Abstract
Uniformity of deposited thickness in electroplating processes is vital to the realization of desirable surface qualities in many products. The thickness distribution of deposits is affected by numerous factors, such as the arrangement and shapes of auxiliary cathodes, anodes, and shields as well as the detailed configuration of the electroplating process. Deposit thickness reflects the amount of ions transported from anodes to cathodes, particularly to the object being plated, although auxiliary cathodes are sometimes placed to prevent excess plating in certain areas of the product, as are shields that impede current flow. This study presents a topology optimization method for achieving uniform deposition thickness, applied to the design of anodes placed in an electroplating bath. The proposed method is based on level set–based topology optimization and FEM is used to analyze the electrochemical field. The shapes and arrangement of anodes are expressed with respect to ion sources using level set functions. The uniformity of the current density on a cathode is employed as an objective function, since current density is nearly proportional to the thickness of the resulting electroplating. To stabilize the optimization process, the Kreisselmeier–Steinhauser function is used. The magnitude of the current density on the cathode is set as a constraint so that it does not fall below a certain value, to avoid lengthy plating times that would occur if the current density were too low. Numerical examples are presented to confirm the utility of the proposed method and the results demonstrate that the proposed method can obtain appropriate shapes and arrangements of anodes.
Similar content being viewed by others
References
Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393. https://doi.org/10.1016/j.jcp.2003.09.032. http://www.sciencedirect.com/science/article/pii/S002199910300487X
Aoki S, Amaya K (1997) Optimization of cathodic protection system by BEM. Eng Anal Bound Elem 19(2):147–156. https://doi.org/10.1016/S0955-7997(97)00019-2. http://www.sciencedirect.com/science/article/pii/S0955799797000192. Optimization and Sensitivity Analysis
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2. http://www.sciencedirect.com/science/article/pii/0045782588900862
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654. https://doi.org/10.1007/s004190050248
Blais M, Désilets M, Lacroix M (2013) Optimization of the cathode block shape of an aluminum electrolysis cell. Appl Thermal Eng 58(1):439–446. https://doi.org/10.1016/j.applthermaleng.2013.04.040. http://www.sciencedirect.com/science/article/pii/S1359431113003116
Bruns T (2007) Topology optimization of convection-dominated, steady-state heat transfer problems. Int J Heat Mass Transfer 50(15):2859–2873. https://doi.org/10.1016/j.ijheatmasstransfer.2007.01.039. http://www.sciencedirect.com/science/article/pii/S0017931007001330
Diaz AR, Kikuchi N (1992) Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Methods Eng 35(7):1487–1502. https://doi.org/10.1002/nme.1620350707
Hestenes MR (1969) Multiplier and gradient methods. J Optim Theory Appl 4(5):303–320. https://doi.org/10.1007/BF00927673
Ho Yoon G, Young Kim Y (2005) The element connectivity parameterization formulation for the topology design optimization of multiphysics systems. Int J Numer Methods Eng 64(12):1649–1677. https://doi.org/10.1002/nme.1422
Iga A, Nishiwaki S, Izui K, Yoshimura M (2009) Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection. Int J Heat Mass Transfer 52:2721–2732. https://doi.org/10.1016/j.ijheatmasstransfer.2008.12.013
Igawa S (1971) Cathodic polarization characteristics of chromium plating bath. J Metal Finish Soc Jpn 22 (1):17–21. https://doi.org/10.4139/sfj1950.22.17
Isakari H, Kuriyama K, Harada S, Yamada T, Takahashi T, Matsumoto T (2014) A topology optimisation for three-dimensional acoustics with the level set method and the fast multipole boundary element method. Mech Eng J 1(4):CM0039–CM0039. https://doi.org/10.1299/mej.2014cm0039. http://ci.nii.ac.jp/naid/130004684523/en/
Ishizuka N, Noguchi Y, Yamada T, Izui K, Nishiwaki S (2017) Topology optimization for unification deposit thickness on electroplating process. Transactions of the JSME (in Japanese) advpub. https://doi.org/10.1299/transjsme.17-00185
John SV, Vasudevan T (1999) Improving the deposit distribution during electroforming of complicated shapes. Bull Electrochem 15(5-6):202–204
Joo Oh Y, Hyo Chung S, seung Lee M (2004) Optimization of thickness uniformity in electrodeposition onto a patterned substrate. Mater Trans 45(10):3005–3010. https://doi.org/10.2320/matertrans.45.3005
Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. International Federation of Active Controls Syposium on Computer-Aided Design of Control Systems, Zurich, Switzerland
Newman J, Thomas-Alyea KE (2004) Electrochemical Systems, 3rd Edition, Wiley-Interscience
Ohara K (1999) Computer simulation of current distribution in electrolytic cells. J Surf Finish Soc Jpn 50 (5):416–420. https://doi.org/10.4139/sfj.50.416
Poroch-Seritan M, Gutt S, Gutt G, Cretescu I, Cojocaru C, Severin T (2011) Design of experiments for statistical modeling and multi-response optimization of nickel electroplating process. Chem Eng Res Des 89(2):136–147. https://doi.org/10.1016/j.cherd.2010.05.010. http://www.sciencedirect.com/science/article/pii/S0263876210001772
Powell M (1969) A method for nonlinear constraints in minimization problems. Academic Press, pp 283–298
Rockafellar RT (1973) The multiplier method of Hestenes and Powell applied to convex programming. J Optim Theory Appl 12(6):555–562. https://doi.org/10.1007/BF00934777
Song X, Diaz ARA, Nicholas JD (2013) A 2D model for shape optimization of solid oxide fuel cell cathodes. Struct Multidiscip Optim 47(3):453–464. https://doi.org/10.1007/s00158-012-0837-x
Watanabe K (1999) Chromium plating. J Surf Finish Soc Jpn 50(2):149–154. https://doi.org/10.4139/sfj.50.149
Yaji K, Yamada T, Yoshino M, Matsumoto T, Izui K, Nishiwaki S (2014) Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions. J Comput Phys 274(Supplement C):158–181. https://doi.org/10.1016/j.jcp.2014.06.004. http://www.sciencedirect.com/science/article/pii/S0021999114004112
Yamada T, Nishiwaki S, Izui K, Yoshimura M, Takezawa A (2009) A structural optimization method incorporating level set boundary expressions based on the concept of the phase field method. Trans Jpn Soc Mech Eng Ser A 75(753):550–558. https://doi.org/10.1299/kikaia.75.550. http://ci.nii.ac.jp/naid/110007226514/en/
Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199(45–48):2876–2891. https://doi.org/10.1016/j.cma.2010.05.013. http://www.sciencedirect.com/science/article/pii/S0045782510001623
Yamasaki S, Nomura T, Kawamoto A, Sato K, Nishiwaki S (2011) A level set-based topology optimization method targeting metallic waveguide design problems. Int J Numer Methods Eng 87(9):844–868. https://doi.org/10.1002/nme.3135
Yang JM, Kim DH, Zhu D, Wang K (2008) Improvement of deposition uniformity in alloy electroforming for revolving parts. Int J Mach Tools Manuf 48(3-4):329–337. https://doi.org/10.1016/j.ijmachtools.2007.10.006
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Responsible Editor: Seonho Cho
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Replication of results
The codes used to generate the results are company property and cannot be shared.
Rights and permissions
About this article
Cite this article
Ishizuka, N., Yamada, T., Izui, K. et al. Topology optimization for unifying deposit thickness in electroplating process. Struct Multidisc Optim 62, 1767–1785 (2020). https://doi.org/10.1007/s00158-020-02574-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-020-02574-8