HERMESH : a geometrical domain composition method in computational mechanics

With this thesis we present the HERMESH method which has been classified by us as a a composition domain method. This term comes from the idea that HERMESH obtains a global solution of the problem from two independent meshes as a result of the mesh coupling. The global mesh maintains the same number of degrees of freedom as the sum of the independent meshes, which are coupled in the interfaces via new elements referred to by us as extension elements. For this reason we enunciate that the domain composition method is geometrical. The result of the global mesh is a non-conforming mesh in the interfaces between independent meshes due to these new connectivities formed with existing nodes and represented by the new extension elements. The first requirements were that the method be implicit, be valid for any partial differential equation and not imply any additional effort or loss in efficiency in the parallel performance of the code in which the method has been implemented. In our opinion, these properties constitute the main contribution in mesh coupling for the computational mechanics framework. From these requirements, we have been able to develop an automatic and topology-independent tool to compose independent meshes. The method can couple overlapping meshes with minimal intervention on the user's part. The overlapping can be partial or complete in the sense of overset meshes. The meshes can be disjoint with or without a gap between them. And we have demonstrated the flexibility of the method in the relative mesh size. In this work we present a detailed description of HERMESH which has been implemented in a high-performance computing computational mechanics code within the framework of the finite element methods. This code is called Alya. The numerical properties will be proved with different benchmark-type problems and the manufactured solution technique. Finally, the results in complex problems solved with HERMESH will be presented, clearly showing the versatility of the method.En este trabajo presentamos el metodo HERMESH al que hemos catalogado como un método de composición de dominios puesto que a partir de mallas independientes se obtiene una solución global del problema como la unión de los subproblemas que forman las mallas independientes. Como resultado, la malla global mantiene el mismo número de grados de libertad que la suma de los grados de libertad de las mallas independientes, las cuales se acoplan en las interfases internas a través de nuevos elementos a los que nos referimos como elementos de extensión. Por este motivo decimos que el método de composición de dominio es geométrico. El resultado de la malla global es una malla que no es conforme en las interfases entre las distintas mallas debido a las nuevas conectividades generadas sobre los nodos existentes. Los requerimientos de partida fueron que el método se implemente de forma implícita, sea válido para cualquier PDE y no implique ningún esfuerzo adicional ni perdida de eficiencia para el funcionamiento paralelo del código de altas prestaciones en el que ha sido implementado. Creemos que estas propiedades son las principales aportaciones de esta tesis dentro del marco de acoplamiento de mallas en mecánica computacional. A partir de estas premisas, hemos conseguido una herramienta automática e independiente de la topología para componer mallas. Es capaz de acoplar sin necesidad de intervención del usuario, mallas con solapamiento parcial o total así como mallas disjuntas con o sin "gap" entre ellas. También hemos visto que ofrece cierta flexibilidad en relación al tamaños relativos entre las mallas siendo un método válido como técnica de remallado local. Presentamos una descripción detallada de la implementación de esta técnica, llevada a cabo en un código de altas prestaciones de mecánica computacional en el contexto de elementos finitos, Alya. Se demostrarán todas las propiedades numéricas que ofrece el métodos a través de distintos problemas tipo benchmark y el método de la solución manufacturada. Finalmente se mostrarán los resultados en problemas complejos resueltos con el método HERMESH, que a su vez es una prueba de la gran flexibilidad que nos brinda

Transport processes in fractured porous media

112 stranThis habilitation thesis summarizes author's theoretical work related to development of the Flow123d simulator. This includes especially methods and algorithms for solving Darcy ow problems in saturated and unsaturated fractured porous media. A model with semi-discrete fractures called mixed dimension model is derived at the beginning. Then the abstract model for advection-di usion equation is applied to the Darcy ow. The mixed-hybrid formulation of the Darcy ow mixed dimension problem is presented followed by its discretization using Raviart-Thomas nite elements. An analytical solution to a test single fracture problem is supplied which allows veri cation of the model's implementation. Finally, the BDDC method is applied to obtain a scalable solver of the linear systems arising from the problem's discretization. Subsequently, new developments for the non-conforming mixed meshes are presented. Four methods with common strategy are used to introduce a coupling between equations living on the intersecting nite element meshes of di erent dimension. Further a family of e cient algorithms for computing mesh intersections is presented. Final chapter is devoted to the Richards' equation and modi cation of the mixed-hybrid scheme in order to satisfy discrete maximum principle. This is of particular importance for the Richards' equation where short time steps are often necessary which leads to strong oscillations for the schemes that violate DMP