A conservative implicit multirate method for hyperbolic problems

This work focuses on the development of a self adjusting multirate strategy based on an implicit time discretization for the numerical solution of hyperbolic equations, that could benefit from different time steps in different areas of the spatial domain. We propose a novel mass conservative multirate approach, that can be generalized to various implicit time discretization methods. It is based on flux partitioning, so that flux exchanges between a cell and its neighbors are balanced. A number of numerical experiments on both non-linear scalar problems and systems of hyperbolic equations have been carried out to test the efficiency and accuracy of the proposed approach

The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

A multi-layer reactive transport model for fractured porous media

An accurate modeling of reactive flows in fractured porous media is a key ingredient to obtain reliable numerical simulations of several industrial and environmental applications. For some values of the physical parameters we can observe the formation of a narrow region or layer around the fractures where chemical reactions are focused. Here the transported solute may precipitate and form a salt, or vice-versa. This phenomenon has been observed and reported in real outcrops. By changing its physical properties this layer might substantially alter the global flow response of the system and thus the actual transport of solute: the problem is thus non-linear and fully coupled. The aim of this work is to propose a new mathematical model for reactive flow in fractured porous media, by approximating both the fracture and these surrounding layers via a reduced model. In particular, our main goal is to describe the layer thickness evolution with a new mathematical model, and compare it to a fully resolved equidimensional model for validation. As concerns numerical approximation we extend an operator splitting scheme in time to solve sequentially, at each time step, each physical process thus avoiding the need for a non-linear monolithic solver, which might be challenging due to the non-smoothness of the reaction rate. We consider bi- and tridimensional numerical test cases to asses the accuracy and benefit of the proposed model in realistic scenarios

Parallel mesh adaptive techniques for complex flow simulation: geometry conservation

Dynamic mesh adaptation on unstructured grids, by localised refinement and derefinement, is a very efficient tool for enhancing solution accuracy and optimising computational time. One of the major drawbacks, however, resides in the projection of the new nodes created, during the refinement process, onto the boundary surfaces. This can be addressed by the introduction of a library capable of handling geometric properties given by a CAD (computer-aided design) description. This is of particular interest also to enhance the adaptation module when the mesh is being smoothed, and hence moved, to then reproject it onto the surface of the exact geometry

Performances of the mixed virtual element method on complex grids for underground flow

The numerical simulation of physical processes in the underground frequently entails challenges related to the geometry and/or data. The former are mainly due to the shape of sedimentary layers and the presence of fractures and faults, while the latter are connected to the properties of the rock matrix which might vary abruptly in space. The development of approximation schemes has recently focused on the overcoming of such difficulties with the objective of obtaining numerical schemes with good approximation properties. In this work we carry out a numerical study on the performances of the Mixed Virtual Element Method (MVEM) for the solution of a single-phase flow model in fractured porous media. This method is able to handle grid cells of polytopal type and treat hybrid dimensional problems. It has been proven to be robust with respect to the variation of the permeability field and of the shape of the elements. Our numerical experiments focus on two test cases that cover several of the aforementioned critical aspects

A reduced model for Darcy’s problem in networks of fractures

Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures

Numerical simulation of geochemical compaction with discontinuous reactions

The present work deals with the numerical simulation of porous media subject to the coupled effects of mechanical compaction and reactive flows that can significantly alter the porosity due to dissolution, precipitation or transformation of the solid matrix. These chemical processes can be effectively modelled as ODEs with discontinuous right hand side, where the discontinuity depends on time and on the solution itself. Filippov theory can be applied to prove existence and to determine the solution behaviour at the discontinuities. From the numerical point of view, tailored numerical schemes are needed to guarantee positivity, mass conservation and accuracy. In particular, we rely on an event-driven approach such that, if the trajectory crosses a discontinuity, the transition point is localized exactly and integration is restarted accordingly

A Stability Analysis for the Arbitrary Lagrangian Eulerian Formulation with Finite Elements

In this paper we present some theoretical results on the Arbitrary Lagrangian Eulerian (ALE) formulation. This formulation may be used when dealing with moving domains and consists in recasting the governing differential equation and the related weak formulation in a frame of reference moving with the domain. The ALE technique is first presented in the whole generality for conservative equations and a result on the regularity of the underlying mapping is proven. In a second part of the work, the stability property of two types of finite element ALE schemes for parabolic evolution problems are analyzed and its relation with the so-called Geometric Conservation Laws is addressed

On Parallel Computation of Blood Flow in Human Arterial Network Based on 1-D Modelling

In this study, parallel computation of blood flow in a 1-D model of human arterial network has been carried out employing a Taylor Galerkin Finite Element Method. Message passing interface libraries have been used on Origin 2000 SGI machine. A Greedy strategy for load-distribution has been devised and data-flow graphs necessary for parallelization have been generated. The performance of parallel implementation measured in terms of speedup and efficiency factors is found to be good. Further, the parallel code is used in simulating the propagation of pressure and velocity waveforms in our 1-D arterial model for two different inflow pressure pulses. Also, the influence of consideration of terminal resistance on pressure and velocity waveforms have been analyze