spatial field mapping

In PDE simulations, mesh transformation operations—such as refinement, coarsening, extrusion, element-type conversion, and filtering—are typically implemented via scattered, error-prone, and hard-to-maintain ad hoc code. To address this, we propose a lightweight, table-driven, general-purpose mesh transformation paradigm. Our framework unifies topological and geometric transformations using index tables and sparse mappings as core abstractions, supports MPI parallelism and memory locality optimization, and is implemented atop PETSc. Unlike legacy case-specific hardcoded approaches, our method significantly improves maintainability, extensibility, and cross-developer comprehensibility. Experimental evaluation on realistic simulation workloads demonstrates 3–8× throughput improvement and >90% reduction in transformation-related errors. The implementation has been integrated into the PETSc mainline and released as open-source software to the broader scientific computing community.

This work addresses the challenge of preserving mapping continuity and bijectivity during complex remeshing processes, where conventional data transfer methods often induce geometric or attribute distortions. The authors propose a composite mapping framework based on local bijective atlases, enhanced by a Shared Scaffold structure that guarantees global bijectivity. The approach is generalized to support a variety of remeshing operations and, for the first time, enables the construction of bijective mappings on 3D tetrahedral remeshings by innovatively integrating Steinitz’s theorem with Maxwell–Cremona lifting theory. This framework facilitates precise tracking of geometric entities—including points, curves, and surfaces—across remeshing sequences, significantly improving fidelity in high-precision applications such as texture transfer and volumetric simulation.

Existing quad-mesh extraction methods rely on idealized mesh-preserving parameterizations, yet real-world inputs often exhibit unmodeled geometric and topological deviations, severely compromising robustness. This work systematically characterizes typical deviation patterns from mesh preservation in non-ideal parameterizations for the first time. We propose a discrete topological operation sequence modeling framework that explicitly translates continuous mapping distortions into computable, verifiable discrete operations—including vertex splitting, edge flipping, and face re-partitioning. Our framework unifies the description of deviation causes and corresponding correction pathways, providing a principled theoretical foundation for algorithm design. Experiments demonstrate that the proposed method significantly improves fault tolerance against noise, geometric distortion, and topological inconsistencies. It achieves more stable and higher-quality quad-mesh extraction across multiple benchmark datasets.

This paper addresses spline interpolation on compact Riemannian manifolds—such as spheres and cylinders—i.e., non-Euclidean geometric domains. We propose the first general, smooth spline modeling framework applicable to arbitrary compact Riemannian manifolds. Our method formulates interpolation within a Gaussian Markov random field (GMRF) kriging framework, with numerical solution achieved via discretization of the Laplace–Beltrami operator and finite-element approximation. Key contributions include: (1) the first extension of spline interpolation systems to general compact Riemannian manifolds; (2) support for locally anisotropic covariance modeling; and (3) incorporation of domain deformation to enhance geometric adaptability. Experiments on spherical and cylindrical domains demonstrate substantial improvements over classical spherical harmonic methods, with superior robustness and broad applicability across diverse manifold geometries.

Three-dimensional (3D) mappings are fundamental in computational mechanics (CAE), computer graphics, and medical imaging; however, conventional vertex-coordinate-based representations struggle to simultaneously ensure geometric fidelity and intuitive, controllable editing. To address this, we propose the first theoretically rigorous and computationally tractable 3D quasiconformal representation—extending the Beltrami coefficient to three dimensions—to characterize local scaling distortion in a mathematically sound manner. We further design an invertible reconstruction algorithm that stably and accurately recovers the original mapping from its distortion representation. Our approach integrates 3D quasiconformal theory, partial differential equation (PDE)-based modeling, and numerical optimization. Experiments demonstrate that our method significantly outperforms state-of-the-art alternatives in 3D mapping reconstruction, keyframe interpolation, and compression—achieving superior accuracy, robustness, and editability while preserving theoretical guarantees.

This work addresses the problem of automatically generating low-distortion, orthogonal quadrilateral surface meshes that satisfy user-specified feature alignment and sizing constraints. The authors propose a novel approach based on integrable orthogonal frame fields, where the symmetry of the frames is implicitly modeled using three-dimensional orthogonally decomposable (odeco) tensors. Within a finite element framework, the method jointly optimizes area and stretch distortion while enforcing shear-free orthogonality constraints. A key contribution is the extension of two-dimensional odeco integrability to three dimensions, coupled with an automatic singularity placement strategy that ensures global integrability without manual intervention or greedy heuristics. Experimental results demonstrate that the method consistently outperforms existing techniques on both smooth surfaces and complex CAD models, achieving significantly reduced mesh distortion under strict sizing control.

This study addresses the challenge of achieving both accuracy and conservation in field variable transfer within black-box multiphysics coupling when source mesh information is unavailable. The authors propose a novel field transfer method based on stochastic approximation Galerkin projection, which, for the first time, integrates stochastic approximation into a black-box coupling framework. This approach enables asymptotic conservation and high accuracy without requiring access to the source mesh. By overcoming the limitations of conventional radial basis function and mesh intersection methods, and leveraging GPU parallel acceleration (NVIDIA A100), the proposed technique demonstrates superior performance on both standard domains and the LTX fusion reactor mesh—exhibiting lower conservation error, higher accuracy, and computational cost comparable to that of mesh intersection methods.

This work addresses the high computational cost of geometric mapping under spatially varying fields at high resolutions by proposing a resolution-agnostic neural surrogate model. The method operates without reliance on fixed grids or ground-truth solution labels, leveraging coordinate-augmented multi-resolution field encoding to predict mapping positions at arbitrary point sets. A geometry-aware unsupervised loss is formulated by integrating variational energy, diffusion equilibration, and quasiconformal theory. Experimental results demonstrate that the approach achieves efficient, accurate, and resolution-flexible geometric parameterization in both quasiconformal mapping and density equilibration tasks, significantly enhancing computational efficiency and generalization capability.

This work proposes a novel surface representation framework that addresses the lack of effective smooth interpolation methods for non-quadrilateral faces in arbitrarily topologized closed meshes. By integrating local polygonal quadratic interpolation with rational curve parameterization, the method constructs smoothly connected quadrilateral patches and introduces a specialized rational-curve-based parameterization strategy for triangular and general polygonal faces. Through sub-patch blending and surface stitching techniques, the approach achieves globally C¹-continuous, high-quality surface reconstruction. Notably, this is the first unified framework capable of effectively handling smooth interpolation across faces with arbitrary numbers of edges, significantly enhancing both the quality and flexibility of surface generation for complex-topology meshes.