Algorithm Developement For Adaptive Grid Generation Using Galerkin Finite Element Method

The adaptive grid concept is developed in order to achieve more accurate and stable results for the numerical simulations of Partial Differential Equations (PDEs). There are two main types of strategies used for adaptive grids: local refinement by increasing the number of elements (h-refinement) and deforming grids where the number of elements remains fixed (r-refinement). The h-refinement method requires inserting additional elements and nodes in a certain region of the mesh. This significantly affects the software data structure and makes programming more difficult. To maintain the same data structure, the method must delete the same number of nodes from another region. Again, this complicates the program. However, the deforming grid method works by simply moving the nodes of an existing mesh. The proposed method formulates this deformation as an unsteady problem where the position of the nodes can be determined from their velocities. For each time step, the node movement is controlled by a user defined error indicator or user desired target node distribution. The number of nodes and elements remains constant for each time step and this greatly simplifies the program structure compared to the r-method. During the deformation process, the element shape changes to achieve the desired distribution and solution accuracy at each time step. The current research work focuses on the development of the algorithm for unstructured meshes and its application to the solution of unsteady elliptic PDEs solved by finite element (unstructured). The successful development of the Adaptive Grid Generation is achieved with the implementation of Galerkin Finite Element Method on Grid Deformation Method for unstructured and the results are validated with the structured grids results. It is also implemented on few model fluid problems.