# ALGORITHM TO MAINTAIN LINEAR ELEMENT IN 3D LEVEL SET TOPOLOGY OPTIMIZATION

### Christopher J. Brampton, Alicia H. Kim, James L. Cunningham

#### Abstract

In level set topology optimization the boundary of the structure is defined by level set function values stored at the nodes of a regular gird of simple bilinear elements. By changing the level set function values according to optimization sensitivities the boundary of the structure is moved to create an optimal structure. However it is possible for the boundary to cut an element more than once; violating the linear element assumptions resulting in insufficient nodal information for the optimization sensitivity calculations. To resolve this the local boundary of the structure is moved so that each element is only cut once. In 2D where a square element mesh is used an element cut twice times is altered by moving one of the boundaries within the element to intercept the node closest to it removing the extra cut from the element. In 3D where a voxel mesh is used the process of moving the boundary within an element is more complicated due to the greater number of boundary cuts possible and the effect that it can have on neighbouring elements. An algorithm is developed which allows the boundary within a 3D element to be moved with these considerations taken into account.

#### References

- Allaire G., Jouve F., Toader AM., 2004. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics; 194(1): 363-393.
- Dunning P D., Kim H A., 2011. Investigation and improvement of sensitivity computation using the area-fraction weighted fixed grid FEM and structural optimization. Finite Elements in Analysis and Design; 47: 933-941.
- Jang G., Kim Y Y., 2005. Sensitivity analysis for fixedgrid shape optimization by using oblique boundary curve approximation. International Journal of Solids and Structures; 42: 3591-3609.
- Wang S Y., Lim K Y., Khoo B C., Wang M Y., 2007, An extended level set method for shape and topology optimization. Journal of Computational Physics; 221: 395-421.
- Wang S., Wang M Y., 2006, A moving superimposed finite element method for structural topology optimization. International Journal for Numerical Methods in Engineering; 65:1892-1922.
- Challis V J., 2010, A discrete level-set topology optimization code written in Matlab. Structural and Multidisciplinary Optimization; 41(3): 453-464.
- Belytschko T., Parimi C., Moës N., Sukumar N., Usui S., 2003, Structured extended finite element methods for solids defined by implicit surfaces. International Journal for Numerical Methods in Engineering; 56; 609-635.
- Wei P., Wang M Y., Xing X., 2010, A study on X-FEM in continuum structural optimization using a level set model. Computer-Aided Design; 42: 708-719.
- Bourke P., 1994. Polygonising a Scalar Field. http:// paulbourke.net
- Michell A. G. M., 1904. The limits of economy of material in frame-structures; Royal Society Philosophical Magazine; 8: 589-597

#### Paper Citation

#### in Harvard Style

J. Brampton C., H. Kim A. and L. Cunningham J. (2012). **ALGORITHM TO MAINTAIN LINEAR ELEMENT IN 3D LEVEL SET TOPOLOGY OPTIMIZATION** . In *Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 1: SADM, (ICPRAM 2012)* ISBN 978-989-8425-98-0, pages 341-350. DOI: 10.5220/0003860003410350

#### in Bibtex Style

@conference{sadm12,

author={Christopher J. Brampton and Alicia H. Kim and James L. Cunningham},

title={ALGORITHM TO MAINTAIN LINEAR ELEMENT IN 3D LEVEL SET TOPOLOGY OPTIMIZATION},

booktitle={Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 1: SADM, (ICPRAM 2012)},

year={2012},

pages={341-350},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003860003410350},

isbn={978-989-8425-98-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 1: SADM, (ICPRAM 2012)

TI - ALGORITHM TO MAINTAIN LINEAR ELEMENT IN 3D LEVEL SET TOPOLOGY OPTIMIZATION

SN - 978-989-8425-98-0

AU - J. Brampton C.

AU - H. Kim A.

AU - L. Cunningham J.

PY - 2012

SP - 341

EP - 350

DO - 10.5220/0003860003410350