temporal não pode aumentar além de certos limites, bem aquém daquele da restrição CFL, sem provocar sérios problemas de precisão nos resultados numéricos e consequente aparecimento de resultados não físicos. Portanto, mesmo na formulação semi-implícita, uma restrição no passo temporal é aplicada. Neste trabalho, analisamos e implementamos no sistema FREEFLOW2D uma estratégia de adaptação do passo temporal de maneira a garantir a estabilidade e a precisão utilizando o maior passo temporal possível. A eficiência e robustez da técnica incorporada à formulação implícita do GENSMAC são demonstradas na solução de problemas bidimensionais complexos com superfícies livres e baixo número de Reynolds, incluindo os problemas do inchamento do extrudado e jet flow / The simulation of free surfaces flows has gained importance in the recent years due to the many practical applications of this type of flow. Among the many numerical methods available for the simulation of fluid flows, there is GENSMAC, which is a numerical technique to simulate Newtonian flows with free surfaces. The implementation of semi-implicitmethods for the temporal discretization of the Navier-Stokes equations allowed a significant loosening in the time step restriction, reducing considerably the computational cost of the simulation of flows with Re 1. But, even with the semi-implicit methods, the time step cannot increase beyond certain limits, well below the CFL restriction, without causing serious accuracy problems in the numerical results and the consequent appearance of non-physical results. Therefore, even in the semi-implicit formulation, a time step restriction is applied. In this work, we analyse and implement in the FREEFLOW2D system a strategy for adaptive time-stepping in order to ensure stability and precision while using the largest possible time step. The efficiency and robustness of the technique incorporated to the implicit formulation of GENSMAC are demonstrated in the solution of two-dimensional complex problems with free surfaces and low Reynolds numbers, including the swelling of the extrudate and jet flow problems