Advanced Isogeometric Design-through-analysis Concepts
Final Report Abstract
Research objectives: The original goal of isogeometric analysis (IGA) was to support a more tightly connected interaction between CAD and finite element tools by using the same smooth and higher-order spline basis functions for the representation of the geometry and the approximation of solution fields. Since then IGA has developed into an innovative computational mechanics technology, offering a range of new perspectives and opportunities that go far beyond the geometric point of view of analysis. The objective of this project is to explore some of these opportunities in the context of isogeometric collocation and immersed boundary methods. Isogeometric collocation: We manifested the computational advantages of isogeometric collocation methods by an extensive efficiency study. We showed that collocation can be orders of magnitude faster than standard FEM and isogeometric Galerkin to achieve a specified level of accuracy. We derived a weighted collocation scheme that allows for mesh adaptivity based on hierarchical refinement of NURBS. Reduced quadrature in IGA: We investigated the use of Bézier element-based reduced quadrature for smooth quadratic and cubic spline discretizations, based on various tensor-product and monomial rules, and examined their rank-sufficiency, accuracy, and spectral properties. We found several stable and fully accurate rules for smooth spline elements that offer significant gains in computational efficiency compared to standard full Gauss quadrature. Collocated FEM: Motivated by our experience with IGA collocation, we transferred some of the collocation ideas to standard C0 finite element discretizations. Collocated FEM uses the Kronecker δ property of Lagrange basis functions at Gauss-Lobatto quadrature points and a symmetrization concept based on the averaging of the weighted residual and ultra-weak formulations. We showed that due to a well-behaved discrete spectrum and inexpensive point evaluations collocated FEM has the potential to open the door for fast higher-order explicit dynamics at acceptable critical time steps. These results are beyond the scope of the original proposal. Immersed boundary methods: We identified the geometry error in cut elements as a variational crime. We showed that the approximation of the physical domain and its boundary in cut elements needs to have at least the same order of accuracy as the approximation of the solution fields to maintain optimal rates of convergence. We developed an immersed fluid-structure interaction technique based on the augmented Lagrangian. We used this famework with stabilized isogeometric elements for the fluid and isogeometric Kirchhoff-Love shell elements for the structure to simulate the cyclic dynamics of a bioprosthetic heart valve. We showed that the immersed concept eliminates the need for remeshing and naturally handles the topology changes that appear as a result of the contact of the leaflets.
Publications
- Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Computer Methods in Applied Mechanics and Engineering, 267:170–232, 2013
D. Schillinger, J.A. Evans, A. Reali, M.A. Scott, T.J.R. Hughes
- Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 277:1–45, 2014
D. Schillinger, S.J. Hossain, T.J.R. Hughes
(See online at https://doi.org/10.1016/j.cma.2014.04.008)
