Aproximación del transporte de contaminantes en aguas someras mediante elementos finitos de alto orden
- Autores: Angel Villota
- Directores de la Tesis: Ramón Codina Rovira (dir. tes.)
- Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2020
- Idioma: español
- Materias:
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- Resumen
The objective of the doctoral thesis is to perform the numerical approximation of the transient convection-diffusion-reaction (CDR) vector equation in 2D with high order finite elements, quadratic, cubic and fourth order, using stabilized finite element methods of the type VMS (Variational Multi-Scale) such as ASGS and OSS, testeds in recent years to solve the transient vector equation of CDR when there is the phenomenon of dominant convection or reaction and aggravated by the nonlinearity of either the convective term or the term reaction.
The standard Galerkin finite element method applied to the CDR transient scalar equation presents instabilities in the solution when the convective and reaction terms are dominant versus the diffusive term. We solve this difficulty by two finite element methods based on subscales, these are the known methods called ASGS (Algebraic Sub-Grid Scale) and OSS (Orthogonal Subscale Stabilization), which basically consist of decomposing the unknown continuous scalar variable into two components, one than is resolved in the finite element space and the other that cannot be captured by the finite element mesh and therefore belongs to another function space that we call subscale space. It is precisely the choice of the subscale space that imposes the difference between the ASGS and OSS methods.
We will experiment with the stabilization parameter suggested in the literature for linear elements, making an extension of the same parameter to take into account the interpolation order to deal with high order finite elements. Likewise, in the calculation of the subscale with fourth order triangular elements for the OSS method, we have proposed the modification of the standard triangular element in order to have a closed integration rule with the integration points in the nodes. As for temporal and spatial discretization, we first discretize in time, and then for each instant of time we make the spatial approximation and stabilization including the temporal derivative in said stabilization.
We also present the approximation and stabilization of the transient vector equation of CDR for solving problems with more than one variable. As in the scalar case, the standard Galerkin method presents instabilities when the diffusive term is small in relation to the convective and reaction terms, and that in some problems it may be aggravated by the nonlinearity of these terms. The considering equal interpolation for all the variables, the design of the diagonal matrix of stabilization parameters, the determination of the space of the subscales, the inclusion of the time derivatives in the stabilization and the treatment of nonlinearity are aspects to be considered in the ASGS and OSS formulations.
To confirm the robustness of the analyzed high-order finite element methods, several mesh convergence tests have been performed with known analytical solutions, as well as boundary layer tests for the CDR transient scalar equation, examples of the approximation of the motion of a fluid in shallow water, such as the flow through an elliptical obstacle and the flow of a dam rupture, the transport of a pollutant in a square cavity, the distribution of the transport of a pollutant in the Gulf of Creus and in the mouth of the Guadalquivir river, and the distribution of population density in the predator-prey model. These are some examples that confirm the robustness of the stabilized formulations presented with high order finite elements to solve of the general transient convection-diffusion-reaction vector equation including non-linearity in terms of convection or reaction.
