Efficient CAD-integrated isogeometric analysis of trimmed solids

This publication presents a robust and efficient approach for fully CAD-integrated analyses of solids, which aims to reduce the current modeling effort for static and transient problems, including implicit and explicit dynamic simulations. Generating high-quality finite element meshes of solid structures is still a time- and labor-intensive process. Since embedded methods do not require sophisticated boundary-fitted meshes, they have gained popularity in recent years. However, most approaches tend to be computationally expensive due to numerous integration points, especially within trimmed elements. Moreover, their practical applicability in explicit dynamics is often limited because the classically used C⁰ continuous discretization field combined with trimming leads to infeasible time steps. In the following, we present methodologies addressing both of these shortcomings. The basic idea is to embed a three-dimensional object into a uniform C^p−1 continuous B-Spline cuboid, where the solid boundary representation provided by CAD is used as trimming surfaces to distinguish between material and void domain. Our primary focus is on constructing highly efficient quadrature rules for both trimmed and full knot spans, which accelerates required matrix formations and, in particular, drastically reduces the simulation times of explicit transient analyses. To fully exploit the potential of the B-Spline bases employed, first- and second-order reduced integration schemes are investigated in addition to optimal quadrature constructions. Despite the appearance of arbitrarily shaped domains, trimmed knot spans are evaluated at most with the same number of integration points as required for full Gaussian quadrature while maintaining optimal convergence in the energy norm. For full knot spans, savings in the number of quadrature points beyond 90% with respect to full Gaussian quadrature are achieved without observing any degradation in accuracy. The proposed methodologies are critically assessed based on scientific benchmarks of increasing complexity and a detailed industrial example, completing the design-through-analysis workflow by performing postprocessing operations directly on the deformed solid CAD model.