This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

New insights into large eddy simulation

We describe new benchmark settings for the rigorous evaluation of different methods for fluid-structure interaction problems. The configurations consist of laminar incompressible channel flow around an elastic object which results in self-induced oscillations of the structure. Moreover, characteristic flow quantities and corresponding plots are provided for a quantitative comparison.

/pdf/proposal-for-numerical-benchmarking-of-fluid-structure-5e6j2wbnnf.pdf

Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow

Benchmark configurations for quantitative validation and comparison of incompressible interfacial flow codes, which model two-dimensional bubbles rising in liquid columns, are proposed. The benchmark quantities: circularity, center of mass, and mean rise velocity are defined and measured to monitor convergence toward a reference solution. Comprehensive studies are undertaken by three independent research groups, two representing Eulerian level set finite-element codes and one representing an arbitrary Lagrangian-Eulerian moving grid approach.

/pdf/quantitative-benchmark-computations-of-two-dimensional-36jsjlc58r.pdf

Quantitative benchmark computations of two-dimensional bubble dynamics

https://biblio.ugent.be/publication/533365/file/533369.pdf

Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction

We investigate a new method of solving the problem of fluid-structure interaction of an incompressible elastic object in laminar incompressible viscous flow. Our proposed method is based on a fully implicit, monolithic formulation of the problem in the arbitrary Lagrangian-Eulerian framework. High order FEM is used to obtain the discrete approximation of the problem. In order to solve the resulting systems a quasi-Newton method is applied with the linearized systems being approximated by the divided differences approach. The linear problems of saddle-point type are solved by a standard geometric multigrid with local multilevel pressure Schur complement smoothers. 1 Overview We consider the problem of viscous fluid flow interacting with an elastic body which is being deformed by the fluid action. Such a problem is encountered in many real life applications of great importance. Typical examples of this type of problem are the areas of aero-elasticity, biomechanics or material processing. For example, a good mathematical model for biological tissue could be used in such areas as early recognition or prediction of heart muscle failure, advanced design of new treatments and operative procedures, and the understanding of atherosclerosis and associated problems. Other possible applications include the development of virtual reality programs for training new surgeons or designing new operative procedures (see [12]). 1.1 Fluid structure models in biomechanics There have been several different approaches to the problem of fluid-structure interaction. Most notably these include the work of [13, 14, 15, 16] where an immersed boundary method was developed and applied to a three-dimensional model of the heart. In this model they consider a set of one-dimensional elastic fibers immersed in a three-dimensional fluid region and using a parallel supercomputer they were able to model the pulse of the heart ventricle. ⋆ This work has been supported by German Reasearch Association (DFG), Reasearch unit 493. A fluid-structure model with the wall modelled as a thin shell was used to model the left heart ventricle in [3, 4] and [18, 17]. In [7, 8] a similar approach was used to model the flow in a collapsible tube. In these models the wall is modelled by two-dimensional thin shells which can be modified to capture the anisotropy of the muscle. In reality the thickness of the wall can be significant and very important. For example in arteries the wall thickness can be up to 30% of the diameter and its local thickening can lead to the creation of an aneurysm. In the case of heart ventricle the thickness of the wall is also significant and also the direction of the muscle fibers changes through the wall. 1.2 Theoretical results The theoretical investigation of fluid structure interaction problems is complicated by the need of mixed description. While for the solid part the natural view is the material (Lagrangian) description, for the fluid it is the spatial (Eulerian) description. In the case of their combination some kind of mixed description (usually referred to as the arbitrary Lagrangian-Eulerian description or ALE) has to be used which brings additional nonlinearity into the resulting equations. In [10] a time dependent, linearized model of interaction between a viscous fluid and an elastic shell in small displacement approximation and its discretization is analyzed. The problem is further simplified by neglecting all changes in the geometry configuration. Under these simplifications by using energy estimates they are able to show that the proposed formulation is well posed and a global weak solution exists. Further they show that an independent discretization by standard mixed finite elements for the fluid and by nonconforming discrete Kirchhoff triangle finite elements for the shell together with backward or central difference approximation of the time derivatives converges to the solution of the continuous problem. In [19] a steady problem of equilibrium of an elastic fixed obstacle surrounded by a viscous fluid is studied. Existence of an equilibrium state is shown with the displacement and velocity in C and pressure in C under assumption of small data in C and domain boundaries of class C. A numerical solution of the resulting equations of the fluid structure interaction problem poses a great challenge since it includes the features of nonlinear elasticity, fluid mechanics and their coupling. The easiest solution strategy, mostly used in the available software packages, is to decouple the problem into the fluid part and solid part, for each of those parts to use some well established method of solution then the interaction is introduced as external boundary conditions in each of the subproblems. This has an advantage that there are many well tested finite element based numerical methods for separate problems of fluid flow and elastic deformation, on the other hand the treatment of the interface and the interaction is problematic. The approach presented here treats the problem as a single continuum with the coupling automatically taken care of as internal interface, which in our formulation does not require any special treatment. 2 Continuum description Let Ω ⊂ R be a reference configuration of a given body. Let Ωt ⊂ R 3 be a configuration of this body at time t. Then a one-to-one, sufficiently smooth mapping χΩ of the reference configuration Ω to the current configuration χΩ : Ω × [0, T ] 7→ R , (1) describes the motion of the body, see figure 1. The mapping χΩ depends on the choice of the reference configuration Ω which can be fixed in a various ways. Here we think of Ω to be the initial (stress-free) configuration Ω0. Thus, if not emphasized, we mean by χ exactly χΩ = χΩ0 .

A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics

The goal of this article is to contribute to a more precise discussion of the question whether Lattice-Boltzmann (LB) methods can be regarded as ecient CFD solvers. After a short review of the basic model and recommendable extensions, we compare the accuracy and computational eciency of two research simulation codes based on the LB and the Finite-Element method (FEM) for incompressible laminar two-dimensional flow problems in complex geometries. As LB methods are weakly compressible by nature, we also study the influence of the Mach number on the solution by comparing compressible and incompressible results obtained by the LB code and the commercial code CFX. Our results indicate, that for the quantities studied (lift, drag, pressure drop) our LB prototype is at least competitive for incompressible transient problems, but asymptotically slower for steady-state Stokes flow as the asymptotic algorithmic complexity of the classical LB-method is not optimal compared to the multigrid solvers incorporated in the FEM and CFX code. For the weakly compressible case, the LB approach has a significant wall clock time advantage as compared to CFX. In addition, we demonstrate that the influence of the finite Mach number in LB simulations of incompressible flow is easily underestimated.

Lattice-Boltzmann, Finite Element and Finite Volume Methods for laminar Flows

https://www.mathematik.tu-dortmund.de/lsiii/cms/papers/GellerKrafczykToelkeTurekHron2004.pdf

Benchmark computations based on Lattice-Boltzmann, Finite Element and Finite Volume Methods for laminar Flows

In his seminal paper on fluid motion, Stokes developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure. Such an assumption is particul...

Jaroslav Hron

Papers

Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow

A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics

Lattice-Boltzmann, Finite Element and Finite Volume Methods for laminar Flows

Benchmark computations based on Lattice-Boltzmann, Finite Element and Finite Volume Methods for laminar Flows

Simple flows of fluids with pressure–dependent viscosities