Isogeometric Analysis and Iterative Solvers for Shear Bands

Numerical modeling of shear bands present several challenges, primarily due to strain softening, strong nonlinear multiphysics coupling, and steep solution gradients with fine solution features. In general it is not known a priori where a shear band will form or propagate, thus adaptive refinement is sometimes necessary to increase the resolution near the band. In this work we first explore the use of isogeometric analysis for shear band problems by constructing and testing several combinations of NURBS elements for a mixed finite element shear band formulation. Owing to the higher order continuity of the NURBS basis, fine solution features such as shear bands can be resolved accurately and efficiently without adaptive refinement. The results are compared to a mixed element formulation with linear functions for displacement and temperature and Pian–Sumihara shape functions for stress. We find that an element based on high order NURBS functions for displacement, temperature and stress, combined with gauss point sampling of the plastic strain leads to attractive results in terms of rate of convergence, accuracy and cpu time. This element is implemented with a Bbar strain projection method and is shown to be nearly locking free. Second we develop robust parallel preconditioners to GMRES in order to solve the Jacobian systems arising at each time step of the problem efficiently. The main idea is to design Schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands. We develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands. The proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as GMRES/ILU and LU direct solvers. Nonlinear and linear iterations counts as well as CPU times and computational speedups are reported and it is shown that the proposed preconditioners are robust, efficient and outperform traditional state of the art solvers. Finally, we extend the preconditioners to further take advantage the physics of the problem. That is most of the deformation and plasticity is localized in a narrow band while out of this domain only small deformations and minor plasticity is observed. Hence, a preconditioner that decomposes the domain and concentrate more effort in the shear band domain while reusing information away from the band may lead to a significantly improved computational performance. To this end, we first propose a schur complement strategy which takes advantage of the gauss point history variables conveniently. Then, a general overlapping domain decomposition procedure is performed, partitioning the domain into so called 'shear band subdomain' and a 'healthy subdomain', which is used to precondition the Schur complement system. The shear band subdomain preconditioner is then solved exactly with an LU solver while the healthy subdomain preconditioner is only solved once in the elastic region and reused throughout the simulation. This localization awareness approach is shown to be very efficient and leads to an attractive solver for shear bands

Non-invasive multigrid for semi-structured grids

Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid