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Two‐scale topology optimization, combined with the design of microstructure families with a broad range of effective material parameters, is widely used in many fabrication applications to achieve a target deformation behavior for a variety of objects. The main idea of this approach is to optimize the distribution of material properties in the object partitioned into relatively coarse cells, and then replace each cell with microstructure geometry that mimics these material properties. In this paper, we focus on adapting this approach to complex shapes in situations when preserving the shape's surface is essential. Our approach extends any regular (i.e. defined on a regular lattice grid) microstructure family to complex shapes, by enriching it with tiles adapted to the geometry of the cut‐cell. We propose a fully automated and robust pipeline based on this approach, and we show that the performance of the regular microstructure family is only minimally affected by our extension while allowing its use on 2D and 3D shapes of high complexity. [ABSTRACT FROM AUTHOR]

We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code. We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations. [ABSTRACT FROM AUTHOR]

We present a numerically robust algorithm for computing the constrained Delaunay tetrahedrization (CDT) of a piecewise-linear complex, which has a 100% success rate on the 4408 valid models in the Thingi10k dataset. We build on the underlying theory of the well-known tetgen software, but use a floating-point implementation based on indirect geometric predicates to implicitly represent Steiner points: this new approach dramatically simplifies the implementation, removing the need for ad-hoc tolerances in geometric operations. Our approach leads to a robust and parameter-free implementation, with an empirically manageable number of added Steiner points. Furthermore, our algorithm addresses a major gap in tetgen's theory which may lead to algorithmic failure on valid models, even when assuming perfect precision in the calculations. Our output tetrahedrization conforms with the input geometry without approximations. We can further round our output to floating-point coordinates for downstream applications, which almost always results in valid floating-point meshes unless the input triangulation is very close to being degenerate. [ABSTRACT FROM AUTHOR]

3D printing is revolutionizing manufacturing and is now being considered in the electronics industry. The creation of the first 3D volumetric circuit (3DVC) has created a way to make circuits smaller, lighter, into unconventional form factors and exploit physics like anisotropy more effectively than planar geometries can. While this is exciting, many problems must be solved to make 3DVCs a reality. One of these problems is electromagnetic interference and mutual coupling between components that are expected to be highly problematic in high frequency 3DVCs. Spatially-variant anisotropic metamaterials (SVAMs) could be a solution to overcome this difficulty, but research in this area is not possible without a way to generate SVAMs around multiple components. In this paper, an algorithm is integrated into CAD software that can generate SVAMs for 3D circuits which will enable future studies of SVAMs. [ABSTRACT FROM AUTHOR]

We propose In-Timestep Remeshing, a fully coupled, adaptive meshing algorithm for contacting elastodynamics where remeshing steps are tightly integrated, implicitly, within the timestep solve. Our algorithm refines and coarsens the domain automatically by measuring physical energy changes within each ongoing timestep solve. This provides consistent, degree-of-freedom-efficient, productive remeshing that, by construction, is physics-aware and so avoids the errors, over-refinements, artifacts, per-example hand-tuning, and instabilities commonly encountered when remeshing with timestepping methods. Our in-timestep computation then ensures that each simulation step's output is both a converged stable solution on the updated mesh and a temporally consistent trajectory with respect to the model and solution of the last timestep. At the same time, the output is guaranteed safe (intersection- and inversion-free) across all operations. We demonstrate applications across a wide range of extreme stress tests with challenging contacts, sharp geometries, extreme compressions, large timesteps, and wide material stiffness ranges - all scenarios well-appreciated to challenge existing remeshing methods. [ABSTRACT FROM AUTHOR]

We introduce a novel approach to describe mesh generation, mesh adaptation, and geometric modeling algorithms relying on changing mesh connectivity using a high-level abstraction. The main motivation is to enable easy customization and development of these algorithms via a declarative specification consisting of a set of per-element invariants, operation scheduling, and attribute transfer for each editing operation. We demonstrate that widely used algorithms editing surfaces and volumes can be compactly expressed with our abstraction, and their implementation within our framework is simple, automatically parallelizable on shared-memory architectures, and with guaranteed satisfaction of the prescribed invariants. These algorithms are readable and easy to customize for specific use cases. We introduce a software library implementing this abstraction and providing automatic shared-memory parallelization. [ABSTRACT FROM AUTHOR]

We introduce a code generator that converts unoptimized C++ code operating on sparse data into vectorized and parallel CPU or GPU kernels. Our approach unrolls the computation into a massive expression graph, performs redundant expression elimination, grouping, and then generates an architecture-specific kernel to solve the same problem, assuming that the sparsity pattern is fixed, which is a common scenario in many applications in computer graphics and scientific computing. We show that our approach scales to large problems and can achieve speedups of two orders of magnitude on CPUs and three orders of magnitude on GPUs, compared to a set of manually optimized CPU baselines. To demonstrate the practical applicability of our approach, we employ it to optimize popular algorithms with applications to physical simulation and interactive mesh deformation. [ABSTRACT FROM AUTHOR]

We describe a method for the generation of seamless surface parametrizations with guaranteed local injectivity and full control over holonomy. Previous methods guarantee only one of the two. Local injectivity is required to enable these parametrizations' use in applications such as surface quadrangulation and spline construction. Holonomy control is crucial to enable guidance or prescription of the parametrization's isocurves based on directional information, in particular from cross-fields or feature curves, and more generally to constrain the parametrization topologically. To this end we investigate the relation between cross-field topology and seamless parametrization topology. Leveraging previous results on locally injective parametrization and combining them with insights on this relation in terms of holonomy, we propose an algorithm that meets these requirements. A key component relies on the insight that arbitrary surface cut graphs, as required for global parametrization, can be homeomorphically modified to assume almost any set of turning numbers with respect to a given target cross-field. [ABSTRACT FROM AUTHOR]

We propose Deep Estimators of Features (DEFs), a learning-based framework for predicting sharp geometric features in sampled 3D shapes. Differently from existing data-driven methods, which reduce this problem to feature classification, we propose to regress a scalar field representing the distance from point samples to the closest feature line on local patches. Our approach is the first that scales to massive point clouds by fusing distance-to-feature estimates obtained on individual patches. We extensively evaluate our approach against related state-of-the-art methods on newly proposed synthetic and real-world 3D CAD model benchmarks. Our approach not only outperforms these (with improvements in Recall and False Positives Rates), but generalizes to real-world scans after training our model on synthetic data and fine-tuning it on a small dataset of scanned data. We demonstrate a downstream application, where we reconstruct an explicit representation of straight and curved sharp feature lines from range scan data. We make code, pre-trained models, and our training and evaluation datasets available at https://github.com/artonson/def. [ABSTRACT FROM AUTHOR]

The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of this benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance. As a specific application of our geometry and benchmark dataset, we explore the question of relative advantages of unstructured (triangular/ tetrahedral) and structured (quadrilateral/hexahedral) discretizations. We observe that for Lagrange-type elements, while linear tetrahedral elements perform poorly, quadratic tetrahedral elements perform equally well or outperform hexahedral elements for our set of problems and currently available mesh generation algorithms. This observation suggests that for common problems in structural analysis, thermal analysis, and low Reynolds number flows, high-quality results can be obtained with unstructured tetrahedral meshes, which can be created robustly and automatically. We release the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure to enable statistically significant comparisons between different FE methods, which we hope will be helpful in the development of new meshing and FEA techniques. [ABSTRACT FROM AUTHOR]

We introduce the first exact root parity counter for continuous collision detection (CCD). That is, our algorithm computes the parity (even or odd) of the number of roots of the cubic polynomial arising from a CCD query. We note that the parity is unable to differentiate between zero (no collisions) and the rare case of two roots (collisions). Our method does not have numerical parameters to tune, has a performance comparable to efficient approximate algorithms, and is exact. We test our approach on a large collection of synthetic tests and real simulations, and we demonstrate that it can be easily integrated into existing simulators. [ABSTRACT FROM AUTHOR]

We describe an efficient algorithm to compute a discrete metric with prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the boundary of a mesh. The metric is (discretely) conformally equivalent to the input metric. Its construction is based on theory developed in [Gu et al. 2018b] and [Springborn 2020], relying on results on hyperbolic ideal Delaunay triangulations. Generality is achieved by considering the surface's intrinsic triangulation as a degree of freedom, and particular attention is paid to the proper treatment of surface boundaries. While via a double cover approach the case with boundary can be reduced to the case without boundary quite naturally, the implied symmetry of the setting causes additional challenges related to stable Delaunay-critical configurations that we address explicitly. We furthermore explore the numerical limits of the approach and derive continuous maps from the discrete metrics. [ABSTRACT FROM AUTHOR]

We introduce a large-scale benchmark for continuous collision detection (CCD) algorithms, composed of queries manually constructed to highlight challenging degenerate cases and automatically generated using existing simulators to cover common cases. We use the benchmark to evaluate the accuracy, correctness, and efficiency of state-of-the-art continuous collision detection algorithms, both with and without minimal separation. We discover that, despite the widespread use of CCD algorithms, existing algorithms are (1) correct but impractically slow; (2) efficient but incorrect, introducing false negatives that will lead to interpenetration; or (3) correct but over conservative, reporting a large number of false positives that might lead to inaccuracies when integrated in a simulator. By combining the seminal interval root finding algorithm introduced by Snyder in 1992 with modern predicate design techniques, we propose a simple and efficient CCD algorithm. This algorithm is competitive with state-of-the-art methods in terms of runtime while conservatively reporting the time of impact and allowing explicit tradeoff between runtime efficiency and number of false positives reported. [ABSTRACT FROM AUTHOR]

We introduce the first implicit time-stepping algorithm for rigid body dynamics, with contact and friction, that guarantees intersection-free configurations at every time step. Our algorithm explicitly models the curved trajectories traced by rigid bodies in both collision detection and response. For collision detection, we propose a conservative narrow phase collision detection algorithm for curved trajectories, which reduces the problem to a sequence of linear CCD queries with minimal separation. For time integration and contact response, we extend the recently proposed incremental potential contact framework to reduced coordinates and rigid body dynamics. We introduce a benchmark for rigid body simulation and show that our approach, while less efficient than alternatives, can robustly handle a wide array of complex scenes, which cannot be simulated with competing methods, without requiring per-scene parameter tuning. [ABSTRACT FROM AUTHOR]

Establishing a consistent normal orientation for point clouds is a notoriously difficult problem in geometry processing, requiring attention to both local and global shape characteristics. The normal direction of a point is a function of the local surface neighborhood; yet, point clouds do not disclose the full underlying surface structure. Even assuming known geodesic proximity, calculating a consistent normal orientation requires the global context. In this work, we introduce a novel approach for establishing a globally consistent normal orientation for point clouds. Our solution separates the local and global components into two different sub-problems. In the local phase, we train a neural network to learn a coherent normal direction per patch (i.e., consistently oriented normals within a single patch). In the global phase, we propagate the orientation across all coherent patches using a dipole propagation. Our dipole propagation decides to orient each patch using the electric field defined by all previously orientated patches. This gives rise to a global propagation that is stable, as well as being robust to nearby surfaces, holes, sharp features and noise. [ABSTRACT FROM AUTHOR]

We introduce a robust and automatic algorithm to convert linear triangle meshes with feature annotated into coarse tetrahedral meshes with curved elements. Our construction guarantees that the high-order meshes are free of element inversion or self-intersection. A user-specified maximal geometrical error from the input mesh controls the faithfulness of the curved approximation. The boundary of the output mesh is in bijective correspondence to the input, enabling attribute transfer between them, such as boundary conditions for simulations, making our curved mesh an ideal replacement or complement for the original input geometry. The availability of a bijective shell around the input surface is employed to ensure robust curving, prevent self-intersections, and compute a bijective map between the linear input and curved output surface. As necessary building blocks of our algorithm, we extend the bijective shell formulation to support features and propose a robust approach for boundary-preserving linear tetrahedral meshing. We demonstrate the robustness and effectiveness of our algorithm by generating high-order meshes for a large collection of complex 3D models. [ABSTRACT FROM AUTHOR]

We present an efficient, trivially parallelizable algorithm to compute offset surfaces of shapes discretized using a dexel data structure. Our algorithm is based on a two-stage sweeping procedure that is simple to implement and efficient, entirely avoiding volumetric distance field computations typical of existing methods. Our construction is based on properties of half-space power diagrams, where each seed is only visible by a half-space, which were never used before for the computation of surface offsets. The primary application of our method is interactive modeling for digital fabrication. Our technique enables a user to interactively process high-resolution models. It is also useful in a plethora of other geometry processing tasks requiring fast, approximate offsets, such as topology optimization, collision detection, and skeleton extraction. We present experimental timings, comparisons with previous approaches, and provide a reference implementation in the supplemental material, which can be found on the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TVCG.2019.2945961. [ABSTRACT FROM AUTHOR]

3-axis milling enables cheap and precise fabrication of target objects from precut slabs of materials such as wood or stone. However, the space of directly millable shapes is limited since a 3-axis mill can only carve a height-field (HF) surface during each milling and their size is bounded by the slab dimensions, one of which, the height, is typically significantly smaller than the other two for many typical materials. Extending 3-axis milling of precut slabs to general arbitrarily-sized shapes requires decomposing them into bounded-height 3-axis millable parts, or slices, which can be individually milled and then assembled to form the target object. We present DHFSlicer, a novel decomposition method that satisfies the above constraints and significantly reduces both milling time and material waste compared to alternative approaches. We satisfy the fabrication constraints by partitioning target objects into double height-field (DHF) slices, which can be fabricated using two milling passes: the HF surface accessible from one side is milled first, the slice is then flipped using appropriate fixtures, and then the second, remaining, HF surface is milled. DHFSlicer uses an efficient coarse-to-fine decomposition process: It first partitions the inputs into maximally coarse blocks that satisfy a local DHF criterion with respect to per-block milling axes, and then cuts each block into well-sized DHF slices. It minimizes milling time and material waste by keeping the slice count small, and maximizing slice height. We validate our method by embedding it within an end-to-end DHF milling pipeline and fabricating objects from slabs of foam, wood, and MDF; demonstrate that using the obtained slices reduces milling time and material waste by 42% on average compared to existing automatic alternatives; and highlight the benefits of DHFSlicer via extensive ablation studies. [ABSTRACT FROM AUTHOR]

We introduce an algorithm to convert a self-intersection free, orientable, and manifold triangle mesh T into a generalized prismatic shell equipped with a bijective projection operator to map T to a class of discrete surfaces contained within the shell whose normals satisfy a simple local condition. Properties can be robustly and efficiently transferred between these surfaces using the prismatic layer as a common parametrization domain. The combination of the prismatic shell construction and corresponding projection operator is a robust building block readily usable in many downstream applications, including the solution of PDEs, displacement maps synthesis, Boolean operations, tetrahedral meshing, geometric textures, and nested cages. [ABSTRACT FROM AUTHOR]

Sparse matrix computations are among the most important computational patterns, commonly used in geometry processing, physical simulation, graph algorithms, and other situations where sparse data arises. In many cases, the structure of a sparse matrix is known a priori, but the values may change or depend on inputs to the algorithm. We propose a new methodology for compile‐time specialization of algorithms relying on mixing sparse and dense linear algebra operations, using an extension to the widely‐used open source Eigen package. In contrast to library approaches optimizing individual building blocks of a computation (such as sparse matrix product), we generate reusable sparsity‐specific implementations for a given algorithm, utilizing vector intrinsics and reducing unnecessary scanning through matrix structures. We demonstrate the effectiveness of our technique on a benchmark of artificial expressions to quantitatively evaluate the benefit of our approach over the state‐of‐the‐art library Intel MKL. To further demonstrate the practical applicability of our technique we show that our technique can improve performance, with minimal code changes, for mesh smoothing, mesh parametrization, volumetric deformation, optical flow, and computation of the Laplace operator. [ABSTRACT FROM AUTHOR]

We propose an octree‐based algorithm to tessellate the interior of a closed surface with hexahedral cells. The generated hexahedral mesh (1) explicitly preserves sharp features of the original input, (2) has a maximal, user‐controlled distance deviation from the input surface, (3) is composed of elements with only positive scaled jacobians (measured by the eight corners of a hex [SEK*07]), and (4) does not have self‐intersections. We attempt to achieve these goals by proposing a novel pipeline to create an initial pure hexahedral mesh from an octree structure, taking advantage of recent developments in the generation of locally injective 3D parametrizations to warp the octree boundary to conform to the input surface. Sharp features in the input are bijectively mapped to poly‐lines in the output and preserved by the deformation, which takes advantage of a scaffold mesh to prevent local and global intersections. The robustness of our technique is experimentally validated by batch processing a large collection of organic and CAD models, without any manual cleanup or parameter tuning. All results including mesh data and statistics in the paper are provided in the additional material. The open‐source implementation will be made available online to foster further research in this direction. [ABSTRACT FROM AUTHOR]

Most additive manufacturing processes fabricate objects by stacking planar layers of solidified material. As a result, produced parts exhibit a so-called staircase effect, which results from sampling slanted surfaces with parallel planes. Using thinner slices reduces this effect, but it always remains visible where layers almost align with the input surfaces. In this research we exploit the ability of some additive manufacturing processes to deposit material slightly out of plane to dramatically reduce these artifacts. We focus in particular on the widespread Fused Filament Fabrication (FFF) technology, since most printers in this category can deposit along slightly curved paths, under deposition slope and thickness constraints. Our algorithm curves the layers, making them either follow the natural slope of the input surface or on the contrary, make them intersect the surfaces at a steeper angle thereby improving the sampling quality. Rather than directly computing curved layers, our algorithm optimizes for a deformation of the model which is then sliced with a standard planar approach. We demonstrate that this approach enables us to encode all fabrication constraints, including the guarantee of generating collision-free toolpaths, in a convex optimization that can be solved using a QP solver. We produce a variety of models and compare print quality between curved deposition and planar slicing. [ABSTRACT FROM AUTHOR]

We propose a robust 2D meshing algorithm, TriWild, to generate curved triangles reproducing smooth feature curves, leading to coarse meshes designed to match the simulation requirements necessary by applications and avoiding the geometrical errors introduced by linear meshes. The robustness and effectiveness of our technique are demonstrated by batch processing an SVG collection of 20k images, and by comparing our results against state of the art linear and curvilinear meshing algorithms. We demonstrate for our algorithm the practical utility of computing diffusion curves, fluid simulations, elastic deformations, and shape inflation on complex 2D geometries. [ABSTRACT FROM AUTHOR]

Tutte embedding is one of the most common building blocks in geometry processing algorithms due to its simplicity and provable guarantees. Although provably correct in infinite precision arithmetic, it fails in challenging cases when implemented using floating point arithmetic, largely due to the induced exponential area changes. We propose Progressive Embedding, with similar theoretical guarantees to Tutte embedding, but more resilient to the rounding error of floating point arithmetic. Inspired by progressive meshes, we collapse edges on an invalid embedding to a valid, simplified mesh, then insert points back while maintaining validity. We demonstrate the robustness of our method by computing embeddings for a large collection of disk topology meshes. By combining our robust embedding with a variant of the matchmaker algorithm, we propose a general algorithm for the problem of mapping multiply connected domains with arbitrary hard constraints to the plane, with applications in texture mapping and remeshing. [ABSTRACT FROM AUTHOR]

We propose a stretch-sensing soft glove to interactively capture hand poses with high accuracy and without requiring an external optical setup. We demonstrate how our device can be fabricated and calibrated at low cost, using simple tools available in most fabrication labs. To reconstruct the pose from the capacitive sensors embedded in the glove, we propose a deep network architecture that exploits the spatial layout of the sensor itself. The network is trained only once, using an inexpensive off-the-shelf hand pose reconstruction system to gather the training data. The per-user calibration is then performed on-the-fly using only the glove. The glove's capabilities are demonstrated in a series of ablative experiments, exploring different models and calibration methods. Comparing against commercial data gloves, we achieve a 35% improvement in reconstruction accuracy. [ABSTRACT FROM AUTHOR]

We introduce an integrated meshing and finite-element method pipeline enabling solution of partial differential equations in the volume enclosed by a boundary representation. We construct a hybrid hexahedral-dominant mesh, which contains a small number of star-shaped polyhedra, and build a set of high-order bases on its elements, combining triquadratic B-splines, triquadratic hexahedra, and harmonic elements. We demonstrate that our approach converges cubically under refinement, while requiring around 50% of the degrees of freedom than a similarly dense hexahedral mesh composed of triquadratic hexahedra. We validate our approach solving Poisson's equation on a large collection of models, which are automatically processed by our algorithm, only requiring the user to provide boundary conditions on their surface. [ABSTRACT FROM AUTHOR]