Introduction

The topic of homogenization has come up several times in this technical blog.

(2) Adidas 3DP Shoes Explanation: Design of Mechanical Metamaterials that Convert Compressive Forces into Shear Deformations

Homogenizing Mechanical Metamaterial

This article introduces the method of homogenization of lattice structures by numerical experiments and visualization of the results in terms of elastic bodies and fluids (channel).

This article is the second part and introduces homogenization of channels

The first part is here：(1) Homogenization of Lattice Structures

Note that while this article introduces homogenization of lattices as an Additive Manufacturing prerequisite,
In client work, it is common to utilize metamaterial structural design techniques while assisting in the development of designs that can be mass-produced at the client's manufacturing facility.

Homogenization of channels by numerical experiments

The porous model is commonly known as a homogenization method for channels.
This presentation will introduce the process of homogenizing periodic channels using the porous model.

It is known that the flow velocity \(\boldsymbol{v}\) in the porous media is expressed by Darcy's law and is proportional to the pressure gradient and inversely proportional to the viscosity coefficient \(\mu\) at low Reynolds numbers.

\(\begin{align*}
    {\boldsymbol{v}}=-\frac{\kappa}{\mu}\boldsymbol{\nabla}p
\end{align*}\)

There is also the Darcy-Forchheimer equation, which adds an inertial contribution to Darcy's law and expresses the pressure drop as the sum of a viscous term proportional to the velocity and an inertial term proportional to the square of the velocity.

\(\begin{align*}
    \nabla p=-\frac{\mu}{\kappa}v-\frac{\rho}{\kappa'}v^2
\end{align*}\)

relating velocity and pressure gradient can be taken as governing equations as they are, or they can be put into the pressure source term of the Navier-Stokes equation.
In this case, the pressure drop (pressure gradient) in a periodic flow is formulated by putting it into the pressure source of the Navier-Stokes equation as follows.

\(\begin{align*}
    (\boldsymbol{v}\cdot\nabla) \boldsymbol{v}
    =-\boldsymbol{\nabla} p - \mu\Delta \boldsymbol{v} - \mathbf{\alpha}\boldsymbol{v} - \beta|\boldsymbol{v}|\boldsymbol{v}
\end{align*}\)

\(\alpha\) and \(beta\) are the matrix parameters of the porous model, and for simplicity we consider a steady uncompressed flow.

Numerical Experiments Fitting Porous Models

The homogenization process is presented using a 2-dimensional lattice as an example.

The unit structure can be taken out as follows

After setting the periodic boundary of the boundary velocity of this unit structure, a pressure gradient is added in each direction for analysis as shown in the figure below.

Some of the results of the analysis are illustrated in the following figures.

Thus, velocity fields are obtained for pressure gradients in each direction.

Here, the unit structure is analyzed and the results are visualized, but because periodic boundary conditions are used, the analysis is based on a situation in which the flow path repeats itself periodically.

In the analysis results for each pressure gradient, the apparent velocity is estimated by averaging the velocities, including the non-flow path portion.

The apparent velocity obtained from the analysis results, along with the input pressure gradient, is mapped with the direction and magnitude of the red arrow as shown below.

From the pressure gradient and apparent velocity results obtained from this numerical analysis, the matrix parameters \(\alpha\) and \(beta\) of the porous model are calculated.
In the formulation, it is explained that the pressure drop due to the periodic channel is included in the pressure source term of the Navier-Stokes equation, but based on the idea that the pressure source due to the homogenized periodic channel is sufficiently larger than the viscous and inertial terms in the Navier-Stokes equation, the Darcy-Forchheimer equation is used for fitting.

\(\begin{align*}
    \boldsymbol{\nabla} p = - \mathbf{\alpha}\boldsymbol{v} - \beta|\boldsymbol{v}|\boldsymbol{v}
\end{align*}\)

First, try fitting using only data with pressure gradients of 500 Pa/m or less.
The apparent velocity is input based on the analysis results, and \(\alpha\) and \(\beta\) are calculated by linear regression as values to predict the pressure drop. The plot is as follows.

When the pressure drop is small and the velocity is slow (small Reynolds number), the flow could be fitted by the porous model with some accuracy.

Next, let us try fitting using all the data.

The maximum error is nearly 30%, which is not a very accurate fitting.
The larger the pressure gradient, the larger the velocity and Reynolds number, and the larger the relative contribution of inertia.
In the lattice structure homogenized in this study, inertia can significantly change the flow pattern, and the contribution of inertia is anisotropic.

Therefore, it is believed that the range of the formulation of the porous model in this case was not sufficient to represent the flow, resulting in a large error.
When actually performing homogenization, one should be careful about the applicable range of the homogenization model to be handled.

Porous model analysis

The following three channels are analyzed using a porous model fitted with data for pressure drop less than 500 Pa/m.
The first and second channels have different lattice structure orientations.

The following is the result of an analysis with a constant pressure difference between the inlet and outlet and a comparison of the non-homogenized and homogenized results.
Although there are some differences due to the small number of iterations of the unit structure, it is clear that the porous model is able to reproduce the flow.

Visualization of ease of flow in each direction of lattice structure

We visualized the homogenized elastic matrix in the first part of this article.
Using the same visualization technique, we will try to visualize the ease of flow in each direction of the lattice structure in a sufficiently slow flow situation.

As shown in the figure below, each flow direction is represented by a point in 2-D or 3-D space separated from the origin by a distance corresponding to the pressure drop in each direction.
By connecting those points to form a line or a plane, the ease of flow in each direction is visualized.

Below we visualize the ease of flow in each direction for several lattices.

2D Lattice

Homogenize the 2D lattice homogenized above.

3D Lattice 1

The first figure below shows the wall surface of the channel, the second shows an example of the analysis results, and the third illustrates the ease of flow in each direction.

3D Lattice 2

As in 3D Lattice 1, the walls of the channel and the ease of flow in each direction are illustrated.

3D Lattice 3

As in 3D Lattice 1 and 2, the walls of the channel and the ease of flow in each direction are illustrated.

Finally

In the first part of this article, homogenization by elastic matrices of structures and methods of visualizing elastic matrices were introduced, as well as homogenization by porous models of channels and visualization of flow ease in each direction.
Homogenization is used in various fields and helps not only to reduce computational cost but also to understand phenomena.
If you are interested, please try homogenization!