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AHPCRC Bulletin: Summer 1994 - Volume 4 Number 4

Automatic Mesh Generation and Update Strategies in 3D Flow Simulations

Andrew Johnson, AHPCRC

Today's parallel supercomputers are allowing engineers and scientists solve a wide range of complex, real-world problems at scales that were considered impossible only a few years ago. The finite element method can be employed for flow simulations with an accurate representation of the geometric complexities involved in the actual application, provided that the method is accompanied with a versatile and efficient geometric modeling and mesh generation capability. This author, a Ph. D. student advised by Tayfun Tezduyar in Aerospace Engineering and Mechanics at the University of Minnesota, has developed a preliminary version of a fully automatic, 3D mesh generation system.

Quite often, for a specific class of problems, a special-purpose mesh generator based on a variable set of user-specified geometric parameters can be designed and used effectively. This approach has some obvious advantages. a) The cost for generating the mesh will be much lower compared to the cost for using an automatic mesh generator. b) The user can exercise more control on the structure of the mesh generated. c) In problems with moving boundaries and interfaces, such as free surfaces, two-liquid interfaces, moving mechanical components, and fluid- structure and fluid-particle interactions, the mesh needs to be updated every time step to accommodate the changes in the spatial domain. Ideally we would like to update the mesh, as much as we can, by just moving the mesh with minimal frequency of remeshing (generating a new set of nodes and elements). A mesh generated by a special-purpose mesh generator can be moved by using a special- purpose mesh moving scheme in which nodes are moved according to an explicitly defined rule. This moving will be easier and less costly compared to using an automatic mesh moving scheme that would be unavoidable if the initial mesh has been generated using an automatic mesh generator.

Although special-purpose mesh generation and mesh moving strategies are rather limited in handling complex geometries and spatial domain changes with complex modes and large magnitudes, such strategies have been employed by Tezduyar's research group for simulation of several practical flow problems such as the deployment and gliding of parafoils and two trains passing each other in a tunnel.

Automatic mesh generators, on the other hand, though more costly to use, especially in complex 3D problems, provide much more flexibility in mesh generation. They involve little or no assumption on the shape of the domain or the objects being modeled. Because of that, a mesh can be generated over almost any geometry. Although the users cannot specify exactly the type of refinement they may wish to have in all areas of the domain, they can still exercise some control over the kind of mesh created. The actual input to these mesh generators are information on the boundaries, either an actual surface mesh composed of triangles, or some mathematical definition of the boundary like Bézier surfaces.

The automatic mesh generation system developed at the AHPCRC has three components. The first component is an interactive, 3D geometric modeler which runs on workstations. By using this program, the user can construct a geometric model interactively by viewing it on the workstation, and the program has many features which help the user during this modeling stage. The geometric model is constructed by using Bézier quadrilateral and triangular surfaces.

The second component is an automatic surface mesh generator. This program takes the geometric model as input, and creates a surface mesh consisting of triangular elements over each Bézier surface. The level of refinement previously specified by the user at the edges of each Bézier surface determines the refinement within the interior of these surfaces.

The surface mesh is used as input to the 3D automatic mesh generator which uses Delaunay-Vorono methods with edge-swapping techniques. The level of refinement in the interior of the 3D mesh is determined from the level of refinement in the surface mesh. Meshes over interior boundaries which are not actually part of any physical boundary may be used to help control refinement in the interior of the domain. Also, the automatic mesh generator has the capability of creating thin, structured layers of elements around solid surfaces. This feature, which gives the user more control over the kind of mesh generated near such surfaces, is desirable in analysis of viscous flows.

Many applications have been or are being currently studied using this mesh generation system. For example, this author has created meshes for an automobile and a fighter aircraft. Other examples are meshes for a concept car, a semi truck, a stealth aircraft, and a M1 tank, created by the AHPCRC 1994 Summer Institute students Joshua Simer and Eric Engebretsen, and AHPCRC undergraduate research assistant Matt Litke. Marek Behr, an Assistant Professor at the AHPCRC, is using this system in hydrodynamics computations in a collaborative effort with researchers from Waterways Experiment Station. The ram air parafoil studies carried out at the AHPCRC in collaboration with Natick RDEC is also making use of this mesh generation system by using unstructured meshes created by AHPCRC undergraduate research assistant Chris Waters. All of these applications are computed on the Thinking Machines CM-5 and Cray T3D parallel supercomputers using the flow codes developed at the AHPCRC by Tezduyar's research group.

The largest mesh that has been created by using this mesh generator is the one for an automobile. A geometric model needed to be created for only half of the car since the geometry is assumed to be symmetric. This geometric model, which is quite detailed, contains 1,823 points, 457 Bézier edges, and 261 Bézier surfaces. The geometric model includes wheels, recessed headlights, rearview mirrors, and a spoiler. The surface mesh for this geometry contains 35,133 nodes and 70,589 triangular elements. The 3D mesh generated by the automatic mesh generator has 226,383 nodes and 1,400,744 tetrahedral elements. This half mesh was then reflected about the symmetry plane to complete the mesh to a full one. The full mesh contains 447,180 nodes and 2,801,488 elements. The mesh includes thin, structured layers of elements around the automobile.

Figure 1. Flow past an automobile at 55 miles per hour. Streamlines color-coded with the magnitude of the velocity.
The finite element analysis using this mesh consisted of simulation of incompressible flow past the automobile at 55 miles per hour with a modified Smagorinsky turbulence model. For the efficient use of the CM-5 resources, a semi- discrete, matrix-free version of the fluid flow code was used to obtain the solution. The simulation was actually carried out under two different conditions. In one simulation, we took into account the actual road conditions. We implemented this by keeping the car stationary, moving the road with the free-stream velocity imposed at the upstream boundary, and letting the tires spin with a speed corresponding to this velocity. In the second simulation, we assumed wind-tunnel conditions. This was implemented by keeping the car and the tires stationary, and by imposing the free-stream velocity at the upstream boundary. In this implementation, we assume zero shear stress conditions on the road surface. The forces that were computed on the automobile compare well with those expected in automobile aerodynamics. The drag coefficient was calculated to be 0.455 under road conditions and 0.354 for wind tunnel conditions. Figure 1 shows the streamlines (color-coded with the magnitude of fluid velocity) around the automobile for simulation under road conditions.

Another simulation we carried out which relies heavily on automatic mesh generation is the study of multiple spheres (1-5) falling in a liquid-filled tube. There are several features in this simulation which make it quite complicated.

First, because the spheres are moving freely as they react to the fluid forces, there is the requirement that there may be any number of spheres at any location inside the tube. A special version of the automatic mesh generation system has been developed to handle this case. Given the size of the tube, the number and size of the spheres and their location, and other refinement information, this mesh generator creates the mesh needed. The meshes include thin, structured layers of elements around each sphere.

Second, we need to accommodate the motion of the spheres. The locations of the spheres are unknown and need to be determined as part of the overall solution. Because the motion is unknown and an automatic mesh generator is being used, the only option in moving the mesh to accommodate the motion of the spheres is to use an automatic mesh moving scheme. We previously have developed such a scheme at the AHPCRC. In this method, the motion of the mesh is modeled by letting the nodal displacements be governed by the equations of elasticity, with the boundary conditions for these equations specified based on the motion of the spheres.

Third, if the displacements of the spheres become to large, the automatic mesh moving scheme may lead to unacceptable levels of distortion in the mesh. When this happens, a new mesh (with a new set of nodes and elements) for the current configuration of the spheres has to be created with the automatic mesh generator. The solution then has to be projected from the old mesh on to the new one. Computations can then proceed by using the new mesh and by moving that mesh with the automatic mesh moving algorithm.

In one of the simulations we carried out, there are two spheres starting in a staggered configuration. As the spheres fall, the trailing sphere is attracted to the low-pressure region in the wake of the leading sphere. The two spheres eventually collide, then separate and fall side by side throughout the rest of the simulation. Figure 2 shows the spheres at five different instants during the simulation.

Figure 2. (left): Two spheres falling in a liquid-filled tube. The spheres at five different instants during the simulation.

Figure 3. (right): Five spheres falling in a liquid-filled tube. The spheres at five different instants during the simulation.

In a simulation with five spheres, initially the spheres are arranged in a slightly jumbled pentagon configuration. As the spheres fall, the five spheres rearrange themselves into an exact pentagon configuration also with all spheres aligned vertically. Figure 3 shows the spheres at five different instants during the simulation.