Manufacturing of free-form geometries using 5-axis Computer Numerically Controlled (CNC) machining brings challenges in path- and motion-planning as one typically wants to minimize the manufacturing time of the object under consideration, while keeping the machining error within fine machining tolerances that ranges in tens of microns. We propose an optimization-based pipeline that, for a given toroidal and/or cylindrical flat-end cutter, simultaneously optimizes its milling paths together with its local positioning represented by the rotation and tilt functions. The proposed strategy is validated on a variety of benchmark surfaces, with different hyperparameters for the objective function and initial conditions, showing that our results provide high-quality approximations of free-form geometries using by-construction non-colliding motions of the given tool. • Optimize toolpaths, rotation, and tilt together for 5-axis toroidal/flat tools. • Evolution-based framework (CMA-ES) for the multiobjective optimization problem. • Optimizes path distribution to improve accuracy and reduce overall length. • Validated on various benchmarks, including industrial free-form geometries.

We propose a computational framework for motion planning for 5-axis CNC machining of free-form surfaces. Given a reference surface, a set of contact paths on it, and a shape of a toroidal cutting tool as input, the proposed algorithm designs the tool motions that are by construction locally and globally collision-free, and offers a trade-off between approximation quality and tool wear using an optimization-based framework. The proposed algorithm first quickly constructs 2D time-tilt configuration spaces along each contact path, detecting regions that are collision-free. The configuration spaces are then merged into a single time-tilt configuration space to find a global tilt function to control the overall motion of the tool. An initial collision-free tilt function in B-spline form is first estimated and then optimized to minimize the machining error while distributing the tool wear as uniformly as possible along the entire cutting edge of the tool while staying in the collision-free region. Our algorithm is validated on both synthetic free-form surfaces and industrial benchmarks, showing that one can considerably reduce the tool wear without degrading the machining accuracy.

Abstract A gerotor pump is a two‐piece mechanism where two rotational components, interior and exterior, engage each other via a rotational motion to transfer a fluid in a direction parallel to their rotational axes. A natural question arises on what shape of the gerotor is the optimal one in the sense of maximum fluid being pumped for a unit of time, given the constraint of a fixed material needed to manufacture the pump. As there is no closed‐formula to answer this question, we propose a new algorithm to design and optimize the shape of gerotor pumps to be as efficient as possible. The proposed algorithm is based on a fast construction of the envelope of the interior component and subsequent optimization. We demonstrate our algorithm on a benchmark gerotor and show that the optimized solution increases the estimated flowrate by 16%. We also use our algorithm to study the effect of the number of teeth on the cavity area of a gerotor.

Existing flank milling path-planning methods typically lead to tiny gaps or overlaps between neighboring paths, which causes artifacts and imperfections in the workpiece. We propose a new multi-strip path-planning method for 5-axis flank milling of free-form surfaces which targets G 1 (tangent-plane) continuity of the neighboring strips along shared boundaries. While for some geometries one cannot achieve G 1 continuity and high approximation quality at the same time, our optimization framework offers a good trade-off between machining accuracy in terms of distance error and the G 1 connection of neighboring strips. We demonstrate our algorithm on synthetic free-form surfaces as well as on industrial benchmark datasets, showing that we are able to meet fine industrial tolerances and simultaneously significantly reduce the kink angle of adjacent strips, and consequently to improve the surface finish in terms of smoothness.

CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between "dual" classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software.

We investigate a class of stream surfaces that expand in time as much as possible. Given a vector field, we look for seed curves that locally propagate in time in a stretch-maximizing manner, i.e., curves that infinitesimally expand most progressively. We show that such a curve is generically unique at every point in an incompressible flow and offers a very good initial guess for a stretch-maximizing stream surface. With the application of efficient fluid advection–diffusion in mind, we optimize fluid injection towards optimal advection and show several examples on benchmark datasets.

We investigate the close correspondence between barycentric coordinates and barycentric kernels from the point of view of the limit process when finer and finer polygons converge to a smooth convex domain. We show that any barycentric kernel is the limit of a set of barycentric coordinates and prove that the convergence rate is quadratic. Our convergence analysis extends naturally to barycentric interpolants and mappings induced by barycentric coordinates and kernels. We verify our theoretical convergence results numerically on several examples.

Abstract We study the difficult problem of deciding if parts of a freeform surface can be generated, or approximately generated, by the motion of a planar profile through space. While this task is basic for understanding the geometry of shapes as well as highly relevant for manufacturing and building construction, previous approaches were confined to special cases like kinematic surfaces or “moulding” surfaces. The general case remained unsolved so far. We approach this problem by a combination of local and global methods: curve analysis with regard to “movability”, curve comparison by common substring search in curvature plots, an exhaustive search through all planar cuts enhanced by quick rejection procedures, the ordering of candidate profiles and finally, global optimization. The main applications of our method are digital reconstruction of CAD models exhibiting sweep patches, and aiding in manufacturing freeform surfaces by pointing out those parts which can be approximated by sweeps.

We present a framework for exploring topologically unique quadrangulations of an input shape. First, the input shape is segmented into surface patches. Second, different topologies are enumerated and explored in each patch. This is realized by an efficient subdivision-based quadrangulation algorithm that can exhaustively enumerate all mesh topologies within a patch. To help users navigate the potentially huge collection of variations, we propose tools to preview and arrange the results. Furthermore, the requirement that all patches need to be jointly quadrangulatable is formulated as a linear integer program. Finally, we apply the framework to shape-space exploration, remeshing, and design to underline the importance of topology exploration.

Abstract We discuss the theory, discretization, and numerics of curves which are evolving such that part of their shape, or at least their curvature as a function of arc length, remains unchanged. The discretization of a curve as a smooth sequence of circular arcs is well suited for such purposes, and allows us to reduce evolution of curves to the evolution of a control point collection in a certain finite‐dimensional shape space. We approach this evolution by a 2‐step process: linearized evolution via optimized velocity fields, followed by optimization in order to exactly fulfill all geometric side conditions. We give applications to freeform architecture, including “rationalization” of a surface by congruent arcs, form finding and, most interestingly, non‐static architecture.

We present an algorithm which robustly computes the intersection curve(s) of an under-constrained piecewise polynomial system consisting of n equations with n + 1 unknowns. The solution of such a system is typically a curve in Rn+1. This work extends the single solution test of [6] for a set of algebraic constraints from zero dimensional solutions to univariate solutions, in Rn+1. Our method exploits two tests: a no loop test (NLT) and a single component test (SCT) that together isolate and separate domains D where the solution curve consists of just one single component. For such domains, a numerical curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting curve is guaranteed. Several possible application of the solver, like 3D trisector curves or kinematic simulations in 3D are discussed.