February  2009, 23(1&2): 165-183. doi: 10.3934/dcds.2009.23.165

Levelsets and anisotropic mesh adaptation

1. 

UPMC Univ Paris 06, UMR 7598, Laboratoire J.L. Lions, F-75005 Paris, France, France

2. 

Universidad de Chile, UMI 2807, Centro de Modelamiento Matem´atico, Santiago, Chile

Received  December 2007 Revised  May 2008 Published  September 2008

In this paper, we focus on the problem of adapting dynamic triangulations during numerical simulations to reduce the approximation errors. Dynamically evolving interfaces arise in many applications, such as free surfaces in multiphase flows and moving surfaces in fluid-structure interactions. In such simulations, it is often required to preserve a high quality interface discretization thus posing significant challenges in adapting the triangulation in the vicinity of the interface, especially if its geometry or its topology changes dramatically during the simulation. Our approach combines an efficient levelset formulation to represent the interface in the flow equations with an anisotropic mesh adaptation scheme based on a Riemannian metric tensor to prescribe size, shape and orientation of the elements. Experimental results are provided to emphasize the effectiveness of this technique for dynamically evolving interfaces in flow simulations.
Citation: Vincent Ducrot, Pascal Frey, Alexandra Claisse. Levelsets and anisotropic mesh adaptation. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 165-183. doi: 10.3934/dcds.2009.23.165
[1]

Prabhat Mishra, Vikas Gupta, Ritesh Kumar Dubey. A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021029

[2]

Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

[3]

Maciek Korzec, Andreas Münch, Endre Süli, Barbara Wagner. Anisotropy in wavelet-based phase field models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1167-1187. doi: 10.3934/dcdsb.2016.21.1167

[4]

Xiaoxue Gong, Ying Xu, Vinay Mahadeo, Tulin Kaman, Johan Larsson, James Glimm. Mesh convergence for turbulent combustion. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4383-4402. doi: 10.3934/dcds.2016.36.4383

[5]

Sepideh Mirrahimi. Adaptation and migration of a population between patches. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 753-768. doi: 10.3934/dcdsb.2013.18.753

[6]

Diana M. Thomas, Ashley Ciesla, James A. Levine, John G. Stevens, Corby K. Martin. A mathematical model of weight change with adaptation. Mathematical Biosciences & Engineering, 2009, 6 (4) : 873-887. doi: 10.3934/mbe.2009.6.873

[7]

Henry Adams, Lara Kassab, Deanna Needell. An adaptation for iterative structured matrix completion. Foundations of Data Science, 2021  doi: 10.3934/fods.2021028

[8]

Birol Yüceoǧlu, ş. ilker Birbil, özgür Gürbüz. Dispersion with connectivity in wireless mesh networks. Journal of Industrial & Management Optimization, 2018, 14 (2) : 759-784. doi: 10.3934/jimo.2017074

[9]

Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637

[10]

Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems & Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052

[11]

Shenglong Hu. A note on the solvability of a tensor equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021146

[12]

Rongjie Lai, Jiang Liang, Hong-Kai Zhao. A local mesh method for solving PDEs on point clouds. Inverse Problems & Imaging, 2013, 7 (3) : 737-755. doi: 10.3934/ipi.2013.7.737

[13]

Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521

[14]

Francisco J. Ibarrola, Ruben D. Spies. A two-step mixed inpainting method with curvature-based anisotropy and spatial adaptivity. Inverse Problems & Imaging, 2017, 11 (2) : 247-262. doi: 10.3934/ipi.2017012

[15]

Laura M. Smith, Andrea L. Bertozzi, P. Jeffrey Brantingham, George E. Tita, Matthew Valasik. Adaptation of an ecological territorial model to street gang spatial patterns in Los Angeles. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3223-3244. doi: 10.3934/dcds.2012.32.3223

[16]

Mina Youssef, Caterina Scoglio. Mitigation of epidemics in contact networks through optimal contact adaptation. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1227-1251. doi: 10.3934/mbe.2013.10.1227

[17]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A leaky integrate-and-fire model with adaptation for the generation of a spike train. Mathematical Biosciences & Engineering, 2016, 13 (3) : 483-493. doi: 10.3934/mbe.2016002

[18]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001

[19]

Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial & Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078

[20]

Yiju Wang, Guanglu Zhou, Louis Caccetta. Nonsingular $H$-tensor and its criteria. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1173-1186. doi: 10.3934/jimo.2016.12.1173

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]