An Extended Finite Element Method for a Two-Phase Stokes problem

Transcription

1 XFEM project An Extended Finite Element Method for a Two-Phase Stokes problem P. Lederer, C. Pfeiler, C. Wintersteiger Advisor: Dr. C. Lehrenfeld August 5, 2015 Contents 1 Problem description Physics Bubble Curvature Weak formulation Pressure enrichment Pressure enrichment Q x h Ghost penalty Velocity enrichment Velocity enrichment V x h Nitsche XFEM formulation The weighted average Choice of λ Numerical examples First example Second example Conclusion 16 1

2 The theory of this work is based on the notes [Leh15] of the XFEM - lecture in the summer term 2015 held by Dr. C. Lehrenfeld. In this project we consider an approximation of a two-phase stokes problem where the interface is not resolved by the triangulation. In section 1 we discuss the problem and the physical intepretation of the occuring equations by presenting two model problems. Depending on the parameters, the solution can have weak and strong discontinuities. Therefore new approximation spaces are designed in the section 2 and 3. In the last section we present numerical examples, where an exact solution exists and analyze the convergence of the new methods. 1 Problem description We consider an open domain Ω R 2, which is separated into two subdomains Ω 1 and Ω 2 by the interface = Ω 1 Ω 2. We assume that one phase is completely surrounded by the other, i.e. Ω =. Let {T h } h>0 be a family of shape regular meshes of Ω. We only consider simplicial meshes T h. The elements of such a mesh T h are simplices, i.e. triangles in 2D, and it holds h = max {h T T T h }, where h T := diam(t ) is the local element size. In general the interface does not coincide with the boundaries of the elements of T h. We call the interface unfitted w.r.t. the mesh T h. We assume that the resolution close to the interface is sufficiently high, such that can be resolved by the triangulation, in the sense that, if T =: T, then T can be represented as the graph of a function on a planar cross-section of T. Further the resolution close to the interface should be sufficiently high, such that T h c T for all T cut by, and with a constant c > 0 independent of h and the cut position of. We consider the two-phase Stokes problem on the domain Ω, which results as div (µ i D(u)) + p = ρ i g in Ω i, i = 1, 2, (1a) div (u) = 0 in Ω i, i = 1, 2, (1b) u = 0 on, (1c) µd(u) n + pn = f on, (1d) u = 0 on Ω, (1e) with the domainwise constant density ρ, domainwise constant viscosity µ, given right hand side g [L 2 (Ω)] 2 and surface tension force f [L 2 ()] 2. D(u) denotes the symmetric gradient D(u) := u + ( u). 1.1 Physics Before we introduce two examples of a two phase Stokes problem, we first take a closer look at the equations (1a) - (1e) and the occuring material parameters ρ, µ. Equation (1a) is the so called momentum conservation equation and can be derived by balancing the momentums in and on a control volume under the assumptions of a Newtonian fluid. Another important fact for the derivation of (1a) is that we consider a creeping flow, which means that the advective inertial forces are small compared to the viscous forces. The second equation (1b) can be derived from the conservation of mass and (1e) is a given boundary condition. The new aspects compared to an one-phase stokes problem, are the equations (1c) and (1d), which will be called the interface conditions. We are looking for continuous velocity fields what leads to (1c). Equation (1d) describes the momentum balance at the interface. The stress from the domains Ω 1 and Ω 2 2

3 Ω 1 : µ 1 = 1 ρ i g = ( ) 0 0 Ω 2 : µ 2 = 100 ( ) 0 ρ i g = 1 : f = ( ) 0 0 Figure 1: A bubble in a surrounding fluid balances with the force f on the interface. A typical f would be the surface tension force. To classify the fluid in Ω i we have two material parameters: The dynamic viscosity µ i which describes the resistance due to deformation by the stress tensor and the density ρ i. To complete our equations we consider the gravitational force ρ i g as right hand side, which acts as a pushing force on the fluid domains Bubble In the first example (Figure 1) we want to simulate a bubble with a very small density ρ 1 (for example air) surrounded by a fluid with a higher density ρ 2 (e.g. water). Also the viscosities are chosen differently with µ 1 = 1 and µ 2 = 100. The force on the interface is set to zero. Due to the different viscosities we expect a jump in the pressure to satisfy the interface condition (1d). In Figure 2 and 3 the velocity and pressure (turned 180 degrees) fields are displayed. We observe, that the gravitational force drags the mass of the outer fluid down and therefore the lighter bubble rises up and the displayed velocity field is produced. Far away from the bubble the pressure is essentially the hydrostatic pressure and increases almost linear. Closer to the bubble (Ω 2 ), we see that the pressure is still increasing towards the bottom, but due to the interaction with the bubble the increase is no longer linear. Furthermore, we see that we get a discontinuity across the interface. Only at two points (cf. Figure 3) the pressure is continuous as the normal component of D(u) vanishes there and thus equation (1e) demands p = 0. Note that the pressure gradient (from bottom to top) is not a driving force in this example! Curvature In the second example (Figure 4) we demonstrate how the surface tension force affects the velocity field. The surface tension can be modeled as σκn, where κ is the mean curvature in each point on the interface and σ is a material parameter which controls the influence of the curvature. The curvature κ has different impacts depending on the substance system. For instance the resulting surface tension force is higher for an air bubble than for an oil bubble 3

4 Figure 2: Velocity plot for example µd(u) n + pn = 0 Figure 3: Pressure plot for example

5 Ω 1 : µ 1 = 1 ρ i g = ( ) 0 0 Ω 2 : µ 2 = 1 ρ i g = ( ) 0 0 : f = σκn Figure 4: Rounded square and considered paramters for example (with the same gemoetry). When we look at this example (see Figure 4 for the considered geometrical configuration and the parameters. Note: ρ and µ are globally constant), one can see that the curvature is much higher in the corners as on the straight sides (where it is actual zero). In Figure 5 we see how this affects the velocity. In the corners the flow is directed towards the center of the bubble and on the straight sides outwards. This goes hand in hand with the physical expectations, as the bubble tends to a spherical shape if the surface tension is the dominant force. 1.2 Weak formulation We multiply (1a) by v V = [H 1 (Ω)] 2 and (1b) by q Q = L 2 (Ω). Partial integration and the interface condition (1d), then lead to the weak formulation. Approximating the spaces V and Q by V h and Q h leads to the discrete problem: Find (u, p) V h Q h, such that 1 2 µ i (D(u), D(v)) Ωi (p, div(v)) Ωi = ρ i (g, v) Ωi + (f, v) v V h, (2a) i=1,2 i=1,2 (q, div(u)) Ωi = 0 q Q h, (2b) i=1,2 where (a, b) Ωi := Ω i a b dx. The choice of the finite element spaces V h and Q h with respect to (1c) and (1e) will be discussed in the following sections. b 2 Pressure enrichment 2.1 Pressure enrichment Q x h A popular choice for the velocity-pressure space V h Q h is the Taylor-Hood element, which is LBB-stable. One chooses globally continuous piecewise polynomials of degree k for the velocity, 5

6 Figure 5: Velocity plot for example and globally continuous piecewise polynomials of degree k 1 for the pressure } V h := {v [C(Ω)] 2 v T [P k (T )] 2 for all T T h, { } Q h := q C(Ω) q T P k 1 (T ) for all T T h. In this work we consider the element with k = 2. However, due to the surface tension force f at the interface, the pressure solution p will have jump discontinuities which can not be resolved by Q h very well. In particular it holds inf p h p L p h Q 2 (Ω) h1/2 p H 1 (Ω 1,2 ), (3) h where. 2 H k (Ω 1,2 ) :=. 2 H k (Ω 1 ) +. 2 H k (Ω 2 ). This estimate is sharp, even if p is arbitrarily smooth on the subdomains Ω 1, Ω 2, as long as it has a jump across the interface. The situation can not be improved by choosing piecewise polynomials of higher degree for Q h. Therefore we introduce the enriched space Q h := Q h Q x h, where Qx h is chosen as follows: Define J := {1,..., n}, where n = dim(q h ), and let (ϕ i ) i J be the nodal basis in Q h. Let J := {j J supp ϕ j } be the index set of those basis functions in Q h which have their support intersected by. Let χ 2 be the characteristic function with χ 2 (x) = 0 for x Ω 1 and χ 2 (x) = 1 for x Ω 2. Using this, for j J we introduce the so-called enrichment function Φ j (x) := χ 2 (x) χ 2 (x j ), where x j is the vertex with index j corresponding to ϕ j. We introduce the new basis functions ϕ j := Φ jϕ j, j J. Now we can define Q x h := span { ϕ j j J }. 6

7 With the velocity-pressure space V h Q h, jump discontinuities of the pressure at the interface can be resolved, and so (3) can be improved to inf p h p L p h Q 2 (Ω) hk p H k+1 (Ω 1,2 ). (4) h (a) p h Q h (b) p h Q h Figure 6: Solution without and with pressure space enrichment using triangulation and interface position as in Figure 7a. Enriching the pressure space leads to a new problem. The discrete inf-sup condition obtained by the LBB-stable Taylor-Hood element is no longer given. In particular, with the enriched space Q h, the discretization error and the conditioning of the linear system are no longer independent of the position of the interface, c.f. Figure 6/7. y y interface interface x x (a) Almost no bad cut positions (b) Many bad cut positions Figure 7: Triangulation and interface positions 2.2 Ghost penalty To fix the problems observed, we introduce the so-called ghost penalty stabilization: Let F h be the set of facets corresponding to the elements of T h. Define Ω i,h := Ω i {T T h T } 7

8 } and let Fi,h {F := F h F Ω i,h be the set of facets which have some part in Ω i,h. Then, the ghost penalty term can be defined as J(p, q) := γ h 3 ( E i p n, E i q n ) F, (5) i=1,2 F Fi,h for (p, q) Q h Q h, where E i is the canonical extension operator (on the discrete space Q h ).The stabilization parameter γ is O(1). In particular, the term J(p, q) is independent of the cut position within the elements which results in the robustness of the method, c.f. Figure 8. The problem without the ghost penalty stabilization was, that the enriched space Q h was to big to fulfill the LBB condition. With the new stabilization the LBB condition is still not valid, but by bounding the gradients close to the interface, one can show the discrete inf-sup condition of the big system. (a) without ghost penalty (b) with ghost penalty Figure 8: p h Q h using triangulation and interface position as in Figure 7b. 3 Velocity enrichment 3.1 Velocity enrichment V x h In the past section we enriched the pressure space and obtained Q h, s.t. jumps in the pressure solution at the interface can be resolved. Now, due to (1d), if µ 1 µ 2, the velocity solution u will have weak discontinuities, i.e. kinks at the interface. Those discontinuities can not be resolved very well by the approximation space V h. In particular, for a function u which has a kink at the interface, we have the approximation error estimate inf v h u L v h V 2 (Ω) h3/2 u H 2 (Ω 1,2 ), (6) h which again, is sharp, even if u is arbitrarily smooth on the subdomains Ω 1, Ω 2. Increasing the piecewise polynomial degree in V h again will not cure the situation. To achieve better results, we introduce the enriched velocity approximation space Vh := V h Vh x. The space Vx h is constructed from V h similarly as Q x h has been constructed from Q h. The only difference is that the polynomial degree of V h is one degree higher than the degree of Q h and thus we can not refer to a nodal basis for the construction of Vh x : Therefore let again J := {1,..., n}, where n = dim(v h ) and (ϕ i ) i J a basis of V h. Now define J := {j J supp ϕ j } the index 8

9 set of basis functions which are non trivial on Ω 1 and non trivial on Ω 2. Let χ 2 again be the indicator function with χ 2 (x) = 0 for x Ω 1 and χ 2 (x) = 1 for x Ω 2. Now define ϕ j := χ 2ϕ j for j J and V x h := span { ϕ j j J }. Note, that while it is irrelevant for the theoretical construction of Vh whether we choose ϕ j := χ 2 ϕ j or ϕ j := (1 χ 2)ϕ j as new basis function for j J, it does make a difference for the arising linear system and the computational structure. For details on the right choice we refer to the lecture notes [Leh15]. 3.2 Nitsche XFEM formulation We start with a few simple calculations, the result of which facilitate the introduction of the considered Nitche discretization. Consider the term (f, v) in (2a) and equation (1d), (f, v) = σ(u, p)n v ds, where σ i (u, p) := µ i D(u i ) + Ip i. Since n 1 = n 2, we get σn v = σ 1 n 1 v 1 + σ 2 n 2 v 2 = σ 1 n 1 v 2 + σ 2 n 2 v 2 + σ 1 n 1 (v 1 v 2 ) = (σ 1 σ 2 )n }{{} 1 v 2 + σ 1 n 1 (v 1 v 2 ) =: A =f but a similar computation shows σn v = σ 1 n 1 v 1 + σ 2 n 2 v 1 + σ 2 n 2 (v 2 v 1 ) = (σ 1 σ 2 )n }{{} 1 v 1 + σ 2 n 1 (v 1 v 2 ) =: B =f Now since A = B it holds for all 0 κ 1, κ 2 1 with κ 1 + κ 2 = 1, that A = κ 1 A + κ 2 B = B. This observation shows σn v = κ 1 A + κ 2 B = f (κ 1 v 2 + κ 2 v 1 ) + (κ 1 σ 1 + κ 2 σ 2 )n 1 (v 1 v 2 ) = f (κ 1 v 2 + κ 2 v 1 ) + {σ(u, p) } n 1 v, where the weighted average is defined as {σ } := κ 1 σ 1 + κ 2 σ 2. We obtain (f, v) = N c ((u, p), (v, q)) + (f, κ 1 v 2 + κ 2 v 1 ), where N c ((u, p), (v, q)) := {σ(u, p) } n 1 v ds. We add the symmetrical counterpart N c ((v, q), (u, p)) to preserve symmetry of the bilinear form. This is consistent, due to u = 0 on for the solution u. By introducing the enriched space Vh we now can resolve kinks and jumps in the velocity. As the velocity has to be continuous, we add another symmetric (stabilization) term N s (u, v) := λ {µ } h u v ds, 9

10 which penalizes jumps in the velocity. Note that N s (u, v) = 0 for the solution u. The parameter λ has to be chosen accordingly, which will be discussed in section 3.4. We define the Nitsche term N((u, p), (v, q)) := N c ((u, p), (v, q)) + N c ((v, q), (u, p)) + N s (u, v) (7) and arrive at the Nitsche XFEM-formulation (with ghost penalty stabilization): Find (u, p) Vh Q h, such that 1 2 µ i (D(u), D(v)) Ωi (p, div(v)) Ωi + N((u, p), (v, q)) i=1,2 + (q, div(u)) Ωi + J(p, q) i=1,2 = ρ i (g, v) Ωi + (f, κ 1 v 2 + κ 2 v 1 ). i=1,2 (8) for all (v, q) V h Q h. This formulation is consistent for any valid choice of κ 1, κ 2. However, the stability of the method does depend on the precise choice of κ 1, κ The weighted average We choose the Heaviside weighting for κ 1, κ 2. This means κ i T = κ T,i, where for T T h, let { 1 if Ω 1 T > Ω 2 T κ T,1 := 0 else, (9) κ T,2 := 1 κ T,2. In particular this choice of weights guarantees, that for i = 1, 2, v Vh and all T T h κ 2 T,i T h D(v) 2 ds c tr T i D(v) 2 dx (10) holds, with a constant c tr > 0 only depending on the polynomial degree k = 2, c κ and c. This is important for the stability of the method. c κ is a fixed constant such that κ 2 i c κ T i T holds for all T T h, i = 1, 2. Since we chose the Heaviside weighting, this holds with c κ := Choice of λ For a sufficiently large λ stability of the discretization can be ensured. The drawback of choosing λ unnecessarily large is an increase in the condition number. However it is sufficient to choose λ > c λ := max {1, 4c tr } (for details see [Leh15]). In Figure 9 the differences between using and not using a velocity enrichment can be observed. Without Vh x the kink can not be approximated very well. 10

11 (a) (u h, p h ) V h Q h (b) (u h, p h ) V h Q h Figure 9: Approximation of the velocity kink 4 Numerical examples The results in the following section were calculated using the software package Netgen/NGSolve and the extension ngsxfem, which was presented in the lecture. For the new methods for the two-phase stokes problem we implemented some further extensions in the ngsxfem package. To observe the expected convergence rates, we consider an example given in [KGR15] where the exact solution is given by ( ) y u(x, y) = α(r)e r2 where r = x x 2 + y 2 { µ 1 1 if r < r α(r) = µ (µ 1 1 µ 1 2 )er2 r 2 if r r { p(x, y) = x 3 c if x Ω 1 +, 0 else where Ω := ( 1, 1) 2, Ω 1 := B 2/3 is a ball with radius r = 2/3 and a given force on the interface f = cn. Note that the coefficient α(r) is continuous and either constant (if µ 1 = µ 2 ) or not constant (if the viscosities are different), which will produce a kink in the velocity. The force f = cn will only act on the pressure and will produce a discontinuity across the interface. By that we mean that the pressure can be decomposed p = p 1 + p 2 with p 2 = χ 2 c and p 1 the solution of the problem without surface tension force. 4.1 First example In the first example we choose µ 1 = µ 2 = 1 and c = 0.5 (cf. Figure 10). 11

12 (a) velocity (b) pressure Figure 10: Solutions of the first example Due to (3) the L 2 -error of the pressure only converges with order 1/2 using no enrichment. If we use the pressure space Q h, we observe a quadradic convergence (using piecewise linears for the pressure + enrichment) (cf. Figure 11) Q h V h Ghost + Ghost h 1/2 h Figure 11: pressure error p p h L 2 (Ω) As expected we lose one order of convergence in the H 1 -error and can see that without the ghost penalty, the gradient of the pressure can not be controlled (cf. Figure 12) Q h V h + Ghost Q h V h Ghost h Figure 12: pressure error p p h H 1 (Ω) 12

13 Although we have no kinks in the velocity, we see, using no enrichment V x h, that the L2 -error of the velocity only converges with order 3/2 as the L 2 pressure error converges only with order 1/2, consider interface condition (1d) (cf. Figure 13). If the pressure space is enriched, we observe the optimal order 3 independent of the choice for the velocity space Q h V h + Ghost + Ghost h 3/2 h Figure 13: velocity error u u h L 2 (Ω) In Figure 14 we see one order less than in Figure 13, as expected for the H 1 norm Q h V h Ghost + Ghost h Figure 14: velocity error u u h H 1 (Ω) 13

14 4.2 Second example In the second example we choose µ 1 = 1, µ 2 = 10 and c = 0.5 (cf. Figure 15). (a) velocity (b) pressure Figure 15: Solutions of the second example Similar as in Figure 11 we observe the convergence rate 1/2 without enrichment, but also in the case (and also with the ghost penalty) the rate is not optimal, due to the coupling between the pressure and the velocity in equation (1d). Note, that in this example we have a kink in the velocity across the interface, which means that a bad approximation for the velocity impacts the pressure approximation (cf. Figure 17 and Figure 16). Figure 16: Pressure with and without velocity enrichment 14

15 Q h V h Ghost + Ghost h 1/2 h Figure 17: pressure error p p h L 2 (Ω) The H 1 -error shows the same situation as in Figure 17 but we see that the ghost penalty is (also in the case of a velocity enrichment) important to get an optimal order as the ghost penalty penalizes large gradients in the pressure close to the interface (cf. Figure 18) Q h V h + Ghost Q h V h Ghost h Figure 18: pressure error p p h H 1 (Ω) 15

16 In Figure 19 and 20 the L 2 - and H 1 -error of the velocity are shown. As expected due to (6), if no velocity enrichment is used, we get no better convergence rate than 3/2 respectively 1/2. Including an enrichment for the velocity and the pressure, we get the optimal rates Q h V h + Ghost + Ghost h 3/2 h Figure 19: velocity error u u h L 2 (Ω) Q h V h Ghost Q h V h Ghost h 1/2 h Figure 20: velocity error u u h H 1 (Ω) 5 Conclusion With the combination of the already known ghost penalty stabilization for the pressure enrichment and the new derived Nitsche formulation for the velocity enrichment one is now able to resolve occuring jumps and/or weak discontinuities. The numerical examples show that the optimal convergence rates can be observed. An open question is a proof for the discrete inf-sup condition although we did not observe problems in our numerical examples what indicates that the condition should be valid. Another open question is how the new method can be extended to time dependent problems, as for example the method of lines is not straight forward for velocities with weak discontinuities. 16

17 References [KGR15] Matthias Kirchhart, Sven Gross, and Arnold Reusken. Analysis of an xfem discretization for stokes interface problems Institut für Geometrie und praktische Mathematik ( [Leh15] Christoph Lehrenfeld. Extended finite element methods for interface problems, lecture notes TU Wien. 17