Overview of the Current Implementation of PDSC

So far, there have only several well-developed DSM C codes which mentioned in Section 1.2.2. It is very helpful to develop a general-purpose parallel DSM C code for simulating problems of rarefied gas flows. Thus, development of this numerical solver is the main subject of this thesis. To make the PDSC friendlier to use, an applicable preprocessor and postprocessor are necessary. Figure 2.2 shows the overview of the planed PDSC, which includes a graphic-based preprocessor, main numerical solver and a postprocessor. They are described in the next sections in tern.

2.4.1 MuS T Visual Preprocessor

A graphical user interface (GUI), which names M uST Visual Preprocessor (Fig.

2.3(b)), is developed by Professor Wu’s group [73]. It is used to ease the parameter settings of boundary conditions, initial conditions and parallel processing because these procedures are tedious and complicated as can be shown in Fig. 2.3(a). This preprocessor can be either used for cell-based (e.g. DSM C) or for node-based (Finite Element M ethod) numerical methods. Firstly, the M uST Visual Preprocessor will ask for mesh connectivity information, which is easy to obtain from mesh generators. The mesh types can be quadrilaterals, triangles, hexahedrons, tetrahedrons, prisms and pyramids. Secondly, the preprocessor will transfer mesh format into graph and process the initial partition according to the weighting of each cell. Thirdly, the processor can create the boundary surfaces automatically and we can assign the boundary types very easily to create the input files for the PDSC.

2.4.2 PDSC

Parallel DSM C Code (PDSC) is the main solver developed in this thesis, which utilizes unstructured tetrahedral mesh. Figure 2.4 is the features of PDSC and brief introduction is listed in the following paragraphs.

Unstructured Tetrahedral M esh

Reasons of PDSC using unstructured tetrahedral mesh are: (a) it can be easily used for flows with complicated boundary conditions, (b) parallel processing can be easier implemented via graph-partitioning technique, which can handle irregular inter-processor boundary of dynamic domain decomposition, (c) it can be coupled with unstructured node-based numerical method (e.g. N-S equations).

According to these advantages of using unstructured mesh, a special particle ray-tracing technique has to be designed to efficiently track the particle movement for the special grid system, unstructured grid, which we use in the current study. Briefly speaking, the movement of a particle is determined by the velocity and initial position of the particle. If the intersecting face is an I/O boundary, the particle will be removed.

If not, then process the interaction according to the specified wall boundary condition.

The details of particle ray-tracing techniques of two- and three-dimensional domain are described in Ref. [8, 74].

Collision Cross-Section Data

As mentioned in Section 2.2, the variable soft sphere (VSS) model can reproduce the viscosity and diffusion coefficients correctly. The relevant parameters of using VSS

model for the DSM C method can be found in Bird’s book [10, 11]. This reference provides some usual gaseous species. When the flow involves some special species, it has problem to obtain the relevant parameters of the VSS molecular model. To overcome this problem, a quantum chemistry method is proposed to calculate the intermolecular energy surface according to the distance between molecules [75]. Then the simulated intermolecular energy potential is fitted through the Lennard-Jones (L-J) potential to obtain the constants. Based on these constants and gas kinetic theory, the transport coefficients, which are viscosity and diffusion coefficients, are derived. Finally, the parameters of the VSS model are derived by fitting these computed coefficients to those derived from the VSS model.

Pressure Boundary Treatment

In order to perform accurate simulation for inflow/outflow pressure boundaries, general procedure for treating these conditions by using the concept of particles flux conservation is developed in PDSC [76]. This function is useful for applications of micro-manifold, micro-nozzle and slider air bearing.

Unstructured Adaptive M esh with Variable Time-Step Scheme

To obtain accurate simulated results, two- and three-dimensional h-refinement adaptive mesh with variable time-step scheme is developed [39, 40]. Some parameters are used to determine the adaptive level and a simple cell quality control can prevent the creation of high aspect ratio cells. This module is not only valid for PDSC, but also suitable for other numerical simulators. The detail of adaptive mesh refinement and variable time-step scheme can be found in Chapter 3.

Parallel with Dynamic Domain Decomposition

To save the enormous computational cost of the standard DSM C code, a parallel DSM C with dynamic domain decomposition. M essage passing interface (M PI) is used for data communication. This function can automatically repartition the graph domain according to the loading of each processor, which is the particle number of each cell, to achieve the load balancing of the simulation. It also can be used for other particle simulation and equation solvers. The detail of this feature is presented in Chapter 4.

Conservative Weighting Scheme

When the flow involving trace particle species, the simulation needs lots of simulated particles to satisfy the DSM C limit, which will lead to immense computational time. A weighting scheme is developed to deal with this kind of flows [77]. The basic concept is assigning the lower weight for trace particle species to create

more simulated particles. This method does not use particle cloning and destroying to avoid the statistical error. The detail of conservative weighting scheme is introduced in Chapter 5.

M olecular Chemical Reaction

Finally, PDSC also has the function to simulate flows with chemical reactions.

Chemistry is important and needs to be considered when the flow velocity and temperature is very high. The chemistry in PDSC is developed with help of Professor I.

D. Boyd at the University of M ichigan in United States. It has dissociation, exchange and recombination reactions in PDSC. Chapter 6 is a section of detailed introduction and validation of this feature.

2.4.3 Postprocessor

After the DSM C solver (PDSC), it needs a postprocessor to view the result of the simulation. The output data of PDSC can be transferred from cell-based data into node-base data easily and then importing into both Tecplot and Grapher for displaying purpose.