FEM‐BEM coupling for the exterior Stokes problem with non‐conforming finite elements and an application to small droplet deformation dynamics

Abstract

SUMMARYA non-conforming, discontinuous Galerkin finite element-boundary element coupling procedure is presented for the exterior planar Stokes problem. The novel coupled formulation is developed using that for the conforming case as a guide to the introduction of extra mortar variables used to couple a discontinuous interior finite element solution with a continuous exterior boundary element solution. Convergence results for the new scheme are presented, for a range of different interior penalties, on computational domains discretized with regular structured meshes.To illustrate an application, the excitations required to model two-phase droplet deformations in an extensional flow, under simple surface tension, with the new scheme are also presented. For a selection of different drop viscocities and exterior flows, with and without a rotational component, the progression to a steady-state deformation of initially undeformed circular drops is calculated and the results compared with those from both a conforming FEM-BEM equivalent scheme and from a small perturbation analysis where available.

Non‐conforming finite elements for axisymmetric charged droplet deformation dynamics and Coulomb explosions

“…With the triangulation T h of Ω into elements K ∈ T h allowing the union of all element edges and the union of those edges both coincident with Γ h and interior to , the outer boundary to be defined , the standard notation P 1 NC for Crouzeix–Raviart NC shape functions with a linear velocity, and P 0 piecewise constant pressure allow the definition of the following standard finite element spaces for the velocity and pressure solutions in Ω …”

“…The vector ( T r , T z ) refers to the surface tangential direction unit vector such that ( n r , n z ) × ( T r , T z ) = 1 and the average, 〈〈 · 〉〉, and jump, [[ · ]], operators are as defined in . Note that the integrals involving these operators have not been ‘scaled’ with an r to avoid problems associated with vanishing jump penalization of the flow field around the axis of rotational symmetry.…”

“…Note that such ‘solution insensitive’ penalty methods, providing the historical background for the current work , and for which the convergence can be examined relatively easily with the steady, time‐independent, elliptic Stokes problem, provide an alternative to the classic up‐winding/down‐winding ‘solution sensitive’ schemes commonly used for time‐dependent flow problems but yet to be investigated by the author for the present problem.…”

“…With the triangulation T h of into elements K 2 T h allowing the union of all element edges h A D [ K2T h ¹@Kº and the union of those edges both coincident with h and interior to h 0 D A n h , the outer boundary to be defined [24], the standard notation P NC 1 for Crouzeix-Raviart [21] NC shape functions with a linear velocity, and P 0 piecewise constant pressure allow the definition of the following standard finite element spaces for the velocity and pressure solutions in…”

“…To avoid spurious mass buildup along inter-element edges and to keep the tractions applied by two different elements either side of such shared edges equal and opposite, it is necessary to introduce extra terms to penalize any velocity and traction 'jumps' across these 'internal' edges [24,25]. This is a usual requirement of NC methods and is typically parameterized by a scalar penalty parameterˇ.…”