A “poor man’s” approach for high-resolution three-dimensional topology design for natural convection problems

Abstract

This paper treats topology optimization of natural convection problems. A simplified model is suggested to describe the flow of an incompressible fluid in steady state conditions, similar to Darcy's law for fluid flow in porous media. The equations for the fluid flow are coupled to the thermal convection-diffusion equation through the Boussinesq approximation. The coupled non-linear system of equations is discretized with stabilized finite elements and solved in a parallel framework that allows for the optimization of high resolution three-dimensional problems. A density-based topology optimization approach is used, where a two-material interpolation scheme is applied to both the permeability and conductivity of the distributed material. Due to the simplified model, the proposed methodology allows for a significant reduction of the computational effort required in the optimization. At the same time, it is significantly more accurate than even simpler models that rely on convection boundary conditions based on Newton's law of cooling. The methodology discussed herein is applied to the optimization-based design of three-dimensional heat sinks. The final designs are formally compared with results of previous work obtained from solving the full set of Navier-Stokes equations. The results are compared in terms of performance of the optimized designs and computational cost. The computational time is shown to be decreased to around 5−20% in terms of core-hours, allowing for the possibility of generating an optimized design during the workday on a small computational cluster and overnight on a high-end desktop.

A Review of Topology Optimisation for Fluid-Based Problems

“…Ramalingom et al [159] applied their previous method to multi-objective optimisation of mixed and natural convection in a asymmetrically-heated vertical channel. Pollini et al [160] extended the work of Asmussen et al [157] to large-scale three-dimensional problems, producing results comparable to those of Alexandersen et al [151] with a computational time reduction of 80-95% in terms of core-hours.…”

“…For species transport, there are four papers [93,98,105,107]. For conjugate heat transfer, there are eight in forced convection [110,117,119,122,126,129,132,135] and six in natural convection [150,151,[153][154][155]160]. The fluid-structure interaction category counts four papers [164,168,172,173], but it must be noted that, common for all, the three-dimensional design freedom is severely limited, as the design domain is restricted in the third dimension.…”

“…It can be seen that they follow the same trend, which is reflected in the fact that the percentage of three-dimensional papers has been close to constant since 2010 at approximately 30%. In January 2020, there has been a single three-dimensional paper [160] out of 5, making it 20% so far.…”

“…While 31% of papers treat three-dimensional problems, many of these suffer from either: being very small in the third dimension [107,122]; having severely restricted design freedom in the third dimension [164,168,172,173]; and using very coarse discretisations [14,23,46,56,132,150]. Truly three-dimensional problems inherently carry a large computational cost with them, and this is discussed in a number of papers, where high performance parallel computing [15,25,33,55,76,82,126,129,151,153,160], graphics processing units [119,173] and adaptive meshes [61] have been proposed as solutions.…”

“…Another recent example uses laminar flow to approximate turbulent flow in a multifidelity approximation framework [148]. For natural convection, a potential-like model is derived by reducing the Navier-Stokes equations by assuming the buoyancy term to be dominant [157,160].…”