Annie Raoult

Applications of level set methods in computational biophysics

We describe in this paper two applications of Eulerian level set methods to fluid-structure problems arising in biophysics. The first one is concerned with three-dimensional equilibrium shapes of phospholipidic vesicles. This is a complex problem, which can be recast as the minimization of the curvature energy of an immersed elastic membrane, under a constant area constraint. The second deals with isolated cardiomyocyte contraction. This problem corresponds to a generic incompressible fluid-structure coupling between an elastic body and a fluid. By the choice of these two quite different situations, we aim to bring evidence that Eulerian methods provide efficient and flexible computational tools in biophysics applications.

ELASTIC LIMIT OF SQUARE LATTICES WITH THREE-POINT INTERACTIONS

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

Homogenization of hexagonal lattices

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using

A nonlinearly elastic homogenized constitutive law for the myocardium

\Gamma

Nonlinear elasticity of cross-linked networks

-convergence.

Quasiconvex envelopes in nonlinear elasticity

Publisher Summary This chapter proposes usage of a micro-macro approach for deriving a constitutive law for the myocardium. It uses a discrete homogenization technique, which relies on the smallness of the myocytes with respect to the whole of the myocardium. This provides different approach to determine a constitutive law for the myocardium, this technique of modeling trusses was used in civil engineering, for both: small displacements framework, and for large displacements. The method starts from the geometrical description of the myocytes arrangement and from their mechanical behavior. In the present chapter, the myocytes arrangement is modeled by a network of elastic bars linked by their ends. It is assumed that the traction-compression law of each bar, which gives the tension in terms of the length, is nonlinearly elastic. The technique takes advantage of the specific geometrical arrangement of the myocytes and considers their individual mechanical behavior as given. This method is a discrete analog of the classical homogenization method for continuous media.

Foreword to the Proceedings of the Congrès SMAI 2013

Cross-linked semiflexible polymer networks are omnipresent in living cells. Typical examples are actin networks in the cytoplasm of eukaryotic cells, which play an essential role in cell motility, and the spectrin network, a key element in maintaining the integrity of erythrocytes in the blood circulatory system. We introduce a simple mechanical network model at the length scale of the typical mesh size and derive a continuous constitutive law relating the stress to deformation. The continuous constitutive law is found to be generically nonlinear even if the microscopic law at the scale of the mesh size is linear. The nonlinear bulk mechanical properties are in good agreement with the experimental data for semiflexible polymer networks, i.e., the network stiffens and exhibits a negative normal stress in response to a volume-conserving shear deformation, whereby the normal stress is of the same order as the shear stress. Furthermore, it shows a strain localization behavior in response to an uniaxial compression. Within the same model we find a hierarchy of constitutive laws depending on the degree of nonlinearities retained in the final equation. The presented theory provides a basis for the continuum description of polymer networks such as actin or spectrin in complex geometries and it can be easily coupled to growth problems, as they occur, for example, in modeling actin-driven motility.

The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity

We give several examples of modeling in nonlinear elasticity where a quasiconvexification procedure is needed. We first recall that the three-dimensional Saint Venant-Kirchhoff energy fails to be quasiconvex and that its quasiconvex envelope can be obtained by means of careful computations. Second, we turn to the mathematical derivation of slender structure models: an asymptotic procedure using T-convergence tools leads to models whose energy is quasiconvex by construction. Third, we construct an homogenized quasiconvex energy for square lattices.

Discrete homogenization in graphene sheet modeling

Every two years since 2001, the Societe de Mathematiques Appliquees et Industrielles (SMAI) co-organizes, jointly with one or several French laboratories, a conference entitled Congres SMAI 2001 + 2n which gathers together the Congres National d’Analyse Numerique (CANUM) and the annual conferences of the five thematic groups of the SMAI, namely the GAMNI (Groupe pour l’Avancement des Methodes Numeriques de l’Ingenieur), MAIRCI (Mathematiques Appliquees, Informatique, Reseaux, Industrie), MAS (Modelisation Aleatoire et Statistique), MODE (Mathematiques de l’Optimisation et de la Decision), and SIGMA (Signal, Image, Geometrie, Modelisation, Approximation). The Congres SMAI 2001 + 2n are concerned with a wide spectrum of topics going from theoretical to numerical issues.