Algebraic pressure segregation methods for the incompressible Navier-Stokes equations

This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled

Algebraic fractional-step schemes with spectral methods for the incompressible Navier-Stokes equations

3noThe numerical investigation of a recent family of algebraic fractional-step methods for the solution of the incompressible time-dependent Navier–Stokes equations is presented. These methods are improved versions of the Yosida method proposed in [A. Quarteroni, F. Saleri, A. Veneziani, Factorization methods for the numerical approximation of Navier–Stokes equations Comput. Methods Appl. Mech. Engrg. 188(1–3) (2000) 505–526; A. Quarteroni, F. Saleri, A. Veneziani, J. Math. Pures Appl. (9), 78(5) (1999) 473–503] and one of them (the Yosida4 method) is proposed in this paper for the first time. They rely on an approximate LU block factorization of the matrix obtained after the discretization in time and space of the Navier–Stokes system, yielding a splitting in the velocity and pressure computation. In this paper, we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method, with the aim of investigating the impact of the splitting on the global accuracy of the computation.reservedmixedP. GERVASIO; F. SALERI; A. VENEZIANIGervasio, Paola; F., Saleri; A., Venezian

순응 및 비순응 유한요소를 이용한 공동 구조에서의 유체 유동 수치 해석

학위논문 (박사)-- 서울대학교 대학원 : 협동과정 계산과학전공, 2014. 8. 신동우.This thesis presents a numerical method for solving the incompressible flow in a square cavity without smoothing the corner singularities. Since nonconforming finite element method can avoid vertex degree of freedom, the values at the upper corners of the cavity are not required to solve the problem. By taking this advantage it is possible to compute accurate numerical solution of the cavity flow without any modification of the problem. The stable nonconforming P1-P0 pair used to solve the incompressible flow problem. DSSY finite elements are added to elements which are on the top corners in the cavity to obtain a more accurate approximation of the boundary condition. Numerical solutions by using conforming finite element are computed for the purposes of comparison. The numerical results are compared with those in the literature and show good agreement. Numerical results computed by using the stable nonconforming P1-P0 pair show excellent accuracy.Contents Abstract i Chapter 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 2 Preliminaries 8 2.1 Finite element discretization . . . . . . . . . . . . . . . . . . . . 8 2.2 The stable nonconforming P1-P0 element pair . . . . . . . . . . 10 2.2.1 The P1-nonconforming quadrilateral element . . . . . . 10 2.2.2 The piecewise constant element . . . . . . . . . . . . . . 12 2.2.3 The stable cheapest finite element pair . . . . . . . . . . 13 Chapter 3 Numerical methods for the discretized Navier-Stokes problems 14 3.1 Iterative solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 Krylov subspace methods . . . . . . . . . . . . . . . . . 19 3.1.2 Uzawa method . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Preconditiong . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.1 Algebraic multigrid preconditioner . . . . . . . . . . . . 25 3.2.2 Block preconditioners for saddle point problems . . . . . 30 3.3 Test problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Algebraic multigrid preconditioner . . . . . . . . . . . . 35 3.3.2 The stationary Stokes problem . . . . . . . . . . . . . . 37 Chapter 4 Numerical simulation of lid driven cavity flow 39 4.1 Lid driven square cavity flow problem . . . . . . . . . . . . . . 39 4.2 Indicators for accuracy . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Implementation of the stable P1 NC-P0 element . . . . . . . . . . 43 4.4 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 5 Conclusion 85 국문초록 93 감사의 글 94Docto

Efficient Numerical Methods for Magnetohydrodynamic Flow

This dissertation studies efficient numerical methods for approximating solu-tions to viscous, incompressible, time-dependent magnetohydrodynamic (MHD) flows and computing MHD flows ensembles. Chapter 3 presents and analyzes a fully discrete, decoupled efficient algorithm for MHD flow that is based on the Els¨asser variable formulation, proves its uncondi-tional stability with respect to the timestep size, and proves its unconditional con-vergence. Numerical experiments are given which verify all predicted convergence rates of our analysis, show the results of the scheme on a set of channel flow problems match well the results found when the computation is done with MHD in primitive variables, and finally illustrate that the scheme performs well for channel flow over a step. In chapter 4, we propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep, yet maintains un-conditional stability with respect to timestep size. The scheme is optimally accu-rate in space, and behaves like second order in time in practice. The proposed method chooses θ ∈ [0, 1], dependent on the viscosity ν and magnetic diffusiv-ity νm, so that unconditionally stability is achieved, and gives temporal accuracy O(∆t2 + (1 − θ)|ν − νm|∆t). In practice, ν and νm are small, and so the method be-haves like second order. We show the θ-method provides excellent accuracy in cases where usual BDF2 is unstable. Chapter 5 proposes an efficient algorithm and studies for computing flow en-sembles of incompressible MHD flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through an efficient algo-rithm that, at each time step, uses the same coefficient matrix for each of the J system solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Els¨asser variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments. This work concludes with chapter 6, which proposes, analyzes and tests high order algebraic splitting methods for MHD flows. The key idea is to applying Yosida-type algebraic splitting to the incremental part of the unknowns at each time step. This reduces the block Schur complement by decoupling it into two Navier-Stokes-type Schur complements, each of which is symmetric positive definite and the same at each time step. We prove the splitting is third order in ∆t, and if used together with (block-)pressure correction, is fourth order. A full analysis of the solver is given, both as a linear algebraic approximation, and as a finite element discretization of an approximation to the un-split discrete system. Numerical tests are given to illustrate the theory and show the effectiveness of the method. Finally, conclusions and future works are discussed in the final chapter

Numerical approximation of cardiac electro-fluid-mechanical models:coupling strategies for large-scale simulation

The mathematical modeling of the heart involves several challenges, which are intrinsically related to the complexity of its function. A satisfactory cardiac model must be able to describe a wide range of different processes, such as the evolution of the transmembrane potential in the myocardium, the deformation caused by the muscles contraction, and the dynamics of the blood inside the heart chambers. In this work, we focus on the coupling of the electrophysiology, the active and the passive mechanics, and the fluid dynamics of the blood in the left ventricle (LV) of the human heart. The models describing the previously mentioned processes are called âsingle core modelsâ, and can be regarded as the building blocks of an âintegrated modelâ. In this thesis, we first review the isolated single core mathematical models for the description of the LV function, and discuss their space and time discretizations with particular emphasis on the coupling conditions. We consider both implicit and semi-implicit schemes for the time discretization. The fully discretized single core problems thus obtained are then combined to define integrated electromechanics and electrofluidmechanics problems. We then focus on the numerical coupling strategy for the electromechanics solver in the framework of the active strain formulation. First, we propose a monolithic strategy where the discretized core models are solved simultaneously; then, several novel segregated strategies, where the discretized core models are solved sequentially, are proposed and systematically compared with each other. The segregated strategies are obtained by exploiting a Godunov splitting scheme, which introduces a first order error on the solution. We show that, while the monolithic approach is more accurate and more stable for relatively large timesteps, segregated approaches allow to solve the integrated problem much more efficiently in terms of computational resources. Moreover, with segregated approaches, it is possible to use different timesteps for the different core models in a staggered fashion, thus further improving the computational efficiency of the schemes. The monolithic and the segregated strategies for the electromechanics are used to solve a benchmark problem with idealized geometry: the results are then compared in terms of accuracy and efficiency. We numerically confirm that the segregated strategies are accurate at least of order one. In light of the results obtained, we employ the proposed strategies to simulate the electromechanics of a subject-specific LV for a full heartbeat. We simulate both healthy and pathological scenarios: in the latter case, we account for an ischemic necrosis of the tissue and analyze several clinical indicators such as pressure-volume loops and the end systolic pressure-volume relationship. Finally, we use the proposed strategies to simulate the electrofluidmechanics of a realistic LV during the systolic phase of the heartbeat. When defining the integrated cardiac models, we establish a preprocess pipeline aimed at preparing geometries and data for both idealized and subject-specific simulations. The pipeline is succesfully used for the setting up of large scale simulations in a high performance computing framework, where the (strong and weak) scalability of the proposed coupling strategies is assessed

Spectral element approximation of the incompressible Navier-Stokes equations in a moving domain and applications

In this thesis we address the numerical approximation of the incompressible Navier-Stokes equations evolving in a moving domain with the spectral element method and high order time integrators. First, we present the spectral element method and the basic tools to perform spectral discretizations of the Galerkin or Galerkin with Numerical Integration (G-NI) type. We cover a large range of possibilities regarding the reference elements, basis functions, interpolation points and quadrature points. In this approach, the integration and differentiation of the polynomial functions is done numerically through the help of suitable point sets. Regarding the differentiation, we present a detailed numerical study of which points should be used to attain better stability (among the choices we present). Second, we introduce the incompressible steady/unsteady Stokes and Navier-Stokes equations and their spectral approximation. In the unsteady case, we introduce a combination of Backward Differentiation Formulas and an extrapolation formula of the same order for the time integration. Once the equations are discretized, a linear system must be solved to obtain the approximate solution. In this context, we consider the solution of the whole system of equations combined with a block type preconditioner. The preconditioner is shown to be optimal in terms of number of iterations used by the GMRES method in the steady case, but not in the unsteady one. Another alternative presented is to use algebraic factorization methods of the Yosida type and decouple the calculation of velocity and pressure. A benchmark is also presented to access the numerical convergence properties of this type of methods in our context. Third, we extend the algorithms developed in the fixed domain case to the Arbitrary Lagrangian Eulerian framework. The issue of defining a high order ALE map is addressed. This allows to construct a computational domain that is described with curved elements. A benchmark using a direct method to solve the linear system or the Yosida-q methods is presented to show the convergence orders of the method proposed. Finally, we apply the developed method with an implicit fully coupled and semi-implicit approach, to solve a fluid-structure interaction problem for a simple 2D hemodynamics example

Parallel Algorithms for the Solution of Large-Scale Fluid-Structure Interaction Problems in Hemodynamics

This thesis addresses the development and implementation of efficient and parallel algorithms for the numerical simulation of Fluid-Structure Interaction (FSI) problems in hemodynamics. Indeed, hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the arterial wall. The simulation of fluid-structure interaction problems requires the approximation of a coupled system of Partial Differential Equations (PDEs) and the set up of efficient numerical solution strategies. Blood is modeled as an incompressible Newtonian fluid whose dynamics is governed by the Navier-Stokes equations. Different constituive models are used to describe the mechanical response of the arterial wall; specifically, we rely on hyperelastic isotropic and anistotropic material laws. The finite element method is used for the space discretization of both the fluid and structure problems. In particular, for the Navier-Stokes equations we consider a semi-discrete formulation based on the Variational Multiscale (VMS) method. Among a wide range of possible solution strategies for the FSI problem, here we focus on strongly coupled monolithic approaches wherein the nonlinearities are treated in a fully implicit mode. To cope with the high computational complexity of the three dimensional FSI problem, a parallel solution framework is often mandatory. To this end, we develop a new block parallel preconditioner for the coupled linearized FSI system obtained after space and time discretization. The proposed preconditioner, named FaCSI, exploits the factorized form of the FSI Jacobian matrix, the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations, and the use of a SIMPLE preconditioner for unsteady Navier-Stokes equations. In FSI problems, the different resolution requirements in the fluid and structure physical domains, as well as the presence of complex interface geometries make the use of matching fluid and structure meshes problematic. In such situations, it is much simpler to deal with discretizations that are nonconforming at the interface, provided however that the matching conditions at the interface are properly fulfilled. In this thesis we develop a novel interpolation-based method, named INTERNODES, for numerically solving partial differential equations by Galerkin methods on computational domains that are split into two (or several) subdomains featuring nonconforming interfaces. By this we mean that either a priori independent grids and/or local polynomial degrees are used to discretize each subdomain. INTERNODES can be regarded as an alternative to the mortar element method: it combines the accuracy of the latter with the easiness of implementation in a numerical code. The aforementioned techniques have been applied for the numerical simulation of large-scale fluid-structure interaction problems in the context of biomechanics. The parallel algorithms developed showed scalability up to thousands of cores utilized on high performance computing machines

Algorithms for fluid-structure interaction problems arising in hemodynamics

We discuss in this thesis the numerical approximation of fluid-structure interaction (FSI) problems with a particular concern (albeit not exclusive) on hemodynamics applications. Firstly, we model the blood as an incompressible fluid and the artery wall as an elastic structure. To solve the coupled problem, we propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, hence reducing the computational cost. This approach leads to two different families of methods which extend to FSI problems schemes that were previously adopted for pure fluid problems. The algorithms derived from inexact factorization methods are compared with other schemes based on two preconditioners for the FSI system. The first one is the classical Dirichlet-Neumann preconditioner, which has the advantage of modularity (i.e. it allows to reuse existing fluid and structure codes with minimum effort). Unfortunately, its performance is very poor in case of large added-mass effect, as it happens in hemodynamics. Alternatively, we consider a non-modular approach which consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ILUT preconditioner. The system is then solved by a Krylov method. The drawback of this procedure is the loss of modularity. Independently of the preconditioner, the efficiency of semi-implicit algorithms is highlighted. All the methods are tested on two and three-dimensional blood-vessel systems. The algorithm combining the non-modular ILUT preconditioner with Krylov methods proved to be the fastest. However, modular and inexact factorization based methods should not be disregarded because they can considerably benefit from code parallelization, unlike the ILUT-Krylov approach. Finally, we improve the structure model by representing the vessel wall as a linear poroelastic medium. Our non-modular approach and the partitioned procedures arising from a domain decomposition viewpoint are extended to fluid-poroelastic structure interactions. Their numerical performance are analyzed and compared on simplified blood-vessel systems

Multi space reduced basis preconditioners for parametrized partial differential equations

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs. In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation. The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system. We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency. We apply our strategy to a broad range of problems described by parametrized PDEs: (i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations. Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results. Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques. Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling