Luca Heltai

The deal.II Library, Version 8.4

Abstract This paper provides an overview of the new features of the finite element library deal.II version 8.5.

Reverse engineering the euglenoid movement

Euglenids exhibit an unconventional motility strategy amongst unicellular eukaryotes, consisting of large-amplitude highly concerted deformations of the entire body (euglenoid movement or metaboly). A plastic cell envelope called pellicle mediates these deformations. Unlike ciliary or flagellar motility, the biophysics of this mode is not well understood, including its efficiency and molecular machinery. We quantitatively examine video recordings of four euglenids executing such motions with statistical learning methods. This analysis reveals strokes of high uniformity in shape and pace. We then interpret the observations in the light of a theory for the pellicle kinematics, providing a precise understanding of the link between local actuation by pellicle shear and shape control. We systematically understand common observations, such as the helical conformations of the pellicle, and identify previously unnoticed features of metaboly. While two of our euglenids execute their stroke at constant body volume, the other two exhibit deviations of about 20% from their average volume, challenging current models of low Reynolds number locomotion. We find that the active pellicle shear deformations causing shape changes can reach 340%, and estimate the velocity of the molecular motors. Moreover, we find that metaboly accomplishes locomotion at hydrodynamic efficiencies comparable to those of ciliates and flagellates. Our results suggest new quantitative experiments, provide insight into the evolutionary history of euglenids, and suggest that the pellicle may serve as a model for engineered active surfaces with applications in microfluidics.

NUMERICAL STRATEGIES FOR STROKE OPTIMIZATION OF AXISYMMETRIC MICROSWIMMERS

We propose a computational method to solve optimal swimming problems, based on the boundary integral formulation of the hydrodynamic interaction between swimmer and surrounding fluid and direct constrained minimization of the energy consumed by the swimmer. We apply our method to axisymmetric model examples. We consider a classical model swimmer (the three-sphere swimmer of Golestanian et al.) as well as a novel axisymmetric swimmer inspired by the observation of biological micro-organisms.

Numerical stability of the finite element immersed boundary method

The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully nonlinearly coupled formulation for the study of fluid structure interactions. Many numerical methods have been introduced to reduce the difficulties related to the nonlinear coupling between the structure and the fluid evolution. However numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates of the variational formulation of the immersed boundary method. A two-dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly. We present some numerical tests that show good accordance between the observed stability behavior and the one predicted by our results.

Optimally swimming stokesian robots

We study self-propelled stokesian robots composed of assemblies of balls, in dimensions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability , and its proof relies on applying Chows theorem in an analytic framework, similar to what has been done in [4] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [4] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail.

Variational implementation of immersed finite element methods

Dirac-δ distributions are often crucial components of the solid–fluid coupling operators in immersed solution methods for fluid–structure interaction (FSI) problems. This is certainly so for methods like the immersed boundary method (IBM) or the immersed finite element method (IFEM), where Dirac-δ distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-δ distributions, either formally or practically. This has been shown in the finite element immersed boundary method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-δ distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non-matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling

Abstract The interaction between thin structures and incompressible Newtonian fluids is ubiquitous both in nature and in industrial applications. In this paper we present an isogeometric formulation of such problems which exploits a boundary integral formulation of Stokes equations to model the surrounding flow, and a non linear Kirchhoff–Love shell theory to model the elastic behavior of the structure. We propose three different coupling strategies: a monolithic, fully implicit coupling, a staggered, elasticity driven coupling, and a novel semi-implicit coupling, where the effect of the surrounding flow is incorporated in the non-linear terms of the solid solver through its damping characteristics. The novel semi-implicit approach is then used to demonstrate the power and robustness of our method, which fits ideally in the isogeometric paradigm, by exploiting only the boundary representation (B-Rep) of the thin structure middle surface.

Benchmarking the immersed finite element method for fluid-structure interaction problems

We present an implementation of a fully variational formulation of an immersed method for fluid-structure interaction problems based on the finite element method. While typical implementation of immersed methods are characterized by the use of approximate Dirac delta distributions, fully variational formulations of the method do not require the use of said distributions. In our implementation the immersed solid is general in the sense that it is not required to have the same mass density and the same viscous response as the surrounding fluid. We assume that the immersed solid can be either viscoelastic of differential type or hyperelastic. Here we focus on the validation of the method via various benchmarks for fluid-structure interaction numerical schemes. This is the first time that the interaction of purely elastic compressible solids and an incompressible fluid is approached via an immersed method allowing a direct comparison with established benchmarks.

A stable and adaptive semi-Lagrangian potential model for unsteady and nonlinear ship-wave interactions

Abstract We present an innovative numerical discretization of the equations of inviscid potential flow for the simulation of three-dimensional, unsteady, and nonlinear water waves generated by a ship hull advancing in water. The equations of motion are written in a semi-Lagrangian framework, and the resulting integro-differential equations are discretized in space via an adaptive iso-parametric collocation boundary element method, and in time via implicit backward differentiation formulas (BDF) with adaptive step size and variable order. When the velocity of the advancing ship hull is non-negligible, the semi-Lagrangian formulation (also known as Arbitrary Lagrangian Eulerian formulation or ALE) of the free-surface equations contains dominant transport terms which are stabilized with a streamwise upwind Petrov–Galerkin (SUPG) method. The SUPG stabilization allows automatic and robust adaptation of the spatial discretization with unstructured quadrilateral grids. Preliminary results are presented where we compare our numerical model with experimental results on a Wigley hull advancing in calm water with fixed sink and trim.

Stability Results and Algorithmic Strategies for the Finite Element Approach to the Immersed Boundary Method

The immersed boundary method is both a mathematical formulation and a numerical method for the study of fluid structure interactions. Many numerical schemes have been introduced to reduce the difficulties related to the non-linear coupling between the structure and the fluid evolution, however numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates for the variational formulation of the immersed boundary method. A two dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier-Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution (BE/BE), and we present a computable CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly (FE/BE).