Topology Physics Model

The Topology Physics Model simulates the effects of solid material distributed within a fluid. In the context of topology optimization, it is the topology physics model that provides an approximation of the evolving solid material within the otherwise fluid domain.

When the Topology Physics Model is used in conjunction with the adjoint solver, it produces the derivative of the user-specified adjoint cost functions with respect to the material distribution (as shown in Eqn. (5140)). However, if the optimization is done with an external optimizer, the model can be used in absence of the adjoint solver.

The blockage in the flow is modeled using Brinkman Penalization, which treats the solid as porous media with a very small porosity. From an equation perspective, the penalization adds a source term to the momentum equation that forces the velocity to zero in the solid regions of the domain:

Figure 1. EQUATION_DISPLAY

\frac{\partial ρ v}{\partial t} + \nabla\cdot (ρ v \otimes v) = - \nabla\cdot σ - α (1 - χ) v

(5137)

where $α$ is the Brinkman Penalization Magnitude, which must be sufficiently large that it reduces the velocity within the solid to below an acceptable tolerance. The only disadvantage of selecting too large a value is that convergence can suffer. In Simcenter STAR-CCM+ the default value is 1e7.

$χ$ defines the material distribution within the domain as follows:

Figure 2. EQUATION_DISPLAY

χ = {\begin{matrix} 1 x \in Ω_{f l u i d} \\ 0 x \in Ω_{s o l i d} \end{matrix}

(5138)

Fluid cells are cells with $χ = 1$ and solid cells are cells with $χ = 0$ .

When flow energy is activated, the domain is treated like a porous media in thermal equilibrium. As such, the only additional change in the energy equation is to include the thermal conductivity of the solid. The effective conductivity is given as:

Figure 3. EQUATION_DISPLAY

k_{e f f} = χ k_{f l u i d} + (1 - χ) k_{s o l i d}

(5139)

For low speed flows, such as laminar flows, the results obtained using this model match with a fully resolved geometry (fluid plus surrounding porous medium). For higher Reynolds numbers, the accuracy degrades but objective predictions still reflect the trends of the fully resolved geometry.

By activating the adjoint solver, the treatment produces the derivation of all the user-defined cost functions with respect to the $χ$ field. This derivative is naturally computed as part of the adjoint framework but is roughly equal to:

Figure 4. EQUATION_DISPLAY

\frac{ⅆ L}{ⅆ χ} = \frac{\partial L}{\partial χ} + Λ_{m o m e n t u m}^{T} α v V - Λ_{E}^{T} \frac{\partial R}{\partial k_{e f f}} (k_{f l u i d} - k_{s o l i d})

(5140)

where $Λ_{m o m e n t u m}^{T}$ is the adjoint of momentum and $Λ_{E}^{T}$ is the adjoint of energy. $R$ is the residual of the energy equation.

For turbulent topology optimization simulations, the turbulent viscosity of K-Epsilon and K-Omega turbulence models is scaled:

Figure 5. EQUATION_DISPLAY

μ_{t, s c a l e d} = {\begin{matrix} 1.0 E - 10 μ_{t} & i f & χ < 0.99 \\ μ_{t} & i f & χ \geq 0.99 \end{matrix}

(5141)