Mechanical metamaterials enable customizing the elastic properties of physical objects by altering their fine-scale structure. A broad gamut of effective material properties can be produced even from a single fabrication material by optimizing the geometry of a periodic microstructure tiling. Past work has extensively studied the capabilities of microstructures in the small-displacement regime, where periodic homogenization of linear elasticity yields computationally efficient optimal design algorithms. However, many applications involve flexible structures undergoing large deformations for which the accuracy of linear elasticity rapidly deteriorates due to geometric nonlinearities. Design of microstructures at finite strains involves a massive increase in computation and is much less explored; no computational tool yet exists to design metamaterials emulating target hyperelastic laws over finite regions of strain space. We make an initial step in this direction, developing algorithms to accelerate homogenization and metamaterial design for nonlinear elasticity and building a complete framework for the optimal design of planar metamaterials. Our nonlinear homogenization method works by efficiently constructing an accurate interpolant of a microstructure's deformation over a finite space of macroscopic strains likely to be endured by the metamaterial. From this interpolant, the homogenized energy density, stress, and tangent elasticity tensor describing the microstructure's effective properties can be inexpensively computed at any strain. Our design tool then fits the effective material properties to a target constitutive law over a region of strain space using a parametric shape optimization approach, producing a directly manufacturable geometry. We systematically test our framework by designing a catalog of materials fitting isotropic Hooke's laws as closely as possible. We demonstrate significantly improved accuracy over traditional linear metamaterial design techniques by fabricating and testing physical prototypes.

Basket weaving is a traditional craft for creating curved surfaces as an interwoven array of thin, flexible, and initially straight ribbons. The three-dimensional shape of a woven structure emerges through a complex interplay of the elastic bending behavior of the ribbons and the contact forces at their crossings. Curvature can be injected by carefully placing topological singularities in the otherwise regular weaving pattern. However, shape control through topology is highly non-trivial and inherently discrete, which severely limits the range of attainable woven geometries. Here, we demonstrate how to construct arbitrary smooth free-form surface geometries by weaving carefully optimized curved ribbons. We present an optimization-based approach to solving the inverse design problem for such woven structures. Our algorithm computes the ribbons' planar geometry such that their interwoven assembly closely approximates a given target design surface in equilibrium. We systematically validate our approach through a series of physical prototypes to show a broad range of new woven geometries that is not achievable by existing methods. We anticipate our computational approach to significantly enhance the capabilities for the design of new woven structures. Facilitated by modern digital fabrication technology, we see potential applications in material science, bio- and mechanical engineering, art, design, and architecture.

Abstract We introduce a construction of subspaces of the spaces of tangential vector, n‐vector, and tensor fields on surfaces. The resulting subspaces can be used as the basis of fast approximation algorithms for design and processing problems that involve tangential fields. Important features of our construction are that it is based on a general principle, from which constructions for different types of tangential fields can be derived, and that it is scalable, making it possible to efficiently compute and store large subspace bases for large meshes. Moreover, the construction is adaptive, which allows for controlling the distribution of the degrees of freedom of the subspaces over the surface. We evaluate our construction in several experiments addressing approximation quality, scalability, adaptivity, computation times and memory requirements. Our design choices are justified by comparing our construction to possible alternatives. Finally, we discuss examples of how subspace methods can be used to build interactive tools for tangential field design and processing tasks.

We introduce the Reduced Immersed Method ( RIM ) for the real-time simulation of two-way coupled incompressible fluids and elastic solids and the interaction of multiple deformables with (self-)collisions. Our framework is based on a novel discretization of the immersed boundary equations of motion , which model fluid and deformables as a single incompressible medium and their interaction as a unified system on a fixed domain combining Eulerian and Lagrangian terms. One advantage for real-time simulations resulting from this modeling is that two-way coupling phenomena can be faithfully simulated while avoiding costly calculations such as tracking the deforming fluid-solid interfaces and the associated fluid boundary conditions. Our discretization enables the combination of a PIC/FLIP fluid solver with a reduced-order Lagrangian elasticity solver. Crucial for the performance of RIM is the efficient transfer of information between the elasticity and the fluid solver and the synchronization of the Lagrangian and Eulerian settings. We introduce the concept of twin subspaces that enables an efficient reduced-order modeling of the transfer. Our experiments demonstrate that RIM handles complex meshes and highly resolved fluids for large time steps at high framerates on off-the-shelf hardware, even in the presence of high velocities and rapid user interaction. Furthermore, it extends reduced-order elasticity solvers such as Hyper-Reduced Projective Dynamics with natural collision handling.

Abstract The spectrum and eigenfunctions of the Laplace‐Beltrami operator are at the heart of effective schemes for a variety of problems in geometry processing. A burden attached to these spectral methods is that they need to numerically solve a large‐scale eigenvalue problem, which results in costly precomputation. In this paper, we address this problem by proposing a fast approximation algorithm for the lowest part of the spectrum of the Laplace‐Beltrami operator. Our experiments indicate that the resulting spectra well‐approximate reference spectra, which are computed with state‐of‐the‐art eigensolvers. Moreover, we demonstrate that for different applications that comparable results are produced with the approximate and the reference spectra and eigenfunctions. The benefits of the proposed algorithm are that the cost for computing the approximate spectra is just a fraction of the cost required for numerically solving the eigenvalue problems, the storage requirements are reduced and evaluation times are lower. Our approach can help to substantially reduce the computational burden attached to spectral methods for geometry processing.

We present a method for the real-time simulation of deformable objects that combines the robustness, generality, and high performance of Projective Dynamics with the efficiency and scalability offered by model reduction techniques. The method decouples the cost for time integration from the mesh resolution and can simulate large meshes in real-time. The proposed hyper-reduction of Projective Dynamics combines a novel fast approximation method for constraint projections and a scalable construction of sparse subspace bases. The resulting system achieves real-time rates for large sub-spaces enabling rich dynamics and can resolve general user interactions, collision constraints, external forces and changes to the materials. The construction of the hyper-reduced system does not require user-interaction and refrains from using training data or modal analysis, which results in a fast preprocessing stage.

We introduce a variational approach for modeling n -symmetry vector and direction fields on surfaces that supports interpolation and alignment constraints, placing singularities and local editing, while providing real-time responses. The approach is based on novel biharmonic and m -harmonic energies for n -fields on surface meshes and the integration of hard constraints to the resulting optimization problems. Real-time computation rates are achieved by a model reduction approach employing a Fourier-like n -vector field decomposition, which associates frequencies and modes to n -vector fields on surfaces. To demonstrate the benefits of the proposed n -field modeling approach, we use it for controlling stroke directions in line-art drawings of surfaces and for the modeling of anisotropic BRDFs, which define the reflection behavior of surfaces.

Abstract We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier‐type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominentcotandiscretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a spline‐type editor for modelling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for real‐time modelling of tangential vector fields.

Abstract We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ℝnto the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space.

Abstract Splines are part of the standard toolbox for the approximation of functions and curves in ℝd. Still, the problem of finding the spline that best approximates an input function or curve is ill‐posed, since in general this yields a “spline” with an infinite number of segments. The problem can be regularized by adding a penalty term for the number of spline segments. We show how this idea can be formulated as an ℓ0‐regularized quadratic problem. This gives us a notion of optimal approximating splines that depend on one parameter, which weights the approximation error against the number of segments. We detail this concept for different types of splines including B‐splines and composite Bézier curves. Based on the latest development in the field of sparse approximation, we devise a solver for the resulting minimization problems and show applications to spline approximation of planar and space curves and to spline conversion of motion capture data.