Article
Computer Science, Interdisciplinary Applications
Minmiao Wang, Pankaj Jagad, Anil N. Hirani, Ravi Samtaney
Summary: We propose a discretization scheme for incompressible two-phase flows based on discrete exterior calculus (DEC). By extending our physically-compatible exterior calculus discretization for single phase flow, we are able to simulate immiscible two-phase flows with discontinuous changes in fluid properties across the interface. Our scheme transforms the two-phase incompressible Navier-Stokes equations and conservative phase field equation into the framework of exterior calculus, and obtains the discrete counterpart by using discrete differential forms and operators. We demonstrate the effectiveness and versatility of our scheme through various test cases, showing excellent boundedness, mass conservation, convergence and the ability to handle large density and viscosity ratios as well as surface tension.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mechanics
Pieter D. Boom, Odysseas Kosmas, Lee Margetts, Andrey P. Jivkov
Summary: This study presents a direct formulation of linear elasticity of cell complexes based on discrete exterior calculus, which calculates the relations between displacement differences and internal forces to simulate macroscopic elastic behavior. Numerical simulations have validated the accuracy and reliability of this formulation in several classical problems with known solutions.
INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES
(2022)
Article
Computer Science, Interdisciplinary Applications
Markus Kivioja, Sanna Monkola, Tuomo Rossi
Summary: This paper presents a reliable numerical method and efficient GPU-accelerated implementation for the time integration of the three-dimensional Gross-Pitaevskii equation. The method utilizes discrete exterior calculus and offers more versatile spatial discretization compared to traditional methods. The implementation achieves significant speedups on the GPU and is further parallelized to multiple GPUs.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Multidisciplinary Sciences
Volodymyr Sushch
Summary: In this study, we discuss the discretization of the two-dimensional de Rham-Hodge theory using a discrete exterior calculus framework. We present discrete analogues of the Hodge-Dirac and Laplace operators that capture key geometric aspects of the continuous counterpart. We also provide and prove a discrete version of the Hodge decomposition theorem, and define and compute the cohomology groups in the case of combinatorial torus.
Article
Computer Science, Software Engineering
Alexander Schier, Reinhard Klein
Summary: Discrete exterior calculus (DEC) is a numerical method that discretizes partial differential equations on meshes, ensuring the accuracy of important integral theorems. However, current methods have limitations on the types of meshes they can handle, excluding those with concyclic triangle pairs. Our paper proposes an approach to overcome this limitation by defining DEC operators for concyclic polygons, allowing the use of arbitrary triangulations with concyclic triangle pairs.
COMPUTER AIDED GEOMETRIC DESIGN
(2023)
Article
Computer Science, Interdisciplinary Applications
Jean-Paul Caltagirone
Summary: Discrete mechanics is proposed as an alternative to fluid mechanics equations, deriving the equation of motion based on Galileo's intuitions, Galilean equivalence, and relativity. This approach allows for the treatment of surface discontinuities and two-phase flows, showcasing that addressing jump conditions does not affect the precision of the resolution method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Engineering, Multidisciplinary
Sanna Monkola, Joona Raty
Summary: Discrete exterior calculus is a promising discretization method for photonic crystal waveguides, offering efficient handling of nonhomogeneous computational domains and curved surfaces. In this study, we present a two-dimensional discretization method for photonic crystal waveguides using discrete exterior calculus and demonstrate its advantages through numerical experiments.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2023)
Article
Mathematics
Matthias Hieber, Hideo Kozono, Anton Seyfert, Senjo Shimizu, Taku Yanagisawa
Summary: In this article, the Helmholtz-Weyl decomposition in three dimensional exterior domains is established for 1 < r < infinity within the L-r-setting. It has been proven that for a given L-r-vector field u, the decomposition is unique only when 1 < r < 3, with the proof relying on an L-r-variational inequality.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Computer Science, Software Engineering
Pieter D. Boom, Andrey P. Jivkov, Lee Margetts
Summary: This article presents a new high-performance computing software for accelerating research into geometric formulations of solid mechanics based on discrete exterior calculus (DEC). The software integrates the DEC library ParaGEMS into the parallel finite-element (FE) code ParaFEM, enabling efficient modeling of non-smooth material processes. The tool is validated using miniApps and exhibits excellent scaling and parallel efficiency.
Article
Computer Science, Interdisciplinary Applications
Pieter D. Boom, Ashley Seepujak, Odysseas Kosmas, Lee Margetts, Andrey Jivkov
Summary: A new library for massively parallel computations with 3D domains using discrete exterior calculus (DEC) is developed, which offers efficient handling of heterogeneities and discontinuities. The library is able to analyze steady-state and transient physical processes driven by scalar or vector gradients.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Marta D'Elia, Mamikon Gulian, Tadele Mengesha, James M. Scott
Summary: This study investigates the analytical foundations of nonlocal vector calculus and demonstrates its potential applications in various fields. The research rigorously proves the identities of nonlocal vector calculus and develops a weighted fractional Helmholtz decomposition for smooth vector fields.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Tiago Novello, Joao Paixao, Carlos Tomei, Thomas Lewiner
Summary: This paper introduces the concept of discrete line fields and derives some basic results through the study of discrete line fields, including the Euler-Poincare formula, Morse-Smale decomposition, and topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.
TOPOLOGY AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
Jonni Lohi
Summary: The paper introduces a systematic approach based on discrete exterior calculus for implementing higher order Whitney forms in numerical methods. Algorithms are provided for refining the mesh, solving the coefficients of the interpolant, and evaluating the interpolant. The algorithms demonstrate generality and potential applicability in various methods and scenarios.
NUMERICAL ALGORITHMS
(2022)
Article
Computer Science, Interdisciplinary Applications
Bhargav Mantravadi, Pankaj Jagad, Ravi Samtaney
Summary: We propose a new hybrid method combining discrete exterior calculus (DEC) and finite difference (FD) to simulate three-dimensional Boussinesq convection in spherical shells with internal heating and basal heating. DEC is used to calculate the surface flows, while FD is utilized for discretization in the radial direction. The grid used in this method eliminates issues such as coordinate singularity and grid non-convergence near the poles.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Physics, Multidisciplinary
Rustem N. Garifullin, Ismagil T. Habibullin
Summary: In this article, the integrability of differential-difference lattices of hyperbolic type is investigated with a focus on constructing generalized symmetries. A method for solving functional equations using characteristic Lie-Rinehart algebras of semi-discrete models is proposed, leading to a classification method for integrable semi-discrete lattices. An interesting result includes a new example of an integrable equation, the semi-discrete analogue of the Tzizeica equation.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Article
Mathematics, Applied
Melissa Yeung, David Cohen-Steiner, Mathieu Desbrun
Article
Computer Science, Software Engineering
Max Budninskiy, Ameera Abdelaziz, Yiying Tong, Mathieu Desbrun
COMPUTER AIDED GEOMETRIC DESIGN
(2020)
Article
Computer Science, Software Engineering
Wei Li, Yixin Chen, Mathieu Desbrun, Changxi Zheng, Xiaopei Liu
ACM TRANSACTIONS ON GRAPHICS
(2020)
Article
Computer Science, Software Engineering
Lois Paulin, Nicolas Bonneel, David Coeurjolly, Jean-Claude Iehl, Antoine Webanck, Mathieu Desbrun, Victor Ostromoukhov
ACM TRANSACTIONS ON GRAPHICS
(2020)
Article
Computer Science, Software Engineering
Fernando de Goes, Andrew Butts, Mathieu Desbrun
ACM TRANSACTIONS ON GRAPHICS
(2020)
Article
Computer Science, Software Engineering
Kai Bai, Wei Li, Mathieu Desbrun, Xiaopei Liu
Summary: The article proposes a novel learning approach for dynamically upsampling smoke flows based on a training set of coarse and fine resolution flows. The network constructs a corresponding dictionary during training and is able to provide accurate upsampling through fast evaluation.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Computer Science, Software Engineering
Jiong Chen, Florian Schaefer, Jin Huang, Mathieu Desbrun
Summary: The article introduces an efficient preconditioning method for large-scale and ill-conditioned sparse linear systems, using techniques such as incomplete Cholesky factorization, fine-to-coarse ordering, multiscale sparsity pattern, and conjugate gradient solver. This approach outperforms existing carefully-engineered libraries for graphics problems involving bad mesh elements and/or high contrast of coefficients. The core concepts are supported by theoretical foundations, linking operator-adapted wavelets to Cholesky factorization and multiscale analysis.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Computer Science, Software Engineering
Wei Li, Daoming Liu, Mathieu Desbrun, Jin Huang, Xiaopei Liu
Summary: This article proposes a kinetic model coupling the Navier-Stokes equations with a conservative phase-field equation to provide a general multiphase flow solver. The resulting algorithm is embarrassingly parallel, conservative, far more stable than current solvers, and general enough to capture typical multiphase flow behaviors.
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
(2021)
Article
Computer Science, Software Engineering
Kai Bai, Chunhao Wang, Mathieu Desbrun, Xiaopei Liu
Summary: This paper presents a simple and effective method for spatio-temporal upsampling of fluid simulation using a dictionary-based approach. The neural network approach can accurately reproduce the visual complexity of turbulent flows from coarse velocity fields, demonstrating efficiency and generalizability for predicting high-resolution turbulence details.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Computer Science, Software Engineering
Chaoyang Lyu, Wei Li, Mathieu Desbrun, Xiaopei Liu
Summary: This paper proposes an efficient and versatile approach for simulating two-way fluid-solid coupling in the kinetic fluid simulation framework, addressing the challenges of reproducing the interaction between fluids and solids. The novel hybrid approach introduced in the paper ensures a robust and plausible treatment of turbulent flows near moving solids, significantly reducing boundary artifacts. Simple GPU optimizations are also presented to achieve a higher computational efficiency than existing methods.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Computer Science, Software Engineering
Wei Li, Yihui Ma, Xiaopei Liu, Mathieu Desbrun
Summary: This paper proposes a new solver for coupling the incompressible Navier-Stokes equations with a conservative phase-field equation to simulate multiphase flows. The resulting solver shows efficiency, versatility, and reliability in dealing with large density ratios, high Reynolds numbers, and complex solid boundaries.
ACM TRANSACTIONS ON GRAPHICS
(2022)
Article
Computer Science, Software Engineering
Jiayi Wei, Jiong Chen, Damien Rohmer, Pooran Memari, Mathieu Desbrun
Summary: This paper presents a new robust-statistics approach for denoising pointsets, preserving sharp features by using line processes and offering robustness to noise and outliers. Our method deduces a geometric denoising strategy through robust and regularized tangent plane fitting, obtained numerically for efficiency and reliability. We use line processes to identify inliers vs. outliers and to detect the presence of sharp features.
COMPUTER GRAPHICS FORUM
(2023)
Article
Computer Science, Software Engineering
Wei Li, Mathieu Desbrun
Summary: This paper presents an improved numerical simulation method that can accurately simulate complex fluid phenomena and effectively handle fluid-solid coupling. It introduces a series of numerical improvements in momentum exchange, interfacial forces, and two-way coupling to reduce simulation artifacts and expand the types of fluid-solid coupling that can be efficiently simulated. The benefits of the solver are demonstrated through challenging simulation results and comparisons to previous work and real footage.
ACM TRANSACTIONS ON GRAPHICS
(2023)
Article
Mathematics, Applied
Rui Wang, Rundong Zhao, Emily Ribando-Gros, Jiahui Chen, Yiying Tong, Guo-Wei Wei
Summary: The introduction of persistent spectral graph theory expands the multi-scale paradigm of topological data analysis and geometric analysis. The harmonic spectra constructed from the persistent Laplacian matrices provide topological invariants such as persistent Betti numbers, while the non-harmonic spectra offer additional geometric analysis of the data shape.
FOUNDATIONS OF DATA SCIENCE
(2021)
Article
Mathematics, Applied
Jiahui Chen, Rundong Zhao, Yiying Tong, Guo-Wei Wei
Summary: The evolutionary de Rham-Hodge method proposed in this work provides a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds, with a focus on compact manifolds with 2-manifold boundaries. The proposed method introduces three sets of unique evolutionary Hodge Laplacians to generate topology-preserving singular spectra, revealing topological persistence and geometric progression during manifold evolution. Extensive numerical experiments validate the potential of the paradigm for data representation and shape analysis, particularly in challenging cases such as protein B-factor predictions where existing biophysical models fail.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2021)