Topology optimization in thermal-fluid flow using the lattice Boltzmann method
Introduction
The application of structural optimization methods for fluid dynamics problems has been an attractive area of research for engineers and mathematicians. In the field of structural optimization for fluid dynamics problems, the most widely used approach is based on shape optimization, a kind of structural optimization whose main concept is that the geometry of a solid domain can be optimized, to improve performance, by moving the boundary between the fluid and solid domains. In this research field, Pironneau pioneered a shape optimization method for fluid dynamics problems and constructed the basic mathematical theory for obtaining minimum drag body profiles under Stokes flow [1]. The reader is referred to a recent monograph by Mohammadi and Pironneau [2] in which a variety of shape optimization methods are discussed. However, shape optimization methods only allow changes in boundary shapes, so feasible design modifications are limited. This limitation can be overcome by applying topology optimization, which allows the creation of new holes in the design domain during the optimization process, providing enhanced degrees of freedom compared with shape optimization. In fluid flow topology optimization problems, these newly created holes may be either new fluid or solid domains.
Topology optimization in structural mechanics problems is a well-documented field, since Bendsøe and Kikuchi first proposed the so-called homogenization design method [3]. The basic idea of topology optimization is the introduction of an extended design domain, the so-called fixed design domain, and the replacement of the optimization problem with a material distribution problem, using the characteristic function. The homogenization design method has been applied to a variety of optimization problems, and the density approach, also called the solid isotropic material with penalization (SIMP) method [4], is another currently used topology optimization method. Recent developments in the field of topology optimization have been categorized in a review paper by Sigmund and Maute [5].
Based on the concept of the density approach, Borravall and Petersson [6] pioneered a topology optimization method for minimum power dissipation in Stokes flow problems, where the material distribution in the fixed design domain is represented as either the presence of a fluid or a solid domain. The basic idea of this research is that the fixed design domain is composed of a porous medium governed by the theory of Darcy's law. Thus, the solid and fluid domains in the fixed design domain are represented as porous media of either high or low porosity, and the no-slip boundary condition along the fluid–solid interface can be implicitly satisfied. Based on this methodology, Gersborg-Hansen et al. [7] proposed a topology optimization method for a steady-state fluid flow governed by the Navier–Stokes equations (NSE). Deng et al. [8], and Kreissl et al. [9], extended the steady-state approach to unsteady flow problems. In addition, Evgrafov [10] constructed a mathematical theory to assess the well-posedness of topology optimization problems for the minimization of power dissipation under the incompressible viscous flow.
In most previous research in topology optimization for fluid dynamics problems, the finite element method (FEM) is employed to discretize the flow field governed by the NSE. Although the FEM is a well-studied and accurate approach based on the theory of variational formulation, the computational cost for calculating the flow field dramatically increases when a large-scale computational domain is treated. In particular, due to the inf–sup condition, a high-order shape function must be generally used for the approximation of fluid velocity [11]. Furthermore, an iterative computation, which incurs massive computational cost in large-scale incompressible viscous fluid problems, is necessary to correct the velocity and pressure values so that the conservations of mass and momentum are satisfied.
As an alternative way to solve the flow field, the lattice Boltzmann method (LBM) has become an attractive scheme in the research field of computational fluid dynamics [12], [13], [14], [15]. Comparing the coding used in the LBM with that used in conventional schemes such as the finite element and finite volume methods, LBM algorithm is much simpler since the fluid flow is solved using a time evolution equation, the so-called lattice Boltzmann equation (LBE), and the LBM offers a further advantage, its scalability for complex flow problems such as porous, miscible, and immiscible fluid flow problems. In addition, the LBM enables one to avoid the numerical treatment of iterative computation for the correction of fluid velocity and pressure in the incompressible viscous fluid flow. Topology optimization methods that use the LBM are therefore suitable for dealing with large-scale and complex flow optimization problems.
Pingen et al. [16] proposed a topology optimization using the LBM, and obtained optimal configurations similar to those using a conventional approach proposed by Borrvall and Petersson [6]. Based on the study by Pingen et al., Makhija et al. [17] extended this approach to a scalar transport problem using the multi-relaxation time LBM [18] for the maximization problem of mixing efficiency in a mixing device. In addition, Kreissl et al. [19] proposed a topology optimization method using the LBM for a fluid–solid interaction problem for the design of microchannel devices. However, in the above methodologies based on the research of Pingen et al., a large-scale, hence unwieldy, asymmetric matrix is used to solve the adjoint problem during each iteration of the optimization process. Although a specific method for solving this matrix system was later proposed by Pingen et al. [20], this optimization method is not well suited to more complex large-scale flow problems.
To solve identification problems with the LBM, Tekitek et al. [21] proposed a methodology using the adjoint lattice Boltzmann equation (ALBE), and Krause et al. [22], [23] recently proposed the so-called adjoint lattice Boltzmann method (ALBM). The basic idea of these approaches is that both the state and adjoint fields are solved using the LBM, which can make use of highly efficient algorithms due to the similarity of the locality properties, and the design sensitivities can therefore be obtained without the use of matrix operations. Due to the different ways in which the adjoint sensitivity analysis is conducted, these approaches can be classified in two categories: 1) methods based on a discrete adjoint approach using the ALBE, in which the sensitivity analysis is conducted using discrete equations and the adjoint equation is therefore derived as a discrete equation, the so-called ALBE, and 2) the ALBM, which is based on a continuous adjoint approach in which the sensitivity analysis is conducted using the continuous Boltzmann equation with the Bhatnagar–Gross–Krook (BGK) approximation [24]. In the latter approach, the adjoint equation is therefore obtained as a continuous equation, whose formulation is similar to that of the Boltzmann equation, which is then discretized using the LBM. Krause et al. [22], [23] investigated the parallel performance of the ALBM and demonstrated that the ALBM is exceptionally useful for obtaining efficient parallel implementations, when compared with already well-established schemes.
In the research field of topology optimization, Yaji et al. [25] proposed a topology optimization method based on the ALBM for solving the pressure drop minimization problem that aims to obtain optimal configurations of the two- and three-dimensional flow channels. On the other hand, Liu et al. [26] successfully applied the ALBE in a topology optimization method for a minimum power dissipation problem. Concerning an additional unique topology optimization approach using the LBM, Yonekura and Kanno [27] recently proposed a topology optimization method for a minimum power dissipation problem in which two computational steps, the gradient optimization algorithm and the LBE, are synchronized so that an optimal configuration is rapidly obtained.
In the ALBM [22], [23], the use of the continuous Boltzmann equation prevents the use of the high accuracy boundary conditions that are generally used in the LBM, since the LBM boundary conditions are formulated using discrete particle velocities. Liu et al. [26] recently asserted that most of the boundary conditions for the ALBE had to be defined a posteriori, due to the use of a discrete adjoint approach. Although no-slip or periodic boundary conditions naturally influence the adjoint boundary conditions, their formulation is the same as those of the equations of the state problem, whereas other boundary conditions that are commonly used in fluid flow analysis, such as prescribed velocity or pressure boundary conditions (e.g., [28]), obviously require different formulations than those of the adjoint boundary conditions, due to the complex definition of these boundary conditions. Since various boundary conditions for the LBM [29], [30], [31] are provided, enabling analysis of diverse fluid flow problems, previous optimization methods employing the LBM must be expanded so that any desired boundary condition of the LBM can be treated in the optimization problem. In other words, the adjoint boundary conditions should be theoretically derived under the framework of sensitivity analysis based on the adjoint variable method.
To overcome the problem of how to incorporate the LBM boundary conditions in optimization problems, we propose a new sensitivity analysis based on the ALBM, in which we use the discrete velocity Boltzmann equation with the BGK approximation. Since the discrete velocity Boltzmann equation incorporates discrete particle velocities but continuous space and time, the various boundary conditions for the LBM can be easily introduced, and the adjoint equation can be analytically derived and discretized based on the strategy used in the ALBM [22], [23].
Here, we apply the proposed methodology to isothermal- and thermal-fluid flow optimization problems in which prescribed flow velocity, pressure, temperature, and adiabatic boundary conditions are treated as representative boundary conditions in the LBM. Details of the sensitivity analysis dealing with these boundary conditions are provided to confirm the applicability of the proposed sensitivity analysis. Based on our new formulations, we construct a topology optimization method for the design of a flow channel in which the pressure drop minimization and heat exchange maximization problems are formulated. In the following sections, the basic concept of the LBM is discussed first. Next, the topology optimization problems are formulated for the pressure drop minimization and heat exchange maximization problems, and the procedures used in each sensitivity analysis based on the discrete velocity Boltzmann equation are described in detail. The numerical implementations and optimization algorithms are then explained and, finally, we introduce several numerical examples to confirm the utility of the proposed method.
Section snippets
Basic equation
We now discuss the concept of the LBM that will be applied here to an incompressible viscous fluid while considering the temperature field. In the following, we use the dimensionless variables defined in Appendix A. The basic idea of the LBM is that the fluid regime is represented as an aggregation of fictitious particles, which makes it possible to obtain macroscopic variables such as the fluid velocity, pressure, and temperature, from the moments of the velocity distribution functions that
Topology optimization
Consider a structural optimization problem to determine the boundary of a design domain, Ω, in which the objective function that expresses the intended performance of the target system is to be minimized, or maximized, based on optimization theories. The basic concept of topology optimization is the introduction of a fixed design domain D that includes the original design domain, i.e., , and the use of the characteristic function χ in order to replace the original structural optimization
Sensitivity analysis based on the discrete velocity Boltzmann equation
Based on the ALBM [22], [23], we now consider the strategy for deriving the design sensitivities for the optimization problems discussed in Section 3. The ALBM is based on the use of the LBM to compute the fluid flow in fluid optimization problems, and its key idea is that design sensitivities are derived using the adjoint variable method in which the Boltzmann equation is employed to formulate the Lagrangian. Based on the continuous adjoint approach, the adjoint equation is derived as an
Optimization algorithm
The optimization algorithm of the proposed method is now described.
- Step 1.
The initial values of the state, adjoint, and design variables are set throughout the fixed design domain D.
- Step 2.
The LBE for the pressure drop minimization problem, or the heat exchange maximization problem, is calculated until a steady-state condition is satisfied.
- Step 3.
If the criteria of the objective functional and inequality constraint are satisfied, an optimal configuration is obtained and the optimization is
Numerical examples
Here, we confirm the utility of our proposed method. In all the numerical examples, the reference length and speed are defined as the inlet width and inlet mean velocity magnitude, respectively. Using the kinematic viscosity in Eq. (16), the thermal diffusivity in Eq. (17), and the dimensionless values of the inlet length L and mean velocity magnitude U, the Reynolds number Re and the Prandtl number Pr are defined as For all numerical examples, the initial conditions for the
Conclusion
This paper proposed a topology optimization method using the LBM incorporating a new sensitivity analysis based on the discrete velocity Boltzmann equation. The presented method was applied to pressure drop minimization and heat exchange maximization problems. We achieved the following:
- (1)
Two topology optimization problems were formulated: 1) pressure drop minimization problem, and 2) heat exchange maximization problem. The design sensitivities for these optimization problems were derived based on
Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers 14J02008, 26820032.
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