A monolithic approach to fluid-structure interaction based on a hybrid Eulerian-ALE fluid domain decomposition involving cut elements

A novel method for complex fluid-structure interaction (FSI) involving large structural deformation and motion is proposed. The new approach is based on a hybrid fluid formulation that combines the advantages of purely Eulerian (fixed-grid) and Arbitrary-Lagrangian-Eulerian (ALE moving mesh) formulations in the context of FSI. The structure - as commonly given in Lagrangian description - is surrounded by a fine resolved layer of fluid elements based on an ALE-framework. This ALE-fluid patch, which is embedded in an Eulerian background fluid domain, follows the deformation and motion of the structural interface. This approximation technique is not limited to Finite Element Methods, but can can also be realized within other frameworks like Finite Volume or Discontinuous Galerkin Methods. In this work, the surface coupling between the two disjoint fluid subdomains is imposed weakly using a stabilized Nitsche's technique in a Cut Finite Element Method (CutFEM) framework. At the fluid-solid interface, standard weak coupling of node-matching or non-matching finite element approximations can be utilized. As the fluid subdomains can be meshed independently, a sufficient mesh quality in the vicinity of the common fluid-structure interface can be assured. To our knowledge the proposed method is the only method (despite some overlapping domain decomposition approaches that suffer from other issues) that allows for capturing boundary layers and flow detachment via appropriate grids around largely moving and deforming bodies and is able to do this e.g. without the necessity of costly remeshing procedures. In addition it might also help to safe computational costs as now background grids can be much coarser. Various FSI-cases of rising complexity conclude the work.

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07/30/2018

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1 Introduction

Fluid-structure interaction (FSI) problems involving large structural movements and deformations are of significant interest in various fields of engineering and applied sciences. However, an important prerequisite for achieving reliable results, especially for flows at higher Reynolds numbers, is an appropriate mesh resolution in the boundary layer. The latter is mandatory in order to capture the wall normal gradients around the wet structure surface accurately. An insufficient mesh quality at the fluid-structure interface likely results in an overall corrupted solution of the coupled problem.

The essential advantage of the established Arbitrary-Lagrangian-Eulerian (ALE)-based FSI-approach, which goes back to [1, 2, 3, 4, 5, 6], is that the mesh knows about the position of the structure within the fluid domain, such that for example the mesh can be refined towards the interface area. However, the pre-processing of appropriate high quality meshes that satisfy often extreme requirements in the boundary layer is difficult and time-consuming, and large structural motions can heavily distort the fluid mesh. While this might not be a billing argument along with some meshes, it becomes crucial for example along with boundary layer meshes that not only have extreme aspect ratios but are also placed in the region with the highest deformations and hence are very vunerable. Hence, costly remeshing and mesh-updating procedures have to be considered that again are particularly challenging, e.g. in connection with boundary layer meshes. In summary, also for such approaches optimality of a fluid mesh around the structure can often not be preserved.

The shortcoming of ALE based FSI schemes to deal with large and complex motions was the motivation for the development of an alternative class of FSI approaches, known as fixed-grid methods. Such methods sparked quite some interest in recent years. For an overview of some approaches, the reader shall be referred to [7]. Following a pure fixed-grid approach, the entire fluid domain is described in an Eulerian framework. Since structural mesh and fluid mesh are not required being fitted at the common interface, they seem particularly suitable for large deformation FSI [8, 9, 10]. But unlike in the classical ALE based FSI

approach, an a priori mesh refined around the wet surface can hardly be achieved. A rather straightforward solution would be a local, adaptive mesh refinement and coarsening combined with error estimator-based and/or heuristics-based refinement indicators, as described in, e.g.,

[11, 12]. Though, such an adaptive approach becomes rather inefficient for 3D problems involving large motion of the structural surface, since large-sized regions have to be refined for several levels and the mesh updates may have to be accomplished frequently throughout the simulation. Furthermore, common refinement algorithms operate in all spatial directions, which would destroy the inherent grading of boundary layer meshes towards the solid body. Other attempts to relax strong restrictions with regards to interface rotations have been suggested, e.g., in [13] based on sliding mesh techniques. Another interesting variant of sliding interface-fitted meshes is the so called shear-slip mesh update method introduced in [14] that reconnects nodes in the element layer next to the interface. An interesting method for such FSI problems utilizing an ALE formulation of embedded boundary methods was proposed by [15], where non-interface-fitted embedded meshes are rigidly translated and/or rotated to track the rigid component of the dynamic body motion.

A highly advantageous approach which drastically simplifies meshing around structures and perfectly suites for the creation of refinements in FSI-interface normal direction consists in utilizing domain decomposition for the fluid field. The idea of utilizing two independent overlapping fluid meshes allows to combine ALE and Eulerian based fluid techniques in the vicinity of the solid and the far-field, respectively. Chimera schemes are an example for an iterative coupling method based on an overlapping fluid decomposition, which were introduced originally for mesh generation and for the simulation of flows around rigid bodies (see for example [16, 17, 18]). An extension to problems including flexible structures has been presented in [8]. However, Chimera-like couplings have some drawbacks. In order to obtain a converged solution after iterating between the fluid domains, an overlapping zone of two subsequent fluid domains has to be present. This introduces an additional iteration set over the overlapping fluid grids in order to obtain the final fluid solution. Beside this additional cost, the overlapping domain has to be large enough to achieve a converged solution between the subdomains and this is again particularly cumbersome when highly refined boundary layer meshes should be coupled coarse background grids.

To overcome such shortcomings, a powerful technique consists in utilizing a composite of overlapping grids, where the solution in the background mesh is cut-off at the artificial fluid-fluid interface. The latter is defined as the trace of an embedded grid. This discretization technique is not limited to finite element based schemes, but can be realized in finite volume frameworks as well, even though FEM is chosen in the present work. The application of such a fluid discretization concept for FSI has been considered in a series of works [11, 7, 19, 20]. In an FSI setting, the structure is surrounded by a moving layer of fine ALE-fluid elements, which is then embedded into the fixed-grid Eulerian background fluid grid - motivating the designation hybrid Eulerian-ALE approach. While the structure moves and deforms, the boundary layer mesh follows the deformation of the structural surface - the near surface flow is captured appropriately. However, in order to apply such fluid patches in complex FSI problems, it is crucial to satisfy high demands on the coupling of the separate background and embedded fluid subdomains along the shared fluid-fluid interface. While classical Lagrange-multiplier based couplings show severe restrictions with regards to a reasonable choice of discrete function spaces and in particular requires a careful choice of the multiplier space, stabilized schemes are often more powerful. A stabilized stress-based Lagrange-multiplier method for coupling the fluid phases involving cut elements has been presented first in [19]. To overcome restricting limitations with regards to the location of the embedded fluid patch within the background mesh, a stable and optimal convergent Nitsche-based coupling method has been presented by [21]. The latter method is based on the Cut Finite Element Method (CutFEM) [22], which dates back to the eXtended Finite Element Method (see [23, 24, 25, 20, 26] for various flow applications). The fluid-fluid coupling is enforced weakly employing Nitsche’s formulation [27] supported by additional penalty-like stabilization techniques for cut elements - the face-/edge-oriented ghost penalty stabilizations [28, 29]. Advancements of these stabilization techniques, acting on the inter-element jumps of velocity and pressure normal derivatives of cut elements have been made by [30, 31, 32] for the incompressible Navier-Stokes equations. The stabilized embedded fluid formulation introduced in [21] is one prerequisite of our CutFEM based hybrid Eulerian-ALE FSI approach.

For the fluid-structure coupling, different monolithic coupling schemes are available and the coupling between the moving ALE-fluid domain and the structure can be handled in the same way as in traditional ALE based FSI

schemes, i.e.node match of fluid and solid mesh at the common interface. In the simplest case, common interface velocity degrees of freedom can be shared and continuity conditions can be incorporated strongly (see e.g. 

[33, 34]). A more flexible scheme has been proposed in [13], which allows for non-conforming non-overlapping meshes, where the interface conditions are enforced weakly utilizing a dual-mortar Lagrange multiplier method [35]. Over the past years, also Nitsche’s technique (see, e.g., [36]) has been discussed for FSI couplings with under-resolved boundary layer regions. While strong enforcements and therefore exact fulfillment of coupling conditions often result in oscillatory approximations of the boundary-layer solution (see discussions already for pure flow problems in [37, 38]), an automatic relaxation of these constraints is preferable, which, however, still converges to the exact fulfillment with mesh refinement in a consistent sense. Additionally introduced penalty parameters of Nitsche’s method have to be scaled properly in order to be independent of the flow regime and therefore the considered problem setup. As a further advantage of Nitsche’s method over Lagrange-multiplier methods, no additional new multiplier variables are introduced to the system of equations, which from an implementation point of view allows for an easier setup of the monolithic system and simplifies the design of efficient preconditioners.

Due to these reasons, also in this work, a Nitsche-based coupling at the fluid-solid interface is preferred, which can be setup similar to the fluid-fluid coupling. Such coupling techniques have been reviewed in detail in [39] in the context of unfitted CutFEM based FSI approaches and are the second prerequisite of our hybrid Eulerian-ALE FSI scheme.

Central focus of this paper is to highlight the flexibility of this hybrid FSI scheme for vast challenging FSI settings. Even though the fundamental idea of utilizing fluid domain decomposition for FSI has been presented already in previous works [19, 20, 21], to the best of the authors knowledge, its application to fully coupled time-dependent FSI problems has not been presented so far and just indicated in our previous work [39] as an outlook. In the latter publication and references therein, important theoretical and algorithmic ingredients have been already presented, and therefore will be reviewed just briefly for clarity in the present work. Moreover, some algorithmic peculiarities of the hybrid FSI approach will be elucidated. In addition, since a detailed presentation and investigation of more challenging numerical simulations was still missing so far, this is another focus of this publication.

The present paper is organized as follows: In Section 2, we briefly discuss the limitations of discretization concepts for FSI existing so far and propose our hybrid domain decomposition idea including the governing equations for the coupled FSI problem in its strong form. In Section 3, we propose one potential spatial discretization technique. It is based on a CutFEM fluid domain decomposition method and utilizes a Nitsche-type coupling of the fields at the fluid-solid and the fluid-fluid interface, respectively. A semi-discrete stabilized form for the hybrid Eulerian-ALE FSI problem is presented and algorithmic steps for the monolithic solution of the coupled hybrid FSI system are discussed. We demonstrate several numerical examples of increasing complexity in order to verify our method and highlight the capability and the potential of our approach in Section 4. Finally, conclusions are drawn in Section 5.

2 A hybrid Eulerian-ALE fluid-structure interaction approach

2.1 The hybrid domain decomposition idea for fluid-structure interaction

Fluid-structure interaction belongs to the large class of surface-coupled problems. A classical FSI problem consists of two disjoint bulk subdomains, one for the flow  and one for the structure  such that . The different phases interact at the common fluid-structure interface , at which the respective fields are constraint by coupling conditions. In addition, Dirichlet and Neumann boundary conditions for the involved fields need to be imposed at outer boundaries , respectively, to complete the FSI problem, see Figure 1.

Figure 1: FSI problem settings: domains, interface and boundaries.

Common discrete approximations of the structural field use boundary-fitted meshes , whose boundaries fit to the domain at all times . The structural kinematics are then described in a Lagrangian formalism. Discrete approximations concepts of the coupled FSI problem usually differ in the approximation of the flow domain and the respective fields. Most common techniques will be briefly reviewed and discussed in the following. Afterwards, as the last concept, we introduce the hybrid FSI approach.

Classical Arbitrary-Lagrangian-Eulerian (ALE) flow description.

Following an ALE based FSI approach, the fluid subdomain is approximated with a single ALE fluid mesh . The latter is interface-fitted to the wet structural surface. When the structural body moves, the ALE mesh also deforms and as its boundary follows the fluid-solid interface over time. An introduction to the ALE concept can be found, for instance, in the textbook [40].

The classical ALE based approach for FSI captivates through its simplicity and thus is the state-of-the-art in the approximation of FSI

settings. It allows to easily obtain higher-order geometric approximations using isoparametric concepts and the resulting schemes gain from well-established stability and best-approximation properties for the involved partial differential equations modeling continuum mechanics.

Nevertheless, for complex three-dimensional domains, generating high quality computational fluid grids that conform to the domain boundary and are suitable for capturing boundary layers arising for high Reynolds-number flows can be often time-consuming and difficult. In particular, if large structural motions and deformations are present, the quality of moving meshes cannot be guaranteed in general. As the finite elements need to follow the interface in its evolution, the meshes can rapidly distort. Then time consuming remeshing and projection steps have to be performed regularly. A sketch of ALE based approximations of the FSI-problem is given in Figure 4LABEL:sub@fig:computational-mesh-ale-fsi.

Fixed-grid Eulerian flow description.

In contrast to matching-mesh ALE based methods, pure Eulerian-based fixed-grid flow formulations are more flexible. For non-interface-fitted approximations of the flow domain, the solution to the problem is computed only on the active part . As fluid and solid meshes do not necessarily match at the interface , but may overlap, i.e. , such techniques may drastically simplify meshing of the computational domain and can overcome the shortcomings of interface-fitted meshes with regards to large domain motions and deformations. Therefore, such schemes are much more flexible. A visualization is given in Figure 4LABEL:sub@fig:computational-mesh-xfsi.

Nevertheless, in contrast to ALE based boundary-fitted mesh techniques, special measures are required to impose the interfacial constraints, while preserving robustness, stability and accuracy of the resulting numerical scheme becomes more challenging when intersecting grids. As a major drawback of fixed-grid schemes in FSI, sufficient mesh resolution in the vicinity of the boundary layer can be only hardly achieved at reasonable computational costs, since the location of the solid is usually unknown a priori. This, however, is a prerequisite for the quality of the coupled FSI solution approximation.

(a) Classical ALE based FSI
(b) Fixed-grid FSI
Figure 4: Different moving domain approximation techniques for FSI: LABEL:sub@fig:computational-mesh-ale-fsi Classical ALE based moving mesh methods are subjected to strict limitations regarding interface motion and deformation, otherwise the fluid mesh will distort. LABEL:sub@fig:computational-mesh-xfsi Fixed-grid schemes allow for arbitrary motions of the structural body, however, lack a sufficient resolution of the boundary layer in the vicinity of the FSI interface.

A hybrid Eulerian-ALE approach for Fsi.

In our novel hybrid FSI approach, the advantages of the classical moving mesh Arbitrary-Lagrangian-Eulerian (ALE) flow description are combined with that of a pure fixed-grid Eulerian flow description, as will be elaborated subsequently.

In this approach, the whole physical fluid domain is artificially separated into two disjoint domain parts and , i.e. , which are approximated independently by two overlapping fluid meshes and , as visualized in Figure 5. To benefit from the fixed-grid schemes with regards to the treatment of large structural motions, for the flow field which is far from the fluid-solid interface , usually a coarser fixed-grid Eulerian approximation is utilized. Since, coupled FSI problems require a fine-resolved approximation of wall-normal gradients in high-Reynolds-number flows to accurately capture interfacial forces, a fluid patch surrounding the solid body overlaps with the background fluid mesh in a geometrically unfitted way. At the fluid-solid interface , structural mesh and embedded fluid mesh  are chosen interface-fitted and potentially even node-matching. This fitting is preserved for all solid locations, requiring the fluid patch following the structural body in its motion and deformation. This is realized by the use of an overlapping mesh fluid domain decomposition, where can be embedded arbitrarily into . Then, the solid body and its surrounding boundary layer patch can largely move and deform within the background fluid mesh. In doing so, the fluid-fluid interface subdivides the background mesh into an active physical part and an inactive void/fictitious part , where the latter is covered by the embedded fluid patch  and the solid , such that with denoting the union of all background elements .

In this work, the decisive fluid domain separation is accomplished with the help of the Cut Finite Element Method (CutFEM). With this technique, the coupling of the two fluid approximations takes place just at the fluid-fluid interface , and flow is not approximated twice in the overlap zone of the involved fluid meshes. Also in opposite to overlapping domain decomposition, an iteration between the two fluid fields are needed, but they are solved together in a single shot. Details will be provided in Section 3.1,

Formulating the structural motion in a classical fitting-mesh Lagrangian formalism and applying a moving mesh ALE framework to the embedded fluid patch, a technique which is well-established in classical ALE based FSI approaches, allows the patch to follow the body in its movement. Thus an accurate capturing of flow effects at the fluid-structure interface  is guaranteed. Moreover, such fluid patches can be generated much easier than appropriate high quality meshes in classical ALE based FSI schemes. Utilizing the CutFEM based fluid domain decomposition allows for independent fluid patch locations within the background mesh . Fluid-structure interaction involving large solid deformations in high Reynolds-number flows including boundary layer effects in the vicinity of solids can thus be accurately simulated by such a hybrid Eulerian-ALE FSI approach.

Figure 5: The hybrid Eulerian-ALE discretization concept for FSI: Domain partition (left) according to the overlapping mesh setting (right).

2.2 Governing equations for the coupled fluid-structure interaction system

In this work, we consider FSI problems governed by the transient non-linear incompressible Navier-Stokes equations for the flow field and the non-linear structural elastodynamics equations for the solid body, complemented by appropriate boundary conditions, interfacial constraints and initial conditions. Following the hybrid FSI approach, the flow formulation is separated into two parts utilizing Eulerian fixed-grid and ALE moving mesh descriptions, respectively. The single-field contributions are specified below and summarized afterwards.

The solid elastodynamics formulation. Solid mechanics, see (1)–(5) below, are stated in a Lagrangian formalism, where a mapping  allows to express the motion of a material particle  from reference to current configuration, i.e. . The equations are formulated in terms of the unknown displacement field  and its first- and second-order time derivatives, the velocity and acceleration fields and with .

As a suitable strain-measure,

denotes the Green-Lagrange strain tensor and

the deformation gradient tensor. The corresponding second Piola–Kirchhoff stress tensor is defined as , where denotes the Cauchy stresses in spatial coordinates. In this work, we exclusively consider Neo-Hookean materials with based on a strain energy function , with and Lamé parameters and , which can be expressed in terms of Young’s modulus and Poisson’s ratio as . Further, denotes the structural material density in the initial referential configuration, and is the divergence operator with respect to material referential coordinates.

Appropriate Dirichlet and Neumann boundary data (2)–(3), given by , and initial values for structural displacements and velocities (4)-(5), defined as , complement the second order initial boundary value problem. Detailed explanations can be found, e.g., in the textbooks [41, 42].

The incompressible flow formulation. For the description of the flow field, an Eulerian formulation for the fixed background grid  is combined with an ALE formalism for the embedded moving fluid patch . Both subdomain formulations (6)–(7) are written with respect to the current domain configuration based on the more general ALE description. Therein, denotes the generalized ALE convective velocity with , where is the velocity of the respective referential system. The mapping therein tracks the deformation of the observed fluid domain from its initial configuration, for each subdomain independently. While for the moving embedded patch, the grid velocity is usually non-vanishing, for a fixed non-moving background grid it holds , which states the only difference in the embedded and background grid formulation in the two subdomains . An introduction to the ALE technique can be found in, e.g., [6, 4].

Further, denotes an external body force load, the symmetric strain rate tensor and the Cauchy stresses. The dynamic viscosity is denoted with , where and are the kinematic viscosity and fluid density, respectively. Appropriate Dirichlet and Neumann boundary data (8)–(9) are specified at all times  by functions . The initial condition (10) for the flow field is specified as in .

Coupling Conditions. The Cauchy stresses with respect to the current domain configuration defined on fluid and structural side of the interface are as defined above and are denoted with and , respectively. Since structural displacements are expressed with respect to its reference configuration, the temporal mapping  needs to be taken into account. For viscous fluids, i.e. , the kinematic and dynamic interface constraints emerge to continuity conditions (11)–(12) at the fluid-solid interface and to (13)–(14) at the fluid-fluid interface by analogy.

Definition 1 (Strong form of the coupled hybrid Fsi system).

The final hybrid coupled FSI system in its strong form reads: Find solid displacements , defined in the reference configuration, satisfying

(1)
(2)
(3)
(4)
(5)

and flow velocity and dynamic pressure for such that

(6)
(7)
(8)
(9)
(10)

subjected to kinematic and dynamic interface constraints at the fluid-solid interface

(11)
(12)

and equivalently at the artificial fluid-fluid interface

(13)
(14)

3 A Cut Finite Element Method (CutFEM) based hybrid Fsi formulation

In this section, we present one potential spatial discretization technique based on the framework of Finite Element Methods (FEMs). We would like to highlight, however, that our hybrid Eulerian-ALE discretization concept for multiphysics problems is not limited to Finite Element based schemes, but even possible to realize with, for instance, Finite Volume or Discontinuous Galerkin based techniques. Despite the variety of potential applicable finite-dimensional approximation frameworks, a prerequisite for this hybrid concept is to enable finite elements, cells or volumes to get intersected by an overlapping embedded fluid patch, and thus to enable a sharp disjoint domain decomposition of the fluid region.

The hybrid FSI approximation proposed in this work is based on a CutFEM fluid domain decomposition technique developed in our previous work [21] (see also [43] for couplings in elliptic non-moving problems) and on the Nitsche-based weak coupling of the solid phase to the embedded fluid patch solution, as presented in detail in [39]. It should be mentioned however that any fluid solid coupling scheme from fitting, i.e. classical ALE based FSI approaches could be used as well. Combining both interface coupling methods with suitable bulk-stabilized forms on cut background meshes for the transient incompressible Navier-Stokes equations, as developed in [30, 31], provide a highly accurate and robust hybrid FSI approach.

Before providing our spatial discretization of the FSI

problem, it is worthwhile to mention that the number of future approximation techniques based on this hybrid concept are highly diverse and allow for various promising novel concepts. While in the present work, just low-order continuous flow approximations based on Lagrangian finite elements are utilized, our approach also enables to approximate embedded fluid patch and background mesh with different finite element schemes, like for instance, different types and shapes of elements, interpolation functions, continuous and discontinuous or even enriched function spaces. The subsequent presentation of our

CutFEM FSI method is kept short with regards to numerical details, but still reviews the most important ingredients.

3.1 Semi-discrete Nitsche-type hybrid spatial discretization

Corresponding to the disjoint hybrid domain partition , let be the associated space of admissible discrete FSI solutions, consisting of the two flow approximation spaces, according to the background and the embedded mesh, and a structural approximation space.

Either of the incorporated single-mesh fluid approximations consists of a product space for velocity and pressure with boundary conditions (8) assumed enforced strongly. In this work, velocity and pressure are approximated with continuous equal-order interpolations on quadrilateral or hexahedral meshes of polynomial order  on both families of meshes and .

For the structural approximation in reference configuration, let be a family of boundary/interface-fitted quasi-uniform meshes, each approximating . Displacements and velocities are approximated on linearly-interpolated continuous isoparametric finite element spaces

(15)

where are isoparametric mappings to the element parameter space. Taking into account the respective trace values according to the strongly imposed Dirichlet constraints (2), the discrete function spaces for solid displacements  and velocities  result in , , respectively, and the test function space to .

Definition 2 (Semi-discrete Nitsche-type hybrid Fsi formulation).

The Nitsche-type stabilized formulation for the hybrid FSI problem setting reads as follows: for any time , find background fluid velocity and pressure , embedded fluid patch velocity and pressure and solid displacement and velocity such that

(16)

where

(17)
(18)
(19)
(20)

with single-mesh stabilized fluid operators for (see Definitions 3 and 4), a structural finite element approximation (see Definition 5) and Nitsche-type interface coupling terms for the fluid-fluid interface  (see definition 6) and the fluid-solid interface  (see Definition 7), respectively. It needs to be pointed out that due to the homogeneity of the coupling conditions, no right-hand-side terms are present for the Nitsche couplings.

Definition 3 (Semi-discrete stabilized CutFEM based background fluid formulation).

The CutFEM based semi-discrete approximation of the incompressible Navier-Stokes equations on a cut background mesh  reads

(21)

In this work, a residual-based variational multiscale (RBVM) stabilized form (see, e.g., [44]), is utilized to account for different inherent instabilities [45, 46] arising for highly convective dominant flows and due to the use of equal-order interpolations for velocity and pressure. The stabilization comprises SUPG/PSPG and LSIC terms. For cut meshes, additional interface-zone stabilization in terms of the operator  is required, see elaborations in Remark 2.

The RBVM/GP-stabilized form for a fluid mesh  reads in generalized form:

(22)
(23)

where, if unmistakable, the index has been omitted to shorten the formulas. Therein, and . Appropriate piecewise constant stabilization scaling functions are given as

(24)

with the second rank metric tensor and for linearly interpolated finite elements.

The standard Galerkin terms and the GP stabilization operator  are defined as follows:

(25)
(26)
(27)

where in the interface-zone, facets  located next to intersected elements , i.e. , are stabilized by face-jump penalty terms

(28)
(29)
(30)

Therein, it is set and

(31)

Further, denotes the jump of quantities across interior facets  and the subscript  in stabilization scalings indicates to take the mean over quantities from both adjacent elements . For details, the reader is referred to, e.g., [31, 39, 32].

Remark 1.

Since the background mesh  is assumed to remain fixed over time, as exclusively considered throughout this work, the time derivative occurring in (3) simplifies to , i.e. the grid velocity vanishes such that in a pure Eulerian consideration. Note, however, that it might be also an option to allow even the background mesh to move, as considered for instance in an approach shown in [39]. It should be mentioned that all further numerical or algorithmic steps for this have been already addressed in [39], however, are not considered further for simplicity in this paper.

Remark 2 (Stabilization of cut clements).

Due to the intersection of elements by the fluid-fluid interface , additional stabilization measures are required. Ghost-penalty stabilizations  (26), as developed in [31, 30], penalize jumps of normal derivatives of order  across interior facets  in the vicinity of the interface and thus ensure well system conditioning, stability and optimality of the approximation independent of the mesh intersection. For further details on this technique, the reader is referred to, e.g., [47, 48, 28, 29].

Definition 4 (Semi-discrete stabilized Fem based embedded fluid formulation).

The semi-discrete fluid operators for the fitted-mesh approximation of the embedded fluid patch  is comprised in

(32)

where respective operators are as defined in (3) and (23) with , since embedded meshes are uncut and do not require interface zone stabilization.

Remark 3 (Grid velocity for moving fluid patch).

Similar to the Lagrangian formalism for structures, the motion of the fluid grid is tracked in terms of the mapping . Introducing a fluid patch bulk displacement field  describing the computational grid motion, the ALE time derivative can be simply expressed in terms of the fluid domain displacements resulting in for the grid velocity. For their approximation, a classical linear isoparametric finite element concept is utilized.

Since the fluid patch and the solid mesh need to match at  at all times, the fluid domain displacement field  is constraint being equal to the solid displacement field , i.e.

(33)

For the motion of the non-constraint part of the fluid mesh, a pseudo-structure mesh update algorithm as also used in [13] has been used. It is important to appreciate that mesh distortions are much smaller than in standard ALE cases as the outer boundary of this mesh is free to move.

Definition 5 (Structural finite element approximation).

The space semi-discrete fitted-mesh finite element approximation of the structural elastodynamics form (1)–(3) with solid displacements and velocities reads, find such that

(34)

where

(35)
(36)

are evaluated on the discrete referential counterparts , respectively.

Definition 6 (Nitsche-type fluid domain decomposition coupling operators).

For the weak coupling of the two subdomain approximations on and , specified by the coupling constraints (13)–(14), the stabilized Nitsche-type formulation from [21] is applied, which is comprised in the following coupling operator

(37)
(38)
(39)
(40)
(41)