1、 有理曲面的隐式表示。即给定有理曲线、曲面的参数方程，求相应曲线、曲面的隐式方程。传统的方法有结式方法、Groebner基方法、吴消元法等。1995年作者同Sederberg教授提出了一种新的方法--动曲面方法。该方法能将曲面隐式方程写成一个低阶行列式，并且对有基点的曲面仍然有效。但本方法还没有自动化。给出全自动化的隐式化方法是下一步工作的方向。
2、有理曲线、曲面的μ基的计算、构造与应用。1998年作者同David Cox, Tom Sederberg提出了有理曲线μ基的概念，建立了曲线的参数表示与隐式表示之间的一座桥梁，也就是说，拥有了μ基就同时拥有了参数与隐式两种表示。随后，我们研究了μ基的快速算法与性质，并将结果推广到直纹面情形。下一步的工作是有理曲面的构造及其应用。
3、分片代数曲面造型。本课题的主要工作是利用所谓的分片代数曲面（即是由多片低次代数曲面光滑拼接二成的组合曲面）进行曲面造型。我们尝试将分片代数曲面应用于过渡曲面构造与补洞。但如何控制分片代数曲面仍然是一个急待解决的问题。
4、区间分析在几何造型中的应用。区间运算可以保证几何操作的可靠性。我们研究了区间曲线、曲面的降阶问题。我们希望进一步寻求区间运算在几何造型中的其它应用。
5、T网格上的样条曲面及其应用。T网格具有很好的自适应性，基于T网格上的样条曲面可以较好地处理自适应曲面造型，并且在科学工程计算中应该有一定的应用前景。
6、稀疏优化在几何处理中应用。稀疏优化技术可以较好地处理几何问题中的稀疏信息，因此是一个较好的该类问题的处理工具。
[1] Jing Li, Yihang Fu, Falai Chen*. BrepForge: Factorized B-rep Synthesis via Wireframe Composition and Boundary-Conditioned Surface Instantiation[C]//Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers. 2026: 1-11.
[2] Bingru Huang, Falai Chen*. Dimension of bi-degree (d, d) spline spaces with the highest order of smoothness over hierarchical T-meshes[J]. Science China Mathematics, 2026: 1-24.
[3] Yanyan Yang, Lili Wang, Xiaoya Zhai*, Kai Chen, Wenming Wu, Yunkai Zhao, Falai Chen, Ligang Liu, Xiao-Ming Fu*. Guided diffusion for fast inverse design of voxel-based mechanical metamaterials[J]. Smart Materials in Manufacturing, 2026, 4: 100129.
[4] Wenzheng Wu, Kang Wu*, Siyuan Xing, Yifei Li, Xiao-Ming Fu, Falai Chen, Ligang Liu. AI-driven generation of 3D CAD models: A survey[J]. Computational Visual Media, 2026, 12(3): 575-614.
[5] Shicong Zhong, Bingru Huang, Falai Chen. Modified HB-splines[J]. Journal of Computational and Applied Mathematics, 2026: 117467.
[6] Ying Cai, Falai Chen*, Hongmei Kang, Zhimin Zhang. Multilevel methods for PHT-splines[J]. Computer Methods in Applied Mechanics and Engineering, 2025, 446: 118251.
[7] Ruijie Zou, Maodong Pan*, Ye Zheng, Falai Chen, Weihua Tong. MAT-parameterization: Volumetric multi-patch parameterizations of complex domains for isogeometric analysis using MAT-based decomposition[J]. Computer Methods in Applied Mechanics and Engineering, 2025, 445: 118187.
[8] Nan Zheng, Xiaoya Zhai*, Falai Chen. Efficient worst-case topology optimization of self-supporting structures for additive manufacturing[J]. Computer Aided Geometric Design, 2025, 119: 102441.
[9] Zheng Zhan, Wenping Wang, Falai Chen*. Integral parameterization of volumetric domains via deep neural networks[J]. Computer Methods in Applied Mechanics and Engineering, 2025, 441: 117988.
[10] Kai Li, Xiaohong Jia*, Falai Chen. Fast determination and computation of self-intersections for nurbs surfaces[J]. ACM Transactions on Graphics, 2025, 44(2): 1-16.
[11] Hailun Xu, Hongmei Kang*, Falai Chen, Zhimin Zhang. Superconvergent isogeometric collocation with box splines[J]. Computer Methods in Applied Mechanics and Engineering, 2025, 437: 117763.
[12] Yihui Ye, Nan Zheng, Xiaoya Zhai*, Hongmei Kang, Falai Chen. Shape Optimization of Tunable Poisson's Ratio Metamaterials With Disk B‐Splines[J]. International Journal for Numerical Methods in Engineering, 2025, 126(1): e7620.
[13] 陈播, 胡银雷*, 陈发来. 基于 “由粗到细” 思想的钣金废料嵌套问题快速求解算法[J]. 计算机辅助设计与图形学学报, 2025, 37(1): 28-39.
[14] Jing Li, Yihang Fu, Falai Chen*. Dtgbrepgen: A novel b-rep generative model through decoupling topology and geometry[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2025: 21438-21447.
[15] Lingfeng Li, Qiong Pan, Xiaoya Zhai*, Falai Chen. Tailored Functionally Graded Materials design and concurrent topology optimization with implicit fields[J]. Computer Methods in Applied Mechanics and Engineering, 2024, 432: 117371.
[16] Zepeng Wen, Qiong Pan, Xiaoya Zhai*, Hongmei Kang*, Falai Chen. Adaptive isogeometric topology optimization of shell structures based on PHT-splines[J]. Computers & Structures, 2024, 305: 107565.
[17] Yinlei Hu#, Siyuan Wan#, Yuanhanyu Luo#, Yuanzhe Li, Tong Wu, Wentao Deng, Chen Jiang, Shan Jiang, Yueping Zhang, Nianping Liu, Zongcheng Yang, Falai Chen*, Bin Li*, Kun Qu*. Benchmarking algorithms for single-cell multi-omics prediction and integration[J]. Nature Methods, 2024, 21(11): 2182-2194.
[18] Jie Gao, Zepeng Wen, Xiaoya Zhai*, Falai Chen, Hongmei Kang. Concurrent topology optimization of multiscale composites with differentiable microstructures[J]. Computer Methods in Applied Mechanics and Engineering, 2024, 431: 117271.
[19] Qiong Pan, Xiaoya Zhai*, Falai Chen. Density-based isogeometric topology optimization of shell structures[J]. Computer-Aided Design, 2024, 176: 103773.
[20] Jing Li, Juyong Zhang, Falai Chen*. iShapEditing: Intelligent Shape Editing with Diffusion Models[C]//Computer Graphics Forum. 2024, 43(7): e15253.
[21] Wenbing Shao, Falai Chen*. Topologically Correct Intersection Curves of Two Trimmed Quadrics with Tolerance Control[J]. Journal of Systems Science and Complexity, 2024, 37(5): 2207-2239.
[22] Nan Zheng, Xiaoya Zhai*, Jingchao Jiang, Falai Chen. Topology optimization of self-supporting structures for additive manufacturing via implicit b-spline representations[J]. Computer-Aided Design, 2024, 175: 103745.
[23] Qiong Pan, Xiaoya Zhai*, Hongmei Kang, Xiaoxiao Du, Falai Chen. Isogeometric topology optimization of multi-patch shell structures[J]. Computer-Aided Design, 2024, 174: 103733.
[24] Bingru Huang, Falai Chen*. An Algorithm to Compute the μ-Bases of Rational Parametric Surfaces with Respect to One Variable[J]. Communications in Mathematics and Statistics, 2024, 12(3): 523-541.
[25] Zheng Zhan, Wenping Wang, Falai Chen*. Fast parameterization of planar domains for isogeometric analysis via generalization of deep neural network[J]. Computer Aided Geometric Design, 2024, 111: 102313.
[26] Bingru Huang, Falai Chen*. On the stability of the dimensions of spline spaces with highest order of smoothness over T-meshes[J]. Journal of Computational and Applied Mathematics, 2024, 441: 115681.
[27] Zhiguo Yong, Hongmei Kang, Falai Chen. The Weighted Basis for PHT-Splines[J]. Computer Modeling in Engineering & Sciences (CMES), 2024, 138(1).
[28] Zheng Zhan, Wenping Wang, Falai Chen*. Simultaneous boundary and interior parameterization of planar domains via deep learning[J]. Computer-Aided Design, 2024, 166: 103621.
[29] 张珊珊, 潘茂东, 陈发来*. 基于拟共形映射的三角 Bézier 参数化[J]. 计算机辅助设计与图形学学报, 2024, 36(12): 1863-1869.
[30] Xiaoya Zhai, Weiming Wang, Falai Chen, Jun Wu*. Topology optimization of differentiable microstructures[J]. Computer Methods in Applied Mechanics and Engineering, 2024, 418: 116530.
[31] Kai Li, Xiaohong Jia*, Falai Chen. Efficient computation of moving planes for rational parametric surfaces with base points using Dixon resultants[J]. Computer Aided Geometric Design, 2023, 107: 102253.
[32] Wenbing Shao, Falai Chen*. Topological classification of the intersection curves of two quadrics using a set of discriminants[J]. Computer Aided Geometric Design, 2023, 107: 102244.
[33] Maodong Pan, Ruijie Zou, Weihua Tong, Yujie Guo, Falai Chen*. G1-smooth planar parameterization of complex domains for isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2023, 417: 116330.
[34] Xiaoya Zhai, Falai Chen*. A review of process optimization for additive manufacturing based on machine learning[J]. Annals of Applied Mathematics, 2023, 39(4): 493-543.
[35] Xiaoya Zhai, Yundong Gai, Liuchao Jin, Wei-Hsin Liao, Falai Chen, Ping Hu. Isogeometric topology optimization of auxetic materials based on moving morphable components method[J]. Advanced Topics in Mechanics of Materials, Structures and Construction: AToMech1-2023, 2023, 31: 172.
[36] Xiaohan Bao, Weihua Tong, Falai Chen. A spectral segmentation method for large meshes[J]. Communications in Mathematics and Statistics, 2023, 11(3): 583-607.
[37] Nan Zheng, Xiaoya Zhai*, Falai Chen. Topology optimization of self-supporting porous structures based on triply periodic minimal surfaces[J]. Computer-Aided Design, 2023, 161: 103542.
[38] Zheng Zhan, Ye Zheng, Wenping Wang, Falai Chen*. Boundary correspondence for iso-geometric analysis based on deep learning[J]. Communications in Mathematics and Statistics, 2023, 11(1): 131-150.
[39] Ye Zheng, Falai Chen*. Volumetric parameterization with truncated hierarchical B-splines for isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 401: 115662.
[40] Xiaohong Jia, Falai Chen, Shanshan Yao. Singularity computation for rational parametric surfaces using moving planes[J]. ACM Transactions on Graphics (TOG), 2022, 42(1): 1-14.
[41] Maodong Pan, Falai Chen*. Constructing planar domain parameterization with HB-splines via quasi-conformal mapping[J]. Computer Aided Geometric Design, 2022, 97: 102133.
[42] Bin Li#, Wen Zhang#, Chuang Guo#, Hao Xu, Longfei Li, Minghao Fang, Yinlei Hu, Xinye Zhang, Xinfeng Yao, Meifang Tang, Ke Liu, Xuetong Zhao, Jun Lin, Linzhao Cheng, Falai Chen, Tian Xue, Kun Qu*. Benchmarking spatial and single-cell transcriptomics integration methods for transcript distribution prediction and cell type deconvolution[J]. Nature methods, 2022, 19(6): 662-670.
[43] Wenbing Shao, Falai Chen*, Xuefeng Liu*. Robust algebraic curve intersections with tolerance control[J]. Computer-Aided Design, 2022, 147: 103236.
[44] Yisheng Lai, Falai Chen*, Xiaoran Shi, Yu Gao. A constructive approach to implicitizing rational surfaces with LCI base points by moving planes and moving quadrics[J]. Computer Aided Geometric Design, 2022, 92: 102051.
[45] Ye Zheng, Falai Chen*. Volumetric boundary correspondence for isogeometric analysis based on unbalanced optimal transport[J]. Computer-Aided Design, 2021, 140: 103078.
[46] Xiaoya Zhai, Falai Chen*, Jun Wu*. Alternating optimization of design and stress for stress-constrained topology optimization[J]. Structural and Multidisciplinary Optimization, 2021, 64(4): 2323-2342.
[47] Yinlei Hu#, Bin Li#, Wen Zhang, Nianping Liu, Pengfei Cai, Falai Chen*, Kun Qu*. WEDGE: imputation of gene expression values from single-cell RNA-seq datasets using biased matrix decomposition[J]. Briefings in Bioinformatics, 2021, 22(5): bbab085.
[48] Laurent Busé, Falai Chen. Determinantal tensor product surfaces and the method of moving quadrics[J]. Transactions of the American Mathematical Society, 2021, 374(7): 4931-4952.
[49] Yinlei Hu#, Bin Li#, Falai Chen*, Kun Qu*. Single-cell data clustering based on sparse optimization and low-rank matrix factorization[J]. G3, 2021, 11(6): jkab098.
[50] Bingru Huang, Falai Chen*. Computing μ-bases of univariate polynomial matrices using polynomial matrix factorization[J]. Journal of Systems Science and Complexity, 2021, 34(3): 1189-1206.
[51] Yufeng Tian, Xin Li, Falai Chen. Non‐uniform subdivision surfaces with sharp features[C]//Computer Graphics Forum. 2020, 39(6): 232-242.
[52] Zijian Yuan, Yanzhi Song, Falai Chen*, Zhouwang Yang*. Diamond Inclusion Reconstruction[J]. Communications in Mathematics and Statistics, 2020,8: 341–359.
[53] Caijin Zhou, Falai Chen*. EMD Algorithm of Geometric Models and Its Applications Based on Tensor Product B-Spline[J]. Journal of Systems Science and Mathematical Sciences, 2020, 40(6): 941.
[54] Maodong Pan, Falai Chen*, Weihua Tong. Volumetric spline parameterization for isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 359: 112769.
[55] 周财进, 陈发来*. 基于张量积 B 样条的几何模型 EMD 算法及其应用[J]. 系统科学与数学, 2020, 40(6): 941-953.
1、2025.01-2029.12 国家自然科学基金重大项目：融合设计与仿真的下一代几何引擎的核心理论与算法（项目负责人）
2、2025.01-2027.12 国家重点研发计划项目：面向CAD内核的曲线曲面求交理论与软件（课题负责人）
3、2024.01-2027.12 国家自然科学基金面上项目：基于等几何分析的几何建模及其在结构优化设计中的应用
4、2016.01-2019.12 国家自然科学基金面上项目：稀疏优化在几何建模中的应用
5、2011-2015 国家科技部·973计划：数学机械化方法及其在数学化设计制造中的应用
6、2011-2014 国家自然科学基金重点项目：复杂曲面造型理论及其在科学计算中的应用
7、2009.01-2011.12 国家自然科学基金面上项目：T网格上样条曲面的理论及其在几何造型中的应用
8、2003.01-2006.12 国家自然科学基金杰出青年科学基金项目：计算机辅助技术（包括CAD,CAM,CAN,CAT和CIM等）
