No abstract

A novel surface interrogation technique is proposed to compute the intersection of curves with spline surfaces in isogeometric analysis. The intersection points are determined in one-shot without resorting to a Newton-Raphson iteration or successive refinement. Surface-curve intersection is required in a wide range of applications, including contact, immersed boundary methods and lattice-skin structures, and requires usually the solution of a system of nonlinear equations. It is assumed that the surface is given in form of a spline, such as a NURBS, T-spline or Catmull-Clark subdivision surface, and is convertible into a collection of Bézier patches. First, a hierarchical bounding volume tree is used to efficiently identify the Bézier patches with a convex-hull intersecting the convex-hull of a given curve segment. For ease of implementation convex-hulls are approximated with k-dops (discrete orientation polytopes). Subsequently, the intersections of the identified Bézier patches with the curve segment are determined with a matrix-based implicit representation leading to the computation of a sequence of small singular value decompositions (SVDs). As an application of the developed interrogation technique the isogeometric design and analysis of lattice-skin structures is investigated. Although such structures have been common in large-scale civil engineering, current additive manufacturing, or 3d printing, technologies make it possible to produce up to metre size lattice-skin structures with designed geometric features reaching down to submillimetre scale. The skin is a spline surface that is usually created in a computer-aided design (CAD) system and the periodic lattice to be fitted consists of unit cells, each containing a small number of struts. The lattice-skin structure is generated by projecting selected lattice nodes onto the surface after determining the intersection of unit cell edges with the surface. For mechanical analysis, the skin is modelled as a Kirchhoff-Love thin-shell and the lattice as a pin-jointed truss. The two types of structures are coupled with a standard Lagrange multiplier approach.

Knitting is an effective technique for producing complex three‐dimensional surfaces owing to the inherent flexibility of interlooped yarns and recent advances in manufacturing, providing better control of local stitch patterns. Fully yarn‐level modeling of large‐scale knitted membranes is not feasible. Therefore, we use a two‐scale homogenization approach and model the membrane as a Kirchhoff‐Love shell on the macroscale and as Euler‐Bernoulli rods on the microscale. The governing equations for both the shell and the rod are discretized with cubic B‐spline basis functions. For homogenization, we consider only the in‐plane response of the membrane. The solution of the nonlinear microscale problem requires a significant amount of time due to the large deformations and the enforcement of contact constraints, rendering conventional online computational homogenization approaches infeasible. To sidestep this problem, we use a pretrained statistical Gaussian process regression (GPR) model to map the macroscale deformations to macroscale stresses. During the offline learning phase, the GPR model is trained by solving the microscale problem for a sufficiently rich set of deformation states obtained by either uniform or Sobol sampling. The trained GPR model encodes the nonlinearities and anisotropies present in the microscale and serves as a material model for the membrane response of the macroscale shell. The bending response can be chosen in dependence of the mesh size to penalize the fine out‐of‐plane wrinkling of the membrane. After verifying and validating the different components of the proposed approach, we introduce several examples involving membranes subjected to tension and shear to demonstrate its versatility and good performance.

Summary The need to compute the intersections between a line and a high‐order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a noniterative method for computing intersections by solving a matrix singular value decomposition and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one‐shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton‐Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree q of the shape functions and is 2q × 3q for curves and 6q2 × 8q2 for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used high‐order finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitro-project/.