Skip to main content
SpringerLink
Log in

Elliptical Methods for Surface Meshing

  • Conference paper
  • First Online:
Mathematical Modeling and Simulation of Systems (MODS 2021)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 344))

Included in the following conference series:

  • 275 Accesses

Abstract

A variety of engineering processed could be modelled by differential equation with partial derivatives. Typically, differential equations for complex domains are solved by numerical methods. The initial step of such methods often requires a discrete representation of the domain by a grid or a mesh. In many practical cases domains could be combinations of cylindrical or conic shells. Such domains could be meshed using structured or block-structured grids. The general strategy is an adaptation of the grid to the domain’s shape. Such adaptation should process properties of the problem, e.g. forces. In this case the grid should be non-uniform with the increased density of elements in some regions. The elliptical method is a general method for a mesh adaptation. This method uses elliptical differential equations with partial derivatives to handle density of elements over the domain. In this article, we developed the elliptical method for cylindrical and conic shells meshing. Controlling functions are used to manage density of elements in the mesh increasing it near specified coordinate lines.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Xia K, Zhan M, Wan D, Wei GW (2012) Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. J Comput Phys 231(4):1440–1461

    Article  MathSciNet  Google Scholar 

  2. Huang W, Ma JT, Russell RD (2008) A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations. J Comput Phys 227:6532–6552

    Article  MathSciNet  Google Scholar 

  3. Wan DC, Turek S (2007) An efficient multigrid-FEM method for the simulation of solid-liquid two phase flows. J Comput Appl Math 203:561–580

    Article  MathSciNet  Google Scholar 

  4. Wan DC, Turek S (2007) Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows. J Comput Phys 222:28–56

    Article  MathSciNet  Google Scholar 

  5. Tang T (2005) Moving mesh methods for computational fluid dynamics. Contemp Math 383

    Google Scholar 

  6. Oevermann M, Klein R (2006) A cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. J Comput Phys 219:749–769

    Article  MathSciNet  Google Scholar 

  7. Chen T, Strain J (2008) Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems. J Comput Phys 227:7503–7542

    Article  MathSciNet  Google Scholar 

  8. Chern I, Shu YC (2007) A coupling interface method for elliptic interface problems. J Comput Phys 225:2138–2174

    Article  MathSciNet  Google Scholar 

  9. Zhao S, Wei GW, Xiang Y (2005) Dsc analysis of free-edged beams by an iteratively matched boundary method. J Sound Vib 284(1–2):487–493

    Article  Google Scholar 

  10. Yu SN, Wei GW (2007) Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities. J Comput Phys 227:602–632

    Article  MathSciNet  Google Scholar 

  11. Yu SN, Zhou YC, Wei GW (2007) Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J Comput Phys 224(2):729–756

    Article  MathSciNet  Google Scholar 

  12. Zhou YC, Wei GW (2006) On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method. J Comput Phys 219(1):228–246

    Article  MathSciNet  Google Scholar 

  13. Zhou YC, Zhao S, Feig M, Wei GW (2006) High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J Comput Phys 213(1):1–30

    Article  MathSciNet  Google Scholar 

  14. Zhao S (2010) High order matched interface and boundary methods for the helmholtz equation in media with arbitrarily curved interfaces. J Comput Phys 229:3155–3170

    Article  MathSciNet  Google Scholar 

  15. Wu CL, Li ZL, Lai MC (2011) Adaptive mesh refinement for elliptic interface problems using the nonconforming immerse finite element method. Int J Numer Anal Model 8:466–483

    MathSciNet  MATH  Google Scholar 

  16. Yu SN., Geng WH., Wei GW (2007) Treatment of geometric singularities in implicit solvent models. J Chem Phys 126:244108

    Google Scholar 

  17. Selim MM, Koomullil RP (2016) Mesh deformation approaches—a survey. J Phys Math 7:181

    Google Scholar 

  18. Dwight RP (2009) Robust mesh deformation using the linear elasticity equations. J Comput Fluid Dynam 12:401–406

    Google Scholar 

  19. Luke E, Collins E, Blades E (2012) A fast mesh deformation method using explicit interpolation. J Comput Phys 37:586–601

    Article  MathSciNet  Google Scholar 

  20. Maruyama D, Bailly D, Carrier G (2012) High-quality mesh deformation using quaternions for orthogonality preservation. In: 50th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, USA (2012)

    Google Scholar 

  21. Zhou X, Li S (2015) A novel three-dimensional mesh deformation method based on sphere relaxation. J Comput Phys 298:320–336

    Article  MathSciNet  Google Scholar 

  22. Sun S, Lv S, Yuan Y, Yuan M (2016) Mesh Deformation Method Based on Mean Value Coordinates Interpolation. Acta Mech Solida Sin 29:1–12

    Article  Google Scholar 

  23. Witteveen JAS (2010) Explicit and robust inverse distance weighting mesh deformation for CFD. In: 48th AIAA Aerospace sciences meeting including the new horizons forum and aerospace exposition, USA

    Google Scholar 

  24. Boer AD, van der Schoot MS, Bijl H (2007) Mesh deformation based on radial basis function interpolation. J Computers Struct 45:784–795

    Article  Google Scholar 

  25. Michler AK (2011) Aircraft control surface deflection using RBF-based mesh deformation. Int J Numer Meth Eng 88:986–10079

    Article  Google Scholar 

  26. Khalanchuk LV, Choporov SV (2020) Research of non-uniform structured discrete models generation for two-dimensional domains. Visnyk of Zaporizhzhya National University. Physical and Mathematical Sciences 1:106–112

    Google Scholar 

  27. Khalanchuk LV, Choporov SV (2020) Development of a method for constructing irregular meshes based on the differential poisson equation. Appl Quest Math Model Kherson 3(№ 2.2):274–282

    Google Scholar 

  28. Valger SA, Fedorova NN (2012) Primenenie algoritma k adaptatsii raschetnoi setki k resheniu Eilera. Vychislitelnye tehnologii 17(№ 3):24–33

    Google Scholar 

  29. Sosnytska N, Morozov M, Khalanchuk (2020) Modeling of electron state in quantum dot structures. In: 2020 IEEE problems of automated electrodrive. Theory and Practice (PAEP), Kremenchuk, Ukraine, pp 1–5 (2020)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dmytro Motornyi Tavria State Agrotechnological University, Melitopol, Ukraine

    Larysa Khalanchuk

  2. Zaporizhzhya National University, Zaporizhzhya, Ukraine

    Serhii Choporov

Authors
  1. Larysa Khalanchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Serhii Choporov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The Ministry of Education and Science of Ukraine, Kyiv, Ukraine

    Serhiy Shkarlet

  2. Institute of Mathematical Machines and Systems Problems, Ukraine National Academy of Science, Kyiv, Ukraine

    Anatoliy Morozov

  3. VM Glushkov Institute of Cybernetics, Ukraine National Academy of Science, Kyiv, Ukraine

    Alexander Palagin

  4. Tallinn University of Technology, Tallinn, Estonia

    Dmitri Vinnikov

  5. Bulgarian Defense Institute, Sofia, Bulgaria

    Nikolai Stoianov

  6. Institute of Environmental Radioactivity, Fukushima University, Fukushima, Japan

    Mark Zhelezniak

  7. Chernihiv Polytechnic National University, Chernihiv, Ukraine

    Volodymyr Kazymyr

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Khalanchuk, L., Choporov, S. (2022). Elliptical Methods for Surface Meshing. In: Shkarlet, S., et al. Mathematical Modeling and Simulation of Systems. MODS 2021. Lecture Notes in Networks and Systems, vol 344. Springer, Cham. https://doi.org/10.1007/978-3-030-89902-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-89902-8_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-89901-1

  • Online ISBN: 978-3-030-89902-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation