L’articolo descrive un’esperienza didattica svolta in una classe di quarta media attitudinale1 con lo scopo di indagare le convinzioni e le competenze degli allievi in merito all’uso delle ...rappresentazioni semiotiche nella comprensione e risoluzione di una situazione matematica. Il percorso didattico si compone di molteplici attività volte a stimolare l’uso di differenti registri semiotici (linguistico, numerico, algebrico e grafico) nell’attività di risoluzione di un problema. Gli allievi hanno partecipato con motivazione e interesse alle varie attività, apprezzando i relativi riferimenti al mondo reale. I risultati mostrano come accompagnare l’allievo a matematizzare situazioni attraverso la gestione di differenti rappresentazioni dello stesso oggetto matematico favorisca una comprensione più profonda del problema e un utilizzo più consapevole degli strumenti matematici necessari per la sua risoluzione.
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1 In Canton Ticino a partire dalla terza media gli allievi vengono inseriti in corsi base e attitudinale in funzione delle competenze matematiche raggiunte alla fine della seconda.
•Presents four benchmark cases for single-phase flow in three-dimensional fractured porous media.•Investigates the capabilities of DFM methods in handling complexities common to fracture ...networks.•Compares results of 17 different numerical methods, submitted by 11 participating groups.•Is supplemented by a public Git repository containing result data and reproducing Jupyter notebooks.
Flow in fractured porous media occurs in the earth’s subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, four benchmark cases for single-phase flow in three-dimensional fractured porous media are presented. The cases are specifically designed to test the methods’ capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.
We present a novel framework inspired by the Immersed Boundary Method for predicting the fluid-structure interaction of complex structures immersed in laminar, transitional and turbulent flows.
The ...key elements of the proposed fluid-structure interaction framework are 1) the solution of elastodynamics equations for the structure, 2) the use of a high-order Navier–Stokes solver for the flow, and 3) the variational transfer (L2-projection) for coupling the solid and fluid subproblems.
The dynamic behavior of a deformable structure is simulated in a finite element framework by adopting a fully implicit scheme for its temporal integration. It allows for mechanical constitutive laws including inhomogeneous and fiber-reinforced materials.
The Navier–Stokes equations for the incompressible flow are discretized with high-order finite differences which allow for the direct numerical simulation of laminar, transitional and turbulent flows.
The structure and the flow solvers are coupled by using an L2-projection method for the transfer of velocities and forces between the fluid grid and the solid mesh. This strategy allows for the numerical solution of coupled large scale problems based on nonconforming structured and unstructured grids. The transfer between fluid and solid limits the convergence order of the flow solver close to the fluid-solid interface.
The framework is validated with the Turek–Hron benchmark and a newly proposed benchmark modeling the flow-induced oscillation of an inert plate. A three-dimensional simulation of an elastic beam in transitional flow is provided to show the solver's capability of coping with anisotropic elastic structures immersed in complex fluid flow.
Due to its optimal complexity, the multigrid (MG) method is one of the most popular approaches for solving large-scale linear systems arising from the discretization of partial differential ...equations. However, the parallel implementation of standard MG methods, which are inherently multiplicative, suffers from increasing communication complexity. In such cases, the additive variants of MG methods provide a good alternative due to their inherently parallel nature, although they exhibit slower convergence. This work combines the additive multigrid method with the multipreconditioned conjugate gradient (MPCG) method. In the proposed approach, the MPCG method employs the corrections from the different levels of the MG hierarchy as separate preconditioned search directions. In this approach, the MPCG method updates the current iterate by using the linear combination of the preconditioned search directions, where the optimal coefficients for the linear combination are computed by exploiting the energy norm minimization of the CG method. The idea behind our approach is to combine the \(A\)-conjugacy of the search directions of the MPCG method and the quasi \(H_1\)-orthogonality of the corrections from the MG hierarchy. In the numerical section, we study the performance of the proposed method compared to the standard additive and multiplicative MG methods used as preconditioners for the CG method.
Numerical simulations of flow and transport in porous media usually rely on hybrid-dimensional models, i.e., the fracture is considered as objects of a lower dimension compared to the embedding ...matrix. Such models are usually combined with non-conforming discretizations as they avoid the inherent difficulties associated with the generation of meshes that explicitly resolve fractures-matrix interfaces. However, non-conforming discretizations demand a more complicated coupling of different sub-models and may require special care to ensure conservative fluxes. We propose a novel approach for the simulation of flow and transport problems in fractured porous media based on an equi-dimensional representation of the fractures. The major challenge for these types of representation is the creation of meshes which resolve the several complex interfaces between the fractures and the embedding matrix. To overcome this difficulty, we employ a strategy based on adaptive mesh refinement (AMR). The idea at the base of the proposed AMR is to start from an initially uniform coarse mesh and refine the elements which have non-empty overlaps with at least one of the fractures. Iterating this process allows to create non-uniform non-conforming meshes, which do not resolve the interfaces but can approximate them with arbitrary accuracy. We demonstrate that low-order finite element (FE) discretizations on adapted meshes are globally and locally conservative and we suitably adapt an algebraic flux correction technique to ensure the discrete maximum principle. In particular, we show that the notorious conditions on M-matrices have to be adapted to the basis functions defined on non-conforming meshes. Although the proposed applications come from geophysical applications, the obtained results could be applied to any diffusion and transport problems, on both conforming and non-conforming meshes.
Non-linear phase field models are increasingly used for the simulation of fracture propagation problems. The numerical simulation of fracture networks of realistic size requires the efficient ...parallel solution of large coupled non-linear systems. Although in principle efficient iterative multi-level methods for these types of problems are available, they are not widely used in practice due to the complexity of their parallel implementation. Here, we present Utopia, which is an open-source C++ library for parallel non-linear multilevel solution strategies. Utopia provides the advantages of high-level programming interfaces while at the same time a framework to access low-level data-structures without breaking code encapsulation. Complex numerical procedures can be expressed with few lines of code, and evaluated by different implementations, libraries, or computing hardware. In this paper, we investigate the parallel performance of our implementation of the recursive multilevel trust-region (RMTR) method based on the Utopia library. RMTR is a globally convergent multilevel solution strategy designed to solve non-convex constrained minimization problems. In particular, we solve pressure-induced phase-field fracture propagation in large and complex fracture networks. Solving such problems is deemed challenging even for a few fractures, however, here we are considering networks of realistic size with up to 1000 fractures.
Non-linear phase field models are increasingly used for the simulation of fracture propagation models. The numerical simulation of fracture networks of realistic size requires the efficient parallel ...solution of large coupled non-linear systems. Although in principle efficient iterative multi-level methods for these types of problems are available, they are not widely used in practice due to the complexity of their parallel implementation. Here, we present Utopia, which is an open-source C++ library for parallel non-linear multilevel solution strategies. Utopia provides the advantages of high-level programming interfaces while at the same time a framework to access low-level data-structures without breaking code encapsulation. Complex numerical procedures can be expressed with few lines of code, and evaluated by different implementations, libraries, or computing hardware. In this paper, we investigate the parallel performance of our implementation of the recursive multilevel trust-region (RMTR) method based on the Utopia library. RMTR is a globally convergent multilevel solution strategy designed to solve non-convex constrained minimization problems. In particular, we solve pressure-induced phase-field fracture propagation in large and complex fracture networks. Solving such problems is deemed challenging even for a few fractures, however, here we are considering networks of realistic size with up to 1000 fractures.
We present a novel framework inspired by the Immersed Boundary Method for predicting the fluid-structure interaction of complex structures immersed in flows with moderate to high Reynolds numbers. ...The main novelties of the proposed fluid-structure interaction framework are 1) the use of elastodynamics equations for the structure, 2) the use of a high-order Navier-Stokes solver for the flow, and 3) the variational transfer (L2-projection) for coupling the solid and fluid subproblems. The dynamic behavior of a deformable structure is simulated in a finite element framework by adopting a fully implicit scheme for its temporal integration. It allows for mechanical constitutive laws including nonhomogeneous and fiber-reinforced materials. The Navier-Stokes equations for the incompressible flow are discretized with high-order finite differences which allow for the direct numerical simulation of laminar, transitional and turbulent flows. The structure and the flow solvers are coupled by using an L2-projection method for the transfer of velocities and forces between the fluid grid and the solid mesh. This strategy allows for the numerical solution of coupled large scale problems based on nonconforming structured and unstructured grids. The framework is validated with the Turek-Hron benchmark and a newly proposed benchmark modelling the flow-induced oscillation of an inert plate. A three-dimensional simulation of an elastic beam in transitional flow is provided to show the solver's capability of coping with anisotropic elastic structures immersed in complex fluid flow.
Flow in fractured porous media occurs in the earth's subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has ...seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, we present a portfolio of four benchmark cases for single-phase flow in three-dimensional fractured porous media. The cases are specifically designed to test the methods' capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.
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