constant inner o set empty: unstable lled: unstable optimized inner o set empty: stable lled: unstable optimized inner & outer o set empty: stable lled: stable original model © http://www.modelplusmodel.com Figure 1: We introduce a method for reduced-order shape optimization of 2-manifolds that uses offset surfaces to deform the shape. Left: a bottle model is generated using offset surfaces with constant offsets. The resulting object is unable to stand. Center: the offsets are optimized such that the bottle can stand if empty, however, if filled it is unstable. Right: the model is optimized to stand both empty and filled. In order to account for that, offset surfaces are added inside and outside of the original shape. AbstractGiven the 2-manifold surface of a 3d object, we propose a novel method for the computation of an offset surface with varying thickness such that the solid volume between the surface and its offset satisfies a set of prescribed constraints and at the same time minimizes a given objective functional. Since the constraints as well as the objective functional can easily be adjusted to specific application requirements, our method provides a flexible and powerful tool for shape optimization. We use manifold harmonics to derive a reduced-order formulation of the optimization problem, which guarantees a smooth offset surface and speeds up the computation independently from the input mesh resolution without affecting the quality of the result. The constrained optimization problem can be solved in a numerically robust manner with commodity solvers. Furthermore, the method allows simultaneously optimizing an inner and an outer offset in order to increase the degrees of freedom. We demonstrate our method in a number of examples where we control the physical mass properties of rigid objects for the purpose of 3d printing.

We propose a novel type of planar-to-spatial deployable structures that we call elastic geodesic grids. Our approach aims at the approximation of freeform surfaces with spatial grids of bent lamellas which can be deployed from a planar configuration using a simple kinematic mechanism. Such elastic structures are easy-to-fabricate and easy-to-deploy and approximate shapes which combine physics and aesthetics. We propose a solution based on networks of geodesic curves on target surfaces and we introduce a set of conditions and assumptions which can be closely met in practice. Our formulation allows for a purely geometric approach which avoids the necessity of numerical shape optimization by building on top of theoretical insights from differential geometry. We propose a solution for the design, computation, and physical simulation of elastic geodesic grids, and present several fabricated small-scale examples with varying complexity. Moreover, we provide an empirical proof of our method by comparing the results to laser-scans of the fabricated models. Our method is intended as a form-finding tool for elastic gridshells in architecture and other creative disciplines and should give the designer an easy-to-handle way for the exploration of such structures.

In this paper we present a novel method for non-linear shape optimization of 3d objects given by their surface representation. Our method takes advantage of the fact that various shape properties of interest give rise to underdetermined design spaces implying the existence of many good solutions. Our algorithm exploits this by performing iterative projections of the problem to local subspaces where it can be solved much more efficiently using standard numerical routines. We demonstrate how this approach can be utilized for various shape optimization tasks using different shape parameterizations. In particular, we show how to efficiently optimize natural frequencies, mass properties, as well as the structural yield strength of a solid body. Our method is flexible, easy to implement, and very fast.

We propose a novel method for interactive design of well‐fitting body‐supporting surfaces that is driven by the pressure distribution on the body's surface. Our main contribution is an interactive modeling system that utilizes captured body poses and computes an importance field that is proportional to the pressure distribution on the body for a given pose. This distribution indicates where the body should be supported in order to easily hold a particular pose, which is one of the measures of comfortable sitting. Using our approximation, we propose the entire workflow for interactive design of C2 smooth surfaces which serve as seats, or generally, as body supporting furniture for comfortable sitting. Finally, we also provide a design tool for Rhinoceros/Grasshopper that allows for interactive creation of single designs or entire multi‐person sitting scenarios. We also test the tool with design students and present several results. Our method aims at interactive design in order to help designers to create appropriate surfaces digitally without additional empirical design passes.