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[1] Novotny, A., Feijoo, R., Taroco, E. & Padra, C., Topological sensitivity analysis for three-dimensional linear elasticity problem. Computational Methods in Applied Mechanics and Engineering, 196, pp. 4354–4364, 2007. [Crossref]
[2] Barbarosie, C. & Toader, A.M., Shape and topology optimization for periodic problems. part i: The shape and the topological derivative. Structural and Multidisciplinary Optimization, 40, pp. 381–391, 2010. [Crossref]
[3] Hassan, E., Topology optimization of metallic antennas. IEEE Transactions on Antennas and Propagation, 62(5), pp. 2488–2500, 2014. [Crossref]
[4] Andreasen, C.S. & Sigmund, O., Topology optimization of fluid-structure-interaction problems in poroelasticity. Computer Methods in Applied Mechanics and Engineering, 258, pp. 55–62, 2013. [Crossref]
[5] Alexandersen, J., Aage, N., Andreasen, C.S. & Sigmund, O., Topology optimisation for natural convection problems. International Journal for Numerical Methods in Fluids, 76, pp. 699–721, 2014. [Crossref]
[6] Melchels, F.P.W., Feijen, J. & Grijpma, D.W., A review on stereolithogra-phy and its applications in biomedical engineering. Biomaterials, 31(24), pp. 6121–6130, 2010. [Crossref]
[7] Gross, B., Erkal, J., Lockwood, S., Chen, C. & Spence, D., An evaluation of 3d printing and its potential impact on biotechnology and the chemical sciences. Analytical Chemistry, 86(7), pp. 3240–3253, 2014. [Crossref]
[8] Azegami, H., Shimoda, M., Katamine, E. & Wu, Z., A domain optimization technique for elliptic boundary value problems. Computer Aided Optimization Design of Structures IV, Structural Optimization, eds S. Hernandez, M. El-Sayed & C. Brebbia, Computational Mechanics Publications: Southampton, 1995.
[9] Allaire, G., Jouve, F. & Toader, A.M., Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 194(1), pp. 363–393, 2004. [Crossref]
[10] Allaire, G., Bonnetier, E., Francfort, G. & Jouve, F., Shape optimization by the homogenization method. Numerische Mathematik, pp. 27–68, 1997. [Crossref]
[11] Aage, N., Andreassen, E. & Lazarov, B.S., Topology optimization using petsc: An easyto-use, fully parallel, open source topology optimization framework. Structural and Multidisciplinary Optimization, 51, pp. 565–572, 2014. [Crossref]
[12] Marczak, R., Optimization of elastic structures using boundary element and a topological- shape sensitivity formulation. Mechanics of Solids in Brazil, Brasilian Society of Mechanical Sciences and Engineering, pp. 279–293, 2007.
[13] Bertsch, C., Cisilino, A., Langer, S. & Reese, S., Topology optimization of 3d elastic structures using boundary elements. Proceeding of Applied Mathematics and Mechanics, 8, pp. 10771–10772, 2008. [Crossref]
[14] Nemitz, N. & Bonnet, M., Topological sensitivity and fmm-accelerated bem applied to 3d acoustic inverse scattering. Engineering Analysis with Boundary Elements, 32, pp. 957–970, 2008. [Crossref]
[15] Ostanin, I., Zorin, D. & Oseledets, I., Toward fast topological-shape optimization with boundary elements. preprint arXiv, (1503.02383), pp. 1–5, 2015.
[16] Ostanin, I., Mikhalev, A., Zorin, D. & Oseledets, I., Engineering optimization with the fast boundary element method. WIT Transactions on Modelling and Simulation, 61, pp. 175–181, 2015. [Crossref]
[17] Ying, L., Biros, G. & Zorin, D., A kernel-independent adaptive fast multipole algorithm in two and three dimensions. Journal of Computational Physics, 196, pp. 591–626, 2004. [Crossref]
[18] Malhotra D. & Biros, G., Pvfmm: A parallel kernel independent fmm for particle and volume potentials. Communications in Computational Physics, 18(3), pp. 808–830, 2015. [Crossref]
[19] Cruse, T.A., Numerical solutions in three-dimensional elastostatics. International Journal of Solids and Structures, 5, pp. 1259–1274, 1969. [Crossref]
[20] Saad, Y. & Schultz, M.H., Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3), pp. 856–869, 1986. [Crossref]
[21] Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H. & Zhang, H., PETSc users manual. Technical Report ANL-95/11 - Revision 3.7, Argonne National Laboratory, 2016.
[22] Balay, S., Gropp, W.D., McInnes, L.C. & Smith, B.F., Efficient management of parallelism in object oriented numerical software libraries. Modern Software Tools in Scientific Computing, eds E. Arge, A.M. Bruaset & H.P. Lang-tangen, Birkhauser Press, pp. 163–202, 1997.
[23] Bendsoe, M.P. & Sigmund, O., Topology Optimization: Theory, Methods and Applications, Springer Science & Business Media, 2013.
[24] Herrmann, L.R., Laplacian-isoparametric grid generation scheme. Journal of the Engineering Mechanics Division, 102(5), pp. 749–907, 1976.
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Open Access
Research article

Parallel Optimization with Boundary Elements and Kernel Independent Fast Multipole Method

igor ostanin,
denis zorin,
ivan oseledets
Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, 143026 Moscow, Russia
International Journal of Computational Methods and Experimental Measurements
|
Volume 5, Issue 2, 2017
|
Pages 154-162
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
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Abstract:

We propose a new framework for topology optimization based on the boundary element discretization and kernel-independent fast multipole method (KIFMM). The boundary value problem for the considered partial differential equation is reformulated as a surface integral equation and is solved on the domain boundary. Volume solution at selected points is found via surface integrals. At every iteration of the optimization process, the new boundary is extracted as a level set of a topological derivative. Both surface and volume solutions are accelerated using KIFMM. The obtained technique is highly universal, fully parallelized, it allows achieving asymptotically the best performance with the optimization iteration complexity proportional to a number of surface discretization elements. More-over, our approach is free of the artifacts that are inherent for finite element optimization techniques, such as “checkerboard” instability. The performance of the approach is showcased on few illustrative examples.

Keywords: Kernel-independent fast multi-pole method, Topological-shape optimization

1. Introduction

2. Method

3. Numerical Results

4. Conclusions and Future Work

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Acknowledgments

Authors gratefully acknowledge the financial support from Russian National Foundation under the grant 15-11-00033. I.O acknowledges the financial support from the Russian Foundation of Basic Research under grant 16-31-60100.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References
[1] Novotny, A., Feijoo, R., Taroco, E. & Padra, C., Topological sensitivity analysis for three-dimensional linear elasticity problem. Computational Methods in Applied Mechanics and Engineering, 196, pp. 4354–4364, 2007. [Crossref]
[2] Barbarosie, C. & Toader, A.M., Shape and topology optimization for periodic problems. part i: The shape and the topological derivative. Structural and Multidisciplinary Optimization, 40, pp. 381–391, 2010. [Crossref]
[3] Hassan, E., Topology optimization of metallic antennas. IEEE Transactions on Antennas and Propagation, 62(5), pp. 2488–2500, 2014. [Crossref]
[4] Andreasen, C.S. & Sigmund, O., Topology optimization of fluid-structure-interaction problems in poroelasticity. Computer Methods in Applied Mechanics and Engineering, 258, pp. 55–62, 2013. [Crossref]
[5] Alexandersen, J., Aage, N., Andreasen, C.S. & Sigmund, O., Topology optimisation for natural convection problems. International Journal for Numerical Methods in Fluids, 76, pp. 699–721, 2014. [Crossref]
[6] Melchels, F.P.W., Feijen, J. & Grijpma, D.W., A review on stereolithogra-phy and its applications in biomedical engineering. Biomaterials, 31(24), pp. 6121–6130, 2010. [Crossref]
[7] Gross, B., Erkal, J., Lockwood, S., Chen, C. & Spence, D., An evaluation of 3d printing and its potential impact on biotechnology and the chemical sciences. Analytical Chemistry, 86(7), pp. 3240–3253, 2014. [Crossref]
[8] Azegami, H., Shimoda, M., Katamine, E. & Wu, Z., A domain optimization technique for elliptic boundary value problems. Computer Aided Optimization Design of Structures IV, Structural Optimization, eds S. Hernandez, M. El-Sayed & C. Brebbia, Computational Mechanics Publications: Southampton, 1995.
[9] Allaire, G., Jouve, F. & Toader, A.M., Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 194(1), pp. 363–393, 2004. [Crossref]
[10] Allaire, G., Bonnetier, E., Francfort, G. & Jouve, F., Shape optimization by the homogenization method. Numerische Mathematik, pp. 27–68, 1997. [Crossref]
[11] Aage, N., Andreassen, E. & Lazarov, B.S., Topology optimization using petsc: An easyto-use, fully parallel, open source topology optimization framework. Structural and Multidisciplinary Optimization, 51, pp. 565–572, 2014. [Crossref]
[12] Marczak, R., Optimization of elastic structures using boundary element and a topological- shape sensitivity formulation. Mechanics of Solids in Brazil, Brasilian Society of Mechanical Sciences and Engineering, pp. 279–293, 2007.
[13] Bertsch, C., Cisilino, A., Langer, S. & Reese, S., Topology optimization of 3d elastic structures using boundary elements. Proceeding of Applied Mathematics and Mechanics, 8, pp. 10771–10772, 2008. [Crossref]
[14] Nemitz, N. & Bonnet, M., Topological sensitivity and fmm-accelerated bem applied to 3d acoustic inverse scattering. Engineering Analysis with Boundary Elements, 32, pp. 957–970, 2008. [Crossref]
[15] Ostanin, I., Zorin, D. & Oseledets, I., Toward fast topological-shape optimization with boundary elements. preprint arXiv, (1503.02383), pp. 1–5, 2015.
[16] Ostanin, I., Mikhalev, A., Zorin, D. & Oseledets, I., Engineering optimization with the fast boundary element method. WIT Transactions on Modelling and Simulation, 61, pp. 175–181, 2015. [Crossref]
[17] Ying, L., Biros, G. & Zorin, D., A kernel-independent adaptive fast multipole algorithm in two and three dimensions. Journal of Computational Physics, 196, pp. 591–626, 2004. [Crossref]
[18] Malhotra D. & Biros, G., Pvfmm: A parallel kernel independent fmm for particle and volume potentials. Communications in Computational Physics, 18(3), pp. 808–830, 2015. [Crossref]
[19] Cruse, T.A., Numerical solutions in three-dimensional elastostatics. International Journal of Solids and Structures, 5, pp. 1259–1274, 1969. [Crossref]
[20] Saad, Y. & Schultz, M.H., Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3), pp. 856–869, 1986. [Crossref]
[21] Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H. & Zhang, H., PETSc users manual. Technical Report ANL-95/11 - Revision 3.7, Argonne National Laboratory, 2016.
[22] Balay, S., Gropp, W.D., McInnes, L.C. & Smith, B.F., Efficient management of parallelism in object oriented numerical software libraries. Modern Software Tools in Scientific Computing, eds E. Arge, A.M. Bruaset & H.P. Lang-tangen, Birkhauser Press, pp. 163–202, 1997.
[23] Bendsoe, M.P. & Sigmund, O., Topology Optimization: Theory, Methods and Applications, Springer Science & Business Media, 2013.
[24] Herrmann, L.R., Laplacian-isoparametric grid generation scheme. Journal of the Engineering Mechanics Division, 102(5), pp. 749–907, 1976.
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Ostanin, I., Zorin, D., & Oseledets, I. (2017). Parallel Optimization with Boundary Elements and Kernel Independent Fast Multipole Method. Int. J. Comput. Methods Exp. Meas., 5(2), 154-162. https://doi.org/10.2495/CMEM-V5-N2-154-162
I. Ostanin, D. Zorin, and I. Oseledets, "Parallel Optimization with Boundary Elements and Kernel Independent Fast Multipole Method," Int. J. Comput. Methods Exp. Meas., vol. 5, no. 2, pp. 154-162, 2017. https://doi.org/10.2495/CMEM-V5-N2-154-162
@research-article{Ostanin2017ParallelOW,
title={Parallel Optimization with Boundary Elements and Kernel Independent Fast Multipole Method},
author={Igor Ostanin and Denis Zorin and Ivan Oseledets},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2017},
page={154-162},
doi={https://doi.org/10.2495/CMEM-V5-N2-154-162}
}
Igor Ostanin, et al. "Parallel Optimization with Boundary Elements and Kernel Independent Fast Multipole Method." International Journal of Computational Methods and Experimental Measurements, v 5, pp 154-162. doi: https://doi.org/10.2495/CMEM-V5-N2-154-162
Igor Ostanin, Denis Zorin and Ivan Oseledets. "Parallel Optimization with Boundary Elements and Kernel Independent Fast Multipole Method." International Journal of Computational Methods and Experimental Measurements, 5, (2017): 154-162. doi: https://doi.org/10.2495/CMEM-V5-N2-154-162
OSTANIN I， ZORIN D， OSELEDETS I. Parallel Optimization with Boundary Elements and Kernel Independent Fast Multipole Method[J]. International Journal of Computational Methods and Experimental Measurements， 2017， 5(2)： 154-162. https://doi.org/10.2495/CMEM-V5-N2-154-162