Monotonic solution of heterogeneous anisotropic diffusion problems

Anisotropic problems arise in various areas of science and engineering, for example groundwater transport and petroleum reservoir simulations. The pure diffusive anisotropic time-dependent transport problem is solved on a finite number of nodes, that are selected inside and on the boundary of the given domain, along with possible internal boundaries connecting some of the nodes. An unstructured triangular mesh, that attains the Generalized Anisotropic Delaunay condition for all the triangle sides, is automatically generated by properly connecting all the nodes, starting from an arbitrary initial one. The control volume of each node is the closed polygon given by the union of the midpoint of each side with the "anisotropic" circumcentre of each final triangle. A structure of the flux across the control volume sides similar to the standard Galerkin Finite Element scheme is derived. A special treatment of the flux computation, mainly based on edge swaps of the initial mesh triangles, is proposed in order to obtain a stiffness M-matrix system that guarantees the monotonicity of the solution. The proposed scheme is tested using several literature tests and the results are compared with analytical solutions, as well as with the results of other algorithms, in terms of convergence order. Computational costs are also investigate

Comparison of different 2nd order formulations for the solution of the 2D groundwater flow problem over irregular triangular meshes

Mixed and Mixed Hybrid Finite Elements (MHFE) methods have been widely used in the last decade for simulation of groundwater flow problem, petroleum reservoir problems, potential flow problems, etc. The main advantage of these methods is that, unlike the classical Galerkin approach, they guarantee local and global mass balance, as well the flux continuity between inter-element sides. The simple shape of the control volume, where the mass conservation is satisfied, makes also easier to couple this technique with a Finite Volume technique in the time splitting approach for the solution of advection-dispersion problems. In the present paper, a new MHFE formulation is proposed for the solution of the 2D linear groundwater flow problem over domain discretized by means of triangular irregular meshes. The numerical results of the modified MHFE procedure are compared with the results of a modified 2 nd spatial approximation order Finite Volume (FV2) formulation [2], as well as with the results given by the standard MHFE method. The FV2 approach is equivalent to the standard MHFE approach in the case of isotropic medium and regular or mildly irregular mesh, but has a smaller number of unknowns and better matrix properties. In the case of irregular mesh, an approximation is proposed to maintain the superior matrix properties of the FV2 approach, with the consequent introduction of a small error in the computed solution. The modified MHFE formulation is equivalent to the standard MHFE approach in both isotropic and heterogeneous medium cases, using regular or irregular computational meshes, but has a smaller number of unknowns for given mesh geometr

Monotonic solution of flow and transport problems in heterogeneous media using Delaunay unstructured triangular meshes

Transport problems occurring in porous media and including convection, diffusion and chemical reactions, can be well represented by systems of Partial Differential Equations. In this paper, a numerical procedure is proposed for the fast and robust solution of flow and transport problems in 2D heterogeneous saturated media. The governing equations are spatially discretized with unstructured triangular meshes that must satisfy the Delaunay condition. The solution of the flow problem is split from the solution of the transport problem and it is obtained with an approach similar to the Mixed Hybrid Finite Elements method, that always guarantees the M-property of the resulting linear system. The transport problem is solved applying a prediction/correction procedure. The prediction step analytically solves the convective/reactive components in the context of a MAST Finite Volume scheme. The correction step computes the anisotropic diffusive components in the context of a recently proposed Finite Elements scheme. Massa balance is locally and globally satisfied in all the solution steps. Convergence order and computational costs are investigated and model results are compared with literature on

Anisotropic potential of velocity fields in real fluids: Application to the MAST solution of shallow water equations

In the present paper it is first shown that, due to their structure, the general governing equations of uncompressible real fluids can be regarded as an "anisotropic" potential flow problem and closed streamlines cannot occur at any time. For a discretized velocity field, a fast iterative procedure is proposed to order the computational elements at the beginning of each time level, allowing a sequential solution element by element of the advection problem. Some closed circuits could appear due to the discretization error and the elements involved in these circuits could not be ordered. We prove in the paper that the total flux of these not ordered elements goes to zero by refining the computational mesh and that it is possible to order all the remaining elements by neglecting the minimum inter-element flux inside each circuit, with a very small resulting error.The methodology is then applied to the solution of the 2D shallow water equations. The governing Partial Differential Equations are discretized over a generally unstructured triangular mesh, which attains the generalised Delaunay property. Solution is obtained applying a prediction-correction time step procedure. The prediction problem is solved applying a MArching in Space and Time (MAST) procedure, where the computational elements are required to be ordered and explicitly solved. In the correction step, a large linear well-conditioned system is solved. Model results are compared with experimental data and other numerical literature results. Computational costs have been estimated and the convergence order has been investigated according to a known exact solution. © 2013 Elsevier Ltd

Inserimento di restringimenti e ponti in un modello diffusivo 2D di acque basse

Gli effetti su una corrente, causati dalle pile di un ponte, o più in generale da strutture che riducono la sezione trasversale dell’alveo, risultano di particolare interesse per le variazioni idrometriche che comportano alla corrente stessa. Nonostante i numerosi studi teorici e sperimentali di letteratura, l’attuale modellistica numerica diffusiva non integra la presenza di tali manufatti nelle proprie tecniche risolutive. Nella presente memoria viene presentata la metodologia implementata nel modello diffusivo bidimensionale FLOW2D per la valutazione del rigurgito provocato da restringimenti della sezione trasversale, nonché dalla presenza delle campate. I profili di rigurgito ottenuti con il modello proposto, in alcuni casi test, sono stati confrontati con le soluzioni esatte e con i profili di un modello completo. Inoltre, i risultati della metodologia proposta si sono dimostrati in buon accordo con i dati di pieno campo rilevati in un venturimetro