Elliptic partial differential equations are ubiquitous in graphics and engineering, but remain challenging to solve on complex or evolving geometries. Traditional discretization schemes (e.g., FEM/FDM) provide stable, globally coupled solutions but require heavy meshing or extreme refinement to accurately resolve geometric detail. In contrast, grid-free Monte Carlo methods (e.g., Walk on Spheres/Stars) adapt naturally to arbitrary geometry and offer massive parallelism, but rely on long random walks whose variance grows rapidly, particularly in the presence of Neumann boundaries, leading to slow convergence. We introduce a hybrid approach that combines the geometric flexibility of Monte Carlo estimation with deterministic global solves that do not introduce additional stochastic error. Our method decomposes the domain into simple, regular subdomains and uses Monte Carlo to estimate local first-passage solution operators (Poisson kernels), where walk lengths and variance are inherently controlled by the reduced spatial scale. These local operators are assembled into a sparse global system whose solution is obtained via a deterministic linear solve that exactly replaces simulating discrete random walks throughout the domain. This global solve trades stochastic variance for a fixed, resolution-dependent discretization bias, yielding stable and reusable solution operators. As a result, our method attains accurate, geometry-aware solutions even on coarse discretizations, and enables efficient solves and re-solves by computing and updating only the local operators affected by the geometry and its changes. We evaluate the approach on complex two-dimensional domains, benchmarking accuracy and convergence against standard grid-free and grid-based baselines, and demonstrate applications to microstructure simulation and flow-based path planning and streamline visualization.

The miniaturization of shell structures presents a versatile and complex challenge, bridging geometry with diverse practical applications. In this paper, we introduce a novel approach for computing origami crease patterns to compress arbitrary 3D shell objects. First, we employ the adapted Material Point Method (MPM) to simulate the compression of a target surface and obtain an initial folded configuration. Since MPM produces overly smooth curved surfaces, their crease patterns are unsuitable for practical origami fabrication. We then propose a novel Folding Line Extraction (FLE) method that optimizes these smoothed surfaces to extract folding lines that achieve the target compression with minimal deformation and stretching outside the crease lines. This method produces smooth curved folding lines. Fabrication and experimental validation of the extracted patterns demonstrate their effectiveness and applicability in real-world scenarios.

Processing visual data often involves small adjustments or sequences of changes, e.g., image filtering, surface smoothing, and animation. While established graphics techniques like normal mapping and video compression exploit redundancy to encode such small changes efficiently, the problem of encoding small changes to neural fields—neural network parameterizations of visual or physical functions—has received less attention. We propose a parameter-efficient strategy for updating neural fields using low-rank adaptations (LoRA). LoRA, a method from the parameter-efficient fine-tuning LLM community, encodes small updates to pre-trained models with minimal computational overhead. We adapt LoRA for instance-specific neural fields, avoiding the need for large pre-trained models and yielding lightweight updates. We validate our approach with experiments in image filtering, geometry editing, video compression, and energy-based editing, demonstrating its effectiveness and versatility for representing neural field updates.

Generative models have recently demonstrated impressive capabilities in producing high-quality 3D shapes from a variety of user inputs (e.g., text or images). However, generated objects often lack physical integrity. We introduce PhysiOpt, a differentiable physics optimizer designed to improve the physical behavior of 3D generative outputs, enabling them to transition from virtual designs to physically plausible, real-world objects. While most generative models represent geometry as continuous implicit fields, physics-based approaches often rely on the finite element method (FEM), requiring ad hoc mesh extraction to perform shape optimization. In addition, these methods are typically slow, limiting their integration in fast, iterative generative design workflows. Instead, we bridge the representation gap and propose a fast and effective differentiable simulation pipeline that optimizes shapes directly in the latent space of generative models using an intuitive and easy-to-implement differentiable mapping. This approach enables fast optimization while preserving semantic structure, unlike traditional methods relying on local mesh-based adjustments. We demonstrate the versatility of our optimizer across a range of shape priors, from global and part-based latent models to a state-of-the-art large-scale 3D generator, and compare it to a traditional mesh-based shape optimizer. Our method preserves the native representation and capabilities of the underlying generative model while supporting user-specified materials, loads, and boundary conditions. The resulting designs exhibit improved physical behavior, remain faithful to the learned priors, and are suitable for fabrication. We demonstrate the effectiveness of our approach on both virtual and fabricated objects.

We present a computational approach for designing freeform structures that can be rapidly assembled from initially flat configurations by a single string pull. The target structures are decomposed into rigid spatially varied quad tiles that are optimized to approximate the user-provided surface, forming a flat mechanical linkage. Our algorithm then uses a two-step method to find a physically realizable string path that controls only a subset of tiles to smoothly actuate the structure from flat to assembled configuration. We initially compute the minimal subset of tiles that are required to be controlled with the string considering the geometry of the structure and interaction among the tiles. We then find a valid string path through these tiles that minimizes friction, which will assemble the flat linkage into the target 3D structure upon tightening a single string. The resulting designs can be easily manufactured with computational fabrication techniques such as 3D printing, CNC milling, molding, etc. in flat configuration that, in addition to manufacturing, facilitates storage and transportation. We validate our approach by developing a series of physical prototypes and showcasing various application case studies, ranging from medical devices, space shelters, to architectural designs.

Bayesian optimization has been successfully applied to optimize black-box functions where the number of evaluations is severely limited. However, in many real-world applications, it is hard or impossible to know in advance which designs are feasible due to some physical or system limitations. These issues lead to an even more challenging problem of optimizing an unknown function with unknown constraints. In this paper, we observe that in such scenarios optimal solution typically lies on the boundary between feasible and infeasible regions of the design space, making it considerably more difficult than that with interior optima. Inspired by this observation, we propose BE-CBO, a new Bayesian optimization method that efficiently explores the boundary between feasible and infeasible designs. To identify the boundary, we learn the constraints with an ensemble of neural networks that outperform the standard Gaussian Processes for capturing complex boundaries. Our method demonstrates superior performance against state-of-the-art methods through comprehensive experiments on synthetic and real-world benchmarks. Code available at: https://github.com/yunshengtian/BE-CBO

Many science, engineering, and design optimization problems require balancing the trade-offs between several conflicting objectives. The objectives are often black-box functions whose evaluations are time-consuming and costly. Multi-objective Bayesian optimization can be used to automate the process of discovering the set of optimal solutions, called Pareto-optimal, while minimizing the number of performed evaluations. To further reduce the evaluation time in the optimization process, testing of several samples in parallel can be deployed. We propose a novel multi-objective Bayesian optimization algorithm that iteratively selects the best batch of samples to be evaluated in parallel. Our algorithm approximates and analyzes a piecewise-continuous Pareto set representation. This representation allows us to introduce a batch selection strategy that optimizes for both hypervolume improvement and diversity of selected samples in order to efficiently advance promising regions of the Pareto front. Experiments on both synthetic test functions and real-world benchmark problems show that our algorithm predominantly outperforms relevant state-of-the-art methods. Code is available at https://github.com/yunshengtian/DGEMO.

We present RoboGrammar , a fully automated approach for generating optimized robot structures to traverse given terrains. In this framework, we represent each robot design as a graph, and use a graph grammar to express possible arrangements of physical robot assemblies. Each robot design can then be expressed as a sequence of grammar rules. Using only a small set of rules our grammar can describe hundreds of thousands of possible robot designs. The construction of the grammar limits the design space to designs that can be fabricated. For a given input terrain, the design space is searched to find the top performing robots and their corresponding controllers. We introduce Graph Heuristic Search - a novel method for efficient search of combinatorial design spaces. In Graph Heuristic Search, we explore the design space while simultaneously learning a function that maps incomplete designs (e.g., nodes in the combinatorial search tree) to the best performance values that can be achieved by expanding these incomplete designs. Graph Heuristic Search prioritizes exploration of the most promising branches of the design space. To test our method we optimize robots for a number of challenging and varied terrains. We demonstrate that RoboGrammar can successfully generate nontrivial robots that are optimized for a single terrain or a combination of terrains.

We present X-shells , a new class of deployable structures formed by an ensemble of elastically deforming beams coupled through rotational joints. An X-shell can be assembled conveniently in a flat configuration from standard elastic beam elements and then deployed through force actuation into the desired 3D target state. During deployment, the coupling imposed by the joints will force the beams to twist and buckle out of plane to maintain a state of static equilibrium. This complex interaction of discrete joints and continuously deforming beams allows interesting 3D forms to emerge. Simulating X-shells is challenging, however, due to unstable equilibria at the onset of beam buckling. We propose an optimization-based simulation framework building on a discrete rod model that robustly handles such difficult scenarios by analyzing and appropriately modifying the elastic energy Hessian. This real-time simulation method forms the basis of a computational design tool for X-shells that enables interactive design space exploration by varying and optimizing design parameters to achieve a specific design intent. We jointly optimize the assembly state and the deployed configuration to ensure the geometric and structural integrity of the deployable X-shell. Once a design is finalized, we also optimize for a sparse distribution of actuation forces to efficiently deploy it from its flat assembly state to its 3D target state. We demonstrate the effectiveness of our design approach with a number of design studies that highlight the richness of the X-shell design space, enabling new forms not possible with existing approaches. We validate our computational model with several physical prototypes that show excellent agreement with the optimized digital models.

Deployable structures are physical mechanisms that can easily transition between two or more geometric configurations; such structures enable industrial, scientific, and consumer applications at a wide variety of scales. This paper develops novel deployable structures that can approximate a large class of doubly-curved surfaces and are easily actuated from a flat initial state via inflation or gravitational loading. The structures are based on two-dimensional rigid mechanical linkages that implicitly encode the curvature of the target shape via a user-programmable pattern that permits locally isotropic scaling under load. We explicitly characterize the shapes that can be realized by such structures---in particular, we show that they can approximate target surfaces of positive mean curvature and bounded scale distortion relative to a given reference domain. Based on this observation, we develop efficient computational design algorithms for approximating a given input geometry. The resulting designs can be rapidly manufactured via digital fabrication technologies such as laser cutting, CNC milling, or 3D printing. We validate our approach through a series of physical prototypes and present several application case studies, ranging from surgical implants to large-scale deployable architecture.

We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications.