Numerical Weather Prediction (NWP) and climate models parametrize the effects of boundary-layer turbulence as a diffusive process, dependent on a diffusion coefficient, which appears as nonlinear terms in the governing equations. In the advection dominated zone of the boundary layer and in the free atmosphere, the air flow supports different wave motions, with the fastest being the sound waves. Time integrations of these terms, in both zones, need to be implicit otherwise they impractically restrict the stable time step sizes. At the same time, implicit schemes may lose accuracy compared to explicit schemes in the same level, which is due to dispersion error associated with these schemes. Furthermore, the implicit schemes need iterative approaches like the Newton-Raphson method. Therefore, the combination of implicit and explicit methods, called IMEX or semi-implicit, has extensively attracted attention. In the combined method, the linear part of the equation as well as the fast wave terms are treated by the implicit part and the rest is calculated by the explicit scheme. Meanwhile, minimizing the dissipation and dispersion errors can enhance the performance of time integration schemes, since the stability and accuracy will be restricted by these inevitable errors. Hence, the target of this thesis is to increase the stability range, while obtaining accurate solutions by using IMEX and semi-implicit time integration methods. Therefore, a comprehensive effort has been made toward minimizing the numerical errors to develop new Runge-Kutta schemes, in IMEX and semi-implicit forms, to temporally integrate the governing equations in the atmospheric field so that the stability is extended and accuracy is improved, compared to the previous schemes. At the first step, the A-stability and the Strong Stability Preserving (SSP) optimized properties were compared as two essential properties of the time integration schemes. It was shown that both properties attempt to minimize the dissipation and dispersion errors, but in two different aspects. The SSP optimized property focuses on minimizing the errors to increase the accuracy limits, while the A-stability property tries to extend the range of stability. It was shown that the combination of both properties is essential in the field of interest. Moreover, the A-stability property was found as an essential property to accelerate the steady state solutions. Afterward, the dissipation and dispersion errors, generated by three-stage second order IMEX Runge-Kutta scheme were minimized, while the proposed scheme, so called IMEX-SSP2(2,3,2) enjoys the A-stability and SSP properties. A practical governing equation set in the atmospheric field, so called compressible Boussinesq equations set, was calculated using the new IMEX scheme and the results were compared to one well-known IMEX scheme in the literature, i.e. ARK2(2,3,2), which is an abbreviation of Additive Runge-Kutta. Note that, the ARK2(2,3,2) was compared to various types of IMEX Runge-Kutta schemes and it was found as the more efficient scheme in the atmospheric fields (Weller et al., 2013). It was shown that the IMEX-SSP2(2,3,2) could improve the accuracy and extend the range of stable time step sizes as well. Through the van der Pol test case, it was shown that the ARK2(2,3,2) with L-stability property may decline to the first order in the calculation of stiff limit, while IMEX-SSP2(2,3,2), with A-stability property, is able to retain the assigned second order of accuracy. Therefore, it was concluded that the L-stability property, due to restrictive conditions associated with, may weaken the time integration’s performance, compared to the A-stability property. The ability of the IMEX-SSP2(2,3,2) was proved in solving different case, which is the inviscid Burger equation in spherical coordinate system by using a realistic initial condition dataset. In the next step, it was attempted to maximize the non-negativity property associated with the numerical stability function of three-stage third order Diagonally Implicit Runge-Kutta (DIRK) schemes. It was shown that the non-negativity has direct relation with non-oscillatory behaviors. Two new DIRK schemes with A- and L-stability properties, respectively, were developed and compared to the SSP(3,3), which obtains the SSP optimized property in the same class of DIRK schemes. The SSP optimized property was found to be more beneficial for the inviscid (advection dominated) flows, since in the von Neumann stability analysis, the SSP optimized property provides more nonnegative region for the imaginary component of the stability function. However, in most practical cases, i.e. the viscous (advection diffusion) flows, the nonnegative property is needed for both real and imaginary components of the stability function. Therefore, the SSP optimized property, individually, is not helpful, unless mixed with the A-stability property. Meanwhile, the A- and L-stability properties were compared as well. The intention is to find how these properties influence the DIRK schemes’ performances. The A-stability property was found as preserving the SSP property more than the L-stability property. Moreover, the proposed A-stable scheme tolerates larger Courant Friedrichs Lewy (CFL) number, while preserving the accuracy and non-oscillatory computations. This fact was proved in calculating different test cases, including compressible Euler and nonlinear viscous Burger equations. Finally, the time integration of the boundary layer flows was investigated as well. The nonlinearity associated with the diffusion coefficient makes the implicit scheme impractical, while the explicit scheme inefficiently limits the stable time step sizes. By using the DIRK scheme, a new semi-implicit approach was proposed, in which the diffusion coefficient at each internal stage is calculated by a weight-averaged combination of the solutions at current internal stage and previous time step, in which the time integration can benefit from both explicit and implicit advantages. As shown, the accuracy was improved, which is due to engaging the explicit solutions and the stability was extended due to taking advantages of implicit scheme. It was found that the nominated semi-implicit method results in less dissipation error, more accurate solutions and less CPU time usage, compared to the implicit schemes, and it enjoys larger range of stable time steps than other semi-implicit approaches in the literature.

No description available.

No / Two numerical particle models, the Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) methods, coupled with a sub-particle scale (SPS) turbulence model, are presented to simulate free surface flows. Both SPH and MPS methods have the advantages in that the governing Navier¿Stokes equations are solved by Lagrangian approach and no grid is needed in the computation. Thus the free surface can be easily and accurately tracked by particles without numerical diffusion. In this paper different particle interaction models for SPH and MPS methods are summarized and compared. The robustness of two models is validated through experimental data of a dam-break flow. In addition, a series of numerical runs are carried out to investigate the order of convergence of the models with regard to the time step and particle spacing. Finally the efficiency of the incorporated SPS model is further demonstrated by the computed turbulence patterns from a breaking wave. It is shown that both SPH and MPS models provide a useful tool for simulating free surface flows

Modelling in cardiac electrophysiology results in a complex system of partial differential equations (PDE) describing the propagation of the electrical wave in the heart muscle coupled with a highly nonlinear system of ordinary differential equations (ODE) describing the ionic activity in the cardiac cells. This system forms the widely accepted bidomain model or its slightly simpler version, the monodomain model. To a large extent, the stiffness of the whole model depends on the choice of the ionic model, which varies in terms of complexity and realism. These simulations require accurate and, depending on the ionic model used, possibly very stable numerical methods. At this time, solving these models numerically requires CPU time of around one day per heartbeat. Therefore, it is necessary to use the most efficient method for these simulations. This research focuses on the comparison and analysis of several time-stepping methods: explicit or semi-implicit, operator splitting, deferred correction and Rush-Larsen methods. The goal is to find the optimal method for the ionic model used. For our analysis, we used the monodomain model but our results apply to the bidomain model as well. We compare the methods for three ionic models of varying complexity and stiffness: the Mitchell-Schaeffer models with only 2 variables, the more realistic Beeler-Reuter model with 8 variables, and the stiff and very complex ten Tuscher-Noble-Noble-Panfilov (TNNP) models with 17 variables. For each method, we derived absolute stability criteria of the spatially discretized monodomain model and verified that the theoretical critical time steps obtained closely match the ones in numerical experiments. Convergence tests were also conducted to verify that the numerical methods achieve an optimal order of convergence on the model variables and derived quantities (such as speed of the wave, depolarization time), and this in spite of the local non-differentiability of some of the ionic models. We looked at the efficiency of the different methods by comparing computational times for similar accuracy. Conclusions are drawn on the methods to be used to solve the monodomain model based on the model stiffness and complexity, measured respectively by the most negative eigenvalue of the model's Jacobian and the number of variables, and based on strict stability and accuracy criteria.

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第22047号 / 工博第4628号 / 新制||工||1722(附属図書館) / 京都大学大学院工学研究科社会基盤工学専攻 / (主査)教授 後藤 仁志, 教授 細田 尚, 准教授 KHAYYER,Abbas / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DGAM

Two numerical particle models, the Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) methods, coupled with a sub-particle scale (SPS) turbulence model, are presented to simulate free surface flows. Both SPH and MPS methods have the advantages in that the governing Navier¿Stokes equations are solved by Lagrangian approach and no grid is needed in the computation. Thus the free surface can be easily and accurately tracked by particles without numerical diffusion. In this paper different particle interaction models for SPH and MPS methods are summarized and compared. The robustness of two models is validated through experimental data of a dam-break flow. In addition, a series of numerical runs are carried out to investigate the order of convergence of the models with regard to the time step and particle spacing. Finally the efficiency of the incorporated SPS model is further demonstrated by the computed turbulence patterns from a breaking wave. It is shown that both SPH and MPS models provide a useful tool for simulating free surface flows.

O concreto é um dos materiais de construção civil mais versáteis, podendo se adaptar a formas diversas quando em seu estado fresco e resistindo a grandes cargas de compressão em seu estado rígido. No entanto, em estruturas mais densamente armadas ou com geometrias mais complexas, pode-se apresentar dificuldades para a moldagem, causando falhas no preenchimento da forma, o que reduz a capacidade resistente da peça e sua vida útil. Neste trabalho foi utilizado o método de partículas lagrangeanas Moving Particle Semi-Implicit (MPS) como base para um simulador para o estudo do comportamento do escoamento de pastas e argamassas cujo comportamento pode se aproximado por modelos reológicos como Bingham e Herschel-Bulkley. Foram propostos módulos para a simulação da viscosidade não-newtoniana, variação térmica no processo de cura e modelagem de turbulência. Para modelar a variação de viscosidade de um fluido não-newtoniano foi utilizado o modelo de Herschel-Bulkley. O modelo de Herschel-Bulkley possui uma singularidade para taxas de deformação muito pequenas, que resulta em valores de viscosidade infinitas, dificuldade contornada pela solução proposta por Papanastasiou (1987). Na modelagem térmica foram analisados dois modelos de dissipação, sendo um original do método e outro calculado por meio do divergente do gradiente utilizando os modelos de partículas e que teria melhores resultados para o cálculo da dissipação térmica. Também foi modelada a convecção térmica, utilizando para isso a hipótese de Boussinesq. A reação de hidratação do concreto foi modelada utilizando uma equação do tipo Hill para representar a elevação de temperatura obtida por meio um ensaio adiabático. Para complementar as simulações, foi utilizado o modelo de turbulência Detached Eddy Simulation (DES), baseado no método Large Eddy Simulation (LES), que utiliza um modelo de parede para simular a interação do fluido. Para a implementação deste modelo de turbulência foi proposto um algoritmo para o cálculo da distância da partícula de fluido à parede. Este algoritmo utiliza estruturas de dados já existentes no método de partículas de modo que sua execução requer muito menos recursos que a busca binária. Apesar do trabalho ter se guiado pela simulação do concreto fresco, que é um material particularmente complexo, outros escoamentos encontrados na engenharia civil também podem ser beneficiados pelo método, como os estudos do escoamento em sistemas prediais de água e esgoto, do escoamento e prevenção de erosão em rios e córregos, do escorregamento de encostas, dos reatores para depuração de esgotos, entre outros. / The concrete is one of the more versatile civil construction materials, it can adapt to various forms when in its fresh state while resisting to high compression loads in its rigid state. However, in some cases like densely reinforced concrete structures and complex geometry concrete structures can present issues to the casting, and failure in proper form filling can occur. These failures can reduce the resistance and the lifetime of the structure. This work used a simulator based on the lagrangean particle method called Moving Particle Semi-Implicit (MPS) to study the concrete behavior in its distinctive characteristics. Also, this work proposed modules to simulate the non-Newtonian viscosity, the thermal process of concrete cure and to model the turbulent flow. To model the variation of viscosity of a non-Newtonian fluid, the Herschel-Bulkley model, which relates the shear stress with the strain rate, was applied. The Herschel-Bulkley model has a singularity at low strain rates, which results in infinite viscosities. To overcome this issue, Papanastasiou (1987) proposed a modification in the model in order to eliminate the singularity. For the thermal modeling, two models for thermal dissipation were compared, the original method and other calculated from the divergence of gradient using the differential operators for the particle model and that could present improved results for the thermal dissipation calculation. Also, the thermal convection was modeled using the Boussinesq hypothesis. The hydration reaction of the concrete was modeled using a Hill type equation to reproduce the temperature elevation. In addition, a Detached Eddy Simulation (DES) based turbulence model with a simple wall model in the interaction of wall and fluid was applied in the simulations. To improve the turbulence model, an algorithm to calculate the distance between fluid and the nearest wall particle was proposed. The algorithm uses already calculated information from particles so that the execution requires much less resources than a binary search. Although the work has been focused on to the modeling of fresh concrete simulation, which is a particularly complex material, other flows found in civil engineering can also be benefited by the method, such as studies of drainage in water and sewage systems, drainage and prevention of erosion into rivers and streams, prevention and damage mitigation of landslides, reactors for sewage treatment among many others.

O concreto é um dos materiais de construção civil mais versáteis, podendo se adaptar a formas diversas quando em seu estado fresco e resistindo a grandes cargas de compressão em seu estado rígido. No entanto, em estruturas mais densamente armadas ou com geometrias mais complexas, pode-se apresentar dificuldades para a moldagem, causando falhas no preenchimento da forma, o que reduz a capacidade resistente da peça e sua vida útil. Neste trabalho foi utilizado o método de partículas lagrangeanas Moving Particle Semi-Implicit (MPS) como base para um simulador para o estudo do comportamento do escoamento de pastas e argamassas cujo comportamento pode se aproximado por modelos reológicos como Bingham e Herschel-Bulkley. Foram propostos módulos para a simulação da viscosidade não-newtoniana, variação térmica no processo de cura e modelagem de turbulência. Para modelar a variação de viscosidade de um fluido não-newtoniano foi utilizado o modelo de Herschel-Bulkley. O modelo de Herschel-Bulkley possui uma singularidade para taxas de deformação muito pequenas, que resulta em valores de viscosidade infinitas, dificuldade contornada pela solução proposta por Papanastasiou (1987). Na modelagem térmica foram analisados dois modelos de dissipação, sendo um original do método e outro calculado por meio do divergente do gradiente utilizando os modelos de partículas e que teria melhores resultados para o cálculo da dissipação térmica. Também foi modelada a convecção térmica, utilizando para isso a hipótese de Boussinesq. A reação de hidratação do concreto foi modelada utilizando uma equação do tipo Hill para representar a elevação de temperatura obtida por meio um ensaio adiabático. Para complementar as simulações, foi utilizado o modelo de turbulência Detached Eddy Simulation (DES), baseado no método Large Eddy Simulation (LES), que utiliza um modelo de parede para simular a interação do fluido. Para a implementação deste modelo de turbulência foi proposto um algoritmo para o cálculo da distância da partícula de fluido à parede. Este algoritmo utiliza estruturas de dados já existentes no método de partículas de modo que sua execução requer muito menos recursos que a busca binária. Apesar do trabalho ter se guiado pela simulação do concreto fresco, que é um material particularmente complexo, outros escoamentos encontrados na engenharia civil também podem ser beneficiados pelo método, como os estudos do escoamento em sistemas prediais de água e esgoto, do escoamento e prevenção de erosão em rios e córregos, do escorregamento de encostas, dos reatores para depuração de esgotos, entre outros. / The concrete is one of the more versatile civil construction materials, it can adapt to various forms when in its fresh state while resisting to high compression loads in its rigid state. However, in some cases like densely reinforced concrete structures and complex geometry concrete structures can present issues to the casting, and failure in proper form filling can occur. These failures can reduce the resistance and the lifetime of the structure. This work used a simulator based on the lagrangean particle method called Moving Particle Semi-Implicit (MPS) to study the concrete behavior in its distinctive characteristics. Also, this work proposed modules to simulate the non-Newtonian viscosity, the thermal process of concrete cure and to model the turbulent flow. To model the variation of viscosity of a non-Newtonian fluid, the Herschel-Bulkley model, which relates the shear stress with the strain rate, was applied. The Herschel-Bulkley model has a singularity at low strain rates, which results in infinite viscosities. To overcome this issue, Papanastasiou (1987) proposed a modification in the model in order to eliminate the singularity. For the thermal modeling, two models for thermal dissipation were compared, the original method and other calculated from the divergence of gradient using the differential operators for the particle model and that could present improved results for the thermal dissipation calculation. Also, the thermal convection was modeled using the Boussinesq hypothesis. The hydration reaction of the concrete was modeled using a Hill type equation to reproduce the temperature elevation. In addition, a Detached Eddy Simulation (DES) based turbulence model with a simple wall model in the interaction of wall and fluid was applied in the simulations. To improve the turbulence model, an algorithm to calculate the distance between fluid and the nearest wall particle was proposed. The algorithm uses already calculated information from particles so that the execution requires much less resources than a binary search. Although the work has been focused on to the modeling of fresh concrete simulation, which is a particularly complex material, other flows found in civil engineering can also be benefited by the method, such as studies of drainage in water and sewage systems, drainage and prevention of erosion into rivers and streams, prevention and damage mitigation of landslides, reactors for sewage treatment among many others.

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.