Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Topology Optimization Based on Explicit Geometry Description

  • Xu GuoEmail author
  • Weisheng Zhang
  • Zongliang Du
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_248-1



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.


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.

In order to circumvent the aforementioned difficulties, some topology optimization approaches based on explicit geometry description have been proposed recently. The central idea of these approaches is to use some structural components as the basic building blocks of topology optimization and adopt the parameters for describing their geometries as design variables. Figure 1 illustrates the basic idea of the so-called Moving Morphable Component (MMC)-based explicit topology optimization approach (Guo et al. 2014) schematically. In this approach, a set of structural components with explicit geometry description is initially deployed in the design domain, then optimization algorithm is applied to find the optimized sizes, shapes, and layout of the components. Finally, an optimized structural topology can be obtained through the deforming, overlapping, and merging of these components. Besides components made of solid materials, void can also be viewed as a specific type of structural component and the so-called Moving Morphable Void (MMV)-based explicit topology optimization approach, where a set of voids is used as the basic building blocks of topology optimization, had also been developed in the literature (Zhang et al. 2017c). Besides the MMC/MMV approaches mentioned above, the readers are referred to Norato et al. (2015) and Hur et al. (2017) for other forms of explicit geometry-based topology optimization approaches.
Fig. 1

The basic idea of the MMC-based topology optimization approach

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.


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

In the MMC-based approach, the material distribution of a structure can be descripted in the following form:
$$ \left\{\begin{array}{l}{\chi}^{\mathrm{s}}\left(\boldsymbol{x}\right)>0,\, \mathrm{if}\, \boldsymbol{x}\in {\Omega}^{\mathrm{s}},\\ {}{\chi}^{\mathrm{s}}\left(\boldsymbol{x}\right)=0,\, \mathrm{if}\, \boldsymbol{x}\in \partial {\Omega}^{\mathrm{s}},\\ {}{\chi}^{\mathrm{s}}\left(\boldsymbol{x}\right)<0,\, \mathrm{if}\, \boldsymbol{x}\in \mathrm{D}\backslash \left({\Omega}^{\mathrm{s}}\cup \partial {\Omega}^{\mathrm{s}}\right),\end{array}\right. $$
respectively. In the above equations, D represents a prescribed design domain. Ωs ⊂ D is a subset of D comprised by n components made of solid material. As shown in Guo et al. (2014), χs(x) =  max(χ1(x), …, χn(x)) with χi(x) denoting the topology description function (TDF) of the i-th component. For two-dimensional (2D) case, the function χi(x) can be adopted as:
$$ {\chi}_i\left(x,y\right)=1-{\left(\frac{x^{{\prime}}}{a_i}\right)}^p-{\left(\frac{y^{\prime}}{b_i\left({x}^{\prime}\right)}\right)}^p, $$
$$ \left\{\begin{array}{l}{x}^{\prime}\\ {}{y}^{\prime}\end{array}\right\}=\left[\begin{array}{l}\cos {\theta}_i\, \sin {\theta}_i\\ {}-\sin {\theta}_i\, \cos {\theta}_i\end{array}\right]\left\{\begin{array}{l}x-{x}_{0i}\\ {}y-{y}_{0i}\end{array}\right\} $$
and p is a relatively large even integer number (p = 6 is often adopted in the MMC approach). In the above equations, the symbols (x0i, y0i),  ai, bi(x′) and θi 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 bi(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:
$$ {b}_i\left({x}^{\prime}\right)=\frac{t_i^1+{t}_i^2}{2}+\frac{t_i^2-{t}_i^1}{2{a}_i}{x}^{\prime }, $$
where \( {t}_i^1 \) and \( {t}_i^2 \) are parameters used to describe the thicknesses of the component.
For three-dimensional (3D) case, the following TDF can be used to characterize the region occupied by the i-th component:
$$\begin{aligned} {\chi}_i\left(x,y,z\right)&=1-{\left(\frac{x^{\prime }}{L_i^1}\right)}^p-{\left(\frac{y^{\prime }}{g_i\left({x}^{\prime}\right)}\right)}^p\\ &\quad-{\left(\frac{z^{\prime }}{f_i\left({x}^{\prime },{y}^{\prime}\right)}\right)}^p \end{aligned}$$
$$ \left\{\begin{array}{l}{x}^{\prime}\\ {}{y}^{\prime}\\ {}{z}^{\prime}\end{array}\right\}=\left[\begin{array}{ccc}{R}_{11}& {R}_{12}& {R}_{13}\\ {}{R}_{21}& {R}_{22}& {R}_{23}\\ {}{R}_{31}& {R}_{32}& {R}_{33}\end{array}\right]\left\{\begin{array}{l}x-{x}_{0i}\\ {}y-{y}_{0i}\\ {}z-{z}_{0i}\end{array}\right\} $$
$$ \left[\begin{array}{ccc}{R}_{11}& {R}_{12}& {R}_{13}\\ {}{R}_{21}& {R}_{22}& {R}_{23}\\ {}{R}_{31}& {R}_{32}& {R}_{33}\end{array}\right]=\left[\begin{array}{ccc}{c}_b\cdot {c}_t& -{c}_b\cdot {s}_t& {s}_b\\ {}{s}_a\cdot {s}_b\cdot {c}_t+{c}_a\cdot {s}_t& -{s}_a\cdot {s}_b\cdot {s}_t+{c}_a\cdot {c}_t& -{s}_a\cdot {c}_b\\ {}-{c}_a\cdot {s}_b\cdot {c}_t+{s}_a\cdot {s}_t& {c}_a\cdot {s}_b\cdot {s}_t+{s}_a\cdot {c}_t& {c}_a\cdot {c}_b\end{array}\right], $$
respectively. In the above equations, \( {s}_a=\sin \alpha, \, {s}_b=\sin \beta, \, {s}_t=\sin \theta, \, {c}_a=\sqrt{1-{s}_a^2},\, {c}_b=\sqrt{1-{s}_b^2} \) and \( {c}_t=\sqrt{1-{s}_t^2} \) with α, β, and θ denoting the rotation angles of the component from a global coordinate system Oxyz to the local coordinate system Oxyz′, respectively. The central coordinate and the half-length of the component are represented by the coordinate (x0i, y0i,z0i) and \( {L}_i^1 \), respectively. Furthermore, the functions gi(x′) and fi(x′, y′) in the above equation are used to describe the thickness profiles of the component in y′ and z′ directions, respectively. The functions gi(x′) and fi(x′, y′) can be simply chosen as
$$ {g}_i\left({x}^{\prime}\right)={L}_i^2,\, {f}_i\left({x}^{\prime },{y}^{\prime}\right)={L}_i^3. $$
The readers are referred to Fig. 2 for the reference of the above components descriptions.
Fig. 2

The geometry description of a structural component

With use of the above expressions, the region \( {\Omega}_i^{\mathrm{s}} \) occupied by the i-th component (composed of solid material) can be described as:
$$ \left\{\begin{array}{l}{\chi}_i\left(\boldsymbol{x}\right)>0,\, \mathrm{if}\, \boldsymbol{x}\in {\Omega}_i^{\mathrm{s}},\\ {}{\chi}_i\left(\boldsymbol{x}\right)=0,\, \mathrm{if}\, \boldsymbol{x}\in \partial {\Omega}_i^{\mathrm{s}},\\ {}{\chi}_i\left(\boldsymbol{x}\right)<0,\, \mathrm{if}\, \boldsymbol{x}\in \mathrm{D}\backslash \left({\Omega}_i^{\mathrm{s}}\cup \partial {\Omega}_i^{\mathrm{s}}\right).\end{array}\right. $$

Optimization Formulation

Under the above geometry representation scheme, the layout of a structure can be solely determined by a design vector D = ((D1)Τ, …,(Di)Τ, …, (Dn)Τ)Τ, where Di 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.

Under the MMC-based solution framework, a typical topology optimization problem can be formulated as follows:
$$ {\displaystyle \begin{array}{l}\mathrm{Find}\, \boldsymbol{D}={\left({\left({\boldsymbol{D}}^1\right)}^{\mathrm{T}},\dots, {\left({\boldsymbol{D}}^i\right)}^{\mathrm{T}},\dots, {\left({\boldsymbol{D}}^n\right)}^{\mathrm{T}}\right)}^{\mathrm{T}}\\ {}\operatorname{Minimize}\, I=I\left(\boldsymbol{D}\right)\\ {}\mathrm{S}.\mathrm{t}.\\ {}{g}_k\left(\boldsymbol{D}\right)\le 0,\, k=1\dots, m,\\ {}\boldsymbol{D}\subset {\boldsymbol{U}}_{\boldsymbol{D}},\end{array}} $$
where I(D), gk,  k = 1,…, m are the objective function/functional and constraint functions/functionals. \( {\mathbf{U}}_{\boldsymbol{D}} \) is the admissible set of D. 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:
$$\begin{aligned} \mathrm{Find}\, \boldsymbol{D}&={\left({\left({\boldsymbol{D}}^1\right)}^{\mathrm{T}},\dots, {\left({\boldsymbol{D}}^i\right)}^{\mathrm{T}},\dots, {\left({\boldsymbol{D}}^n\right)}^{\mathrm{T}}\right)}^{\mathrm{T}},\\ &\boldsymbol{u}(x)\in {\boldsymbol{H}}^1\left({\Omega}^{\mathrm{s}}\right)\\ &\operatorname{Minimize}\, C=\underset{\mathrm{D}}{\int }H\left({\chi}^{\mathrm{s}}\left(\boldsymbol{x};\boldsymbol{D}\right)\right)\boldsymbol{f}\cdot \boldsymbol{u}\mathrm{dV}\\ &\qquad\qquad\qquad+\underset{\Gamma_t}{\int}\boldsymbol{t}\cdot \boldsymbol{u}\mathrm{dS}\\ &\mathrm{S}.\mathrm{t}.\\ &\underset{D}{\int }H\left({\chi}^{\mathrm{s}}\left(\boldsymbol{x};\boldsymbol{D}\right)\right)\boldsymbol{\varepsilon} \left(\boldsymbol{u}\right):\mathbb{E}:\boldsymbol{\varepsilon} \left(\boldsymbol{v}\right) \mathrm{dV}\\ &\quad=\underset{D}{\int }H\left({\chi}^{\mathrm{s}}\left(\boldsymbol{x};\boldsymbol{D}\right)\right)\boldsymbol{f}\cdot \boldsymbol{v} \mathrm{dV}+\underset{\Gamma_t}{\int}\boldsymbol{t}\cdot \boldsymbol{v}\mathrm{d}\mathrm{S},\\ &\quad\forall \boldsymbol{v}\in {U}_{\mathrm{ad}}\\ &\underset{\mathrm{D}}{\int }H\left({\chi}^{\mathrm{s}}\left(\boldsymbol{x};\boldsymbol{D}\right)\right)\mathrm{dV}\le \overline{V},\\ &\boldsymbol{D}\subset {U}_{\boldsymbol{D}},\\ &\boldsymbol{u}=\overline{\boldsymbol{u}},\, \mathrm{on}\, {\Gamma}_u,\end{aligned} $$
where D, f, t, u, ε =  sym (∇u) and \( \overline{\boldsymbol{u}} \) are the design domain, the body force density, the prescribed surface traction on Neumann boundary Γu, the displacement field, the linear strain tensor, and the prescribed displacement on Dirichlet boundary Γt, respectively. The symbol H = H(s) denotes the Heaviside function with H = 1 if s > 0 and H = 0 otherwise. 𝔼 = Es/(1 + v)[𝕀+vs/(1 − 2vs)δ ⊗ δ] is the fourth order elasticity tensor of the isotropic solid material with Es, vs, 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 v and \( \overline{V} \) is the upper limit of the volume of the available solid material.

Sensitivity Analysis

In the section, the sensitivity analysis of the objective and constraint functions/functionals under the MMC-based solution framework will be discussed. Generally speaking, the well-established adjoint approach can be used to obtain the corresponding sensitivity information. The sensitivity of a general structural shape-related functional I = I(D) with respect to a design variable a associated with χi (i.e., the TDF of the i-th component) can be written as
$$ \frac{\partial I}{\partial a}=\underset{\mathrm{D}}{\int }f\left(\boldsymbol{u},\boldsymbol{w}\right)\frac{\partial {H}_{\varepsilon}\left({\chi}_i\right)}{\partial a}\mathrm{dV},\, i=1,\dots, n, $$
where u and w are the primary and adjoint displacement fields, Hε(x) is the regularized Heaviside function (Zhang et al. 2016b), respectively. When I is the structural compliance, it yields that f(u, w) = 𝔼:ε(u) : ε(w) with w = −u while f(u, w) = 1 when I is the volume of the structure. Furthermore
$$ \frac{\partial {H}_{\varepsilon}\left({\chi}_i\right)}{\partial a}={\delta}_{\varepsilon}^i\left({\chi}^{\mathrm{s}}\right)\frac{\partial {\chi}_i}{\partial a} $$
with δε(s) = dHε(s)/ds 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 χi 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.


In order to illustrate the effectiveness of the MMC-based method for topology optimization, a torsion beam example shown in Fig. 3 is considered. The geometry of the design domain, the boundary condition, and the external load are all depicted in Fig. 3. Four loads are imposed on four vertices of the right side of the design domain, respectively. The goal of this problem is to minimize the structural compliance considering the volume constraint (it is assumed V ≤ 0.15 × |D| = 28.8 in this example).
Fig. 3

The torsion beam example

The initial designs of the problem shown in Fig. 4 consist of 128 components. The total number of design variables in the MMC is only 1152, while there are about 98,304 design variables in traditional methods (if the design domain is discretized by 96 × 32 × 32 meshes). Figure 5 plots the corresponding optimized structures obtained with use of MMC-based method. It is worth noting that the optimized structures obtained with MMC-based approach are actually pure black-and-white and contain no grey elements which are unavoidable in traditional approaches especially for 3D problems. Furthermore, since the optimized structures are described by a set of parameters of geometric meanings, the final results can import to CAD system directly as shown in Fig. 6.
Fig. 4

The initial design for the torsion beam example by the MMC approach

Fig. 5

Optimized structure for the torsion beam example obtained by the MMC approach

Fig. 6

Optimized structure obtained by the MMC approach plotted in CAD system

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).


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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering MechanicsInternational Research Center for Computational Mechanics, Dalian University of TechnologyDalianChina

Section editors and affiliations

  • Pablo Andres Munoz Rojas
    • 1
  1. 1.Universidade do Estado de Santa Catarina – UDESCJoinvilleBrazil