266 research outputs found

    A global method for coupling transport with chemistry in heterogeneous porous media

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    Modeling reactive transport in porous media, using a local chemical equilibrium assumption, leads to a system of advection-diffusion PDE's coupled with algebraic equations. When solving this coupled system, the algebraic equations have to be solved at each grid point for each chemical species and at each time step. This leads to a coupled non-linear system. In this paper a global solution approach that enables to keep the software codes for transport and chemistry distinct is proposed. The method applies the Newton-Krylov framework to the formulation for reactive transport used in operator splitting. The method is formulated in terms of total mobile and total fixed concentrations and uses the chemical solver as a black box, as it only requires that on be able to solve chemical equilibrium problems (and compute derivatives), without having to know the solution method. An additional advantage of the Newton-Krylov method is that the Jacobian is only needed as an operator in a Jacobian matrix times vector product. The proposed method is tested on the MoMaS reactive transport benchmark.Comment: Computational Geosciences (2009) http://www.springerlink.com/content/933p55085742m203/?p=db14bb8c399b49979ba8389a3cae1b0f&pi=1

    Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

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    The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

    Nonlinear multiscale viscosity methods and time integration schemes for solving compressible Euler equations

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    Este trabalho apresenta duas formulações do método de elementos finitos, utilizando estabilização multiescala, para resolver o sistema de equações de Euler compressíveis bidimensionais em variáveis conservativas. O espaço submalha é definido através de funções polinomiais que se anulam na fronteira dos elementos, conhecidas como funções bolha, permitindo o uso de um complemento de Schur local para definir o problema das escalas resolvidas. Esse procedimento resulta em uma metodologia numérica que permite variações temporais das escalas não resolvidas. As formulações propostas neste trabalho são baseadas em resíduo e consideram viscosidade artificial agindo em todas as escalas de discretização. Na primeira formulação um operador não linear é adicionado sobre todas as escalas, já na segunda formulação diferentes operadores não lineares são incluídos sobre as escalas macro e micro. A eficiência das novas formulações são avaliadas através de estudos numéricos, comparando-as com outras formulações, tais como os métodos SUPG combinado com o operador de captura de choque YZBeta e CAU. Outra contribuição que este trabalho apresenta diz respeito ao avanço no tempo, uma vez que métodos baseados em densidade sofrem com efeitos indesejados em escoamento com baixa velocidade, o que inclui convergência lenta e perda de acurácia. Devido a esse fenômeno, a técnica de precondicionamento local é aplicada às equações no caso contínuo. Uma alternativa para resolver esta deficiência consiste em utilizar esquemas de avanço no tempo com propriedade de decaimento como L-estabilidade. Com esse intuito é proposto um esquema preditor-corretor baseado em Backward Differentiation Formulas (BDF) cuja predição é realizada através de extrapolação

    Nonlinear fluid-structure interaction problem. Part I: implicit partitioned algorithm, nonlinear stability proof and validation examples

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    International audienceIn this work we consider the fluid-structure interaction in fully nonlinear setting, where different space discretization can be used. The model problem considers finite elements for structure and finite volume for fluid. The computations for such interaction problem are performed by implicit schemes, and the partitioned algorithm separating fluid from structural iterations. The formal proof is given to find the condition for convergence of this iterative procedure in the fully nonlinear setting. Several validation examples are shown to confirm the proposed convergence criteria of partitioned algorithm. The proposed strategy provides a very suitable basics for code-coupling implementation as discussed in Part II

    Numerical Methods for PDE Constrained Optimization with Uncertain Data

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    Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization. The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods

    Parallel Algorithms for Time and Frequency Domain Circuit Simulation

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    As a most critical form of pre-silicon verification, transistor-level circuit simulation is an indispensable step before committing to an expensive manufacturing process. However, considering the nature of circuit simulation, it can be computationally expensive, especially for ever-larger transistor circuits with more complex device models. Therefore, it is becoming increasingly desirable to accelerate circuit simulation. On the other hand, the emergence of multi-core machines offers a promising solution to circuit simulation besides the known application of distributed-memory clustered computing platforms, which provides abundant hardware computing resources. This research addresses the limitations of traditional serial circuit simulations and proposes new techniques for both time-domain and frequency-domain parallel circuit simulations. For time-domain simulation, this dissertation presents a parallel transient simulation methodology. This new approach, called WavePipe, exploits coarse-grained application-level parallelism by simultaneously computing circuit solutions at multiple adjacent time points in a way resembling hardware pipelining. There are two embodiments in WavePipe: backward and forward pipelining schemes. While the former creates independent computing tasks that contribute to a larger future time step, the latter performs predictive computing along the forward direction. Unlike existing relaxation methods, WavePipe facilitates parallel circuit simulation without jeopardizing convergence and accuracy. As a coarse-grained parallel approach, it requires low parallel programming effort, furthermore it creates new avenues to have a full utilization of increasingly parallel hardware by going beyond conventional finer grained parallel device model evaluation and matrix solutions. This dissertation also exploits the recently developed explicit telescopic projective integration method for efficient parallel transient circuit simulation by addressing the stability limitation of explicit numerical integration. The new method allows the effective time step controlled by accuracy requirement instead of stability limitation. Therefore, it not only leads to noticeable efficiency improvement, but also lends itself to straightforward parallelization due to its explicit nature. For frequency-domain simulation, this dissertation presents a parallel harmonic balance approach, applicable to the steady-state and envelope-following analyses of both driven and autonomous circuits. The new approach is centered on a naturally-parallelizable preconditioning technique that speeds up the core computation in harmonic balance based analysis. The proposed method facilitates parallel computing via the use of domain knowledge and simplifies parallel programming compared with fine-grained strategies. As a result, favorable runtime speedups are achieved
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