Introduction

In recent years, a type of structural optimization method called topology optimization has begun to be applied to actual products, with applications spanning a wide range of fields such as elasticity, vibration, heat transfer, and thermal fluid dynamics.

Topology optimization methods have been proposed in various forms through its history [1-5]. When applying topology optimization to practical problems, it is important to choose the appropriate method based on the manufacturing method of the product, the physical phenomena involved, and the objectives, the consideration of which require a deep understanding of each method.

This article introduces an overview of representative methods including homogenization method, density method, level set method, and Moving Morphable Components (MMC), which has been actively researched in recent years. Particularly, as there are few explanatory articles on MMC, we would like to delve into the mathematical expressions. Additionally, we will implement the density method and MMC in MATLAB to observe the differences in results.

Sensitivity Analysis

In mathematical optimization, various methods exist, including gradient-based methods that utilize sensitivity with respect to the design variables, evolutionary methods inspired by biological evolution, exhaustive methods that randomly explore the design space, and black-box optimization that finds optimal solutions solely based on input-output pairs without utilizing the form or gradients of the objective function.

In this article, we will utilize gradient-based methods due to their ease of implementation and efficient exploration. Gradient-based methods utilize sensitivity with respect to the design variables, often likened to descending a mist-covered slope on a mountain. By using information about the value of the objective function \(f(x)\) and its gradient \(\frac{df}{dx}\) at the current design variable \(x\), the method gradually moves towards directions where the objective function decreases, aiming to find the minimum value.
Sensitivity analysis means calculating the sensitivity (gradient) of the objective function with respect to the design variables and analyzing how the objective function behaves when the design variables change.

If the objective function is explicitly described in terms of the design variables, sensitivity can be directly calculated through differentiation. If not, sensitivity is approximated by providing a small value to the design variables, which is known as numerical differentiation.

\(\frac{df(x)}{dx}
≃\frac{f(x+dx)-f(x-dx)}{2dx}\)

In the program implemented later, the density method uses direct differentiation because the objective function is explicitly represented by the design variables, while MMC uses numerical differentiation because the objective function is not explicitly represented by the design variables.

Numerical differentiation has the advantage of being able to calculate sensitivity independently of the objective function, but it has the disadvantage of increasing computational cost due to the need to evaluate the objective function multiple times.

Overview of typical topology optimization methods

Topology Optimization Based on Homogenization Method

Topology optimization based on the homogenization method is a technique that defines periodically arranged microstructures within the design domain and optimizes macrostructural properties by controlling parameters that govern these microstructures as design variables.

The calculation is formulated so that the mechanical properties of the homogenized structure match those of the actual structure with microstructure.

At the time when the method was proposed, manufacturing microstructures was challenging. However, recent advancements in Additive Manufacturing have made it possible to manufacture microstructures using various materials.

Here, we present a prototype model of lattice structure obtained through topology optimization based on the homogenization method.

It can be observed that the topology is represented by the sparsity of the microstructure. Topology optimization based on the homogenization method is used when optimizing structural components using fine structures.

For those who want to learn more about homogenization, please refer to our past technical articles.

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

Homogenizing Mechanical Metamaterial

(1) Homogenization of lattice structure

(2) Homogenization of lattice structure

Density Method

Density method is a technique where normalized material densities ranging from 0 to 1 are assigned as design variables to each finite element used in the analysis, with the concept of optimizing the distribution of these densities.

Within the optimization framework, penalties are applied to intermediate densities to encourage a binary distribution closer to 0 and 1. Nonetheless, intermediate densities may still occur, particularly at boundary regions, so a threshold is eventually set to determine the presence or absence of material. This method has capability to represent relatively complex topologies with high degrees of freedom and is implemented in many general-purpose CAE software.

However, akin to representing topology with pixelated artwork where each finite element serves as a pixel, obtaining smooth topology requires a large number of finite elements, leading to increased computational costs.

In addition, overly complex topologies may be created, and topologies may need to be deformed to fit the manufacturing methods. It is not uncommon for solutions to converge to local optima that are initially unintelligible structurally, requiring interpretation of the topology and iterative refinement of the initial values to find the optimal solution.

Additionally, the checkerboard pattern phenomenon, where material presence alternates, poses a known challenge. Although such patterns may seem structurally weak, they are often evaluated as having high rigidity in numerical analysis. Filtering sensitivity or density is one approach to mitigate this issue.

Topology Optimization Based on Level Set Method

Topology optimization based on the level set method employs a scalar function called the level set function as the design variable, distributed within the design domain. The zero level set of this function defines the material boundary, evolving gradually to generate topology.

Boundary evolution introduces a pseudo-time concept, driving sensitivity with respect to the objective function.

Unlike methods such as the density method that represent topology with finite element unit pixelation, this method allows for clear representation of topologies with free-form boundaries.

In actual implementation, the boundaries of the level set function are updated at every finite element node throughout the design domain, so the number of design variables increases in proportion to the number of nodes.

For further details on the theory and applications of these three methods, please refer to references [6,7], as well as numerous web resources available.

Moving Morphable Components (MMC)

Moving Morphable Components (MMC) is a method based on the concept of representing topology through the movement, rotation, enlargement, reduction, and superposition of predefined elements called "constituent elements," which are placed in the design space separately from the finite elements used in the analysis.

The number of constituent elements placed as initial condition can be arbitrarily determined, allowing control over the complexity of the topology generated according to manufacturing methods and design constraints.

The representation of topology and the finite elements used for analysis are independent, allowing for clear representation of topology even with a coarse mesh compared to density methods. The number of design variables depends only on the number of constituent elements, regardless of the number of finite elements.

Furthermore, since the dimensions of the constituent elements are treated as design variables, it is possible to directly control the dimensions of the members within the framework of optimization. This is useful when there are constraints on the minimum or maximum dimensions of members for manufacturing reasons.

However, it is important to note that the initial values can influence the converged solution. Design variables consist of the center coordinates, longitudinal and transverse dimensions, and rotation angle of the constituent elements, and optimization must be performed repeatedly adjusting their initial values.

Additionally, since MMC represents topology using a set of predefined elements with determined shapes, it may not be suitable for problems requiring representation of complex and smooth shapes, such as heat transfer or flow path problems. However, research aiming to enable MMC to represent curved and freeform shapes is underway, and various applications to different problems can be expected in the future.

Mathematical Description of MMC Constituent Elements

Based on this concept, one may wonder how to mathematically represent the regions where constituent elements exist in the design space. Let's explain this question below.

As mentioned earlier, the \(i\)th constituent element \(d_i\) consists of five design variables: the center coordinates \((X_0, Y_0)\), the longitudinal dimension \(L\), the transverse dimension \(t\), and the rotation angle \(θ\).

\(\begin{align*}
        d_{i}
        =
       \left[
            \begin{array}{ccc}
                 X_{0} & Y_{0} & L& t & θ\\
            \end{array}
        \right]
    \end{align*}\)

Using these five design variables, the existence of constituent elements can be determined by the implicit function \(φ\), which is defined as:

\(φ(X,Y)>0⇔(X,Y)∈Ω\)

\(φ(X,Y)=0⇔(X,Y)∈∂Ω\)

\(φ(X,Y)<0⇔(X,Y)∈D/Ω\)

Where,

\(φ(X,Y)=1-\left(\frac{cosθ(X-X_0)+sinθ(Y-Y_0)}{L/2}\right)^m-\left(\frac{-sinθ(X-X_0)+cosθ(Y-Y_0)}{t/2}\right)^m\)

\(m\) is an even constant. Now, let's break down the equation line by line.

First, lines 1 to 3 indicate that positive values of \(φ\) represent regions where constituent elements exist, \(φ = 0\) represents the boundaries of constituent elements, and negative values of \(φ\) represent regions where constituent elements do not exist. Thus, in MMC, the existence of constituent elements is implicitly expressed by the positive and negative values of the implicit function \(φ\).

Next, line 4 describes what specific function \(φ\) is based on.

First, let's consider the equation of an ellipse with major axis \(a\) and minor axis \(b\):

\(\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1\)

An extension of this ellipse concept is the hyperellipse (lame curve). A hyperellipse is a function that extends the power part of an ellipse to take even values other than 2, gradually approaching a rectangle as the value of \(m\) is increased. In the calculation perfoemed later, we set \(m=6\).

\(\left(\frac{x}{a}\right)^m+\left(\frac{y}{b}\right)^m=1\)

\(⇔1-\left(\frac{x}{a}\right)^m-\left(\frac{y}{b}\right)^m=0\)

The equation of this superellipse represents a closed curve, so by treating \(x\) and \(y\) as functions, it is possible to use the value of \(ψ\) to indicate the inside, outside, and boundary of the closed curve.

\(ψ(x,y)=1-\left(\frac{x}{a}\right)^m-\left(\frac{y}{b}\right)^m\)

Here, \(ψ\) is a function defined in the local coordinate system \(x\) and \(y\) fixed at the center of the hyperellipse. To translate and rotate the hyperellipse, we transform \(ψ\) to a global coordinate system of coordinates, which we define as \(φ\).

\(\begin{bmatrix} x \\ y \end{bmatrix}=
\begin{bmatrix} 
cosθ & sinθ \\ -sinθ & cosθ
\end{bmatrix}
\begin{bmatrix} 
X-X_0 \\ Y-Y_0
\end{bmatrix}=
\begin{bmatrix} 
cosθ(X-X_0)+sinθ(Y-Y_0) \\ -sinθ(X-X_0)+cosθ(Y-Y_0)
\end{bmatrix}\)

We have now derived the topology description function.

In the subsequent calculations, we input the coordinates of each finite element node into the \(X\) and \(Y\) of the implicit function \(φ\), and examine the presence of constituent elements. Then, based on the \(φ\) values ​​of the four nodes surrounding each finite element, we determine the rigidity of the finite element composed by those nodes and perform the analysis.

Topology optimization implementation

Optimization Problem Setup

As a benchmark, topology optimization was solved with the following problem setup:

• Boundary conditions: Cantilever
• Load conditions: Vertical load at the center of the free end
• Objective: Minimization of strain energy
• Constraint: 40% volume constraint of the design domain

The optimization iterations were conducted up to the 200th iteration regardless of convergence status.

Optimization Results by Density Method

The initial and optimized topologies by the density method are shown below.

The fine mesh with 120 × 60 elements allows for a clearer representation of topology.

Let's try reducing the element divisions to 60×30. As a result, we can observe a decrease in the resolution of the topology representation.

Furthermore, let's explore the results without sensitivity filtering. We can observe the checkerboard pattern in which elements are arranged alternately.

Optimization Results by MMC

The optimized topology by MMC is shown below, where the fill of constituent elements is removed for better visibility. For this case, 16 constituent elements were placed diagonally to evenly cover the design domainas the initial condition, following previous research methodologies.

To understand how the optimized topology was generated, results at iteration 1, 10, and 30 are shown below.

It's evident that each constituent element gradually conforms to the optimized topology over iterations.

As mentioned earlier, since MMC's topology representation is independent of the number of finite elements, the topology can be clearly represented even with a coarse mesh. Let's examine the results with a mesh of 60×30 elements.

It's evident that similar topology trends can be clearly represented even with a coarser mesh, validating the independence of topology representation from the number of mesh divisions.

Next, let's compare the results of both methods from the perspectives of the generated topology and computational cost.

Comparison of Generated Topology

We'll compare the topologies generated by both methods. While the density method can represent topologies without restrictions, allowing for smooth topology representations, MMC, which represents topology using a collection of rectangular constituent elements, yields less refined topologies. Although structural features are consistent between the results of the density method and MMC, the MMC topology appears less refined, exhibiting discontinuities in the connections between constituent elements and unnecessary protrusions.

The lack of constraints on the continuity between constituent elements and the dependency of constituent element variations solely on sensitivity may contribute to such differences. Minute features like the protrusions shown in the figure may have minimal effects on sensitivity, thus not being recognized as areas needing modification during optimization.

Although both methods produced similar shapes because the benchmark problem solved in this article mainly consisted of linear shapes, if the optimization problem involves more complex curved shapes, the disparity in topology representations between the density method and MMC would be more pronounced.

Comparison of Computational Costs

Let's compare the computational costs of the density method and MMC for mesh sizes of 120×60 and 60×30 elements. Following table summarizes the optimization method, the number of finite elements, design variables, and required computation time. Relative differences in computation time from the density method for 120×60 elements are also shown.

The number of design variables for the density method corresponds to the number of finite elements, i.e., 7200 and 1800 for mesh sizes of 120×60 and 60×30, respectively. For MMC, the number of design variables remains constant at 80, regardless of the number of finite elements. MATLAB's timing feature was used to measure the computation time until the 200th iteration of optimization.

As shown in the table, there is a significant difference in computation time between the density method and MMC. Even for the density method with a lower resolution of 60×30 elements, it required approximately twice the computational cost of MMC with a mesh size of 120×60.

Thus, while the density method offers greater freedom in topology representation, MMC excels in terms of computational cost. It's essential to consider these aspects when choosing between the two methods.

For example, if simple topologies are desired due to manufacturing constraints or to obtain a rough idea of the optimized topology, MMC or similar simplified methods may be suitable. Conversely, for detailed understanding of complex topology features, the density method or full-size topology optimization may be preferable.

Conclusion

In this article, we introduced four topology optimization methods: homogenization method, density method, level set method, and MMC, highlighting their characteristics. We then compared the results of optimization by the density method and MMC, both of which we implemented. Each method has its strengths and weaknesses, and the choice between them depends on manufacturing methods, physical phenomena, and design constraints.

We focused on explaining topology optimization methods in this article. However, when applying these methods to real-world problems, the most critical factors include:

• How to model the design object and formulate it as an optimization problem
• How to interpret the optimization results and shape them into tangible products

In real-world applications, meaningless topologies may be generated depending on the modeling and method selection. Moreover, directly using topology optimization results as design outcomes remains challenging. Thus, collaboration between computers and humans is necessary to shape the optimized topology into tangible products.

Our strength lies precisely in this area, as we possess various knowledge and expertise in modeling, method selection, interpretation of results, and product shaping. Please feel free to explore our technical articles on topology optimization:

Heat Sink Design Case Study by Thermal Fluid Topology Optimization

Design Example : Lower Arm for Automobile

Although we addressed a simple benchmark problem related to structural strength in this article, we would like to introduce the differences in optimization results when applying various topology optimizations to real-world problems someday.

We encourage you to implement topology optimization methods yourself and compare the results. It will lead to new discoveries and deepen your understanding of the subject.

Reference

[1] Bendsøe, M. P., and Kikuchi, N., “Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 71, No. 2, 1988, pp. 197–224.

[2] Dorn, W., Gomory, R., and Greenberg, H., “Automatic Design of Optimal Structures,” Journal de Mecanique, Vol. 3, No. 6, 1964, pp. 25–52.

[3] Bendsøe, M. P., “Optimal Shape Design as a Material Distribution Problem,” Structural Optimization, Vol. 1, No. 4, 1989, pp. 193–202.

[4] Sethian, J. A., and Wiegmann, A., “Structural Boundary Design via Levelset and Immersed Interface Methods,” Journal of Computational Physics, Vol. 163, No. 2, 2000,

pp. 489–528.

[5] Guo, X., Zhang, W. S., and Zhong, W. L., “Doing Topology Optimization Explicitly and Geometrically-A New Moving Morphable Components Based Framework,” Journal of Applied Mechanics, Vol. 81, No. 8, 2014, p. 081009.

[6] 日本計算工学会 編,「トポロジー最適化」,丸善出版,2013.

[7] 近藤 継男, 矢地 謙太郎, 西脇 眞二 共著,「トポロジー最適化の基礎 弾性体ならびに熱流体工学諸問題への応用のために」,コロナ社,2024.

[8] Zhang, Z., Yuan, J., Zhang, J., and Guo, X., “A New Topology Optimization Approach based on Moving Morphable Components (MMC) and the Ersatz Material Model,” Structural and Multidisciplinary Optimization, Vol. 53, No. 6, 2016, pp. 1243–1260.

[9] Guo, X., Zhang, W., Zhang, J., and Yuan, J., “Explicit Structural Topology Optimization Based on Moving Morphable Components (MMC) with Curved Skeletons,” Computer Methods in Applied Mechanics and Engineering, Vol. 310, 2016, pp. 711–748.

[10] Otsuka, K., Dong, S., Kuzuno, R., Sugiyama, H., Makihara, K., “Moving Morphable Multi Components Introducing Intent of Designer in Topology Optimization,” AIAA Journal, Vol. 61, No. 4, pp. 1720–1734.