A parallel Schur complement solver for the solution of the adjoint steady-state lattice Boltzmann equations: application to design optimisation

Abstract

We study the numerical performance of a parallel Schur complement method towards the solution of the steady-state lattice Boltzmann equations on large meshes as needed for parametric design optimisation problems, in particular topology optimisation. For the gradient-based topology optimisation framework a porosity model is used to continuously transition from fluid to solid. Deriving the sensitivity equations for the lattice Boltzmann method (LBM), we identify the Jacobian of the LBM fixed-point formulation, which poses the key computational challenge in LBM topology optimisation due to its large size. We show that the Schur complement method can be used to decompose the fixed-point Jacobian, making the solution of the LBM sensitivity equations more memory efficient, in particular for 2D problems. The effectiveness of the overall lattice Boltzmann based topology optimisation framework is illustrated with the optimal design of a micro valve, posted by a dual-objective optimisation problem.

Keywords:

• lattice Boltzmann method
• design optimisation
• topology optimisation
• sensitivity analysis
• Schur complement method
• fixed-geometry valve
• Tesla valve

Acknowledgements

The authors acknowledge the support of the National Science Foundation under Grant No. DMI-0348759 and the additional support of the National Science Foundation in providing Travel Support to the ICMMES conference. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organisation.

Notes

^1. A 28 node Linux cluster with 4 × 2.0 Ghz Opteron processors and 8GB memory per node.

^2. The image was rotated for comparison purposes.

^3. SIMP = solid isotropic material with penalisation.