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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.

 

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.

 

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. 

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. 

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. 

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. 

Modern fabrication methods have greatly simplified manufacturing of complex free-form shapes at an affordable cost, and opened up new possibilities for improving functionality and customization through automatic optimization, shape optimization in particular. However, most existing shape optimization methods focus on single parts. In this work, we focus on supporting shape optimization for assemblies, more specifically, assemblies that are held together by contact and friction. Examples of which include furniture joints, construction set assemblies, certain types of prosthetic devices and many other. To enable this optimization, we present a framework supporting robust and accurate optimization of a number of important functionals, while enforcing constraints essential for assembly functionality: weight, stress, difficulty of putting the assembly together, and how reliably it stays together. Our framework is based on smoothed formulation of elasticity equations with contact, analytically derived shape derivatives, and robust remeshing to enable large changes of shape, and at the same time, maintain accuracy. We demonstrate the improvements it can achieve for a number of computational and experimental examples. 

Interval computation is widely used in Computer Aided Design to certify computations that use floating point operations to avoid pitfalls related to rounding error introduced by inaccurate operations. Despite its popularity and practical benefits, support for interval arithmetic is not standardized nor available in mainstream programming languages. We propose the first benchmark for interval computations, coupled with reference solutions computed with exact arithmetic, and compare popular C and C++ libraries over different architectures, operating systems, and compilers. The benchmark allows identifying limitations in existing implementations, and provides a reliable guide on which library to use on each system for different CAD applications. We believe that our benchmark will be useful for developers of future interval libraries, as a way to test the correctness and performance of their algorithms.