Truncation Error Based Mesh Adaptation and its Application to Multi-Mesh CFD
Jackson V., Charles Wilson
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One of the largest sources of error in a CFD simulation is the discretization error. One of the least computationally expensive ways of reducing the discretization error in a simulation is by performing mesh adaptation. In this work, the mesh adaptation processes are driven by the truncation error, which is the local source of the discretization error. Because this work is focused on methods for structured grids, r-adaptation is used as opposed to h-adaptation. A new method for performing the r-adaptation based on an optimization process is developed and presented here. This optimization process was applied to simple 1D and 2D Euler problems as a method of testing the approach. The mesh optimization approach is compared to the more common equidistribution approach to determine which produces more accurate results as well as the costs associated with each. It is found that the optimization process is able to reduce the truncation error than equidistribution. However, in the 2D cases optimization does not reduce the discretization error sufficiently to warrant the significant costs of the approach. This indicates that the much cheaper equidistribution process provides a cost-effective manner to reduce the discretization error in the solution. Further, equidistribution is able to achieve the bulk of the potential reductions in discretization error possible through r-adaptation. This work also develops a new framework for reducing the cost of performing truncation error based r-adaptation. This new framework also addresses some of the issues associated with r-adaptation. In this framework, adaptation is performed on a coarse mesh where it is faster to perform, creating a mapping function for this mesh, and finally evaluating this mapping at a fine enough mesh to meet the error target. The framework is used for 2D Euler and 2D laminar Navier-Stokes problems and shown to be the most cost-effective way to meet a desired error target. Finally, the multi-mesh CFD method is introduced and applied to a wide variety of problems from quasi-1D nozzle to 2D laminar and turbulent boundary layers. The multi-mesh method allows the system of equations to be solved on a system of meshes. With this method, each equation is solved on a mesh that is adapted specifically for it, meaning that more accurate solutions for each equation can be obtained. This work shows that, for certain problems, the multi-mesh approach is able to achieve more accurate results in less time compared to using a single mesh.
General Audience Abstract
Computational fluid dynamics (CFD) describes a method of numerically solving equations that attempt to model the behavior of a fluid. As computers have become cheaper and more powerful and the software has become more capable, CFD has become an integral part of the engineering process. One of the goals of the field is to be able to bring these higher fidelity simulations into the design loop earlier. Ideally, using CFD earlier in the design process would allow design engineers to create new innovative designs with less programmatic risk. Likewise, it is also becoming necessary to use these CFD tools later in the final design process to replace some physical experiments which can be expensive, unsafe, or infeasible to run. Both of these goals require the CFD codes to meet the accuracy requirements for the results as fast as possible. This work discusses several different methods for improving the accuracy of the simulations as well as ways of obtaining these more accurate results for the cheapest cost. In CFD, the governing equations modeling the flow behavior are solved on a computer. As a result, these continuous differential equations must be approximated as a system of discrete equations, so that they can be solved on a computer. These approximations result in discretization error, the difference between the exact solutions to the discrete and continuous equations, which is typically the largest type of numerical error in a CFD solution. The source of the discretization error is the truncation error, which is composed of the terms left out of the approximations made when discretizing the continuous equations. Thus, if the truncation error can be reduced, the discretization error in the solution should also be reduced. In this work, several different ways of reducing this truncation error through mesh adaptation are discussed, including the use of optimization methods. These mesh optimization methods are compared to a more common way of performing adaptation, namely equidistribution. It is determined that equidistribution is able to reduce the discretization error by a similar amount while being significantly faster than mesh optimization. This work also presents a framework for making the adaptation process faster overall by performing the adaptation on a coarse mesh and then refining the mesh enough to meet the error tolerance for the application. This framework was the cheapest method investigated to meet a given error target. This work also introduces a new technique called multi-mesh CFD, which allows each equation (conservation of mass, momentum, energy, etc.) to be solved on a separate mesh. This allows each equation to be solved on a mesh that is specifically adapted for it, resulting in a more accurate solution. Here, it is shown that, for certain problems, the multi-mesh technique is able to obtain a solution with lower error than only using a single mesh. This work also shows that these more accurate results can be obtained in less time using multiple meshes than on a single mesh.
- Doctoral Dissertations