Regina C. Almeida

An adaptive Petrov-Galerkin formulation for the compressible Euler and Navier-Stokes equations

In this paper we combine a stable Petrov-Galerkin formulation for the compressible Euler and Navier-Stokes equations with an h-adaptive remeshing refinement, including directional stretching and stretching ratio in the mesh regeneration procedure. The result is an accurate and efficient scheme appropriated to solve problems presenting shocks and boundary-layers.

A stable Petrov-Galerkin method for convection-dominated problems

In this paper, a new Petrov-Galerkin formulation for solving convection-dominated problems is presented. The method developed achieves the quasi-optimal convergence rates when the solution is regular and provides the necessary stability to avoid spurious oscillations when strong gradients are present. Such important properties allow the use of p refinement to improve the solution in regions with discontinuities because of the stability engendered by the new Petrov-Galerkin method. In this matter, a proper evaluation of the intrinsic time scale function, appearing in the design of this method, is crucial to guarantee the required accuracy.

Iterative local solvers for distributed Krylov-Schwarz method applied to convection-diffusion problems

Abstract Nowadays, supercomputers can be used to solve large-scale problems that come from simulation of industrial or research problems. However, those machines are usually inacessible to most industries and university laboratories around the world. In this work we present an iterative solver, a Krylov-Schwarz Method (KSM), to be used in a collection of workstations under PVM. The subdomain problems are solved by using many methods in order to show how the choice of the local solvers affects the overall performance of the distributed KSM.

Numerical analysis of the nonlinear subgrid scale method

This paper presents the numerical analysis of the Nonlinear Subgrid Scale (NSGS) model for approximating singularly perturbed transport models. The NSGS is a free parameter subgrid stabilizing method that introduces an extra stability only onto the subgrid scales. Thisnew feature comes from the local control yielded by decomposing the velocity field into the resolved and unresolved scales. Such decomposition is determined by requiring the minimum of the kinetic energy associated to the unresolved scales and the satisfaction of the resolved scale model problem at element level. The developed method is robust for a wide scope of singularly perturbed problems. Here, we establish the existence and uniqueness of the solution, and provide an a priori error estimate. Convergence tests on two-dimensional examples are reported. Mathematical subject classification: Primary: 65N12; Secondary: 74S05.

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

Abstract In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.

Performance of a Euler solver using a distributed system

Publisher Summary This chapter presents a distributed solver for solving computational fluid dynamics (CFD problems), in particular the compressible Euler equations using a stable Petrov Galerkin method written in entropy variables, designed using a space-time finite element formulation. It focuses on an Euler solver to be used in a collection of workstations under parallel virtual machine (PVM) that can overcome the use of direct methods on a single processor. The chapter presents a distributed implementation of the generalized minimal residual method GMRES (k) with right pre-conditioning, for which the computational cost, depends on the dimension of the Krylov Space (k) and on the interface of the domains. The consistent approximate upwind (CAU) method for the compressible Euler equations employing entropy variables is presented. The numerical results show that distributed solver is appropriate for distributed systems like a collection of workstations connected by a token-ring or bus based network. The chapter concludes that this type of approach can be efficient and it can be superior or faster than the usual solvers even for the small size problems solved in the chapter.

A Influencia da Diferenciação Fenotípica na Dinâmica do Crescimento Tumoral

O crescimento tumoral e resultado de mecanismos nao lineares complexos que ocorrem em diversas escalas de tempo e espaco. A modelagem computacional pode ajudar na compreensao de tais mecanismos, assim como auxiliar no desenvolvimento de terapias efetivas. Neste trabalho, desenvolvemos um modelo hibrido avascular que integra tres escalas espaciais (tecidual, celular e molecular) com o objetivo de estudar a influencia da resposta regulatoria intra-celular nos processos de migracao e proliferacao. Essas repostas resultam de reacoes bioquimicas de sinalizacao molecular iniciadas por estimulos extracelulares. Experimentos computacionais sao realizados para demonstrar a importância da diferenciacao fenotipica na dinâmica do crescimento tumoral.

Calibração de um Modelo de Crescimento Tumoral Avascular

Neste trabalho utilizamos um modelo matematico simples para representar o crescimento de tumores avasculares. Atraves de dados da literatura realizamos a calibracao do modelo proposto tendo como base a abordagem Bayesiana. Nesta abordagem, as distribuicoes a posteriori dos parâmetros sao obtidas utilizando duas tecnicas: amostragem via algoritmo de Metropolis-Hastings, um metodo de Monte Carlo via cadeias de Markov, e amostragem via grade fixa. Ambas conduziram a resultados satisfatorios para os experimentos realizados e sao por natureza mais informativas que os metodos tradicionais de regressao.

Modelo Hı́brido para o Crescimento Tumoral do Carcinoma Avascular

crescimento tumoral e resultado de uma serie de complexos fenomenos que ocorrem em multiplas escalas de tempo e espaco. Na escala sub-celular ocorre uma variedade de cascatas de reacoes moleculares que regulam as atividades celulares. Na escala da celula, destacam-se os mecanismos de agregacao, adesao e sinalizacao entre celulas e entre os componentes do microambiente. Fenomenos tipicos dos meios continuos ocorrem na escala do tecido, tais como difusao de nutrientes, fatores de crescimento, etc. Eventos que ocorrem em uma escala interferem com os que ocorrem em outras e vice-versa, de modo que o entendimento destes mecanismos e fundamental para a compreensao da doenca e para o desenvolvimento de terapias. Neste trabalho, desenvolvemos um modelo hibrido que representa fenomenos que ocorrem nas escalas tecidual e celular. A escala celular e descrita atraves de um modelo baseado em agentes (MBA), que possibilita tratar cada celula individualmente e descrever seu comportamento no microambiente. Na escala do tecido representamos a dispersao de nutrientes no meio atraves de uma equacao diferencial parcial de difusao-reacao. Sem perda de generalidade, este modelo e usado para descrever o crescimento tumoral do carcinoma avascular e experimentos computacionais sao realizados para demonstrar o potencial uso da metodologia desenvolvida.

Hierarchical Models of Tumor Growth

We propose in this work a simple framework to build a hierarchical family of tumor growth models by selecting a subset of the most important parameters of our base model with respect to the evolution of the tumor volume. The importance of each parameter is identified through a model-free sensitivity analysis technique, the elementary effects (EE), due to its simplicity and low computational cost. This model framework encompasses the essential hypotheses and the limited set of important parameters acquired from the sensitivity analysis. In this way, we are able to create a family of models described by at least the same essential conditions and parameters but with different complexities regarding the number of parameters used. Numerical experiments are conducted to show the reasoning behind the hierarchical developed family of tumor growth modes. The modeling framework in this manner provides a powerful way for studying a model itself or either its simplification or extension. The framework can also be tailored to form the basis for future models, incorporating new processes and phenomena.