 Research Article
 Open Access
Numerical simulation of solid deformation driven by creeping flow using an immersed finite element method
 Thomas Rüberg^{1, 2}Email author and
 José Manuel Garcí Aznar^{1}
https://doi.org/10.1186/s4032301600610
© Rüberg and Aznar. 2016
 Received: 17 December 2015
 Accepted: 12 February 2016
 Published: 15 March 2016
Abstract
An immersed finite element method for solid–fluid interaction is presented with application focus on highly deformable elastic bodies in a Stokes flow environment. The method is based on a global balance equation which combines the solid and fluid momentum balances, the fluid mass balance and, in weak form, the interface conditions. By means of an Updated Lagrangian description for finite elasticity, only one analysis mesh is used, where the solid particles are backtracked in order to preserve the deformation history. The method results in a full coupling of the solidfluid system which is solved by an exact Newton method. The location of the material interface is captured by a signed distance function and updated according to the computed displacement increments and the help of an explicit surface parameterisation; no bodyfitted volume meshes are needed. Special emphasis is placed on the accurate integration of finite elements traversed by the interface and the related numerical stability of the shape function basis. A number of applications for compressible NeoHookean solids subject to creeping flow are presented, motivated by microfluidic experimentation in mechanobiology.
Keywords
 Immersed finite elements
 Fluid–solid interaction
 Updated Lagrangian method
 Computational mechanobiology
 Nitsche’s method
Background
In a large class of biological and biomedical problems, the interaction of fluid flow and highly deformable solids plays an important role. Consider, as examples, the biological response of cells to a mechanical stimulus [1], the flow of capsules in a narrowed tube [2], the deformation of thinwalled blood vessels [3], or the motion of red blood cells [4]. Not only the mechanical response of cells and tissues in a fluid environment is of great interest, but also the biological implications of the mechanical environment (that is, mechanobiology). It is, for instance, suggested that fluid shear stresses can control the phenotype of a living cell: to regulate the cell’s differentiation or proliferation, or simply damage it, see for example [5] for endothelial cells. To this end, microfluidic experimentation [6] has proven a powerful tool for the in vitro analysis of such phenomena because it allows to precisely control the mechanical and chemical environment of individual cells or aggregates thereof. Nevertheless, this methodology is expensive and timeconsuming. Therefore, computational mechanobiology [7] often proves to be a promising alternative to assist the experimentation. For the considered field of applications, inertia forces are negligible (flow with very low Reynolds numbers, \( Re \ll 1\)) and the ratio of elastic to viscous stresses is by orders of magnitude smaller than in systems akin to aeroelasticity (for example, heaving wings [8] and insect flight [9]). For the modelling of living cells in a continuum mechanical framework, the most common approaches are liquidfilled, elastic membranes or (visco)elastic solids. Both descriptions have their merits and both can be supported or rejected by experiments [10]. In this work, we consider a solid domain occupied by a homogeneous compressible NeoHookean material that is subject to Stokes flow. This class of problems yields an initial, but generic approach to above range of applications with the potential of future extension.
Over the last decades a large quantity of methods has been developed for the numerical analysis of fluidstructure interaction problems (see, for instance, the journal issues [11–13] dedicated entirely to this topic, or the monograph [14]). Broadly speaking, these can be classified into monolithic methods, e.g. [15], and iterative coupling approaches, e.g. [16, 17]. Whereas in the former class of methods a fully coupled system of equations is formulated, in the latter class the equations of fluid and solid domains are solved independently and provide the boundary conditions (velocities or forces) to the respectively other domain. Although this iterative approach bears the potential of higher numerical efficiency and software modularity, the stability of the iteration process often requires severe restrictions of the size of the time step or subiterations [18]. In order to avoid this intricacy, the approach presented here is monolithic.
Another important classification is to divide the methods into those using bodyfitted meshes and their counterpart, immersed methods [19]. A bodyfitted mesh conforms to the geometry of the problem, this means that the interface between solid and fluid is solved and tracked by the mesh [20], see also [21] for spacetime formulation. The advantage of this approach is a simplified implementation of interface conditions and has been successfully applied to a wide range of applications: biomedical (for instance, airways and arteries [3, 22]), parachutes [23, 24], or the isogeometric analysis of wind turbines [25]. Nevertheless, the use of bodyfitted meshes has the downside that large solid deformations can lead to a virtual destruction of the analysis mesh. Even though this problem is addressed by the ALE formalism [26], there are situations where a bodyfitted mesh cannot be maintained. Tedious remeshing and solution mapping techniques are often the consequence of the severe mesh distortions and are avoided by the immersed techniques. The most prominent approach with nonbodyfitted meshes is the immersed boundary method [27] and tailored towards elastic surfaces subject to a flow environment. It has been successfully applied to simulations of blood flow and the deformation of red blood cells [28]. Moreover, a finite element counterpart has been developed in [29]. Despite its great success, the numerical stability of the immersed boundary method is subject to severe restrictions of the time step [27] due to its mainly explicit character. Apart from the mentioned techniques which rely domain discretisations, the use of boundary integral equations is a popular alternative [30]. Due to the surfaceonly description of the flow field, the numerical cost can be reduced significantly in this class of methods. See, for instance, the simulation of vesicle flow [31, 32] where an unbounded fluid domain and liquidfilled membranes are used as model problem. Nevertheless, there are restrictions on the use of boundary integral equations: they rely on a homogeneous Stokes flow and any extension to nonlinear flow behaviour is not straightforward; moreover, the robust implementation of singular integral equations is a demanding task. The application target of the aforementioned methods is mostly the interaction between a fluid and a reduced solid, such as a membrane or a shell. Here we focus on voluminous solids instead as, for instance, in [33, 34], where an explicit finite difference method is presented for this problem class.
The here presented approach falls into the class of finite element methods with immersed boundaries and the spatial discretisation does not conform with the location of the solidfluid interface. Henceforth, we refer to our method as an immersed finite element method even though the term has already been employed in [29] (see also, [35]) for a finite element analogy to the immersed boundary method of [27], which basically consists of an explicit coupling scheme where structural forces are imposed on the fluid mesh and the computed velocities modify the structural configuration. The method presented in [33] and typically the immersed boundary method do not sharply resolve the interface but work with a transition zone between the two media that depends on the grid resolution. The method proposed here works with a socalled sharp interface representation [19] and is based on the immersed bspline finite element method proposed in [36]; see also [37, 38] for finite element methods with embedded interfaces. Therein, the application to viscous fluid flow with moving boundaries has been targeted and later carried over to the partitioned analysis of fluidstructure interaction in [8]. In this latter reference, the structure is resolved by a Lagrangian mesh and the fluid with a Eulerian mesh in which an implicit geometry representation is superimposed in order to capture the interface. Here, this idea is pushed further by using only one analysis mesh for both solid and fluid, in the spirit of the methods developed in [39, 40], which are based on a fully Eulerian description, see also [41–43]. The drawback of fully Eulerian approaches for the solid is that displacement and velocity become independent fields that are coupled by an additional advection equation which needs to be solved [42]. In order to avoid this additional equation and have solid velocities as an extra field, we choose an Updated Lagrangian method [44] combined with particle tracking, as advertised in [45] for plasticity problems with large deformations. Moreover, this choice of expressing the solid equilibrium in the latest known configuration avoids the need of shape derivatives [46] in the linearisation process for the used Newton method. Under the assumption of small velocities, the time derivatives are completely neglected from the field equations and only appear through the interface condition between solid and fluid. This condition is weakly incorporated using Nitsche’s method [47], see also [8, 46] for this technique in the context of an iterative coupling method. A signed distance function is used to capture the location of the interface as a level set [48] for the field computation. At the end of every time step an explicit surface parameterisation is generated and moved with the computed displacement increment. Note that we combine explicit and implicit geometry descriptions; the implicit version is used for a quick interface detection at the beginning of each simulation step. Falling back to a temporary explicit description allows to accurately trace the interface location without the need of additional advection equations as common in level set methods [48]. We thereby avoid possible distortions of the surface mesh when using an entirely explicit surface description.
Several test applications demonstrate the potential of the method. First, the convergence behaviours of the updated Lagrangian method for a solidonly problem and a fully coupled fluidsolid interaction problem are studied. Next, the cases of a solid subject to shear flow, the flow through a constricted pipe section and driven cavity flow are analysed numerically. Based on the computed field data, several quantities of interest are studied, such as the shape variation of the immersed solid, particle motion, Cauchy stresses in fluid and solid, and fluid velocity patterns.
The outline of this article is as follows. In “Fluid–solid interaction” section, the basic balance equations of solid and fluid are presented and a global solid–fluid balance equation that incorporates the interface conditions weakly is derived. Other than customary, we first discretise this expression in time and linearise it before introducing the spatial discretisation in “Immersed finite element method” section. More specifically, in “Updated Lagrangian method and linearisation” section the linearisation is based on an Updated Lagrangian formalism that avoids the need of shape derivatives. Regarding the immersed finite element discretisation, a special emphasis is placed on the integration and numerical stability related to the elements which are traversed by the interface in “Cut elements” section. Moreover, the displacement history is maintained with a particle tracking algorithm presented in “Interface update and particle tracking” section. Finally, in “Example applications” section several example applications are presented which demonstrate the potential of the new method.
Fluid–solid interaction
We begin by deriving our coupling formulation from the balances of momentum and mass of the solid and the fluid parts, respectively. Here, we restrict ourselves to the case of very low Reynolds numbers such that inertia terms and fluid advection can be safely discarded. This simplification is justified by the fact that viscous forces are significantly larger than inertial forces [49].
Static balance laws
Here, \(\varvec{\sigma }^s\) and \(\varvec{\sigma }^f\) are the Cauchy stresses in the solid and the fluid, respectively, and we have the primal field variables solid displacement \(\varvec{d}\), fluid velocity \(\varvec{u}\) and fluid pressure p; all functions of the spatial coordinate \(\varvec{x}\) and time t. The spatial divergence operation is symbolised by \(\nabla \cdot ()\). Equations (1) and (2) are the quasistatic momentum balances in solid and fluid, and (3) is the balance of mass for an incompressible fluid.
Equations (1), (2), (3) together with the material laws, (4) and (5), and the interface and boundary conditions, (6) and (7), completely describe the problem. Note that even though we work with the static balance laws (neglected inertia), the problem is timedependent due to the change of the configurations: \(\Omega ^s(t)\) and \(\Omega (t)\) are functions of \(\varvec{d}\) and thus of time. Due to this dependency, the problem is nonlinear in addition to the nonlinearity given by the solid stresses (4). Moreover, the first condition in (6) couples solid with fluid velocities and thus renders the solid subproblem timedependent.
Coupling formulation
Here, a global balance law for fluid and solid is derived which does not rely on essential boundary conditions. This means that the interface conditions (6) are not directly fulfilled by the choice of the finite element basis, which will be later introduced in Section “Immersed finite element method”. Its main advantage is to allow for a configuration of solid and fluid that changes independently of the used finite element mesh. Nevertheless, the conditions imposed on \(\partial \Omega \) for the outer fluid boundary are treated in the classic way by imposing \(\varvec{u}= \bar{\varvec{u}}\) essentially and \(\varvec{\sigma }^{f} \varvec{n}= \varvec{0}\) naturally. Since this boundary does not move, there is no disadvantage of this standard approach.
Time semidiscretisation
The fluid–solid balance (12) as derived in the previous section is timedependent and nonlinear. Typically, such expressions are discretised by following the concept of the method of lines: the discretisation in space leads to a system of (nonlinear) ODEs which is than tackled by a time discretisation. Here, we reverse this order and make use of what is referred to as Rothe’s method [56]. The aim of this work is to use a fixed, stationary finite element mesh in which the fluidsolid domain configuration moves freely. Nevertheless the finite element spaces vary in function of this configuration and, for this reason, it is preferred to begin with time discretisation and linearisation before finally applying a spatial discretisation with in finite elements as outlined in “Immersed finite element method” section.
Updated Lagrangian method and linearisation
At first, consider the three solid domain configurations occurring in the Updated Lagrangian Method: the initial configuration \(\Omega ^s_0\), the latest known configuration \(\Omega ^s_n\) and the unknown configuration \(\Omega ^s_{n+1}\), see Fig. 2. Coordinates in these configurations are denoted with the same subscript, for instance, \(\varvec{x}_n \in \Omega _n\). The maps are between two configurations, \(\Omega _a\) and \(\Omega _b\) are denoted by \(\varvec{\varphi }_a^b: \Omega _a \rightarrow \Omega _b\). For instance, the coordinate \(\varvec{x}_{n+1}\) results from either mapping from the initial or the latest known configuration, i.e., \(\varvec{x}_{n+1} = \varvec{\varphi }^{n+1}_0( \varvec{x}_0 ) = \varvec{\varphi }^{n+1}_n( \varvec{x}_n )\). At last, the deformation gradients are defined as \(\varvec{F}_a^b = \partial \varvec{x}_b / \partial \varvec{x}_a\) and are maps between the respective tangent spaces. We refer to [44, 50, 53] for more details on large elastic deformations and the notion of configurations.
Immersed finite element method
The global fluid–solid balance equation (12) and its timediscretised and linearised version (26) are perfectly suited for an immersed finite element method [19]. Note that here immersed solely refers to the fact that the interface location is independent of the finite element mesh and we do not refer to the method of [29]. Observe that Dirichlet boundary conditions only appear on the boundary \(\partial \Omega \) which is fixed in space and time, and that the interface conditions (6) have been incorporated in a weak form. This implies that the finite element spaces need only be equipped with essential boundary conditions on \(\partial \Omega \) but are not affected by the specific location of the interface \(\Gamma \).
Figure 3 shows the configuration of Fig. 1 with an immersed finite element discretisation. A simple structured grid is filling the domain \(\Omega \) without awareness of the current location of \(\Gamma _n\). More precisely, the figure shows the respective discretisation of fluid and solid in the left and right pictures. Obviously, the elements which are traversed by the interface \(\Gamma _n\) have a physical and a fictitious side such that the solution is welldefined until the boundary. These fictitious element parts are highlighted in Fig. 3.
Implicit geometry representation
Cut elements
After the calculation of the intersection points, piecewise linear elements form the surrogate interface \(\Gamma _n^h\) in the solution process. Effectively all terms that contain integrals over \(\Gamma _n\) are expressed as integrals over \(\Gamma _n^h\). In addition, the subregions of the cut element which belong to the fluid (\(\Omega _n^f\)) and the solid (\(\Omega _n^s\)) side are now polygons or polytopes for which a standard numerical integration is in general not possible. Therefore, these shapes are subdivided into triangles or tetrahedrons on which Gauß quadrature rules [57, 59] are used, as it is common in the implementation of XFEM [60]. The integration on the cut elements is therefore carried out with the same accuracy as the volume elements strictly inside the domain. See [61] for an overview of the numerical implementation of integration over cut elements and an alternative approach based on the divergence theorem and surface integration. Alternatively, in [62] an approach for explicit time integration is given in which a surrogate boundary is instead thereby avoiding the cut element integration.
At last, the stability of the finite element basis used in (27) needs to be considered. Consider Fig. 3 and let us assume that all grid points hold degrees of freedom \(\varvec{d}_{n,i}\), \(\varvec{u}_{n,j}\), and \(p_{n,k}\). In fact, the quadratic shape functions used for the fluid velocity \(\varvec{u}\) lead to additional degrees of freedom, but they are ignored in this discussion for sake of simplicity. These degrees of freedom can be classified by the intersection of their support with the domain of integration. Let \(N_i(\varvec{x})\) be any shape function of (27) and \(S_i = {{\mathrm{supp}}}(N_i)\) its support, that is \(N_i(\varvec{x}) = 0\) for all \(\varvec{x}\notin S_i\). Focusing on the discretisation of the solid displacements (the fluid side is treated analogously), the degrees of freedom are classified as follows: inactive if \(S_i \cap \Omega _n^s = \emptyset \); critical if \(S_i \cap \Omega _n^s \ne \emptyset \) and \(S_i \cap \Omega _n^s < \epsilon h\), using some predefined threshold \(\epsilon \); active otherwise. Note that this categorisation is dependent on the time instant \(t_n\) and therefore the used finite element spaces change between the time instants.
Interface update and particle tracking

Extract a surface mesh \(\tilde{\Gamma }_n\) from the current level set data \({{\mathrm{dist}}}_{\Gamma _n}\) (Eq. (28) and the construction in Fig. 4) which delivers an explicit description of the interface

Evaluate at every node \(\varvec{x}_i^\Gamma \) of this surface mesh the current displacement increment and update the coordinate of that node.$$\begin{aligned} \varvec{x}^\Gamma _i \leftarrow \varvec{x}^\Gamma _i + (\varvec{d}_{n+1}  \varvec{d}_n)(\varvec{x}^\Gamma _i). \end{aligned}$$(36)

The thereby updated surface mesh becomes \(\Gamma _{n+1}\) and is used for the the signed distance function (29) in the next step.
Fluid–solid coupling
 Geometry immersion :

Compute the signed distance function (28) based on the given surface mesh \(\Gamma _n\).
 Fluid–solid balance :

Solve problem (12) with the interface term (16) by a Newton method as shown in expression (26). After convergence, the new solid displacement \(\varvec{d}_{n+1}\) and the fluid state variables \(\varvec{u}_{n+1}\) and \(p_{n+1}\) are known. The spatial discretisation is performed as described in this section.
 Geometry update :

Extract a surface mesh \(\tilde{\Gamma }_n\) from the signed distance function. Update the node locations of this mesh based on the displacement increment \((\varvec{d}_{n+1}  \varvec{d}_n)\) taking into account the rescaling as explained above. This yields the new interface location \(\Gamma _{n+1}\) and implies the solid and fluid domain locations.
 Backtrack nodal locations :

For every finite element node in the new solid domain \(\Omega _{n+1}^s\), find its previous location in \(\Omega ^s_n\). Transfer the field variables from the previous to the new location.
Example applications
For the finite element method, we use lowestorder TaylorHood elements (\(Q_2/Q_1\)) for the fluid (velocity/pressure) and \(Q_1\) elements for the solid displacement [57]. An Euler backward time integration is used with constant step size \(\Delta t\). The coupling parameter \(\gamma \) is chosen as \(\gamma = \alpha \mu ^f / h\), where h is the characteristic element size, see Eq (35). If not indicated otherwise, \(\alpha =1\) is the default choice. The convergence criterion for the Newton method (26) is the \(\ell ^2\) norm of the vector representing the displacement increment \(\Delta \varvec{d}\) divided by the number of solid degrees of freedom. The tolerance in all computations is chosen as \(10^{10}\) and at most three iterations are observed. The same tolerance is used for the Newton method (40) in the coordinate backtracking. Note that in the examples the effective mesh size can be arbitrarily small, while the interface moves through the finite element mesh. But due to the stabilisation, as presented in “Cut elements” section, the size of the intersection of the shape function support with the integration domain does not shrink to zero but remain of the order of h. For this reason, the use of the element size h of the embedding domain grid is a valid mesh characteristic.
Before focusing on the fluid–solid applications, a pure solid example is considered. The aim of this example is to demonstrate the viability and accuracy of the devised Updated Lagrangian method for the solid domain based on the immersed finite element method with particle backtracking.
Updated Lagrangian method
In a first, preliminary example, the accuracy of the updated Lagrangian method, as introduced in section “Updated Lagrangian method and linearisation”, together with the particle backtracking of “Interface update and particle tracking section” is assessed. Therefore, a problem only consisting of a solid domain without fluid interaction is devised. The undeformed geometry of the solid domain is depicted in the left picture of Fig. 7. Other than in the derivation of the presented fluid–solid coupling method and the remaining examples, the boundary of the solid domain \(\Omega ^s\) partially overlaps with the boundary \(\partial \Omega \) of the embedding domain. Along this overlap the displacements are set to zero by employing essential boundary conditions. Moreover, the initial solid domain \(\Omega ^s_0\) has subdomain \(\tilde{\Omega }^s_{0}\) in which a constant downward body force \(\varvec{f}_0 =  \tfrac{n}{2} \varvec{e}_2\) is applied with \(0\le n \le 4\) as the number of the load step. The region of the applied body force is darker in the picture. The solid is hyperelastic according to (41) with parameters \(E=100\) and \(\nu =0.3\).
For a more quantitative comparison, the right graph in Fig. 7 shows the modulus of the measured displacement \(\varvec{u}^*\) for the two approaches and various mesh sizes. Clearly, both approaches converge to very similar numerical values of the measured displacement. Note that the presented method for the solid part is tailored towards the analysis large deformation problems. In a linearised setting, the distinction between the configurations as shown in Fig. 2 does not make sense and there would be no need for the update of a configuration. For this reason, analytic solutions for the convergence study are highly complicated and we therefore rely on numerical reference solutions. Moreover, the methods we compare in this section operate in different configurations such that we are restricted to compare the results at individual points.
Shear flow–convergence analysis
To this end, we consider the problem as described in Fig. 9, a fully immersed hyperelastic solid domain \(\Omega ^s\) of depicted shape whose material points in the dark red circle are held fixed. The radius of the upper and lower circular boundaries of the solid is 0.15, whereas the circle of fixed material points is half as big with a radius of 0.075. The distance between the centres of the circular boundaries is 0.5 and the whole fluid box has the dimensions \(1.5 \times 1.5\). The surrounding fluid box is subject to a prescribed shear flow and leads to a bending of the solid body. The fluid is Newtonian with viscosity \(\mu ^f=1\) and the parameters for the hyperelastic material as in (41) are \(E=1000\) and \(\nu =0.3\). For the simulation 10 time steps with a step size \(\Delta t = 0.2\) are used which yield the deformed shape as shown in the middle picture of Fig. 9 and which corresponds to a quasistatic state.
As reference solution a fine grid of \(480\times 480\) elements, i.e. \(h=0.003125\), is used. The right graphic in Fig. 9 shows the \(L_2\)norm errors of the distance function \({{\mathrm{dist}}}_\Gamma \) and the velocity field \(\varvec{u}\) for various grid sizes h with respect to the chosen reference at the final step of the simulation at \(t=2\). One can see the both measured errors behave similarly. They have an approximately linear decay for coarse grids followed by a order higher than linear. The final order, which appears to be more than quadratic, is clearly owed to the choice of reference solution and does not claim to be a characteristic of the method. Due to the choice of the time stepping method, see (17), the convergence order is impeded and does not reach the quadratic behaviour as linear finite elements for linear static problems commonly exhibit [57]. Moreover, one has to bear in mind that the solution of nonlinear systems in every time step by the Newton method (26) and the particle backtracking (40) contribute to the overall error of the method.
Shear flow–parameter study
Parameters used in twodimensional shear flow problem
Parameter  Standard value  Variations 

N  50  25, 75, 100, 200 
\(\Delta t\)  0.1  0.05, 0.2, 0.5 
\(\alpha \)  1  0.01, 0.1, 10, 100 
E  50  2, 5, 10, 20 
\(\nu \)  0.3  — 
\(\mu ^f\)  1  0.1, 0.5, 2, 4 
f  1  0.2, 0.5, 2, 5, 10 
R  0.6  0.2, 0.4, 0.8, 1.0 
L  1.4  1.0, 1.8, 2.4, 3.0 
At first the method parameters N, \(\Delta t\) and \(\alpha \) are considered. It turned out that the multiplier \(\alpha \) of the boundary term as in the paragraph leading to Equation (35) did not show any noticeable influence on the monitored quantities, and is thus omitted from the rest of the discussion. The left picture of Fig. 11 shows the eccentricity e plotted over the time of analysis \(0 \le t \le 20\) for the variation of N, the number of elements per direction. One can clearly see that there is a fluctuation of the values throughout time and the amplitude of this fluctuation diminishes with increasing values of N. This becomes more clear, when looking at the timeaveraged values of e and\(\theta \) in the right picture. Here \(\bar{a}\) denotes the average of the quantity a over the time interval \(5 \le 20\). The values of the average angle and the average eccentricity tend towards a specific value with increasing N, see Fig. 11. Similar observations are made for the time step variations.
Finally, we consider for the softest solid with \(E=2\) the variation of the stress components \(\sigma _{ij}\) in the fluid and the solid domains at some time instants. Figure 16 displays contour plots of the stress components for four different times. It has to be emphasises the plotted stress components refer to the Cartesian coordinate axis and not the principal axes of the deformed solid. Note that there are strong indications that stresses regulate substantial biological processes in living cells [5]. The proposed method allows for future applications in which a detailed stress analysis is required for a deeper insight in such processes.
Flow in a narrowed tube
Parameter variations in constricted pipe flow
Parameter  Value  Variation 

\(\delta \)  0.33  0.25, 0.3, 0.4 
E  100  50, 200, 500 
Above observations are quantified in Figs. 21 , 22, where the enclosed area A of the solid body and the velocity are plotted versus the current position and for all parameter variations. The initial shrinkage due to the fluid pressure is clear visible in Fig. 21 and more pronounced for lower values of E (softer material) and lower values of \(\delta \) (higher fluid pressure). The area then stays constant while the solid travels towards the narrower section. When entering the transition region it begins to shrink more until reaching a minimum size when approximately entering the narrow section. The travelled distance is measured by the location of the centroid of the solid body with respect to its initial location. When inside the narrow section, the solid gets stretched and increases in area.
The velocity of the solid body’s centroid is shown in Fig. 22 for all parameter choices. At the begin of the simulation the solid catches the velocity of the surrounding fluid and this increases when approaching the transition to the narrower section. Once entered this final part, the velocity stays approximately constant.
Finally, in Fig. 23 for a few parameter combinations the velocity streamlines are shown together with the fluid pressure and the surface traction \(\varvec{\sigma }\varvec{n}\). For the computation of the streamlines, the discrete velocity according to (17) has been used inside the solid domain. Clearly, the flow pattern do not differ significantly among the displayed images. But the fluid pressure is higher in case of a larger solid stiffness E as it is necessary in order to sufficiently deform the solid body. In case of a smaller pipe diameter the fluid pressures are obviously larger. Accordingly, the distribution of the surface traction becomes higher for larger values of E and smaller values of \(\delta \).
This example is concluded by a comment on the advantage of immersed finite elements. The most common technique for the fullycoupled analysis of fluid–solid interaction is the Arbitrary LagrangianEulerian (ALE) technique [26] in which the fluid mesh is deformed in order to accommodate for the solid deformation and to maintain a usable analysis mesh. Although a powerful method, it is expected that the here presented example is not directly accessible by an ALE method. Figs. 18, 19, 20 clearly show that the initial fluid mesh would be highly distorted when the solid body enters the narrow section of the pipe. Tedious remeshing and solution mapping techniques are required in an ALE approach.
Threedimensional shear flow
Figure 24 displays the initial configuration and the final deformed shapes for the two chosen material parameters. As in the twodimensional case, the solid assumes an elliptic shape that is inclined in the plane of the shear flow. In Fig. 25, the principal moments of inertia, computed from the \(3\times 3\)matrix with coefficients (43), are shown for a time interval \(0 \le t \le 2.5\) and the two choices of E. The spheres flatten as discussed above and rotate about the axis perpendicular to the plane of shear by approximate angles \(0.20\pi \) for \(E=10\) and \(0.17\pi \) for \(E=5\). As seen in Fig. 25, the moment of inertia around this outofplane axis stays almost constant throughout the simulation and is similar for both stiffnesses. The other two moments clearly deviate from their initial values as the elliptic shape in the plane of shear is formed.
Conclusion
A new approach for the numerical analysis of the interaction between viscous fluid flow and highly deformable solids has been presented. The method builds up on previous works, such as [8, 36] for the analysis of fluid flow around moving and highly flexible boundaries. Derived from the basic balance equations for the quasistatic equilibrium of solid and fluid, the interface conditions are incorporated weakly and a global solidfluid balance law is obtained. The formalism of an updated Lagrangian method is used for the description of the solid constituent. Its equilibrium is thus expressed in the latest known configuration and the deformation history maintained by particle tracking between the newly computed and the previous state of the solid. By means of this choice, the advection equations and the shape derivatives in the linearisation of a fully Eulerian method are avoided. The analysis is carried out on one mesh which stays fixed in space and time.
For the spatial discretisation an immersed finite element method is employed with the mesh independent of the location of the solid–fluid configuration. By using a signed distance function, the location of the interface is given implicitly to the finite element solver. The difficulty of an accurate quadrature of the elements which are crossed by the interface is as much addressed as the possible illconditioning of the system of equations due to the small support of shape functions on such elements. Once the nonlinear iterations of the fully coupled fluid–solid system are converged and the new equilibrium has been found, the configuration is updated. To this end, an explicit representation of the surface is recovered from the level set and this is updated by means of the displacement increments along the surface. Finally, every mesh node of the solid domain is tracked back to its previous location in order to transfer the displacement history.
Due to the choice of a monolithic fluid–solid coupling, the method is unconditionally stable. Moreover, the full linearisation in the Updated Lagrangian framework leads to a fast convergence within every time step. Several example applications in two and three spatial dimensions are presented and the influence of all parameters of the method are studied. Being tailored towards the analysis of cell motility and microfluidic experimentation, we consider shear flow examples which reveal the basic characteristics of liquidfilled vesicles, such as tumbling and tanktreading behaviour. Moreover, the passing of a deformable object through a narrowed tube of diameter smaller than the body is analysed and the trajectories of an elastic solid in the vortex of a driven cavity flow. Especially, the former application provides an important step in the direction of computational analysis of cell migration in confined spaces [67].
In view of these examples, we highlight the following features of the devised method. Based on simple balance laws, virtually any material law of solid and fluid constituents can be incorporated. Especially, active behaviour, like growth or selfpropulsion, is a feasible extension. Moreover, the restriction to a quasistatic equilibrium is not essential and an adaption of the method to fully dynamic solid–fluid interaction is straightforward. The developed immersed finite element method operates with a single analysis mesh for solid and fluid that is not subject to any deformation. This is particularly useful for the analysis of the narrowed tube and the driven cavity examples in which the use of a bodyfitted mesh is not possible in any standard way. Finally, we aim to emphasise that the here presented approach is virtually meshfree: even though a volume finite element mesh is employed it is not subject to any geometrical restrictions and its generation is a trivial task.
Declarations
Authors' contributions
TR derived the mathematical model for the immersed FE method and carried out the numerical implementation. JMGA defined the conception of the underlying physical models. Both authors were fully involved in the preparation of this manuscript and the interpretation of the results. All authors read and approved the final manuscript.
Acknowledgements
The support of the European Research Council (ERC), through project ERC2012StG 306751, and of the Spanish Ministry of Economy and Competitiveness, through project DPI201238090C0301 (partly financed by the European Union through the European Regional Development Fund), is gratefully acknowledged.
Competing interests
The authors declare that they have no competing interests.
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