The application and development of numerical method for moving boundary and complex geometry problems

Various methods have recently been introduced to alleviate the difficulties associated with simulating moving boundary problems. Among them, the fictitious domain method is explored and methods to mitigate the errors associated with this technique are presented in this work. A summary of moving mesh strategies currently in use in commercial computational fluid dynamics software is first provided. Significant issues associated with moving grid techniques such as the interpolation errors, the human cost for mesh generation, and the quality of the resulting mesh are discussed by solving, with the help of a commercial code, a variety of fluid dynamics problems associated with in-cylinder flows of internal combustion engines. The Lagrange multiplier/fictitious domain method is then discussed by solving simple heat transfer problems. The fictitious domain method is based on the use of Lagrange multipliers that do not match with an underlying mesh. Such an approach introduces errors on the adjacent nodes and a parametric study of various factors affecting the quality of the results is performed. Two approaches for reducing the errors are proposed. A first method uses modified boundary conditions to reduce the errors on the adjacent nodes. The method is based on a predictor/corrector scheme and is called the "fictitious constraint" method. A second method consists of simply modifying the shape of the boundary by matching the complex boundary with the underlying mesh. Improvements in the solutions by using these techniques are illustrated with the help of one-dimensional and two-dimensional problems.