The discontinuities due to the discretization lead to some challenges. First, the normal direction of the contact surface is not steady because the discrete surface is only C0 continuous. One might smooth the normal vector field. Second, the question of contact enforcement has to be cleared. Contact forces can be modeled with either a Lagrange multiplier method or a penalty formulation to prevent penetration. Third, there must be developed a integration scheme which is able to handle the non-steady boundary. Last, there is a strong discontinuity in measuring the penetration, where different criteria for enabling or disabling contact can be found (active set strategy). In this work, different approaches to solve this tasks are presented and brought into context.The mortar finite element method describes the coupling of non-matching discretization of two subdomains. This coupling is realized by enforcing a relation between the two domains with the help of Lagrange multipliers. It is very characteristic that the geometric and the kinetic conditions are fulfilled in weak form. This method for coupling subdomains can also be applied on contact problems. The treatment of frictionless contact and solutions for the discretization issues are topic of this work. Contact FormulationVirtual work of contact forces We define a contact boundary Γ C with contact force density t (1) N on the non-mortar surface and t(2) N on the mortar surface and the normal vector n. It is more or less arbitrary which subdomain is treated as mortar or non-mortar domain. The problem of defining an appropriate normal vector is discussed in [3]. The virtual work of contactN n · δu (i) dΓ corresponds to the Neumann boundaries. For contact surfaces, the equilibrium t (1) N dγ (1) C = −t (2) N dγ (2) C and γ (1) C = γ(2) C = γ C in normal direction has to be fulfilled. If we use this equilibrium and focus on non-mortar force density t (1) N , we can formulate the virtual work of contact forces as shown in (1).Karush-Kuhn-Tucker condition Beside the virtual work of contact forces, we need Karush-Kuhn-Tucker conditions. These describe the main properties of mechanical contact and can be written as g(X, t) ≥ 0, tN ≤ 0 and t (1) N g(X, t) = 0. Once again, the continuous and strong condition g(X, t) ≥ 0 can be written in a weak formulation (2).It is also possible to retrieve the two conditions (1) and (2) by defining a variation of the contact potential Π C (u) =Contact Enforcement What is left is the contact enforcement (which means the definition of t(1) N ). There is the possibility to use a penalty method with t (1) N = εg(X, t) (and the corresponding base potential Π C = − 1 2 Γ C ε g(X, t) 2 dΓ ). The more common way (as mainly proposed for the mortar method) is the Lagrange method with t (1) N = −λ N (base potential Π C = − Γ C λ N g(X, t)dΓ ). Naturally, one may also use the augmented Lagrange method as some kind of method in between. By insertion of this definitions for t (1) N into (1) and (2), the base equations for the finite element method are ...
We will present a comparison between two formulations of the normal vector field for contact algorithms based on the mortar method. First the non steady normal field is discussed. The non steadiness is a result of the C 0 continuity of the boundary discretization. This is the common result if one discretize the domain with classical finite element methods. Second we will present results for a special normal field distribution. We average the nodal normal vector of two ascending edges and interpolate this nodal normal throughout the edges. We have implemented both methods and present comparisons based on numerical experiments. .Discontinuous mortar side normal vectors ( Figure 1a) The proposed mortar side normal was presented in [1] and also in other publications. As the normal vector is normal to its adjacent edge on the mortar side it is obvious that the direction of the normal vector jumps on start and end nodes of a mortar edge.Continuously averaged non-mortar side normal vectors (Figure 1b 2 Problems related to C 0 only surfaces in contact mechanics Oscillation of projection For the nearest point projection we have to determine the corresponding edge for a given projection point. This edge might change during the Newton-Raphson procedure from one iteration to the next. This situation is shown in figure 2b. If the "best" projection edge changes back again in the next iteration the projection starts to oscillate and no convergence can be reached. This problem is not influenced by averaging.
Past computational studies of planet-induced vortices have shown that the dust asymmetries associated with these vortices can be long-lived enough that they should be much more common in mm/sub-mm observations of protoplanetary discs, even though they are quite rare. Observed asymmetries also have a range of azimuthal extents from compact to elongated even though computational studies have shown planet-induced vortices should be preferentially elongated. In this study, we use 2-D and 3-D hydrodynamic simulations to test whether those dust asymmetries should really be so long-lived or so elongated. With higher resolution (29 cells radially per scale height) than our previous work, we find that vortices can be more compact by developing compact cores when higher-mass planets cause them to re-form, or if they are seeded by tiny compact vortices from the vertical shear instability (VSI), but not through dust feedback in 3-D as was previously expected in general. Any case with a compact vortex or core(s) also has a longer lifetime. Even elongated vortices can have longer lifetimes with higher-mass planets or if the associated planet is allowed to migrate, the latter of which can cause the dust asymmetry to stop decaying as the planet migrates away from the vortex. These longer dust asymmetry lifetimes are even more inconsistent with observations, perhaps suggesting that discs still have an intermediate amount of effective viscosity.
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