Skip to main content
SpringerLink
Log in

E(FG)2: a new fixed-grid shape optimization method based on the element-free galerkin mesh-free analysis

Taking large steps in shape optimization

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

We propose a shape optimization method over a fixed grid. Nodes at the intersection with the fixed grid lines track the domain’s boundary. These “floating” boundary nodes are the only ones that can move/appear/disappear in the optimization process. The element-free Galerkin (EFG) method, used for the analysis problem, provides a simple way to create these nodes. The fixed grid (FG) defines integration cells for EFG method. We project the physical domain onto the FG and numerical integration is performed over partially cut cells. The integration procedure converges quadratically. The performance of the method is shown with examples from shape optimization of thermal systems involving large shape changes between iterations. The method is applicable, without change, to shape optimization problems in elasticity, etc. and appears to eliminate non-differentiability of the objective noticed in finite element method (FEM)-based fictitious domain shape optimization methods. We give arguments to support this statement. A mathematical proof is needed.

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

Similar content being viewed by others

References

  • Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256

    Article  MathSciNet  MATH  Google Scholar 

  • Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  MATH  Google Scholar 

  • Bobaru F (2001) Meshless methods in shape optimization of linear elastic and thermoelastic solids. Ph.D. dissertation, Cornell University, Ithaca, NY

  • Bobaru F, Mukherjee S (2001) Shape sensitivity analysis and shape optimization in planar elasticity using the element-free Galerkin method. Comput Methods Appl Mech Eng 190(32–33):4319–4337

    Article  MATH  Google Scholar 

  • Bobaru F, Mukherjee S (2002) Meshless approach to shape optimization of linear thermoelastic solids. Int J Numer Methods Eng 53(4):765–796

    Article  Google Scholar 

  • Bobaru F, Rachakonda S (2004a) Boundary layer in shape optimization of convective fins using a meshfree approach. Int J Numer Methods Eng 60(7):1215–1236

    Article  MATH  Google Scholar 

  • Bobaru F, Rachakonda S (2004b) Optimal shape profiles for cooling fins of high and low conductivity. Int J Heat Mass Transfer 47(23):4953–4966

    Article  MATH  Google Scholar 

  • Chen JS, Wang HP (2000) New boundary condition treatments in meshless computation of contact problems. Comput Methods Appl Mech Eng 187:441–468

    Article  MATH  Google Scholar 

  • Duarte CAM (1996) The HP cloud method. Ph.D. dissertation, University of Texas at Austin, Austin, Texas, USA

  • Farlow JO, Thompson CV, Rosner DE (1976) Plates of stegosaurus. Forced convection heat loss fins? Science 192:1123–1125

    Article  Google Scholar 

  • Garcia-Ruiz MJ, Steven GP (1999) Fixed grid finite elements in elasticity problems. Eng Comput 16(2–3):145–164

    MATH  Google Scholar 

  • Grindeanu I, Choi KK, Chen JS, Chang KH (1999) Shape design optimization of hyperelastic structures using a meshless method. AIAA J 37(8):990–997

    Article  Google Scholar 

  • Haslinger J, Mäkinen RAE (2003) Introduction to shape optimization. SIAM, Philadelphia

    MATH  Google Scholar 

  • Haslinger J, Maitre JF, Tomas L (2001a) Fictitious domains methods with distributed Lagrange multipliers Part I: application to the solution of elliptic state problems. Math Models Methods Appl Sci 11(3):521–547

    Article  MathSciNet  MATH  Google Scholar 

  • Haslinger J, Maitre JF, Tomas L (2001b) Fictitious domains methods with distributed Lagrange multipliers Part II: application to the solution of shape optimization problems. Math Models Methods Appl Sci 11(3):549–563

    Article  MathSciNet  MATH  Google Scholar 

  • Kim NH, Choi KK, Botkin ME (2002) Numerical method for shape optimization using meshfree method. Struct Multidisc Optim 24(6):418–429

    Article  Google Scholar 

  • Mäkinen RAE, Rossi T, Toivanen J (2000) A moving mesh fictitious domain approach for shape optimization problems. Mathematical Modelling and Numerical Analysis 34(1):31–45

    Article  MathSciNet  MATH  Google Scholar 

  • Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318

    Article  MATH  Google Scholar 

  • Norato J, Haber R, Tortorelli D, Bendsoe MP (2004) A geometry projection method for shape optimization. Int J Numer Methods Eng 60(14):2289–2312

    Article  MathSciNet  MATH  Google Scholar 

  • Rabczuk T, Belytschko T (2005) Adaptivity for structured meshfree particle methods in 2D and 3D. Int J Numer Methods Eng 63(11):1559–1582

    Article  MathSciNet  MATH  Google Scholar 

  • Rachakonda S (2003) Optimal shape design of thermal systems with meshfree methods over a fixed grid. Master thesis, University of Nebraska-Lincoln, Lincoln, Nebraska, USA

  • Turk G, O’Brien JF (2002) Modelling with implicit surfaces that interpolate. ACM Trans Graph 21(4):855–873

    Article  Google Scholar 

  • White FM (1988) Heat and mass transfer. Addison-Wesley

Download references

Author information

Author notes

  1. Srinivas Rachakonda

    Present address: Olin Neuropsychiatry Research Center, Hartford, CT, 06106, USA

Authors and Affiliations

  1. Department of Engineering Mechanics, University of Nebraska-Lincoln, Lincoln, NE, 68516-0526, USA

    Florin Bobaru & Srinivas Rachakonda

Authors
  1. Florin Bobaru
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Srinivas Rachakonda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Florin Bobaru.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bobaru, F., Rachakonda, S. E(FG)2: a new fixed-grid shape optimization method based on the element-free galerkin mesh-free analysis. Struct Multidisc Optim 32, 215–228 (2006). https://doi.org/10.1007/s00158-006-0018-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-006-0018-x

Keywords

Search

Navigation