Astrid Bunge
Marc Alexa
Mario Botsch
ABSTRACT
The Laplace-Beltrami operator is one of the essential tools in geometric processing. It allows us to solve numerous partial differential equations on discrete surface and volume meshes, which is a fundamental building block in many computer graphics applications. Discrete Laplacians are typically limited to standard elements like triangles or quadrilaterals, which severely constrains the tessellation of the mesh. But in recent years, several approaches were able to generalize the Laplace Beltrami and its closely related gradient and divergence operators to more general meshes. This allows artists and engineers to work with a wider range of elements which are sometimes required and beneficial in their field. This course, which extends the state-of-the-art report by Bunge and Botsch [2023], discusses the different constructions of these three ubiquitous differential operators on arbitrary polygons and polyhedra and analyzes their individual advantages and properties in common computer graphics applications.
- Marc Alexa, Philipp Herholz, Maximilian Kohlbrenner, and Olga Sorkine-Hornung. 2020. Properties of Laplace Operators for Tetrahedral Meshes. Computer Graphics Forum 39, 5 (2020).Google Scholar
- Marc Alexa and Max Wardetzky. 2011. Discrete Laplacians on General Polygonal Meshes. ACM Transactions on Graphics 30, 4 (2011), 102:1--102:10.Google ScholarDigital Library
- Boris Andreianov, Mostafa Bendahmane, and Florence Hubert. 2013. On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems. Computational Methods in Applied Mathematics 13, 4 (2013), pp. 369--410. Google ScholarCross Ref
- Boris Andreianov, Mostafa Bendahmane, Florence Hubert, and Stella Krell. 2012. On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality. IMA Journal of Numerical Analysis 32, 4 (2012), pp.1574--1603. Google ScholarCross Ref
- Douglas N. Arnold, Richard S. Falk, and Ragnar Winther. 2006. Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15 (2006), 1--155.Google ScholarCross Ref
- Prusty Aurojyoti, Piska Raghu, Amirtham Rajagopal, and Jn Reddy. 2019. An n-sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates. Internat. J. Numer. Methods Engrg. 120 (2019), 1071--1107.Google ScholarCross Ref
- Lourenço Beirão da Veiga, Franco Brezzi, and Luisa Donatella Marini. 2013. Virtual Elements for Linear Elasticity Problems. SIAM J. Numer. Anal. 51, 2 (2013), 794--812.Google ScholarDigital Library
- Lourenço Beirão da Veiga, Franco Dassi, and Alessandro Russo. 2017. High-order Virtual Element Method on polyhedral meshes. Computers and Mathematics with Applications 74 (2017), 1110--1122.Google ScholarDigital Library
- J.E. Bishop. 2009. Simulating the Pervasive Fracture of materials and Structures Using Randomly Close Packed Voronoi Tessellations. Computational Mechanics 44, 4 (2009), 455--471.Google ScholarCross Ref
- J.E. Bishop. 2014. A displacement-based finite element formulation for general polyhedra using harmonic shape functions. Internat. J. Numer. Methods Engrg. 97 (2014), 1--31.Google ScholarCross Ref
- Mario Botsch, Robert Sumner, Mark Pauly, and Markus Gross. 2006. Deformation Transfer for Detail-Preserving Surface Editing. In Proc. of Vision, Modeling and Visualization. 357--364.Google Scholar
- Franco Brezzi, Konstantin Lipnikov, Mikhail Shashkov, and Valeria Simoncini. 2007. A new Discretization Methodology for Diffusion Problems on Generalized Polyhedral Meshes. Computer Methods in Applied Mechanics and Engineering 196, 37 (2007), 3682--3692. Google ScholarCross Ref
- Franco Brezzi, Konstantin Lipnikov, and Valeria Simoncini. 2005. A Family of Mimetic Finite Difference Methods on Polygonal and Polyhedral Meshes. Mathematical Models and Methods in Applied Sciences 15, 10 (2005), 1533--1551.Google ScholarCross Ref
- Astrid Bunge and Mario Botsch. 2023. A Survey on Discrete Laplacians for General Polygonal Meshes. Computer Graphics Forum 42, 2 (2023), 521--544. Google ScholarCross Ref
- Astrid Bunge, Mario Botsch, and Marc Alexa. 2021. The Diamond Laplace for Polygonal and Polyhedral Meshes. Computer Graphics Forum 40, 5 (2021), 217--230.Google ScholarCross Ref
- Astrid Bunge, Philipp Herholz, Misha Kazhdan, and Mario Botsch. 2020. Polygon Laplacian Made Simple. Computer Graphics Forum 39, 2 (2020), 303--313. Astrid Bunge, Philipp Herholz, Olga Sorkine-Hornung, Mario Botsch, and Michael Kazhdan. 2022. Variational Quadratic Shape Functions for Polygons and Polyhedra. ACM Trans. Graph. 41, 4, Article 54 (2022), 14 pages. Google ScholarDigital Library
- Renjie Chen and Craig Gotsman. 2016. On pseudo-harmonic barycentric coordinates. Computer Aided Geometric Design 44 (2016), 15--35. Google ScholarDigital Library
- Yanqing Chen, Timothy A. Davis, William W. Hager, and Sivasankaran Rajamanickam. 2008. Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate. ACM Trans. Math. Softw. 35, 3 (2008), 1--14.Google ScholarDigital Library
- Ming Chuang, Linjie Luo, Benedict J. Brown, Szymon Rusinkiewicz, and Michael Kazhdan. 2009. Estimating the Laplace-Beltrami Operator by Restricting 3D Functions. Computer Graphics Forum 28, 5 (2009), 1475--1484.Google ScholarCross Ref
- Yves Coudière and Florence Hubert. 2011. A 3D Discrete Duality Finite Volume Method for Nonlinear Elliptic Equations. SIAM J. Sci. Comput. 33, 4 (2011), 1739--1764.Google ScholarDigital Library
- Keenan Crane. 2019. The n-dimensional cotangent formula. https://www.cs.cmu.edu/~kmcrane/Projects/Other/nDCotanFormula.pdf.Google Scholar
- Keenan Crane, Clarisse Weischedel, and Max Wardetzky. 2013. Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow. ACM Transactions on Graphics 32, 5 (2013), 152:1--152:11.Google ScholarDigital Library
- Fernando de Goes, Andrew Butts, and Mathieu Desbrun. 2020. Discrete Differential Operators on Polygonal Meshes. ACM Transactions on Graphics 39, 4 (2020), 110:1--110:14.Google ScholarDigital Library
- Fernando de Goes, Mathieu Desbrun, Mark Meyer, and Tony DeRose. 2016. Subdivision Exterior Calculus for Geometry Processing. ACM Transactions on Graphics 35, 4 (2016), 133:1--133:11.Google ScholarDigital Library
- Mathieu Desbrun, Anil N. Hirani, Melvin Leok, and Jerrold E. Marsden. 2005. Discrete Exterior Calculus. arXiv: Differential Geometry (2005). Google ScholarCross Ref
- Mathieu Desbrun, Mark Meyer, Peter Schröder, and Alan H. Barr. 1999. Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow. In Proceedings of ACM SIGGRAPH. 317--324.Google Scholar
- K. Domelevo and P. Omnés. 2005. A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. Math. Model. Numer. Anal. (M2AN) 39, 6 (2005), 1203--1249.Google Scholar
- G.M. Dusinberre. 1961. Heat-transfer Calculations by Finite Differences. International Textbook Company.Google Scholar
- G. M. Dusinberre. 1955. Heat transfer calculations by numerical methods. Journal of the American Society for Naval Engineers 67, 4 (1955), 991--1002. Google ScholarCross Ref
- Gerhard Dziuk. 1988. Finite Elements for the Beltrami operator on arbitrary surfaces. In Partial Differential Equations and Calculus of Variations, Stefan Hildebrandt and Rolf Leis (Eds.). Springer Berlin Heidelberg, 142--155. Google ScholarCross Ref
- Graeme Fairweather and Andreas Karageorghis. 1998. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics 9, 1 (1998), 69--95.Google ScholarCross Ref
- Michael S. Floater. 2003. Mean value coordinates. Computer Aided Geometric Design 20, 1 (2003), 19--27.Google ScholarDigital Library
- Michael S. Floater. 2015. Generalized Barycentric Coordinates and Applications. Acta Numerica 24 (2015), 161--214. Google ScholarCross Ref
- Richard Franke. 1979. A critical comparison of some methods for interpolation of scattered data. Technical Report. Naval Postgraduate School.Google Scholar
- Gene H. Golub and Charles F. Van Loan. 1996. Matrix Computations. Johns Hopkins University Press.Google ScholarDigital Library
- Gaël Guennebaud, Benoît Jacob, et al. 2010. Eigen v3. http://eigen.tuxfamily.org.Google Scholar
- Philipp Herholz, Jan Eric Kyprianidis, and Marc Alexa. 2015. Perfect Laplacians for Polygon Meshes. Computer Graphics Forum 34, 5 (2015), 211--218.Google ScholarDigital Library
- F. Hermeline. 2000. A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160, 2 (2000), 481--499.Google ScholarDigital Library
- F. Hermeline. 2009. A Finite Volume Method for Approximating 3D Diffusion Operators on General Meshes. J. Comput. Phys. 228, 16 (2009), 5763--5786.Google ScholarDigital Library
- Klaus Hildebrandt, Konrad Polthier, and Max Wardetzky. 2006. On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces. Geometriae Dedicata 123 (2006), 89--112. Google ScholarCross Ref
- W. Höhn and H. D. Mittelmann. 1981. Some remarks on the discrete maximum-principle for finite elements of higher order. Computing 27, 2 (1 June 1981), 145--154. Google ScholarCross Ref
- K. Hormann and N. Sukumar. 2008. Maximum Entropy Coordinates for Arbitrary Polytopes. Computer Graphics Forum 27, 5 (2008), 1513--1520.Google ScholarDigital Library
- T.J.R. Hughes. 2012. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications.Google Scholar
- Pushkar Joshi, Mark Meyer, Tony DeRose, Brian Green, and Tom Sanocki. 2007. Harmonic Coordinates for Character Articulation. ACM Trans. Graph. 26, 3 (2007), 71--81. Google ScholarDigital Library
- Tao Ju, Scott Schaefer, and Joe Warren. 2005. Mean Value Coordinates for Closed Triangular Meshes. ACM Transactions on Graphics 24, 3 (2005), 561--566.Google ScholarDigital Library
- Dilip Krishnan, Raanan Fattal, and Richard Szeliski. 2013. Efficient Preconditioning of Laplacian Matrices for Computer Graphics. ACM Transactions on Graphics 32, 4, Article 142 (2013), 15 pages. Google ScholarDigital Library
- J.M. Lee. 1997. Riemannian Manifolds: An Introduction to Curvature. Springer New York.Google Scholar
- Konstantin Lipnikov, Gianmarco Manzini, and Mikhail Shashkov. 2014. Mimetic finite difference method. J. Comput. Phys. 257 (2014), 1163--1227.Google ScholarDigital Library
- Richard MacNeal. 1949. The Solution of Partial Differential Equations by Means of Electrical Networks. Ph. D. Dissertation. California Institute of Technology.Google Scholar
- G. Manzini, A. Russo, and N. Sukumar. 2014. New perspectives on polygonal and polyhedral finite element methods. Mathematical Models and Methods in Applied Sciences 24 (2014), 1665--1699.Google ScholarCross Ref
- Sebastian Martin, Peter Kaufmann, Mario Botsch, Martin Wicke, and Markus Gross. 2008. Polyhedral Finite Elements Using Harmonic Basis Functions. Computer Graphics Forum 27, 5 (2008), 1521--1529.Google ScholarDigital Library
- M. Meyer, M. Desbrun, P. Schröder, and A. H. Barr. 2003. Discrete Differential-Geometry Operators for Triangulated 2-Manifolds. In Visualization and Mathematics III, Hans-Christian Hege and Konrad Polthier (Eds.). Springer-Verlag, 35--57.Google Scholar
- Ean Tat Ooi, Chongmin Song, Francis Tin-Loi, and Zhenjun Yang. 2012. Polygon scaled boundary finite elements for crack propagation modelling. Internat. J. Numer. Methods Engrg. 91, 3 (2012), 319--342. Google ScholarCross Ref
- Ulrich Pinkall and Konrad Polthier. 1993. Computing discrete minimal surfaces and their conjugates. Experim. Math. 2 (1993), 15--36.Google ScholarCross Ref
- El Houssaine Quenjel, Mazen Saad, Mustapha Ghilani, and Marianne Bessemoulin-Chatard. 2020. Convergence of a positive nonlinear DDFV scheme for degenerate parabolic equations. Calcolo 50, 19 (2020). Google ScholarDigital Library
- Bastian E. Rapp. 2017. Chapter 31 - Finite Volume Method. In Microfluidics: Modelling, Mechanics and Mathematics. Elsevier, Oxford, 633--654. Google ScholarCross Ref
- Steven Rosenberg. 1997. The Laplacian on a Riemannian Manifold. Cambridge University Press, 1--51. Google ScholarCross Ref
- Teseo Schneider, Yixin Hu, Jérémie Dumas, Xifeng Gao, Daniele Panozzo, and Denis Zorin. 2018. Decoupling Simulation Accuracy from Mesh Quality. ACM Transactions on Graphics 37, 6 (2018), 280:1--280:14.Google ScholarDigital Library
- Teseo Schneider, Yixin Hu, Xifeng Gao, Jeremie Dumas, Denis Zorin, and Daniele Panozzo. 2022. A Large Scale Comparison of Tetrahedral and Hexahedral Elements for Solving Elliptic PDEs with the Finite Element Method. ACM Transactions on Graphics 41, 3 (2022), 23:1--23:14.Google ScholarDigital Library
- Nicholas Sharp. 2021. Intrinsic Triangulations in Geometry Processing. PhD thesis. Carnegie Mellon University. Google ScholarCross Ref
- Alireza Tabarraei and N. Sukumar. 2006. Application of Polygonal Finite Elements in Linear Elasticity. International Journal of Computational Methods 03 (2006), 503--520.Google ScholarCross Ref
- Alireza Tabarraei and N. Sukumar. 2008. Extended Finite Element Method on Polygonal and Quadtree Meshes. Computer Methods in Applied Mechanics and Engineering 197 (2008), 425--438. Google ScholarCross Ref
- Max Wardetzky. 2008. Convergence of the Cotangent Formula: An Overview. Birkhäuser Basel, Basel, 275--286. Google ScholarCross Ref
- Max Wardetzky, Saurabh Mathur, Felix Kälberer, and Eitan Grinspun. 2007. Discrete Laplace operators: No free lunch. In Proceedings of Eurographics Symposium on Geometry Processing. 33--37.Google Scholar
- Hassler Whitney. 1957. Geometric Integration Theory. Princeton University Press.Google Scholar
- Martin Wicke, Mario Botsch, and Markus Gross. 2007. A Finite Element Method on Convex Polyhedra. Computer Graphics Forum 26, 3 (2007), 355--364.Google ScholarCross Ref
- O.C. Zienkiewicz, R.L. Taylor, and J.Z. Zhu. 2013. Chapter 8 - The Patch Test, Reduced Integration, and Nonconforming Elements. In The Finite Element Method: its Basis and Fundamentals (7th edition ed.), O.C. Zienkiewicz, R.L. Taylor, and J.Z. Zhu (Eds.). Butterworth-Heinemann, Oxford, 257--284. Google ScholarCross Ref
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- Discrete Laplacians for General Polygonal and Polyhedral Meshes
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