A consistent approach for fluid-structure-contact interaction based on a porous flow model for rough surface contact

Simulation approaches for fluid-structure-contact interaction, especially if requested to be consistent even down to the real contact scenarios, belong to the most challenging and still unsolved problems in computational mechanics. The main challenges are twofold - one is to have a correct physical model for this scenario, and the other one is to have a numerical method that is capable of working and being consistent down to a zero gap. And when analyzing such challenging setups of fluid-structure interaction that include contact of submersed solid components it gets obvious that the influence of surface roughness effects is essential for a physical consistent modeling of such configurations. To capture this system behavior, we present a continuum mechanical model which is able to include the effects of the surface microstructure in a fluid-structure-contact interaction framework. An averaged representation for the mixture of fluid and solid on the rough surfaces, which is of major interest for the macroscopic response of such a system, is introduced therein. The inherent coupling of the macroscopic fluid flow and the flow inside the rough surfaces, the stress exchange of all contacting solid bodies involved, and the interaction between fluid and solid is included in the construction of the model. Although the physical model is not restricted to finite element based methods, a numerical approach with its core based on the Cut Finite Element Method (CutFEM), enabling topological changes of the fluid domain to solve the presented model numerically, is introduced. Such a CutFEM based approach is able to deal with the numerical challenges mentioned above. Different test cases give a perspective towards the potential capabilities of the presented physical model and numerical approach.

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1 Introduction

The interaction of two surfaces with fluid in between is of great interest in various engineering and biomechanical applications. Any surface, no matter if manufactured or naturally occurring, is not completely smooth but has a microstructure with an average roughness height, which is magnitudes smaller than the object size itself. As soon as the height of the fluid film between these surfaces is in the same order of magnitude as surface asperities, surface roughness has an essential impact on the physical response of such a system. Occurring leakage, lubrication, friction, and wear are of great importance for the performance of valves, bearings, gears, and tires, for example.

The challenge to develop a consistent physical model for fluid-structure interaction including contact lies in the multiscale nature of the considered problem. While at the macroscopic scale the fluid physics are governed by the well-known Navier-Stokes equations supplemented by a no-slip condition on the fluid-structure interface, this does not necessarily hold true when solid bodies come into contact. Solving simply this macroscopic physical model does not lead to contact of submersed smooth solid bodies due to the increasing viscous stress when the fluid gap gets smaller. Therefore a finite fluid gap is retained, which is not in agreement with the observation of contacting bodies. A deeper insight into the underlying physical process of contacting submersed bodies allows to identify two ways out of this dilemma.

First, the limit of absolutely smooth contacting solid bodies is considered. Taking into account this assumption, the fluid gap between approaching solid bodies will fall below the validity limit of the macroscopic fluid dynamic models. As a first consequence a transition of the no-slip interface condition to a slip interface condition (including in general a surface roughness dependent sliplength) into the so called “Slip-Flow Regime” (see e.g. [1]), while retaining the validity of the Navier-Stokes equations in the fluid domain, can be observed. Therefore this local relaxation of the tangential no-slip condition has to be incorporated in the physical model, while retaining the no-slip condition for the remaining part of the interface. This results in a consistent physical model to consider contact of smooth solid bodies with fluid between the contacting surfaces. It should be pointed out, that the only exception is the case of two parallel plates, where due to the required acceleration of the fluid mass no contact occurs in finite time.

Second, taking into account that any real surface has a microstructure, this contacting process can change fundamentally. If single asperities come into contact before the validity of the classical fluid equations is lost, the contacting process described first cannot hold anymore. From a macroscopic point of view, contact is enabled in this case via a fluid mass transfer from the fluid domain into the rough microstructure. Therefore considering the rough microstructure of contacting surfaces can be essential for a consistent fluid-structure-contact interaction model. This is the case when the characteristic roughness height is larger than the limiting size of classical fluid equations, which depends on the considered type of fluid and its molecular mean free path length. In the following we will focus on the modeling of this second process as it is the relevant one for many problems of interest.

To analyze and predict such tribological systems for thin fluid films, the Reynolds equation [2], which can be derived from the Navier-Stokes equation by utilizing assumptions valid for thin film flows, is widely used. To incorporate the effect of surface roughness without resolving the surfaces, an averaged Reynolds equation is often used to solve for the averaged fluid pressure, see e.g. [3, 4, 5, 6, 7]. In [8, 9]

, it is shown that significantly fewer degrees of freedom are required for solving the homogenized equations compared to the direct equations in order to obtain the pressure field between rough surfaces. A framework to consider the effects of deformation of structural bodies interacting via a thin fluid film was presented in

[10, 11].

A comparison of numerical solutions for the full spatially discretized fluid momentum and continuity equations and the Reynolds approach, tested for a problem setup with valid thin film approximation presented in [12], shows that there is no significant deviation of the results between both approaches. Nevertheless, with increasing film size this result does not hold any longer, as the underlying geometrical assumptions of the Reynolds equation become invalid. In this case, the solution of the full fluid equations seems to be absolutely essential, even though the computational cost is higher due to the increased number of degrees of freedom as compared to the Reynolds approach.

In fluid-structure interaction problems, structures of arbitrary shape are deformed by the fluid traction acting on the interfaces. Herein, the geometry of the fluid domain and all corresponding interface or boundary conditions, which are given by the structural boundaries and outer boundaries, are not known in advance. Therefore, fluid equations, like the Navier-Stokes equations, are solved spatially resolved without including information concerning the fluid domain or boundary/interface conditions beforehand and are coupled to the structural deformation. Details on such methods can be found e.g. in [13, 14, 15], and in [16, 17, 18, 19, 20, 21] with attempts to treat contact of submersed bodies. However, none of these methods is able to meet the challenges mentioned above concerning physical modeling and practicability of the numerical approach. While several of those approaches are numerically inconsistent down to a zero gap or cannot represent the fluid stress discontinuity (especially the discontinuity of the fluid pressure) across considered slender bodies, also surface roughness is typically not considered and the “no-slip” condition is applied on the overall fluid-structure interface. This is only a reasonable assumption as long as the distance between these surfaces is large compared to the microstructure of them.

As soon as contact between solid bodies cannot be ruled out, there is no lower limit for the size of the gap between the surfaces involved. Therefore, effects arising from the roughness of these surfaces can even dominate the macroscopic overall physical response. Two examples are the analysis of leakage in seals or the opening pressure in valves. Another example arising in physical modeling is the “no-collision” paradox that contact between smooth surfaces with “no-slip” cannot occur in an incompressible, viscous fluid in finite time (see [22, 23]); this is contrary to the macroscopic observation. For “slip” interface conditions on the colliding surfaces (shown in e.g. [24]) or non-smooth surfaces (analyzed in e.g. [25, 26]), contact is possible. A physical explanation for this paradox is the lack of consideration of the surface microstructure. As soon as the microscopic roughness is treated, solid-solid contact can occur [27].

This aspect highlights the importance of considering the effects of surface roughness to extend the validity within a fluid-structure interaction framework for specific configurations. However, a direct resolution of the microstructure with a computational discretization is not practicable for engineering applications, as the focus of interest is mostly on averaged quantities such as the average velocity in the fluid gap of a valve or the average pressure in the fluid gap of a bearing. With such computed averaged quantities, predictions for global quantities such as the leakage flow of a valve or the load capacity of a bearing can be deduced. Resolving the potentially complex fluid flow between single roughness asperities is generally not necessary for these types of applications besides the fact that the exact microstructure is not known at all in most cases.

To reduce the high computational demands arising from a fully-resolving computation, as well as the necessity to know the exact microstructure, we propose a homogenized model to include the average physical behavior of the roughness layer into the fluid-structure-contact interaction framework. While this has not been done for real FSI so far, similar ideas have been successfully applied to consider roughness within Reynolds equation based formulations. Homogenizing a domain which consists of a fluid-filled deformable microstructure results in a poroelastic, fluid-saturated averaged domain. The basic idea of modeling surface roughness as a porous layer can already be found in [28, 29]. Our novel approach is based on a similar idea but also works for general fluid-structure interaction problems, applicable for arbitrary shaped domains, including finite deformations of the solid, topological changes of the fluid domain and rearrangement of all interface conditions.

Averaging the surface roughness as a poroelastic layer still allows us to account for many physically essential effects on a macroscopic level, thus in an averaged sense. Examples are the fluid pressure distribution between contacting bodies, stresses exchanged between contacting solids, the deformation of the roughness layer and the resulting fluid to solid fraction within the layer. Having this physical information available allows us to also include additional models to treat specific physical phenomena of the general problem of colliding bodies in fluid such as, for instance, friction of mixed lubrication contact or wear, which, however, are not in the focus of this contribution.

In addition to the physical modeling, a potential numerical approximation of the proposed model will be presented. It is based on spatial finite element discretizations for the structural, the fluid, as well as the poroelastic domain. It should be pointed out that this is just one possible discretization technique which allows for approximating the proposed physical model. Alternative approaches, for example based on the finite volume framework, might be also possible. In the present work, contact occurring between structural bodies is handled by the dual mortar contact approach [30] for finite deformations [31]. Topological changes in the fluid domain due to the occurring contact are enabled by the Cut-Finite Element Method (CutFEM) [32, 33, 34] applied to the Navier-Stokes equations. Interface coupling conditions are introduced weakly by Nitsche-based methods [35, 36, 37]. Finally, a so-called monolithic scheme [38, 39, 15, 40] is applied to solve the global system of equations, which is beneficial in the case of a strong interaction between all involved domains.

The paper is organized as follows: In Section 2, we depict the porous flow based model for rough fluid-structure-contact interaction, followed by the presentation of all governing equations for the different physical domains in Section 3. In Section 4, the coupling conditions on all interfaces between the occurring physical domains, as well as the interactions of all involved interfaces, are discussed. Section 5 presents the numerical method applied to solve different exemplary problem configurations. Three configurations, including a leakage test, a rough surface contacting stamp and a non-return valve, are presented and analyzed in Section 6. Finally, a short summary and conclusion is given in Section 7.

2 Rough surface contact model

A typical configuration of fluid-structure-contact interaction (FSCI) problems, with fluid domain and structural bodies occupying domain , is shown in Figure 1 (left). Herein, solid surfaces do not include any information about their present microstructure. This simplification is a good approximation regarding interfaces between structural bodies and fluid like , where the distance to the next interface or boundary is large compared to the average height of surface asperities. In this case, the influence of surface roughness on the physical response of the fluid-structure interaction system is completely negligible. However, for interfaces , where the surfaces of solid bodies approach each other and the size of the fluid gap in between can get very small or even vanish, this argumentation does not hold anymore. In the following, the term “gap” or “fluid gap” is exclusively used to specify the normal distance between approaching or contacting surfaces. As soon as the gap is in the same order of magnitude as the roughness height (see Figure 1 (right)), effects caused by the microstructure of the surfaces will start to influence the macroscopic physical behavior of the system. Finally, when first surface asperities start to contact, assuming smooth surfaces is definitely far off from the physical response of such an FSCI system. As the prediction of effects dominated by rough surfaces such as, e.g. leakage or lubrication is of great importance, a model to include surface roughness of these contacting surfaces into an FSCI framework in an efficient computational way will be presented in the following.

Figure 1: Typical configuration of an FSCI problem (left). Real surface geometry including the microstructure of rough bodies before contact (right).

The domain of interest consists of a part of the fluid domain between the contacting surfaces, as well as the structural domain in the neighborhood of the rough surface (see Figure 2). The overall domain is composed of the remaining solid domain , fluid domain , and the fluid filled poroelastic domain .

We propose to model this region in a homogenized manner, an approach well known from porous media mechanics, see e.g. [41, 42, 43, 44, 45].

homogenization
Figure 2: Detailed view of a potential microstructure between the contacting surfaces (left), represented by a homogenized poroelastic domain with porosity in the proposed model (right).

We obtain a poroelastic layer, which describes the fluid flow and the structural elastodynamics in an averaged sense in every point of the poroelastic domain . Herein, the porosity specifies the ratio of fluid volume inside the poroelastic layer. The model describing this poroelastic layer should be able to represent the influence of structural deformation on the fluid flow and vice versa. Furthermore, the deformation of the roughness layer, arising from applied external stress or external deformation on the outer boundaries of the poroelastic layer, has to be considered. A coupling of the poroelastic layer to the outer fluid flow has to be guaranteed as well. We propose to add a poroelastic layer (at least on one of the contacting interfaces) to all interfaces , which potentially come into contact. As the influence of this porous layer is negligibly small if bodies are not close to contact, a porous layer for all fluid structure interfaces does not seem to be necessary.

Figure 3: Proposed rough surface contact model for fluid-structure interaction, including all computationally essential domains and interfaces between these domains ; not in contact (left), in contact (right).

In Figure 3, a schematic of the final model for the rough surface FSCI is presented, where all involved physical domains with different physical principles and the interfaces between these domains are shown. In the following sections, the governing equations for the whole domain , which is split into three resulting domains, namely the structural domain , the fluid domain and the poroelastic domain with appropriate boundary conditions on the outer boundary , are presented. It should be pointed out that the fluid as well as the structural domain no longer include the volume occupied by the rough surface, as this is represented by the poroelastic model. All interfaces occurring between these domains and the appropriate coupling conditions will be presented in Section 4. These are the interfaces between the fluid domain and structural domain, the interfaces between the fluid domain and poroelastic domain, and the interfaces , between the poroelastic domain and the structural domain.

In contrast to the widely used Reynolds equation in elastohydrodynamic lubrication (EHL), where the no-slip condition is incorporated directly, the averaging procedure does not introduce any boundary conditions onto the fluid flow. This allows the incorporation of spatial and temporal changing boundary/coupling conditions during the computation. As a result, a continuous in time representation of the different states of colliding solid bodies in surrounding fluid is possible.

To illustrate this feature of the presented model, the impact of two submerged bodies serves as an example. As long as the distance between both structural bodies is large, the influence of the poroelastic layer is small and a “free-flowing” fluid similar to classical fluid-structure interaction problems is described. Once the gap between both contacting surfaces is in the same order of magnitude as the roughness height amplitude, a continuously increasing part of the fluid volume flow passes through the poroelastic layer. Further reduction of the fluid gap until first surface asperities start to contact directly is represented by contact between the poroelastic layer and the solid body. As soon as all of the fluid in the gap is inside the microstructure of the surfaces, the entire fluid flow passes through the poroelastic layer. Further increase of the contact pressure will result in progressing deformation of the poroelastic layer, which results in a reduction of the fluid fraction inside the layer. This goes along with an increase of fluid resistance and therefore reduced flow or “leakage rate” for a specific fluid pressure gradient in the flow direction. Finally, for high loads in relation to the structural stiffness, the fluid fraction will approach the vanishing limit, whereby only the structural part is left between the contacting surfaces. It should be mentioned that this exemplary sequence of steps which was described in detail, occurs just in the case that small variations in the fluid pressure and velocity far-fields arise during the approaching process of both structural bodies. If this is not the case, different pathways will be passed, for instance when an increasing fluid pressure deforms the poroelastic layer such that the contained fluid fraction is increased, which finally can lead to a “lift-off” of the contacting surfaces.

3 Governing equations for all involved physical domains

The following section presents the governing equations as well as appropriate boundary conditions for the different physical domains, i.e. structure, fluid and poroelastic domain.
The time interval of interest should be bounded by the initial point in time and the end time . Below, all quantities with additional index are described in the undeformed material configuration. The “hat” symbol indicates prescribed time-dependent quantities at boundaries and in the domains.

Finally, specifies prescribed initial quantities at the initial point in time.

3.1 Structural domain

The well-known initial boundary value problem (IBVP) for non-linear elastodynamics (see e.g. [46]) is described by

(1)
(2)

Therein,

denotes the displacement vector, which describes the motion of a material point (with position

at initial time ) due to deformation of the elastic body to the current position . Next, is the structural density in material configuration, is the material divergence operator, the deformation gradient,

the second Piola-Kirchhoff stress tensor, and

the body force per unit mass. The nonlinear, compressible material behavior is given by the hyperelastic strain energy function , which expresses the stress-strain relation. Therein, is the Green-Lagrange strain tensor. For the interaction with other physical fields described in the current configuration, the Cauchy stress is given by , with being the determinate of the deformation gradient. Further, adequate initial conditions with a prescribed displacement field and velocity field are defined

(3)

Finally, to complete the description of the structural problem, suitable boundary conditions on the outer boundary must be specified with the predefined displacement on Dirichlet boundaries and the given traction on Neumann boundaries :

(4)

Therein, is the outward-pointing unit normal vector on the boundary. Still missing are the conditions on the subset of the structural boundary where the structural domain is coupled to the other fields. They are not part of the outer boundary of the FSCI problem and will be discussed in Sections 4.1 and 4.3 .

3.2 Fluid domain

Transient, incompressible, viscous flow should be considered in the fluid domain. Considering that the balance of mass and momentum is given by the Navier-Stokes equations (see e.g. [47])

(5)
(6)

Herein, and are the fluid velocity and pressure, is the constant fluid density, the dynamic viscosity, the strain-rate tensor, and the body force per unit mass. Again, adequate initial conditions with the given initial velocity field are prescribed:

(7)

To finalize the description of the fluid problem, boundary conditions on the outer boundary are considered. Subsequently, the fluid velocity on Dirichlet boundaries or the fluid traction on Neumann boundaries is prescribed. Here, is the Cauchy stress and the outward-pointing unit normal vector to the boundary

(8)

Again, conditions on the boundary subset , which is not part of the outer boundary , will be discussed in Sections 4.1 and 4.2.

3.3 Poroelastic domain

As presented in Section 2, modeling the fluid-saturated rough surface domain as homogenized poroelastic media (see e.g. [43]), leads to specific requirements on the applied model. The following governing equations were developed and successfully applied in [44, 45, 48, 49]

and are capable to represent all essential physical effects, such as incompressible flow on the micro-scale, finite deformations of the poroelastic matrix, deformation dependent and variable porosity, as well as arbitrary strain energy functions for the skeleton. Due to the high flexibility of this formulation for the homogenized roughness layer, a broad range of different microstructures as well as material behavior of fluid and structures are applicable. Nevertheless, the focus of this contribution should be on including general modeling of rough surface contact into a fluid-structure interaction framework. A brief outline of parameter estimation for specific rough surfaces is provided in a following remark.

In the presented formulation, macroscopic flow through the deformable porous media is modeled by a Darcy flow based equation (10). The balance of mass and momentum of the fluid phase in the current configuration, the balance of mass (included implicitly, see [44, 45]) and momentum for the whole poroelastic mixture (consisting of fluid and solid) in material configuration on macroscopic scale can be expressed as:

(9)
(10)
(11)

These equations are valid on the macro-scale and represent an average microscopic state in the poroelastic media. Fluctuations due to the microstructure are not represented. Therefore, all quantities occurring in equations (9)-(11) also describe the average state from a macroscopic view. Herein, is the porosity already introduced in Section 2, the velocity and the pressure of the fluid phase, the macroscopic displacement of the poroelastic domain analogous to Section 3.1, the body force acting on the embedded fluid per unit mass, the spatial permeability of the poroelastic matrix with being the corresponding material permeability. Further, is the macroscopic averaged initial density of the solid phase with the associated averaged initial density in the solid domain. is the body force acting on the poroelastic mixture per unit averaged solid mass, the macroscopic deformation gradient, the determinant of the macroscopic deformation gradient, and the homogenized second Piola-Kirchhoff stress tensor. The material behavior of the poroelastic mixture is given by the macroscopic strain energy function , whereas accounts for the strain energy due to macroscopic deformation of the solid phase, arises from the volume change of the solid phase due to changing fluid pressure, and finally guarantees positive porosity of the poroelastic model (see [44, 45]). Using the Green-Lagrange strain tensor as the strain measure gives two constitutive relations to complete the system of equations for poroelasticity:

(12)
(13)

The necessary initial conditions for the poroelastic problem are:

(14)

Here, , , and are the initial displacement, initial solid phase velocity, porosity, and fluid velocity field, respectively. To complete the problem description of poroelasticity, adequate boundary conditions on the outer boundary have to be prescribed:

(15)
(16)

Therein, is the scalar normal fluid velocity of the Darcy-like flow on Dirichlet boundaries , the traction in normal direction on Neumann boundaries , the displacement of the poroelastic domain on Dirichlet boundaries , and the traction acting onto the poroelastic mixture on Neumann boundaries , with being the outward-pointing unit normal vector on the boundary. Conditions on the still not considered part of the poroelastic boundary will be presented in Sections 4.2 and 4.3. For further details on this poroelastic formulation, the reader is referred to [44, 45, 48].

Remark (A brief outline on the estimation of the poroelastic material parameters).

In the following, a computationally assisted way to determine a proper set of parameters of the poroelastic layer for a specific roughness layer is presented. To estimate the material parameters of the poroelastic layer for specific surfaces, the macroscopic material behavior of both contacting solid bodies as well as a resolved, representative microstructure geometry of the rough surfaces is required. First, performing direct contact simulations without any fluid pressure contribution for characteristic parts of the resolved rough surfaces, allows us to specify the material parameters of strain energy function of the poroelastic layer. In general, a comparison to the bulk material shows that the initial tangent stiffness of the poroelastic layer is smaller due to the deformation of single contacting asperities. Increasing contact stress leads to a rapid increase in the tangent stiffness, as the fluid fraction decreases. This behavior is reflected by an increased non-linearity of the strain energy function compared to the bulk material. Measuring the void space allows the determination of the porosity . Furthermore, by including a predefined normal load, which represents the fluid pressure, on the contact interface of the direct contact simulations, the correlation of fluid pressure and the solid phase compression can be analyzed. This allows us to identify the parameters of the strain energy function . To specify the permeability numerically, a resolved computation of the fluid flow between the rough surfaces can be consulted. Performing these fluid flow simulations for several deformation states, allows us to specify the relation of permeability and porosity . Besides this proposed computational approach, experimental determination for single parameters or the whole set of parameters is also possible.

4 Interfacial coupling constraints

In this section, appropriate coupling conditions on the interfaces occurring between all physical domains are discussed independently of their interaction, followed by an examination of the change of these different interface coupling conditions in context of the entire FSCI problem. In the following, is a uniquely chosen unit normal vector on each considered interface.

4.1 Interface between fluid domain and structural domain

The interface between viscous fluid domain and structural domain needs to fulfill the dynamic stress balance (17) as well as the no-slip condition (18):

(17)
(18)

4.2 Interface between fluid domain and poroelastic domain

On the interface between viscous fluid and the poroelastic domain , the following conditions need to be fulfilled (see [50, 51] for details):

(19)
(20)
(21)
(22)

Herein, (19) is the dynamic stress balance in current configuration between the Cauchy stresses of fluid and the entire poroelastic mixture, with being the Cauchy stress of the poroelastic mixture. Additionally, the dynamic stress balance (20) between the normal components of viscous fluid and the fluid inside of the poroelastic layer is required. Mass balance on the interface leads to a kinematic constraint in normal direction (21). Finally, the kinematic constraint in tangential direction for the viscous fluid is still missing. Equation (22) is the so-called Beavers-Joseph condition [52], which proposes a proportionality (with factor ) of the viscous fluid shear stress and the relative velocity slip in tangential direction between the adjacent fluids on both sides of the interface. Therein, the tangential plane is specified by two tangential vectors , orthogonal to vector . It should be mentioned that in many cases a simplified condition can be used, where the seepage velocity is neglected as proposed in [53]. Analyses on the Beavers-Joseph condition can be found in [54, 55, 56, 51].

4.3 Interface between poroelastic domain and structural domain in contact case

The interfaces between poroelastic layer and structural domain consist of a matching part and the contacting subset (see Figure 3 (right)), which is only non-zero in the contact case. Here we want to focus onto the contacting part only, as the physical and numerical treatment of interfaces such as can be found in the domain decomposition literature (see e.g. [57]) for the coupling of displacement degrees of freedom and [48] for the impermeability constraint. Considering frictionless contact, the following conditions need to be fulfilled:

(23)
(24)
(25)
(26)

Herein, (23) enforces the zero gap between the structural body and porous layer, with being the position on the interface and the projection of in direction onto the interface . Equation (24) represents the mass flow balance on the interface, where the relative fluid flow has to vanish due to the impermeability of the solid. Finally, (25) and (26) reflect the dynamic equilibrium between poroelastic and solid domain in normal and tangential direction, respectively.

4.4 Change of interface conditions in the coupled problem

In the previous sections, interfaces and their corresponding conditions were considered to be independent of each other. It is obvious that occurring contact between poroelastic layer and structural domain modifies not only the “active” contact interface , but also the interfaces between fluid and poroelastic/structural domain . The union of all three interfaces is given by the current configuration of the solid and poroelastic domain, in particular of the respective parts of the outer boundaries . The criteria specifying the different interface types on the interface , are given by the following Karush-Kuhn-Tucker conditions, which need to be fulfilled:

(27)
(28)
(29)

Condition (27) guarantees that there is always a positive gap or no gap between potentially contacting bodies. Additionally, (28) restricts the minimal stress by compression transferred directly between the solid bodies to the surrounding fluid normal traction. Herein, (e.g. or ) is the traction difference between total contact traction and the ambient fluid stress. This condition is a result of the assumption that the time interval for the formation of a fully covering fluid film on top of the rough microstructure is negligible and this process does not need to be modeled. As soon as the local (e.g. at single asperities) structural stress is smaller than the fluid stress, this fluid film will develop. In Figure 4, a schematic visualization of this process is given. Finally, (29) enforces exclusively either a zero gap between both interfaces (see equation (27)) or a vanishing relative traction (see equation (28)).

fully covering fluid filmfully covering fluid film
Figure 4: Detailed view on the condition transition point . Arrows indicate the process of generation (left) or degeneration (right) of a fully covering fluid film, depending on the traction difference between total contact traction and the ambient fluid stress.

The subset of the interface , where the first condition (27) is exactly zero and therefore fulfills (23), is the contact interface between poroelastic domain and structural domain . The remaining parts, where the second condition (28) is exactly zero, are the interfaces between fluid domain and structural domain or between fluid domain and poroelastic domain , depending on if it is a subset of the structural or poroelastic boundary.

In the following paragraph, the continuity of the formulation at the point or line of changing interface conditions (marked by the black cross in Figure 5) is discussed. To ensure continuity of the problem at this specific position, the conditions of all adjacent interfaces have to be fulfilled simultaneously. To prove this, we assume one type of interface conditions (e.g. contact) plus the criteria for transition to hold and verify the fulfillment of the other conditions (e.g. fluid-structure and fluid-poroelastic conditions). We analyze the fulfillment of the different interface conditions in normal direction at the transition points from a contact interface () to non-contact interfaces ( and ) and vice versa. Those points are formed by the intersection of the interfaces , and and they satisfy the transition conditions, stated as conditions (27) and (28) satisfied equal to zero. In order to enable a continuous transition of interface types, combining conditions on the contact interface (conditions (23)-(26)) with the transition conditions has to fulfill conditions on the interfaces and (conditions (17)-(18) and (19)-(22)) by default (see upper red path in Figure 5). Therefore, let’s assume that the conditions of contact are satisfied and even for a vanishing fluid film we consider fluid state vectors (, ) and implicitly the corresponding fluid stress () to result in continuous fulfillment of all interface conditions. Then, for the normal components of velocity and normal traction difference, the following relations hold:

(30)

(*1) = normal velocity of emerging fluid film equals contact interface normal velocity

(31)

(*2) = normal fluid stress of emerging fluid film in balance with fluid pressure in poroelastic layer

As a consequence of (30) and (31) the conditions on the interfaces and (conditions (17)-(18) and (19)-(21)) in normal direction are fulfilled naturally. For the change from the non-contact interfaces and to the contact interface this can be shown analogously (see lower blue path in Figure 5).

(23)-(26) (27), (28)(17)-(18)(19)-(22)
Figure 5: Visualization of the continuous transition of normal conditions at the point which is marked by the black cross. Following the upper red path indicates the equivalence of the contact conditions by making use of the transition conditions and the normal conditions on the interfaces and . Following the lower blue path indicates the equivalence of the conditions on the interfaces and in normal direction by making use of the transition conditions and the contact conditions on .
Remark (Continuity of the formulation for changing interface conditions in tangential direction).

Considering frictionless contact at the contact interface in combination with the no-slip condition at the interface and the Beavers-Joseph condition at the interface leads to a non-continuous change in the tangential component of the interface conditions. A continuous change between contact and non-contact interfaces for frictionless contact would require zero tangential stress (“full-slip” conditions) close to the contacting zone on the interfaces and as well. Nevertheless, contrary to the continuity of the normal component of interface conditions, the continuity of tangential components turned out to be less essential in our numerical computations.

It should be pointed out that for a pure structural contact case with vanishing fluid, conditions (27), (28), and (29) reduce to the classical contact conditions, as the relative traction equals the absolute traction for vanishing fluid pressure.

5 Discretization and solution approach

In this section, the applied methods for solving the coupled rough FSCI problem are presented. The discretization of the continuous problem described in the previous Sections 3 and 4 is based on the Finite Element Method. First, the spatially discretized semi-discrete formulations for the structural domain, the fluid domain as well as the poroelastic layer are given. Topological changes of the fluid domain in the rough FSCI problem due to occurring contact of surfaces are enabled via a CutFEM applied onto the fluid domain. The embedded interfaces between fluid domain and poroelastic domain together with the interfaces between fluid domain and structural domain are imposed by Nitsche-based methods. To incorporate contact into the formulation, the dual mortar method is used. Finally, all contributions are considered in one global system and solved monolithically.

As the focus of this contribution should not be on the specific numerical methods, these are just presented briefly, since further details can be found in the referenced literature. In the following sections, all quantities, including the primary unknowns, the test functions in the weak form as well as the domains and interfaces are discretized in space. No additional index is added to these discrete quantities since this double meaning of notation is accepted for the sake of simplicity of presentation. Below, the expressions and denote the inner product integrated in the domain and on the boundary/interface , respectively.

5.1 Weak forms of domains

The weak forms of the structural, fluid, and poroelastic domains can be derived from equations (1), (5) - (6) and (9) - (11), respectively: