There is a wide variety of CFD grid types including Cartesian, structured, unstructured and hybrids, as well as, numerous methodologies of combining these to reduce the time required to generate high-quality grids around complex configurations. If the grid methodologies were implemented in different codes, they should be written in such a way as to obtain the maximum performance from the available computer resources. A common interface should also be required to allow for ease of use. However, it is very time consuming to develop, maintain and add extra functionally to different codes. This paper examines the possibility of taking an existing CFD solver, the Helicopter Multi-Block (HMB) CFD method, and implementing a new grid type while reusing as much as possible the original code base. The paper presents some of the challenges encountered in extending the code which was written for a single mesh type, to a more flexible solver that is still computationally efficient but can cope with a variety of grid types.

A fully implicit numerical method, based upon a combination of adaptively refined hierarchical meshes and geometric multigrid, is presented for the simulation of binary alloy solidification in three space dimensions. The computational techniques are presented for a particular mathematical model, based upon the phase-field approach, however their applicability is of greater generality than for the specific phase-field model used here. In particular, an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step. It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids. In effect this paper presents an extension of earlier research in two space dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully three-dimensional problems. This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions, obtained using an explicit scheme. The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.

Explicit time stepping schemes for the immersed boundary method require very small time steps in order to maintain stability. Solving the equations that arise from an implicit discretization is difficult. Recently, several different approaches have been proposed, but a complete understanding of this problem is still emerging. A multigrid method is developed and explored for solving the equations in an implicit-time discretization of a model of the immersed boundary equations. The model problem consists of a scalar Poisson equation with conformation-dependent singular forces on an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, but the efficiency gain is limited. The multigrid method alone may not be an effective solver, but when used as a preconditioner for Krylov methods, the speed-up over the explicit-time method is substantial. For example, depending on the constitutive law for the boundary force, with a time step 100 times larger than the explicit method, the implicit method is about 15-100 times more efficient than the explicit method. A very attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundary points.