A Stabilized Finite Element Framework for Anisotropic Adaptive Topology Optimization of Incompressible Fluid Flows

Abstract

This paper assesses the feasibility of performing topology optimization of laminar incompressible flows governed by the steady-state Navier–Stokes equations using anisotropic mesh adaptation to achieve a high-fidelity description of all fluid–solid interfaces. The present implementation combines an immersed volume method solving stabilized finite element formulations cast in the variational multiscale (VMS) framework and level-set representations of the fluid–solid interfaces, which are used as an a posteriori anisotropic error estimator to minimize interpolation errors under the constraint of a prescribed number of nodes in the mesh. Numerical results obtained for several two-dimensional problems of power dissipation minimization show that the optimal designs are mesh-independent (although the convergence rate does decreases as the number of nodes increases), agree well with reference results from the literature, and provide superior accuracy over prior studies solved on isotropic meshes (fixed or adaptively refined).

1. Introduction

Fluid flow topology optimization is the process of finding the best path for a fluid to flow in a prescribed design domain to maximize a measure of performance under a set of design constraints, for instance, to minimize dissipation subject to a constant volume of fluid. This approach originates from solid mechanics [1,2], where it has matured into a powerful, reliable, and increasingly available tool for engineers in the early stages of complex structural design processes at the component level [3,4]. It has since spread to a variety of other physics modeled after partial differential equations (see Refs. [5,6] for surveys of the evolving methods and applications, and Ref. [7] for a recent literature review within the context of fluid flow problems). Topology optimization has a mathematical foundation built on iterative analysis and design update steps, often steered by gradient evaluations. What stands out (compared to the size and shape optimization methods it has emerged from) is the great design freedom, which allows for the generation of non-intuitive designs from arbitrary initial guesses, possibly meeting conflicting requirements and complex interdependencies between design parameters and system responses.

We leave aside here explicit boundary methods that represent the fluid–solid interface using edges or faces of a body-fitted mesh and have limited flexibility in handling complicated topological changes. The prevalent classes of methods for topology optimization are the density and level-set methods. Density methods rely on a Brinkman penalization of the solid domain, where the flow is modeled as a fictitious porous material with very low permeability [1,8,9]. They manage drastic topological changes, as the gradient information (or sensitivity) is distributed over a large part of the domain, but can lead to spurious or leaking flows if the penalization factor is not well calibrated since the velocity and pressure fields are computed in the entire domain (both the solid and fluid regions). Level-set methods conversely model the solid boundaries using iso-contours of a level-set function [10,11,12]. They lack a nucleation mechanism to create new holes due to the sensitivities being located only at the solid–fluid interface, which is often mitigated using initial designs with many holes. However, they can easily handle complicated topological changes (e.g., merging or cancellation of holes) and allow for well-defined, crisp interface representations while avoiding the intermediate material phases (grayscales) and mesh-dependent spatial oscillations of the interface geometry (staircasing) often encountered in density methods [13].

The norm in topology optimization is to employ fixed finite element meshes with close-to-uniform element size, small enough so that all relevant physical phenomena are reliably captured, but not so small that the cost of performing the optimization becomes unaffordable. A recent trend has been to use adaptive remeshing techniques to maintain a competitive computational cost. Such an approach consists of generating a coarse base grid and then adding recursively finer and finer subgrids in the regions requiring higher resolution. This process repeats either until a maximum level of refinement is reached or until the local truncation error drops below a certain tolerance for more sophisticated implementations endowed with error estimation routines. Within the context of fluid flow problems, particular emphasis has been placed on (but not limited to) adaptive meshing refinement (AMR) schemes using both density [14,15] and level-set methods [16,17]; see also [18] for an application to phase-field methods (another class of interface capturing schemes that remain less popular due to the higher computational cost and difficulty of numerically discretizing the biharmonic phase-field equation) and [19,20,21] for recent efforts applying a different remeshing scheme to a combination of level-set functions and adaptive body-conforming meshes.

However, there is still ample room for progress, as almost all adaptive algorithms applied so far to fluid flow topology optimization support only isotropic size maps. Fluid dynamics, conversely, involves convection-dominated phenomena for which anisotropic meshes are highly desirable [22], especially in the vicinity of the solid boundaries, where the fluid velocity exhibits steep gradients in the wall-normal direction, and skin friction plays a defining role. The premise of this study is that the ability to generate highly stretched elements in boundary layer regions can substantially increase the accuracy of the geometric representation, compared to what is often seen in the topology optimization of flow problems, and naturally convey the said accuracy to the numerical solution without sophisticated interpolation or discretization techniques. We note that this is all perfectly in line with the recommendations made in [7] to improve upon the current state of the art. Nonetheless, our literature review did not reveal any other study combining anisotropic mesh adaptation and fluid flow topology optimization, besides the density-based optimization of Stokes flows in Ref. [15], possibly because of the notorious difficulty of finding spatial discretization schemes that meet the level of robustness required by automatic anisotropic mesh adaptation.

This research intends to fill the gap by introducing a novel numerical framework for the topology optimization of Navier–Stokes flows. The latter combines level-set methods and anisotropic mesh adaptation to handle arbitrary geometries immersed in an unstructured mesh. The Navier–Stokes system is solved using a variational multiscale (VMS) stabilized finite element method supporting elements of aspect ratio up to the order of 1000:1 [23]. The same numerical method is used to solve the adjoint Navier–Stokes system underlying the sensitivity analysis needed to evolve the level-set function. The metric map, providing both the size and stretching of mesh elements in very condensed information data, is derived from the level set. An a posteriori anisotropic error estimator is then used to minimize the interpolation error under the constraint of a prescribed number of nodes in the mesh. The latter can be adjusted over the course of the optimization, meaning that the base grid can be either refined or coarsened on demand. This contrasts with AMR, whose total number of mesh elements cannot be controlled, and whose mesh cannot be coarsened further than its base configuration. Since it reduces the cost of modeling the solid material away from the interface, such an approach is expected to achieve further speed-ups while also helping improve the manufacturability of the optimal design, which remains an issue, as most classical topology optimization methods render organic designs that can be difficult to translate into computer-aided design models.

The paper is organized as follows. The governing equations are formulated in Section 2. The anisotropic mesh adaptation algorithm and the immersed, stabilized finite element numerical framework used to perform the design update step are described in Section 3 and Section 4, respectively. The details of the implemented topology optimization algorithm are provided in Section 5. Finally, numerical experiments showcasing the potential of the approach on two-dimensional power dissipation minimization problems are presented in Section 6, with particular attention paid to highlighting the improved accuracy and mesh independence of the obtained solutions.

2. Immersed Model for Fluid Flow Topology Optimization

In the following, we denote by $\Omega$ a fixed, open bounded domain in ${\mathbb{R}}^d$ (with d being the space dimension), with the boundary $\partial \Omega$ oriented with an inward-pointing normal vector $\mathbf{n}$. Throughout this study, $\Omega ={\Omega }_f\cup {\Omega }_s$ is the disjoint reunion of two domains: ${\Omega }_f$ and ${\Omega }_s$ (for simplicity, we refer to ${\Omega }_f$ as the fluid domain, and to ${\Omega }_s$ as the solid domain, although we also fill ${\Omega }_s$ with a fluid for numerical convenience, as further explained below). The two domains are separated by an interface $\Gamma ={\Omega }_f\cap {\Omega }_s$, whose position we seek to optimize with respect to a certain measure of performance, here, a cost function J to minimize.

2.1. State Equations

Mathematically, the problem is characterized by a set of physical variables determined as the solutions of partial differential equations, which are themselves derived from modeling considerations. Here, the flow motion in the fluid domain ${\Omega }_f$ is modeled using the steady incompressible Navier–Stokes equations

$$\begin{array}{cccc}\hfill \nabla ·\mathbf{u}=& 0\hfill & & \mathrm{in}{\Omega }_f,\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill \rho \mathbf{u}·\nabla \mathbf{u}=& -\nabla p+\nabla ·\left(2\mu \epsilon \right(\mathbf{u}\left)\right)\hfill & & \mathrm{in}{\Omega }_f,\hfill \end{array}$$

(2)

where $\mathbf{u}$ is the velocity, p is the pressure, $\epsilon \left(\mathbf{u}\right)=(\nabla \mathbf{u}+\nabla {\mathbf{u}}^T)/2$ is the rate of deformation tensor, and we assume constant fluid density $\rho$ and dynamic viscosity $\mu$. The fluid domain boundary $\partial {\Omega }_f$ is split into the (wall) interface $\Gamma$, the inlet ${\Gamma }_i$, i.e., the combined boundary of all surfaces where fluid enters the domain, and the outlet ${\Gamma }_o$, i.e., the combined boundary of all surfaces where the fluid leaves the domain. Open flow boundary conditions are appended in the form of a prescribed velocity at the inlet, zero pressure and viscous stress conditions at the outlet, and zero velocity at the interface, hence

$$\begin{array}{cccc}\hfill \mathbf{u}=& {\mathbf{u}}_i\hfill & & \mathrm{on}{\Gamma }_i,\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill p\mathbf{n}=\mu \epsilon \left(\mathbf{u}\right)·\mathbf{n}=& \mathbf{0}\hfill & & \mathrm{on}{\Gamma }_o,\hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill \mathbf{u}=& \mathbf{0}\hfill & & \mathrm{on}\Gamma .\hfill \end{array}$$

(5)

2.2. Adjoint-Based Sensitivity Analysis

We assume in the following that the cost function (i) can be formulated as a surface integral over the domain boundary (rather than its interior), and (ii) does not depend on the flow quantities on the wall, which is most often true in topology optimization. It is thus expressed as integrals over all or any part of the inlet and/or outlet, i.e.,

$$\begin{array}{c}\hfill J_s={\int }_{{\Gamma }_i\cup {\Gamma }_o}J\mathrm{d}s.\end{array}$$

(6)

The problem of minimizing the cost function subject to Navier–Stokes as state equations is tackled using the continuous adjoint method. The reader interested in the technicalities of the method is referred to [24]. One first forms the Lagrangian

$$\begin{array}{c}\hfill \mathcal{L}={\int }_{{\Gamma }_i\cup {\Gamma }_o}J\mathrm{d}s-{\int }_{{\Omega }_f}\tilde{p}\nabla ·\mathbf{u}\mathrm{d}v-{\int }_{{\Omega }_f}\tilde{\mathbf{u}}·(\rho \mathbf{u}·\nabla \mathbf{u}+\nabla p-\nabla ·\left(2\mu \epsilon \left(\mathbf{u}\right)\right)\mathrm{d}v,\end{array}$$

(7)

featuring the adjoint velocity $\tilde{\mathbf{u}}$ as the Lagrange multiplier for the momentum Equation (2) and the adjoint pressure $\tilde{p}$ as the Lagrange multiplier for the continuity Equation (1). One then seeks to decompose the variation of $\mathcal{L}$ due to a change in the interface position into individual variations with respect to the adjoint, state, and design variables. The variation with respect to the adjoint variables

$$\begin{array}{c}\hfill {\delta }_{(\tilde{\mathbf{u}},\tilde{p})}\mathcal{L}=-{\int }_{{\Omega }_f}\delta \tilde{p}\nabla ·\mathbf{u}\mathrm{d}v-{\int }_{{\Omega }_f}\delta \tilde{\mathbf{u}}·(\rho \mathbf{u}·\nabla \mathbf{u}+\nabla p-\nabla ·\left(2\mu \epsilon \left(\mathbf{u}\right)\right)\mathrm{d}v,\end{array}$$

(8)

is trivially zero, as long as $(\mathbf{u},p)$ is a solution to the above Navier–Stokes equations, in which case, $\mathcal{L}=J_s$. After integrating by parts, the variation with respect to the state variables is

$$\begin{array}{cc}\hfill {\delta }_{(\mathbf{u},p)}\mathcal{L}=& {\int }_{{\Omega }_f}(\nabla ·\tilde{\mathbf{u}})\delta p\mathrm{d}v+{\int }_{{\Omega }_f}(-\rho \mathbf{u}·\nabla \tilde{\mathbf{u}}+\rho \nabla \mathbf{u}{}^T·\tilde{\mathbf{u}}-\nabla \tilde{p}-\nabla ·\left(2\mu \epsilon \left(\tilde{\mathbf{u}}\right)\right))·\delta \mathbf{u}\mathrm{d}v\hfill \\ \hfill +& {\int }_{{\Gamma }_i\cup {\Gamma }_o}{\partial }_{\mathbf{u}}J·\delta \mathbf{u}\mathrm{d}s+{\int }_{\partial {\Omega }_f}(\tilde{p}\mathbf{n}+2\mu \epsilon \left(\tilde{\mathbf{u}}\right)·\mathbf{n}+\rho (\mathbf{u}·\mathbf{n})\tilde{\mathbf{u}})·\delta \mathbf{u}\mathrm{d}s\hfill \\ \hfill -& {\int }_{{\Gamma }_i\cup {\Gamma }_o}{\partial }_{p}J\mathbf{n}·(-\delta p\mathbf{n}+2\mu \epsilon \left(\delta \mathbf{u}\right)·\mathbf{n})\mathrm{d}s-{\int }_{\partial {\Omega }_f}\tilde{\mathbf{u}}·(-\delta p\mathbf{n}+2\mu \epsilon \left(\delta \mathbf{u}\right)·\mathbf{n})\mathrm{d}s,\hfill \end{array}$$

(9)

on account of the viscous stress being purely tangential in incompressible flows. At this stage, adjoint equations and boundary conditions are designed to ensure ${\delta }_{(\mathbf{u},p)}\mathcal{L}=0$, which requires the domain and boundary integrals to vanish individually in (9). Keeping in mind that we work here under the assumption of a fixed interface (since the design variable is constant), we obtain the linear, homogeneous problem

$$\begin{array}{cccc}\hfill \nabla ·\tilde{\mathbf{u}}=& 0\hfill & & \mathrm{in}{\Omega }_f,\hfill \end{array}$$

(10)

$$\begin{array}{cccc}\hfill -\rho \mathbf{u}·\nabla \tilde{\mathbf{u}}+\rho \nabla \mathbf{u}{}^T·\tilde{\mathbf{u}}=& \nabla \tilde{p}+\nabla ·\left(2\mu \epsilon \left(\tilde{\mathbf{u}}\right)\right)\hfill & & \mathrm{in}{\Omega }_f,\hfill \end{array}$$

(11)

driven by the non-homogeneous boundary conditions

$$\begin{array}{cccc}\hfill \tilde{\mathbf{u}}=& -{\partial }_{p}J\mathbf{n}\hfill & & \mathrm{on}{\Gamma }_i,\hfill \end{array}$$

(12)

$$\begin{array}{cccc}\hfill \tilde{p}\mathbf{n}+2\mu \epsilon \left(\tilde{\mathbf{u}}\right)·\mathbf{n}+\rho (\mathbf{u}·\mathbf{n})\tilde{\mathbf{u}}=& -{\partial }_{\mathbf{u}}J\hfill & & \mathrm{on}{\Gamma }_o,\hfill \end{array}$$

(13)

$$\begin{array}{cccc}\hfill \tilde{\mathbf{u}}=& \mathbf{0}\hfill & & \mathrm{on}\Gamma ,\hfill \end{array}$$

(14)

associated with (3)–(5). Note that the minus sign ahead of the first term of the adjoint momentum Equation (11) reflects the reversal in directionality due to the non-normality of the linearized evolution operator [25]. By expressing the interface normal deformation after [26] as

$$\begin{array}{c}\hfill \delta \mathbf{u}=\beta \nabla \mathbf{u}·\mathbf{n},\end{array}$$

(15)

the variation with respect to the design variable (now encompassing the domain deformation) is ultimately computed as

$$\begin{array}{c}\hfill {\delta }_{\beta }{J}_s\equiv {\delta }_{\beta }\mathcal{L}={\int }_{\Gamma }\beta (\tilde{p}\mathbf{n}+2\mu \epsilon \left(\tilde{\mathbf{u}}\right)·\mathbf{n})·(\nabla \mathbf{u}·\mathbf{n})\mathrm{d}s={\int }_{\Gamma }\beta \mu (\nabla \tilde{\mathbf{u}}·\mathbf{n})·(\nabla \mathbf{u}·\mathbf{n})\mathrm{d}s,\end{array}$$

where the last equality stems from the incompressibility of the state and adjoint solutions [24]. This enables efficient design update schemes via first-order gradient descent methods, as the second term in the integrand is the desired sensitivity to a displacement $\beta$ at some specific point of the interface. For instance, the simplest steepest-descent algorithm implemented herein moves down the cost function in the direction of the steepest slope using

$$\begin{array}{c}\hfill \beta =-\mu (\nabla \tilde{\mathbf{u}}·\mathbf{n})·(\nabla \mathbf{u}·\mathbf{n}),\end{array}$$

(16)

up to a positive multiplicative factor to control the step taken in the gradient direction.

2.3. Level-Set Representation of the Interface

The level-set method is used here to localize and capture the interface between the fluid and solid domains using the zero iso-value of a smooth level-set function, classically, the signed distance function defined as

$$\begin{array}{c}\hfill \phi \left(\mathbf{x}\right)=\left\{\begin{array}{cc}-\mathrm{dist}(\mathbf{x},\Gamma )\hfill & \mathrm{if}\mathbf{x}\in {\Omega }_f,\hfill \\ 0\hfill & \mathrm{if}\mathbf{x}\in \Gamma ,\hfill \\ \mathrm{dist}(\mathbf{x},\Gamma )\hfill & \mathrm{if}\mathbf{x}\in {\Omega }_s,\hfill \end{array}\right.\end{array}$$

(17)

with the convention that $\phi <0$ in the fluid domain. Once the sensitivity analysis has output a displacement $\beta$ in the direction of the steepest slope, the position of the level set is updated by solving a transport equation with a normal velocity $\beta \mathbf{n}/\Delta \tau$, where $\Delta \tau$ is a pseudo-time step to convert from displacement to velocity, which has no physical relevance since we are not concerned by the absolute displacement of a given point on the interface, only by its relative displacement with respect to its neighbors. This equation is posed in the whole domain $\Omega$ because the normal vector recovered at the interface as $\mathbf{n}=\nabla \phi /\left|\right|\nabla \phi \left|\right|$ can be easily extended to $\Omega$ using (17). The main problem with this approach is that the level set after transport is generally no longer a distance function, which is especially problematic when a specific remeshing strategy, depending on the distance property, is used at the interface (as is the case in this study). As a result, the distance function needs to be reinitialized, which is performed here using a coupled convection-reinitialization method, wherein the level-set function is automatically reinitialized during the resolution of the transport equation. In practice, the signed distance function is cut off using a hyperbolic tangent filter, as defined by

$$\begin{array}{c}\hfill \varphi =Etanh\left(\frac{\phi }{E}\right),\end{array}$$

(18)

where E is the cut-off thickness so that the metric property is asymptotically satisfied in the vicinity of the zero iso-value. This filtered level set is then evolved by solving the auto-reinitialization equation

$$\begin{array}{c}\hfill {\partial }_{\tau }\varphi +{\mathbf{a}}_{\tau }·\nabla \varphi =S,\end{array}$$

(19)

where we note that

$$\begin{array}{c}\hfill {\mathbf{a}}_{\tau }=\frac{\beta }{\Delta \tau }\mathbf{n}+\frac{\lambda }{\Delta \tau }sgn\left(\varphi \right)\frac{\nabla \varphi }{\left|\right|\nabla \varphi \left|\right|},S=\frac{\lambda }{\Delta \tau }sgn\left(\varphi \right)\left(1-{\left(\frac{\varphi }{E}\right)}^2\right),\end{array}$$

(20)

and $\lambda$ is a parameter homogeneous to a length, set to the mesh size $h_{\perp }$ in the direction normal to the interface. Such an approach is shown in [27,28,29] to reduce the computational cost and ensure better mass conservation compared to the classical Hamilton–Jacobi method, in which both steps are performed in succession. Moreover, since the filtered level set defined in (18) is bounded, Dirichlet boundary conditions $\varphi =\pm E$ can be easily appended to Equation (19) to explicitly design the fluid and solid sub-regions of $\partial \Omega$.

3. Anisotropic Mesh Adaptation

A primitive pseudo-code of the procedure for solving the topology optimization problem posed in Section 2 is provided in the Algorithm 1 below.

Algorithm 1 Simplified update scheme
Require: Anisotropic mesh adapted to initial level-set function
1:	loop
2:	Compute state
3:	Compute adjoint
4:	Compute cost function sensitivity
5:	Set displacement in the direction of the steepest slope
6:	Update level set
7:	Generate anisotropic mesh adapted to new level set

This algorithm repeats until a maximum number of iterations or a convergence threshold has been reached. In a nutshell, this is performed here using a finite element immersed numerical framework that combines the implicit representation of the different domains, level-set description of the interface, and anisotropic remeshing capabilities. For the sake of readability, the mesh adaptation algorithm, whose implementation in the context of fluid flow topology optimization makes for the main novelty of this study, is presented here as a stand-alone section. We then walk through each of the other steps in Section 4 and review the various problems involved and the numerical methods used to solve them.

3.1. Construction of an Anisotropic Mesh

The main idea of anisotropic, metric-based mesh adaptation is to generate a uniform mesh (with unit-length edges and regular elements) in a prescribed Riemannian metric space, but anisotropic and well adapted (with highly stretched elements) in the Euclidean space. Assuming that, in the context of metric-based adaptation methods, controlling the interpolation error suffices to master the global approximation error, the objective can be formulated as finding the mesh, made up of at most $N_n$ nodes, that minimizes the linear interpolation error in the $L^1$ norm. Following the approach of [30,31], an edge-based error estimator combined with a gradient recovery procedure is used to compute, for each node, a metric tensor that prescribes a set of anisotropic directions and stretching factors along these directions, without any direct information from the elements or any underlying interpolation. The optimal stretching factor field is obtained by solving an optimization problem using the equidistribution principle under the constraint of a fixed number of nodes in the mesh, after which a new mesh is generated using the procedure described in [32], which is based on a topological representation of the computational domain.

3.2. Edge Error Estimate

Given a mesh ${\Omega }_h$ of the domain $\Omega$, we denote by ${\mathbf{x}}^{ij}$ the edge connecting a given node ${\mathbf{x}}^i$ to ${\mathbf{x}}^j\in \Sigma \left(i\right)$, where $\Sigma \left(i\right)$ is the set of nodes connected to ${\mathbf{x}}^i$, and the number of these nodes is noted as $|\Sigma (i\left)\right|$. Also, given a regular analytical (scalar) function $\psi$ defined on $\Omega$ and its P1 finite element approximation ${\psi }_h$ computed on ${\Omega }_h$, we follow [30] and estimate the interpolation error along the edge ${\mathbf{x}}^{ij}$ as the projection along the edge of the second derivative of $\psi$. This is obtained by projecting along the edge a Taylor expansion of the gradient of $\psi$ at ${\mathbf{x}}^j$ to yield

$$\begin{array}{c}\hfill {\epsilon }_{ij}=|{\mathbf{g}}^{ij}·{\mathbf{x}}^{ij}|,\end{array}$$

(21)

where the i and j superscripts indicate the nodal values at nodes ${\mathbf{x}}^i$ and ${\mathbf{x}}^j$, respectively, ${\mathbf{g}}^i=\nabla \psi \left({\mathbf{x}}^i\right)$ is the exact value of the gradient at ${\mathbf{x}}^i$, and ${\mathbf{g}}^{ij}={\mathbf{g}}^j-{\mathbf{g}}^i$ is the variation of the gradient along the edge. Although Equation (21) involves only the values of the gradient at the edge extremities and can thus be evaluated without resorting to resource-expensive Hessian reconstruction methods, this, however, requires the gradient of $\psi$ to be known and continuous at the nodes, which, in turn, requires full knowledge of $\psi$. Meanwhile, only the linear interpolate ${\psi }_h$ is known in practice, whose gradient is piecewise constant and discontinuous from element to element, although its projection along the edges is continuous since it depends only on the nodal values of the field.

A recovery procedure is thus used to build a continuous gradient estimator defined directly at the nodes. It is shown in [30] that a suitable error estimate preserving second-order accuracy is obtained by substituting the reconstructed gradient for the exact gradient into (21), to yield

$$\begin{array}{c}\hfill {\epsilon }_{ij}=|{\overline{\mathbf{g}}}^{ij}·{\mathbf{x}}^{ij}|,\end{array}$$

(22)

where ${\overline{\mathbf{g}}}^{ij}={\overline{\mathbf{g}}}^j-{\overline{\mathbf{g}}}^i$, and we denote by ${\overline{\mathbf{g}}}^i$ the recovered gradient of ${\psi }_h$ at node ${\mathbf{x}}^i$. The latter is defined in a least-square sense as

$$\begin{array}{c}\hfill {\overline{\mathbf{g}}}^i=\underset{\mathbf{g}\in {\mathbb{R}}^d}{argmin}\sum _{j\in \Sigma \left(i\right)}{|(\mathbf{g}-\nabla {\psi }_h)·{\mathbf{x}}^{ij}|}^2,\end{array}$$

(23)

for which an approximate solution using the nodal values as the sole input is shown in [30] to be

$$\begin{array}{c}\hfill {\overline{\mathbf{g}}}^i={\left({\mathbf{X}}^i\right)}^{-1}·\sum _{j\in \Sigma \left(i\right)}({\psi }_h\left({\mathbf{x}}^j\right)-{\psi }_h\left({\mathbf{x}}^i\right)){\mathbf{x}}^{ij},\end{array}$$

(24)

where ${\mathbf{X}}^i$ is the length distribution tensor defined as

$$\begin{array}{c}\hfill {\mathbf{X}}^i=\frac{1}{|\Sigma (i\left)\right|}\sum _{j\in \Sigma \left(i\right)}{\mathbf{x}}^{ij}\otimes {\mathbf{x}}^{ij},\end{array}$$

(25)

that yields an average representation of the distribution of the edges sharing an extremity.

3.3. Metric Construction

In order to relate the error indicator ${\epsilon }_{ij}$ defined in (22) to a metric suitable for mesh adaptation purposes, we introduce the stretching factor $s_{ij}$ as the ratio between the length of the edge ${\mathbf{x}}^{ij}$ after and before the adaptation. The metric at node ${\mathbf{x}}^i$ is sought to generate the unit-stretched edge length in the metric space, that is,

$$\begin{array}{c}\hfill {\left(s_{ij}{\mathbf{x}}^{ij}\right)}^T·\mathbf{M}{}^i·\left(s_{ij}{\mathbf{x}}^{ij}\right)=1,\forall j\in \Sigma \left(i\right),\end{array}$$

(26)

for which an approximate least-square solution is shown in [30] to be

$$\begin{array}{c}\hfill {\mathbf{M}}^i={\left(\frac{d}{|\Sigma (i\left)\right|}\sum _{j\in \Sigma \left(i\right)}{s}_{ij}^2{\mathbf{x}}^{ij}\otimes {\mathbf{x}}^{ij}\right)}^{-1},\end{array}$$

(27)

provided the nodes in $\Sigma \left(i\right)$ form at least d non-co-linear edges with ${\mathbf{x}}^i$ (which is the case if the mesh is valid). The metric solution of (27) is ultimately computed by setting a target total number of nodes $N_n$. Assuming a total error equidistributed among all edges, the stretching factor is shown in [31] to be

$$\begin{array}{c}\hfill s_{ij}={\left(\frac{{\displaystyle \sum _i{N}_i\left(1\right)}}{N_n}\right)}^{\frac{2}{d}}{\epsilon }_{ij}^{-1/2},\end{array}$$

(28)

where $N_i\left(1\right)$ is the number of nodes generated in the vicinity of node ${\mathbf{x}}^i$ for a unit error, given by

$$\begin{array}{c}\hfill N_i\left(1\right)={\left(\det \left(\frac{d}{|\Sigma (i\left)\right|}\sum _{j\in \Sigma \left(i\right)}{\epsilon }_{ij}^{1/2}\frac{{\mathbf{x}}^{ij}}{|{\mathbf{x}}^{ij}|}\otimes \frac{{\mathbf{x}}^{ij}}{|{\mathbf{x}}^{ij}|}\right)\right)}^{-1/2}.\end{array}$$

(29)

3.4. Summary

In order to simplify and clarify the presentation, the main steps needed for metric construction at the nodes are summarized in the Algorithm 2 below.

Algorithm 2 Anisotropic mesh adaptation algorithm
Require: Anisotropic adapted mesh
1:	Set number of nodes $N_n$
2:	Compute ${\psi }_h$ on current mesh
3:	for each node ${\mathbf{x}}^i$ do
4:	Compute length distribution tensor ${\mathbf{X}}^i$ using (25)
5:	Compute nodal recovered gradient. ${\overline{\mathbf{g}}}^i$ using (24)
6:	for all edges ${\mathbf{x}}^{ij}$ do
7:	Compute edge recovered gradient ${\overline{\mathbf{g}}}^{ij}$
8:	Compute edge-based error ${\epsilon }_{ij}$ using (22)
9:	Compute stretching factor $s_{ij}$ using (28)
10:	Compute metric $\mathbf{M}{}^i$ using (27)
11:	Generate new mesh using local improvement in the neighborhood of the nodes and edges [32]
12:	Interpolate ${\psi }_h$ on new mesh

In this algorithm, classical linear interpolation from one mesh to another is applied.

3.5. Level-Set-Based Adaptation Criteria

In practice, the variable used for the purpose of error estimation is the filtered level set defined in (18), as it satisfies the metric property in a thin layer around the interface (in particular, it preserves the zero iso-value of $\phi$, which is the only relevant information for mesh adaptation purposes), but avoids the unnecessary adaption of the mesh further away from the interface (where the interpolation error is close to zero due to $\left|\right|\nabla \varphi \left|\right|\sim 0$). This means that the criterion for mesh adaptation is purely geometric, i.e., the same mesh is pre-adapted around the fluid–solid interface and then used to compute all quantities needed to perform the next design update step. The flexibility of the proposed mesh adaptation technique is illustrated in Figure 1 and Figure 2, where a solid circle, square, and regular pentagram defined by the level-set functions have been immersed close to the boundary of a square cavity filled with fluid to assess the capability of handling different features (angles, singular points, curvatures) even under drastic conditions. Four meshes made up of 500, 1000, 2500, and 5000 nodes are considered, each of which comes in two flavors: one structured and the other anisotropic, adapted to the level set. On the one hand, the adapted meshes exhibit the expected orientation and deformation of the mesh elements, whose longest edges are parallel to the solid boundaries. On the other hand, they are naturally and automatically coarsened in smooth regions where the filtered level set is constant while being extremely refined near the interface. Also, the transition is finer with an anisotropic adaptive mesh, allowing for maintaining very good accuracy even with a low number of nodes, as evidenced in Figure 2 by the zero iso-value of the level sets. More quantitative results are available in [33], where it is shown that at least ten times more elements are required in a structured mesh to achieve the same accuracy, as measured by computing the total perimeter and area of the three immersed objects.

Nonetheless, it is worth mentioning that this approach also supports more complex adaptation criteria featuring physical quantities, thus providing the ability to dynamically adapt the mesh during the simulations. The common method used to adapt a mesh to several variables is to combine the metrics corresponding to each individual variable using metric intersection algorithms, which is known to incur a relatively high computational cost and result in a potentially non-unique, suboptimal outcome. Conversely, the present approach allows for directly building a unique metric from a multi-component error vector that combines the level set and any relevant flow quantity of interest, as Definition (22) can be easily extended to account for several sources of error [34]. Indeed, if we consider $\mathit{\psi }=({\psi }_1,{\psi }_2,\dots ,{\psi }_p)$ as a vector consisting of p scalar variables, it becomes straightforward that the error is now a vector ${\mathit{\epsilon }}_{ij}=({\epsilon }_{ij,1},{\epsilon }_{ij,2},\dots ,{\epsilon }_{ij,p})$, whose $L^2$ norm can serve as a simple error value for the edge from which to compute the stretching factor (28), and ultimately, the metric solution of (27). For instance, the $2d+3$-sized nodal vector field defined as

$$\begin{array}{c}\hfill {\mathit{\psi }}_h\left({\mathbf{x}}^i\right)=\left(\frac{{\varphi }_h^i}{{\displaystyle \underset{j\in \Sigma \left(i\right)}{max}{\varphi }_h^j}},\frac{u_{h_{k\in \{1\dots d\}}}^i}{\left|\right|{\mathbf{u}}_h^i\left|\right|},\frac{||{\mathbf{u}}_h^i||}{{\displaystyle \underset{j\in \Sigma \left(i\right)}{max}\left|\right|{\mathbf{u}}_h^j\left|\right|}},\frac{{\tilde{u}}_{h_{k\in \{1\dots d\}}}^i}{\left|\right|{\tilde{\mathbf{u}}}_h^i\left|\right|},\frac{||{\tilde{\mathbf{u}}}_h^i||}{{\displaystyle \underset{j\in \Sigma \left(i\right)}{max}\left|\right|{\tilde{\mathbf{u}}}_h^j\left|\right|}}\right),\end{array}$$

(30)

can be used to combine adaptivity with respect to the norm and direction of the state and adjoint velocity vectors, in addition to the level set. Because all fields are normalized by their respective global maximum, a field much larger in magnitude cannot dominate the error estimator, meaning that the variations of all variables are fairly taken into account. This benefits problems involving more complex physics (e.g., turbulence, heat transfer, fluid-structure interaction, multiple phases, possibly interacting with each another), all the more so in the context of topology optimization, as the difference in the spatial supports of the state and adjoint quantities, resulting from the non-normality of the linearized evolution operator [35], may otherwise yield conflicting requirements in terms of the regions of the computational domain most in need of refinement.

4. Computational Methods

This section is devoted to the stabilized finite element numerical framework used to compute all solutions of interest on anisotropic adapted meshes and perform the design update steps. For the sake of simplicity in the notations, and as long as it does not lead to ambiguity, we omit in what follows the distinction between all continuous variables (e.g., domains, solutions, operators) and their discrete finite element counterparts, as well as the dependency of all variables on the iteration of the optimization process.

4.1. Immersed Volume Method

The immerse volume method (IVM) [36,37] is used to combine the fluid and solid phases of the problem into a single fluid with variable material properties (density and viscosity). This amounts to solving the state and adjoint equations identical to those introduced in Section 2, but formulated not only in the fluid domain ${\Omega }_f$ but also in the whole domain $\Omega$, with phase-dependent density and viscosity fields adequately interpolated over a small layer around the interface, and otherwise equal to their fluid and solid values. Note that the thickness of the interpolation layer is user-defined and does not increase in size during the optimization, unlike the homogenization method or any other generalized material method. Using the level-set function (17) as a criterion for anisotropic mesh adaptation ensures that individual material properties can be distributed as accurately and smoothly as possible over the smallest possible thickness around the interface. This is classically done through linear interpolation between the fluid and solid values, using a smooth Heaviside function computed from the level set to avoid discontinuities by creating an interface transition with a thickness of a few elements. This approach is simpler than the Ersatz material approach [38], which adds a Brinkman penalization term to the Navier–Stokes equations and has clear connections to density-based methods through the material distribution [17]. It is especially relevant to thermal coupling problems, as having composite conductivity and specific heat means that the amount of heat exchanged at the interface then proceeds solely from the individual material properties on either side of it, and removes the need for a heat-transfer coefficient. However, for the pure flow problems tackled here, it suffices to use constant density and viscosity (equal to the fluid values) and to set the velocity to zero at all grid nodes located inside the solid domain ${\Omega }_s$. Compared to using a very high solid-to-fluid viscosity ratio to ensure that the velocity is zero in the solid domain, this can be seen as a hard penalty that prevents the fluid from leaking across the immersed interface. The latter holds numerically because anisotropic mesh adaptation precisely aligns the mesh element edges along the interface. This ensures that the latter does not arbitrarily intersect the mesh elements, which would otherwise compromise the accuracy of the finite element approach.

4.2. Variational Multiscale Modeling

The convective terms in the incompressible Navier–Stokes and level-set transport equations may cause spurious node-to-node velocity oscillations. Furthermore, the equal-order linear/linear approximations used for the velocity and pressure variables, albeit very desirable due to their simplicity of implementation and affordable computing cost (especially for 3D applications), may give rise to spurious pressure oscillations. To prevent these numerical instabilities, here, we solve stabilized formulations cast in the variational multiscale (VMS) framework that enhance the stability of the Galerkin method via a series of additional integrals over the element interior. The basic idea is to split all quantities into coarse- and fine-scale components, corresponding to different levels of resolution, and to approximate the effect of the fine scale (that cannot be resolved by the finite element mesh) on the coarse scale via consistently derived residual-based terms.

4.2.1. Navier–Stokes Equations

In practice, the state solution is computed by time-stepping the unsteady Navier–Stokes equations with large time steps to accelerate convergence toward a steady state. The stopping criterion here is for two consecutive time steps to differ by less than ${10}^{-6}$ in the $L^{\infty }$ norm. In order to deal with the time dependency and nonlinearity of the momentum equation, the transport time of the time scale is assumed to be much smaller than that of the coarse scale. In return, the fine-scale contribution to the transport velocity is neglected, and the fine scale is not tracked over time, although it is driven by the coarse-scale, time-dependent residuals and, therefore, does vary over time in a quasi-static manner. In-depth technical and mathematical details, together with extensive discussions regarding the relevance of the approximations, can be found in [39]. Ultimately, the coarse-scale variational problem is formulated as

$$\begin{array}{cc}\hfill & {\int }_{\Omega }(\rho {\partial }_t\mathbf{u}+\rho \mathbf{u}·\nabla \mathbf{u})·\mathbf{w}\mathrm{d}v+{\int }_{\Omega }2\mu \epsilon \left(\mathbf{u}\right):\epsilon \left(\mathbf{w}\right)\mathrm{d}v-{\int }_{\Omega }p(\nabla ·\mathbf{w})\mathrm{d}v+{\int }_{\Omega }(\nabla ·\mathbf{u})q\mathrm{d}v\hfill \\ & -\sum _{k=1}^{N_e}{\int }_{{\Omega }_k}{\tau }_1^{}{\mathbf{r}}_1^{}·(\rho \mathbf{u}·\nabla \mathbf{w})\mathrm{d}v-\sum _{k=1}^{N_e}{\int }_{{\Omega }_k}{\tau }_1^{}{\mathbf{r}}_1^{}·\nabla q\mathrm{d}v-\sum _{k=1}^{N_e}{\int }_{{\Omega }_k}{\tau }_2^{}{r}_2^{}(\nabla ·\mathbf{w})\mathrm{d}v=0,\hfill \end{array}$$

(31)

where we have considered a discretization of $\Omega$ into $N_e$ non-overlapping elements (triangles or tetrahedrons), ${\Omega }_k$ is the domain occupied by the kth element, and ${\mathbf{r}}_1^{}$ and $r_2^{}$ are the momentum and continuity residuals

$$\begin{array}{c}\hfill -{\mathbf{r}}_1^{}=\rho {\partial }_t\mathbf{u}+\rho \mathbf{u}·\nabla \mathbf{u}+\nabla p,-r_2^{}=\nabla ·\mathbf{u},\end{array}$$

(32)

whose second derivatives vanish since we use linear interpolation functions. Finally, ${\tau }_1^{}$ and ${\tau }_2^{}$ are ad hoc stabilization coefficients, computed on each element after [37,40] as

$$\begin{array}{c}\hfill {\tau }_1^{}=\frac{1}{\rho {\left({\tau }_t^2\left(u\right)+{\tau }_d^2\right)}^{1/2}},{\tau }_2^{}=\frac{h^2}{{\tau }_1^{}},\end{array}$$

(33)

with convection (transport) and diffusion-dominated limits defined as

$$\begin{array}{c}\hfill {\tau }_t\left(u\right)=c_t\frac{u}{h},{\tau }_d=c_d\frac{\mu }{\rho h^2}.\end{array}$$

(34)

where u is a characteristic norm of the velocity on the element, computed as the average $L^2$ norm of the nodal element velocities. Also, h is the element size, computed as its diameter in the direction of the velocity to support using anisotropic meshes with highly stretched elements [41], and $c_{t,d}$ are algorithmic constants taken as $c_t=2$ and $c_d=4$ for linear elements [40]. Equation (31) is discretized with a first-order-accurate time-integration scheme that combines semi-implicit treatment of the convection term; implicit treatment of the viscous, pressure, and divergence terms; and explicit treatment of the stabilization coefficients. All linear systems are preconditioned using a block Jacobi method, supplemented by an incomplete LU factorization, and solved using the GMRES iterative algorithm, with a tolerance threshold set to ${10}^{-6}$.

4.2.2. Adjoint Navier–Stokes Equations

The application of the stabilized formulation, as described above, to the adjoint Navier–Stokes equations yields the following coarse-scale variational problem

$$\begin{array}{cc}\hfill & {\int }_{\Omega }(-\rho \mathbf{u}·\nabla \tilde{\mathbf{u}}+\rho \nabla {\mathbf{u}}^T·\tilde{\mathbf{u}})·\mathbf{w}\mathrm{d}v+{\int }_{\Omega }2\mu \epsilon \left(\tilde{\mathbf{u}}\right):\epsilon \left(\mathbf{w}\right)\mathrm{d}v+{\int }_{\Omega }\tilde{p}(\nabla ·\mathbf{w})\mathrm{d}v+{\int }_{\Omega }(\nabla ·\tilde{\mathbf{u}})q\mathrm{d}v\hfill \\ \hfill & -\sum _{k=1}^{N_e}{\int }_{{\Omega }_k}{\tilde{\tau }}_1^{}{\tilde{\mathbf{r}}}_1^{}·(-\rho \mathbf{u}·\nabla \mathbf{w})\mathrm{d}v-\sum _{k=1}^{N_e}{\int }_{{\Omega }_k}{\tilde{\tau }}_1^{}{\tilde{\mathbf{r}}}_1^{}·\nabla q\mathrm{d}v-\sum _{k=1}^{N_e}{\int }_{{\Omega }_k}{\tilde{\tau }}_2^{}{\tilde{r}}_2^{}(\nabla ·\mathbf{w})\mathrm{d}v\hfill \\ & -{\int }_{{\Gamma }_o}\rho (\mathbf{u}·\mathbf{n})(\tilde{\mathbf{u}}·\mathbf{w})\mathrm{d}s={\int }_{{\Gamma }_o}{\partial }_{\mathbf{u}}J·\mathbf{w}\mathrm{d}s.\hfill \end{array}$$

(35)

The associated momentum and continuity residuals read

$$\begin{array}{c}\hfill -{\tilde{\mathbf{r}}}_1^{}=-\rho \mathbf{u}·\nabla \tilde{\mathbf{u}}+\rho \nabla {\mathbf{u}}^T·\tilde{\mathbf{u}}-\nabla \tilde{p},-{\tilde{r}}_2^{}=\nabla ·\tilde{\mathbf{u}},\end{array}$$

(36)

and the stabilization coefficients are computed on each element after [42] as

$$\begin{array}{cc}\hfill {\tilde{\tau }}_1^{}=\frac{1}{{\left({\tau }_t^2\left(u\right)+{\tau }_d^2+{\tau }_r^2\right)}^{1/2}}.{\tilde{\tau }}_2^{}& ={\tau }_2^{},\hfill \end{array}$$

(37)

Note that ${\tau }_r$ is an additional component that corresponds to the reaction-dominated limit, which stems from the $\rho \nabla {\mathbf{u}}^T·\tilde{\mathbf{u}}$ term that describes the production of adjoint perturbations. It is defined as

$$\begin{array}{c}\hfill {\tau }_r=\rho \nabla u,\end{array}$$

(38)

where $\nabla u$ is a characteristic norm of $\nabla \mathbf{u}$ on the element, computed as the average $L^2$ norm of the nodal velocity gradients. It is important to note that the adjoint stabilization coefficients depend solely on $\mathbf{u}$, not $\tilde{\mathbf{u}}$, because the adjoint flow field is transported at (minus) the state velocity. Note also that Equation (35) features boundary terms evaluated at the outlet because the integration by part of the pressure and viscous terms unveils a boundary term

$$\begin{array}{c}\hfill {\int }_{\partial \Omega }(\tilde{p}\mathbf{n}+2\mu \epsilon \left(\tilde{\mathbf{u}}\right)·\mathbf{n})·\mathbf{w}\mathrm{d}s=-{\int }_{{\Gamma }_o}(\rho (\mathbf{u}·\mathbf{n})\tilde{\mathbf{u}}+{\partial }_{\mathbf{u}}J)·\mathbf{w}\mathrm{d}s,\end{array}$$

(39)

due to the adjoint boundary condition (13).

Equation (35) is fully implicitly integrated, except for the outflow boundary term that needs to be treated explicitly for implementation convenience. Even though the last computed adjoint solution (pertaining to the previous design) is used to evaluate the boundary term, this simple scheme has been found to converge to identical shapes and cost function minimum compared to solving iteratively with relaxed sub-iterations. Due to the linearity of Equations (10) and (11), this, in turn, cuts down the numerical effort, as only one single linear system needs to be solved at each update step, for which we use a BCGS iterative algorithm with a tolerance threshold set to ${10}^{-12}$ and LU factorization as a preconditioner.

4.2.3. Interface Update Scheme Using the Convective Level-Set Method

The auto-reinitialization level-set problem (19) is solved using an SUPG method [43,44], whose stabilization proceeds from that of the ubiquitous convection–diffusion-reaction equation [45,46]. The associated variational problem is formulated as

$$\begin{array}{c}\hfill {\int }_{\Omega }({\partial }_{\tau }\varphi +{\mathbf{a}}_{\tau }·\nabla \varphi )\xi \mathrm{d}v-{\int }_{{\Omega }_k}{\tau }_3^{}{r}_3^{}{\mathbf{a}}_{\tau }·\nabla \xi \mathrm{d}v={\int }_{\Omega }S\xi \mathrm{d}v,\end{array}$$

(40)

with the residual

$$\begin{array}{c}\hfill -r_3^{}={\partial }_{\tau }\varphi +{\mathbf{a}}_{\tau }·\nabla \varphi -S,\end{array}$$

(41)

and the stabilization coefficient

$$\begin{array}{c}\hfill {\tau }_3^{}=\frac{1}{{\tau }_t\left(a_{\tau }\right)}.\end{array}$$

(42)

It can be easily checked that all terms scale as $1/\Delta \tau$, so we can set $\Delta \tau =1$ without any loss of generality because the solution is ultimately independent of the pseudo-time-step value. As the convection velocity ${\mathbf{a}}_{\tau }$ depends on the main unknown $\varphi$, Equation (40) is solved with the semi-implicit treatment of the convection term and explicit treatment of the source term and stabilization coefficients. All linear systems are solved using the GMRES algorithm with incomplete LU factorization as a preconditioner and a tolerance threshold set to ${10}^{-8}$.

5. Numerical Implementation

5.1. Geometrical Constraints

Fluid flow topology optimization is generally performed under geometrical constraints, typically constant or upper bounded surfaces and/or volumes, to avoid the two extreme cases of the solid domain clogging the entire design domain (as in pressure-drop minimization problems) or disappearing altogether (as in drag minimization problems). This is usually performed by adding penalty terms to the Lagrangian, each of which consists of an empirical penalty parameter multiplied by a measure of the constraint’s violation, and whose variations with respect to the state and design variables snowball into the derivation of the adjoint problem and the cost function’s sensitivity. Here, the constraint of a constant volume of fluid $V_{target}$ is applied a posteriori, i.e., we solve the unconstrained problem presented in Section 2, in the sense that no penalty term is added to the Lagrangian, although the optimization remains subject to Navier–Stokes as state equations. Once the convective level-set method presented in Section 4.2.3 has updated the interface position, a first pass of anisotropic mesh adaptation is performed, after which the volume of the fluid domain is computed as

$$\begin{array}{c}\hfill V_{\phi }={\int }_{\Omega }{H}_{ϵ}\left(\phi \right)\mathrm{d}v,\end{array}$$

(43)

where $H_{ϵ}$ is the smoothed Heaviside function on the fluid domain defined as

$$\begin{array}{c}\hfill H_{ϵ}\left(\phi \right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phi <-ϵ,\hfill \\ {\displaystyle \frac{1}{2}\left(1-\frac{\phi }{ϵ}-\frac{1}{\pi }sin\left(\pi \frac{\phi }{ϵ}\right)\right)}\hfill & \mathrm{if}\left|\phi \right|\le ϵ,\hfill \\ 0\hfill & \mathrm{if}\phi >ϵ,\hfill \end{array}\right.\end{array}$$

(44)

and $ϵ$ is a regularization parameter set to $2h_{\perp }$. A simple dichotomy approach is then used to optimize a constant deformation $\delta \phi$ meant to enlarge ($\delta \phi <0$) or shrink ($\delta \phi >0$) the fluid domain until the difference $|V_{\phi +\delta \phi }-V_{target}|$ between the actual and target volumes drops below a certain tolerance. At this point, we cut off $\phi +\delta \phi$ and perform a second pass of mesh adaptation. Two points are worth mentioning. First, because each offset changes the min-max values of the truncation, the above procedure requires knowledge of the level set $\phi$, in addition to the filtered level set $\varphi$. A brute force algorithm, therefore, initially performs a complete reconstruction of the distance function from the zero iso-value of $\varphi$, as only the filtered level set (not the level set) is evolved during the convection-reinitialization step. Second, only small deformations are considered so that no intermediate mesh adaptation passes are required. By doing so, the total cost is essentially that of performing the second pass of the mesh adaptation, as further discussed below.

5.2. Steepest-Descent Update Rule

In practice, the displacement used to perform the update step is defined as

$$\begin{array}{c}\hfill \beta =-\theta \frac{\mu (\nabla \tilde{\mathbf{u}}·\mathbf{n})·(\nabla \mathbf{u}·\mathbf{n}){\chi }_{\Gamma }^{}\left(\mathbf{x}\right)}{{\displaystyle \underset{\Omega }{max}\mu (\nabla \tilde{\mathbf{u}}·\mathbf{n})·(\nabla \mathbf{u}·\mathbf{n}){\chi }_{\Gamma }^{}\left(\mathbf{x}\right)\prod _l^{}\zeta \left(\right||\mathbf{x}-{\mathbf{x}}_s^l\left|\right|)}},\end{array}$$

(45)

where $\theta >0$ is a descent factor that controls the step taken in the gradient direction, and ${\chi }_{\Gamma }^{}$ and $\zeta$ are activation functions between 0 and 1 that ensure that the design is fittingly updated only in the relevant regions of the computational domain. More details are as follows:

• ${\chi }_{\Gamma }^{}$ is a binary filter that returns a value of 1 only at nodes within a distance E of the interface. This is because the normal vector in a level-set framework is recovered as $\mathbf{n}=\nabla \varphi /\left|\right|\nabla \varphi \left|\right|$, so the displacement is non-zero in the whole fluid domain, even far from the interface where $\mathbf{n}$ has a unit norm because $\left|\right|\nabla \varphi \left|\right|$ only tends asymptotically to zero. In return, the update step can break down numerically at nodes nearly equidistant from two subparts of the interfaces (for instance, the centerline of a channel).
• $\zeta$ is a smooth filter assigning a 0 value to a position ${\mathbf{x}}_s^{}\in \partial \Omega$, which is singled out prior to optimization, because the flow there may be driven to a singularity, and ill-defined velocity gradients may cause large, unphysical displacements. Such singularities can be dealt with numerically by appending fluid/solid Dirichlet boundary conditions to the level-set convection-reinitialization problem. Nonetheless, they must not be included in the normalization step to avoid forcing excessively small displacements along the remaining part of the interface, thereby considerably slowing down the convergence rate of the iterative optimization process. Here, we use hyperbolic tangent filters

  $$\begin{array}{c}\hfill \zeta \left(r\right)=\frac{1}{2}+\frac{1}{2}tanh\left({\alpha }_{s}tan\left(-\frac{\pi }{2}+\frac{\pi }{2}\frac{r}{r_s^{}+{ϵ}_{s1}}+{ϵ}_{s2}\right)\right),\end{array}$$

  (46)

  increasing from 0 to 1 within a distance of $2r_s^{}$ from the singularity, with $r_s^{}$ a transition radius such that

  $$\begin{array}{c}\hfill 4r_s^{}<\underset{l,m}{min}\left|\right|{\mathbf{x}}_s^l-{\mathbf{x}}_s^m\left|\right|,\end{array}$$

  (47)

  to prevent overlaps, where ${\alpha }_s$ is a steepness parameter that controls the sharpness of the transition, and ${ϵ}_{s1,2}$ are small regularization parameters to avoid local discontinuities.

Ultimately, the above filtering and normalization steps ensure that the level set is updated using a displacement that is non-zero only in a thin layer of thickness E about the interface, minus a certain number of spheres of radius $r_s^{}$ centered on the singularities.

5.3. Descent Factor

It follows from Equation (45) that the descent factor $\theta$ physically represents the maximum displacement amplitude over the update region of interest. However, in practice, the actual numerical displacement, estimated from the difference between the zero iso-value of the filtered level set before and after transport, has been found to be well below its theoretical value. This is because the state and adjoint velocities are forced to zero in the solid domain. Hence, the displacement, being driven by the velocity gradients, is also zero everywhere in the solid, except in a very narrow region about the interface, typically a couple of elements thick. As a result, it is not possible to explicitly control the displacement achieved numerically at each iteration. A simple scheme for doing so would have been to repeatedly evolve the interface with a small descent factor until the difference between the cumulated and target displacements drops below a certain tolerance. However, the interface can be evolved only once per update step, as the gradient information is lost if the displacement happens to be in the direction of the solid (for the same reason mentioned above). We thus tune the descent factor manually on a case-by-case basis for the achieved displacement to be slightly smaller than the cut-off thickness. This has been found to be a satisfactory trade-off between accuracy and numerical effort, as the number of iterations required for convergence remains very affordable, and the position of the evolved interface is accurately tracked. Displacements larger than the cut-off thickness, conversely, move the level set into regions of the computational domain lacking the proper mesh refinement, which has been found to ultimately affect the accuracy of the interface representation.

5.4. General Algorithm

Figure 3 shows a flowchart of the implemented topology optimization algorithm, in which anisotropic mesh adaptation is key to capturing the interface with the highest precision possible. Note that as a consequence of the level-set-based technique used to enforce the volume of fluid constraint, convergence is achieved not when the displacement is identically zero (as would be the case using a penalized Lagrangian approach) but when the displacement is uniform along the interface. However, this cannot be easily performed on the fly; rather, we iterate until a maximum number of iterations has been reached and evaluate convergence a posteriori (see Section 7).

6. Numerical Benchmarks

This section assesses the accuracy and efficiency of the numerical framework through three examples of two-dimensional $(d=2)$ topology optimization problems recently considered in the fluid mechanics literature. It is thus worth insisting that the novelty lies not in the associated optimal designs themselves but in the accuracy to which the optimal interfaces are captured in the simulation model.

6.1. Preliminaries

All examples feature either a single inlet or multiple identical inlets of width $e_i$, and either a single outlet or multiple identical outlets of width $e_o$. Parabolic flow profiles normal to the boundary are prescribed at all inlets, as defined by

$$\begin{array}{c}\hfill {\mathbf{u}}_i=\frac{3q_i}{2e_i}\left(1-{\left(\frac{2r}{e_i}\right)}^2\right)\mathbf{n},\end{array}$$

(48)

where $q_i$ is the injected volumetric flow rate (the same for all inlets), and r is the distance from the inlet centerline. For each case, the sole control parameter is the Reynolds number defined as $\mathrm{Re}=\rho q_i/\mu$, which amounts to using the inlet width and mean inlet velocity as the reference length and velocity scales. The cost function used to minimize is the net inward flux of total pressure through the boundaries, taken as a measure of the total power dissipated by a fluid dynamic device. Since the orientation of the normal $\mathbf{n}$ yields ${\mathbf{u}·\mathbf{n}|}_{{\Gamma }_i}>0$ and ${\mathbf{u}·\mathbf{n}|}_{{\Gamma }_o}<0$, this can be expressed in the form of (6) using

$$\begin{array}{c}\hfill J=p_{tot}(\mathbf{u}·\mathbf{n})=(p+\frac{1}{2}\rho (\mathbf{u}·\mathbf{u}))(\mathbf{u}·\mathbf{n}),\end{array}$$

(49)

from which the derivatives needed to complete the derivation of the adjoint boundary conditions can be deduced as

$$\begin{array}{c}\hfill {\partial }_{p}J=\mathbf{u}·\mathbf{n},{\partial }_{\mathbf{u}}J=p_{tot}\mathbf{n}+\rho (\mathbf{u}·\mathbf{n})\mathbf{u}.\end{array}$$

(50)

The remaining practical implementation details are as follows:

• All design domains are initialized with spherical solid inclusions coming in various sizes, adjusted for the initial volume of fluid to match the target within the desired tolerance. This essentially removes the need to create new holes by a dedicated nucleation mechanism. The admissible error on the target volume is set to $1\%$ (actually, in the case where $d=2$, this refers to the cross-sectional area or volume-per-unit length in the third dimension, but we chose to retain the volume terminology for the sake of generality).
• Leads of length $l_c$, appended normally to the boundary, are used to systematically convey the fluid into and out of the design domain. This ensures numerical consistency, as the exact problem formulation may vary depending on the case, the reference, and the problem dimensionality, and it is not always clear whether such leads are included in the design domain (which they are here, although they are not considered in the volume constraint, the definition of the target volume, or the computation of the volume of fluid).
• Since the reference design domains (without the leads) consist of square and rectangular cavities, the singular points excluded from the displacement normalization step are the sharp intersections between the leads and the boundary of the cavities (without it being a consequence of explicitly representing the leads, as the same procedure has been found to be suitable without such appendage).
• The leads are excluded from the displacement normalization step, for which we simply add to the max argument of (45) a binary filter, returning a value of 0 at all nodes located inside the pipes. This is implemented to avoid slowing down the convergence rate of the iterative optimization process, as the maximum displacement would otherwise be located in the leads (because the easiest way to minimize the dissipated power is to suppress the flow by having the solid entirely clogging the leads).
• Boundary conditions are appended to the auto-reinitialization level-set equation in the form of fluid at the inlet and outlet, and solid everywhere else.
• All meshes have been checked to ensure they have an element-to-node ratio close to 2 (as it should be for dense meshes made up of triangular elements). The mesh information is thus documented below in terms of its equivalent number of elements $N_{el}=2N_n$ to facilitate a comparison with the available literature.

6.2. Design of a Pipe Bend

First, we consider the design of a pipe bend, which is a standard example for topology optimization in fluid dynamics [14,17,47,48,49,50] used to provide a first verification and characterization of the method. All relevant problem parameters are provided in

Table 1. Numerical parameters for the pipe-bend, four-terminal-device, and double-pipe-topology optimization problems.

$\Omega =[0;1]\times [0;1]$	$[0;0.7]\times [0;1]$	$[0;1.5]\times [0;1]$	Design domain
$V_{target}=0.25$	0.28	0.5	Target volume of fluid
$\mathrm{Re}=2$	»	»	Reynolds number
$q_i=0.0266$	»	$0.0222$	Injected volumetric flow rate
$e_i=0.2$	»	$0.166$	Inlet width
$e_o=0.2$	»	$0.166$	Outlet width
$l_c=0.1$	0.4	0.1	Conveying pipes length
$N_n$ = 30,000	»	40,000	No. mesh nodes
$N_{el}$ = 60,000	»	80,000	No. mesh elements
$h_{\perp }=0.0001$	»	»	Min. interface normal mesh size
$\Delta t=0.1$	»	»	CFD Numerical time step
$E=0.005$	»	»	Level-set cut-off thickness
$\left|\delta \phi \right|=0.001$	»	»	Initial volume recovery offset
$r_s^{}=0.0125$	»	»	Transition radius
${\alpha }_s=2.1$	»	»	Sharpness parameter
$({ϵ}_{s1},{ϵ}_{s2})=(0.0005,0.005)$	»	»	Regularization parameters

. The design domain is a square cavity with a unit length, which has one inlet (left side) and one outlet (bottom side)—see Figure 4. The aim is to determine the optimal design of the pipe bend that connects the inlet to the outlet and minimizes the dissipated power while subject to the constraint that the fluid must occupy $25\%$ of the cavity, which is the same volume as a quarter torus fitting exactly into the inlet and outlet areas.

A total of 400 iterations are run with 60,000 mesh elements, as illustrated in Figure 5 by the anisotropic adapted mesh, zero iso-value of the level-set function, and velocity norm of a selected sample. The method is found to easily handle the multiple topological changes (e.g., merging or cancellation of holes) occurring over the course of the optimization. Also, consistently with the results in Section 3, all meshes exhibit the expected refinement and deformation, with coarse and regular elements away from the interface between the solid and fluid (all the more so in the solid domain, where only a few tens of elements are used), but fine, extremely stretched elements on either side of the interface, allowing the velocity to smoothly transition to zero across the boundary layer (see the close-up in Figure 6). In return, the interfaces are sharply captured, not only at optimality but also during all stages of the optimization. This represents a major improvement in the accuracy of the geometric representation compared to the available recent literature, as even traditional (isotropic) adaptive mesh refinement techniques have been shown to yield quality issues (such as staircase effects) in smoothly curved regions. Ultimately, we obtain an almost straight channel that is nearly identical to that documented in [47], albeit with a higher resolution. This is because most energy is dissipated by shear at low Reynolds numbers, so an optimal flow pipe is preferably as short and wide as possible. The obtained results are further discussed in Section 7, with a particular emphasis on the convergence rate and sensitivity of both the optimal and the optimization path to the number of nodes.

6.3. Design of a Four-Terminal Device

Our second numerical example deals with the minimization of power dissipation in a four-terminal device [51]. This is a follow-up to the previous pipe-bend problem in which the cavity features a rectangular cavity with a unit height and an aspect ratio of 0.7:1. It has two inlets and two outlets, distributed antisymmetrically on the left and right sides to level up the complexity (see Figure 7 for an illustration of the configuration and Table 1 for the remaining problem parameters). The aim is to determine the optimal design that connects the inlets to the outlets, subject to the constraint that the fluid must occupy $40\%$ of the cavity, which is the same volume as two straight parallel pipes fitting into the upper and lower pairs of the inlet/outlet.

A total of 300 iterations are run with $\mathrm{60,000}$ mesh elements (see Figure 8, which shows the anisotropic adapted mesh, zero iso-value of the level-set function, and velocity norm of a selected sample collected over the course of the optimization). All adapted meshes are especially reminiscent of their pipe-bend counterparts, with coarse, regular elements away from the interface and fine, elongated elements on either side of the interface (see Figure 9) allowing for an accurate representation of the boundary layers at all stages of the optimization, even in the leads). Ultimately, we obtain a pair of U-turns connecting each inlet to the outlet on the same side of the design domain. This is consistent with the literature results, showing that the U-turn solution is favored over the simpler parallel channels solution at aspect ratios larger than 0.6:1 [14,17,51]; only the present solution is captured with superior accuracy. This is again because optimal pipes at low Reynolds numbers are preferably short and wide, and the cost of bending the fluid stream is low, given that most fluid flows in the (shorter) inner region.

6.4. Design of a Double Pipe

In the third numerical example, we consider the double-pipe problem, another benchmark for fluid topology optimization [47,50,52,53], whose parameters are provided in Table 1. The design domain is a rectangular cavity with a unit height and an aspect ratio of 3:2. It has two inlets (left side) and two outlets (right side)—see Figure 10. The aim is to determine the optimal design of the double pipe that connects the inlets to the outlets and minimizes the dissipated power, subject to the constraint that the fluid must occupy $33.3\%$ of the cavity, which is the same volume as two straight parallel pipes fitting into the upper and lower pairs of the inlet/outlet.

A total of 3000 iterations are run with 80,000 mesh elements (due to the larger design domain), during which time the design goes through several complex stages, all accurately represented on anisotropic adapted meshes, as evidenced by the selected sample shown in Figure 11 and Figure 12. Ultimately, the optimal design resembles a single-ended wrench, with the two inlet pipes connecting to a wider pipe in the center of the domain. This wider pipe then connects to a single outlet (either the upper or lower outlet since the set-up exhibits horizontal reflectional symmetry). Since the optimal flow pipe at low Reynolds numbers is preferably short and wide, this represents a better trade-off between transporting fluid the shortest way and transporting it in the widest possible pipe. Note that the obtained solution differs from that of the double-ended wrench documented in [47,50,52], in which the center pipe ultimately connects to the two outlets. This is because the authors prescribe parabolic flow profiles at both the inlets and outlets. The flow is thus forced to exit via both outlets, whereas it can exit via a single outlet under the more physical zero-pressure/zero-viscous-stress condition used here, which allows for saving the cost of pipe splitting [53]. The number of iterations for this case is larger by one order of magnitude compared to those of the pipe-bend and four-terminal problems, which can be easily explained by the fact that the optimization must bypass the basin of attraction of the double-ended wrench, which remains a local minimizer. This is all the more difficult because the cost function of both minimizers differs by only 10%. However, we show in Section 7 that this particular feature is ultimately very sensitive to the number of nodes used to perform the mesh adaptation.

7. Discussion

7.1. Computational Efficiency

Figure 13 presents the detailed timing results obtained by averaging 300 dedicated update steps performed using the parameters compiled in Table 1; 100 steps for each case presented in Section 6.2–Section 6.4. As could have been expected, the cost of an iteration is dominated by that of computing the state solution. This takes about 10 Navier–Stokes iterations, representing 40% of the total cost, which can be scaled down substantially in the context of steady-state problems using an iterative Newton-like method. Otherwise, the cost of performing the two passes of mesh adaptation represents about a cumulative 40% of the total cost. Meanwhile, the cost of both geometrically reinitializing the signed-distance function level set and optimizing the volume constraint offset is very affordable (less than 1% in total, with 4-5 dichotomy iterations needed to reach the desired accuracy of $1\%$). Such conclusions presumably carry over to any other problem of the same dimensionality, tackled with comparable parameters.

7.2. Convergence and Mesh Dependency

Since we perform a fixed number of iterations here, convergence is assumed when the sliding average over the 10 latest cost functional values is less than a prescribed error set to 2% of the cost functional average over the final 50 iterations. The reason is twofold. First, the cost function keeps varying even after convergence because the mesh slightly changes between consecutive iterations, and so does the volume of fluid as long as the deviation from the target does not exceed the admissible error. Second, assuming convergence simply when the relative difference between two successive cost functional values is less than a prescribed error has been found to yield premature convergence to the double-ended wrench local minimizer of the double-pipe problem. Note that all data discussed below pertain to a single optimization run. Rigorously speaking, convergence is best assessed by averaging results over multiple independent runs, as mesh adaptation is not a deterministic process (the outcome depends on the processors and number of processors used, and any initial difference propagates over the course of the optimization because the meshes keep being adapted at each iteration), but we have found very little variability by doing so.

Exhaustive convergence data are provided in

Table 2. Convergence data for the pipe-bend, four-terminal-device, and double-pipe topology optimization problems. All cost function values are made non-dimensional using the inlet width and mean inlet velocity (equivalently, using $\rho q_i^3/e_i^2$ as the reference cost functional value).

Convergence Iter.	Cost Function	No. Mesh Elements
295	33.1	80,000
306	32.7	60,000
212	32.9	40,000
148	32.1	20,000
153	68.9	80,000
129	69.1	60,000
104	69.0	40,000
68	68.6	20,000
2460	68.6	105,000
1750	67.6	80,000
2130	68.2	55,000
1594	67.0	25,000

for all three cases reported above. Putting the obtained results in a broader context is not easy because convergence is rarely documented in the literature, and even when it is, the key factors explicitly affecting convergence (e.g., initial shape, convergence criterion, and threshold) are not. In practice, our literature review did not reveal any other study putting all these levels of information together. Here, the pipe-bend problem converges within 306 iterations, which is well above the convergence iteration reported in the seminal paper by Borrvall and Peterson [47], lying in a range from 64 (using 2500 mesh elements) to 85 (using 10,000 mesh elements). A first explanation is that all designs in the aforementioned reference were evaluated on the same isotropic mesh, hence the descent factor was not constrained by the thickness of the level set and larger values could be used to speed up convergence. Another possibility, further discussed below, is that most studies in the literature rely on a limited number of elements in a range from 5000 to 20,000. Conversely, we purposely use a much larger value to equally assess all steps of the optimization but ultimately slows down the convergence rate.

A first important point is that such a large number of nodes is mostly useful during the early stage of optimization, where the many solid inclusions dramatically increase the surface of the interfaces that needs to be captured. In practice, this has been found to decrease significantly after the first dozens of iterations (by a factor of 3–10 depending on the case). See Figure 14a, which shows the surface area computed over the first 200 iterations as

$$\begin{array}{c}\hfill S_{\phi }={\int }_{\Omega }{\delta }_{ϵ}\left(\phi \right)\mathrm{d}v,\end{array}$$

(51)

where ${\delta }_{ϵ}$ is the Dirac function

$$\begin{array}{c}\hfill {\delta }_{ϵ}\left(\phi \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2ϵ}\left(1+cos\left(\pi \frac{\phi }{ϵ}\right)\right)}\hfill & \mathrm{if}\left|\phi \right|\le ϵ,\hfill \\ 0\hfill & \mathrm{if}\left|\phi \right|>ϵ,\hfill \end{array}\right.\end{array}$$

(52)

smoothed using the same regularization parameter $ϵ$ as the Heaviside function (44). A second important point is that the anisotropic mesh adaptation algorithm refines the mesh in the hierarchical importance of the level-set gradient. If new geometrical features appear in the solution (associated with high gradients), the mesh is automatically coarsened in regions with lower gradients and refined near the newly emerging features. If the number of nodes is large, as has been the case so far, the decrease in the interface surface area allows for resolving finer, more complex patterns without degrading the accuracy in other parts of the design domain because the coarsened regions are actually over-resolved. This is evident through the progressive mesh refinement in the fluid domain in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, as more and more elements become available to improve the mesh in other regions of the domain. If the number of nodes is small, all essential features of the solution will remain well captured (albeit to a slightly lower accuracy), but the finest, most intricate topologies will be smoothed out. This is expected to yield faster convergence because the sensitivity will have fewer overshoots and the displacement will be more homogeneously distributed over the interface.

Confirmation comes from additional runs performed on both denser and coarser meshes. The look-alike design samples documented in Figure 15, Figure 16 and Figure 17 indicate that all runs follow the same optimization path, with smaller details being captured as the number of elements increases. Just as important is the fact that all optimal solutions are independent of the mesh size. This means that the ability of the method to represent smaller and smaller features does not result in smaller and smaller features being represented in the optimal designs, as can occur in the stiffness optimization of mechanical structures [54] (see also [47] for proof that total power dissipation minimization is well posed in this respect). For the pipe-bend and four-terminal-device problems, the expected behavior is observed, as coarser-mesh runs converge substantially faster. For instance, the pipe bend with 20,000 elements converges within 165 iterations, which is lower by about 45% compared to using 60,000 elements. If a less restrictive convergence threshold of 5% is enforced, this drops to 102, which is only slightly above the 85 iterations in [47]. This improvement carries over to the four-terminal-device problem, whose run with 20,000 elements converges within 68 iterations, which is lower by about 50% compared to using 60,000 elements (this further drops to 61 using a convergence threshold of 5%). Note that in both cases, coarser does not equate to coarse, as the convergence information compiled in Table 2 shows that the coarsest meshes actually resolve the optimal interface with excellent accuracy.

Meanwhile, convergence for the double pipe ends up being almost arbitrary, and the algorithm has difficulties in finding the optimal topology due to the characteristics of the cost function landscape. The convergence history in Figure 14d shows that the run with 55,000 elements does indeed converge faster to the double-ended wrench solution minimizer, but then needs more iterations to ultimately reach the single-ended wrench global minimizer, so convergence is ultimately slower than when using 80,000 elements. Interestingly, the run with 25,000 elements successfully bypasses the local minimizer because the lack of elements does not allow for representing the complexity prevailing in the early stage of the optimization. This ends up quickly breaking the horizontal reflectional symmetry, but the convergence rate ultimately remains comparable to that with 80,000 elements, which raises the possibility that the wrench solutions are actually flat minimizers.

7.3. Application to a Simplified Flow Distributor Problem

Finally, we consider a practical application of the developed framework with the simplified flow distributor problem shown in Figure 18. The design domain for this case is a rectangular cavity with an aspect ratio of 0.4:0.5, widening through four consecutive steps of aspect ratio 0.2:0.1, resulting in a stair shape with an overall aspect ratio of 1.2:1.3. It features a single inlet on the left and six identical outlets on the right. The aim is to determine the optimal design that connects the inlet to the outlets and minimizes the dissipated power, subject to the constraint that the fluid must occupy 40% of the cavity, and the flow must be distributed evenly over the multiple outlet orifices for each outlet to have 1/6 of the fluid flow entering through the inlet. Since the zero-pressure outflow condition does not force the inlet to connect to all the outlets (as has been assessed in Section 6.4 for the double-pipe problem), we use the modified cost function

$$\begin{array}{c}\hfill J=(1-\omega )p_{tot}(\mathbf{u}·\mathbf{n})+\frac{\omega }{2}\left|\right|\mathbf{u}-{\mathbf{u}}_{target}{\left|\right|}^2,\end{array}$$

(53)

where $\omega$ is a scalar-valued factor that weighs the priority given to either the power dissipation or uniformity of the outflow distribution, and ${\mathbf{u}}_{target}$ is a target parabolic velocity distribution, whose outlet centerline velocity is adjusted to ensure that the mass flow exiting through the outlets precisely matches that entering through the inlet. In doing so, the theoretical framework developed in Section 2 can be applied directly, provided that the adjoint boundary conditions are updated accordingly using

$$\begin{array}{c}\hfill {\partial }_{p}J=(1-\omega )\mathbf{u}·\mathbf{n},{\partial }_{\mathbf{u}}J=(1-\omega )p_{tot}\mathbf{n}+(1-\omega )\rho (\mathbf{u}·\mathbf{n})\mathbf{u}+\omega (\mathbf{u}-{\mathbf{u}}_{target}).\end{array}$$

(54)

The entire domain for this case is meshed with 50,000 elements, with the remaining parameters provided in

Table 3. Numerical parameters for the flow distributor problem.

$\Omega =[0;1.2]\times [0;1.3]$	Design domain
$V_{target}=0.4$	Target volume of fluid
$\mathrm{Re}=1$	Reynolds number
$q_i=0.08$	Injected volumetric flow rate
$e_i=0.12$	Inlet width
$e_o=0.1$	Outlet width
$l_{c,i}=0.4$	Inlet conveying the pipe length
$l_{c,o}=0.3$	Outlet conveying the pipe length
$N_n$= 25,000	No. of mesh nodes
$N_{el}$= 50,000	No. of mesh elements

. All other parameters are identical to those in Table 1. Note that a large weight, $\omega =0.999$, is used here to achieve comparable orders of magnitude for both the power dissipation and uniformity contributions to the cost function. This yields the optimal duct shown in Figure 19, which delivers most of the fluid in the center area of the cavity before evenly distributing it to the outlet channels (within a 5% accuracy) via a fine comb-like structure. The obtained solution has all the attributes of a power dissipation optimal solution: an initial large straight pipe that ultimately divides into a near-perfect symmetrical network of six short pipes to minimize the cost of bending the fluid stream. It differs from that documented in [55] for a similar problem because the authors did not consider power dissipation in their cost function and imposed only an upper bound on the volume of fluid. Despite this difference, it showcases the potential of the method for providing smooth solutions to engineering problems of practical interest. For instance, it could be applied to the design of compact and lightweight heat exchangers, such as those widely used in air-conditioning (the design domain being representative of the refrigerant distributor section), or microfluidics, where minimizing dissipation while maintaining an even fluid distribution in all branches of a network is of great interest to improve the performance of lab-on-a-chip devices.

8. Conclusions

The present study proves that it is feasible to perform topology optimization of Navier–Stokes flows using anisotropic meshes adapted under the constraint of a fixed number of nodes. The proposed approach combines a level-set method for representing the boundary of the fluid domain through the zero iso-value of a signed distance function and stabilized formulations of the state, adjoint, and level-set transport equations cast in the variational multiscale (VMS) framework. The method has been shown to allow for drastic topology changes during the optimization process. Nonetheless, the main advantage over existing methods is the ability to capture all interfaces to a very high degree of accuracy using adapted meshes whose anisotropy matches that of the numerical solutions. This provides hope that the method can facilitate the transition to manufacturable CAD models that closely resemble the optimal topology.

The method has been tested on several examples of power dissipation minimization in two dimensions. The obtained optimal designs are identical to those of the reference literature results, which assess the relevance of the present implementation for designing fluidic devices, as further illustrated by a simplified engineering case optimizing a flow distributor to minimize power dissipation while maintaining even flow distribution at multiple outlets. All optimal designs are shown to be mesh-independent, although the convergence rate does decrease as the number of nodes increases, despite the method being able to resolve smaller and smaller geometrical features. Exhaustive computational efficiency data are reported with the hope of fostering future comparisons, but it is worth emphasizing that we did not seek to optimize the said efficiency, for instance, using an iterative Newton-like method to compute all state solutions, which takes up the bulk of the computational time. The obtained results show the difficulty of determining the global minimum when two strong minima are competing, which simply reflects the fact that gradient-based algorithms are easily trapped in local optima, all the more so when applied to stiff nonlinear problems (gradient-free methods are better equipped in this regard, but can be more complex to implement and use). Future work will be aimed at considering a multi-component adaptation criterion to take into account the difference in the spatial supports of the state and adjoint solutions and extending the present method to more general two- and three-dimensional cases, including fluid–thermal coupling problems.