Numerical Approximation Of Internal Discontinuity Interface Problems

This work focuses on the finite element discretization of boundary value problems whose solution features either a discontinuity or a discontinuous conormal derivative across an interface inside the computational domain. The interface is characterized via a level set function. The discontinuities are accounted for by using suitable extension operators whose numerical implementation requires a very low computational effort. After carrying out the error analysis, numerical results to validate our approach are presented in one, two, and three dimensions.

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Claudia Maria Colciago, Simone Deparis

This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method. The gains obtained in term of CPU time are of three orders of magnitude.

2018

Numerical approximation of the unsteady Navier-Stokes equations with an application to the flow past a square cylinder

Francesco Casalegno

The goal of this project is to numerically solve the Navier-Stokes equations by using different numerical methods with particular emphasis on solving the problem of the flow past a square cylinder. In particular, we use the finite element method based on P2/P1 elements for the velocity and pressure fields for the spatial approximation, while the backward Euler method (with semi-implicit treatment of the nonlinear term) for the time discretization. We firstly test the numerical schemes on a benchmark problem with known exact solution. Then, we discuss in detail the advantages, in terms of computational costs, in using the algebraic Chorin-Temam method with additional implementation improvements. We finally investigate the problem of the two-dimensional flow past a square cylinder, focusing our attention on the range 0.1-300 for the Reynolds number (Re). We describe the two different regimes associated to the steady and the unsteady flows and we remark as the latter is in fact due to a Hopf bifurcation of the system. We also discuss the relation between the Strouhal and Reynolds numbers concluding that the Strouhal number attains its maximum value in the range 169-170 for the Reynolds number. In particular, a cubic model is proposed, showing very good matching with observed data and a better fitting than other models available in literature.

2013

TimeEvolver: A program for time evolution with improved error bound

Sebastian Zell

We present TimeEvolver, a program for computing time evolution in a generic quantum system. It relies on well-known Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix iH, where His the Hamiltonian, with an initial vector v. The fact that His Hermitian makes it possible to provide an easily computable bound on the accuracy of the Krylov approximation. Apart from effects of numerical roundoff, the resulting a posteriori error bound is rigorous, which represents a crucial novelty as compared to existing software packages such as Expokit[1]. On a standard notebook, TimeEvolverallows to compute time evolution with adjustable precision in Hilbert spaces of dimension greater than 10(6) Program summary Program Title: TimeEvolver CPC Library link to program files: https://doi.org/10.17632/vvwvng9w36.1 Code Ocean capsule: https://codeocean.com/capsule/8431379 Developer's repository link: https://github.com/marco-michel/TimeEvolver Licensing provisions: MIT Programming language: C++ Supplementary material: An example which demonstrates the computation of time evolution in a concrete physical system. Nature of problem: Computing time evolution in a generic physical quantum system can be reduced to the numerical task of calculating exp(-iHt)v. Here His the Hamiltonian matrix, which is large and sparse, icorresponds to the imaginary unit, tdenotes time and the vector vrepresents the initial state. A program is needed to perform this computation efficiently. Since the use of approximation methods is unavoidable, it is important to quantify as rigorously as possible the resulting error. Moreover, in order to facilitate the application to various problems in physics, additional functionalities are needed, in particular for forming the Hamiltonian matrix from a more abstract representation of the Hamiltonian operator. Solution method: The program employs known Krylov subspace methods for calculation the exponential of the large sparse matrix (- iHt) times the vector v. The Arnoldi algorithm is used to form the Krylov subspace and exponentiation of the resulting small matrix is achieved by diagonalization. The fact that (-iHt) is anti-Hermitian makes it possible to calculate the error of the Krylov approximation in terms of an easily-computable integral formula. This allows to choose a maximal size of the time step, after which the method is restarted and a new Krylov subspace is formed, while respecting an adjustable error bound. It is rigorous up to inaccuracies of a one-dimensional numerical integral and effects of finite machine precision, for which we also give an estimate. All linear algebra operations are performed with the Intel (R) Math Kernel Library and Boost is used for numerical integration. The methods for deriving the Hamiltonian matrix rely on a hashtable representation of Hilbert space. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).. Additionally, we provide routines for deriving the matrix Hfrom a more abstract representation of the Hamiltonian operator.