A New Approach to Mesh Adaptation Procedure Using Linear Elasticity for Geometries Undergoing Large Displacements

The equations of linear elasticity have been extensively used in computational fluid dynamics to deform meshes for moving boundary simulations, shape design optimization, solution adaptive refinement, and for the construction of higher-order discetizations in finite-element schemes. Inherently this method does not have any mechanism to control the quality of the mesh, since it represents the structural response to prescribed surface deflections. Unfortunately, this method does not prevent the possibility of generating negative volumes in the mesh when large deformations take place. In the current work, two approaches are examined in an attempt to mitigate this shortcoming. In the first approach, a source term is added to each node in the mesh. These source terms are chosen as design variables, and an optimization strategy is utilized to improve mesh quality. The second approach represents a modification to an existing method whereby each element in the mesh may be considered as a different material. Entries in the constitutive relations are then selected as the design variables. In this approach the number of design variables is extremely large and, thus, very computationally expense. To alleviate some of this computational burden, the design variables are selected from a subset of the elements in the mesh. In both approaches presented, the cost function is defined as a function of the mesh quality, and a limited memory BFGS optimization scheme used to minimize this function. Results for two dimensional test cases are presented; however, the concept can be easily applied to three dimensional meshes and practical problems.