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

Abstract

a b s t r a c tA discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice Boltzmann method (LBM) with multiple-relaxation-times (MRT) is developed. The lattice Boltzmann fluid solver is supplemented by a porosity model using a Darcy force. The continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels. The adjoint LBM equation, which is used to compute the gradient of the optimization objective with respect to the design variables, is derived in moment space and found to be as simple as the original LBM. The moment based spatial momentum derivatives used to express the discrete objective functional (cost function) have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field. This is particularly useful if dissipation is a design criterion as demonstrated in this paper. The method is validated by a detailed comparison with results obtained by Borrvall et al. for Stokes flow. While their approach is only valid for Stokes flow (i.e. very low Reynolds numbers) our approach in its present form can in principle be applied for flows of different Reynolds numbers just like the Navier-Stokes equation based approaches. This point is demonstrated with a bending pipe example for various Reynolds numbers.

A Review of Topology Optimisation for Fluid-Based Problems

“…Yaji et al [36] formulated a level set based topology optimisation of steady flow using the LBM and a continuous adjoint sensitivity analysis based on the Boltzmann equation. Liu et al [37] presented a density-based method based on the LBM using a discrete adjoint sensitivity analysis posed in the moment space. Yonekura and Kanno [38] suggested to use transient information for steady state optimisation using the LBM by modifying the design during a single transient solve.…”

“…It is clear that FEM is the most widely used discretisation method with 134 papers or 76%. The next most used method is LBM at 28 papers [14,20,22,24,26,[36][37][38]44,55,65,69,75,77,78,84,97,108,113,118,119,126,131,163,168,172,173,186] or 16%. PM is the least used method with only a single paper [79].…”

Topology optimization of unsteady flow problems using the lattice Boltzmann method

“…Alternatively, it is possible to exploit the local nature of the lattice Boltzmann equation to derive explicit expressions for the Lagrange multipliers in each time step, thereby bypassing the need for matrix routines entirely; in this way the adjoint problem can be solved on the same time scale as the forward problem. A detailed derivation of a discrete adjoint lattice Boltzmann method has already been given by Liu et al [36], who applied it to steady-state fluid topology optimization problems. Below we give an outline of the derivation of the explicit adjoint expressions, including the treatment of the boundary conditions described in section 2.2, which to our knowledge have not been covered before.…”

Universal Differential Equations for Scientific Machine Learning

“…Previous research has shown that the discrete adjoint approach is more stable than continuous adjoints in some cases [43,39,44,45,46,47] while continuous adjoints have been demonstrated to be more stable in others [48,45] and can reduce spurious oscillations [49,50,51]. This trade-off between discrete and continuous adjoint approaches has been demonstrated on some equations as a trade-off between stability and computational efficiency [52,53,54,55,56,57,58,59,60].…”