Transformation methods for the integration of singular and near-singular functions in XFEM

Resumo

This doctoral thesis addresses the problem of numerical integration of singular and near-singular functions, in two and three dimensions, using variable transformation methods. It includes the analysis of transformations with a geometric purpose, i.e., they map the physical domain onto a parent, standard domain, and transformations of an algebraic nature, with the purpose of softening the (near-)singularities in the integrand. Transformations used to map the physical element onto the parent domain are described in chapter 2. The most general case of a degenerate isoparametric map, such that it is homogeneous in one of its variables is presented, and its equivalence to the polar transformation is justified in the two-dimensional case. These maps induce a factorization of certain types of integral kernels into a radial and an angular part, allowing a separate, specific treatment of each factor. The two-dimensional singular integration problem is examined in chapter 3. The radial kernel is completely regularized by means of a new scheme that removes its singularity. Regarding the angular kernel, it is shown to have the same form as the one-dimensional near-singular kernel, and thus the same set of transformations can be successfully applied to both kernels. The two-dimensional near-singular kernel is the subject of chapter 4. Whilst the treatment of the angular kernel is exactly the same as in chapter 3, the radial kernel admits a whole new set of regularizing maps, taking advantage of the linear factor in the Jacobian of the degenerate isoparametric transformation. The generalization of the problem to adjacent triangles, in which the source point lies outside the integration domain is also considered. The extension of the singular integration to three-dimensional domains is covered in chapter 5. The treatment of the radial kernel is very similar as in chapter 3, whereas the bivariate angular kernel, restricted to the boundary of the bidimensional angular domain, behaves very similarly to the near-singular one dimensional kernel, and yet the same set of softening transformations as in chapter 3 and chapter 4 can be suitable re-utilized in this situation. Lastly, chapter 6 presents a proof of the optimal form of the well-known cubic transformation, employed as one of the most common alternatives to regularize the angular kernel in the three previous chapters. All proposed methods have been extensively tested from the numerical point of view, showing that they are able to outperform the existing methods for a broad variety of situations.