Abstract
In this paper, we present a framework for multiscale topology optimization of fluid flow devices. The objective is to minimize dissipated power, subject to a desired contact area. The proposed strategy is to design optimal microstructures in individual finite element cells, while simultaneously optimizing the overall fluid flow. In particular, parameterized super-shapes are chosen here to represent microstructures since they exhibit a wide range of permeability and contact area. To avoid repeated homogenization, a finite set of these super-shapes are analyzed a priori and a variational auto-encoder (VAE) is trained on their fluid constitutive properties (permeability), contact area, and shape parameters. The resulting differentiable latent space is integrated with a coordinate neural network to carry out a global multiscale fluid flow optimization. The latent space enables the use of new microstructures that were not present in the original dataset. The proposed method is illustrated using numerous examples in 2D.
References
Aage N, Poulsen TH, Gersborg-Hansen A, Sigmund O (2008) Topology optimization of large scale stokes flow problems. Struct Multidiscip Optim 35:175–180
Alexandersen J (2023) A detailed introduction to density-based topology optimisation of fluid flow problems with implementation in matlab. Struct Multidiscip Optim 66:12
Alexandersen J, Andreasen CS (2020) A review of topology optimisation for fluid-based problems. Fluids 5:29
Allaire G, Kohn RV (1993) Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Euro J Mech A. Solids 12:839–878
Allaire G, Bonnetier E, Francfort G, Jouve F (1997) Shape optimization by the homogenization method. Numerische Mathematik 76:27–68
Allaire G, Geoffroy-Donders P, Pantz O (2019) Topology optimization of modulated and oriented periodic microstructures by the homogenization method. Comput Math Appl 78:2197–2229
Andreasen C S. Multiscale topology optimization of solid and fluid structures (DTU Mechanical Engineering, 2011)
Andreassen E, Andreasen CS (2014) How to determine composite material properties using numerical homogenization. Comput Mater Sci 83:488–495
Barr AH (1984) Global and local deformations of solid primitives. ACM Siggraph Comput Graph 18:21–30
Bixler GD, Bhushan B (2012) Bioinspired rice leaf and butterfly wing surface structures combining shark skin and lotus effects. Soft Matter 8:11271–11284
Bixler GD, Bhushan B (2013) Fluid drag reduction and efficient self-cleaning with rice leaf and butterfly wing bioinspired surfaces. Nanoscale 5:7685–7710
Bocanegra Evans H, Gorumlu S, Aksak B, Castillo L, Sheng J (2016) Holographic microscopy and microfluidics platform for measuring wall stress and 3d flow over surfaces textured by micro-pillars. Sci Rep 6:1–12
Bochev P, Lehoucq RB (2005) On the finite element solution of the pure neumann problem. SIAM RevA 47:50–66
Borrvall T, Petersson J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41:77–107
Chan Y-C, Da D, Wang L, Chen W (2022) Remixing functionally graded structures: data-driven topology optimization with multiclass shape blending. Struct Multidiscip Optim 65:135
Chandrasekhar A, Suresh K (2021) Tounn: Topology optimization using neural networks. Struct Multidiscip Optim 63:1135–1149
Chandrasekhar A, Suresh K (2022) Approximate length scale filter in topology optimization using fourier enhanced neural networks. Comput-Aided Design 150:103277
Chandrasekhar A, Sridhara S, Suresh K (2021) Auto: a framework for automatic differentiation in topology optimization. Struct Multidiscip Optim 64:4355–4365
Chandrasekhar A, Sridhara S, Suresh K (2023) Graded multiscale topology optimization using neural networks. Adv Eng Softw 175:103359
Choi J-W et al (2002) An integrated microfluidic biochemical detection system for protein analysis with magnetic bead-based sampling capabilities. Lab Chip 2:27–30
Coelho PG, Fernandes PR, Guedes JM, Rodrigues HC (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidiscip Optim 35:107–115
Dede EM et al (2022) Measurement of low reynolds number flow emanating from a turing pattern microchannel array using a modified bernoulli equation technique. Exp Thermal Fluid Sci 139:110722
Dede EM, Zhou Y, Nomura T (2020) Inverse design of microchannel fluid flow networks using turing pattern dehomogenization. Struct Multidiscip Optim 62:2203–2210
DeSalvo, G J, Swanson J A. ANSYS Engineering Analysis System: User’s Manual (Swanson Analysis Systems, 1979)
Doersch, C (2016) Tutorial on variational autoencoders. ar**v preprintar**v:1606.05908
Du T et al (2020) Functional optimization of fluidic devices with differentiable stokes flow. ACM Trans Graph (TOG) 39:1–15
Fan ZH et al (1999) Dynamic dna hybridization on a chip using paramagnetic beads. Anal Chem 71:4851–4859
Feppon F (2024) Multiscale topology optimization of modulated fluid microchannels based on asymptotic homogenization. Comput Methods Appl Mech Eng 419:116646
Fougerolle YD, Gribok A, Foufou S, Truchetet F, Abidi MA (2005) Boolean operations with implicit and parametric representation of primitives using r-functions. IEEE Trans Vis Comput Graph 11:529–539
Garcke H, Hecht C. in A phase field approach for shape and topology optimization in stokes flow 103–115 (Springer, 2015)
Geng D, Wei C, Liu Y, Zhou M (2022) Concurrent topology optimization of multi-scale cooling channels with inlets and outlets. Struct Multidiscip Optim 65:234
Gersborg-Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30:181–192
Gielis J (2003) A generic geometric transformation that unifies a wide range of natural and abstract shapes. Am J Bot 90:333–338
Gillies S et al (2022). Shapely. https://doi.org/10.5281/zenodo.7428463
Gladstone R J, Nabian M A, Keshavarzzadeh V, Meidani H (2021) Robust topology optimization using variational autoencoders. ar**v preprintar**v:2107.10661
Glorot X, Bengio Y. Understanding the difficulty of training deep feedforward neural networks, 249–256 (JMLR Workshop and Conference Proceedings, 2010)
Groen JP, Sigmund O (2018) Homogenization-based topology optimization for high-resolution manufacturable microstructures. Int J Numer Methods Eng 113:1148–1163
Guest JK, Prévost JH (2006) Topology optimization of cree** fluid flows using a darcy-stokes finite element. Int J Numer Methods Eng 66:461–484
Guo D et al (2013) Multiphysics modeling of a micro-scale stirling refrigeration system. Int J Thermal Sci 74:44–52
Hankins SN, Zhou Y, Lohan DJ, Dede EM (2023) Generative design of large-scale fluid flow structures via steady-state diffusion-based dehomogenization. Sci Rep 13:14344
Haubner J, Neumann F, Ulbrich M (2023) A novel density based approach for topology optimization of stokes flow. SIAM J Sci Comput 45:A338–A368
Hayes MA, Polson NA, Phayre AN, Garcia AA (2001) Flow-based microimmunoassay. Anal Chem 73:5896–5902
Heaton J (2018) Ian goodfellow, yoshua bengio, and aaron courville: Deep learning: The mit press, 2016, 800 pp, isbn: 0262035618. Genetic Programming and Evolvable Machines19, 305–307
Higgins, I (2016) et al.beta-vae: Learning basic visual concepts with a constrained variational framework
Huang X et al (2018) Review on optofluidic microreactors for artificial photosynthesis. Beilstein J Nanotechnol 9:30–41
Jensen KE (2018) Topology optimization of stokes flow on dynamic meshes using simple optimizers. Comput Fluids 174:66–77
Jiang G, Harrison DJ (2000) mrna isolation in a microfluidic device for eventual integration of cdna library construction. Analyst 125:2176–2179
Jung T, Lee J, Nomura T, Dede EM (2022) Inverse design of three-dimensional fiber reinforced composites with spatially-varying fiber size and orientation using multiscale topology optimization. Composite Struct 279:114768
Kingma D P, Welling M (2013) Auto-encoding variational bayes. ar**v preprint ar**v:1312.6114
Kingma D P, Welling M, et al. (2019) An introduction to variational autoencoders. Foundations and Trends® in Machine Learning12, 307–392
Lang P, Paluszny A, Zimmerman R (2014) Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions. J Geophys Res 119:6288–6307
Lauder GV et al (2016) Structure, biomimetics, and fluid dynamics of fish skin surfaces. Phys Rev Fluids 1:060502
Lee J et al (2021) Design of spatially-varying orthotropic infill structures using multiscale topology optimization and explicit de-homogenization. Additive Manuf 40:101920
Li L et al (2014) High surface area optofluidic microreactor for redox mediated photocatalytic water splitting. Int J Hydrogen Energy 39:19270–19276
Li H et al (2022) Topology optimization for lift-drag problems incorporated with distributed unstructured mesh adaptation. Struct Multidiscip Optim 65:222
Li Y et al (2022) Fluidic topology optimization with an anisotropic mixture model. ACM Trans Graphics (TOG) 41:1–14
Liakopoulos AC (1965) Darcy’s coefficient of permeability as symmetric tensor of second rank. Hydrol Sci J 10:41–48
Li D, Dai N, Tang Y, Dong G, Zhao Y F (2019) Design and optimization of graded cellular structures with triply periodic level surface-based topological shapes. Journal of Mechanical Design141
Liu Y-J et al (2007) A micropillar-integrated smart microfluidic device for specific capture and sorting of cells. Electrophoresis 28:4713–4722
Liu J et al (2022) A marker-and-cell method for large-scale flow-based topology optimization on gpu. Struct Multidiscip Optim 65:125
Ma C et al (2022) Compliance minimisation of smoothly varying multiscale structures using asymptotic analysis and machine learning. Comput Methods Appl Mech Eng 395:114861
Maas A L, Hannun A Y, Ng A Y, et al.Rectifier nonlinearities improve neural network acoustic models, Vol. 30, 3 (Atlanta, Georgia, USA, 2013)
Moran M, Wesolek D, Berhane B, Rebello K (2004) Microsystem cooler development, 5611
Nagrath S et al (2007) Isolation of rare circulating tumour cells in cancer patients by microchip technology. Nature 450:1235–1239
Nguyen CHP, Choi Y (2021) Multiscale design of functionally graded cellular structures for additive manufacturing using level-set descriptions. Struct Multidiscip Optim 64:1983–1995
Nomura T et al (2019) Inverse design of structure and fiber orientation by means of topology optimization with tensor field variables. Composites Part B 176:107187
Oliphant T E. et al.Guide to numpy Vol. 1 (Trelgol Publishing USA, 2006)
Padhy R K, Chandrasekhar A, Suresh K (2023) Fluto: Graded multi-scale topology optimization of large contact area fluid-flow devices using neural networks. Engineering with Computers 1–17
Pantz O, Trabelsi K (2008) A post-treatment of the homogenization method for shape optimization. SIAM J Control Opt 47:1380–1398
Paszke A. et al. in Pytorch: An imperative style, high-performance deep learning library (eds Wallach, H. et al.) Advances in Neural Information Processing Systems 32 8024–8035 (Curran Associates, Inc., 2019). http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf
Pereira A, Talischi C, Paulino G H, M Menezes I F, Carvalho MS (2016) Fluid flow topology optimization in polytop: stability and computational implementation. Structural and Multidisciplinary Optimization 54: 1345–1364
Rahaman N, et al.On the spectral bias of neural networks, 5301–5310 (PMLR, 2019)
Rautela M, Senthilnath J, Huber A, Gopalakrishnan S (2022) Towards deep generation of guided wave representations for composite materials. IEEE Transactions on Artificial Intelligence
Rozvany G I. Structural design via optimality criteria: the Prager approach to structural optimization Vol. 8 (Springer Science & Business Media, 2012)
Sanchez-Palencia E (1980) Fluid flow in porous media. Non-homogeneous media and vibration theory 129–157
Schmidhuber J (2015) Deep learning in neural networks: An overview. Neural Networks 61:85–117
Shen C, Hou L, Zhang E, Lin J (2018) Topology optimization of three-phase interpolation models in darcy-stokes flow. Struct Multidiscip Optim 57:1663–1677
Svanberg K (1987) The method of moving asymptotesl, Äîa new method for structural optimization. Int J Numer Methods Eng 24:359–373
Takezawa A, Zhang X, Kato M, Kitamura M (2019) Method to optimize an additively-manufactured functionally-graded lattice structure for effective liquid cooling. Additive Manuf 28:285–298
Takezawa A, Zhang X, Kitamura M (2019) Optimization of an additively manufactured functionally graded lattice structure with liquid cooling considering structural performances. Int J Heat Mass Transf 143:118564
Tancik M et al (2020) Fourier features let networks learn high frequency functions in low dimensional domains. Adv Neural Inform Process Syst 33:7537–7547
Vasilev I, Slater D, Spacagna G, Roelants P, Zocca V. Python Deep Learning: Exploring deep learning techniques and neural network architectures with Pytorch, Keras, and TensorFlow (Packt Publishing Ltd, 2019)
Vianna RS, Cunha AM, Azeredo RB, Leiderman R, Pereira A (2020) Computing effective permeability of porous media with fem and micro-ct: An educational approach. Fluids 5:16
Wang L et al (2020) Deep generative modeling for mechanistic-based learning and design of metamaterial systems. Comput Methods Appl Mech Eng 372:113377
Wang L et al (2022) Data-driven multiscale design of cellular composites with multiclass microstructures for natural frequency maximization. Composite Struct 280:114949
Wang Y, Sun S, Yu B (2013) On full-tensor permeabilities of porous media from numerical solutions of the navier-stokes equation. Adv Mech Eng 5:137086
Wang Y, Xu H, Pasini D (2017) Multiscale isogeometric topology optimization for lattice materials. Comput Methods Appl Mech Eng 316:568–585
Wang L, Tao S, Zhu P, Chen W (2021) Data-driven topology optimization with multiclass microstructures using latent variable gaussian process. Journal of Mechanical Design143
Watts S, Arrighi W, Kudo J, Tortorelli DA, White DA (2019) Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design. Struct Multidiscip Optim 60:1887–1920
White DA, Arrighi WJ, Kudo J, Watts SE (2019) Multiscale topology optimization using neural network surrogate models. Comput Methods Appl Mech Eng 346:1118–1135
Wiker N, Klarbring A, Borrvall T (2007) Topology optimization of regions of darcy and stokes flow. Int J Numer Methods Eng 69:1374–1404
Wright S J (2006) Numerical optimization
Wu T (2019) Topology Optimization of Multiscale Structures Coupling Fluid, Thermal and Mechanical Analysis. Ph.D. thesis, Purdue University Graduate School
Wu J, Sigmund O, Groen JP (2021) Topology optimization of multi-scale structures: a review. Struct Multidiscip Optim 63:1455–1480
**a L, Breitkopf P (2014) Concurrent topology optimization design of material and structure within fe2 nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542
Zhao R, Zhao J, Wang C (2022) Stress-constrained multiscale topology optimization with connectable graded microstructures using the worst-case analysis. Int J Numer Methods Eng 123:1882–1906
Zheng L, Kumar S, Kochmann DM (2021) Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy. Comput Methods Appl Mech Eng 383:113894
Zhou S, Li Q (2008) Design of graded two-phase microstructures for tailored elasticity gradients. J Mater Sci 43:5157–5167
Zhou Y, Lohan DJ, Zhou F, Nomura T, Dede EM (2022) Inverse design of microreactor flow fields through anisotropic porous media optimization and dehomogenization. Chem Eng J 435:134587
Zhou Z, Zhu Y, Guo X (2023) Machine learning based asymptotic homogenization and localization: Predictions of key local behaviors of multiscale configurations bearing microstructural varieties. Int J Numer Methods Eng 124:639–669
Zhu Y et al (2016) Prediction and characterization of dry-out heat flux in micropillar wick structures. Langmuir 32:1920–1927
Acknowledgements
The University of Wisconsin, Madison Graduate School supported this work. The authors extend their thanks to Subodh Subedi for his assistance in the 3D printing process.
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Padhy, R.K., Suresh, K. & Chandrasekhar, A. TOMAS: topology optimization of multiscale fluid flow devices using variational auto-encoders and super-shapes. Struct Multidisc Optim 67, 119 (2024). https://doi.org/10.1007/s00158-024-03835-6
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DOI: https://doi.org/10.1007/s00158-024-03835-6