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] Shicong Zhong, Bingru Huang, Falai Chen. Modified HB-splines[J]. Journal of Computational and Applied Mathematics, 2026, 485: 117467.
[2] Ying Cai, Falai Chen*, Hongmei Kang, Zhimin Zhang. Multilevel methods for PHT-splines[J]. Computer Methods in Applied Mechanics and Engineering, 2025, 446: 118251.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] Qiong Pan, Xiaoya Zhai*, Falai Chen. Density-based isogeometric topology optimization of shell structures[J]. Computer-Aided Design, 2024, 176: 103773.
[16] Jing Li, Juyong Zhang, Falai Chen*. iShapEditing: Intelligent Shape Editing with Diffusion Models[C]//Computer Graphics Forum. 2024, 43(7): e15253.
[17] 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.
[18] 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.
[19] 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.
[20] Bingru Huang, Falai Chen*. An Algorithm to Compute the -Bases of Rational Parametric Surfaces with Respect to One Variable[J]. Communications in Mathematics and Statistics, 2024, 12(3): 523-541.
[21] 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.
[22] 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.
[23] Zhiguo Yong, Hongmei Kang, Falai Chen. The Weighted Basis for PHT-Splines[J]. Computer Modeling in Engineering & Sciences (CMES), 2024, 138(1).
[24] Zheng Zhan, Wenping Wang, Falai Chen*. Simultaneous boundary and interior parameterization of planar domains via deep learning[J]. Computer-Aided Design, 2024, 166: 103621.
[25] 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.
[26] Xiaoya Zhai, Weiming Wang, Falai Chen, Jun Wu*. Topology optimization of differentiable microstructures[J]. Computer Methods in Applied Mechanics and Engineering, 2024, 418: 116530.
[27] 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.
[28] 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.
[29] 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.
[30] 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.
[31] 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.
[32] Xiaohan Bao, Weihua Tong, Falai Chen. A spectral segmentation method for large meshes[J]. Communications in Mathematics and Statistics, 2023, 11(3): 583-607.
[33] 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.
[34] 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.
[35] 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.
[36] 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.
[37] Maodong Pan, Falai Chen*. Constructing planar domain parameterization with HB-splines via quasi-conformal mapping[J]. Computer Aided Geometric Design, 2022, 97: 102133.
[38] 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.
[39] Wenbing Shao, Falai Chen*, Xuefeng Liu*. Robust algebraic curve intersections with tolerance control[J]. Computer-Aided Design, 2022, 147: 103236.
[40] 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.
[41] Ye Zheng, Falai Chen*. Volumetric boundary correspondence for isogeometric analysis based on unbalanced optimal transport[J]. Computer-Aided Design, 2021, 140: 103078.
[42] 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.
[43] 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.
[44] 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.
[45] 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.
[46] 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.
[47] Yufeng Tian, Xin Li, Falai Chen. Non‐uniform subdivision surfaces with sharp features[C]//Computer Graphics Forum. 2020, 39(6): 232-242.
[48] Zijian Yuan, Yanzhi Song, Falai Chen*, Zhouwang Yang*. Diamond Inclusion Reconstruction[J]. Communications in Mathematics and Statistics, 2020,8: 341–359.
[49] 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.
[50] Maodong Pan, Falai Chen*, Weihua Tong. Volumetric spline parameterization for isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 359: 112769.
[51] 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.
[52] Weiming Wang, Tuanfeng Y Wang, Zhouwang Yang*, Ligang Liu, Xin Tong, Weihua Tong, Jiansong Deng, Falai Chen, Xiuping Liu. Cost-effective printing of 3D objects with skin-frame structures[J]. ACM Transactions on Graphics (ToG), 2013, 32(6): 1-10.
[53] Jiansong Deng*, Falai Chen*, Xin Li, Changqi Hu, Weihua Tong, Zhouwang Yang, Yuyu Feng. Polynomial splines over hierarchical T-meshes[J]. Graphical models, 2008, 70(4): 76-86.
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）
