# Topology Optimization Based on Explicit Geometry Description

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_248-1

## Synonyms

## Definition

Topology optimization based on explicit geometry description is defined as a structural topology optimization paradigm where structural topology/geometry is described in an explicit way. The so-called Moving Morphable Components/Voids (MMC/MMV) method, geometry projection method, and B-spline based topological derivative method can all be ascribed to this solution paradigm. Since these methods have the potential to reduce the number of design variables associated with numerical optimization and establish a direct link with the computer aided design/engineering (CAD/CAE) systems, recent years witnessed a growing interest in developing topology optimization methods based on explicit geometry description.

## Introduction

Traditional topology optimization approaches, for example, Solid Isotropic Material with Penalization approach (SIMP) (Bendsøe 1989; Zhou and Rozvany 1991) and level set approach (LSM) (Wang et al. 2003; Allaire et al. 2004), are established based on implicit geometry description, where optimized structural topology is *extracted* from a binary pixel/voxel image or the nodal values of a level set function. Although remarkable achievements have been made with these approaches, it is worth noting that the numerical implementation of these approaches often leads to a large number of design variables, and it is not an easy task to establish a seamless link between the optimized results obtained by these methods with CAD/CAE systems. Moreover, it is also not straightforward to consider geometry-related objective/constraint functions in these approaches.

Compared with the implicit geometry-based topology optimization framework, the explicit geometry-based solution framework has the following advantages: (1) Direct link with the computer aided design (CAD) modeling systems, since the geometries of the whole structural topology are described explicitly by a set of parameters. (2) Capability of integrating shape, size, and topology optimization or even structural type optimization in a unified framework. (3) Great potential to share the merits of both Lagrangian and Eulerian topology optimization approaches. (4) The optimized structures obtained are pure black-and-white and there is no need to introduce special techniques to eliminate numerical instabilities such as checkerboard phenomenon and mesh-dependent solutions. (5) Great potential to reduce the computational efforts associated with topology optimization.

These distinct advantages over other topology optimization approaches based on implicit geometry description render the MMC/MMV-based explicit topology optimization approaches become a hot topic in topology optimization field. Nowadays, numerous MMC/MMV-based methods have been developed to solve topology optimization problems based on explicit geometry description.

## Theory

In this section, the theoretical aspects of the MMC approach, which is a representative explicit geometry-based topology optimization approach, will be introduced.

### Geometry Description

^{s}⊂ D is a subset of D comprised by

*n*components made of solid material. As shown in Guo et al. (2014),

*χ*

^{s}(

*) = max(*

**x***χ*

_{1}(

*), …,*

**x***χ*

_{n}(

*)) with*

**x***χ*

_{i}(

*) denoting the topology description function (TDF) of the*

**x***i*-th component. For two-dimensional (2D) case, the function

*χ*

_{i}(

*) can be adopted as:*

**x***p*is a relatively large even integer number (

*p*= 6 is often adopted in the MMC approach). In the above equations, the symbols (

*x*

_{0i},

*y*

_{0i}),

*a*

_{i},

*b*

_{i}(

*x*′) and

*θ*

_{i}denote the coordinate of the center, the half-length, the variable half width, and the inclined angle (measured from the horizontal axis anticlockwisely) of the

*i*-th component. It should be noted that the variation of the width of the component

*b*

_{i}(

*x*′) is measured with respect to local coordinate system and can take different forms (Guo et al. 2016), such as the linearly varying thicknesses as follows:

*i*-th component:

*α*,

*β*, and

*θ*denoting the rotation angles of the component from a global coordinate system

*Oxyz*to the local coordinate system

*O*′

*x*′

*y*′

*z*′, respectively. The central coordinate and the half-length of the component are represented by the coordinate (

*x*

_{0}

_{i},

*y*

_{0}

_{i,}

*z*

_{0}

_{i}) and \( {L}_i^1 \), respectively. Furthermore, the functions

*g*

_{i}(

*x*′) and

*f*

_{i}(

*x*′,

*y*′) in the above equation are used to describe the thickness profiles of the component in

*y*′ and

*z*′ directions, respectively. The functions

*g*

_{i}(

*x*′) and

*f*

_{i}(

*x*′,

*y*′) can be simply chosen as

*i*-th component (composed of solid material) can be described as:

### Optimization Formulation

Under the above geometry representation scheme, the layout of a structure can be solely determined by a design vector * D* = ((

**D**^{1})

^{Τ}, …,(

**D**^{i})

^{Τ}, …, (

**D**^{n})

^{Τ})

^{Τ}, where

**D**^{i}contains the design variables associated with the

*i*-th component. It can be observed that under the MMC-based solution framework, a topology optimization problem, which intends to seek the optimal material distribution in a prescribed design domain, is transformed to a shape optimization problem.

*I*(

*),*

**D***g*

_{k},

*k*= 1,…,

*m*are the objective function/functional and constraint functions/functionals. \( {\mathbf{U}}_{\boldsymbol{D}} \) is the admissible set of

*. For example, if structures are designed to minimize the structural compliance under the volume constraint of available solid material, the corresponding problem formulation can be specified as:*

**D***,*

**f***,*

**t***,*

**u***= sym (∇*

**ε***) and \( \overline{\boldsymbol{u}} \) are the design domain, the body force density, the prescribed surface traction on Neumann boundary Γ*

**u**_{u}, the displacement field, the linear strain tensor, and the prescribed displacement on Dirichlet boundary Γ

_{t}, respectively. The symbol

*H*=

*H*(

*s*) denotes the Heaviside function with

*H*= 1 if

*s*> 0 and

*H*= 0 otherwise. 𝔼 =

*E*

^{s}/(1 +

*v*)[𝕀+

*v*

^{s}/(1 − 2

*v*

^{s})

**δ**⊗

**δ**] is the fourth order elasticity tensor of the isotropic solid material with

*E*

^{s},

*v*

^{s}, 𝕀, and

**δ**denoting the Young’s modulus as well as the Poisson’s ratio of the solid material, symmetric part of the fourth order identity tensor, and the second order identity tensor, respectively. The symbol \( {U}_{ad}=\left\{\boldsymbol{v}|\boldsymbol{v}\in {\boldsymbol{H}}^1\left({\Omega}^{\mathrm{s}}\right),\, \boldsymbol{v}=\boldsymbol{0}\, \mathrm{on}\, {\Gamma}_u\right\} \) represents the admissible set of virtual displacement vector

*and \( \overline{V} \) is the upper limit of the volume of the available solid material.*

**v**### Sensitivity Analysis

*I*=

*I*(

*) with respect to a design variable*

**D***a*associated with

*χ*

_{i}(i.e., the TDF of the

*i*-th component) can be written as

*and*

**u***are the primary and adjoint displacement fields,*

**w***H*

_{ε}(

*x*) is the regularized Heaviside function (Zhang et al. 2016b), respectively. When

*I*is the structural compliance, it yields that

*f*(

*,*

**u***) = 𝔼:*

**w***(*

**ε***) :*

**u***(*

**ε***) with*

**w***= −*

**w***while*

**u***f*(

*,*

**u***) = 1 when*

**w***I*is the volume of the structure. Furthermore

*δ*

_{ε}(

*s*) = d

*H*

_{ε}(

*s*)/d

*s*denoting the regularized Dirac delta function and \( {\delta}_{\varepsilon}^i\left({\chi}^{\mathrm{s}}\right)=\min \left({\delta}_{\varepsilon}\left({\chi}_i\right),\, {\delta}_{\varepsilon}\left({\chi}^{\mathrm{s}}\right)\right) \). The expressions of the sensitivities of

*χ*

_{i}with respect to each design variable are trivial and will not be repeated here. The readers are referred to Guo et al. (2014) and Zhang et al. (2016b) for the details.

## Examples

*V*≤ 0.15 × |D| = 28.8 in this example).

## Some Extensions

Topology optimization based on explicit geometry description has received more and more attention since the MMC approach was proposed by Guo et al. (2014). As a dual method of MMC, the MMV method was also developed by Zhang et al. (2017c). Takalloozadeh and Yoon (2017) proposed a topological derivative based method under the MMC-based solution framework. Due to the big potential in reducing the computational cost and providing explicit geometry information, MMC and MMV methods had also been extended to solve 3D topology optimization problems in Zhang et al. (2017a, b). Bujny et al. (2018) proposed to solve the crashing problem under the MMC framework with use of evolutionary algorithms. Zhang et al. (2016a), Hoang and Jang (2017), and Guo et al. (2017) solved the length scale control problem and the overhang angle control problem in additive manufacturing by the MMC/MMV methods, respectively. A successfully generalization of the MMC method was proposed by Sun et al. (2018a, b). In these works, the authors took the full advantages of the MMC method to solve topology optimization problems in flexible multibody systems. Inspired by the MMC method, Deng and Chen (2016) developed a connected morphable components (CMC) method to design flexible structures. Recently, Hou et al. (2017) introduced the isogeometric analysis scheme into the MMC-based solution framework to obtain higher accuracy structural response analysis. Zhang et al. (2018b) proposed a MMC-based method for solving the topology optimization problem with multiple materials. Topology optimization problem considering stress constraints was also discussed under the MMV-based framework in Zhang et al. (2018a).

## Conclusions

Compared with the traditional SIMP and LSM approaches for structural topology optimization where a *fixed* ground structure is adopted, the MMC/MMV-based explicit topology optimization approaches actually represent a new type of paradigm for structural topology optimization using *adaptive* ground structures. These approaches, in some sense, revival the classical shape optimization since in principle we can now use Lagrangian description-based shape optimization methods to solve topology optimization problems under the MMC/MMV-based solution framework. Possible future directions for developing explicit topology optimizations can be summarized as follows: (1) Topology optimization considering uncertainty. Since one has explicit boundary description in the MMC/MMV approaches, it is straightforward to consider the perturbation of structural boundary (possibly due to manufacturing error) by simply allowing the uncertainty of shape parameters describing the profiles of the components/voids. It is also more natural to considered fail-safe design (Zhou and Fleury 2016) in the MMC-based solution framework. (2) Data driven topology optimization. Since in the MMC/MMV-based solution framework the number of design variables is relatively small, the computational time associated with supervised learning or network construction can also be saved substantially as shown in Lei et al. (2019). (3) Topology optimization via hybrid explicit/implicit approaches. As demonstrated in Liu et al. (2018), the explicit MMC-based approach can be degenerated into the classical SIMP approach if multidomain strategy is employed. Therefore, it is highly promising to develop some hybrid approaches which can take both advantages of the explicit and implicit geometry description-based solution frameworks.

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