We employ adaptive mesh refinement, implicit time stepping, a nonlinear multigrid solver and parallel computation, to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt substantially cooled below its freezing point at the microscale. Underlying physical molecular forces are captured at this scale by a specification of the energy field. The time rate of change of the temperature, alloy concentration and an order parameter to govern the state of the material (liquid or solid) is controlled by the diffusion parameters and variational derivatives of the energy functional. The physical problem is important to material scientists for the development of solid metal alloys and, hitherto, this fully coupled thermal problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques this problem is now shown here to be tractable at appropriate resolution with relatively moderate computational resources. Figure 1: Snapshot of the solid-liquid interface for a typical dendrite. This image was obtained from a simulation with Le = 40, ∆ = 0.525 and ∆x = 0.78.The computational techniques we employ are: use of very fine meshing in the region around the moving boundary where phase field and solute field resolution is critical, and coarse meshing away from the boundary where only the slowly changing temperature field requires resolution; implicit time stepping to allow much larger time steps than would otherwise be possible; nonlinear smoothing in conjunction with a nonlinear multi-grid solver; and parallel processing with up to 1024 cores as the simulation progresses. The combination of all of these techniques allows an almost optimal solution process to be developed, in which the number of degrees of freedom is evolved with the dendrite, to maintain the required resolution as the interface grows, and the solution time at each time step is approximately proportional to the number of degrees of freedom. Furthermore, the use of a parallel implementation ensures that sufficient primary memory is available to support a mesh resolution which is fully converged whilst maintaining a tractable solution time.The particular phase field model we employ is an extension of [3], and is based on the three dimensional thermalphase field model of [4] and two dimensional thermal-solutal phase field model of [5]. One feature of the physical problem is that it is purely dissipative, or entropy increasing, as all natural relaxational phenomena are. The resulting PDEs are of Allen-Cahn [6] and Carn-Hilliard type [7]. That is to say, the model involves time derivatives of the three fields coupled to forms involving variational derivatives of some functional -typically the free energy functional. As the dendrite grows the free energy reduces monotonically with time but never achi...

SUMMARYIn this paper a high order Discontinuous Galerkin method is used to solve steady-state isothermal line contact elastohydrodynamic lubrication problems. This method is found to be stable across a wide range of loads and is shown to permit accurate solutions using just a small number of degrees of freedom provided suitable grids are used. A comparison is made between results obtained using this proposed method and those from a very large finite difference calculation in order to demonstrate excellent accuracy for a typical highly loaded test problem.

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SUMMARYThe computation of numerical solutions to elastohydrodynamic lubrication problems is only possible on fine meshes by using a combination of multigrid and multilevel techniques. In this paper, we show how the parallelization of both multigrid and multilevel multi-integration for these problems may be accomplished and discuss the scalability of the resulting code. A performance model of the solver is constructed and used to perform an analysis of the results obtained. Results are shown with good speed-ups and excellent scalability for distributed memory architectures and in agreement with the model.