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This article presents a detailed introduction to density-based topology optimisation of fluid flow problems. The goal is to allow new students and researchers to quickly get started in the research area and to skip many of the initial steps, often consuming unnecessarily long time from the scientific advancement of the field. This is achieved by providing a step-by-step guide to the components necessary to understand and implement the theory, as well as extending the supplied MATLAB code. The continuous design representation used and how it is connected to the Brinkman penalty approach, for simulating an immersed solid in a fluid domain, are illustrated. The different interpretations of the Brinkman penalty term and how to chose the penalty parameters are explained. The accuracy of the Brinkman penalty approach is analysed through parametric simulations of a reference geometry. The chosen finite element formulation and the solution method are explained. The minimum dissipated energy optimisation problem is defined and how to solve it using an optimality criteria solver and a continuation scheme is discussed. The included MATLAB implementation is documented, with details on the mesh, pre-processing, optimisation and post-processing. The code has two benchmark examples implemented and the application of the code to these is reviewed. Subsequently, several modifications to the code for more complicated examples are presented through provided code modifications and explanations. Lastly, the computational performance of the code is examined through studies of the computational time and memory usage, along with recommendations to decrease computational time through approximations.

This paper demonstrates that there is much more to gain from topology optimization of heat sinks than what is described by the so-called pseudo 3D models. The utilization of 3D effects, even for microchannel heat sinks is investigated and compared to state-of-the art industrial designs, for a microelectronic application. Furthermore, the use of design restrictions in the optimization framework demonstrates that the performances of microchannel heat sinks are highly dependent on the ability to provide complex refrigerant distribution and intricate flow paths through the heat sink. The topology optimized microchannel heat sinks are exported from a voxel mesh to bodyfitted mesh using Trelis Sculpt and imported into a commercial CFD software. A systematic comparison with the state-of-the art industrial design shows that the temperature elevation of the microelectronic chip can be reduced by up to 70%, using a 3D topology optimized microchannel heat sink. Restricting the design freedom, for example, by limiting the solid features to be unidirectional downgrades the performances of the optimized microchannel heat sinks but still outperforms the reference case, for a similar design complexity.

This paper proposes a density-based topology optimization method for natural convection problems using the lattice Boltzmann method (LBM). As the LBM can be developed as a completely explicit scheme, its attractive features over the traditional ones, such as the finite element method, are (1) suitability for solving unsteady flow problems and (2) scalability for large-scale parallel computing. We develop an LBM code for solving unsteady natural convection problems and provide its sensitivity analysis based on the so-called adjoint lattice Boltzmann method. Notably, the adjoint equation is derived from the discrete particle velocity Boltzmann equation and can be solved similarly to the original LBM concerning unsteady natural convection problems. We first show that the proposed method can produce similar results to the previous work in a steady-state natural convection problem. We then demonstrate the efficacy of the proposed method through 2D numerical examples concerning unsteady natural convection. As a large-scale problem, we tackle a 3D unsteady natural convection problem on a parallel supercomputer. All the developed codes written in C++ are available at https://github.com/PANFACTORY/PANSLBM2.git .

Modeling of fluid flows in density-based topology optimization forms a longstanding challenge. Methods based on the Navier–Stokes equations with Darcy penalization (NSDP equations) are widely used in fluid topology optimization. These methods use porous materials with low permeability to represent the solid domain. Consequently, they suffer from flow leakage in certain areas. In this work, the governing equations for solid/fluid density-based topology optimization are reevaluated and reinterpreted. The governing equations are constructed using the volume averaged Navier–Stokes (VANS) equations, well known in the field of porous flow modeling. Subsequently, we simplify, interpret and discretize the VANS equations in the context of solid/fluid topology optimization, and analytically derive lower bounds on the Darcy penalization to sufficiently prevent flow leakage. Based on both the NSDP and VANS equations, two flow solvers are constructed using the Finite Volume method. Their precision and the lower bound on the Darcy penalization are investigated. Subsequently, the solvers are used to optimize flow channels for minimal pressure drop, and the resulting designs and convergence behavior are compared. The optimization procedure using the VANS equations is found to show less tendency to converge to inferior local optima for more precise flow solutions and is less sensitive to its parameter selection.

The latest updates on the Vertex Morphing technique for large optimization problems are shown in this work. Discussions about the challenges of node-based shape optimization in academic and industrial applications are included. The adaptive Vertex Morphing technique is demonstrated, which is easy to use in practice and allows the full exploitation of the potential of node-based shape optimization to find new designs in large-scale applications. We also show an efficient optimization method to handle different physical responses with many geometrical constraints. A state-of-the-art example of industrial importance supports the work.

Solving conjugate heat transfer design problems is relevant for various engineering applications requiring efficient thermal management. Heat exchange between fluid and solid can be enhanced by optimizing the system layout and the shape of the flow channels. As heat is transferred at fluid/solid interfaces, it is crucial to accurately resolve the geometry and the physics responses across these interfaces. To address this challenge, this work investigates for the first time the use of an eXtended Finite Element Method (XFEM) approach to predict the physical responses of conjugate heat transfer problems considering turbulent flow. This analysis approach is integrated into a level set-based optimization framework. The design domain is immersed into a background mesh and the geometry of fluid/solid interfaces is defined implicitly by one or multiple level set functions. The level set functions are discretized by higher-order B-splines. The flow is predicted by the Reynolds Averaged Navier–Stokes equations. Turbulence is described by the Spalart–Allmaras model and the thermal energy transport by an advection–diffusion model. Finite element approximations are augmented by a generalized Heaviside enrichment strategy with the state fields being approximated by linear basis functions. Boundary and interface conditions are enforced weakly with Nitsche’s method, and the face-oriented ghost stabilization is used to mitigate numerical instabilities associated with the emergence of small integration subdomains. The proposed XFEM approach for turbulent conjugate heat transfer is validated against benchmark problems. Optimization problems are solved by gradient-based algorithms and the required sensitivity analysis is performed by the adjoint method. The proposed framework is illustrated with the design of turbulent heat exchangers in two dimensions. The optimization results show that, by tuning the shape of the fluid/solid interface to generate turbulence within the heat exchanger, the transfer of thermal energy can be increased.

This paper presents a concurrent topology optimization approach to design multi-scale fluid channels, inlets, and outlets for efficient cooling capabilities. An approximate but low-cost thermofluidic model is used to predict the temperature distributions, in which the Darcy model is utilized to simulate the fluid flow. A novel boundary-based penalty approach is employed to facilitate a continuous layout optimization of inlets and outlets. The numerical homogenization is employed to predict the effective permeability and conductivity properties of predefined micro-channel unit cells. Besides, surrogate models are established to characterize the effective properties at the sub-level scale for cheap computational costs. Numerical examples show that the optimized multi-scale channels exhibit better cooling performances compared to the mono-scale cooling channels with the same pressure drop. In addition, the pressure drop of the concurrent design of the mono-scale channels, inlets, and outlets is lower than that of the design with predefined inlets and outlets.

The paper is concerned with a node-based, gradient-driven, continuous adjoint two-phase flow procedure to optimize the shapes of free-floating vessels and discusses three topics. First, we aim to convey that elements of a Cahn–Hilliard formulation should augment the frequently employed Volume-of-Fluid two-phase flow model to maintain dual consistency. It is seen that such consistency serves as the basis for a robust primal/adjoint coupling in practical applications at huge Reynolds and Froude numbers. The second topic covers different adjoint coupling strategies. A central aspect of the application is the floating position, particularly the trim and the sinkage, that interact with a variation of hydrodynamic loads induced by the shape updates. Other topics addressed refer to the required level of density coupling and a more straightforward—yet non-frozen—adjoint treatment of turbulence. The third part discusses the computation of a descent direction within a node-based environment. We will illustrate means to deform both the volume mesh and the hull shape simultaneously and at the same time obey technical constraints on the vessel’s displacement and its extensions. The Hilbert-space approach provides smooth shape updates using the established coding infrastructure of a computational fluid dynamics algorithm and provides access to managing additional technical constraints. Verification and validation follow from a submerged 2D cylinder case. The application includes a full-scale offshore supply vessel at Re = 3 × 10 8 and Fn = 0.37 . Results illustrate that the fully parallel procedure can automatically reduce the drag of an already pre-optimized shape by 9–13% within ≈ O (10,000-30,000) CPUh depending on the considered couplings and floatation aspects.

This paper proposes an integrated two-step strategy for an optimal design of liquid-cooled channel layout based on the moving morphable component (MMC)-density approach. The proposed strategy intends to take the advantage of both the MMC approach for its high flexibility in searching a physically reasonable layout and the density approach for its better capacity of topology description. On the basis of the above-mentioned strategy, an intermediate layout is obtained through MMC approach and further optimized as initial solution of density approach step. Through density approach step, the final layout shows smoother boundary while retaining reasonable feature size. The original contributions of this paper are as follows: (i) An assembled quadratic Bézier curves component is proposed to describe the largely curved channel with limited numbers of optimization variables and computation order. (ii) Benefited from explicit geometric description, adaptive mesh refinement (AMR) is applied in MMC approach step for the first time. The application of AMR, from the numerical point of view, has two key ingredients to be highlighted: (i) the accuracy of solution in fluid–solid boundary region can be ensured with relatively limited computational cost. (ii) The contradiction that the difference step of MMC updating needs to be both as small as possible and integer multiple of the mesh size can be avoided. The performance of our methodology is demonstrated by numerical examples aiming for maximal heat exchange with power dissipation constraint. The main finding reveals that the proposed strategy can offer reasonable channel layout with better thermal performance, compared with conventional density approach. The whole numerical implementation relies on OpenFOAM and PETSc open-source software packages.

This paper presents an educational code for topology optimization controlling the trajectory of particle in steady-state laminar fluid. To control the trajectory of particle, fluid motion is optimized by the fluid topology optimization. The one-way forward analysis between fluid and particle and the adjoint sensitivity analysis are formulated and implemented in the framework of MATLAB. The Navier–Stokes equation is solved by the finite element method with Newton–Raphson iteration and Newton’s equation for the analysis of transient particle motion is solved by the Newmark scheme. In the present paper, the educational code is attached in the supplementary material. Throughout the code, the optimization problems considering particle trajectory can be solved. The code consists of the finite element analysis of a fluid, transient analysis of a particle suspended in fluid, and computation of the adjoint sensitivity analysis. This code can be easily expanded for complex particle fluid problems. Several benchmark problems are presented that control the velocity and position of a particle and separate multiple particles suspended in a fluid.