A comparison between a collocation and weak implementation of the rigid-body motion constraint on a particle surface

In this work we implemented and compared two different methods to impose the rigid-body motion constraint on a solid particle moving inside a fluid. We consider a fictitious domain method to easily manage the particle motion. Since the solid as well as fluid inertia is neglected, the particle can be discretized through its boundary only. The rigid-body motion is imposed via Lagrange multipliers on the boundary. In the first method, such constraints are imposed in discrete points on the boundary (collocation) whereas in the second one the constraint is imposed in a weak way on elements dividing the particle surface.Two test problems, i.e. a spherical and an ellipsoidal particle in a sheared Newtonian fluid, are chosen to compare the methods. In both cases, the analysis is carried out in 2D as well as 3D.The results show that for the collocation method an optimal number of collocation points exists leading to the smallest error. However, small variations in the optimal value can generate large deviations. In the weak implementation, the error is only mildly affected by the number of elements used to discretize the particle boundary and by the Lagrange multipliers interpolation space. A further analysis is carried out to study the effect of an approximated integration of weak constraints.A comparison between the two methods showed that the same accuracy can be achieved by using less constraints if the weak discretization is used. Finally, the rigid-body motion imposed via weak constraints leads to better conditioned linear systems.