Source-linked AI summary
Large scale three-dimensional topology optimisation of heat sinks cooled by natural convection
Joe Alexandersen, Ole Sigmund, Niels Aage
TL;DR
The paper addresses physically consistent three-dimensional topology optimisation of heat sinks cooled by natural convection at large computational scale. It solves a fully coupled nonlinear thermofluid model in a parallel finite-element framework and finds that higher Grashof numbers produce compact, complex, multi-branched designs.
Problem
Constant convection coefficients cannot reliably represent changing geometries, cavities, and velocities during topology optimisation, motivating the full conjugate heat-transfer problem.
Method
Density-based topology optimisation solves the fully coupled nonlinear thermofluid model using a PETSc-based parallel framework for large-scale three-dimensional problems.
Results
Optimised structures transition from simple branches conducting heat toward cold boundaries at low Grashof numbers to compact multi-branched structures at higher Grashof numbers.
Takeaways & Limitations
The fully coupled methodology recovers physical effects while avoiding artificial convection assumptions and non-physical internal cavities.
Takeaways & Limitations
The optimisation is nonconvex, so designs are at best local minima and depend on the initial design and continuation strategy.
Abstract
from arXiv · showhide
This work presents the application of density-based topology optimisation to the design of three-dimensional heat sinks cooled by natural convection. The governing equations are the steady-state incompressible Navier-Stokes equations coupled to the thermal convection-diffusion equation through the Bousinessq approximation. The fully coupled non-linear multiphysics system is solved using stabilised trilinear equal-order finite elements in a parallel framework allowing for the optimisation of large scale problems with order of 40-330 million state degrees of freedom. The flow is assumed to be laminar and several optimised designs are presented for Grashof numbers between $10^3$ and $10^6$. Interestingly, it is observed that the number of branches in the optimised design increases with increasing Grashof numbers, which is opposite to two-dimensional optimised designs.
1. Introduction
The paper develops large-scale three-dimensional topology optimisation for natural-convection heat sinks using a fully coupled thermofluid model. Its parallel implementation targets physically consistent designs and computationally demanding nonlinear systems.
- Natural convection passively cools objects because temperature-induced density gradients cause fluid motion.
- Predetermined constant convection coefficients are difficult to justify because optimisation changes geometries, cavities, velocities, and fluid interactions.
- The methodology therefore solves the full conjugate heat-transfer problem to capture convective effects physically.
- The article presents large-scale three-dimensional results and addresses computational issues in nonlinear and linear system solves.
- PETSc and a parallel topology-optimisation framework enable truly large-scale conjugate heat-transfer problems.
2. Governing equations
The governing model unifies fluid and solid regions through coefficient-controlled dimensionless equations. It couples incompressible flow, heat transport, buoyancy, material properties, and heat generation under steady laminar assumptions.
- The equations combine Navier–Stokes and convection–diffusion physics in a unified domain partitioned into fluid and solid subdomains.
- The model assumes constant properties, incompressible steady flow, and neglected viscous dissipation while incorporating buoyancy through the Boussinesq approximation.
- The effective coefficients α(x), K(x), and s(x) represent impermeability, thermal conductivity, and volumetric heat-source distributions.
- The volumetric heat source is active within a predefined subdomain ω contained in the solid domain.
- The Grashof number measures the ratio of buoyancy to viscous forces and distinguishes diffusion-dominated low-Gr flow from convection-dominated high-Gr flow.
3. Optimisation formulation
The optimisation distributes solid and fluid through interpolated material properties while minimising thermal compliance under a volume constraint. Filtering, MMA, and continuation stabilise the nonconvex design process.
- 3.1. Interpolation functions: The continuous design field γ(x) represents pure fluid at γ(x)=1, solid at γ(x)=0, and interpolated material properties between these endpoints.
- 3.1. Interpolation functions: Effective conductivity and impermeability are interpolated to satisfy endpoint behavior and control intermediate design values.
- 3.2. Optimisation problem: A PDE-based density filter regularises the design field, and the method of moving asymptotes solves the optimisation problem.
- 3.3. Continuation scheme: Continuation varies selected parameters to stabilise optimisation and improve results, using five prescribed steps.
- 3.3. Continuation scheme: The continuation sequence increases qf to penalise intermediate conductivity values and initially keeps maximum effective impermeability α relatively low for stability.
- 3.3. Continuation scheme: α increases by two orders of magnitude in the final steps to further reduce velocities in solid regions.
- 3.3. Continuation scheme: Because the problem is nonconvex, designs are at best local minima and depend on the initial design and continuation strategy.
4. Finite element formulation
The formulation uses stabilised trilinear hexahedral finite elements with elementwise constant design variables. Its monolithic discretisation preserves temperature and flux continuity across fluid–solid interfaces.
- The governing equations use stabilised trilinear hexahedral finite elements, while the design field is represented by elementwise constant variables.
- The monolithic discretisation ensures continuity of temperature and fluxes across fluid–solid interfaces.
5. Numerical implementation
The implementation solves the coupled nonlinear systems arising in large-scale three-dimensional topology optimisation using PETSc-based parallel solvers. The nonlinear state problem uses damped Newton iterations, while the costly linearised systems use a parallel Krylov method.
- Solver framework: PETSc provides the implementation framework for the discretised finite-element equations, including parallel, nonlinear, and linear solver components.The framework also supports preconditioners and structured mesh handling.
- Nonlinear solution: The nonlinear system is solved with a damped Newton method whose damping coefficient is selected by fitting residual norms along the Newton step.The residual is evaluated at the current point, halfway through the step, and at the full step.
- Linear solution: The linearised systems dominate computational cost because the problems are both large scale and three-dimensional.These systems arise within the Newton scheme.
- Linear solution: An unsymmetric linear system is solved with a fully parallelised iterative Krylov subspace solver to make large-scale problems tractable.The solver is designed for reduced wall-clock time and simplicity using PETSc core components.
6. Results
The study evaluates a heat sink in a closed cubic cavity across Grashof numbers and mesh scales, assessing solver performance and the resulting three-dimensional designs. Increasing Grashof number shifts designs from long conductive branches toward shorter, more complex structures with greater vertical interfaces, while the solver remains applicable at large scale.
- 6.1. Problem setup: The problem places a volumetrically heated solid source beneath a design domain in a closed cubic cavity, with cooling fluid passing beneath the domain.The source represents an electronics chip, and the design domain occupies 5% volume fraction in the quarter-symmetric computational domain.
- 6.2. Parallel performance: The parallel-performance study averages results over 250 design cycles for Gr = 10^3 and Gr = 10^6, while mesh-resolution tests collect average F-GMRES iterations.The mesh study uses 250, 500, and 1000 design iterations for its three resolutions.
- 6.2. Parallel performance: Increasing problem size increases computational complexity, but the moderate growth in F-GMRES iterations supports applying the solver to large-scale natural-convection problems.The study explicitly concludes that the solver is not numerically scalable, while still being applicable to large-scale cases.
- 6.3. Varying Grashof number: For Gr = 10^3 to Gr = 10^6, optimised heat sinks change from long conducting branches toward shorter branches, higher vertical interfaces, and increased design complexity.The shift reflects a transition from diffusion-dominated heat transfer toward convection-dominated transfer.
- 6.3. Varying Grashof number: In three dimensions, increasing Grashof number reverses the two-dimensional trend: additional branches can improve heat transfer by increasing vertical surface area.At Gr = 10^6, branches are positioned to keep the structure open from below and form vertical walls.
- 6.3. Varying Grashof number: Designs optimised for particular flow conditions outperform the other designs at their specified Grashof numbers.This comparison is reported as a crosscheck of the objective-function values.
7. Discussion and conclusion
The paper demonstrates large-scale three-dimensional topology optimisation for natural-convection heat sinks and identifies how optimised structures change with Grashof number. The study also defines scope boundaries and computational extensions for future work.
- 7. Discussion and conclusion: More than 300 million degrees of freedom and almost 30 million design variables are handled on regular grids using a PETSc-based parallel implementation.The methodology uses a fully coupled non-linear thermofluidic model for three-dimensional heat-sink optimisation.
- 7. Discussion and conclusion: Higher Grashof numbers produce complex, compact, multi-branched structures that maximise convection heat transfer, whereas diffusion-dominated cases use simpler branches directed toward cold outer boundaries.The designs transition from conductive branches toward complex structures with increased convection-oriented surface area.
- 7. Discussion and conclusion: The example is primarily academic, while future work targets real-life applications, irregular meshes, multiple orientations, transient problems, and boundary-layer modelling accuracy.These directions mark the current study’s practical and modelling scope.
- 7. Discussion and conclusion: The paper reports both computational and actual Grashof numbers for the presented designs.The distinction is documented in Appendix A rather than treated as a single reported Grashof-number measure.
Appendix A. Computational versus actual Grashof number
The appendix distinguishes an imposed computational Grashof number from an actual Grashof number calculated after optimisation using the realised temperature difference.
- Appendix A. Computational versus actual Grashof number: The computational Grashof number uses an a priori reference temperature difference, whereas the actual Grashof number uses the computed temperature difference between the heat source and walls.Only one temperature is prescribed directly, while the heat source is volumetric, so the realised difference is not known beforehand.
- Appendix A. Computational versus actual Grashof number: The actual Grashof number can support experimental studies and future comparisons, and fine-mesh designs have lower values because smaller length scales permit lower thermal compliance.The appendix lists actual Grashof numbers for the designs presented.
Appendix B. Stabilisation parameters
The appendix describes stabilisation parameters and their derivatives for the finite-element formulation, enabling consistent Jacobians, adjoints, and design sensitivities.
- Appendix B. Stabilisation parameters: PSPG and SUPG stabilisation are used for the weak form, with both parameters defined through the same approximate min-function.The thermal SUPG stabilisation receives analogous definitions and derivations.
- Appendix B. Stabilisation parameters: The limit factors use characteristic element size and element velocity degrees of freedom, with the first factor simplified using centroid evaluation and one Gauss point.This produces a constant stabilisation factor within each element for the first limit factor.
- Appendix B. Stabilisation parameters: Derivatives of the stabilisation factors with respect to velocity and design fields are included to form consistent Jacobians, adjoints, and design sensitivities.The derivatives are required separately for state-field and design-field consistency.
- Appendix B. Stabilisation parameters: One to two orders of magnitude greater sensitivity accuracy than finite differences has been observed when these derivatives are included.Despite improved sensitivity accuracy, significant differences were not observed in optimisation behaviour or final designs, and general guarantees are absent.