Introduction

This article is the second part of a explanation and demonstration of Implicit Modeling.

1. Explanation of Implicit Modeling, modeling by TPMS (Triply Periodic Minimal Surface)
   
   https://nature-architects.com/blog/1115/
2. Implicit modeling with geometry as an argument

In the second part, modeling by implicit functions with geometries as arguments will be explained and demonstrated.

The heat exchange structure shown in the youtube video below is modeled using implicit functions based on our patented method of designing partition structures from the feature geometry of the flow path (our patented method, JP patent No.7207640 ).

Modeling by implicit functions with a geometry as an argument

The implicit function of the corrugated plate shown in Chapter 1 had \(k_x,k_y,k_z,\phi_x,\phi_y,t\) as arguments to determine the function.

\(F(x,y,z|k_x,k_y,k_z,\phi_x,\phi_y,t)=|\cos (k_xx+\phi_x)+\cos (k_yy+\phi_y) +k_zz|-t<0\)

Here, \(k_x,k_y,k_z,\phi_x,\phi_y\) were scalar quantities, but you can also define an implicit function with this argument as the geometries.

For example, consider the following implicit function representation with the geometry \(g_1\) as an argument.

\(F(x,y,z|g_1,t)=\text{Distance}(p_1,g_1)-t<0
\\
p_1=\text{Point}(x,y,z)\)

If we define \(\text{Point}(x,y,z)\) as a point with coordinates `(x,y,z)$ and \(\text{Distance}(p_1,g_1)\) as a function representing the distance between point \(p_1\) and geometry \(g_1\), this implicit function expression generates a structure with thickness in the geometry \(g_1\).

Also, the Schwarz D embedded in Gyroid with the shape of N, which I had put in a gif at the end of Chapter 1, was also modeled with an implicit function based on the geometry \(g_1\) as follows.

\(\begin{aligned}
F(x,y,z|g_1,t)&=|a(x,y,z|g_1)F_{Gyroid}+(1-a(x,y,z|g_1))F_{Diamond}+s|-t<0
\\
a(x,y,z|g_1)&=\frac{a_0}{1+\exp(-\alpha d(x,y,z|g_1))}
\\
d(x,y,z|g_1)&=\text{Distance}(p_1,{g_1}_{boundary})\text{sign}(x,y,z|g_1)
\\
p_1&=\text{Point}(x,y,z)
\\
\text{sign}(x,y,z|g_1)&=
\left\{
\begin{matrix}
1 & (\text{if point }(x,y,z) \text{ is contained in } g_1)
\\
-1 &(\text{else})
\end{matrix}
\right.
\end{aligned}\)

Thus, more complex modeling can be done by using implicit functions with geometries as arguments.

The implicit functions with a geometry as an argument shown so far involve relatively heavy calculation, as the evaluation of the value involves estimating the distance between the geometry and the point.
Therefore, complex implicit functions can take a long time to generate structures.
If you would like to experiment with geometry-based implicit function modeling, we recommend that you start with simple implicit functions or with reduced resolution for structure generation.

Implicit function modeling combining multiple geometries

We can also model by implicit functions that combine multiple geometries.

We will demonstrate with the example of an implicit function defined by the following set of two classified geometries \(G_1,G_2\).

\(F(x,y,z|G_1,G_2,t)=|\text{Distance}(p_1,G_1)-\text{Distance}(p_1,G_2)+s|-t<0
\\
p_1=\text{Point}(x,y,z)\)

Here, \(\text{Distance}(p_1,G_1)\) is defined as the distance between point \(p_1\) and the nearest geometry to \(p_1\) in the set of geometries \(G_1\).

This implicit function will generate a partition structure between the sets of geometries \(G_1,G_2\).

Modeling with two straight lines

Defining two geometries with parallel straight lines generates a structure that separates the lines.

\(G_1\) : one straight line (blue line)

\(G_2\) : one straight line (green line)

Modeling with two spirals

Defining the two geometries as a spiral with the same axis and a half-period shift generates a spiral septum structure.

\(G_1\) : one spiral (blue line)

\(G_2\) : one spiral (green line)

Modeling with square lattice

Defining two sets of geometries based on the skeleton of a square lattice generates a geometry similar to Schwarz P.

\(G_1\) : Square lattice skeleton (blue line)

\(G_2\) : \(G_1\) shifted by half a period in each direction (green line)

Modeling with K4 lattice

Defining two sets of geometries based on the skeleton of the K4 lattice generates a geometry similar to Gyroid.

\(G_1\) : Skeleton of K4 lattice (blue line)

\(G_2\) : Mirror reversal of \(G_1\) (green line)

Modeling of heat exchanger with implicit function

The implicit function introduced in "Implicit function modeling combining multiple geometries" was to generate a partition structure between the sets of geometries \(G_1, G_2\).

Using this feature, a two-fluid heat exchanger (a structure that exchanges heat between two fluids) can be modeled using the aforementioned implicit function by defining two sets of geometries as the feature geometry of the flow path.

The actual modeling is shown below.

Conclusion

In Chapter 1, we have explained implicit modeling and demonstrated modeling with TPMS, and in Chapter 2, we have explained and demonstrated implicit modeling with geometries as arguments.

Implicit functions allow for complex modeling based on mathematical formulas, and defining implicit functions in combination with geometry allows for a wider range of modeling.

The development of 3D printing technology has made it possible to create complex shapes, and modeling with implicit functions is being used industrially, including in heat exchangers.

We hope that through this article, you have gained some interest in implicit function modeling.