Consiglio Nazionale delle Ricerche. Istituto di Matematica Applicata e Tecnologie Informatiche PUBBLICAZIONI

Transcription

1 Consiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche PUBBLICAZIONI D. Boffi, L. Gastaldi, L. Heltai, C.S. Peskin A NOTE ON THE HYPER-ELASTIC FORMULATION OF THE IMMERSED BOUNDARY METHOD N. 39-PV 2006

2 Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R. Sede di Pavia Via Ferrata, PAVIA (Italy) Tel Fax Sezione di Genova Via De Marini, 6 (Torre di Francia) GENOVA (Italy) Tel Fax Sezione di Milano Via E. Bassini, MILANO (Italy) Tel Fax URL:

3 A note on the Hyper-Elastic Formulation of the Immersed Boundary Method 1 Daniele Boffi a, Lucia Gastaldi b, Luca Heltai a,2, Charles S. Peskin c a Dipartimento di Matematica F. Casorati, Via Ferrata 1, I Pavia, Italy b Dipartimento di Matematica, Via Valotti 9, I Brescia, Italy c Courant Institute of Mathematical Sciences, 251, Mercer Street, 10012, New York, NY, United States Abstract The immersed boundary (IB) method is both a mathematical formulation and a numerical method for fluid-structure interaction problems, where immersed incompressible viscoelastic bodies or boundaries interact with an incompressible fluid. Previous formulations of the IB method were not able to treat appropriately thick materials modeled by hyper-elastic constitutive laws due to the lack of appropriate transmission conditions between the immersed body and the surrounding fluid. We present a derivation of the IB method which takes into account in an appropriate way the missing term. The derivation presented in this paper starts from the separation of the stress in the principle of virtual work. The stress is divided in its fluid-like and solid-like components and each of this two terms is treated in its natural framework, i.e. the Eulerian framework for the fluid-like behavior and the Lagrangian one for the elastic-like one. We present a brief overview on how the IB method can be used in conjunction with standard formulations of continuum mechanics models for incompressible elastic materials and we present some significant numerical experiments. Key words: immersed boundary method, fluid-structure interaction, hyper-elasticity. addresses: daniele.boffi@unipv.it (Daniele Boffi), gastaldi@ing.unibs.it (Lucia Gastaldi), luca.heltai@unipv.it (Luca Heltai ), peskin@cims.nyu.edu (Charles S. Peskin). 1 This work was partially supported by IMATI-CNR, Pavia. 2 The author was partially supported by a Fulbright grant.

4 1 Introduction One of the main difficulties that arises when dealing with elasticity and fluid structure interaction problems is the fact that fluids and elastic materials have a different natural framework. The usual way of characterizing a fluid motion is the Eulerian framework, where the system is described using the velocity and pressure fields. On the other hand, when dealing with elasticity, it is customary to express the stress as a function of the displacements of the material particles from their reference, or Lagrangian, position, which is not directly available in the Eulerian formulation. The Immersed Boundary method gives one way to link the two frameworks together and deploy the strengths of both formulations at the same time. The original IB formulation was intended to simplify the study of the interaction between thin membranes undergoing large deformations and fluids described by the Navier- Stokes equations and was later applied to more general fluid structure interaction systems. The original IB method relies on the finite difference method, with respect to both the Eulerian and the Lagrangian variables and uses an approximation of the Dirac delta distribution in order to pass information between the two frameworks. The (discrete) conversion from the Lagrangian to the Eulerian coordinates is called force spreading, while the passage from the Eulerian to the Lagrangian framework is called velocity interpolation. Spreading can be seen as the adjoint operator of interpolation. Peskin shows in [20] that the proposed discretization of the interaction equations preserves mass, momentum, angular momentum, torque and power, ensuring that in the conversion between the two frameworks no spurious creation or destruction of mass, momentum or energy is induced by the numerical approximation. In the case of uniform constant mass density of the system, it is also possible to prove that the complete space-discrete problem (motion and interaction equations) is conservative, meaning that momentum and energy are conserved. In [20], the IB formulation was derived from the principle of least action for an hyper-elastic material filling the entire space. This formulation can be extended in a straightforward manner to problems involving immersed bodies which do not occupy the entire space. When this is done, however, a restriction not explicitly recognized in [20] is that σ s n should be equal to zero at the interface between the elastic material and the fluid in which it is immersed, where σ s is the stress tensor derived from the elastic energy function of the immersed material and where n is normal to the interface. This condition is automatically satisfied, for example, if the elastic material is comprised entirely of elastic fibers which are not cut by the interface. If one wants to treat more general elasticity models via the IB method, 2

5 some extra care has to be dedicated to the transmission conditions between the immersed body and the surrounding fluid. The formulation presented in [20] lacks in the treatment of such conditions. In this paper we show how it is possible to use the IB method also in the case of thick bodies modeled by an hyper-elastic constitutive law and we present an IB formulation where an extra term is added to the original formulation to take into account the transmission conditions between the solid and the surrounding fluid in a way which is appropriate for the elastic model under study. Section 2 presents briefly the concept behind the IB method. We propose a formulation of the IB method based on hyper-elastic materials, whose properties are summarized in section 3. We will concentrate ourselves to the cases of isotropic and anisotropic incompressible elasticity using both a Neo-Hookean elasticity model and a fiber-like model, both for thick materials as well as for thin membranes of co-dimension one. We introduce the variational and finite element formulation of the IB method in sections 4 and 5 while in section 6 we describe the fully discrete problem. Some numerical experiments are presented in section 7 and we draw some conclusions in section 8. 2 The Immersed Boundary Method We study a region Ω of R d containing a fluid and an immersed solid material. In particular we are interested in the interaction between the two when the fluid is described by the Navier-Stokes equations and the solid is described by a viscous hyper-elastic model. We identify the material particles contained in Ω with the set ω R d of their position in a reference configuration and we suppose that the subset B ω R d identifies the elastic body. The coordinate system in this Lagrangian framework is given by the vector variable s. When s B it marks a solid particle, while if s ω\b it identifies a fluid particle. We fix a coordinate system for the physical space where the position variable x is used to determine a fixed point in space. For simplicity we assume that during the entire motion all the material particles (both solid and fluid) remain contained in the region Ω R d. The relationship between the two different frameworks is given by the mappings X :ω [0,T] Ω B t q :Ω [0,T] ω B, (1) 3

6 ω X(s, t) Ω B N B t n P P t Fig. 1. Evolution of a continuum of material. representing in the first case the trajectory of a material particle (when s is fixed) or the mapping between the reference configuration and the current one (when t is fixed) and the inverse mapping that associates with each point x Ω the material point that happens to be there (as in Figure 1). The following identities hold x = X(q(x,t),t), s = q(x(s,t),t), (2) and we assume that at any given time the mapping X(s,t) is invertible, which implies that the deformation gradient F αi := ( s X(s,t) ) αi = X α,i(s,t) = X α(s,t) s i, (3) has a non zero determinant. We assume it to be positive at time t = 0, and therefore at any subsequent time, i.e. F = det F > 0. (4) Functions having x as the space variable are usually called spatial or Eulerian functions, while the ones having s as the space variable are called material or Lagrangian functions. We define the velocity field u by and its total time derivative by u(x,t) = X t (s,t) s=q(x,t), (5) Du Dt (x,t) = 2 X t (s,t) 2 s=q(x,t) = u (x,t) + u(x,t) u(x,t), (6) t which describe respectively the velocity and the acceleration of the particle that happens to be at the point x at time t. To ease the reading, we will try to maintain throughout the paper the following notation: capital letters refer to functions whose domain is the Lagrangian one (e.g. 4

7 X(s, t)), while lower case letters refer to functions whose domain is the Eulerian or spatial one (e.g. u(x,t)). Boldface letters are vector fields (e.g. u) and thick capital or greek letters are tensor fields respectively in the Lagrangian coordinate system (e.g. P) or in the Eulerian one (e.g. σ). The symbol I will always denote the identity matrix, regardless of the domain in which it appears. Whenever there could be confusion, we will explicitly write the variable names and indices, with the additional convention that latin subscript indices vary in the Lagrangian domain while greek subscript indices vary in the Eulerian one. Summation over repeated indices will always be understood. Occasionally we will need to treat jumps across interfaces, for which we borrow the following standard notation: consider an interface dividing the space in two sides (denoted by the superscript + and ). The jump of an Eulerian (resp. Lagrangian) quantity a (resp. A) across the interface is defined as: [a] = a + n + + a n [A] = A + N + + A N, where n ± (resp. N ± ) are normals to the interface pointing outward in each case from the + or region. We will use the same convention for scalars, vectors and tensors, provided that the correct multiplication with the normals are used (namely scalar products or matrixvector products in the vector and tensor cases respectively). We remark that with this convention the jump of a scalar field is a vector parallel to the normals, the jump of a vector field is a scalar value while the jump of a tensor field is a vector in general not parallel to the normals. The equations that govern continuum mechanics can be derived from the integral form of the conservation of linear and angular momenta (see, for example, [10]). If we consider an arbitrary smooth portion P of ω evolving as P t = X(P,t), these laws can be expressed through a continuous surface force density t(x,n) as P t ρ Du Dt (x,t) dx = P t x ρ Du Dt (x,t) dx = t(x,n,t) da + b(x,t) dx P t P t x t(x,n,t) da + x b(x,t) dx, P t P t (7a) (7b) where b are external forces and ρ is the Eulerian mass density. The hypothesis that the surface force density depends only on the normal n at each x is usually referred to as Cauchy s hypothesis. A necessary and sufficient condition for equations (7) to be satisfied is that for any of the considered P and for each time t there exists a symmetric stress tensor σ = σ T, called Cauchy stress tensor, such that t(x,n,t) = σ(x,t)n for any unit 5

8 vector n and such that the following is satisfied (principle of virtual work) ρ Du P t Dt v dx + σ : vdx = σn v da + b v dx, (8) P t P t P t for any virtual displacement or test function v regular enough for the integrals in (8) to make sense, where we used the standard notation σ : v = σ αβ v α,β = σ αβ v α x β. Under some additional regularity assumptions, this is equivalent to the strong formulation of the equations of motion ρ Du Dt = σ + b in Ω. (9) Most of the equations of continuum mechanics can be described via an appropriate choice of the Cauchy stress tensor, however its analytical expression can become cumbersome in Eulerian form for elasticity problems. The IB method addresses this issue by separating the Cauchy stress into two parts, one whose natural framework is the Eulerian one and the other one which is more easily described in Lagrangian form, treating each part in its natural framework. Specifically we separate the Cauchy stress as σ f in Ω\B t σ = (10) σ f + σ s in B t where σ f describes the fluid-like behavior of the material (typically modeled with the same Navier-Stokes stress tensor both in the solid and in the fluid) and we specialize σ s to contain the elastic part of the stress present in the solid material. In elasticity it is convenient to express the stress in the reference (Lagrangian) variables. This is usually achieved via the d d tensor field P (referred to as the first Piola-Kirchhoff stress tensor), defined in such a way that for any arbitrary smooth portion P of ω evolving as P t = X(P,t) it holds σn da = PN da, (11) P t P where N is the outer normal to the region P in the Lagrangian coordinates (as in Figure 1) while da and da are the area differentials in dimension d 1. The first Piola-Kirchhoff stress gives the force per unit reference area (d = 3) or length (d = 2), expressed in the physical space, and its point-wise expression is given by P(s,t) = F(s,t) σ(x(s,t),t) F T (s,t), (12) 6

9 where F T is the inverse transpose of F. We separate the first Piola-Kirchhoff stress in its fluid-like and solid-like components as we did with the Cauchy s stress P f in ω\b P = P f + P s in B, (13) where it is understood that (12) is to be applied separately on each of the two stress components. With this separation of the stress and neglecting external forces, the principle of virtual work reads ρ Du P t Dt v dx + σ f : vdx + σ s : vdx = P t P t B t σ f n v da + σ f n v da. (14) P t P t B t Using (11) and (12) we can treat the elastic part of the stress in its natural Lagrangian framework: ρ Du P t Dt v dx + σ f : vdx + P s : s V ds = P t P B σ f n v da + P t where we defined V(s, t) = v(x(s, t)). P B P f N V da, (15) Note that the region P may contain both solid and fluid particles. If we integrate by parts and use the identity (P B) = ( P B) (P B), (16) the principle of virtual work (15) can be rewritten as P t (ρ Du Dt σ f) v dx = P B ( s P s ) V ds P B P s N V da. (17) In our formulation the fluid-like stress σ f exists within the fluid as well as within the solid body, where the total state of stress depends on both σ f and P s. The reason for the use of such a formulation is the fact that the response of an elastic material to deformations is easily described as a function of the deformation gradient F, i.e. P s = P s (F), which is not directly available in Eulerian coordinates. It is 7

10 possible to get around this limitation by adding an evolution equation for the deformation gradient F in Eulerian coordinates (see, for example, [16]), however we are interested in maintaining the elastic stress in its natural Lagrangian framework. This can be achieved in a formal way, avoiding explicit change of variables, through the defining property of the d-dimensional Dirac delta distribution V(s,t) = v(x(s,t)) = v(x)δ(x X(s,t)) dx s P. (18) P t We define the interior force density G, supported throughout the elastic material, and the transmission force density T, supported on the boundary of the elastic material, as G(s,t) = s P s (s,t), (19) T(s,t) = P s (s,t)n(s). (20) Via the definition (18) we can rewrite the volumetric Lagrangian part of (17) as B P ( s P s ) V ds = ( s P s (s,t)) B P v(x)δ(x X(s,t)) dx ds P t = ( s P s (s,t))δ(x X(s,t)) ds v(x) dx P t B P = ( s P s (s,t))δ(x X(s,t)) ds v(x) dx P t B = g(x,t) v(x) dx, P t (21) where g(x,t) = B G(s,t)δ(x X(s,t)) ds. (22) Similarly we can write P s (s,t)n(s) V(s,t) da = B P = P s (s,t)n(s) B P v(x)δ(x X(s,t)) dx da P t = P s (s,t)n(s)δ(x X(s,t)) da v(x) dx P t B P = P s (s,t)n(s)δ(x X(s,t)) da v(x) dx P t B = t(x,t) v(x) dx, P t where t(x,t) = B (23) T(s,t)δ(x X(s,t)) ds. (24) 8

11 Note that the two terms g and t have a totally different character. The former is a function and represents the interior force density generated by the body, while the latter is a singular distribution that takes into account the transmission conditions between the body and the surrounding fluid. In particular, the singular term t has the character of a layer of Dirac delta distributions lying along the boundary of the body, whose action on any smooth test function v is defined as < t(t),v >= Ω t(x,t) v(x) dx = B T(s,t) v(x(s,t)) da. (25) Even though the singular term t has not been included in previous IB computations involving thick immersed bodies, it is interesting to note that the form of equation (24) is precisely the one that arises in IB computations for an immersed elastic membrane lying along B (except that the computation of T(s,t) would in that case be different), so the IB method already contains the apparatus needed to deal with this new term. The arbitrariness of both P and v implies that we can formally rewrite the equations of motion in a point-wise manner as ρ Du Dt σ f = g + t = G(s,t)δ(x X(s,t)) ds + B B T(s, t)δ(x X(s, t)) da in Ω. (26) The equations for incompressible fluid structure interaction systems derive from adding an incompressibility constraint to (26) and modeling the fluid part of the Cauchy stress via the Navier-Stokes stress σ f = pi + 2ηD(u) = pi + η( u + ( u) T ), (27) where η is the viscosity and p is the Lagrange multiplier associated with the incompressibility constraint u = 0 and it is sought in the space L 2 0(B t ) (square summable functions with zero mean value). For the sake of simplicity, we assume that ρ and η are the same constants both in the solid and in the fluid. We notice that in general the separation of the stress (10) can be performed in a non unique way. In this paper we concentrate only on the case where the solid is a viscous incompressible hyper-elastic material which can be described by the same constant viscosity and the same constant mass density of the surrounding fluid. In this case the stress in the solid can be seen as the superimposition of the Navier- Stokes stress (27) and of a hyper-elastic stress which may or may not contain some pressure terms due to eventual potential energy associated with change in volume of the material. Our reading of the pressure p as a Lagrange multiplier for the incompressibility 9

12 constraint suggests however a way to render unique the division of the stress in its two components, by discarding from the elastic part of the stress P s all the informations related to changes in volume, which are prevented by the divergence free constraint on the velocity field u. The immersed boundary method accounts to the resolution by finite differences (with the introduction of a suitable approximation of the Dirac delta distribution) of the following problem. Problem 1 Find u, p and X which satisfy: ρ ( ) u t + u u η u + p = g + t in Ω ]0,T[ (28a) u = 0 in Ω ]0,T[ (28b) g(x,t) = G(s,t)δ(x X(s,t)) ds in Ω ]0,T[ (28c) B t(x,t) = T(s,t)δ(x X(s,t)) da in Ω ]0,T[ (28d) B G(s,t) = s P s (s,t) in B ]0,T[ (28e) T(s,t) = P s (s,t)n(s) in B ]0,T[ (28f) X (s,t) = u(x(s,t),t) t in B ]0,T[ (28g) u(x,t) = 0 on Ω ]0,T[ (28h) u(x, 0) = u 0 (x) in Ω (28i) X(s, 0) = X 0 (s) in B. (28j) Conditions (28h) and (28i) represent boundary and initial conditions relative to the Navier-Stokes equation (28a)-(28b); other boundary conditions could also be used. The last equation (28j) is the initial condition for (28g) which drives the motion of the immersed structure. In particular the equations of motion described in problem 1 are equivalent to studying separately Navier-Stokes equations in the region Ω\B t and the incompressible visco-elasticity equations on the region B t, coupling the two problems via the continuity of the stress along the boundary B t of the visco-elastic region: which implies [σ] = σ + f n+ + (σ f + σ s )n = 0 on B t, (29) F [σ f ] = T on B t, (30) where we denoted with the visco-elastic region and with + the fluid region. One of the advantages of this formulation is the unified treatment of singular forces supported on immersed boundaries of dimension d 1 via the use of a Dirac delta 10

13 distribution. In this case it is possible to consider the same formulation as in problem 1 where B is a manifold of dimension m = d 1. This feature is the one that gave the name to the immersed boundary method. To avoid the treatment of d 2 dimensional terms in the formulation, we will only consider this case when B t is a closed manifold of co-dimension 1, i.e. the transmission term t vanishes, and g takes on the singular character that t had in the case of a thick immersed body. In this case the deformation gradient F is an d m matrix, and it no longer makes sense to talk about its determinant. We will refer to the condition F > 0 as of F having maximum rank, i.e. if m = 1 and d = 2 X F = s (s,t) > 0, (31) while, if m = 2 and d = 3 X F = (s,t) X (s,t) > 0. (32) s 1 s 2 We will not distinguish between (4), (31) or (32), and we will always write F > 0, intending one or the other based on the dimension m of the Lagrangian domain, with the meaning that any portion of the reference domain with positive measure (length if m = 1, area if m = 2 and volume if m = 3) is transformed by the mapping X(s,t) into a portion with positive measure into the spatial domain, with F being the local ratio between the reference and the deformed m measure. For the singular problem of an immersed elastic membrane, the total Cauchy stress is no longer continuous on the entire domain Ω as we have introduced a membrane which is able to sustain a finite force on an infinitely thin support. This results in a jump in the stress along the membrane B t and problem 1 can be rewritten using a standard formulation by considering separately the fluid equations on both sides ( + or this time both refer to fluid regions) of the immersed membrane and coupling them via the jump conditions in particular (see, for example, [23,13]), F [σ] = F [σ f ] = G on B t, (33) F [p] = (n n)g on B t F η [ u] = (I n n)g on B t, (34) where G this time is the force density per unit reference area (d = 3) or length (d = 2) applied by the membrane to the fluid and is the tensor or dyadic product defined as (a b) ij = a i b j. The Eulerian force density g(x,t) is defined exactly as 11

14 before and it still represents the force per unit volume (d = 3) or area (d = 2). In this case (m=d-1) g(x,t) is a singular force supported on the co-dimension 1 viscoelastic domain B t, whose integral on any finite volume (d = 3) or area (d = 2) is finite. The IB method is particularly well suited for efficient numerical simulations in both the thin and thick cases, thanks to the introduction of the Dirac delta distribution which aims to decouple the Lagrangian framework from the Eulerian one, acting as an interpolation between the two spaces. One consequence of this structure is that it is possible to use a fast Navier-Stokes solver on a fixed lattice structure without the need to adapt the fluid mesh to the deforming Lagrangian one. On the other hand, the elastic computations needed in order to obtain stress and strain measurements, which will then translate via the Dirac delta spreading to force densities on the fluid domain, can be carried on in a relatively inexpensive way (compared to the solution of the Navier-Stokes equations) in their natural, Lagrangian, framework. 3 Hyper Elasticity We will limit our discussion only to elastic materials for which it is possible to define a positive potential energy density W(F,s) > 0 associated with the deformation. A material of this kind is usually called hyper-elastic and its properties depend only on its current state and not on the path that it followed to reach its current configuration. For the potential energy to be physically meaningful, it should be independent of rigid motions of the elastic body. To be more precise, W is a function of F (and possibly s, to take into account spatial inhomogeneities of the material) for which Y = RX + U W ( ) ( ) ( ) Y X X s,s = W s RT,s = W s,s where U is a fixed translation vector and R is a rotation. R R d R d s.t. R R T = I, detr > 0, (35) The first Piola-Kirchhoff stress tensor P can be expressed in terms of the potential energy density as (P(s,t)) αi = W ( ) W (s,t) = F αi F (s,t), (36) αi 12

15 where i = 1,...,m and α = 1,...,d. The relationship between the potential energy density W and the generated internal force density G is then given by (G(s,t)) α := P αi,i (s,t) = P αi s i (s,t) = 2 W F αi F βj F βj s i (s,t), (37) where summation is implied over repeated indices. It is intended here that the density is measured with respect to the Lagrangian domain, i.e. G is a force density per unit length, area or volume respectively when m = 1, 2 or 3. Common measures used to quantify the energy associated with the deformation of a material are expressed by the Green and right Cauchy-Green d d strain tensors: and G = 1 2 (FFT I), (38) C = FF T, (39) which are independent of rigid motions of the material in the Eulerian space. If the material is isotropic, then the energy density W is also independent of rotations of the body in the Lagrangian framework, and the constitutive models can be generated via invariants of the strain tensors C or G, such as I C = trace(c), II C = trace(c 2 ) and III C = det(c), which are common choices for such invariants. There is a very large variety of elastic models available to describe different materials. In the general case the potential energy density may have some terms that depend on the compression of the material under study. When treating incompressible visco-elasticity (m = d) from a purely Lagrangian point of view, it is a delicate task to express W in a way that takes into account the incompressibility constraint. This is usually achieved via a penalization technique, introducing some terms in the formulation of W that go to infinity as F goes to zero. On the other hand, if we use the IB method combined with an hyper-elastic model to describe the behavior of our fluid-structure problem, we enforce incompressibility on the entire system (therefore also on the solid body) via the divergence free constraint on the velocity field in a way which is independent from the potential energy density formulation. As a side effect of this fact it is possible to eliminate from the energy density W all the terms which are associated with a change in volume (i.e. all the terms depending on F ). The IB formulation coupled with the most commonly used hyper-elastic models 13

16 generate a system of equation which describes efficiently viscous and incompressible hyper-elastic materials interacting with viscous incompressible fluids. A particularly popular generalization of linear isotropic elasticity to the large strain regime is the so called Neo-Hookean material, which is characterized by a logarithmic resistance to compression due to an additional term in the energy density which depends on F : W = 1 2 µ(trace(c) m) µ ln( F ) λ ln( F )2. (40) With this energy density, the first Piola-Kirchhoff stress tensor becomes P = µ(f F T ) + λ ln( F )F T. (41) Using the IB formulation the potential energy density for an incompressible neohookean material simplifies to W = 1 µ(trace(c) m), (42) 2 as the incompressibility is already taken care of by the divergence free constraint on the velocity in Eulerian form. The stress in this case becomes simply P = µf, (43) making this model particularly appealing in the applications for its simplicity and for the fact that, when combined with the IB method, it becomes linear. In the literature, the most commonly used model in combination with the IB method, is the fiber like model. This model gives a natural way to treat co-dimension one structures immersed in fluids, starting from an energy formulation which is derived from the hookean energy formulation. Consider as a first example the energy formulation used to model thin fibers (m = 1) immersed in a two-dimensional fluid: W(F) = µ 2 F 2, (44) where µ is the stiffness of the fiber and the Piola-Kirchhoff stress tensor can be defined by a scalar tension T = µ F and the versor τ = F/ F tangent to the fibers (note that both P and F in this case are vectors): P = Tτ = µf, G = µ 2 X s 2. (45) 14

17 This formulation is useful also outside the one dimensional setting if we consider a continuum collection of such fibers. It is possible to show that any incompressible linearly elastic material can be modeled via a finite collection of families of fibers along different directions (see TBD). Note here the similarity between the neo-hookean Piola-Kirchhoff stress given in (43) and the fiber formulation of the Piola-Kirchhoff stress given in (45). The only difference between the two models is the dimension of the Lagrangian domain, wich is d in the first case and d 1 in the second, giving rise to a d d matrix for the neo-hookean Piola-Kirchhoff stress and to a d m matrix for the fiber Piola- Kirchhoff stress. If we consider a continuum collection of fibers lying one next to the other on the direction r (where r is a unit vector in the lagrangian space) to form a finite dimensional strip, we obtain an elastic model which represents a totally anisotropic material whose energy density can be expressed by W = µ X 2 r (s,t) 2 = µ (Fr) (Fr), (46) 2 where the term that intervene in the definition of the energy involves only the components of the deformation gradient F along the direction of anisotropy. The most direct discretization of such a model is given by a collection of fibers lying along the direction r with zero length at rest and elastic constant µ dr, where dr is the distance between two consecutive fibers. Similar models were used, for example, to study the blood flow in the heart [21,19] the fluid flow in collapsible elastic tubes [24] or flapping flexible filaments in a flowing soap film [30,31]. 4 Variational Formulation The spatial discretization of the original version of the IB method is done by means of finite differences. The main issue is the computation of the terms in (28c) and (28d) due to the presence of the Dirac delta distributions. This has been realized by the construction of a suitable approximation function δ h, which is nonsingular for each h and approaches δ as h 0 (we refer to [20] for a detailed description of such a construction). We revise the work presented in [2 4,8,7], where a finite element approach to the spatial discretization of the IB method was proposed. Our spatial discretization is based on the use of standard finite elements in the approximation of the Navier- Stokes equation and on the discretization of the immersed boundary by continuous piecewise linear elements. Our approach is different from that of [28,29], since we 15

18 deal with the force terms involving the Dirac delta distribution in a variational way, so that there is no need of approximating it. Let B t denote the immersed elastic structure, that is a m-dimensional body in the d-dimensional domain Ω, then let X(s,t) denote a material point on B t, where the variable s gives the Lagrangian representation and varies in the set B R m as in (1). The variational formulation of problem 1 can then be written as Problem 2 Given u 0 H 1 0(Ω) d and X 0 : B Ω, for all t ]0,T[, find (u(t),p(t)) H 1 0(Ω) d L 2 0(Ω) and X : B ]0,T[ Ω, such that ρ ( ) d (u(t),v) + (u(t) u(t),v) + η( u(t), v) dt ( v,p(t)) =< f(t),v > ( u(t),q) = 0 < f(t),v >= B P(s,t) : s v(x(s,t)) ds v H 1 0(Ω) d (47a) q L 2 0(Ω) (47b) v H 1 0(Ω) d (47c) X (s,t) = u(x(s,t),t) t s B (47d) u(x, 0) = u 0 (x) x Ω (47e) X(s, 0) = X 0 (s) s B. (47f) In (47a) and (47b), (, ) stands for the usual scalar product in L 2 (Ω), while <, > denotes the duality pairing between H 1 (Ω) d and H 1 0(Ω) d. We recall that L 2 0(Ω) is the subset of L 2 (Ω) containing functions with zero mean value. When m < d, equation (47c) has to be intended in the usual weak sense (taking into account the fact that we allow only for configurations where B t is the empty set), i.e. P(s,t) : s v(x(s,t)) ds = ( s P(s,t)) v(x(s,t)) ds. B B We observe that X should satisfy a further condition so that for all s B and t [0,T] we have X(s,t) Ω. However, since we enforce an homogeneous Dirichlet boundary condition on u, equation (47d) implies that if X reaches the boundary of Ω then it remains on Ω for all the successive times. 16

19 5 Space Discretization by Finite Elements Let T h be a subdivision of Ω into triangles or rectangles if d = 2, tetrahedrons or parallelepipeds if d = 3. We then consider two finite dimensional subspaces V h H 1 0(Ω) d and Q h L 2 0(Ω). It is well known that the pair of spaces V h and Q h needs to satisfy the inf-sup condition in order to have existence, uniqueness and stability of the discrete solution of the Navier-Stokes problem (47a)-(47b) (see [12,9]). Next consider a subdivision S h of B into segments, triangles or tetrahedrons (respectively m = 1, 2, 3). We will denote by s j, j = 1,...,M the vertices of the triangulation S h and by T k, k = 1,...,Me the elements of S h. Let S h be the finite element space of piecewise linear d-vectors defined on B as follows S h = {Y C 0 (B; Ω) : Y Ti P 1 (T k ) d, k = 1,...,Me}, (48) where P 1 (T k ) stands for the space of affine polynomials on the element T k. For an element Y S h we shall use also the following notation Y j = Y(s j ) for j = 0,...,M. The first step, in order to introduce the discrete counterpart of Problem 2, is the computation of (47c) for all X S h and for all v V h : < f h (t),v >= B P αi s i (v α (X)) ds, (49) We write the integral as a sum over the elements T k, use the fact that X is linear, so that its derivative and P are constant, and integrate by parts to arrive at: < f h (t),v >= = Me k=1 Me k=1 P αi (v α (X)) ds T k s i T k P αi N i v α (X) da. (50) By defining the set of the edges of the triangulation S h as we can rewrite Equation (50) as: E = T Sh e T, (51) < f h (t),v >= e E e [P] v(x) da, (52) with the convention that, when e B, then [P] = PN, where N is the outer normal to B. 17

20 Notice that the right hand side of (52) is meaningful also when m = 1, since v is continuous as it is required for the elements in V h. The spatial finite element discretization of Problem 2 then reads: Problem 3 Given u 0 V h and X 0 S h, for all t ]0,T[, find (u(t),p(t)) V h Q h and X(t) S h, such that d (u(t),v) + (u(t) u(t),v) + η( u(t), v) dt ( v,p(t)) = [P] v(x) da v V h (53a) e E e ( u(t),q) = 0 q Q h (53b) u(x, 0) = u 0 (x) x Ω (53c) X i t (t) = u(x i(t),t) i = 1,...,M (53d) X i (0) = X 0 i i = 1,...,M. (53e) The choice of the space S h for the solution of the finite element IB method depends highly on the kind of potential energy that needs to be approximated. In particular we will be interested in isotropic and anisotropic incompressible elasticity, both in the singular case of thin membranes (one-dimensional curves immersed in twodimensional fluids or two-dimensional surfaces immersed in three-dimensional fluids) and in the case of thick materials. In the singular case of the one-dimensional curve immersed in a two-dimensional fluid, the source term for the Navier-Stokes equations becomes M [P] v(x) da = [P] v(x j ), (54) e E e j=1 as in this case the faces of the elements of S h are the vertices of the segments, and the integral on the face is given by the point evaluation of the test function in the vertices of the segment. The extremely simple expression of the source term given by (54) can be further exploited in the non-singular anisotropic case, where we thicken the Lagrangian domain but we consider an energy formulation that depends only on one of the two Lagrangian variables. In particular it is possible to approximate a continuous anisotropic material by laying several fibers of the kind used in the singular case one next to each other. Sections 7.3 to 7.6 present a comparison between this fiber-formulation and a more standard finite element approach, where the thickness of the material is ap- 18

21 proximated by two-dimensional continuous P1 triangular elements, and the anisotropy of the material is incorporated in the definition of the first Piola-Kirchhoff stress. We would like to remark here the fact that the two different approximations do not differ from the qualitative point of view, as the energy formulation that is being approximated is the same in both cases. 6 Time Discretization by Finite Differences To solve numerically the fully coupled problem it is necessary to introduce an appropriate time discretization. The natural and simplest choice if one wants to maintain stability in the solution of ODE problems would be the use of an implicit technique, such as the backward Euler method. However in Problem 3, the Navier- Stokes equations (53a)-(53b) are strongly coupled through the source term (52) with the system of ordinary differential equations given by (53d) and an implicit method implies the resolution of a fully nonlinear coupled system of equations at each time step which contains the non linear Navier-Stokes solve as an inner step. A natural alternative to the fully implicit method is the use of a semi-implicit modification. Let t denote the time step and let us indicate by the superscript n an unknown function at time t n = n t, so that the number of time steps needed to reach the final time T is N. In general the full space-time discretization implies three steps: given the approximation X n of X at time n t, we calculate f n, then we find the solution (u n+1,p n+1 ) to the Navier-Stokes equations with the given load and we finally move the immersed boundary, getting X n+1. Problem 4 Given u 0 V h and X 0 S h, set u 0 = u 0 and X 0 = X 0, then for n = 0, 1,...,N 1 Step 1. compute the source term < f n,v >= e E Step 2. find (u n+1,p n+1 ) V h Q h, such that e [P(X n )] v(x n ) da v V h ; ( u n+1 u n ),v + (u n u n,v) + η( u n+1, v) t ( v,p n+1 ) =< f n,v > ( u n+1,q) = 0 v V h q Q h 19

22 Step 3. find X n+1 S h, such that X n+1 i X n i t = u n+1 (X n i ) i = 1,...,M. The stability properties of this approximation were observed first in [27]. An analisys based on the modes of oscillations of a straightened one-dimensional fiber was performed in [26] and the consequences for possible time stepping strategies were further studied in [25]. In [6] the autors propose a stability analysis based on energy estimates for the co-dimension one case in two dimensions, which is further generalized in [11]. 7 Numerical Experiments We introduce now a series of test cases crafted to underline some of the characteristics of the IB method. The nonlinearity of the coupled system together with the non-local nature of the problem itself makes it very difficult to construct simple test cases with an analytical solution. The only cases in which this is possible are static test cases (velocity equal to zero everywhere) where the solution is computed by symmetry and by imposing continuity on the stress tensor. We start our numerical example section with one such a problem. We would like to remark that, notwithstanding its simplicity, the first test case we present is enough to underline the convergence and regularity properties of the IB method for singular problems. In all test cases we considered, the fluid domain is the unit square [0, 1] 2 or the unit cube [0, 1] 3 and, unless otherwise stated, we set the mass density ρ and the viscosity η to be equal to one. We discretized the domain Ω uniformly into subsquares or sub-cubes and we used a finite element discretization based on piecewise continuous bi- or tri-quadratic finite elements for each of the velocity components and piecewise discontinuous linear finite elements for the pressure, also known as the Q2-P1 Stokes finite element pair. The chosen pair of finite elements satisfies the inf-sup condition and is well known for its stability properties and optimal convergence behavior. All the computations were performed on a quadri processor RedHat Linux System, with a code written in C++ and with the support of the deal.ii libraries (see [1] for a technical reference). 20

23 7.1 Two-dimensional inflated balloon The simplest possible proof of concept problem that can be imagined and for which we have an analytical solution is the discretization of a one-dimensional closed membrane filled and immersed into the same two-dimensional fluid. If the configuration of the membrane is chosen to be a circle, then this is an equilibrium configuration, and the exact solution is known. We consider the fiber energy formulation previously introduced: W = µ X 2 s (s,t) 2 = µ (F : F), (55) 2 where F is a 2 1 vector. The first Piola-Kirchhoff stress is given by P = µ X (s,t) = µf. (56) s The configuration of the balloon is a circle with radius R.5 immersed in the middle of the square domain [0, 1] 2, with null initial velocity u and initial Lagrangian representation given by R cos(s/r) +.5 X(s,t) = s [0, 2πR]. (57) R sin(s/r) +.5 Notice here that we parametrized the immersed boundary by arc length. One consequence of this choice is the fact that the force density per unit length of the boundary, if we use the Piola-Kirchhoff stress formulation given by Eq. (56), is the curvature of the circle times the elastic constant of the immersed material, that is µ/r, directed towards the center of the circle along its radii. This configuration is a local minimum for the potential energy (55) thanks to the incompressibility constraint, and we can use the jump conditions expressed in Eq. (34) to derive the exact solution: u(x,t) = 0 x Ω, t ]0,T[ µ(1/r πr) x R p(x,t) = µπr x > R t ]0,T[. (58) Figures 2(a) and 2(b) show the pressure we obtained using the Q2/P1 finite element pair on a uniform grid of squares with immersed boundary parametrized 21

24 (a) Pressure (b) Pressure cutlines Fig. 2. Inflated balloon in a static fluid. Table 1 Convergence table for two dimensional static case. # cells # dofs u u h 0 u u h 1 p p h e e e e e e e e e e e e e e e with mesh size equal to 1/128, time step.005, using a radius R =.25 and elastic constant µ = 1, after 3 seconds of simulation. Table 1 shows the convergence results obtained by refining uniformly the domain Ω after one time step. We can infer here an order of convergence.5 for the velocity in H 1 and for the pressure in L 2 and of order 1.5 for the L 2 norm of the velocity and pressure. This is in accordance with the nature of the right hand side in the Navier-Stokes equations, as the IB source term is a singular distribution which belongs to H 1 2(Ω), leading to a maximum regularity for u in H 3 2(Ω). 7.2 Two-dimensional inflated balloon: dynamic case A dynamical modification of the previous test case can be obtained by changing the initial configuration to one which is not a minimum of the potential energy. In 22

25 Table 2 Convergence table for the two dimensional dynamic case. # cells # dofs u u h 0 u u h 1 p p h e e e e e e e e e e e e e e e particular we set it to be a perturbation of the previous one: R cos(s/r) +.5 X(s,t) = s [0, 2πR], (59) (R + γ) sin(s/r) +.5 where γ is chosen in the open interval ( R,R). In this case we do not know the analytical solution, and we computed the error rates by taking the differences between successively refined grids at time t = 1. The results using the same numerical parameters as in the static case while perturbing with γ =.1 are reported in table 2, while Figures 3(a) and 3(b) present some steps of the evolution of the boundary and the elevation plot of the pressure at the first time step. (a) Evolution of the immersed boundary. (b) Elevation plot of the pressure. Fig. 3. Inflated elliptical balloon immersed in a static fluid. The original IB method in this same test case has been shown to be first order accurate as far as the L 2 norm of the velocity is concerned. The increased convergence rate that can be observed using the finite element formulation is believed to be due to the variational treatment of the Dirac delta distribution. Also in the dynamical case, i.e. when u 0, the observed convergence rates are in accordance with the singular behavior of the right hand side term. 23

26 7.3 Two-dimensional visco-elastic shell A direct generalization to a non-singular case of the previous example is given by thickening the membrane and maintaining the same fiber-like behavior. This can be obtained by considering a long strip of incompressible elastic material whose reference configuration is the region B = [0, 2πR] [0, w]. We suppose the width w of the strip to be small compared to its length, so that the elastic behavior is given only by the deformations along its length. This is the model which is typically used to simulate the behavior of muscle fibers in the IB framework. We consider a totally anisotropic material which responds to deformations only on one direction and whose potential energy density is given by W = µ 2w X 2 (s,t) s 1 = µ 2w F α1f α1. (60) The first Piola-Kirchhoff stress tensor associated with this formulation of the energy is given by P = µ F 11 0 = µ X 1 s 1 0. (61) w F 21 0 w X 2 s 1 0 We assume furthermore that the material is bent on itself and joined together to form an elastic shell, i.e. B is periodic in s 1, and the initial configuration is given by X 0 = R(1 + s 2) cos(s 1 /R) +.5. (62) R(1 + s 2 ) sin(s 1 /R) +.5 Using this energy formulation and the periodicity in s 1 of B, the transmission component of the stress (T = PN) is always zero on B (this is true for any configuration X) and we have that the only force applied by the shell on the fluid is the interior force density G = s P given by G 0 = µ w 2 X s 2 1 = µ w 1 + s 2 R cos(s 1/R) = µ sin(s 1 /R) w where r is the unit vector directed along the radial direction. 1 + s 2 r, (63) R As both the fluid and the material are incompressible, we expect this configuration to remain unchanged if the initial velocity is zero. In particular we expect the force density (63) to generate a pressure field which is constant in the inside and in the 24

27 outside of the shell, and which varies in a linear way inside the material along radial directions. From the fact that we scaled the elasticity constant in a way which takes into account the thickness of the shell, the limit case when w 0 is given by the two-dimensional example we gave in the previous section. The exact solution of the pressure is given by p 0 + µ r R R p(r + c,t) = p 0 + µ 1 (R + w r) R < r R + w w R p 0 R + w < r, (64) where c is the center of the shell, c = (.5,.5), and where r = r, c + r Ω. The constant p 0 is given by p 0 = µπ [ R 2 3w ] (R + w)3, R and it it chosen to make the mean value of p equal to zero. The most direct and simple approximation of this kind of problem is given by a collection of fibers (as in the one-dimensional example above) put one next to the other. We remark here that the collection of fibers has to be intended as a discretization of a continuum of material. In particular the distance between the different fibers has to be smaller than the fluid mesh size, and has to go to zero when the mesh size of the fluid goes to zero. If this restriction is respected, this model behaves like a continuous incompressible anisotropic visco-elastic material. For this example we chose the radius to be R =.25, the width to be w =.0625 and the elasticity constant to be µ = 1. The computed solution after one time step is shown in Figure 7.3, where we made a comparison between the fiber-like discretization of the material and the more standard finite element P1 representation of the shell. In particular we show the comparison between the two meshes in Figures 4(a) and 4(b) and between the two pressure plots in Figures 4(c) and 4(d). In Figure 4(e) we present a cut-line of the computed pressure along the lines y =.2, y =.3, y =.4 and y =.5 while in Figure 4(f) we present an elevetion plot of the obtained pressure, where it is more evident the linear behavior of the pressure inside the material. Table 3 shows the convergence results obtained by refining uniformly the domain Ω. It is possible to see a convergence rate of order 2.5 for the L 2 norm of the velocity and of order 1.5 for the L 2 norm of the pressure and the H 1 norm of the velocity, again showing better convergence properties respect to the original IB method, where the observed convergence rates are of order 2 for the L 2 norm of the 25

28 (a) Fiber Lagrangian mesh. (b) FEM Lagrangian mesh. (c) Fiber case: Pressure. (d) FEM case: Pressure. (e) Pressure cut lines. (f) Pressure elevation plot. Fig. 4. Visco-Elastic static shell. velocity. We observe that in the non singular case we gained one order of convergence in all the computed errors, but we are still suboptimal with respect to the approximation properties of the Q2/P1 finite element pair, for which the maximum convergence 26

29 Table 3 Convergence table for two dimensional thick static case. # cells # dofs u u h 0 u u h 1 p p h e e e e e e e e e e e e e e e rates would be of order 2 for the L 2 norm of the pressure and for the H 1 norm of the velocity and of order 3 for the L 2 norm of the velocity. This again is in accordance with the regularity of the exact solution and with the non-simmetry of the Navier- Stokes problem. In particular the exact pressure is an H 3/2 function, but not H 2, and this deteriorates the convergence properties of both the pressure and the velocity. On the other hand we expect an additional deterioration also of the convergence properties for the L 2 norm of the velocity on higher Reinolds numbers, where the convective term takes more importance than in these simple cases, and we expect the L 2 norm of the velocity to be of the same order of its H 1 norm. 7.4 Two-dimensional visco-elastic shell: dynamic case In this example we keep the same energy formulation as in the previous case and we modify the initial configuration of the immersed structure to be X 0 = R(1 + s 2) cos(s 1 /R) +.5, (65) R(1 + γ + s 2 ) sin(s 1 /R) +.5 where γ is chosen in such a way that 0 < R + γ <.5. This initial configuration is no longer a minimum for the potential energy of the material, and the resulting behavior is no longer static. Figure 5 shows the pressure field where we set the density ρ to be equal to one and the viscosity η to be equal to We use the same values as in the previous example, and we add a perturbation γ =.1. We present only the first complete oscillation. The system will perform approximately four complete oscillations before going to the rest state expressed in the previous example. 27

30 (a) t = 0 (b) t = 0.1 (c) t = 0.2 (d) t = 0.3 (e) t = 0.4 (f) t = 0.5 (g) t = 0.6 (h) t = 0.7 (i) t = 0.8 Fig. 5. Evolution of the pressure field. Visco-elastic shell from a perturbed status. 7.5 Two-dimensional visco-elastic radial shell If we consider the same strip of incompressible elastic material of the previous examples and we make the elastic response be on the other direction, we obtain a problem which is the dual of the previous one, where the transmission term PN is no longer zero while it is zero the term G = s P. The potential energy density in this case is given by W = µ 2w X 2 (s,t) s 2 = µ 2w F α2f α2. (66) The first Piola-Kirchhoff stress tensor associated with this formulation of the energy 28

31 is given by P = µ 0 F 12 = µ 0 X 1 s 2 w 0 F 22 w 0 X 2 s 2, (67) which gives a transmission term T = PN (for the same configuration of the previous example) given by T = ± µ cos(s 1/R) = ± µ r, (68) w sin(s 1 /R) w respectively on the inside and on the outside of the shell. This problem can be solved exactly considering the solution for the one-dimensional example above, to obtain the exact expression of the pressure given by p 0 µ r R R+w p(r + c,t) = p 0 + µ R R < r R + w w R+w p 0 R + w < r, where the value of the constant p 0 this time is simply given by p 0 = µπr. Once again the most direct discretization of this kind of material is given by a collection of fibers uniformly arranged along radial directions. If the distance between two consecutive fibers is small compared to the fluid mesh size, then this model behaves like an anisotropic incompressible visco-elastic material. This kind of discretization however is not numerically stable, due to the fact that there is no cohesion between different fibers and they are free to shrink and slide one with respect to the other in a more accentuated way with respect to the previous example, and a small numerical perturbation would tear the structure apart. A simple remedy is to add a collection of very light fibers also in the perpendicular direction (as in the previous example) to reinforce and stabilize the structure. Figures 6(b) and 6(a) show a numerical simulation based on adaptively refined grids for the fluid solver, where the fluid domain cells are refined based on the presence of Lagrangian particles in them. The technique we used to refine the mesh follows the consideration that most of the discretization error will be concentrated on the interface between the fluid and the immersed body. With the IB method it is not necessary to follow the boundary of the immersed body with the elements of the fluid domain, however an appreciable improvement in the accuracy of the solution is observed if the domain is refined in those regions containing the immersed material. We refine trivially each fluid cell that contains at least a Lagrangian particle by dividing it into four or eight equal sub-squares or sub-cubes respectively. The degrees of freedom associated with the hanging nodes that arise from this kind of (69) 29

32 (a) Pressure and mesh plot. (b) Elevation plot of the pressure. Fig. 6. Anistropic radial shell. local refinement are eliminated from the final system by forcing the continuity of the velocity across cell boundaries. The same results and better stability properties are obtained when using the standard P1 finite element approximation for the Lagrangian mesh. The convergence rates obtained in this case are equivalent to the one-dimensional singular problem. 7.6 Neo-Hookean two-dimensional visco-elastic shell In this example we use the Neo-Hookean formulation of the energy given by W = µ 2 (F : F) m) = µ 2 (F αif αi m), (70) which gives the first Piola-Kirchhoff stress tensor P = µf. (71) In this case both the volumetric term G = s P and the transmission term T = PN are in general different from zero, and we cannot expect convergence rates better than the ones obtained in the membrane case and in the anisotropic radial case. We note here that this model is equivalent to a superimposition of the previous two examples, as the energy formulation (70) is precisely the sum of the two terms (60) and (66). In TBA the authors show how it is possible to recover any incompressible linearly elastic model by a finite collection of families of fibers. Figures 7(a) and 7(b) present the computational mesh and the pressure elevation plot in the case of the neo-hookean formulation for the energy. The results are the 30

33 (a) Pressure and mesh plot. (b) Elevation plot of the pressure. Fig. 7. Neo-hookean static shell. same both using the P1 finite element formulation for the immersed structure or using two different families of fibers laying along perpendicular directions. 7.7 Three dimensional inflated balloon. In this example we present the three dimensional equivalent model of section 7.1. The main difficulty in this case is given by the implementation of the deformation gradient F, which, in our framework, is an d m matrix which maps the deformations from the reference to the deformed configuration. The construction of a suitable reference configuration for a spherical membrane is not trivial, and it is possible to show in particular that any representation mapping a plane to a sphere will have at least one singularity point. To avoid this complication we map the reference configuration in a local way to a surface which is already deformed. We chose to use a sphere as the reference configuration. We propose, as in the two-dimensional case, also a dynamical example, where the initial configuration is not a minimum of the energy, but a perturbation of the sphere into an ellipsoide. This process is equivalent, from the continuous point of view, to substituting the gradient and the divergence in the Lagrangian space with the Riemaniann equivalents on the reference manifold. The energy density is similar to the previous cases, except that now we associate energy to the area of the membrane: W = µ 2 (F : F) = µ 2 F αif αi, (72) where i = 1, 2 and α = 1, 2, 3. Again the formulation for the first Piola-Kirchhoff 31

34 stress tensor is given by P = µf, (73) that is the closed membrane tends to minimize its surface by maintaining the inner volume constant, with a weight which is given by the ratio of the area in the reference configuration and the area in the current deformed configuration. From the numerical point of view, the discretization was obtained by constructing a triangular mesh of a sphere for the reference domain, and by straightening any two neighbor triangles in order to obtain a local map of co-dimension one. This map was then used as the reference configuration in order to evaluate the terms [P] across the edges of any two triangles, which were then integrated against the test functions v to give the right hand side of the Navier-Stokes equations as in (52). Figures 8(a) and 8(b) show the Lagrangian mesh and a cut view of the pressure plot in the equilibrium case when the initial configuration is given by a sphere, while in figure 9 we show the evolution of the membrane when the initial configuration is forced to be an ellipsoide. (a) Membrane mesh. (b) Pressure plot. Fig. 8. Three-dimensional inflated balloon. 7.8 Thick three-dimensional visco-elastic shell. As we did in the two dimensional case, it is possible to obtain a non-singular threedimensional anisotropic visco-elastic model by simply thickening the membrane without adding energy associated to the thickness variable. Similarly to the twodimensional case, we have the following energy formulation, where w is the thickness of the shell and we suppose that the reference configuration can be mapped (possibly only locally) to rectangular regions of the kind [a,b] [c,d] [0,w] and 32

35 (a) t = 0 (b) t = 0.5 (c) t = 1 (d) t = 1.5 Fig. 9. Evolution of a spherical membrane from a perturbed status. the variables s 1, s 2 and s 3 varies along these local coordinate systems: W = µ 2w X 2 X 2 (s,t) + (s,t). (74) s 1 s 2 The first Piola-Kirchhoff stress tensor is then given by P = µ F 11 F 12 0 w F 21 F 22 0 = µ w F 31 F 32 0 X 1 s 1 X 1 X 2 s 1 X 2 X 3 s 1 X 3 s 2 0 s 2 0 s 2 0. (75) A straightforward numerical discretization of this model is given, as in the twodimensional case, by a stack of thin membranes as in the previous example. The same restrictions apply to the three-dimensional case, i.e. we have to arrange the thin membranes sufficiently close to each other in order to obtain a reasonable model of a thick anisotropic visco-elastic material. As already observed in the two-dimensional case, the transmission term PN in this case is always zero and we can expect optimal order of convergence for the 33

36 approximation errors. As in the two-dimensional case, the pressure is a continuous field that varies linearly inside the material along radial directions and is constant inside and outside the shell. Once again the limit case for w 0 is given by the singular problem described above. Figure 10(a) shows a cut view of the pressure field, while Figure 10(b) shows a cut view of the Lagrangian mesh, where it is evident the onion-like configuration of the thick body. (a) Pressure plot. (b) Cut view of the mesh. Fig. 10. Three-dimensional anisotropic shell. 8 Conclusions In this paper we presented a formulation of the IB Method starting from the separation of the Cauchy stress in the principle of virtual work in two components, that we identified as the fluid-like and elastic-like components. The idea behind the IB method is to treat each of these two components in its natural framework, i.e. the Lagrangian one for the elastic behavior of the material and the Eulerian one for its fluid behavior. The advantage of this approach with respect to the more traditional one introduced in [20], lies in the fact that this formulation fits naturally into classical continuum mechanics theories (see, for example, [10]) and permits to adopt the IB method in a natural way for the study of the interaction between viscous hyper-elastic materials described by their Piola-Kirchhoff stress formulation, and viscous flows, described by the standard Navier-Stokes equations. We highlighted how the original IB formulation is inadequate for this general elasticity model in the case of thick materials, due to the lack of a transmission term that imposes the continuity of the total stress along the boundary of the material itself. As a side note, the additional term that appears in the formulation has the 34