Title: Numerical Solution of the Three-dimensional Compressible Flow Author: Martin Kyncl Department: Department of Numerical Mathematics Supervisor: Doc. RNDr. Jiří Felcman, CSc. Abstract: This thesis deals with a fluid flow in 3D in general. The system of the equations, describing the compressible gas flow, is solved numerically, with the aid of the finite volume method. The main purpose is to describe particular boundary conditions, based on the analysis of the incomplete Riemann problem. The analysis of the original initial-value problem shows, that the right hand-side initial condition, forming the Riemann problem, can be partially replaced by the suitable complementary condition. Several modifications of the Riemann problem are introduced and analyzed, as an original result of this work. Algorithms to solve such problems were implemented and used in code for the solution of the compressible gas flow. Numerical experiments documenting the suggested methods are performed. Keywords: compressible fluid flow, the Navier-Stokes equations, the Euler equations, boundary conditions, finite volume method, the Riemann problem, numerical flux, tur- bulent flow

Dans cette thèse, nous nous intéressons à l'étude d'un modèle de transport de sédiments en nous plaçant sous deux angles d'approche différents. L'un concerne la modélisation numérique du problème et propose une méthode de résolution numérique basée sur un solveur de Riemann approché pour le système de Saint-Venant-Exner qui reste un des systèmes les plus répandus pour traiter le transport sédimentaire par charriage. Ce dernier repose sur un couplage du modèle hydraulique de Saint-Venant et du modèle morphodynamique d'Exner. Le point essentiel de la méthode proposée se situe au niveau du traitement du couplage de ce système. En effet, il existe deux stratégies; la première consiste à découpler la résolution de la partie fluide de la partie solide et les faire interagir à des instants donnés alors que la seconde considère une résolution couplée du système en mettant à jour conjointement les grandeurs hydrauliques et solides aux mêmes instants. Se posera alors la question du choix de la stratégie de résolution pour laquelle nous apporterons des éléments de réponses en comparant les deux approches. L'autre se concentre sur la mise en place d'une méthodologie pour l'étude des incertitudes liées au même modèle. Pour ce faire, nous proposons une formulation stochastique du système de Saint-Venant-Exner et nous cherchons à caractériser la variabilité des sorties par rapport aux paramètres d'entrée naturellement aléatoires. Cette première étude révélera la nécessité de revenir à un système de Saint-Venant avec un fond bruité pour étudier la sensibilité des grandeurs hydrauliques par rapport aux perturbations topographiques. / In this thesis, we are interested on the study of a sediment transport model through two different approaches. One of them concerns the numerical modelling of the problem and proposes a numerical problem-solving method based on an approximate Riemann solver for the Saint-Venant-Exner system which is one of the most common model to deal with sedimentary bed-load transport. This last one is based on a coupling between the hydraulic model of Saint-Venant and the morphodynamic model of Exner. The key point of the proposed method is the treatment of the coupling issue. Indeed, there exists two strategies; the first one consists on decoupling the resolution of the fluid part from the solid part and making them interact at fixed times whereas the second one considers a coupled approach to solve the system by jointly updating the hydraulic and solid quantities at same times. We then raise the issue of the choice of the strategy for which we suggest answers by comparing both approaches. The other one focuses on the development of a methodology to study the uncertainties related to the model previously mentioned. To this end, we propose a stochastic formulation of the Saint-Venant-Exner system and we look for characterizing the variabilities of the outputs in relation to the naturally random input parameters. This first study reveals the need for a return to the Saint-Venant system with a perturbed bed to understand the sensitivity of the hydraulic quantities on the topographical perturbations.

Depuis de nombreuses années, le givrage a été identifié comme un danger dans le domaine de l’aéronautique.L’accrétion de givre se produit lorsque des gouttelettes d’eau surfondue se déposent sur une surface, enparticulier le bord d’attaque d’une aile ou la lèvre d’entrée d’air moteur, et gèlent après l’impact. Ceci peutensuite engendrer une dégradation des performances aérodynamiques, un dysfonctionnement des sondes ouencore un endommagement du moteur. C’est pourquoi cette problématique est étudiée avec attention. Lesessais en vol et en soufflerie étant longs et coûteux, la simulation numérique de l’accrétion de givre est devenueun outil nécessaire dans le processus de conception et de certification des avions. Cette thèse s’inscrit dans le contexte de la modélisation 3D de l’accrétion de givre, et plus particulièrement des couches limites dynamique et thermique qui se développent autour du corps givré. Les outils numériques devant être rapides et robustes, l’approche proposée dans cette thèse pour le calcul aérodynamique est une méthode couplée Euler/couche limite intégrale. Ainsi, un modèle intégral est développé pour représenterle développement de la couche limite dynamique. La partie thermique est modélisée soit par une méthodesimplifiée basée sur des approches algébriques, soit par une méthode intégrale. Cette modélisation des coucheslimites dynamique et thermique est valable sur paroi lisse ou rugueuse et permet de fournir notamment lecoefficient de frottement et le coefficient d’échange thermique qui sont nécessaires pour un calcul d’accrétion degivre. Les équations intégrales de couche limite, associées à leurs relations de fermeture, sont ensuite résoluespar une méthode Volumes-Finis sur maillage surfacique non structuré, qui est bien adaptée pour les géométriescomplexes. De plus, des traitements numériques spécifiques sont mis en œuvre pour améliorer la précision dela méthode au voisinage du point d’arrêt et pour rendre le code robuste au passage du décollement.Après la validation de la méthode de couche limite, le code est utilisé dans les chaînes de givrage 2D et 3Dde l’ONERA pour des applications d’accrétion de givre. Ceci permet de montrer l’intérêt de la méthode entermes de robustesse et de précision par rapport aux codes de couche limite habituellement utilisés dans lescodes de givrage actuels. / Icing has since long been identified as a serious issue in the aeronautical world. Ice accretion occurs whensupercooled water droplets impinge on a surface, particularly the leading edge of a wing or an engine inlet, andfreeze after the impingement. This can lead to degradation of aerodynamic performances, sensor malfunctionor engine damage. This is why this issue is being carefully studied. The lengthy and costly flight and windtunnel tests have made numerical simulation of ice accretion a necessary tool in the aircraft design andcertification process. The present work deals with the 3D numerical modeling of ice accretion, and more particularly the modeling of the dynamic and thermal boundary layers that develop around an iced body. Since numerical tools must befast and robust, the approach proposed in this thesis for aerodynamic computation is a coupled Euler/integralboundary layer method. Thus, an integral model is developed to represent the development of the dynamicboundary layer. The thermal part is modeled either by a simplified method based on algebraic approaches,or by an integral method. This modeling of the dynamic and thermal boundary layers is valid on smoothor rough walls and provides the friction coefficient and heat exchange coefficient that are necessary for thecalculation of ice accretion. The integral boundary layer equations, associated with their closure relations,are then solved by a Finite-Volume method on unstructured surface mesh, that is well suited for complexgeometries. In addition, specific numerical treatments are implemented to improve the accuracy of the methodin the vicinity of the stagnation point and to make the code robust to separated boundary layers.After validation of the boundary layer method, the code is used in ONERA’s 2D and 3D icing tools foricing applications. This demonstrates the value of the method in terms of robustness and accuracy comparedto the boundary layer codes more commonly used in current icing tools.

Le Disjoncteur Basse-Tension (DBT) est un appareil classique de la distribution électrique depuis plus de cinquante ans. Mais aujourd'hui, avec l'arrivée de produits bas-coût fabriqués dans les pays émergents, les industriels sont soumis à une forte pression pour développer de nouveaux systèmes moins encombrants, utilisant d'autres matériaux, ou incorporant davantage de fonctionnalités. Cette recherche est très compliquée dans la mesure où le DBT est un système hautement multi-physique (mécanique, thermique, physique des matériaux, physique des plasmas, ...). De fait, le développement de nouveaux produits passe par un processus empirique long et coûteux. Cet effort pourrait être réduit par l'utilisation de modèles prédictifs permettant d'arriver plus vite à un système fonctionnel. De nos jours, avec l'augmentation des moyens de résolution numérique, de plus en plus de travaux portent sur la description multi-physique en 3D du DBT et notamment sur la chambre de coupure ou l'arc électrique est amorcé, se déplace et doit être éteint, l'objet de nôtre étude. Le travail de cette thèse se divise en deux axes complémentaires : le développement d'un modèle fluide 3D en méthode des volumes finis simulant l'arc électrique et son déplacement dans la chambre de coupure; ainsi que la mise en place d'un dispositif expérimental permettant d'analyser le phénomène physique en œuvre. Pour ces deux points la problématique est abordée dans une configuration simplifiée de DBT où l'arc se déplace entre deux rails parallèles dans une chambre parallélépipédique. Basé sur le savoir-faire du groupe AEPPT, un modèle numérique est établi pour simuler le plasma thermique. Les particularités de ce modèle, du fait de l'application, sont la nécessité d'une résolution précise du champ magnétique en utilisant le calcul de Biot & Savart pour les conditions limites ainsi que l'utilisation de méthodes permettant le déplacement et la commutation de l'arc. La validation de ce modèle se fera à géométrie similaire par confrontation avec l'expérience. En s'inspirant de précédents travaux nous avons réalisé une maquette expérimentale composée d'un réacteur faisant office de chambre de coupure et d'un mécanisme permettant l'amorçage de l'arc dans le réacteur par ouverture rotative du contact à vitesse contrôlée. D'autres paramètres modifiables sont la taille du réacteur ainsi que les matériaux qui le constituent. Les diagnostiques disponibles en plus de la mesure de courant et de tension sont l'imagerie rapide et la mesure de pression en différents points de la chambre de coupure. Notre expérience est utile pour la réalisation d'études paramétriques en découplant facilement les paramètres. En outre, par la mise en évidence des phénomènes prépondérants, notre maquette aide à la mise en place du modèle en plus de permettre sa validation expérimentale. Cette thèse est donc une étape cruciale vers la mise en place d'un modèle prédictif. / Low-Voltage Circuit Breakers (LVCB) are classical apparatuses of electrical distribution since more than fifty years. But nowadays, with the outbreak of low-cost products from the developing countries, industry is under a strong stress in order to improve their devices by making them more compact, using different materials or to implement new functionalities. This research is harsh since LVCB are highly multiphysics systems (mechanics, thermal properties, materials, plasma physics, ...). Therefore, developing new products goes through a long and expensive empirical process. Those efforts could be reduced by using predictive models allowing to get faster to a functional device. With the improvements of the numerical solution capacity, there are more and more works toward the 3D multiphysical description of the LVCB, especially on the extinction chamber where the electrical arc is ignited, moved and must be quenched. This is the subject of our work. The study described here is divided in two complementary parts: development of a 3D fluid model with finite volume method simulating the electrical arc and its movement inside the arc extinction chamber; and the set-up of experimental means to analyse this physical phenomenon. For those two points, we use a simplified LVCB configuration with an arc moving between two parallel rails inside a rectangular box chamber. Based on AEPPT's know-how, a numerical model is established to simulate thermal plasma. Particularities on this model, due to the application, are the resolution of Biot&Savart law to calculate precisely the magnetic field for the boundary condition and the development of methods to model the arc roots movement and commutation of the arc from the moving contact to the rail. Validation of this model will be done with the same geometry by confrontation with the experiment. Inspired by precedent works we designed a test apparatus with a reactor representing the extinction chamber of the LVCB and an opening mechanism allowing arc ignition by contact opening at a specified speed. Other parameters such as size of the chamber and materials can be modified. Measurements will include high speed imaging and pressure acquisition in several points of the reactor in addition to the classical current and voltage measurements. This experiment is useful for parametric studies with its easy uncoupling of the parameters. Moreover, by highlighting the dominating phenomena for arc movement, this set-up helps in the build-up of the model over and above the experimental validation.

O presente trabalho trata de construir um modelo computacional utilizando o método dos volumes finitos para malhas não-estruturadas, a fim de se calcular a carga hidráulica num meio poroso, considerando este meio não homogêneo e anisotrópico. A anisotropia é uma característica de muitos materiais encontrados na natureza e depende da propriedade estudada no meio. Primeiramente apresenta-se a dedução da equação do transporte advectivo dispersivo e a formulação matemática para a equação de Laplace, esta última utilizada para o cálculo da carga hidráulica. Em seguida, apresenta-se o algoritmo de solução de um programa computacional em linguagem C++ que permite calcular a velocidade do fluxo em cada face de um volume de controle. Finalmente são feitos vários testes para validação do código computacional utilizado, o que levou a crer que o método utilizado é eficaz para os tipos de malhas testados, apresentando algumas diferenças quanto ao erro da solução. / The present work build a computational model using the finite volumes method for unstructured meshes, with the purpose of calculating the hydraulic load in a porous medium, considering it material non - homogeneous and anisotropic. The Anisotropy is a characteristic of many materials found in the nature and it depends on the property studied in this material. First, we present the deduction of the equation of advective-dispersive transport and the mathematical formulation for the Laplaces equation, this last one used for the calculation of the hydraulic load. Soon afterwards, we present the solution algorithm of a computational program in the C++ language that allows to calculate the speed of the flow in each face of the control volume. Finally several tests for validation of the code are made, which makes it that the plausible to assume method is effective for the types of meshes tested, presenting some differences for the wrong solution.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / A Amazônia exibe uma variedade de cenários que se complementam. Parte desse ecossistema sofre anualmente severas alterações em seu ciclo hidrológico, fazendo com que vastos trechos de floresta sejam inundados. Esse fenômeno, entretanto, é extremamente importante para a manutenção de ciclos naturais. Neste contexto, compreender a dinâmica das áreas alagáveis amazônicas é importante para antecipar o efeito de ações não sustentáveis. Sob esta motivação, este trabalho estuda um modelo de escoamento em áreas alagáveis amazônicas, baseado nas equações de Navier-Stokes, além de ferramentas que possam ser aplicadas ao modelo, favorecendo uma nova abordagem do problema. Para a discretização das equações é utilizado o Método dos Volumes Finitos, sendo o Método do Gradiente Conjugado a técnica escolhida para resolver os sistemas lineares associados. Como técnica de resolução numérica das equações, empregou-se o Método Marker and Cell, procedimento explícito para solução das equações de Navier-Stokes. Por fim, as técnicas são aplicadas a simulações preliminares utilizando a estrutura de dados Autonomous Leaves Graph, que tem recursos adaptativos para manipulação da malha que representa o domínio do problema / Amazon exhibits a variety of scenarios that complement each other. Yearly, part of this ecosystem suffers severe changes in its hydrological cycle, causing flood through vast stretches of the forest. This phenomenon, however, is extremely important for the maintenance of natural cycles. In this context, understanding the Amazonian floodplains dynamics is important to anticipate the effect of unsustainable actions. Under this motivation, this work presents a study of an Amazonian floodplains flow model, based on Navier-Stokes equations, besides other tools that can be applied to the model, thus leading to a new approach to the problem. To discretize the model equations the Finite Volume Method was employed. To solve the associated linear systems the Conjugate Gradient Method technique was chosen. For numerical solution of the equations, the Marker and Cell Method, a explicit solution procedure for Navier-Stokes equations, was employed. All these techniques are applied to preliminary simulations using the Autonomous Leaves Graph algorithm which has adaptive features to manipulate the domain grid problem

The hyperbolic conservation laws model the phenomena of nonlinear waves including discontinuities. The coupled nonlinear equations representing such conservation laws may lead to discontinuous solutions even for smooth initial data. To solve such equations, developing numerical methods which are accurate, robust, and resolve all the wave structures appearing in the solutions is a challenging task. Among several discretization techniques developed for solving hyperbolic conservation laws numerically, Finite Volume Method (FVM) is the most popular. Numerical algorithms, in the framework of FVM, are broadly classified as upwind and central discretization methods. Upwind methods mimic the features of hyperbolic conservation laws very well. However, most of the popular upwind schemes are known to suffer from the shock instabilities. Many upwind methods are heavily dependent on eigen-structure, therefore methods developed for one system of conservation laws are not straightforwardly extended to other systems. On the contrary, central discretization methods are simple, independent of eigen-structure, and therefore, are easily extended to other systems. In the first part of the thesis, a hybrid central discretization method is introduced for Euler equations of gas dynamics. This hybrid scheme is then extended to other hyperbolic conservation laws namely, shallow water equations of oceanography and ideal magnetohydrodynamics equations. The baseline solver for the new hybrid scheme, Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), is an accurate scheme capable of capturing grid aligned steady discontinuities exactly. This central scheme is free from complicated Riemann solvers and therefore is easy to implement. This low diffusive algorithm produces sonic glitches at the expansion regions involving sonic points and is prone to shock instabilities. Therefore it requires an entropy fix to avoid these problems. With the use of entropy fix the exact discontinuity capturing property of the scheme is lost, although sonic glitches and shock instabilities are avoided. The motivation for this work is to develop a numerical method which exactly preserves the steady contacts, is accurate, free of multi-dimensional shock instabilities and yet avoids the entropy fix. This is achieved by constructing a coefficient of numerical diffusion based on pressure gradient sensor. The pressure gradients are known to detect shocks and they vanish across contact discontinuities. This property of pressure sensor is utilized in constructing the coefficient of numerical diffusion. In addition to the numerical diffusion of the baseline solver, a numerical diffusion based on the pressure sensor, scaled by the maximum of eigen-spectrum, is used to avoid shock instabilities. At contact discontinuities, pressure gradients vanish and coefficient of numerical diffusion of MOVERS is automatically retained to capture steady contact discontinuities exactly. This simple hybrid central solver is accurate, captures steady contact discontinuities exactly and is free of multi-dimensional shock instabilities. This novel method is extended to shallow water and ideal magnetohydrodynamics equations in a similar way. In the second part of the thesis, an entropy stable central discretization method for hyperbolic conservation laws is introduced. In a quest for optimal numerical viscosity, development of entropy stable schemes gained importance in recent times. In this work, the entropy conservation equation is used as a guideline to fix the coefficient of numerical diffusion for smooth regions of the flow. At the large gradients, coefficient of numerical diffusion of baseline solver is used. Switch over between smooth and large gradients of the flow is done using limiter functions which are known to distinguish between smooth and high gradient regions of the flow. This simple and stable central scheme termed MOVERS-LE captures grid aligned steady discontinuities exactly and is free of shock instabilities in multi-dimensions. Both the above algorithms are tested on various well established benchmark test problems.

Made available in DSpace on 2016-06-02T19:56:42Z (GMT). No. of bitstreams: 1 3506.pdf: 2271432 bytes, checksum: fc1d05123fb3be32395a9e84b2da5ad8 (MD5) Previous issue date: 2011-02-25 / Universidade Federal de Sao Carlos / This work investigated numerical methods in the solution of mathematical models of fixed-bed reactors. For the reactors modeling and simulation, two numerical methods were used: sequencing method (SM) and finite volume method (FVM). There were also proposed two mathematical models: the pseudo-homogeneous model and the dimensionless one, which is based on the Peclet (Pe) and Biot (Bi) numbers. A horizontal-flow anaerobic immobilized sludge (HAIS) reactor developed in bench scale and after a scale-up, reducing the COD in the wastewater treatment was simulated by sequencing method, varying the numbers of mesh; a tubular fixed-bed reactor with biomass immobilized for the startup period of lactic acid fermentation, also simulated by sequencing method and compared with experimental data; and was also evaluated the precision of sequencing and finite volume methods over the reactor s profile, varying the Peclet e Biot numbers. The models development was based on studies about hydrodynamics and biochemistry kinetics. Both methods described satisfactorily the behavior of the reactors in the performed simulations, but in high values of Peclet, the finite volume method generated inadequacies such as oscillatory responses and over the limit. This paper is an elucidation to sequencing method, which besides its huge range and simplicity, still is not so studied neither known, because it s a recent method. / Este trabalho investigou metodos numericos na solucao de modelos matematicos para reatores de leito fixo. Para a modelagem e simulacao dos reatores, foram utilizados dois metodos numericos: metodo da sequencia (SM) e metodo dos volumes finitos (FVM). Foram propostos dois modelos matematicos: o pseudo-homogeneo e o adimensional, sendo este ultimo baseado nos numeros de Peclet (Pe) e Biot (Bi). Um reator anaerobio horizontal de leito fixo (RAHLF) desenvolvido inicialmente em escala de bancada e posterior aumento de escala na reducao da DQO de aguas residuarias foi simulado pelo metodo da sequencia, variando-se o numero de malhas; um reator tubular de leito fixo com biomassa anaerobia imobilizada no periodo da partida da fermentacao de acido latico, tambem simulado pelo metodo da sequencia e comparado com dados experimentais; e avaliou-se a precisao dos metodos da sequencia e dos volumes finitos sobre o perfil da concentracao de um reator, variando-se os valores de Peclet e Biot. O desenvolvimento dos modelos foi baseado em estudos sobre caracteristicas hidrodinamicas do sistema e de cinetica bioquimica. Ambos os metodos descreveram satisfatoriamente o comportamento dos reatores nas simulacoes realizadas, porem em valores elevados de Peclet, o metodo dos volumes finitos gerou inadequacoes como respostas oscilatorias e superiores ao limite. Este trabalho foi uma elucidacao ao metodo da sequencia, que apesar da sua grande abrangencia e simplicidade, por ser um metodo recente, ainda e pouco estudado e conhecido.

O presente trabalho trata de construir um modelo computacional utilizando o método dos volumes finitos para malhas não-estruturadas, a fim de se calcular a carga hidráulica num meio poroso, considerando este meio não homogêneo e anisotrópico. A anisotropia é uma característica de muitos materiais encontrados na natureza e depende da propriedade estudada no meio. Primeiramente apresenta-se a dedução da equação do transporte advectivo dispersivo e a formulação matemática para a equação de Laplace, esta última utilizada para o cálculo da carga hidráulica. Em seguida, apresenta-se o algoritmo de solução de um programa computacional em linguagem C++ que permite calcular a velocidade do fluxo em cada face de um volume de controle. Finalmente são feitos vários testes para validação do código computacional utilizado, o que levou a crer que o método utilizado é eficaz para os tipos de malhas testados, apresentando algumas diferenças quanto ao erro da solução. / The present work build a computational model using the finite volumes method for unstructured meshes, with the purpose of calculating the hydraulic load in a porous medium, considering it material non - homogeneous and anisotropic. The Anisotropy is a characteristic of many materials found in the nature and it depends on the property studied in this material. First, we present the deduction of the equation of advective-dispersive transport and the mathematical formulation for the Laplaces equation, this last one used for the calculation of the hydraulic load. Soon afterwards, we present the solution algorithm of a computational program in the C++ language that allows to calculate the speed of the flow in each face of the control volume. Finally several tests for validation of the code are made, which makes it that the plausible to assume method is effective for the types of meshes tested, presenting some differences for the wrong solution.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / A Amazônia exibe uma variedade de cenários que se complementam. Parte desse ecossistema sofre anualmente severas alterações em seu ciclo hidrológico, fazendo com que vastos trechos de floresta sejam inundados. Esse fenômeno, entretanto, é extremamente importante para a manutenção de ciclos naturais. Neste contexto, compreender a dinâmica das áreas alagáveis amazônicas é importante para antecipar o efeito de ações não sustentáveis. Sob esta motivação, este trabalho estuda um modelo de escoamento em áreas alagáveis amazônicas, baseado nas equações de Navier-Stokes, além de ferramentas que possam ser aplicadas ao modelo, favorecendo uma nova abordagem do problema. Para a discretização das equações é utilizado o Método dos Volumes Finitos, sendo o Método do Gradiente Conjugado a técnica escolhida para resolver os sistemas lineares associados. Como técnica de resolução numérica das equações, empregou-se o Método Marker and Cell, procedimento explícito para solução das equações de Navier-Stokes. Por fim, as técnicas são aplicadas a simulações preliminares utilizando a estrutura de dados Autonomous Leaves Graph, que tem recursos adaptativos para manipulação da malha que representa o domínio do problema / Amazon exhibits a variety of scenarios that complement each other. Yearly, part of this ecosystem suffers severe changes in its hydrological cycle, causing flood through vast stretches of the forest. This phenomenon, however, is extremely important for the maintenance of natural cycles. In this context, understanding the Amazonian floodplains dynamics is important to anticipate the effect of unsustainable actions. Under this motivation, this work presents a study of an Amazonian floodplains flow model, based on Navier-Stokes equations, besides other tools that can be applied to the model, thus leading to a new approach to the problem. To discretize the model equations the Finite Volume Method was employed. To solve the associated linear systems the Conjugate Gradient Method technique was chosen. For numerical solution of the equations, the Marker and Cell Method, a explicit solution procedure for Navier-Stokes equations, was employed. All these techniques are applied to preliminary simulations using the Autonomous Leaves Graph algorithm which has adaptive features to manipulate the domain grid problem