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1. NURBS-OT: An Advanced Model for Generative Curve Modeling

   https://doi.org/10.1115/1.4066549  

   Yang, Shaoliang; Wang, Jun; Wang, Kang (March 2025, Journal of Mechanical Design)

   Abstract This paper presents NURBS-OT (non-uniform rational B-splines—optimal transport), a new approach in the field of computer graphics and computer-aided design (CAD)/computer-aided manufacturing (CAM) for modeling complex free-form designs like aerodynamic and hydrodynamic structures, traditionally shaped by parametric curves such as Bézier, B-spline, and NURBS. Unlike prior models that used generative adversarial networks (GANs) involving large and complex parameter sets, our approach leverages a much lighter (0.37M versus 5.05M of BézierGAN), theoretically robust method by blending optimal transport with NURBS. This integration facilitates a more efficient generation of curvilinear designs. The efficacy of NURBS-OT has been validated through extensive testing on the University of Illinois Urbana-Champaign (UIUC) airfoil and superformula datasets, where it showed enhanced performance on various metrics. This demonstrates its ability to produce precise, realistic, and esthetically coherent designs, marking a significant advancement by merging classical geometrical techniques with modern deep learning. 

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2. Enhancing Isogeometric Analysis With NURBS-Based Synthesis

   https://doi.org/10.1115/DETC2024-142195  

   Yang, Shaoliang; Wang, Kang; Wang, Jun (August 2024, American Society of Mechanical Engineers)

   Abstract Isogeometric analysis (IGA) is a computational technique that integrates computer-aided design (CAD) with finite element analysis (FEA) by employing the same basis functions for both geometry representation and solution approximation. While IGA offers numerous advantages, such as improved accuracy and efficiency, it also presents several challenges related to geometric modeling. Some of these challenges include accurately representing complex geometries with NURBS (Non-Uniform Rational B-Splines) or other basis functions used in IGA and generating high-quality meshes that conform to the complex geometry represented by NURBS curves/surfaces. This paper introduces an analytical framework to provide a more efficient and theoretically grounded method for generating curvilinear configurations and its analytical solution in IGA, bridging the gap between generated data and its physical representations. This innovative approach is distinguished by integrating the NURBS parameterization in curve generation and providing a corresponding framework to achieve a broader and more accurate explanation of meshes and properties, especially constructing new coordinates and calculating the physical displacements under these conditions. Our model enables the analytical understanding of complex curves from the UIUC airfoil and superformula datasets, demonstrating a deeper dive into simulations. This study signifies a pivotal juncture wherein machine-learning-based complex geometrical formulations are synergistically combined with actual isogeometric analysis. 

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3. ParaValve: An open source framework for parametric design and fluid–structure interaction simulation of bioprosthetic heart valves in patient-specific aortic geometries

   https://doi.org/10.1016/j.cagd.2025.102455  

   Saraeian, Mehdi; Corpuz, Ashton M; Hsu, Ming-Chen; Krishnamurthy, Adarsh (July 2025, Computer Aided Geometric Design)

   Heart valve disease (HVD), a significant cardiovascular complication, is one of the leading global causes of morbidity and mortality. Treatment for HVD often involves medical devices such as bioprosthetic valves. However, the design and optimization of these devices require a thorough understanding of their biomechanical and hemodynamic interactions with patient-specific anatomical structures. Parametric procedural geometry has become a powerful tool in enhancing the efficiency and accuracy of design optimization for such devices, allowing researchers to systematically explore a wide range of possible configurations. In this work, we present a robust framework for parametric and procedural modeling of stented bioprosthetic heart valves and patient-specific aortic geometries. The framework employs non-uniform rational B-splines (NURBS)-based geometric parameterization, enabling precise control over key anatomical and design variables. By enabling a modular and expandable workflow, the framework supports iterative optimization of valve designs to achieve improved hemodynamic performance and durability. We demonstrate its applicability through simulations on bioprosthetic aortic valves, highlighting the impact of geometric parameters on valve function and their potential for personalized device design. By coupling parametric geometry with computational tools, this framework offers researchers and engineers a streamlined pathway toward innovative and patient-specific cardiovascular solutions. 

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4. Enhanced MPM framework with multipatch isogeometric analysis for geotechnical applications

   https://doi.org/10.1016/j.compgeo.2025.107743  

   Alsardi, Abdelrahman; Yerro, Alba; Long, Christopher Curtis (February 2026, Computers and Geotechnics)

   Achieving stable stress solutions at large strains using the Material Point Method (MPM) is challenging due to the accumulation of errors associated with geometry discretization, cell-crossing noise, and volumetric locking. Several simplified attempts exist in the literature to mitigate these errors, including higher-order frameworks. However, the stability of the MPM solution in such frameworks has been limited to simple geometries and the single-phase formulation (i.e., neglecting pore fluid). Although never explored, multipatch isogeometric analysis offers desirable qualities to simulate complex geometries while mitigating errors in the MPM. The degree of required high-order spatial integration has also never been investigated to infer a minimum limit for the stability of the stress solution in MPM. This paper presents a general-purpose numerical framework for simulating stable stresses in porous media, capturing both near incompressibility and multiphase interactions. First, the numerical framework is presented considering Non-Uniform Rational B-splines (NURBS) to perform isogeometric analysis (IGA) in MPM. Additionally, a volumetric strain smoothing algorithm is used to alleviate errors associated with volumetric locking. Second, the manifestation of cell-crossing errors is assessed via a series of problems with orders ranging from linear to cubic interpolation functions. Third, the use of NURBS is investigated and verified for problems with circular geometries. Finally, multipatch analysis is deployed to simulate plane strain and 3D penetration in soils, considering nearly incompressible elastoplastic (total stress) analysis and fully-coupled hydro-mechanical (effective stress) analysis. The stability of the solution is also analyzed for different constitutive models. From the results, it can be concluded that the framework using cubic interpolation functions with strain smoothing is the most convenient, presenting stable stress solutions for a broad range of multiphase geotechnical applications. 

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5. Gradient Dynamics of Shallow Univariate ReLU networks

   Williams, Francis; Trager, Matthew; Silva, Claudio; Zorin, Denis; Panozzo, Daniele; Bruna, Joan (January 2019, Advances in neural information processing systems)

   We present a theoretical and empirical study of the gradient dynamics of overparam- eterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network function. We examine the principal qualitative features of this gradient flow. In particular, we determine conditions for two learning regimes: kernel and adaptive, which depend both on the relative magnitude of initialization of weights in different layers and the asymptotic behavior of initialization coefficients in the limit of large network widths. We show that learning in the kernel regime yields smooth interpolants, minimizing curvature, and reduces to cubic splines for uniform initializations. Learning in the adaptive regime favors instead linear splines, where knots cluster adaptively at the sample points. 

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