Construction of an objective function for optimization-based smoothing

An objective function for optimization-based smoothing is proposed for both linear and quadratic triangular and quadrilateral elements. Unlike currently published objective functions that are used to perform smoothing or untangling separately, this objective function can be used to untangle and smooth a mesh in a single process. The objective function is designed in such a way that it is easy and straightforward to be extended to higher order elements. The objective function has higher order continuous derivatives that make it suitable for optimization techniques. It has been shown empirically that the proposed function only has one minimum. With the integration of the proposed new objective function into our optimization-based smoothing algorithm, our combined Laplacian/optimization smoothing scheme provides us with satisfactory high quality meshes.