Article
Engineering, Multidisciplinary
Juan Cao, Yi Xiao, Yanyang Xiao, Zhonggui Chen, Fei Xue, Xiaodong Wei, Yongjie Jessica Zhang
Summary: This paper proposes a method for constructing quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. The method extends the construction to general polygons and achieves linear to quadratic precision. The paper also proves the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains with geometric constraints.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Todd Arbogast, Chuning Wang
Summary: This paper presents new families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons. These elements are H-1 and H(div) conforming, respectively, and achieve optimal accuracy for any order. They have the minimum number of degrees of freedom while satisfying conformity and accuracy constraints. The convergence properties of the finite elements are demonstrated under certain regularity assumptions and mild restrictions on the construction of supplemental functions. Numerical experiments on various meshes illustrate the performance of these new finite element families.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Alaa Eddine Bensad, Aziz Ikemakhen
Summary: We propose a general method for defining and efficiently computing barycentric coordinates with respect to polygons on the unit sphere. We develop a novel explicit construction to compute the spherical barycentric coordinates from their 2D-Euclidean counterparts and provide families of spherical coordinates for convex and non-convex spherical polygons. We also present an alternative construction for spherical barycentric coordinates using 3D barycentric coordinates for closed triangular meshes, which can be extended to arbitrary dimensions. Our spherical and 3D coordinates have wide applicability in various domains, demonstrated through examples in spherical blending, space deformations, and shape morphing in 3D.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Jorge Marchena-Menendez, Robert C. Kirby
Summary: Serendipity elements provide computational savings with high accuracy, and we have developed methods to address this issue and demonstrated their effectiveness through numerical experiments.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics
Todd Arbogast, Chuning Wang
Summary: This paper introduces new families of direct serendipity and direct mixed finite elements defined on general planar, strictly convex polygons. The finite elements provide optimal approximation while using the minimal degrees of freedom. The paper proposes alternative ways to construct supplemental functions on the element, resulting in better accuracy and robustness in numerical tests.
Article
Computer Science, Software Engineering
Justin Crum, Cyrus Cheng, David A. Ham, Lawrence Mitchell, Robert C. Kirby, Joshua A. Levine, Andrew Gillette
Summary: This paper presents an implementation of the trinuned serendipity finite element family using Firedrake, an open-source finite element package. The new elements can be seamlessly used in problems requiring H-1, H(curl), or H(div)-conforming elements on square or cubic meshes. Comparative numerical experiments show that the trimmed serendipity elements converge at the same rate as traditional tensor product elements, while offering significant savings in time or memory for certain problems.
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
(2022)
Article
Mathematics, Applied
Xinjiang Chen, Yanqiu Wang
Summary: This paper presents an H1-conforming quadratic finite element on convex polygonal meshes using generalized barycentric coordinates, with optimal approximation rates. Two stable discretizations for the Stokes equations are developed using this element, representing extensions of the P2-P0 and Q2-(discontinuous)P1 elements to polygonal meshes. Numerical results are provided to support the theoretical claims.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Engineering, Electrical & Electronic
Yingiu Xia, Chengpu Yu, Chengyang He
Summary: This paper studies the problem of exploratory distributed localization of networked mobile agents in a GPS-Denied 3D environment. The authors provide an analytic solution and develop a distributed algorithm to solve the static and mobile target localization problems. The proposed method exhibits good scalability and effectiveness in different scenarios.
IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS
(2022)
Article
Computer Science, Software Engineering
Alaa Eddine Bensad, Aziz Ikemakhen
Summary: This paper presents the construction of hyperbolic Barycentric coordinates on the hyperbolic plane, including hyperbolic Wachspress, mean values, and discrete harmonic coordinates. These coordinates are unique for points in a hyperbolic triangle and are derived on the Poincare disk model. Furthermore, the paper demonstrates the applications of hyperbolic parameterization, such as hyperbolic deformation and shape morphing.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Computer Science, Interdisciplinary Applications
Christopher G. Albert, Patrick Lainer, Oszkar Biro
Summary: This paper presents a numerical method for solving three-dimensional linear magnetostatic problems by embedding the geometry into a symmetric domain and using Fourier expansion. The developed method allows for a decoupled set of two-dimensional problems, reducing computational complexity and improving efficiency. The approach has important applications in analyzing magnetic plasma confinement devices.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Kent T. Danielson, Robert S. Browning, Mark D. Adley
Summary: This paper evaluates the performances of second-order finite elements for nodal lumped-mass explicit methods in nonlinear solid dynamics, with a particular focus on 10-node serendipity and 15-node Lagrange tetrahedral elements. The results indicate that the inclusion of face and body centroid nodes with Lagrange interpolants, including the 15-node tetrahedron, provides robust overall performance with lumped-mass explicit methods and with contact, while the 10-node tetrahedra also show significant computational reductions compared to their 20-node serendipity hexahedral counterparts.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2021)
Article
Mathematics
Odysseas Kosmas, Pieter Boom, Andrey P. Jivkov
Summary: This article discusses the deformation of solids due to changing boundary conditions, presenting a discrete energy model based on mappings between nodal positions and deformation gradient invariants. The analysis of these deformations is achieved through energy minimization, with constraints eliminated using Lagrange multipliers. The accuracy of the technique is verified through numerical examples, demonstrating its potential in describing solid deformation.
Article
Physics, Mathematical
Yanlong Zhang
Summary: In this article, we propose and analyze a quadratic serendipity finite volume element method for arbitrary convex polygonal meshes based on the idea of serendipity element. We introduce the explicit construction of quadratic serendipity element shape function and select the quadratic serendipity element function space as the trial function space. Furthermore, we construct a family of unified dual partitions for arbitrary convex polygonal meshes, which is crucial to the finite volume element scheme, and present a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom. The optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained under certain geometric assumption conditions and verified by numerical experiments.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Lourenco Beirao da Veiga, Lorenzo Mascotto
Summary: This paper discusses the stability and interpolation properties of serendipity nodal virtual element spaces in two and three dimensions. Specifically, it rigorously proves stability bounds for doff-doff stabilization and demonstrates that the best interpolation error in serendipity nodal spaces is controlled by a best polynomial approximation term, up to constants.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Engineering, Electrical & Electronic
Sajjad Mohammadi, James L. Kirtley, Jeffrey H. Lang
Summary: This article develops a model for a rotary actuator with an elliptically shaped stator curvature to achieve reluctance torque that restores the rotor to the maximum torque per ampere position. The total torque is decomposed into coil torque and reluctance torque, while the rotor is represented by equivalent Amperian currents and the stator geometry is simplified to an ellipse. Field solutions within the ellipse are obtained using Laplace's equation, and the coil torque is determined using the Lorentz force. Reluctance torque is derived using the energy method and differential flux tubes. The proposed model is validated through prototyping and experimental results.
IEEE TRANSACTIONS ON MAGNETICS
(2023)
Article
Computer Science, Theory & Methods
Michael S. Floater, Andrew Gillette
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2017)
Article
Biochemical Research Methods
Muhibur Rasheed, Nathan Clement, Abhishek Bhowmick, Chandrajit L. Bajaj
IEEE-ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS
(2019)
Article
Mathematics, Applied
Ozan Oektem, Chong Chen, Nevzat Onur Domanic, Pradeep Ravikumar, Chandrajit Bajaj
Article
Mathematics, Applied
Andrew Gillette, Michael Holst, Yunrong Zhu
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2017)
Article
Computer Science, Software Engineering
Juan Cao, Yanyang Xiao, Zhonggui Chen, Wenping Wang, Chandrajit Bajaj
COMPUTER AIDED GEOMETRIC DESIGN
(2018)
Article
Mathematics, Applied
Andrew Gillette, Tyler Kloefkorn
MATHEMATICS OF COMPUTATION
(2019)
Article
Computer Science, Software Engineering
Qixing Huang, Zhenxiao Liang, Haoyun Wang, Simiao Zuo, Chandrajit Bajaj
ACM TRANSACTIONS ON GRAPHICS
(2019)
Article
Multidisciplinary Sciences
Soumyajit Gupta, Shachi Mittal, Andre Kajdacsy-Balla, Rohit Bhargava, Chandrajit Bajaj
Article
Computer Science, Software Engineering
Andrew Gillette, Kaibo Hu, Shuo Zhang
BIT NUMERICAL MATHEMATICS
(2020)
Article
Computer Science, Software Engineering
Justin Crum, Joshua A. Levine, Andrew Gillette
COMPUTER-AIDED DESIGN
(2019)
Article
Computer Science, Information Systems
Krzysztof Gajowniczek, Iga Grzegorczyk, Tomasz Zabkowski, Chandrajit Bajaj
Article
Physics, Applied
Alexander A. Demkov, Chandrajit Bajaj, John G. Ekerdt, Chris J. Palmstrom, S. J. Ben Yoo
Summary: Progress in computing architectures is approaching a paradigm shift as traditional computing is reaching its physical limits. Alternative approaches, such as brain-like computation and quantum computing using photons, are being researched. By integrating new materials like oxides with silicon using silicon photonics, more efficient and low power computers can be achieved, leveraging current manufacturing infrastructure.
JOURNAL OF APPLIED PHYSICS
(2021)
Proceedings Paper
Computer Science, Artificial Intelligence
Chandrajit Bajaj, Yi Wang, Tianming Wang
2019 IEEE INTERNATIONAL CONFERENCE ON BIG DATA (BIG DATA)
(2019)
Article
Biochemical Research Methods
Nathan Clement, Muhibur Rasheed, Chandrajit Lal Bajaj
JOURNAL OF COMPUTATIONAL BIOLOGY
(2018)
Article
Computer Science, Software Engineering
Tiago Simoes, Daniel Lopes, Sergio Dias, Francisco Fernandes, Joao Pereira, Joaquim Jorge, Chandrajit Bajaj, Abel Gomes
COMPUTER GRAPHICS FORUM
(2017)