O fenômeno de impacto mecânico entre corpos sólidos está presente em diversas áreas da engenharia. Exemplos atuais deste tipo de problema podem ser encontrados no projeto de elementos de máquinas, sistemas de transporte como containers com material nuclear, tubulações em indústrias químicas, autoveículos e várias outras estruturas que devem obedecer à códigos de segurança estabelecidos por legislações governamentais. Na maioria destes casos, o conhecimento das tensões oriundas do impacto entre os corpos é fundamental para evitarem-se fa-lhas nas estruturas projetadas, predizer danos indesejáveis, diminuir coeficientes de segurança, etc. Neste contexto, é proposta neste trabalho uma metodologia de projeto contra impacto em sistemas mecânicos elásticos baseada nas equações de superfície de tensão máxima, que representam diferentes situações de impacto em uma determinada geometria. O Método dos Elementos Finitos com a integração explícita no tempo é aplicado para resolver o problema dinâmico associado ao impacto. Como exemplos de aplicações são estudados um suporte e um eixo chavetado. / Impact between solid bodies is present in many areas of engineering. Relevant examples of this sort of problem can be found in machine element design, transport systems such as containers for nuclear material, pipes in chemical plants, vehicles and many others structures that should comply with safety codes issued by govern agencies. In the majority of these cases, the knowledge of the stresses due to the impact between the bodies is fundamental to avoid failures on the designed structures, to predict undesired damages, and to decrease safety factors. Therefore, in this work a design methodology for linear mechanical systems submitted to impact is proposed. It is based on the surface of maximum stress which represents different crash situations for a given elastic model. The Finite Element Method with the explicit time integration algorithm is used to solve the associated dynamic problem. Examples are presented such as a bracket and a shaft.

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.

<p>Problems in structural dynamics that involve rapid evolution of the material at multiple scales of length and time are challenging to solve numerically. One such problem is that of a structure un- dergoing fracture, where the material in the vicinity of a crack front may experience high stresses and strains while the remainder of the structure may be unaffected by it. Usually, such problems are solved using numerical methods based on a finite element discretization in space and a finite difference time-stepping scheme to capture dynamic response. Regions of interest within the struc- ture, where high transients are expected, are usually modeled with a fine discretization in space and time for better accuracy. In other regions of the model where the response does not change rapidly, a coarser discretization suffices and helps keep the computational cost down. This variation in spatial and temporal discretization is achieved through domain decomposition and multi-time-step coupling methods which allow the use of different levels of mesh discretization and time-steps in different regions of the mesh.</p>

A plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases it is difficult to satisfy certain physical properties while maintaining high order of accuracy. In this thesis, we develop high-order time-stepping methods that are capable of maintaining stability constraints of the solution, when coupled with suitable spatial discretizations. Such methods are called strong stability preserving (SSP) time integrators, and we mainly focus on perturbed methods that use both upwind- and downwind-biased spatial discretizations. Firstly, we introduce a new family of third-order implicit Runge–Kuttas methods with arbitrarily large SSP coefficient. We investigate the stability and accuracy of these methods and we show that they perform well on hyperbolic problems with large CFL numbers. Moreover, we extend the analysis of SSP linear multistep methods to semi-discretized problems for which different terms on the right-hand side of the initial value problem satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems. Optimal perturbed methods attain augmented monotonicity-preserving step sizes when the different forward Euler conditions are taken into account. On the other hand, we show that optimal SSP additive methods achieve a monotonicity-preserving step-size restriction no better than that of the corresponding non-additive SSP linear multistep methods. Furthermore, we develop the first SSP linear multistep methods of order two and three with variable step size, and study their optimality. We describe an optimal step-size strategy and demonstrate the effectiveness of these methods on various one- and multi-dimensional problems. Finally, we establish necessary conditions to preserve the total variation of the solution obtained when perturbed methods are applied to boundary value problems. We implement a stable treatment of nonreflecting boundary conditions for hyperbolic problems that allows high order of accuracy and controls spurious wave reflections. Numerical examples with high-order perturbed Runge–Kutta methods reveal that this technique provides a significant improvement in accuracy compared with zero-order extrapolation.

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This work deals with the design of a hybrid time integrator that couples spatially explicit and implicit time integrators. In order to cope with the industrial solver of Ariane Group called FLUSEPA, the explicit scheme of Heun and the implicit scheme of Crank-Nicolson are hybridized using the transition parameter : the whole technique is called AION time integration. The latter is studied into details with special focus on spectral behaviour and on its ability to keep the accuracy. It is shown that the hybrid technique has interesting dissipation and dispersion properties while maintaining precision and avoiding spurious waves. Moreover, this hybrid approach is validated on several academic test cases for both convective and diffusive fluxes. And as expected the method is more interesting in term of computational time than standard time integrators. For the extension of this hybrid approach to the temporal adaptive method implemented in FLUSEPA, it was necessary to improve some treatments in order to maintain conservation and acceptable spectral properties. Finally the hybrid time integration was also applied to a RANS/LES turbulent test case with interesting computational time while capturing the flow physics.

Current climate models have a limited ability to increase spatial resolution because numerical stability requires the time step to decrease. I describe initial experiments with two independent but complementary strategies for attacking this "time barrier". First I describe computational experiments exploring the performance improvements from overlapping computation and communication on hybrid parallel computers. My test case is explicit time integration of linear advection with constant uniform velocity in a three-dimensional periodic domain. I present results for Fortran implementations using various combinations of MPI, OpenMP, and CUDA, with and without overlap of computation and communication. Second I describe a semi-Lagrangian method for tracer transport that is stable for arbitrary Courant numbers, along with a parallel implementation discretized on the cubed sphere. It shows optimal accuracy at Courant numbers of 10-20, more than an order of magnitude higher than explicit methods. Finally I describe the development and stability analyses of the time integrators and advection methods I used for my experiments. I develop explicit single-step methods with stability up to Courant numbers of one in each dimension, hybrid explicit-implict methods with stability for arbitrary Courant numbers, and interpolation operators that enable the arbitrary stability of semi-Lagrangian methods.

Time integration methods are necessary for the solution of transient flow problems. In recent years, interest in transient flow problems has increased, leading to a need for better understanding of the costs and benefits of various time integration schemes. The present work investigates two common time integration schemes, namely the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) and the Fractional Step (FS) method. Three two-dimensional, transient, incompressible flow problems are solved using a cell centered, finite volume code. The three test cases are laminar flow in a lid-driven skewed cavity, laminar flow over a square cylinder, and turbulent flow over a square cylinder. Turbulence is modeled using wall functions and the k - ε turbulence model with the modifications suggested by Kato and Launder. Solution efficiency as measured by the effort carried out by the flow equation solver and CPU time is examined. Accuracy of the results, generated using the SIMPLE and FS time integration schemes, is analyzed through a comparison of the results with existing experimental and/or numerical solutions. Both the SIMPLE and FS algorithms are shown to be capable of solving benchmark flow problems with reasonable accuracy. The two schemes differ slightly in their prediction of flow evolution over time, especially when simulating very slowly changing flows. As the time step size decreases, the SIMPLE algorithm computational cost (CPU time) per time step remains approximately constant, while the FS method experiences a reduction in cost per time step. Also, the SIMPLE algorithm is numerically stable for time steps approaching infinity, while the FS scheme suffers from numerical instability if the time step size is too large. As a result, the SIMPLE algorithm is recommended to be used for transient simulations with large time steps or steady state problems while the FS scheme is better suited for small time step solutions, although both time-stepping schemes are found to be most efficient when their time steps are at their maximum stable value.

Time integration methods are necessary for the solution of transient flow problems. In recent years, interest in transient flow problems has increased, leading to a need for better understanding of the costs and benefits of various time integration schemes. The present work investigates two common time integration schemes, namely the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) and the Fractional Step (FS) method. Three two-dimensional, transient, incompressible flow problems are solved using a cell centered, finite volume code. The three test cases are laminar flow in a lid-driven skewed cavity, laminar flow over a square cylinder, and turbulent flow over a square cylinder. Turbulence is modeled using wall functions and the k - ε turbulence model with the modifications suggested by Kato and Launder. Solution efficiency as measured by the effort carried out by the flow equation solver and CPU time is examined. Accuracy of the results, generated using the SIMPLE and FS time integration schemes, is analyzed through a comparison of the results with existing experimental and/or numerical solutions. Both the SIMPLE and FS algorithms are shown to be capable of solving benchmark flow problems with reasonable accuracy. The two schemes differ slightly in their prediction of flow evolution over time, especially when simulating very slowly changing flows. As the time step size decreases, the SIMPLE algorithm computational cost (CPU time) per time step remains approximately constant, while the FS method experiences a reduction in cost per time step. Also, the SIMPLE algorithm is numerically stable for time steps approaching infinity, while the FS scheme suffers from numerical instability if the time step size is too large. As a result, the SIMPLE algorithm is recommended to be used for transient simulations with large time steps or steady state problems while the FS scheme is better suited for small time step solutions, although both time-stepping schemes are found to be most efficient when their time steps are at their maximum stable value.

Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos. / In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.