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Dieser Artikel stellt eine bahnbrechende Methode zur gleichzeitigen Lösung inkompressibler Navier-Stokes-Ströme auf mehreren Oberflächen vor, die in einem dreidimensionalen Massenbereich eingebettet sind. Die Studie geht den Herausforderungen bei der Modellierung von Strömungsphänomenen auf gekrümmten Oberflächen nach und präsentiert ein neuartiges mechanisches Modell und eine Finite-Elemente-Methode, um diese Probleme zu lösen. Zu den Schlüsselthemen zählen die Formulierung der herrschenden Gleichungen in einem koordinatenfreien Rahmen, die Anwendung der Co-Flächenformel zur gleichzeitigen Analyse und die Entwicklung des Bulk Trace FEM für numerische Lösungen. Der Artikel untersucht auch die Durchsetzung tangentialer Geschwindigkeitsfelder und die Anwendung von Strafmethoden zur Stabilisierung. Zahlenbeispiele zeigen Konvergenzraten höherer Ordnung und bestätigen die Methode gegenüber herkömmlichen Oberflächen-FEM-Lösungen. Praktische Anwendungen in Designprozessen und Klimamodellierung werden diskutiert, was das Potenzial dieses Ansatzes in verschiedenen Branchen verdeutlicht. Die Schlussfolgerungen betonen die Wirksamkeit der Methode und skizzieren zukünftige Forschungsrichtungen zur Verbesserung instationärer Navier-Stokes-Lösungen und Stabilisierungstechniken.
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Abstract
A mechanical model and finite element method for the simultaneous solution of Stokes and incompressible Navier–Stokes flows on multiple curved surfaces over a bulk domain are proposed. The two-dimensional surfaces are defined implicitly by all level sets of a scalar function, bounded by the three-dimensional bulk domain. This bulk domain is discretized with hexahedral finite elements which do not necessarily conform with the level sets but with the boundary.The resulting numerical method is a hybrid between conforming and non-conforming finite element methods. Taylor–Hood elements or equal-order element pairs for velocity and pressure, together with stabilization techniques, are applied to fulfil the inf-sup conditions resulting from the mixed-type formulation of the governing equations. Numerical studies confirm good agreement with independently obtained solutions on selected, individual surfaces. Furthermore, higher-order convergence rates are obtained for sufficiently smooth solutions.
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1 Introduction
Flow phenomena are important topics of applied and basic research in several subjects, e.g., physics, chemistry, biology, engineering, and mathematics [1‐4]. The flow models are usually described by partial differential equations (PDEs) such as the Navier–Stokes equations. For flows on curved surfaces, the interaction of the physics, i.e., the flow, which take part on the curved domain, and the geometry of the domain plays a crucial role in the formulation of the mathematical model. Due to more involved definitions of geometric and differential operators in these cases, such models are more advanced than usual models for two- or three-dimensional Euclidean geometries. Flows on curved domains have important applications in nature, see for example [5, 6], and engineering, see [2‐4]. In recent years, flows on curved manifolds which are embedded in some higher-dimensional background space have gained significant attention as can be seen in [7‐12]. Therein, various formulations of the model and approximation methods have been proposed for individual surfaces. In this work, we propose a mechanical model and corresponding finite element method to solve flows on all surfaces over a bulk domain simultaneously.
An overview of different models for surface flows and their derivations is found in [10]. The equations can be formulated based on curvilinear coordinates [3, 10] or in a coordinate-free formulation. The latter one may be based on the tangential differential calculus (TDC) [13‐15]. The TDC can be interpreted as the modern perspective on differential geometry to formulate PDEs on manifolds based on surface differential operators rather than based on local curvilinear coordinate systems. A detailed derivation of the incompressible Navier–Stokes equations on an evolving surface from first principles of (continuum) mechanics and formulated in the TDC-framework is given in [7]. It should be noted that formulations of the Navier–Stokes equations on manifolds which are derived by substituting classical differential operators with their geometric counterparts are not necessarily equivalent to formulations derived from first mechanical principles because there may be (subtle) differences, see [7], page 364. The numerical solution of flows on manifolds comes with additional challenges compared to the classical d-dimensional Euclidean space. One of these is the enforcement of a tangential velocity field.
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Different methods to solve (incompressible Navier–)Stokes flows on surfaces have been proposed in recent years, e.g., finite difference methods in [16], (classical) Surface finite element methods (FEM) in [8, 17], Trace FEM in [12, 18‐20], an FEM with tangential function spaces in [11, 21], and a mesh-free method in [22]. Furthermore, one may distinguish models formulated for velocity and pressure as primal variables and models in which only scalar quantities are sought, e.g., stream-function formulations, see [9]. In all of these methods (incompressible Navier–)Stokes flows on one single surface are considered. In this paper, we simultaneously solve the (incompressible Navier–)Stokes flows on all, i.e., infinitely many, level sets of a scalar function embedded in a three-dimensional bulk domain. We refer to the text books [1, 2, 23] for the FEM in classical fluid dynamics. For an overview about different approaches of finite element methods for PDEs on curved surfaces, the reader is referred to [24].
Next, the concept of the simultaneous analysis of PDEs on manifolds in general and for (incompressible Navier–)Stokes flows on manifolds in particular is introduced which is the novelty of this paper. The manifolds are defined implicitly as level sets \(\Gamma _{\!c}\) of a level-set function \(\phi \) and embedded in a higher-dimensional bulk domain \(\Omega \). The \(c \in {\mathbb {R}}\) is some constant value related to an iso-surface. In a usual Trace FEM context, see [12, 18‐20], the zero-isosurface of a level-set function is considered, i.e., \(c=0\). For an introduction to level-set functions and the level-set method see, e.g., [25, 26]. The manifolds are bounded by the boundary of the bulk domain, hence, the boundaries of the manifolds and the bulk domain are conforming. It is important to emphasise that the level sets do not have to be aligned to the mesh which is used to discretize the bulk domain in the applied FEM. Therefore, this method may be seen as a hybrid of conforming methods, for example the Surface FEM, and fictitious domain methods, such as the Trace FEM, and was labelled Bulk Trace FEM by the authors in [27] in the context of structural mechanics. However, fictitious domain methods often come with additional challenges, among these the need for stabilization and special quadrature in cut elements. These do not apply for the Bulk Trace FEM. Similar approaches have been used in transport problems and diffusion on stationary surfaces in [28] and in [29] on evolving surfaces. A comparison of the solution of elliptic PDEs on all level sets within some bulk domain between phase-field methods and the simultaneous analysis with the FEM is given in [30]. Historically related to the concept of the simultaneous analysis are narrow-band methods, see [31‐35]. One could interpret that the goal of narrow-band methods was to reduce the bulk domain to a minimum around a single surface. This goal was reached with fictitious domain methods where only one level set, i.e., usually the zero-isosurface, is considered. This has been applied to various topics of PDEs on manifolds in computational mechanics, see for example [36‐44]. Although only the solution on one manifold is often required, it is useful to develop methods for the simultaneous solution of all manifolds embedded in a prescribed bulk domain. Possible applications are in the design process where variations in the geometry should be studied to find an optimal design. Furthermore, in (global) weather and climate models, some models individually consider flows on (spherical) layers with some cross-coupling in the normal direction [45, 46], being a natural fit for the conceptual and theoretical framework proposed herein. The authors developed mechanical models and applied corresponding Bulk Trace FEMs in the context of structural mechanics for geometrically non-linear membranes in [27, 47, 48], Reissner–Mindlin shells in [49], and for Timoshenko beams in [50]. An overview of the used models and first results for transport and incompressible flow problems are shown in [51] by the authors of this paper.
In [8], a higher-order Surface FEM for (incompressible Navier–)Stokes flows (on single surfaces) is presented by the authors. A crucial aspect for models of surface flows is the enforcement of the tangentiality of the velocities. In [8], a Lagrange multiplier is used and in this work, a (consistent) penalty method is applied, similar to a single surface in [7, 12, 52]. The governing equations are based on [7]. Herein, we consider stationary Stokes, stationary and instationary Navier–Stokes flows on spatially fixed, two-dimensional curved surfaces \(\Gamma _{\!c}\) embedded in a three dimensional bulk domain \(\Omega \). First, the governing equations are formulated in strong form for each surface \(\Gamma _{\!c}\). In the derivation of the weak form, the co-area formula is applied to formulate a weak form which is suitable for the simultaneous analysis of all embedded level sets in \(\Omega \). Figure 1 shows the conceptual idea of the simultaneous analysis: In Fig. 1a, a bulk domain is depicted in blue and some, arbitrarily selected level sets are depicted in different colours. Figs. 1b and c show an example mesh, highlighting nodes with prescribed velocities.
Fig. 1
A generic example: (a) The volumetric bulk domain in blue, some level sets shown in different colours. In (b) and (c), some example mesh and nodes with no-slip conditions (blue) and those on the inflow (red) are seen
In Sect. 2, the geometric setup, surface differential operators, and divergence theorems are introduced as preliminaries. For each type of considered flow problems, i.e., stationary Stokes, stationary and instationary Navier–Stokes flows, the strong form, the weak forms, and numerical examples are given in Sects. 4, 5, and 6, respectively. In Sect. 7, two possible applications of the proposed method are outlined. The paper ends in Sect. 8 with conclusions and an outlook.
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2 Geometric and mathematical preliminaries
In this section, the geometrical setup of two-dimensional surfaces embedded in a three-dimensional bulk domain is introduced. The differential operators that are used in the formulation of the mechanical models and related numerical methods are defined. Analogous definitions can be found in previous works by the authors, e.g., [27] for the simultaneous solution of geometrically non-linear ropes and membranes and [49] for the simultaneous solution of Reissner–Mindlin shells.
2.1 Geometric setup of embedded surfaces
We consider flow phenomena described by the Stokes and incompressible Navier–Stokes equations on curved two-dimensional surfaces. These surfaces are manifolds with co-dimension 1 embedded in the three-dimensional physical space \({\mathbb {R}}^3\). A three-dimensional bulk domain \(\Omega \subset {\mathbb {R}}^3\) and a level-set function \(\phi \left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) :\Omega \rightarrow {\mathbb {R}}\) are given. Within this bulk domain exists a minimal value \(\phi ^{\min }=\inf \phi \left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) and a maximum value \(\phi ^{\max }=\sup \phi \left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) of the level-set function. The individual manifolds \(\Gamma _{\!c}\) defined by level sets of \(\phi \) with constant level-set values \(c\in {\mathbb {R}}\),
are curved, two-dimensional manifolds, see, Fig. 1a. \(\phi ^{\min }\) and \(\phi ^{\max }\) may be defined as the infimum/supremum of the level-set function inside the bulk domain or as user-defined values to restrict some larger bulk domain to a sub-interval of interest. For bounded surfaces, the boundary of some selected manifold \(\Gamma _{\!c}\) is denoted as \(\partial \Gamma _{\!c}\) and is the intersection curve of the level set \(\Gamma _{\!c}\) with the boundary \(\partial \Omega \) of the bulk domain. In this work, we only consider stationary surfaces and bulk domains, hence, these are fixed in time. The boundary of the bulk domain \(\partial \Omega \) is restricted to the parts of the boundary where \(\phi \left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \ne \phi ^{\min }\) and \(\phi \left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \ne \phi ^{\max }\) for the proper definition of vector fields to be used in the mathematical description of the flow. Another requirement for the geometrical setup to state proper (initial) boundary value problems simultaneously on all level sets is that the embedded surfaces vary smoothly without topology changes within the bulk domain, see [27, 49] for further insights.
2.2 Vector fields and the tangential projector
We start by introducing some geometrical quantities. The normal vector to the level sets \(\Gamma _{\!c}\) can easily be computed via the gradient of the level-set function \(\phi \). The unit normal vector (field) \(\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}(\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}})\) in the whole bulk domain \(\Omega \) is obtained by
using the (classical) gradient of the level-set function \(\phi \). It is shown in Fig. 2, where for clarity, only one selected level set is plotted. The (tangential) projector field \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathbb {R}}^{3\times 3}\) is immediately obtained from the normal vector field as
where \(\mathchoice{\displaystyle \textbf{I}}{\textstyle \textbf{I}}{\scriptstyle \textbf{I}}{\scriptscriptstyle \textbf{I}}\) is the identity matrix in \({\mathbb {R}}^3\). This quantity is crucial to define tangential differential operators in Sec. 2.3. Furthermore, it projects quantities onto the tangent space \(T_{P}\Gamma \) of a curved surface \(\Gamma \) at some point. An arbitrary vector \(\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}} \in {\mathbb {R}}^3\) is projected onto the tangent space as \(\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}_\textrm{t} = \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}\) which will be used frequently in the remainder of this paper. Some important properties of the projector are (i) \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} = \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}^{\textrm{T}}\), (ii) \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} = \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}\), and (iii) \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}} = \mathchoice{\displaystyle \varvec{0}}{\textstyle \varvec{0}}{\scriptstyle \varvec{0}}{\scriptscriptstyle \varvec{0}}\).
Along the boundary \(\partial \Omega \) of the bulk domain, a normal vector (field) \(\mathchoice{\displaystyle \varvec{m}}{\textstyle \varvec{m}}{\scriptstyle \varvec{m}}{\scriptscriptstyle \varvec{m}}(\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}),\,\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\in \partial \Omega \) is defined. For the computation of the simultaneous flows later on, we assume that the bulk domain is discretized by (higher-order) volumetric (i.e., three-dimensional) finite elements. The computation of \(\mathchoice{\displaystyle \varvec{m}}{\textstyle \varvec{m}}{\scriptstyle \varvec{m}}{\scriptscriptstyle \varvec{m}}(\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}})\) on element boundaries is a standard operation in the FEM.
Furthermore, along the boundary \(\partial \Omega \) of the bulk domain lives the tangent vector (field) \(\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) defined as
With the normal vector of the level sets \(\Gamma _{\!c}\) and the tangential vector on \(\partial \Omega \), a co-normal vector field \(\mathchoice{\displaystyle \varvec{q}}{\textstyle \varvec{q}}{\scriptstyle \varvec{q}}{\scriptscriptstyle \varvec{q}}(\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}})\) is defined as
These co-normal vectors play an important role in the formulation of the weak form of the (intitial) boundary value problem as fundamental parts of divergence theorems, see Sec. 2.5. Figure 2 shows normal vectors to the level sets as blue arrows, normal vectors to the boundary of the bulk domain as red arrows, tangential vectors as gray arrows, and co-normal vectors as green arrows.
Fig. 2
Vector fields in the domain \(\Omega \) and on the boundary \(\partial \Omega \) shown on some level set \(\Gamma _{\!c}\) with \(c \in \left[ \phi _{\min }, \phi _{\max }\right] \). The right figure shows a zoom of the left one. Normal vectors \(\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}\) with respect to the level sets \(\Gamma _{\!c}\) in \(\Omega \) are shown in blue. Normal vectors \(\mathchoice{\displaystyle \varvec{m}}{\textstyle \varvec{m}}{\scriptstyle \varvec{m}}{\scriptscriptstyle \varvec{m}}\) with respect to \(\partial \Omega \) are red, tangential vectors \(\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}\) are gray and co-normal vectors \(\mathchoice{\displaystyle \varvec{q}}{\textstyle \varvec{q}}{\scriptstyle \varvec{q}}{\scriptscriptstyle \varvec{q}}\) are green
In the governing equations of flows on surfaces, classical differential operators w.r.t. the embedding three-dimensional space and tangential or surface differential operators w.r.t. the curved, embedded, two-dimensional surfaces (level sets) must be distinguished. A subscript \(\Gamma \) is used for surface quantities, e.g., \(\textrm{div}_{\Gamma }\) for the surface divergence. The resulting coordinate-free definition does not rely on the introduction of (local) curvilinear coordinates. This approach is sometimes labelled Tangential Differential Calculus (TDC), cf. [13]. The surface gradient of a scalar function \(f\!\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) :\Omega \rightarrow {\mathbb {R}}\) is obtained as [7, 8, 13, 29]
where \(\nabla f\) is the classical gradient in the three-dimensional space \({\mathbb {R}}^3\) and \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}\) the projector defined in Eq. (3).
For a vector function \(\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) :\Omega \rightarrow {\mathbb {R}}^{3}\), the directional surface gradient is defined as
Note that the covariant gradient is an in-plane quantity, i.e., \( \nabla _{\Gamma }^{\textrm{cov}}\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}\in T_{P}\Gamma _{\!c}\), while the directional gradient is generally not in the tangent space of \(\Gamma _{\!c}\), i.e., \( \nabla _{\Gamma }^{\textrm{dir}}\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}\notin T_{P}\Gamma _{\!c}\).
The surface divergence of vector-valued functions \(\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) : \Omega \rightarrow {\mathbb {R}}^3\) and second-order tensor-valued functions \(\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) :\Omega \rightarrow {\mathbb {R}}^{3\times 3}\) are defined as
For the formulation of (initial) boundary value problems on curved surfaces the curvature of these domains is an important quantity. We use the Weingarten map [7, 13] to quantify the curvature. It is a symmetric, in-plane tensor defined as
The two non-zero eigenvalues are the principal curvatures, \(\kappa _{1,2} = -\textrm{eig}(\mathchoice{\displaystyle \textbf{H}}{\textstyle \textbf{H}}{\scriptstyle \textbf{H}}{\scriptscriptstyle \textbf{H}})\). The Gauß curvature is obtained as \(K=\kappa _1 \cdot \kappa _2\) and the mean curvature as \(\varkappa = \textrm{tr}(\mathchoice{\displaystyle \textbf{H}}{\textstyle \textbf{H}}{\scriptstyle \textbf{H}}{\scriptscriptstyle \textbf{H}}) = \textrm{div}\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}} = \textrm{div}_{\Gamma } \mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}\) [13, 53].
2.5 Integral theorems
Integral theorems are required to formulate the weak form of (partial) differential equations, needed for the resulting FEM formulation. For the simultaneous analysis of the flow fields on all level sets as proposed in this work, the relation between the integration over all level sets \(\Gamma _{\!c}\) and the integration over the bulk domain \(\Omega \) is given by the co-area formula [28, 30, 54‐56]. The co-area formula for the integration of an arbitrary scalar function \(f\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) over all embedded surfaces \(\Gamma _{\!c}\) in the level-set interval \(\left[ \phi ^{\min },\;\phi ^{\max }\right] \) is defined as
In the co-area formulas, the norm of the classical gradient of the level-set function \(\phi \) is considered on the right hand side in the integration over the bulk domain \(\Omega \) and its boundary \(\partial \Omega \), respectively. It is seen that the co-normal vector \(\mathchoice{\displaystyle \varvec{q}}{\textstyle \varvec{q}}{\scriptstyle \varvec{q}}{\scriptscriptstyle \varvec{q}}\) as defined in Eq. (5) and the normal vector \(\mathchoice{\displaystyle \varvec{m}}{\textstyle \varvec{m}}{\scriptstyle \varvec{m}}{\scriptscriptstyle \varvec{m}}\) on \(\partial \Omega \) occur in the co-area formula, Eq. (13), see [24, 27] for further details.
For the derivation of the weak form, divergence theorems are needed. The divergence theorem for a vector-valued function \(\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}(\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}): \Gamma _{\!c} \rightarrow {\mathbb {R}}^3\) and for a tensor-valued function \(\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}\!\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) : \Gamma _{\!c} \rightarrow {\mathbb {R}}^{3 \times 3}\) on a single surface \(\Gamma _{\!c}\) is defined as [8, 13, 14]
where \(\nabla _{\Gamma }^{\textrm{dir}} \mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}} : \mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}} = \textrm{tr}\left( \nabla _{\Gamma }^{\textrm{dir}} \mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}} \,\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}^{\textrm{T}}\right) \). Note that the mean curvature \(\varkappa \), the normal vector \(\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}\), and the co-normal vector \(\mathchoice{\displaystyle \varvec{q}}{\textstyle \varvec{q}}{\scriptstyle \varvec{q}}{\scriptscriptstyle \varvec{q}}\) are involved. The term which includes the mean curvature \(\varkappa \) on the right hand side vanishes for an in-plane tensor-valued function \(\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}(\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}})\), i.e., \(\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}} = \mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}_{\textrm{t}} = \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}} \,\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \in T_{P}\Gamma _{\!c}\) because \(\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}_{\textrm{t}} \,\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}} = 0\). The combination of this divergence theorem for one surface with the co-area formulas, Eqs. (12) and (13), results in a divergence theorem for all level sets in the bulk domain as [27]
where again, the curvature term vanishes for in-plane tensors \(\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}_\textrm{t}\).
3 Mechanical preliminaries
The governing equations for flows on manifolds given in the following sections are derived from first principles of continuum mechanics. For a detailed derivation of the instationary Navier–Stokes equations on a moving domain we refer to, e.g., [7]. The flow models (for one surface) considered in this work are special cases of those derived in [7] and can also be found in other works, e.g., [8, 52, 57, 58]. A stationary manifold is considered for all flow problems in this work, i.e., the surfaces are fixed in time. In this section, the quantities which are used in the formulation of the [initial] boundary value problems ([I]BVP) given in Sec. 4 to Sec. 6 are introduced.
The velocity field lives in the tangent space of \(\Gamma \), i.e., \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}_\textrm{t}= \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\) with some three-dimensional velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) on the surface. In this work, we use the tangential velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}_\textrm{t}\) in the formulation of the considered models for (Navier–)Stokes flows on manifolds. Therefore, in the formulation of the weak form later on, it is necessary to multiply the test function \(\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) with the projector \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}\). An alternative is to use a general (arbitrary) velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\) which is then constrained to live in the tangent space of \(\Gamma _{\!c}\) using additional Lagrange multipliers, see [7, 8]. The velocity field may be split into a tangential and a normal part as [7]
Furthermore, there is a pressure field \(p\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) and a tangential body force \(\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}_\textrm{t}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) which is often expressed as \(\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}_\textrm{t}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) = \rho \,\mathchoice{\displaystyle \varvec{g}}{\textstyle \varvec{g}}{\scriptstyle \varvec{g}}{\scriptscriptstyle \varvec{g}}_\textrm{t}\) with the density \(\rho \in {\mathbb {R}}^+\) of the fluid and \(\mathchoice{\displaystyle \varvec{g}}{\textstyle \varvec{g}}{\scriptstyle \varvec{g}}{\scriptscriptstyle \varvec{g}}_\textrm{t}= \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}\,[0,0,-9.81]^{\textrm{T}}\) when gravity is considered [8]. Note that the subscript t, i.e., \(\square _\textrm{t}\) indicates tangential quantities, while t stands for time below.
Stress and strain tensors.
A directional and a covariant strain tensor are introduced as [8]
which is the Boussinesq–Scriven surface stress tensor for stationary surfaces [7, 15, 57, 59] and \(\mu \in {\mathbb {R}}^{+}\) is the (constant) dynamic viscosity.
Boundary conditions.
The boundary \(\partial \Gamma _{\!c}\) of a manifold is decomposed into two non-overlapping parts, the Dirichlet boundary \(\partial \Gamma _{\!c,\textrm{D}}\) and the Neumann boundary \(\partial \Gamma _{\!c,\textrm{N}}\). The boundary conditions are given as
with prescribed velocities \(\hat{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}\) along the Dirichlet boundary and
with given tractions \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}_\textrm{t}\) along the Neumann boundary. Note that \(\hat{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}\) and \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}_\textrm{t}\) are in the tangent space of \(\Gamma _{\!c}\), i.e., \(\hat{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}\cdot \mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}=\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}_\textrm{t}\cdot \mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}=0.\)
There are usually no explicit boundary conditions needed for the pressure p. However, if no Neumann boundary is present, i.e., \(\partial \Gamma _{c,\textrm{N}}=\emptyset \) and \(\partial \Gamma _{c,\textrm{D}}=\partial \Gamma \) or in the case of compact manifolds where \(\partial \Gamma =\emptyset \), the pressure is defined up to a constant [8, 60, 61]. For such cases, the pressure is prescribed at a given point on \(\Gamma _{\!c}\) or imposed by the constraint \(\int _{\Omega }p \,\Vert \nabla \phi \Vert \;\textrm{d}\Omega =0\) in the weak form for all level sets within a bulk domain.
Vorticity on manifolds.
A physical quantity which is often computed in the context of flow phenomena is the vorticity \(\mathchoice{\displaystyle \varvec{\omega }}{\textstyle \varvec{\omega }}{\scriptstyle \varvec{\omega }}{\scriptscriptstyle \varvec{\omega }}\). For flows on manifolds it is defined as [8]
The vorticity \(\mathchoice{\displaystyle \varvec{\omega }}{\textstyle \varvec{\omega }}{\scriptstyle \varvec{\omega }}{\scriptscriptstyle \varvec{\omega }}\) is co-linear to the normal vector \(\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}\) which leads to a zero-vector when it is projected onto the tangential space because \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}\,\mathchoice{\displaystyle \varvec{\omega }}{\textstyle \varvec{\omega }}{\scriptstyle \varvec{\omega }}{\scriptscriptstyle \varvec{\omega }}=\mathchoice{\displaystyle \varvec{0}}{\textstyle \varvec{0}}{\scriptstyle \varvec{0}}{\scriptscriptstyle \varvec{0}}\). For this reason, a scalar quantity \(\omega ^{\star }\) is determined which is the signed magnitude of \(\mathchoice{\displaystyle \varvec{\omega }}{\textstyle \varvec{\omega }}{\scriptstyle \varvec{\omega }}{\scriptscriptstyle \varvec{\omega }}\) defined as
Stationary Stokes flow on a manifold in stress-divergence form [7, 8, 60] is formulated in the governing field equations to be fulfilled \(\forall \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\in \Gamma _{\!c}\) as
Function spaces are defined to formulate the weak form of the governing equations. For the weak form on one single manifold, the following function spaces are introduced [8]:
where \({\mathcal {H}}^{1}\) is the Sobolev space of functions with square integrable first derivatives and \({\mathcal {L}}_{2}\) is the Lebesque space. The function space for the pressure \({\mathcal {S}}_{p}^{\Gamma }\) may be replaced by
if no Neumann boundary exists, as described above.
Using the introduced function spaces, the weak form of the stationary Stokes flow on one surface is obtained as usual, that is, by multiplication of the strong form of the governing equations with suitable test functions, i.e., \(\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},\textrm{t}}\) and \(w_p\), and integration over the domain, including the application of the divergence theorem, given in Eq. (14). The resulting continuous weak form is stated as: Given a (constant) shear viscosity \(\mu \in {\mathbb {R}}^+\), body forces \(\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) on \(\Gamma _{\!c}\), and boundary tractions \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) on \(\partial \Gamma _{\!c,\textrm{N}}\), find the velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Gamma }\) and the pressure field \(p\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{p}^{\Gamma }\) such that for all test functions \(\left( \mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}},w_{p}\right) \in {\mathcal {V}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Gamma }\times {\mathcal {V}}_{p}^{\Gamma }\), there holds in \(\Gamma _{\!c}\), see, e.g., [7, 12, 20],
Note that the test and trial functions \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}_\textrm{t}= \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\) and \(\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},\textrm{t}} = \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\), which live in the tangent space of the manifold, are used in the weak form except in the penalty term, introduced later in the discrete weak form, where the (arbitrary) three-dimensional \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\) and \(\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) are used. With the definition of the stress tensor, i.e., Eq. (19), the first term on the left hand side may be written as
In [8] a similar weak form is stated where the tangentiality of the velocity field is enforced by a Lagrange multiplier instead of the penalty method as used herein. Further discussions about weak forms with different strategies to enforce the tangentialty of the velocities with a focus on mathematical details including the definition of the applied function spaces are found in [7]. The weak form given here in Eqs. (31) and (32) is used to obtain the FE solution for one single surface which is extended to all level set surfaces next.
4.3 Weak form for all level sets in a bulk domain
Continuous weak form. To obtain the weak form which is required for the simultaneous solution of all level sets \(\Gamma _{\!c}\) embedded in a bulk domain \(\Omega \), Eqs. (31) and (32) are integrated over the level-set interval from \(\phi ^{\min }\) to \(\phi ^{\max }\). There follows for the weak form of stationary Stokes flow, analogously to Eqs. (31) and (32),
Applying the co-area formula over the domain, i.e., Eq. (12), the double integrals \(\int _{\phi ^{\min }}^{\phi ^{\max }}\int _{\Gamma _{\!c}} \bullet \,\textrm{d}\Gamma \;\textrm{d}c\) over the interval of level sets and the domain of one level set can be converted to an integral over the bulk domain \(\int _{\Omega } \bullet \,\Vert \nabla \phi \Vert \,\textrm{d}\Omega \). With Eq. (13), the procedure is analogous for the boundary terms. Additionally, the following function spaces are introduced:
These function spaces may be seen as the bulk-equivalent over \(\Omega \) compared to those given in Eqs. (27) to (29) w.r.t. individual surfaces \(\Gamma _c\). Now, the weak form of the stationary Stokes flow on all manifolds \(\Gamma _{\!c}\) over the bulk domain \(\Omega \) can be formulated similar to the weak form on one manifold, see Sec. 5.1. Using the introduced function spaces, the trial and test functions \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}_\textrm{t}= \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\) and \(\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},\textrm{t}} = \mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}} \,\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\), respectively, the weak form of the stationary Stokes flow on all surfaces embedded in a bulk domain is obtained as: Given a (constant) shear viscosity \(\mu \in {\mathbb {R}}^+\), body forces \(\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \), and boundary tractions \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) on \(\partial \Omega _{\textrm{N}}\), find the velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Omega }\) and the pressure field \(p\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{p}^{\Omega }\) such that for all test functions \(\left( \mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}},w_{p}\right) \in {\mathcal {V}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Omega }\times {\mathcal {V}}_{p}^{\Omega }\), there holds in \(\Omega \)
Discretization of the weak form. The bulk domain \(\Omega \) is discretized by a conforming, three-dimensional mesh of higher-order tetrahedral or hexahedral Lagrange elements of order \(q_k\). The resulting mesh is an approximation \(\Omega _{q_k}^{h}\) of \(\Omega \) with the nodal coordinates denoted as \(\mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}_{i}\) where \(i=1,\dots ,n_{q_k}\) and \(n_{q_k}\) being the number of nodes in the mesh. Note that the mesh is conforming to the boundary of the bulk domain \(\partial \Omega \), however, it does not have to be aligned with the level sets \(\Gamma _{\!c}\). This is why the resulting FEM can be seen as a hybrid between the conforming Surface FEM (i.e., classical FEM) and non conforming fictitious domain methods, e.g., the Trace FEM.
Finite element spaces of different orders are involved as usual when the FEM is applied to (Navier–)Stokes flow formulated in stress-divergence form (e.g., Taylor-Hood elements [62]). Global \(C^{0}\)-continuous basis functions \(B_{i}^{q_k}\!\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) are introduced with \(i = 1,\ldots ,{n_{q_k}}\). These basis functions span a \(C^{0}\)-continuous finite element space of order \(q_k\) defined as
Note that only the coordinates of the geometry mesh are needed to generate the basis \(\{B_{i}^{q_k}\!\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\!\left( \mathchoice{\displaystyle \varvec{r}}{\textstyle \varvec{r}}{\scriptstyle \varvec{r}}{\scriptscriptstyle \varvec{r}}\right) \right) \}\). Different degrees of polynomial orders are introduced: (i) \(q_{\textrm{geom}}\) for the mesh which describes the geometry of the bulk domain approximately, (ii) \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) for the velocity field, and (iii) \(q_{p}\) for the pressure field, i.e., in Eq. (42) holds \(k\in \left\{ \textrm{geom},\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},p\right\} \). Note that this is only an isoparametric map if \(q_k = q_{\textrm{geom}}\). Analogously to a typical setup in classical FEM for (Navier–)Stokes equations, Taylor–Hood elements where \(q_{p}\) = \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}-1\), c.f., [62], are used. For the geometry, \(q_{\textrm{geom}}\) = \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}+1\) is chosen, c.f., [8]. It is also possible to apply stabilization techniques to obtain a stable FEM with respect to the well-known Ladyzhenskaya–Babuška–Brezzi (LBB) condition [63‐66] in combination with equal element orders for the velocity and pressure. This is discussed in more detail in the context of Navier–Stokes equations below and in a forthcoming publication. Furthermore, the level-set function \(\phi \) is replaced by its interpolation \(\phi ^{h}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {Q}}_{q_{\textrm{geom}}}^{h}\) with prescribed nodal values \({\hat{\phi }}_{i}=\phi \left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}_{i}\right) \). Based on Eq. (42), the following discrete test and trial function spaces are introduced
Discretizing the continuous weak form of the stationary Stokes flow given above, i.e., Eqs. (34) and (35), leads to the discrete weak form which is stated as: Given a shear viscosity \(\mu \in {\mathbb {R}}^+\), penalty parameter \(\alpha \), body forces \(\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \), and boundary tractions \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) on \(\partial \Omega ^h_\textrm{N}\), find the velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Omega ,h}\) and the pressure field \(p^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{p}^{\Omega ,h}\) such that for all test functions \(\left( \mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}^h_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}},w_{p}^h\right) \in {\mathcal {V}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Omega ,h}\times {\mathcal {V}}_{p}^{\Omega ,h}\), there holds in \(\Omega ^h\)
which is closely related to an extension of the weak form proposed in [7] for a single surface. Note that we omit the superscript h on geometric quantities and differential operators, e.g., the normal vector \(\mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}^h \rightarrow \mathchoice{\displaystyle \varvec{n}}{\textstyle \varvec{n}}{\scriptstyle \varvec{n}}{\scriptscriptstyle \varvec{n}}\), the level-set function \(\phi ^h \rightarrow \phi \), and \(\nabla _{\Gamma }^{\textrm{dir},h} \rightarrow \nabla _{\Gamma }^{\textrm{dir}}\), for brevity. The penalty parameter is defined as \(\alpha = h^{-2}\), analgously to [10, 52, 67], with the characteristic mesh size h. Herein, h is defined as the minimum length of the element. This definition applies to all penalty parameters in the remainder of this paper. Furthermore, note that the degrees of freedom (DOFs) for velocities and pressure are usually higher than for a single surface where the flow problem is solved using Surface FEM. However, herein, the flow problem is solved on all embedded surfaces at once. Therefore, a direct comparison of the number of DOFs is difficult because the aim of our method is not to compute the flow on one single surface but on a family of surfaces simultaneously.
This leads to a system of equations with a saddle point structure. The numerical results shown next confirm higher-order convergence rates for the application of the Taylor–Hood elements introduced above.
4.4 Numerical results for stationary Stokes flow
In this section, we show higher-order convergence studies to verify the simultaneous solution method. With a known exact (analytic) velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}^{\textrm{ex}}\), the error \(\varepsilon _{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) in the velocity components is defined as
Furthermore, the integrated residual errors w.r.t. the momentum equation (24), \(\varepsilon _{\textrm{mom}}\), and the continuity equation (25), \(\varepsilon _{\textrm{cont}}\), are evaluated. These error measures do not rely on an analytical solution, hence, these can be used to verify also more advanced test cases where no analytical solutions are known. Similar error measures for (incompressible Navier–) Stokes flow on one surface have been used in [8] and for the simultaneous solution in the context of structural mechanics, e.g., in [27, 49]. For stationary Stokes flow on all embedded manifolds, these error measures are defined as
Note the summation of contributions over element interiors \(\Omega ^{\textrm{el},i}\).
The following numerical example for stationary Stokes flow is inspired by [8] and a summary of the results presented herein is given in [51] by the authors. Axisymmetric surfaces are considered featuring a height of \(L = 3\) and a variable radius of \(r(z,r_0) = r_0 + \nicefrac {1}{5} \,\sin (1+3\,z)\) with \(z \in \left[ 0,L\right] \) and the radius \(r_0 \in \left[ 0.8,1.2\right] \) at the bottom, i.e., at \(z=0\). The outer surface of the bulk domain \(\Omega \) coincides with the axisymmetric surface with radius \(r(z,r_0 = 0.8)\) and \(r(z,r_0 = 1.2)\). At the bottom, i.e., at \(z=0\), and the top, i.e., at \(z=3\), the bulk domain is bounded by horizontal planes, see Fig. 3a. The fluid’s viscosity is \(\mu = 0.1\) and the density is \(\rho = 1\). As in [8] for one surface, the lower boundary at \(z=0\) is the Dirichlet boundary \(\partial \Omega _{\textrm{D}}\) where the inflow in co-normal direction to the surfaces \(\Gamma _{\!c}\) is prescribed and the upper boundary \(\partial \Omega _{\textrm{N}}\) at \(z=3\) is the outflow boundary where zero tractions are applied, see Fig. 3b. The bulk domain \(\Omega \) and some selected surfaces \(\Gamma _{\!c}\) are shown in Fig. 3c.
Fig. 3
Setup for the first numerical example: (a) The level sets which define the bulk domain \(\Omega \), (b) some arbitrary mesh and Dirichlet boundary conditions at the inflow, and (c) the bulk domain in gray and some selected level sets in yellow and the level set with \(r_0=1.0\) in blue
The exact velocity which is required for the convergence study in \(\varepsilon _{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) is computed analogously to [8, 51] as
with \(\hat{r}_0 = r(0,r_0)\) and \(r = r(z,r_0)\). These computed exact velocities are tangent to the surfaces in every point of the bulk domain, analogously to [8] for a single surface. Fig. 4 shows the results of this test case for the velocities and pressure on selected level sets \(\Gamma _{\!c}\) and in comparison to the Surface FEM evaluated for the surface with \(r_0 = 1.0\). For the visualizations of results obtained with the Bulk Trace FEM, physical fields are shown on selected level sets. Therefore, standard methods are used to identify a level set \(\Gamma _c\) within every bulk element in terms of a two-dimensional, linear triangle mesh, see [69, 70]. Then, physical flow quantities (velocity, pressure, vorticities, etc.) are evaluated at the vertices of these reconstructed level-set meshes and visualized using standard tools. Note that the reconstructed meshes are only used for visualization purposes in the post-processing and not needed in the simulation.
Fig. 4
Velocity magnitudes and pressure of the first numerical example are shown. (a) to (d) show results obtained with the Bulk Trace FEM, while in (a) and (b) some arbitrarily selected level sets are shown, (c) and (d) show the surface with \(r_0 = 1.0\). (e) and (f) show results obtained with the Surface FEM where the one considered surface has a radius of 1.0 at \(z=0\) and, therefore, (c) to (f) can be used to compare a Bulk Trace FEM solution with a Surface FEM solution
Fig. 5a shows the results of the error in the velocities \(\varepsilon _{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) which converges with the expected optimal rate of \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}+1\). The energy error \(\varepsilon _e\) converges with \(2\,q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) as seen in Fig. 5b. Furthermore, the residual errors converge with the expected rates \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}-1\) for \(\varepsilon _{\textrm{mom}}\) where second-order derivatives are involved and with \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) for \(\varepsilon _{\textrm{cont}}\) where only first-order derivatives occur. The convergence plots for \(\varepsilon _{\textrm{mom}}\) and \(\varepsilon _{\textrm{cont}}\) are shown in Fig. 5c and d, respectively.
Fig. 5
Convergence results for the simultaneous solution of the axisymmetric test case, (a) \(L_2\)-error of the velocities, (b) energy error, (c) residual error in the momentum equations, and (d) residual error in the continuity equation. The numbers in the legends are the polynomial orders \(\{q_{\textrm{geom}},\, q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}},\,q_p\}\) of the FE function spaces
The well-known Ladyzhenskaya–Babuška–Brezzi (LBB) condition [63‐65] has to be fulfilled for the resulting system of equations, i.e., Eqs. (46) and (47), which is characterized by its saddle-point structure. For recent works on mathematical analysis for Stokes flow on a single surface, see [18, 71, 72]. A rigorous mathematical analysis including a formal proof is beyond the aim of this paper. However, we used a numerical inf-sup test to verify numerically that the applied Taylor–Hood elements are stable for the obtained system of equations. This numerical test cannot replace a definitive analytical proof [73] but it is a good indication that the inf-sup/LBB condition is fulfilled. The following numerical inf-sup test is based on [67, 73‐76]. For Eqs. (46) and (47), the discrete system of equations written in matrix–vector notation follows as
(52)
where \(\mathchoice{\displaystyle \textbf{D}}{\textstyle \textbf{D}}{\scriptstyle \textbf{D}}{\scriptscriptstyle \textbf{D}}\) is a \(m \times m\) diffusion matrix, \(\mathchoice{\displaystyle \textbf{C}}{\textstyle \textbf{C}}{\scriptstyle \textbf{C}}{\scriptscriptstyle \textbf{C}}\) is a \(n \times m\) matrix which represents the continuity equation, \(\underline{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}\) is the sought solution vector of the velocity field and \(\mathchoice{\displaystyle \varvec{p}}{\textstyle \varvec{p}}{\scriptstyle \varvec{p}}{\scriptscriptstyle \varvec{p}}\) the sought solution vector of the pressure field, and \(\hat{\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}}\) is the vector of the applied accelerations on the right hand side. m is the number of unknown velocity components and n is the number of degrees of freedom for the pressure field. The discrete inf-sup condition follows as [67, 76]
with the vectors of coefficients \(\hat{\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}} \in {\mathbb {R}}^m\) and \(\hat{\mathchoice{\displaystyle \varvec{q}}{\textstyle \varvec{q}}{\scriptstyle \varvec{q}}{\scriptscriptstyle \varvec{q}}}\in {\mathbb {R}}^n\) for the corresponding finite element functions, respectively [67, 76]. Furthermore, the \(n \times n\) pressure mass matrix \(\mathchoice{\displaystyle \textbf{M}}{\textstyle \textbf{M}}{\scriptstyle \textbf{M}}{\scriptscriptstyle \textbf{M}}\) is introduced. The Schur complement of the matrix in Eq. (52) is defined as [67]
with the eigenvector \(\mathchoice{\displaystyle \varvec{y}}{\textstyle \varvec{y}}{\scriptstyle \varvec{y}}{\scriptscriptstyle \varvec{y}}\) and the eigenvalues \(\lambda \).
The smallest non-zero eigenvalue \(\lambda ^{\star }\) of Eq.(55) is the squared inf-sup constant \(\beta _h\), i.e., \(\sqrt{\lambda ^{\star }} = \beta _h\). In the numerical inf-sup test, the mesh is refined and the inf-sup constant \(\beta _h\) stays constant with the refinement for stable element pairs for the discretization of velocities and pressure. For non-stable elements, the values for the inf-sup constant decrease. We use the example from Sec. 4.4, keep the boundary conditions but omit the specific material parameters [75] and set these to 1. The applied Taylor–Hood elements with \(k_p = k_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} -1\) and an element pair with equal order elements, i.e., \(k_p = k_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\), which is expected to be unstable, are tested. For the numerical inf-sup test presented herein, elements of order \(k_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} = 2\) are used. The results depicted in Fig. 6 show the expected behaviour of the inf-sup constant for h-refinement of the mesh for both tested element pairs. This result shows that the applied Taylor–Hood elements are expected to be stable with respect to the LBB condition and it is sufficient to apply Taylor–Hood elements for the simultaneous solution of (Navier–)Stokes flows on all surfaces embedded in a bulk domain. However, a detailed mathematical analysis is required in the future to proof this formally.
Fig. 6
Results of the numerical inf-sup test. The blue line shows the results for the Taylor–Hood element which is stable, i.e., the inf-sup constant does not decrease. The black curve shows the inf-sup constants over the mesh refinement for an equal-order element pair for velocity and pressure which is unstable as expected
Stationary Navier–Stokes flow includes a non-linear advection term in the momentum equation. Hence, the strong form of stationary Stokes flow, given in Eqs. (24) and (25), is similar to the strong form of stationary Navier–Stokes flow. While the incompressibility constraint, i.e., Eq. (25) and the boundary conditions remain unchanged, the advection term is added to Eq. (24) leading to [8]
where \(\varrho \in {\mathbb {R}}^+\) is the (constant) fluid density.
5.2 Weak form for one level set
The weak form of the stationary Navier–Stokes equations on one surface is obtained analogously to the weak form of stationary Stokes flow, described in Sec. 4.2. The function spaces are the same as given in Eqs. (27) to (29). Furthermore, the weak form of the incompressiblilty constraint equals Eq. (32). Hence, the only equation which is different in the weak form of stationary Navier–Stokes flow compared to stationary Stokes flow is the momentum equation [8], calculated as
Again, the only change is in the momentum equation while the other equations and function spaces remain unchanged from Sec. 4.2. For the momentum equation follows
The resulting continuous weak form is discretized analogously to the case of the stationary Stokes flow as described in Sec. 4.3. This leads to the discretized weak form of the stationary Navier–Stokes flow for the simultaneous analysis on all level sets which are embedded in a bulk domain: Given a shear viscosity \(\mu \in {\mathbb {R}}^+\), fluid density \(\varrho \in {\mathbb {R}}^+\), penalty parameter \(\alpha \), body forces \(\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \), and boundary tractions \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) on \(\partial \Omega ^h_\textrm{N}\), find the velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Omega ,h}\) and the pressure field \(p^h\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \in {\mathcal {S}}_{p}^{\Omega ,h}\) such that for all test functions \(\left( \mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}^h_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}},w_{p}^h\right) \in {\mathcal {V}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}^{\Omega ,h}\times {\mathcal {V}}_{p}^{\Omega ,h}\), there holds in \(\Omega ^h\)
Picard’s iteration, a fixed-point iteration scheme, is applied to solve the nonlinear system of equations which results due to the advection term in the (stationary) Navier–Stokes equations. For further details on this iteration scheme, see, e.g., Sec. 6.3 in [1]. As an alternative to Picard’s iteration, Newton’s method could be applied.
For the discretization, Taylor–Hood elements, i.e., \(q_{p}\) = \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}-1\), can be applied as in Sec. 4. In addition, element pairs of equal-order for the velocity and pressure discretization, i.e., \(q_{p}\) = \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\) can be used within the PSPG stabilization framework, c.f., [1, 77‐79]. Then, we add the PSPG stabilization term to the left hand side of Eq. (59), defined as
For the geometry discretization we use elements of order \(q_{\textrm{geom}}\) = \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}+1\). The stabilization parameter \(\tau _{\textrm{PSPG}}\) is defined as a function of the element size h similar to [78], for concrete definitions, see Eq. (63) below.
For large Reynolds numbers, the solution, either obtained with Taylor–Hood elements or the PSPG stabilization, may become unstable. In such cases, the SUPG method could be applied [1, 78, 80, 81]. However, in the context of the simultaneous solution of flow problems on all level sets within a bulk domain, this is beyond the scope of this paper.
5.4 Numerical results for stationary Navier–Stokes flow
5.4.1 Flow around an obstacle
This test case is inspired by the two-dimensional bench mark in [82]. First, two-dimensional geometries are embedded in a flat three-dimensional bulk domain. The bulk domain including the embedded surfaces is then mapped to obtain a curved bulk domain with curved embedded surfaces on which the stationary Navier–Stokes flows are simulated. Three different mappings are given here, see Fig. 7 for the resulting geometries. These are used for several examples in the remainder of this paper.
(62)
with \(\bar{s} = -(1+\bar{q}) \,(b-0.205) \,\cos (\pi /6\,(1-a))\), \(\bar{q} = -0.1a^2 + 0.2a\), and \( \tilde{s} = -(1-0.1a^2 + 0.2a) \,(b-0.205) \,\cos (\pi /6\,(1-a))\). a, b, and c are the Cartesian coordinates in the undeformed, i.e., flat domain and x, y, and z are the Cartesian coordinates in which the deformed, i.e., curved domain, is described. The flat bulk domain is defined as the channel \([0,2.20] \times [0,0.41] \times [0,1/3]\) and the obstacle’s radius (the cylinder in the flat 2-dimensional channel) is \(r_b = 0.05\) at the bottom \((c=0)\), \(r_t = 0.06\) at the top \((c=1/3)\), and varies linearly in between. The obstacle is placed slightly unsymmetrically in the channel in y-direction. The level-set function is defined as \(\phi \left( \mathchoice{\displaystyle \varvec{a}}{\textstyle \varvec{a}}{\scriptstyle \varvec{a}}{\scriptscriptstyle \varvec{a}}\right) =c\). Figure 7 visualizes the geometries which are defined by these mappings.
Fig. 7
The three different mappings \(\varphi _i\) with \(i \in [1,2,3]\) as defined in Eq. (62) for the geometry definition of the cylinder flow test case. The bulk domain is shown in light blue and some arbitrarily selected level sets are shown in different colours. Note that the view point is different for each geometry for a better visualization
Figure 8 shows the discretization of the three geometries with some example mesh (coarser than the mesh used for the computation) including the velocity profiles prescribed at the inflow. The red arrows indicate the inflow boundary condition and the blue dots are nodes on the no-slip boundary. Note that the geometry and the inflow velocities (length of the red arrows) are scaled differently for each map in this visualization. The inflow boundary condition is the quadratic velocity profile with \(u_{\textrm{max}} = 1.5\) and \(v=0\) in the flat case and is mapped by the Jacobians of the respective mappings to ensure tangentiality of the velocities at the inflow boundary. At the outflow, traction-free boundary conditions are prescribed. The density is defined as \(\varrho = 1\) and the viscosity is \(\mu = 0.01\). For the computation, the bulk domain \(\Omega \) is discretized by a mesh with 3760 elements. The orders for the geometry, the velocity, and the pressure meshes are \(q_{\textrm{geom}} = 3\), \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} = 2\), and \(q_{p} = 1\), respectively.
Fig. 8
Some mesh with highlighted nodes where velocities are prescribed. Mapping function \(\varphi _1\) in (a), \(\varphi _2\) in (b), and \(\varphi _3\) in (c), respectively. Red arrows are shown for the prescribed velocities at the inflow boundary, blue dots indicate the no-slip condition along the channel walls and the obstacle
The results of this test case for mapping \(\varphi _3\) are shown in Fig. 9; mappings \(\varphi _1\) and \(\varphi _2\) are used in other test cases below. For the other two mappings similar results are obtained but not shown for brevity. To verify that the simultaneous solution produces the same results for selected level sets as indepedent, successive simulations with the Surface FEM (SRF) [8] on these level sets, the differences of the pressure at the front and back nodes of the cylindrical obstacle are compared, see 9b. The selected level sets \(\Gamma _{\!c}\) with \(c=\{0, 1/6, 1/3\}\) are used for the comparison. Furthermore, we visually compare the results for the level set with \(c = 1/6\) for the velocities (Euclidean norm) and the pressure in Figs. 9c and e for the Bulk Trace FEM (BTF) and Figs. 9d and f for the Surface FEM (SRF). Excellent agreements are observed, confirming that the Bulk Trace FEM is successful in simultaneously solving for flow fields on all level sets.
Fig. 9
Results for mapping \(\varphi _3\). The velocity magnitudes obtained with the Bulk Trace FEM on selected level sets are shown in (a) and (c), while in (d), a Surface FEM solution is shown on one level set. The pressure difference at two nodes on the cylindrical obstacle, obtained with the Bulk Trace FEM is shown in (b) and compared to the solution obtained with the Surface FEM. The pressure field obtained with the Bulk Trace FEM is shown in (e) and with the Surface FEM in (f)
This test case is inspired by the bench mark from Ghia et al. [83] and its application to stationary Navier–Stokes flow on one curved surface in [8]. Herein, a flat bulk domain with embedded flat surfaces is mapped into a curved geometry. In the flat bulk domain, each point is defined by the coordinates \(\left( a,b,c\right) \), i.e., \({\mathcal {P}}_{\textrm{flat}} \left( \mathchoice{\displaystyle \varvec{a}}{\textstyle \varvec{a}}{\scriptstyle \varvec{a}}{\scriptscriptstyle \varvec{a}}\right) = \left[ a,b,c\right] ^{\textrm{T}}\). These points are mapped with \(\varphi \) into the Euclidean coordinate system \(\left( x,y,z\right) \), i.e., \({\mathcal {P}}_{\textrm{curved}} \left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) = \left[ x,y,z\right] ^{\textrm{T}} = \varphi \left( {\mathcal {P}}_{\textrm{flat}}\right) \) with
The level-set function is defined as \(\phi \left( \mathchoice{\displaystyle \varvec{a}}{\textstyle \varvec{a}}{\scriptstyle \varvec{a}}{\scriptscriptstyle \varvec{a}}\right) =c\). Figure 10 shows the applied mapping for this test case with \(\alpha = 0.4\) for the bulk domain \(\Omega \) with some embedded level sets \(\Gamma _{\!c}\).
Fig. 10
The map \(\varphi \) for the geometry definition of the driven cavity test case. The bulk domain is shown in light blue and some arbitrarily selected level sets are shown in different colours
The shear viscosity is defined as \(\mu = 0.01\). The Dirichlet boundary conditions along the straight part of the Dirichlet boundary \(\partial \Omega _{\textrm{D},\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},\textrm{s}} \in \left[ x_{\textrm{D}},y_{\textrm{D}}=1,z_{\textrm{D}}\right] \) are \(u(z) = 1-4\,z_{\textrm{D}}\) and \(v=w=0\) to get more variability in the results for each embedded surface. The height along the straight Dirichlet boundary is defined as \(z_{\textrm{D}} \in [0,0.125]\) and \(x_{\textrm{D}} \in [0,1]\) is considered. On the other nodes of the Dirichlet boundary, i.e., \(\partial \Omega _{\textrm{D},\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} \backslash \partial \Omega _{\textrm{D},\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},\textrm{s}}\), no-slip boundary conditions, i.e., \(u=v=w=0\), are prescribed. At the nodes on \(\partial \Omega _{\textrm{N},p} = \left[ 0.5,0,z\right] ^{\textrm{T}}\), the pressure \(p=0\) is prescribed. For the computations shown herein, a mesh of \(40 \times 40 \times 2\) elements is used and shown together with the boundary conditions in Fig. 11.
Fig. 11
The mesh with \(40 \times 40\) elements in x-y-direction and 2 elements in thickness direction from two different perspectives in (a) and (b), respectively. The Dirichlet boundary conditions are shown as red arrows for the prescribed non-zero velocity (for a better visualization the arrow length is scaled and the arrows are only shown on some selected points), the blue dots indicate the no-slip condition. The red triangles show the nodes at which the pressure boundary conditions are prescribed
The results of this test case are given in Fig. 12: (a) shows the velocity magnitudes on some selected level sets. Fig. 12b shows velocity profiles in analogy to [83] and in particular to Fig. 10 in [8]. Along the horizontal centre line, the velocity profile for the velocity component v and along the vertical centre line, the velocity profile for the velocity component u are shown. The lines with the circular markers show the results for a selected surface for which the solution is obtained by a Surface FEM computation using Taylor–Hood elements. The triangular markers indicate the profiles for the Bulk Trace FEM solution obtained with equal-order elements for the velocities and pressure with order \(q_{p}\) = \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} = 2\) and PSPG stabilization. The stabilization parameter is defined in analogy to SUPG/PSPG stabilization for Euclidean geometries [78] as
Note that the norm of the velocities \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}_{\textrm{el,node}}\) is evaluated within each element at each node instead of using an averaged velocity for the entire element because the velocities might be different on every surface which intersects with the element. \(h_{\textrm{el}}\) is the characteristic length of the element. The curves in Fig. 12b belong to the following level sets \(\Gamma _{\!c}\): LSF 1 in the legend refers to \(\Gamma _{\!c=0.0}\), LSF 2 in the legend refers to \(\Gamma _{\!c=0.0625}\), and LSF 3 in the legend refers to \(\Gamma _{\!c=0.125}\) where the index c refers in this context to the height c in the flat domain which is identical to the height z of the straight Dirichlet boundary \(\partial \Omega _{\textrm{D},\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},\textrm{s}}\) of the mapped bulk domain. The curves show that the Surface FEM and Bulk Trace FEM solutions are almost identical on the selected level sets. Fig. 12c shows the velocity magnitudes on the level set \(\Gamma _{\!c=0.0625}\) and Fig. 12d shows the same quantity obtained for this surface in a Surface FEM computation (with Taylor–Hood elements, no stabilization). Comparing these two figures, also verifies that visually the same solution is obtained by both numerical schemes and, therefore, that the Bulk Trace FEM leads to correct solutions for each embedded surface.
Fig. 12
Results for the simultaneous solution of the driven cavity test case. The velocity magnitudes obtained with the Bulk Trace FEM on selected level sets are shown in (a) and (c), while in (d), a Surface FEM solution is shown on one level set. Profiles of the velocity components u along the vertical centre axis and v along the horizontal centre axis are shown for three different surfaces (LSF) in (b)
Instationary Navier–Stokes flow is characterized by the time-dependency of the physical fields, i.e., the velocity \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}} = \mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \) and the pressure \(p = p\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \) are functions of space and time t. Note that the curved surfaces (domains of interest) do not depend on time, i.e., they are fixed. The momentum equation for instationary Navier–Stokes flow is given as [7, 8]
We consider the time interval \(\tau = \left[ 0,T\right] \) and the space-time domain \(\Gamma _{\!c} \times \tau \) in which Eqs. (25) and (64) are solved. Derivatives w.r.t. time are denoted as \({\dot{\square }} = \nicefrac {\partial \square }{\partial t}\). Note that time is indicated by t whereas \(\square _\textrm{t}\) refers to some tangential/in-plane vector/tensor quantities.
Additionally, the boundary conditions, i.e., Eqs. (20) and (21), are extended in time. There are prescribed velocities \(\hat{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \) along \(\partial \Gamma _{\!c,\textrm{D}} \times \tau \) and tractions \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}_\textrm{t}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \) along \(\partial \Gamma _{\!c,\textrm{N}} \times \tau \) over time [8, 51]. Furthermore, the initial condition is defined as
Analogously to Sec. 4.2 and 5.2, the continuous weak form of the instationary Navier–Stokes problem is stated as follows [8]: Given the fluid density \(\varrho \in {\mathbb {R}}^{+}\), viscosity \(\mu \in {\mathbb {R}}^{+}\), body force \(\varrho \,\mathchoice{\displaystyle \varvec{g}}{\textstyle \varvec{g}}{\scriptstyle \varvec{g}}{\scriptscriptstyle \varvec{g}}_\textrm{t}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \) in \(\Gamma _{\!c}\times \tau \), traction \(\hat{\mathchoice{\displaystyle \varvec{t}}{\textstyle \varvec{t}}{\scriptstyle \varvec{t}}{\scriptscriptstyle \varvec{t}}}_\textrm{t}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \) on \(\partial \Gamma _{\!c,\textrm{N}}\times \tau \), and initial condition \(u_{0}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}}\right) \) on \(\Gamma _{\!c}\) at \(t=0\) according to Eq. (65), find the velocity field \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}_\textrm{t}\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \in {\mathcal {S}}^{\Gamma }_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} \times \tau \) and pressure field \(p\left( \mathchoice{\displaystyle \varvec{x}}{\textstyle \varvec{x}}{\scriptstyle \varvec{x}}{\scriptscriptstyle \varvec{x}},t\right) \in {\mathcal {S}}^{\Gamma }_{p} \times \tau \) such that for all test functions \(\left( \mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}},\textrm{t}},w_{p}\right) \in {\mathcal {V}}^{\Gamma }_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\times {\mathcal {V}}^{\Gamma }_{p}\), there holds in \(\Gamma _{\!c}\times \tau \)
Integration over the level-set interval and application of the co-area formula for the domain and the boundary, respectively, leads to the weak form of the instationary Navier–Stokes equations for all level sets embedded in a bulk domain. The continuous weak form is omitted for brevity and we directly give the discrete weak form. Note that the space-time domain for one considered surface was introduced as \(\Gamma _{\!c} \times \tau \). For the case where all surfaces \(\Gamma _{\!c}\) over some bulk domain are considered, the space-time domain is defined as
Note that we again omit the superscript h at geometric quantities and differential operators for brevity. This discrete weak form leads to a system of non-linear semidiscrete equations in time \(t \in \tau \)
with an initial condition \(\underline{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}\left( t=0\right) \). \(\mathchoice{\displaystyle \textbf{T}}{\textstyle \textbf{T}}{\scriptstyle \textbf{T}}{\scriptscriptstyle \textbf{T}}\) is a mass matrix representing the time dependency, \(\mathchoice{\displaystyle \textbf{D}}{\textstyle \textbf{D}}{\scriptstyle \textbf{D}}{\scriptscriptstyle \textbf{D}}\) contains the diffusion part, \(\mathchoice{\displaystyle \textbf{A}}{\textstyle \textbf{A}}{\scriptstyle \textbf{A}}{\scriptscriptstyle \textbf{A}}\) is the advection matrix, \(\mathchoice{\displaystyle \textbf{C}}{\textstyle \textbf{C}}{\scriptstyle \textbf{C}}{\scriptscriptstyle \textbf{C}}\) comes from the continuity equation, and \(\mathchoice{\displaystyle \textbf{G}}{\textstyle \textbf{G}}{\scriptstyle \textbf{G}}{\scriptscriptstyle \textbf{G}}\) represents the penalty term. The vectors \(\underline{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}= [\tilde{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t},\tilde{\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}}_\textrm{t},\tilde{\mathchoice{\displaystyle \varvec{w}}{\textstyle \varvec{w}}{\scriptstyle \varvec{w}}{\scriptscriptstyle \varvec{w}}}_\textrm{t}]^{\textrm{T}}\) and \(\mathchoice{\displaystyle \varvec{p}}{\textstyle \varvec{p}}{\scriptstyle \varvec{p}}{\scriptscriptstyle \varvec{p}}\) contain the sought values for the velocities and the pressure, respectively. Note that \(\tilde{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}= [u_{\textrm{t},1},u_{\textrm{t},2},\ldots ,u_{\textrm{t},n_q}]^{\textrm{T}}\) is the vector of the nodal values of the velocity component \(u_{\textrm{t}}\). This system of equations is advanced in time by the Crank–Nicolson method.
6.4 Numerical results for instationary Navier–Stokes flow
6.4.1 Flow on a torus
This test case is based on similar examples for a single surface which are presented in several publications to verify the approximation of the solution of surface Navier–Stokes equations, e.g., in [84, 85] with finite elements, in [86] with discrete exterior calculus, and in [16] with finite differences. The considered surfaces \(\Gamma _{\!c}\) are tori with a major radius \(R = 2\) and minor radius \(r \in \left[ 0.25,0.75\right] \). The tori are described by the level sets of the function \(\phi = \left( \sqrt{x^2+y^2}-R\right) ^2+z^2-r^2\). The bulk domain \(\Omega \) is the toroidal ring bounded by \(\Gamma _{\!c}\left( r = 0.25\right) \) and \(\Gamma _{\!c}\left( r = 0.75\right) \). Fig. 13 shows the bulk domain in light blue and three different embedded tori \(\Gamma _{\!c}\).
Fig. 13
The bulk domain in light blue and three different torus surfaces in (a) yellow, (b) green, and (c) red
On these compact manifolds, a flow takes place which is initiated by initial conditions of the velocity \(\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}_\textrm{t}= \left[ u,v,w\right] ^{\textrm{T}}\) tangential to the toroidal surfaces and defined on the nodes with coordinates \(\left[ x,y,z\right] ^{\textrm{T}}\) of the bulk domain as
(73)
with \(r_{xy} = x^2+y^2\). This definition of the initial conditions is similar to [16] where single surfaces are considered. The fluid’s density is \(\varrho = 1\) and the viscosity is set to \(\mu = 1\). The time step size in the Crank-Nicolson time stepping scheme is \(\Delta t = 0.1\) for \(\tau = [0,60]\). Taylor–Hood elements of order \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} = 2\), and \(q_{p} = 1\) are used. For the geometry, we use again \(q_{\textrm{geom}} = 3\). Fig. 14 shows the velocity on selected level sets at different times.
Fig. 14
Results for the simultaneous solution of flow on a torus with major radius \(R=2\) and minor radius (a) and (b) \(r = 0.25\), (c) and (d) \(r = 0.5\), and (e) and (f) \(r = 0.75\) at different times. The light blue surface is the boundary surface of the bulk domain \(\Omega \)
As can be seen in Figs. 14 and 15, the flow becomes stationary and the kinetic energy of each surface defined as \(\textrm{E}_{\textrm{kin}} = \int _{\Gamma _{\!c}} \frac{1}{2} \varrho |\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}} |^2 \,\textrm{d}\Gamma \) decreases to a constant value. Fig. 15 shows the kinetic energy normalized by its value at the beginning \(\bar{\textrm{E}}_{\textrm{kin}} = \textrm{E}_{\textrm{kin}} / \textrm{E}_{\textrm{kin}}(t=0)\) over the time for the three surfaces which are also shown in Fig. 14. This shows that the simultaneous solution with the Bulk Trace FEM leads to the same result when compared to solving each surface independently by the Surface FEM.
Fig. 15
The normalized kinetic energy over the time for three selected torus surfaces (LSF 1 to 3), once from the simultaneous solution with the Bulk Trace FEM (BTF) and once obtained for each surface independently with the Surface FEM (SRF)
In this section, we show the simultaneous solution of test cases on curved surfaces which were introduced in Sec. 5.4.1 in the context of stationary Navier–Stokes flow and are inspired by the Schäfer–Turek benchmark for a flat 2-dimensional domain, c.f., [82]. The definition of the geometry is the same as in Sec. 5.4.1, i.e., mappings \(\varphi _1\) to \(\varphi _3\) as given in Eq. (62) and shown in Fig. 7. Furthermore, the boundary conditions and the material parameters except for the viscosity are the same as used for the stationary Navier–Stokes flow above.
The solutions for each surface are strongly nonlinear and, therefore, differ (significantly) from each other. Depending on the geometry and the fluid’s parameters, the solution fields may be not smooth (enough) any more to use the classical (isotropic) Taylor–Hood element pairs for velocity and pressure. One possibility to overcome this problem is to introduce an anisotropic Taylor–Hood element pair. In the anisotropic case, the discretization of the bulk domain with respect to the level sets is as in the isotropic case, i.e., \(q_p = q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}-1\), while in the thickness direction (‘normal’ to the surfaces) the order is \(q_p = q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}\). Fig. 16 shows a comparison of some discretization using isotropic and anisotropic Taylor–Hood elements for a generic geometry. The black dots are the DOFs of the velocity, the blue circles are the DOFs for the pressure and the red dots are the additional DOFs of the pressure which result from the anisotropic order of the elements used in the mesh for the pressure discretization. Applying this concept in the Bulk Trace FEM for the instationary Navier–Stokes flow leads to satisfactory results. However, the disadvantage of the method is that the meshes must be somewhat aligned to the level sets.
Another strategy to overcome this problem and get sufficient results in the simultaneous solutions, is to use element pairs of equal-order for velocity and pressure together with a stabilization scheme, i.e., PSPG stabilization [77] or the Brezzi–Pitkäranta stabilization [87] which is applied in the context of the Trace FEM for one single surface in [18]. For the PSPG stabilization, we add to the left hand side of Eq. (69) the following term
which is to be evaluated as a sum over the element interiors as usual in residual-based stabilization schemes. For the Brezzi–Pitkäranta stabilization [18, 87], we add
Note that for Euclidean geometries, i.e., ‘classical’ Navier–Stokes flows in \({\mathbb {R}}^d\), \(d = \{2,3\}\) (not in a manifold context), the Brezzi–Pitkäranta stabilization is equivalent to the pressure term in the PSPG stabilization. This is not the case in the context of curved surfaces because the projector \(\mathchoice{\displaystyle \textbf{P}}{\textstyle \textbf{P}}{\scriptstyle \textbf{P}}{\scriptscriptstyle \textbf{P}}\) is involved in the definition of the (Boussinesq–Scriven surface) stress tensor, see Eq. 19. The stabilization parameter for the PSPG stabilization is defined as
which is analogously to Eq. (63) and closely related to [78]. For the Brezzi–Pitkäranta stabilization, the parameter is chosen as \(\tau _p = h_\textrm{el}\), see [18].
Fig. 16
The anisotropic mesh concept for Taylor–Hood elements. (a) and (b): mesh pair with classical (isotropic) Taylor–Hood elements; (a) and (c): mesh pair with anisotropic Taylor–Hood elements. The red dots in (c) are the additional DOFs of the anisotropic case. Some level sets within the meshes are shown in grey
For the following computations, the shear viscosity is set to \(\mu = 0.0015\) and mapping \(\varphi _1\) is considered. Fig. 17 shows the pressure difference between the front and the back node at the obstacle over time for mapping \(\varphi _1\) and computations done with iso- and anisotropic Taylor–Hood elements. Note that for the case of anisotropic Taylor–Hood elements, two surfaces more are shown in the figures than for the isotropic Taylor–Hood elements because the anisotropic mesh includes more nodes due to the anisotropic pressure mesh. The bulk domain \(\Omega \) is discretized with 3760 elements. The order for the velocities is \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} = 2\), and for the pressure \(q_{p} = 1\). The geometry is again considered with elements of third order, \(q_{\textrm{geom}} = 3\). The continuous lines show results obtained with the Bulk Trace FEM and the dashed lines show the results obtained with the Surface FEM with Taylor–Hood elements in all subfigures. These plots show that good agreement is obtained. There is some offset between the results of the Bulk Trace FEM and the Surface FEM for the isotropic element pairs. However, this does not significantly change the frequency and the pressure values and, therefore, these figures are still an indication for good agreement between the simultaneous solution and the multiple solutions on individual surfaces. Furthermore, the velocity field, the pressure field, and the vorticity field are shown on one selected surface at time \(t=3\). The results of the simultaneous solutions shown in these figures are obtained with the anisotropic Taylor–Hood elements. More results at different time steps and for the mappings \(\varphi _2\) and \(\varphi _3\) are omitted for brevity.
For lower viscosities, i.e., larger Reynolds numbers, isotropic Taylor–Hood elements may not lead to satisfactory results as seen in Fig. 18. Herein, the viscosity for flows on surfaces defined by mapping \(\varphi _1\) is set to \(\mu = 0.001\). In Fig. 18a, isotropic Taylor–Hood elements are used and it can be clearly seen that these results are not usable due to the differences between the surfaces which result from the strong non-linearities in the solution fields. When anisotropic Taylor–Hood elements or equal-order elements with \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} = q_{p} = 2\) and a stabilization technique are applied, the simultaneously computed results are clearly better. Although there are offsets between the simultaneously and for each surface independently obtained solutions, the amplitudes and the frequencies are similar. Figure 18a, b, c, and d show the same flow situation computed with different methods, i.e., isotropic and anisotropic Taylor–Hood elements, Brezzi–Pitkäranta stabilization, and PSPG stabilization, respectively. For the other mappings the situation is similar for flows with lower viscosity, yet not shown for brevity.
Fig. 17
Results of the instationary flow around an obstacle for mapping \(\varphi _1\). The pressure difference on the obstacle over time is shown in (a) obtained with isotropic and in (b) obtained with anisotropic Taylor–Hood elements. (c) shows the velocity field, (d) the pressure field and (e) the vorticity field obtained with the Bulk Trace FEM on a selected surface at time \(t=3\). (f) shows the vorticity field at the same surface at the same time obtained with the Surface FEM for comparison
Results for Navier–Stokes flow with \(\mu = 0.001\) on geometries implied by mapping \(\varphi _1\). (a) to (d) show the pressure difference on the obstacle over time, where (a) is obtained using (isotropic) Taylor–Hood elements (THE), (b) results from anisotropic THE, in (c) the Brezzi–Pitkäranta (BP) stabilization and in (d) PSPG stabilisation is applied. In (e) the velocity magnitudes and in (f) the vorticity on one selected surface are shown
The simultaneous solution of Stokes and stationary/instationary incompressible Navier–Stokes flows on fixed surfaces is proposed in this work. The main goal of this paper is to introduce the methodology, however, two possible applications are outlined in this section. The first example is a design value search. In the second example, the coupling to flows in three-dimensional bulk domains is skteched. The embedded surfaces represent layers with certain properties. In a long-term vision, this can be applied to models for global numerical weather and climate prediction.
7.1 Design value search
The goal of the design value search is to find the surface, i.e., the level set \(\Gamma _{\!c}\), where a certain prescribed design value is reached. Without loss of generality, the approach is motivated by the following example. We search for a specific given value of some pressure difference \(\Delta p\) at the cylinder of the channel flows of Sec. 5.4.1. Therefore, the results of the simultaneous solution as obtained in Sec. 5.4.1 are employed. Within the pressure mesh, some of the points match the front and back nodes on the cylinder, respectively, enabling a direct interpolation of \(\Delta p\) from the bulk mesh. The task is now to determine the level set \(\Gamma _{\!c}\) at which the target design value for the pressure difference is reached. Therefore, a standard procedure to find given values within a FE solution, e.g., interpolation or a root finding method as the Newton–Raphson method, is applied. Such methods are usually available in FEM codes and, therefore, the design value search can be directly performed when the solution has been obtained with the Bulk Trace FEM as proposed herein. Instead of computing solutions for many surfaces successively, only one simulation is required in the present context. Figure 19 shows the desired surfaces at which the target values occur and the diagram of the pressure difference \(\Delta p\) over the level sets for the mappings \(\varphi _1\) to \(\varphi _3\). The target value is marked with a star within these diagrams. The sought quantities of the pressure differences \(\Delta p\) at the cylinder and the obtained surfaces where these target values occur are summarized in Table 1.
Table 1
Given target values for \(\Delta p\) for different mappings \(\varphi _i\) and obtained level sets \(\Gamma _{\!c}\)
mapping \(\varphi _i\)
target value \(\Delta p\)
c (level-set value)
1
4.70
0.248
2
5.50
0.171
3
6.80
0.243
Fig. 19
Results for the design value search for stationary Navier–Stokes flow. (a) and (b) for mapping \(\varphi _1\), (c) and (d) for mapping \(\varphi _2\), and (e) and (f) for mapping \(\varphi _3\). The star shows the target value for which the design value search is done. (a), (c), and (e) show the pressure field at the level set \(\Gamma _{\!c}\) with value c obtained for the given target value
General circulations on planets are characterized by altitudes or pressure levels over the height of the atmosphere. Hence, layered discretizations are used in numerical weather and climate prediction models to simulate these global circulations at specific layers within the atmosphere [45, 46]. The equations are mostly formulated in spherical coordinates, where the physics which takes place on each spherical surface (this tangential motion is often labelled in the literature as ‘horizontal’ motion) is coupled to physical phenomena in normal (radial) direction. However, in the simulation, these two motions are often separated [45, 46, 88, 89]. For the horizontal motions, spectral methods or spherical harmonics are used, while in normal direction a finite difference or finite element scheme is applied. In [90], a three dimensional fluid simulator for spherical spaces based on layer-by-layer staggered finite difference grids is proposed and applied in computer graphic visualizations. In this method and in the usual climate models, the spherical layers are discrete geometries which compose the domain of interest layer-by-layer. We expect that the method proposed herein, where the surfaces (spheres) are embedded continuously in the three-dimensional domain (spherical shell), is well suited for such applications: The bulk domain is the atmosphere and the individual layers are spherical level sets. Furthermore, the governing equations are formulated in terms of the TDC, hence, these are coordinate independent and models are not restricted to spheres. Therefore, some more sophisticated shapes such as ellipsoids, the geoid, i.e., the real geometry of the earth, or rough terrain over the surface of a planet can be considered straightforwardly. In further research projects, the method proposed herein can be extended to such applications where flows on the surfaces are coupled to flows in the embedding three-dimensional space. Thereby, flows can be coupled in a ‘direct’ way and no consecutive computations of the considered flows are required. Hence, for this type of application, the proposed method may be a natural and efficient choice. Although, the governing equations may have to be adopted, e.g., to the shallow water equations, for numerical weather prediction and climate modelling, the main technical aspects for continuously layered models are already given herein.
Further research is required for an application of this method in sophisticated models for numerical weather prediction or computer graphic visualization. However, we give a simple example to elaborate on this idea further in Fig. 20. The example shows some flow in a channel over an idealized hill. The layer at the bottom is identical with the hilly terrain, while the topmost layer is a horizontal one. The layers in-between change smoothly from the curved surface to the horizontal plane. This geometric setup is inspired by two-dimensional idealized sketches of layers as used in the literature about numerical weather prediction, see for instance chapter 7 in [45], extended to the channel flow herein. Fig. 20a shows the boundary of the channel in blue and some, arbitrarily selected layers in different colours, (b) the velocity field on the boundary of the channel and (c) the surface flow on some selected surfaces.
Fig. 20
The setup and results of the coupled flow example. A three-dimensional channel flow is coupled with flows on layers which are continuously embedded in the channel. (a) the geometry of the channel with the hill at the bottom and some layers (surfaces), (b) the velocity magnitude on the boundary of the channel, and (c) on some selected layers
This example illustrates the coupling of some flow in the whole bulk domain with flow on layers which are continuously embedded in the bulk domain. It only serves as an illustration how the individual modelling of flows in three dimensional bulk domains can be coupled to two-dimensional, curved layers.
8 Conclusions and outlook
The simultaneous solution of Stokes (stationary) and Navier–Stokes flows (stationary and instationary) on multiple curved surfaces is proposed in this work. Therefore, the governing equations for one single surface in a coordinate-free formulation in their weak form are used as a starting point and combined with co-area formulas to obtain a weak form which simultaneously applies for all level sets embedded in a three-dimensional bulk domain. The Bulk Trace FEM is employed to approximate the solution of these weak forms using arbitrary (higher-order) background meshes. These meshes are conforming to the boundary of the surfaces and, hence, drawbacks of the classical Trace FEM for single surface solutions, e.g., small cut-scenarios and, therefore, required stabilization techniques, do not occur in the Bulk Trace FEM. Numerical examples validate the method obtaining higher-order convergence rates for stationary Stokes flow. Good agreements to Surface FEM solutions on selected, individual surfaces for stationary and instationary Navier–Stokes flows are confirmed.
The aim of this work is to introduce a conceptual, theoretical framework for flows on multiple curved surfaces. We envision applications in design processes, outlined in Sec. 7.1, and climate and weather models, where flows on layers are coupled with flows in other directions, see Sec. 7.2. In biomechanics, e.g., cell mechanics, this method may be used when the governing equations are extended to flows on moving surfaces and these are then coupled to simultaneously solved structural membranes, as shown in [27], or shells. Such a setup can then be used to identify the shape and structure of (artificial) lipid bilayers.
Further research shall also focus on improvements for instationary Navier–Stokes equations. Therefore, solution strategies should be obtained to better handle advection-induced instabilities in the flow. A detailed investigation of optimal stabilization parameters for Brezzi–Pitkäranta and PSPG stabilization techniques in the simultaneous solution is desirable.
Declarations
Conflict of interest
The authors declare no potential Conflict of interest.
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Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.
where \(\mathchoice{\displaystyle \textbf{D}}{\textstyle \textbf{D}}{\scriptstyle \textbf{D}}{\scriptscriptstyle \textbf{D}}\) is a \(m \times m\) diffusion matrix, \(\mathchoice{\displaystyle \textbf{C}}{\textstyle \textbf{C}}{\scriptstyle \textbf{C}}{\scriptscriptstyle \textbf{C}}\) is a \(n \times m\) matrix which represents the continuity equation, \(\underline{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}}_\textrm{t}\) is the sought solution vector of the velocity field and \(\mathchoice{\displaystyle \varvec{p}}{\textstyle \varvec{p}}{\scriptstyle \varvec{p}}{\scriptscriptstyle \varvec{p}}\) the sought solution vector of the pressure field, and \(\hat{\mathchoice{\displaystyle \varvec{f}}{\textstyle \varvec{f}}{\scriptstyle \varvec{f}}{\scriptscriptstyle \varvec{f}}}\) is the vector of the applied accelerations on the right hand side. m is the number of unknown velocity components and n is the number of degrees of freedom for the pressure field. The discrete inf-sup condition follows as [67, 76]with the vectors of coefficients \(\hat{\mathchoice{\displaystyle \varvec{v}}{\textstyle \varvec{v}}{\scriptstyle \varvec{v}}{\scriptscriptstyle \varvec{v}}} \in {\mathbb {R}}^m\) and \(\hat{\mathchoice{\displaystyle \varvec{q}}{\textstyle \varvec{q}}{\scriptstyle \varvec{q}}{\scriptscriptstyle \varvec{q}}}\in {\mathbb {R}}^n\) for the corresponding finite element functions, respectively [67, 76]. Furthermore, the \(n \times n\) pressure mass matrix \(\mathchoice{\displaystyle \textbf{M}}{\textstyle \textbf{M}}{\scriptstyle \textbf{M}}{\scriptscriptstyle \textbf{M}}\) is introduced. The Schur complement of the matrix in Eq. (52) is defined as [67]With the Schur complement and the pressure mass matrix, the following eigenvalue problem is formulatedwith the eigenvector \(\mathchoice{\displaystyle \varvec{y}}{\textstyle \varvec{y}}{\scriptstyle \varvec{y}}{\scriptscriptstyle \varvec{y}}\) and the eigenvalues \(\lambda \).
with \(\bar{s} = -(1+\bar{q}) \,(b-0.205) \,\cos (\pi /6\,(1-a))\), \(\bar{q} = -0.1a^2 + 0.2a\), and \( \tilde{s} = -(1-0.1a^2 + 0.2a) \,(b-0.205) \,\cos (\pi /6\,(1-a))\). a, b, and c are the Cartesian coordinates in the undeformed, i.e., flat domain and x, y, and z are the Cartesian coordinates in which the deformed, i.e., curved domain, is described. The flat bulk domain is defined as the channel \([0,2.20] \times [0,0.41] \times [0,1/3]\) and the obstacle’s radius (the cylinder in the flat 2-dimensional channel) is \(r_b = 0.05\) at the bottom \((c=0)\), \(r_t = 0.06\) at the top \((c=1/3)\), and varies linearly in between. The obstacle is placed slightly unsymmetrically in the channel in y-direction. The level-set function is defined as \(\phi \left( \mathchoice{\displaystyle \varvec{a}}{\textstyle \varvec{a}}{\scriptstyle \varvec{a}}{\scriptscriptstyle \varvec{a}}\right) =c\). Figure 7 visualizes the geometries which are defined by these mappings.
The level-set function is defined as \(\phi \left( \mathchoice{\displaystyle \varvec{a}}{\textstyle \varvec{a}}{\scriptstyle \varvec{a}}{\scriptscriptstyle \varvec{a}}\right) =c\). Figure 10 shows the applied mapping for this test case with \(\alpha = 0.4\) for the bulk domain \(\Omega \) with some embedded level sets \(\Gamma _{\!c}\).
with \(r_{xy} = x^2+y^2\). This definition of the initial conditions is similar to [16] where single surfaces are considered. The fluid’s density is \(\varrho = 1\) and the viscosity is set to \(\mu = 1\). The time step size in the Crank-Nicolson time stepping scheme is \(\Delta t = 0.1\) for \(\tau = [0,60]\). Taylor–Hood elements of order \(q_{\mathchoice{\displaystyle \varvec{u}}{\textstyle \varvec{u}}{\scriptstyle \varvec{u}}{\scriptscriptstyle \varvec{u}}} = 2\), and \(q_{p} = 1\) are used. For the geometry, we use again \(q_{\textrm{geom}} = 3\). Fig. 14 shows the velocity on selected level sets at different times.
