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Isogeometric Analysis for Topology Optimization with a Phase Field Model

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Abstract

We consider a phase field model for the formulation and solution of topology optimization problems in the minimum compliance case. In this model, the optimal topology is obtained as the steady state of the phase transition described by the generalized Cahn–Hilliard equation which naturally embeds the volume constraint on the amount of material available for distribution in the design domain. We reformulate the model as a coupled system and we highlight the dependency of the optimal topologies on dimensionless parameters. We consider Isogeometric Analysis for the spatial approximation which facilitates encapsulating the exactness of the representation of the design domain in the topology optimization and is particularly suitable for the analysis of phase field problems. We demonstrate the validity of the approach and numerical approximation by solving two and three-dimensional topology optimization problems.

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Notes

  1. In the standard definition of topology optimization problems in the minimum compliance case, the area/volume constraint is an inequality one, ∫ Ω χ mat   dΩ≤V mat ; see e.g. [13]. However, typically, the optimal design \(\varOmega _{mat}^{*}\) is such that \({V_{mat}^{*}:=\int_{\varOmega} \chi_{mat}^{*} \, d\varOmega\equiv V_{mat}}\) in order to maximize the stiffness of the structure.

  2. For the pure material-void (0–1) designs of Sect. 2.1, the total variation TV mat is defined as \(\mathit{TV}_{mat} = \mathit{TV}(\chi_{mat}) := \sup_{\mathbf{g} \in[C^{1}_{0}(\varOmega )]^{d}, \ \|\mathbf{g}\|_{L^{\infty} \leq1} } \int_{\varOmega} \chi _{mat} \nabla\cdot\mathbf{g} \, d\varOmega\), where \([C^{1}_{0}(\varOmega )]^{d}\) is the vector space of C 1-continuous functions with compact support over Ω. In this case, TV mat coincides with the perimeter of Ω mat .

  3. Other possibilities are the introduction of an augmented Lagrangian functional or the postprocessing filtering of the numerical solution.

  4. Eventually, the constant E 0 could be included among the parameters \(\boldsymbol{\mu}\in\mathcal{D}\); however, the parametric dependence of the penalized objective functional J(ρ;μ) (31) would still be completely represented by only two parameters by means of suitable scalings.

  5. This choice allows a proper balance between the strain energy function ψ E and the bulk and interface energies ψ B and ψ I through all the continuation levels.

  6. The same criterion for the selection of the optimal topological design can be eventually applied in the multiphase approach of Sect. 2.3. However, in this case, the effectivity of the procedure and the selected optimal topology are largely sensitive to the optimization technique used.

  7. When the norm of the residual associated to the solution of the linear system is below the 10−5 threshold, we progressively increase the tolerance on the relative residual. This situation occurs for “large” Δt, which in turn occurs in proximity of the steady state solution, i.e. when the solution of the problem at the previous time step yields a very small residual in the linear system at the current time step. In this case, a fixed tolerance on the relative residual would be too restrictive and would lead to an unnecessary large number of GMRES iterations.

  8. Divergence issues are revealed in the current numerical setting by increasing values in time of the Liapunov objective functional J or by the recursive selection of excessively small values of the adaptive time step Δt n .

  9. In the framework of Isogeometric analysis, we impose the displacement constraints on the degrees of freedom corresponding to the control points on the Dirichlet boundary segments Γ D .

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Acknowledgements

L. Dedè and T.J.R. Hughes were partially supported by the Office of Naval Research under contract number N00014-08-0992. M.J. Borden and T.J.R. Hughes were partially supported by the Army Research Office under contract number W911NF-10-1-0216. M.J. Borden was partially supported by Sandia National Laboratories; Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

The authors also acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper (URL: http://www.tacc.utexas.edu).

The authors acknowledge Prof. R.B. Haber, University of Illinois at Urbana–Champaign, for fruitful discussions and advice.

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  1. Luca Dedè

    Present address: SB–MATHICSE–CMCS, École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

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  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX, 78712, USA

    Luca Dedè, Micheal J. Borden & Thomas J. R. Hughes

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Dedè, L., Borden, M.J. & Hughes, T.J.R. Isogeometric Analysis for Topology Optimization with a Phase Field Model. Arch Computat Methods Eng 19, 427–465 (2012). https://doi.org/10.1007/s11831-012-9075-z

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