Topology optimization of three-phase interpolation models in Darcy-stokes flow

Abstract

This paper extends the topology optimization (TO) methods of fluid flows to design the three-phase (i.e. solid, fluid and porous materials) interpolation scheme. In addition to numerous studies about the optimized layout of regions governed by Darcy-Stokes equations, this paper aims to minimize the pressure attenuation in multiple phase interpolation models. The optimized distribution is obtained by considering both the fluid permeability through the porous media and impenetrable inner walls (solid phase) and neglecting buoyancy and other external body forces. Each material phase is assigned with two design variables that are projected into the element space via the regularized interpolation functions. The solid isotropic material with penalization (SIMP) interpolation functions, which is initially developed for minimizing compliance of multiple structural materials, is applied to TO processes of Darcy–Stokes flow. The fields are divided into the design and non-design domains, and TO layouts are assembled to satisfy the given performance functions. The smoothed Heaviside projection filter and Helmholtz-type Partial Differential Equation (PDE) based filter are utilized to produce discrete solutions in the continuum TO processes. Numerical studies are carried out to verify the proposed interpolation scheme.

Adjoint sensitivity of the optimization problem

The discretized objective function can be augmented with a Lagrange adjoint multiplier λ ^T

$$ \partial \overline{J}\left(\mathbf{u},\boldsymbol{\upchi} \right)=J\left(\mathbf{u},\boldsymbol{\upchi} \right)-{\boldsymbol{\uplambda}}^T\mathbf{R}\left(\mathbf{u},\boldsymbol{\upchi} \right) $$

(25)

where R(u, χ) is the residual form of the discretized Navier-Stokes. Equation (25) can be solved interatively by invoking the Newton-Raphson method. The sensitivity expression is obtained by differentiating the discretized objective function with respect to each component of the design variable vector χ

$$ \frac{\mathrm{d}\overline {J}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\mathrm{d}\boldsymbol{\upchi }}=\frac{\partial J\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \mathbf{u}}\frac{\partial \mathbf{u}}{\partial \boldsymbol{\upchi}}+\frac{\partial J\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}-{\boldsymbol{\uplambda}}^T\left[\frac{\partial \mathbf{R}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \mathbf{u}}\frac{\partial \mathbf{u}}{\partial \boldsymbol{\upchi}}+\frac{\partial \mathbf{R}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}\right] $$

(26)

To isolate the implicit response sensitivities, the above (26) can be separated into explicit and implicit terms. It can be rearranged as

(27)

The unknown implicit response sensitivity ∂u/∂χ, is eliminated from the above equation by defining the Lagrange multiplier λ. Annihilation of the implicit term yields the following adjoint problem for a specialized parameter vector λ.

$$ \frac{\partial J\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \mathbf{u}}-{\boldsymbol{\uplambda}}^T\frac{\partial \mathbf{R}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \mathbf{u}}=0 $$

(28)

where $\partial \overline{J}\left(\mathbf{u},\boldsymbol{\upchi} \right)/\partial \mathbf{u}$ is the adjoint load. Substituting (28) in to (27), the sensitivities become

$$ \frac{\partial \overline{J}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}=\frac{\partial J\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}-{\boldsymbol{\uplambda}}^T\frac{\partial \mathbf{R}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}} $$

(29)

For a sensitivity analysis of the optimization model in (20), the adjoint sensitivity of the pressure drop between the inlet and outlet boundaries can be obtained as:

$$ {\displaystyle \begin{array}{l}\frac{\partial \varDelta p}{\partial \boldsymbol{\upchi}}=\frac{1}{\int_{\varGamma_{in}}1\mathrm{d}s}\bullet \left(\frac{\partial {\int}_{\varGamma_{in}}p\mathbf{n}\bullet \mathrm{d}\mathbf{s}}{\partial \boldsymbol{\upchi}}-{{\boldsymbol{\uplambda}}_{in}}^T\frac{\partial {\mathbf{R}}_{\varGamma_{in}}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}\right)\\ {}-\frac{1}{\int_{\varGamma_{out}}1\mathrm{d}s}\bullet \left(\frac{\partial {\int}_{\varGamma_{out}}p\mathbf{n}\bullet \mathrm{d}\mathbf{s}}{\partial \boldsymbol{\upchi}}-{{\boldsymbol{\uplambda}}_{out}}^T\frac{\partial {\mathbf{R}}_{\varGamma_{out}}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}\right)\end{array}} $$

(30)

where λ _in ^T and λ _out ^T are Lagrange multipliers, and are solved by adjoint (28).

As the fluid channel has the parallel inlet and outlet ports, as well as the same area, it becomes somewhat easy to utilize the pressure drop to evaluate the fluid performance. Then, the description of (30) can be rearranged as:

$$ {\displaystyle \begin{array}{l}\frac{\partial \varDelta p}{\partial \boldsymbol{\upchi}}=\frac{1}{\int_{\varGamma_{in, out}}1\mathrm{d}s}\bullet \left(\frac{\partial {\int}_{\varGamma_{in}}p\mathbf{n}\bullet \mathrm{d}\mathbf{s}}{\partial \boldsymbol{\upchi}}-\frac{\partial {\int}_{\varGamma_{out}}p\mathbf{n}\bullet \mathrm{d}\mathbf{s}}{\partial \boldsymbol{\upchi}}\right)\\ {}-\frac{1}{\int_{\varGamma_{in, out}}1\mathrm{d}s}\bullet \left({{\boldsymbol{\uplambda}}_{in}}^T\frac{\partial {\mathbf{R}}_{\varGamma_{in}}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}-{{\boldsymbol{\uplambda}}_{out}}^T\frac{\partial {\mathbf{R}}_{\varGamma_{out}}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}\right)\end{array}} $$

(31)

The first term of the right side can be evaluated by the Gauss integral relation, having the following transformation:

$$ {\displaystyle \begin{array}{l}\frac{\partial \varDelta p}{\partial \boldsymbol{\upchi}}=\frac{1}{\int_{\varGamma_{in, out}}1\mathrm{d}s}\bullet \left(\frac{\int_{\varOmega}\nabla \bullet p d\varOmega}{\partial \boldsymbol{\upchi}}\right)\\ {}-\frac{1}{\int_{\varGamma_{in, out}}1\mathrm{d}s}\bullet \left({{\boldsymbol{\uplambda}}_{in}}^T\frac{\partial {\mathbf{R}}_{\varGamma_{in}}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}-{{\boldsymbol{\uplambda}}_{out}}^T\frac{\partial {\mathbf{R}}_{\varGamma_{out}}\left(\mathbf{u},\boldsymbol{\upchi} \right)}{\partial \boldsymbol{\upchi}}\right)\end{array}} $$

(32)

in the case where the boundary conditions are symmetrical on all external walls of domain, except for the inlet and outlet. Therefore, the description of pressure drop has been adopted in this work.

At each TO iteration, design variables update according to the values of the objective/constraint functions and their sensitivities to small perturbations of physical design variables. It has been proven that the number of the adjoint functions is small compared to the substantial number of local variations of TO design.