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Knot Optimization for Biharmonic B-splines on Manifold Triangle Meshes

Fei Hou, Ying He, Hong Qin, Aimin Hao
2017 IEEE Transactions on Visualization and Computer Graphics  
Consequently, without explicitly computing their bases, biharmonic B-splines can bypass the Voronoi partitioning and the discretization of bi-Laplacian, enable the computational utilities on any compact  ...  We prove that biharmonic B-splines have an equivalent representation, which is solely based on a linear combination of Green's functions of the bi-Laplacian operator.  ...  This project was partially supported by MOE2013-T2-2-011, RG23/15, NSFC (Grant No. 61300068, 61190120, 61190121, 61190125, 61532002) and NSF (Grant No. IIS-0949467, IIS-1047715 and IIS-1049448).  ... 
doi:10.1109/tvcg.2016.2605092 pmid:27608469 fatcat:2jeowy73y5b77ijn7cxnhi73ge


A.F. Shchepetkin, Shirshov Institute of Oceanology, Russian Academy of Sciences, O.S. Volodko, Moscow Institute of Physics and Technology, Institute of Computational Modeling SB RAS, Siberian Federal University
2018 Journal of Oceanological Research  
of measured data to identify and eliminate (by averaging out) potentially contradictory and unreliable measurements which may cause spurious oscillations of biharmonic spline.  ...  The technique essentially goes along the line of approach of using Green functions to construct biharmonic spline interpolation, which we augment by adding coastline and introduce special preprocessing  ...  Rogozin and A. P.  ... 
doi:10.29006/1564-2291.jor-2018.46(3).5 fatcat:rzt6xdkf45cllezustejysoasu

Smooth Interpolation of Curve Networks with Surface Normals [article]

Tibor Stanko, Stefanie Hahmann, Georges-Pierre Bonneau, Nathalie Saguin-Sprynski
2016 Eurographics State of the Art Reports  
We then introduce a new variational optimization method in which the standard bi-Laplacian is penalized by a term based on the mean curvature vectors.  ...  The normal input increases shape fidelity and allows to achieve globally smooth and visually pleasing shapes.  ...  We introduce a new variational method in which the standard bi-Laplacian is penalized by a term based on the mean curvature vectors.  ... 
doi:10.2312/egsh.20161005 fatcat:c5b2lsoyjndebocpjsfhqqnqg4

Discontinuous Galerkin Isogeometric Analysis for the biharmonic equation

Stephen Edward Moore
2018 Computers and Mathematics with Applications  
We construct B-Spline approximation spaces which are discontinuous across patch interfaces. We present a priori error estimate in a discrete norm and numerical experiments to confirm the theory.  ...  We present and analyze an interior penalty discontinuous Galerkin Isogeometric Analysis (dG-IgA) method for the biharmonic equation in computational domain in R^d with d =2,3.  ...  B-Spline and Isogeometric Analysis We refer the reader to [9] for detailed study on B-splines or NURBS based Galerkin methods.  ... 
doi:10.1016/j.camwa.2018.05.001 fatcat:eo7b2ac3ovbdnmeonx7hk5gyjm

Optimal spectral approximation of 2n-order differential operators by mixed isogeometric analysis

Quanling Deng, Vladimir Puzyrev, Victor Calo
2018 Computer Methods in Applied Mechanics and Engineering  
The mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2p where p is the order of the underlying B-spline space.  ...  An important subclass is the 2n-order PDE, which for n = 25 1, 2, 3 reduces to different classical PDEs including Laplacian, Allen-Cahn, biharmonic, 26 Cahn-Hilliard, Swift-Hohenberg, phase-field crystal  ...  Relative approximation errors for the 1D biharmonic (Bi) equation with Dirichlet boundary conditions and Cahn-Hilliard (CH) and phase-field crystal (PFC) equations with periodic boundary conditions using  ... 
doi:10.1016/j.cma.2018.08.042 fatcat:vi4pwl3zyrbbdjevmt6vese4xm

Modeling n-Symmetry Vector Fields using Higher-Order Energies

Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, Klaus Hildebrandt
2018 ACM Transactions on Graphics  
The approach is based on novel biharmonic and m-harmonic energies for n-fields on surface meshes and the integration of hard constraints to the resulting optimization problems.  ...  We introduce a variational approach for modeling n-symmetry vector and direction fields on surfaces that supports interpolation and alignment constraints, placing singularities and local editing, while  ...  ACKNOWLEDGMENTS We would like to thank Wenzel Jakob, Marco Tarini, Daniele Panozzo, and Olga Sorkine-Hornung for sharing the source code of their Instant Field-Aligned Meshes approach.  ... 
doi:10.1145/3177750 fatcat:h24i7erqbzcmrjv5hkqqsbnfsi

Optimal spectral approximation of 2n-order differential operators by mixed isogeometric analysis [article]

Quanling Deng and Vladimir Puzyrev and Victor Calo
2018 arXiv   pre-print
The mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2p where p is the order of the underlying B-spline space.  ...  This class includes a wide range of differential operators such as those arising in elliptic, biharmonic, Cahn-Hilliard, Swift-Hohenberg, and phase-field crystal equations.  ...  Mixed formulation at discrete level At discrete level, we specify a finite dimensional approximation space V p h ⊂ H 1 0 (Ω) where V p h = span{φ p a } is the span of the B-spline or Lagrange (for FEM)  ... 
arXiv:1806.04286v1 fatcat:j2ksiyi26vajrogv4cnodoenia

Poisson Vector Graphics (PVG) and Its Closed-Form Solver [article]

Fei Hou, Qian Sun, Zheng Fang, Yong-Jin Liu, Shi-Min Hu, Hong Qin, Aimin Hao, Ying He
2017 arXiv   pre-print
Although the harmonic B-spline based solutions are approximate, computational results show that the relative mean error is less than 0.3%, which cannot be distinguished by naked eyes.  ...  In contrast to the conventional finite element method that computes numerical solutions only, our method expresses the solution using harmonic B-spline, whose basis functions can be constructed locally  ...  We also developed a harmonic B-spline based PVG solver that supports random access evaluation, zooming-in of arbitrary resolution and anti-aliasing.  ... 
arXiv:1701.04303v1 fatcat:pbtjgtvr6vgmzmscsjo3u2rylu

Multipatch Discontinuous Galerkin IGA for the Biharmonic Problem On Surfaces [article]

Stephen E. Moore
2020 arXiv   pre-print
By an appropriate discrete norm, we present a priori error estimates for the non-symmetric, symmetric and semi-symmetric interior penalty methods.  ...  We present the analysis of interior penalty discontinuous Galerkin Isogeometric Analysis (dGIGA) for the biharmonic problem on orientable surfaces Ω⊂ℝ^3.  ...  We define the bi-Laplacian operator ∆ 2 Ω := ∆ Ω ∆ Ω with ∆ Ω as the Laplace-Beltrami operator, and the boundary data g 0 , g 1 , g 2 and g 3 are smooth functions.  ... 
arXiv:2012.03425v1 fatcat:m5ngo25gfvfqlei2sfflpmeg6u

Physically based adaptive preconditioning for early vision

Shang-Hong Lai, B.C. Vemuri
1997 IEEE Transactions on Pattern Analysis and Machine Intelligence  
These problems, when discretized, lead to large sparse linear systems.  ...  A preconditioner, based on the membrane spline, or the thin plate spline, or a convex combination of the two, is termed a physically based preconditioner for obvious reasons.  ...  ACKNOWLEDGMENTS This research was supported in part by the National Science Foundation grant ECS-9210648 and the Whitaker Foundation grant.  ... 
doi:10.1109/34.601247 fatcat:cvoxdrzel5b53jp4cgdwcqfp6e

Natural Boundary Conditions for Smoothing in Geometry Processing [article]

Oded Stein and Eitan Grinspun and Max Wardetzky and Alec Jacobson
2017 arXiv   pre-print
The squared Laplacian energy is a popular choice of energy and has a corresponding standard implementation: squaring the discrete Laplacian matrix.  ...  Our analysis begins in the smooth setting and concludes with discretizations using finite-differences on 2D grids or mixed finite elements for triangle meshes.  ...  are the discrete gradient operator, diagonal matrix of triangle areas, discrete matrix divergence operator and discrete mass matrix (see Appendix B).  ... 
arXiv:1707.04348v1 fatcat:gxx2fc6hc5cnjark6nn7mey5je

A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson's Equation on the Disk

Thien Nguyen, Keçstutis Karčiauskas, Jörg Peters
2014 Axioms  
The methods include two classical finite elements, a cotan-formula-based discrete differential geometry approach and four iso-geometric constructions.  ...  This paper outlines and qualitatively compares implementations of seven different methods for solving Poisson's equation on the disk.  ...  Three of the IgA approaches, subdivision, G 1 bi-3/bi-5 and C 1 polar, as well as some extensions of the DDG approach, span the correct space to solve thin shell and biharmonic equations.  ... 
doi:10.3390/axioms3020280 fatcat:em5k4er5lrdtvkcef6vkday7ja

Γ-Convergence of external approximations in boundary value problems involving the bi-Laplacian

Cesare Davini
2002 Journal of Computational and Applied Mathematics  
For certain boundary value problems involving the bi-Laplacian, sequences of discrete functionals are here deÿned and are shown to -converge to the corresponding functionals of the continuous problems.  ...  Thus, we obtain approximation schemes that are nonconforming, but direct, and that can be treated by current algorithms for symmetric and positive deÿnite functionals.  ...  Paroni and Dr. I. Pitacco for many useful discussions and Mr. S. Suraci for providing valuable numerical work.  ... 
doi:10.1016/s0377-0427(01)00525-8 fatcat:s37ggdssozgzlbtlzjimt2vxga

Continuous Fuzzy Transform as Integral Operator [article]

Giuseppe Patanè
2020 arXiv   pre-print
The Fuzzy transform is ubiquitous in different research fields and applications, such as image and data compression, data mining, knowledge discovery, and the analysis of linguistic expressions.  ...  As a generalisation of the Fuzzy transform, we introduce the continuous Fuzzy transform and its inverse, as an integral operator induced by a kernel function.  ...  Acknowledgements We thank the Reviewers for their thorough review and constructive comments, which helped us to improve the technical part and presentation of the revised paper.  ... 
arXiv:2007.13601v1 fatcat:67ffdoqdgfe3nciqjvthzp6jwm

Two-Grid Deflated Krylov Methods for Linear Equations [article]

Ronald B. Morgan, Travis Whyte, Walter Wilcox, Zhao Yang
2020 arXiv   pre-print
Eigenvectors are computed on the coarse grid and used to deflate eigenvalues on the fine grid. GMRES-type methods are first used on both the coarse and fine grids.  ...  While BiCGStab is generally considered to be a non-restarted method, it works well in this context with deflating and restarting.  ...  Bi. Defl. Bi. Defl. Bi. 10 smallest approximate eigenvectors used as starting vectors.  ... 
arXiv:2005.03070v1 fatcat:mcjga7entnblpcqetqmbwklgzu
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