Skip to main content
SpringerLink
Log in

Nodally Integrated Local Maximum-Entropy Approximation-Based Element-Free Galerkin Method for the Analysis of Steady Heat Conduction

  • Research Article-Mechanical Engineering
  • Published:
Arabian Journal for Science and Engineering Aims and scope Submit manuscript

Abstract

This research aims to study the performance of local maximum-entropy approximation (LME)-based element-free Galerkin meshfree (EFG) method and its integration in the heat conduction application. EFG methods have undergone significant development over the past two decades and have come to the forefront to solve partial differential equations. Being non-polynomial functions, LME is smooth and appears to be a viable substitute for the approximation in EFG methods. It possesses weak Kronecker delta property that allows the implementation of essential boundary conditions like FEM. In the present work, stabilized conforming nodal integration (SCNI) and its modified version with additional stability called modified SCNI (MSCNI) is used to perform the integration of LME-based EFG and tested against different discretization node sets. Poisson heat conduction equation with a different set of boundary conditions is chosen to study these integration schemes and compared with several Gaussian integration point schemes. It is found that the 3 or 4 point Gauss integration scheme is optimal for unstructured discretization and MSCNI is optimal for distorted discretization. SCNI and MSCNI are observed to be converging faster than the other methods, irrespective of the grid type.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Abbreviations

\( \emptyset_{i}\) :

Basis functions

h :

Characteristic nodal spacing

H :

Shannon-Jaynes entropy

U :

Second moment about the node of interest

\(f_{\beta }\) :

Minimization functional

\(k_{x} , k_{y}\) :

Thermal conductivities

L 2 :

Error measure norm

n x , n y :

Outward unit normals

N :

Number of nodes

N c :

Number of Delaunay triangular cells

q :

Neumann boundary heat flux, W/m2

\(\dot{Q}\) :

Heat source per unit mass

T :

Unknown scalar in the problem

T h :

Approximation of the trail solution

T 1 :

Scalar value on Dirichlet boundary

w :

Weight function

A L :

Smoothing area

x :

Coordinates of the nodes

x i :

Scattered node set

\(l_{j}\) :

Lengths od sides of Delaunay triangle

\(\Omega\) :

Domain of the problem

\({\text{d}}\Omega ,\Gamma\) :

Boundary of the domain

\(\Gamma_{N}\) :

Neumann boundary conditions imposed boundary

\(\beta\) :

Pareto optimal parameter

\(\gamma\) :

Locality parameter

λ :

Lagrange Multiplier vector

\( \emptyset_{i}\) :

Basis functions

a :

Local Point of interest within the Delaunay triangular cell

cell:

Delaunay triangular cell of interest

i :

Node of interest

j :

Neighbour node

d :

Spatial dimension

num:

Numerical solution

exact:

Exact solution

References

  1. Xue, B.Y.; Wu, S.C.; Zhang, W.H.; Liu, G.R.: A smoothed FEM (S-FEM) for heat transfer problems. Int. J. Comput. Methods 10, 1340001 (2013). https://doi.org/10.1142/S021987621340001X

    Article  MathSciNet  Google Scholar 

  2. Arroyo, M.; Ortiz, M.: Local maximum-entropy approximation schemes : a seamless bridge between finite elements and meshfree methods. Int. J. Numer. Methods Eng. 65, 2167–2202 (2006). https://doi.org/10.1002/nme.1534

    Article  MathSciNet  MATH  Google Scholar 

  3. Belytschko, T.; Lu, Y.Y.; Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994). https://doi.org/10.1002/nme.1620370205

    Article  MathSciNet  MATH  Google Scholar 

  4. Nayroles, B.; Touzot, G.; Villon, P.: Generalizing the finite element method: Diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992). https://doi.org/10.1007/BF00364252

    Article  MathSciNet  MATH  Google Scholar 

  5. Liu, W.K.; Jun, S.; Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 20, 1081–1106 (1995). https://doi.org/10.1002/fld.1650200824

    Article  MathSciNet  MATH  Google Scholar 

  6. Peddavarapu, S.; Raghuraman, S.: Maximum entropy-based variational multiscale element-free Galerkin methods for scalar advection–diffusion problems. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-09845-y

    Article  Google Scholar 

  7. Oñate, E.; Idelsohn, S.; Zienkiewicz, O.C.; Taylor, R.L.: A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW. Int. J. Numer. Methods Eng. 39, 3839–3866 (1996). https://doi.org/10.1002/(SICI)1097-0207(19961130)39:22%3c3839::AID-NME27%3e3.0.CO;2-R

    Article  MathSciNet  MATH  Google Scholar 

  8. Atluri, S.N.; Zhu, T.: A new Meshless Local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998). https://doi.org/10.1007/s004660050346

    Article  MathSciNet  MATH  Google Scholar 

  9. Lin, H.; Atluri, N.; S. : The Meshless Local Petrov–Galerkin (MLPG) method for solving incompressible Navier–Stokes equations. Comput. Model. Eng. Sci. 2, 117–142 (2001)

    MathSciNet  Google Scholar 

  10. Chen, J.-S.; Wu, C.-T.; Yoon, S.; You, Y.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001). https://doi.org/10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A

    Article  MATH  Google Scholar 

  11. Sukumar, N.: Maximum entropy approximation. AIP Conf. Proc. 803, 337–344 (2005). https://doi.org/10.1063/1.2149812

    Article  MATH  Google Scholar 

  12. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948). https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

    Article  MathSciNet  MATH  Google Scholar 

  13. Sukumar, N.; Wright, R.W.: Overview and construction of meshfree basis functions: from moving least squares to entropy approximants. Int. J. Numer. Methods Eng. 70, 181–205 (2007). https://doi.org/10.1002/nme.1885

    Article  MathSciNet  MATH  Google Scholar 

  14. Ortiz, A.; Puso, M.A.; Sukumar, N.: Maximum-entropy meshfree method for incompressible media problems. Finite Elem. Anal. Des. 47, 572–585 (2011). https://doi.org/10.1016/j.finel.2010.12.009

    Article  MathSciNet  Google Scholar 

  15. Ortiz, A.; Puso, M.A.; Sukumar, N.: Maximum-entropy meshfree method for compressible and near-incompressible elasticity. Comput. Methods Appl. Mech. Eng. 199, 1859–1871 (2010). https://doi.org/10.1016/j.cma.2010.02.013

    Article  MathSciNet  MATH  Google Scholar 

  16. Methods, C.; Mech, A.; Ullah, Z.; Coombs, W.M.; Augarde, C.E.: An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems. Comput. Methods Appl. Mech. Eng. 267, 111–132 (2013). https://doi.org/10.1016/j.cma.2013.07.018

    Article  MathSciNet  MATH  Google Scholar 

  17. Methods, C.; Mech, A.; Ortiz-bernardin, A.; Puso, M.A.; Sukumar, N.: ScienceDirect Improved robustness for nearly-incompressible large deformation meshfree simulations on Delaunay tessellations. Comput. Methods Appl. Mech. Eng. 293, 348–374 (2015). https://doi.org/10.1016/j.cma.2015.05.009

    Article  MATH  Google Scholar 

  18. Amiri, F.; Anitescu, C.; Arroyo, M.; et al.: XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Comput. Mech. 53, 45–57 (2014). https://doi.org/10.1007/s00466-013-0891-2

    Article  MathSciNet  MATH  Google Scholar 

  19. Yaw, L.L.; Sukumar, N.; Kunnath, S.K.: Meshfree co-rotational formulation for two-dimensional continua. Int. J. Numer. Methods Eng. 79, 979–1003 (2009). https://doi.org/10.1002/nme.2606

    Article  MathSciNet  MATH  Google Scholar 

  20. Quaranta, G.; Kunnath, S.K.; Sukumar, N.: Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures. Eng. Struct. 1, 1–32

  21. Methods, C.; Mech, A.; Sukumar, N.: Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons. Comput. Methods Appl. Mech. Eng. 263, 27–41 (2013). https://doi.org/10.1016/j.cma.2013.04.009

    Article  MathSciNet  MATH  Google Scholar 

  22. Bompadre, A.; Schmidt, B.; Ortiz, M.: Convergence analysis of meshfree approximation schemes. SIAM J. Numer. Anal. 50, 1344–1366 (2012). https://doi.org/10.1137/110828745

    Article  MathSciNet  MATH  Google Scholar 

  23. Methods, C.; Mech, A.; Bompadre, A.; Perotti, L.E.; Cyron, C.J.; Ortiz, M.: Convergent meshfree approximation schemes of arbitrary order and smoothness. Comput. Methods Appl. Mech. Eng. 221–222, 83–103 (2012). https://doi.org/10.1016/j.cma.2012.01.020

    Article  MathSciNet  MATH  Google Scholar 

  24. Peddavarapu, S.; Srinivasan, R.: Local maximum entropy approximation-based streamline upwind Petrov–Galerkin meshfree method for convection–diffusion problem. J. Braz. Soc. Mech. Sci. Eng. 43, 326 (2021). https://doi.org/10.1007/s40430-021-03038-w

    Article  Google Scholar 

  25. Beissel, S.; Belytschko, T.: Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 139(1–4), 49–74 (1996). https://doi.org/10.1016/S0045-7825(96)01079-1

    Article  MathSciNet  MATH  Google Scholar 

  26. Puso, M.A.; Zywicz, E.; Chen, J.S.: A new stabilized nodal integration approach. In: Griebel, M.; Schweitzer, M.A. (Eds.) Meshfree Methods for Partial Differential Equations III, pp. 207–217. Springer, Berlin (2007)

    Chapter  Google Scholar 

  27. Puso, M.A.; Chen, J.S.; Zywicz, E.; Elmer, W.: Meshfree and finite element nodal integration methods. Int. J. Numer. Methods Eng. 74, 416–446 (2008). https://doi.org/10.1002/nme.2181

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang, T.-H.; Wei, H.; Chen, J.-S.; Hillman, M.C.: RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations. Comput. Part. Mech. 7, 393–433 (2020). https://doi.org/10.1007/s40571-019-00272-x

    Article  Google Scholar 

  29. Hillman, M.; Chen, J.-S.: An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. Int. J. Numer. Methods Eng. 107, 603–630 (2016). https://doi.org/10.1002/nme.5183

    Article  MathSciNet  MATH  Google Scholar 

  30. Quak, W.; Van Den Boogaard, A.H.; González, D.; Cueto, E.: A comparative study on the performance of meshless approximations and their integration. Comput. Mech. 48, 121–137 (2011). https://doi.org/10.1007/s00466-011-0577-6

    Article  MathSciNet  MATH  Google Scholar 

  31. Duan, Q.; Li, X.; Zhang, H.; Belytschko, T.: Second-order accurate derivatives and integration schemes for meshfree methods. Int. J. Numer. Methods Eng. 92, 399–424 (2012). https://doi.org/10.1002/nme.4359

    Article  MathSciNet  MATH  Google Scholar 

  32. Duan, Q.; Gao, X.; Wang, B.; Li, X.; Zhang, H.; Belytschko, T.; Shao, Y.: Consistent element-free Galerkin method. Int. J. Numer. Methods Eng. 99, 79–101 (2014). https://doi.org/10.1002/nme.4661

    Article  MathSciNet  MATH  Google Scholar 

  33. Ortiz-bernardin, A.; Hale, J.S.; Cyron, C.J.: Volume-averaged nodal projection method for nearly-incompressible elasticity using meshfree and bubble basis functions. Comput. Methods Appl. Mech. Eng. 56, 1–52

  34. Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. 108, 171–190 (1957). https://doi.org/10.1103/PhysRev.108.171

    Article  MathSciNet  MATH  Google Scholar 

  35. Kullback, S.; Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)

    Article  MathSciNet  Google Scholar 

  36. Peddavarapu, S.; Venkata, G.; Sunil, S.; Raghuraman, S.: Influence of pareto optimality on the maximum entropy methods. In: AIP Conference Proceedings. p. 020102 (2017). Doi: https://doi.org/10.1063/1.4990255

  37. Sukumar, N.; Moran, B.; Belytschko, T.: The natural element method in solid mechanics. Int. J. Numer. Meth. Eng. 887, 839–887 (1998)

    Article  MathSciNet  Google Scholar 

  38. Garg, S.; Pant, M.: Meshfree methods: a comprehensive review of applications. Int. J. Comput. Methods 15, 1830001 (2017). https://doi.org/10.1142/S0219876218300015

    Article  MathSciNet  MATH  Google Scholar 

  39. Sukumar, N.; Moran, B.; Black, T.; Belytschko, T.: An element-free Galerkin method for three-dimensional fracture mechanics. Comput. Mech. 20, 170–175 (1997). https://doi.org/10.1007/s004660050235

    Article  MATH  Google Scholar 

  40. Wu, X.-H.; Tao, W.-Q.: Meshless method based on the local weak-forms for steady-state heat conduction problems. Int. J. Heat Mass Transf. 51, 3103–3112 (2008). https://doi.org/10.1016/j.ijheatmasstransfer.2007.08.021

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, SASTRA Deemed to be University, Thanjavur, Tamilnadu, 613401, India

    Sreehari Peddavarapu & S. Raghuraman

Authors
  1. Sreehari Peddavarapu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Raghuraman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sreehari Peddavarapu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peddavarapu, S., Raghuraman, S. Nodally Integrated Local Maximum-Entropy Approximation-Based Element-Free Galerkin Method for the Analysis of Steady Heat Conduction. Arab J Sci Eng 47, 8385–8397 (2022). https://doi.org/10.1007/s13369-021-06229-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13369-021-06229-8

Keywords

Search

Navigation