Adjoint lattice Boltzmann equation for parameter identification

Abstract

The lattice Boltzmann equation is briefly introduced using moments to clearly separate the propagation and collision steps in the dynamics. In order to identify unknown parameters we introduce a cost function and adapt control theory to the lattice Boltzmann equation to get expressions for the derivatives of the cost function vs. parameters. This leads to an equivalent of the adjoint method with the definition of an adjoint lattice Boltzmann equation.To verify the general expressions for the derivatives, we consider two elementary situations: a linearized Poiseuille flow to show that the method can be used to optimize parameters, and a nonlinear situation in which a transverse shear wave is advected by a mean uniform flow. We indicate in the conclusion how the method can be used for more realistic situations.

Topology optimization of unsteady flow problems using the lattice Boltzmann method

“…In this paper, we adopt the first approach. The discretize then optimize approach was first used by Tekitek et al [20] for finding optimal parameters for a lattice Boltzmann model, while Krause et al [21] applied the discretize then optimize approach to flow control problems.…”

Discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice Boltzmann method

“…It is interesting to note that the transformation matrix in the adjoint LBM is the transposed inverse of the transformation matrix of the original LBM (similar discoveries were made by Tekitek et al in [16]). We also obtain the transformation for the post-collision state:…”

“…(11)). This phenomenon was first observed by Tekitek et al in [16], and proved again in Krause's continuous approach [24]. It is also interesting to note that equations for the post-collision adjoint moments and distributions n * and g * arise from the variations of the pre-collision primal moments and distributions δm and δf and vice versa.…”

“…The adjoint analysis in the context of the lattice Boltzmann method was pioneered by Tekitek et al for the optimization of relaxation rates of a multiple-relaxation-time LBM [16]. Pingen [17][18][19][20][21][22] conducted a discrete adjoint sensitivity analysis based on LBM for various stationary topology optimization problems.…”

Application of Lattice Boltzmann Method to sensitivity analysis via complex differentiation

“…This study was conducted using a varying porosity model, and an adjoint problem. The adjoint problem technique has very recently been used for LBM in [11]. The classical drawback of adjoint problems is their cost in terms of solution storage when dealing with transient problems.…”