We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of solid boundary conditions. We show that a porosity model is ideally suited for topology optimization purposes and models no-slip boundary conditions with sufficient accuracy when compared to interpolation bounce-back conditions. Augmenting the porous boundary condition with a shaping factor, we define a generalized geometry optimization formulation and derive the corresponding sensitivity analysis for the single relaxation LBM for both topology and shape optimization applications. Using numerical examples, we verify the accuracy of the analytical sensitivity analysis through a comparison with finite differences. In addition, we show that for fluidic topology optimization a scaled volume constraint should be used to obtain the desired “0-1” optimal solutions.
Design optimization of flow domains based on the Navier–Stokes equations has found widespread acceptance. We refer, for example, to the body of work by Jameson [1] on shape optimization for external and internal flows, as well as to the recent monographs by Gunzburger [2], and Mohammadi and Pironneau [3]. Topology optimization for fluids was introduced by Borrvall and Petersson [4]. In recent years it has received increased attention due to its general applicability to a wide range of problems in hydrodynamics and beyond [5], [6], [7], [8], [9], [10], [11] and [12].
The primary difference between shape and topology optimization is illustrated in Fig. 1, showing that shape optimization is limited to varying the location of the boundaries in order to improve the performance of an existing design, while topology optimization can lead to conceptually new design features and layouts without the need for an initial close-to-optimum design. For shape optimization, boundaries are generally parameterized such that a change of parameters leads to a change in the boundary. For material-based topology optimization on the other hand, the geometry of a body is represented via an associated material distribution. In the design domain shown in Fig. 1, each computational node/element is associated with a continuous material description function, which defines if a given node/element contains solid material, fluid, or an intermediate fluid-filled porous material. Here, it is generally the goal to obtain optimal solutions without intermediate material distributions, referred to as “0-1” solutions.
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Fig. 1.
Comparison of shape and topology optimization.
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The authors have recently presented a topology optimization approach based on a lattice Boltzmann fluid solver in order to obtain optimal designs for internal and external flows [12], [13], [14] and [15]. The lattice Boltzmann method (LBM) was chosen to benefit from the inherent use of immersed boundary techniques (IBT), a simple porosity model, and the overall simplicity and versatility of the LBM algorithm. A key component of this LBM-based design optimization solver is the use of an adjoint parameter sensitivity analysis. Parameter sensitivity analysis is not limited to design optimization and is for example also used for reliability analysis, system identification, and reduced order models [16], [17] and [18]. Up to now, only Teritek et al. [17] have applied adjoint sensitivities in an LBM framework, focusing on the identification of two optimal parameters within the multi-relaxation LBM scheme [19] using an adjoint LBM formulation.
In this work, we present the adjoint sensitivity analysis for the single-relaxation LBM scheme in detail, with a focus on the problem formulation for topology optimization, which requires the identification of large numbers of unknown parameters. In addition, the adjoint sensitivity analysis for shape optimization is discussed.
In this study of adjoint parameter sensitivity analysis for the single relaxation lattice Boltzmann method we have placed special emphasis on design optimization and derived the sensitivity equations topology optimization while outlining the sensitivity equations for shape optimization. We discussed the LBM bounce-back and porosity conditions in the framework of local and global immersed boundary techniques, showing that the porosity condition is ideally suited for topology optimization applications. Finally, we verified the analytical adjoint sensitivity analysis and have shown that the introduction of a shaping factor in the porosity condition and its inclusion in the volume constraint yield improved optimal designs consisting of “0-1” solutions.
