# An efficient 3D topology optimization code written in Matlab

- 39k Downloads
- 86 Citations

## Abstract

This paper presents an efficient and compact Matlab code to solve three-dimensional topology optimization problems. The 169 lines comprising this code include finite element analysis, sensitivity analysis, density filter, optimality criterion optimizer, and display of results. The basic code solves minimum compliance problems. A systematic approach is presented to easily modify the definition of supports and external loads. The paper also includes instructions to define multiple load cases, active and passive elements, continuation strategy, synthesis of compliant mechanisms, and heat conduction problems, as well as the theoretical and numerical elements to implement general non-linear programming strategies such as SQP and MMA. The code is intended for students and newcomers in the topology optimization. The complete code is provided in Appendix C and it can be downloaded from http://top3dapp.com.

## Keywords

Topology optimization Matlab Compliance Compliant mechanism Heat conduction Non-linear programming## 1 Introduction

Topology optimization is a computational material distribution method for synthesizing structures without any preconceived shape. This freedom provides topology optimization with the ability to find innovative, high-performance structural layouts, which has attracted the interest of applied mathematicians and engineering designers. From the work of Lucien Schmit in the 1960s (Schmit 1960)—who recognized the potential of combining optimization methods with finite-element analysis for structural design—and the seminal paper by Bendsøe and Kikuchi (1988), there have been more than eleven thousand journal publications in this area (Compendex list as of September 2013), several reference books (Hassani and Hinton 1998; Bendsøe and Sigmund 2003; Christensen and Klarbring 2009), and a number of readily available educational computer tools for Matlab and other platforms. Some examples of such tools include the topology optimization program by Liu et al. (2005) for Femlab, the shape optimization program by Allaire and Pantz (2006) for FreeFem++, the open source topology optimization program ToPy by Hunter (2009) for Python, and the 99-line program for Michell-like truss structures by Sokół (2011) for Mathematica.

For Matlab, Sigmund (2001) introduced the 99-line program for two-dimensional topology optimization. This program uses stiffness matrix assembly and filtering via nested loops, which makes the code readable and well-organized but also makes it slow when solving larger problems. Andreassen et al. (2011) presented the 88-line program with improved assembly and filtering strategies. When compared to the 99-line code in a benchmark problem with 7500 elements, the 88-line code is two orders of magnitude faster. From the same research group, Aage et al. (2013) introduced TopOpt, the first topology optimization App for hand-held devices.

Also for Matlab, Wang et al. (2004) introduced the 199-line program TOPLSM making use of the level-set method. Challis (2010) also used the level-set method but with discrete variables in a 129-line program. Suresh (2010) presented a 199-line program ParetoOptimalTracing that traces the Pareto front for different volume fractions using topological sensitivities. More recently, Talischi et al. (2012a, b) introduced PolyMesher and PolyTop for density-based topology optimization using polygonal finite elements. The use of polygonal elements makes these programs suitable for arbitrary non-Cartesian design domains in two dimensions.

One of the few contributions to three-dimensional Matlab programs is presented by Zhou and Wang (2005). This code, referred to as the 177-line program, is a successor to the 99-line program by Sigmund (2001) that inherits and amplifies the same drawbacks. Our paper presents a 169-line program referred to as top3d that incorporates efficient strategies for three-dimensional topology optimization. This program can be effectively used in personal computers to generate structures of substantial size. This paper explains the use of top3d in minimum compliance, compliant mechanism, and heat conduction topology optimization problems.

The rest of this paper is organized as follows. Section 2 briefly reviews theoretical aspects in topology optimization with focus on the density-based approach. Section 3 introduces 3D finite element analysis and its numerical implementation. Section 4 presents the formulation of three typical topology optimization problems, namely, minimum compliance, compliant mechanism, and heat conduction. Section 5 discusses the optimization methods and their implementation in the code. Section 6 shows the numerical implementation procedures and results of three different topology optimization problems, several extensions of the top3d code, and multiple alternative implementations. Finally, Section 7, offers some closing thoughts. The top3d code is provided in Appendix C and can also be downloaded for free from the website: http://top3dapp.com.

## 2 Theoretical background

### 2.1 Problem definition and ill-posedness

A topology optimization problem can be defined as a binary programming problem in which the objective is to find the distribution of material in a prescribed area or volume referred to as the *design domain*. A classical formulation, referred to as the *binary compliance problem*, is to find the “black and white” layout (i.e., solids and voids) that minimizes the work done by external forces (or compliance) subject to a volume constraint.

The binary compliance problem is known to be ill-posed (Kohn and Strang 1986a, b, c). In particular, it is possible to obtain a non-convergent sequence of feasible black-and-white designs that monotonically reduce the structure’s compliance. As an illustration, assume that a design has one single hole. Then, it is possible to find an improved solution with the same mass and lower compliance when this hole is replaced by two smaller holes. Improved solutions can be successively found by increasing the number of holes and reducing their size. The design will progress towards a *chattering* design within infinite number of holes of infinitesimal size. That makes the compliance problem unbounded and, therefore, ill-posed.

One alternative to make the compliance problem well-posed is to control the perimeter of the structure (Haber and Jog 1996; Jog 2002). This method effectively avoids chattering configurations, but its implementation is not free of complications. It has been reported that the addition of a perimeter constraint creates fluctuations during the iterative optimization process so internal loops need to be incorporated (Duysinx 1997) Op. cit. (Bendsøe and Sigmund 2003). Also, small variations in the parameters of the algorithm lead to dramatic changes in the final layout (Jog 2002).

### 2.2 Homogenization method

Another alternative is to relax the binary condition and include intermediate material densities in the problem formulation. In this way, the chattering configurations become part of the problem statement by assuming a periodically perforated microstructure. The mechanical properties of the material are determined using the homogenization theory. This method is referred to as the *homogenization method* for topology optimization (Bendsøe 1995; Allaire 2001). The main drawback of this approach is that the optimal microstructure, which is required in the derivation of the relaxed problem, is not always known. This can be alleviated by restricting the method to a subclass of microstructures, possibly suboptimal but fully explicit. This approach, referred to as *partial relaxation*, has been utilized by many authors including Bendsøe and Kikuchi (1988), Allaire and Kohn (1993), Allaire et al. (2004), and references therein.

An additional problem with the homogenization methods is the manufacturability of the optimized structure. The “gray” areas found in the final designs contain microscopic length-scale holes that are difficult or impossible to fabricate. However, this problem can be mitigated with *penalization* strategies. One approach is to post-process the partially relaxed optimum and force the intermediate densities to take black or white values (Allaire et al. 1996). This *a posteriori* procedure results in binary designs, but it is purely numerical and mesh dependent. Other approach is to impose *a priori* restrictions on the microstructure that implicitly lead to black-and-white designs (Bendsøe 1995). Even though penalization methods have shown to be effective in avoiding or mitigating intermediate densities, they revert the problem back to the original ill-possedness with respect to mesh refinement.

### 2.3 Density-based approach

An alternative that avoids the application of homogenization theory is to relax the binary problem using a continuous density value with no microstructure. In this method, referred to as the *density-based approach*, the material distribution problem is parametrized by the material density distribution. In a discretized design domain, the mechanical properties of the material element, i.e., the stiffness tensor, are determined using a power-law interpolation function between void and solid (Bendsøe 1989; Mlejnek 1992). The power law may *implicitly* penalize intermediate density values driving the structure towards a black-and-white configuration. This penalization procedure is usually referred to as the *Solid Isotropic Material with Penalization* (SIMP) method (Zhou and Rozvany 1991). The SIMP method does not solve the problem’s ill-possedness, but it is simpler than other penalization methods.

*x*

_{ i }and element Young’s modulus E

_{ i }given by

_{0}is the elastic modulus of the solid material and

*p*is the penalization power (

*p*> 1). A modified SIMP approach is given by

_{min}is the elastic modulus of the void material, which is non-zero to avoid singularity of the finite element stiffness matrix. The modified SIMP approach, as (2), offers a number of advantages over the classical SIMP formulation, as shown in (1), including the independency between the minimum value of the material’s elastic modulus and the penalization power (Sigmund 2007).

*N*

_{ i }is the neighborhood of an element

*x*

_{ i }with volume

*v*

_{ i }, and

*H*

_{ i j }is a weight factor. The neighborhood is defined as

*i*,

*j*) is the distance between the center of element

*i*and the center of element

*j*, and

*R*is the size of the neighborhood or filter size. The weight factor

*H*

_{ i j }may be defined as a function of the distance between neighboring elements, for example

*j*∈

*N*

_{ i }. The filtered density \(\tilde {x_{i}}\) defines a modified (physical) density field that is now incorporated in the topology optimization formulation and the SIMP model as

## 3 Finite element analysis

### 3.1 Equilibrium equation

*i*is interpolated from void to solid as

*ν*is the Poisson’s ratio of the isotropic material. Using the finite element method, the elastic solid element stiffness matrix is the volume integral of the elements constitutive matrix

**C**

_{ i }(x͂

_{ i }) and the strain–displacement matrix

**B**in the form of

*ξ*

_{ e }(

*e*= 1,…,3) are the natural coordinates as shown in Fig. 1, and the hexahedron coordinates of the corners are shown in Table 1. The strain–displacement matrix

**B**relates the strain

**𝜖**and the nodal displacement

**u**,

**𝜖**=

**Bu**. Using the SIMP method, the element stiffness matrix is interpolated as

The eight-node hexahedral element with node numbering conventions

Node | | | |
---|---|---|---|

1 | −1 | −1 | −1 |

2 | +1 | −1 | −1 |

3 | +1 | +1 | −1 |

4 | −1 | +1 | −1 |

5 | −1 | −1 | +1 |

6 | +1 | −1 | +1 |

7 | +1 | +1 | +1 |

8 | −1 | +1 | +1 |

**k**

_{ m }(

*m*= 1,…,6) are 6 × 6 symmetric matrices (see Appendix A). One can also verify that \(\mathbf{k}^{0}_{i}\) is positive definite. The global stiffness matrix

**K**is obtained by the assembly of element-level counterparts

**k**

_{ i },

*n*is the total number of elements. Using the global versions of the element stiffness matrices

**K**

_{ i }and \( \mathbf {k}^{0}_{i}\), (13) is expressed as

**U**(

**x**͂) is the solution of the equilibrium equation

**F**is the vector of nodal forces and it is independent of the physical densities

**x͂**. For brevity of notation, we omitted the dependence of physical densities

**x͂**on the design variables

*x*,

**x͂**=

**x͂**(

**x**).

### 3.2 Numerical implementation

*N*

_{1},…,

*N*

_{8}are ordered in counter-clockwise direction as shown in Fig. 3. Note that the “local” node number (

*N*

_{ i }) does not follow the same rule as the “global” node ID (NID

_{ i }) system in Fig. 2. Given the size of the volume (nelx × nely × nelz) and the global coordinates of node

*N*

_{1}(

*x*

_{1},

*y*

_{1},

*z*

_{1}), one can identify the global node coordinates and node IDs of the other seven nodes in that element by the mapping the relationships as summarized in Table 2.

Illustration of relationships between node number, node coordinates, node ID and node DOFs

Node Number | Node coordinates | Node ID | Node Degree of Freedoms | ||
---|---|---|---|---|---|

x | y | z | |||

| ( | \(\texttt {NID}_{1}^{\dagger }\) | 3∗NID | 3∗NID | 3∗NID |

| ( | NID | 3∗NID | 3∗NID | 3∗NID |

| ( | NID | 3∗NID | 3∗NID | 3∗NID |

| ( | NID | 3∗NID | 3∗NID | 3∗NID |

| ( | \(\texttt {NID}_{5} = \texttt {NID}_{1}+\texttt {NID}_{z}^{\ddagger }\) | 3∗NID | 3∗NID | 3∗NID |

| ( | NID | 3∗NID | 3∗NID | 3∗NID |

| ( | NID | 3∗NID | 3∗NID | 3∗NID |

| ( | NID | 3∗NID | 3∗NID | 3∗NID |

*x*-

*y*-

*z*directions (one element has 24 DOFs). The degrees of freedom are organized in the nodal displacement vector

**U**as

*n*is the number of elements in the structure. The location of the DOFs in

*U*, and consequently

**K**and

*F*, can be determined from the node ID as shown in Table 2.

*x*-

*y*plane (for

*z*= 0), the column vector edofVec contains the node IDs of the first node at each element, and the connectivity matrix edofMat of size nele × 24 containing the node IDs for each element. For the volume in Fig. 2, nelx=4, nely=1, and nelz=2, which results in

The element stiffness matrix KE (size 24 × 24) is obtained from the lk_H8 subroutine (lines 99-146). Matrices iK (size 24 nele × 24) and jK (size nele × 24^{2}), reshaped as column vectors, contain the rows and columns identifying the 24 × 24 × nele DOFs in the structure. The three-dimensional array xPhys (size nely × nelx × nelz) corresponds to the physical densities. The matrix sK (size 24^{2} × nele) contains all element stiffness matrices. The assembly procedure of the (sparse symmetric) global stiffness matrix K (line 71) avoids the use of nested for loops.

## 4 Optimization problem formulation

Three representative topology optimization problems are described in this section, namely: minimum compliance, compliant mechanism synthesis, and heat conduction.

### 4.1 Minimum compliance

**F**is the vector of nodal forces and

**U**(

**x͂**) is the vector of nodal displacements. Incorporating a volume constraint, the minimum compliance optimization problem is

**x͂**=

**x͂**(

**x͂**) are defined by (3),

*n*is the number of elements used to discretize the design domain,

**v**= [

*v*

_{1},…,

*v*

_{ n }]

^{T}is a vector of element volume, and \(\bar {v}\) is the prescribed volume limit of the design domain. The nodal force vector

**F**is independent of the design variables and the nodal displacement vector

**U**(

**x͂**) is the solution of

**K**(

**x͂**)

**U**(

**x͂**) =

**F**.

*v*(

**x͂**) in (18) with respect to the design variable

*x*

_{ e }is given

*v*

_{ i }=

*v*

_{ j }=

*v*

_{ e }=1.

**u**

_{ i }is the element vector of nodal displacements. Since \(\mathbf {k}^{0}_{i}\) is positive definite,

*∂c*(

**x͂**)/

*∂x͂*

_{ i }< 0.

The objective function in (18) is calculated in Line 75. The sensitivities of the objective function and volume fraction constraint with respect to the physical density are given be lines 76-77. Finally, the chain rule as stated in (22) is deployed in lines 79-80.

### 4.2 Compliant mechanism synthesis

**L**is a unit length vector with zeros at all degrees of freedom except at the output point where it is one, and

**U**(

**x͂**) =

**K**(

**x͂**)

^{−1}

**F**.

*c*(

**x͂**) in (29), let us define a global adjoint vector

**U**

_{d}(

**x͂**) from the solution of the adjoint problem

*c*(

**x͂**) with respect to the design variable

*x*

_{ e }is again obtained by the chain rule,

*∂c*(

**x͂**)/

*∂x͂*

_{ i }can be obtained using direct differentiation. The use of the interpolation given by (6) yields an expression similar to the one obtained in (28),

*i*. In this case, \({\partial c({\mathbf {k}}_{i}^{0})}/{\partial \tilde {\mathrm {x}}_{i}} \) may be positive or negative.

Vector Ud (Line 74a) is the dummy load displacement field and vector U (line 74b) is the input load displacement. The codes for the implementation of chain rule are not shown above since they are same as lines 79-80.

### 4.3 Heat conduction

**F**donates the global thermal load vector, and \(\mathbf K({\mathbf {k}}_{i}^{0}) \) donates the global thermal conductivity matrix. For a material with isotropic properties, conductivity is the same in all directions.

**U**(

**x͂**) =

**K**(

**x͂**)

^{−1}

**F**, and

**K**(

**x͂**) is obtained by the assembly of element thermal conductivity matrices \(\mathbf {k}_{i}(\tilde {\mathrm {x}}_{i})\). Following the interpolation function in (6), the element conductivity matrix is expressed as

*k*

_{min}and

*k*

_{0}represent the limits of the material’s thermal conductivity coefficient and \(\mathbf {k}_{i}^{0}\) donates the element conductivity matrix. Note that (34) may be considered as the distribution of two material phases: a good thermal conduction (

*k*

_{0}) and the other a poor conductor (

*k*

_{min}).

## 5 Optimization algorithms

Non-linear programming (NLP) problems, such as minimum compliance (18), compliant mechanism (29), and heat conduction (33), can be addressed using sequential convex approximations such as sequential quadratic programming (SQP) (Wilson 1963) and the method of moving asymptotes (MMA) (Svanberg 1987). The premise of these methods is that, given a current design **x** ^{(k)}, the NLP algorithm is able to find a convex approximation of the original NLP problem from which an improved design **x** ^{(k+1)} can be derived. The nature of the problem’s approximation is determined by the type of algorithm, e.g., quadratic programming (QP) or MMA. An special case of the latter approach, which is historically older than SQP and MMA, is the optimality criterion (OC) method. This method still find its place in topology optimization due to its numerical simplicity and numerical efficiency (Christensen and Klarbring 2009). The following sections presents the implementation of the SQP, MMA, and OC methods to the solution of the minimum compliance topology optimization problems presented in this paper.

### 5.1 Sequential quadratic programming

**x**

^{(k)}and all corresponding active constraints, the QP approximation of the minimum compliance problem in (18) can be expressed as

*c*

^{(k)}is the value of the objective function evaluated at

**x**

^{(k)}and

**d**=

**x**−

**x**

^{(k)}, and

**A**is the matrix of active constraints. The optimality and feasibility conditions of (36) yield

*active set*algorithm (Nocedal and Wright 2006), allows to determine a step \(\tilde {d}^{(k)}\) from the solution of the of system of linear equations in (37) expressed as

*α*

^{(k)}is determined by a line search procedure. The Hessian ∇

^{2}

*c*

^{(k)}can be numerically approximated but, for the problems considered in this paper, one can determined the closed-form expression (see Appendix B), which is given by

| |
---|---|

Choose an initial feasible design Evaluate Identify active constraint matrix A in (36); Solve for Find appropriate step size Set Set |

Finally, Matlab has built-in constrained NLP solver fmincon. The implementation of using fmincon as an optimizer in our top3d program is quite easy, but need some reconstructions of the program (one needs to divide program into different subfunctions, e.g., objective function, constraint function, Hessian function). To further assist on the implementation of an SQP strategy, the reader can find a step-by-step tutorial on our website http://top3dapp.com.

### 5.2 Method of moving asymptotes

**x**

^{(k)}the MMA approximation of the minimum compliance problem in (18) yields to the following linear programming problem:

*k*= 1 and

*k*= 2,

*x*

_{ i }oscillates, the two asymptotes are brought closer to \(x_{i}^{\left (k\right )}\) to have a more conservative MMA approximation. On the other hand, if the signs are same, the two asymptotes are extended away from \(x_{i}^{\left (k\right )}\) in order to speed up the convergence. The MMA algorithm is explained in Algorithm 2.

| |
---|---|

Choose an initial feasible design Update \({L_{i}^{k}}\), and \({U_{i}^{k}}\) using (43); Update \({L_{i}^{k}}\), and \({U_{i}^{k}}\) using (44) and (45); Find appropriate step size Calculate derivate (28); Solve the MMA subproblem (41) to obtain \(\tilde {\mathrm {x}}^{(k+1)}\); Set Set |

The MMA algorithm is available for Matlab (mmasub). The reader may obtain a copy by contacting Prof. Krister Svanberg (http://www.math.kth.se/%7Ekrille/Welcome.html) from KTH in Stockholm Sweden. Although mmasub has total of 29 input and output variables, its implementation for top3d is straightforward. The details can be found at http://top3dapp.com.

### 5.3 Optimality criteria

**0**⩽

**x**⩽

**1**is inactive, then convergence is achieved when the KKT condition

*k*= 1,…,

*n*, where

*λ*is the Lagrange multiplier associated with the constraint

*v*(

**x͂**). This optimality condition can be expressed as

*B*

_{ e }=1, where

*m*is a positive move-limit, and

*η*is a numerical damping coefficient. The choice of

*m*= 0.2 and

*η*= 0.5 is recommended for minimum compliance problems (Bendsøe 1995; Sigmund 2001). For compliant mechanisms,

*η*= 0.3 improves the convergence of the algorithm. The only unknown in (48) is the value of the Lagrange multiplier

*λ*, which satisfies that

*λ*is found by a root-finding algorithm such as the bisection method. Finally, the termination criteria are satisfied when a maximum number of iterations is reached or

*𝜖*is a relatively small value, for example

*𝜖*= 0.01.

| |
---|---|

Choose an initial design FE-analysis using (16) to obtain the corresponding nodal displacement Compute objective function, e.g., compliance put displacement Sensitivity analysis by using the equations as dis- cussed in Section 4; Apply filter techniques, e.g. (3) in Section 2.3 or any other filters like those discussed in Section 6.1.4 Update design variables using (48) to obtain Set |

## 6 Numerical examples

*x*,

*y*, and

*z*directions, volfrac is the volume fraction limit (\(\bar v\)), penal is the penalization power (

*p*), and rmin is filter size (

*R*). User-defined variables are set between lines 3 and 18. These variables determine the material model, termination criteria, loads, and supports. The following examples demonstrate the application of the code to minimum compliance problems, and its extension to compliant mechanism synthesis and heat condition.

### 6.1 Minimum compliance

#### 6.1.1 Boundary conditions

#### 6.1.2 Multiple load cases

*M*is the number of load cases.

#### 6.1.3 Active and passive elements

#### 6.1.4 Alternative filters

In the topology optimization, filters are introduced to avoid numerical instabilities. Different filtering techniques may result different discreteness of the final solutions, and sometimes may even contribute to different topologies. In addition to density filter, in the literatures there are bunch of different filtering schemes. For example, sensitivity filter (Sigmund 1994, 1997), morphology based black and white filters (Sigmund 2007), filtering technique using Matlab built-in function conv2 (Andreassen et al. 2011), filtering based on Helmholtz type differential equations (Andreassen et al. 2011), Heaviside filter (Guest et al. 2004, 2011), and gray scale filter (Groenwold and Etman 2009). All the filters pursue a simple goal to achieve black-and-white structures. Two of them are chosen, which stand for classic and better performance, as well as easy implementation.

### Sensitivity filter

*γ*(=10

^{−3}) is a small number in order to avoid division by zero.

The implementation of the sensitivity filter can be achieved by adding and changing a few lines.

### Gray scale filter

The standard OC updating method is a special case of (52) with *q* = 1. A typical value of *q* for the SIMP-based topology optimization is *q* = 2.

The implementation of the gray scale filter to the code can be done as follows:

*q*should be increased gradually by adding one line after line 68

#### 6.1.5 Iterative solver

Direct solver is a special case by setting the preconditioner (line 72c) to Open image in new window

Time usage of finite element analysis time for different solvers

Mesh size | Direct solver | Iterative solver |
---|---|---|

30 × 10 × 2 | 0.018 sec | 0.129 sec |

60 × 20 × 4 | 0.325 sec | 0.751 sec |

150 × 50 × 10 | 74.474 sec | 22.445 sec |

Time usage of finite element analysis time for different solvers

#### 6.1.6 Continuation strategy

*k*is the iteration number, and

*p*

^{max}is the maximum penalization power.

Though this methodology is not proven to converge to the global optimum, it regularizes the algorithm and allows the comparison of different optimization strategies.

### 6.2 Compliant mechanism synthesis

### 6.3 Heat conduction

The implementation of heat conduction problems is not more complex than the one for compliant mechanism synthesis since the number of DOF per node is one rather than three. Following the implementation of heat conduction problems in two dimensions (Bendsøe and Sigmund 2003), the implementation for three dimension problems is suggested in the following steps.

Also, since there is only one DOF per node in heat condition problems, some variables need to change correspondingly, such as ndof, edofMat.

## 7 Conclusions

This paper presents Matlab the analytical elements and the numerical implementation of an academic three-dimensional structural topology optimization algorithm referred to as top3d. In this topology optimization algorithm, the problem formulation follows a density-based approach with a modified SIMP interpolation for physical densities. The finite element formulation makes use of eight-node hexahedral elements for which a closed-form expression of the element stiffness matrix is derived and numerically implemented. The hexahedral finite elements are used to uniformly discretize a prismatic design domain and solve three related topology optimization problems: minimum compliance, compliant mechanism, and heat conduction problems. For each problem, this paper includes the analytical derivation of the sensitivity coefficients used by three gradient-based optimization algorithms: SQP, MMA, and OC, which is implemented by default. For the implementation of SQP, this paper derives an analytic expression for the second order derivative.

The use of top3d is demonstrated through several numerical examples. These examples include problems with a variety of boundary conditions, multiple load cases, active and passive elements, filters, and continuation strategies to mitigate convergence to a local minimum. The architecture of the code allows the user to map node coordinates of node degrees-of-freedom boundary conditions. In addition, the paper provides a strategy to handle large models with the use of an iterative solver. For large-scale finite-element models, the iterative solver is about 30 times faster than the traditional direct solver. While this implementation is limited to linear topology optimization problems with a linear constraint, it provides a clear perspective of the analytical and numerical effort involved in addressing three-dimensional structural topology optimization problems. Finally, additional academic resources such the use of MMA and SQP are available at http://top3dapp.com.

## References

- Aage N, Nobel-Jørgensen M, Andreasen CS, Sigmund O (2013) Interactive topology optimization on hand-held devices. Struct Multidiscip Optim 47(1):1–6CrossRefGoogle Scholar
- Allaire G (2001) Shape optimization by the homogenization method. Springer, New YorkGoogle Scholar
- Allaire G, Kohn R (1993) Optimal design for minimum weight and compliance in plane-stress using extremal microstructures. Eur J Mech 12(6):839–878zbMATHMathSciNetGoogle Scholar
- Allaire G, Pantz O (2006) Structural optimization with freefem++. Struct Multidiscip Optim 32(3):173–181CrossRefzbMATHMathSciNetGoogle Scholar
- Allaire G, Belhachmi Z, Jouve F (1996) The homogenization method for topology and shape optimization. single and multiple loads case. Eur J Finite Elem 5:649–672zbMATHMathSciNetGoogle Scholar
- Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393CrossRefzbMATHMathSciNetGoogle Scholar
- Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidiscip Optim 43(1):1–16CrossRefzbMATHGoogle Scholar
- Augarde C, Ramage A, Staudacher J (2006) An element-based displacement preconditioner for linear elasticity problems. Comput Struct 84(31–32):2306–2315CrossRefGoogle Scholar
- Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Multidiscip Optim 1(4):193–202CrossRefGoogle Scholar
- Bendsøe MP (1995) Optimization of structural topology shape and material. Springer, New YorkCrossRefGoogle Scholar
- Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224CrossRefGoogle Scholar
- Bendsøe MP, Sigmund O (2003) Topology optimization: theory, method and applications. SpringerGoogle Scholar
- Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459CrossRefzbMATHGoogle Scholar
- Challis VJ (2010) A discrete level-set topology optimization code written in matlab. Struct Multidiscip Optim 41(3):453–464CrossRefzbMATHMathSciNetGoogle Scholar
- Christensen PW, Klarbring A (2009) An introduction to structural optimization. SpringerGoogle Scholar
- Duysinx P (1997) Layout optimization: A mathematical programming approach, DCAMM report. Technical report, Danish Center of Applied Mathematics and Mechanics, Technical University of Denmark, DK-2800 LyngbyGoogle Scholar
- Fleury C (1989) CONLIN: an efficient dual optimizer based on convex approximation concepts. Struct Optim 1(2):81–89CrossRefGoogle Scholar
- Groenwold AA, Etman LFP (2009) A simple heuristic for gray-scale suppression in optimality criterion-based topology optimization. Struct Multidiscip Optim 39(2):217–225CrossRefGoogle Scholar
- Groenwold AA, Etman LFP (2010) A quadratic approximation for structural topology optimization. Int J Numer Methods Eng 82(4):505–524zbMATHMathSciNetGoogle Scholar
- Guest JK, Prevost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254CrossRefzbMATHMathSciNetGoogle Scholar
- Guest JK, Asadpoure A, Ha SH (2011) Elimiating beta-continuation from heaviside projection and density filter algorithms. Struct Multidiscip Optim 44:443–453CrossRefzbMATHMathSciNetGoogle Scholar
- Haber R, Jog C (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11(1):1–12CrossRefGoogle Scholar
- Hassani B, Hinton E (1998) Homogenization and structural topology optimization: theory, practice and software. SpringerGoogle Scholar
- Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49(6)Google Scholar
- Hunter W (2009) Predominantly solid-void three-dimensional topology optimisation using open source software. Master’s thesis, University of StellenboschGoogle Scholar
- Jog C (2002) Topology design of structures using a dual algorithm and a constraint on the perimeter. Int J Numer Methods Eng 54(7):1007–1019CrossRefzbMATHMathSciNetGoogle Scholar
- Kohn R, Strang G (1986a) Optimal design and relaxation of variational problems (part I). Commun Pure Applied Math 39(1):113–137CrossRefzbMATHMathSciNetGoogle Scholar
- Kohn R, Strang G (1986b) Optimal design and relaxation of variational problems (part II). Commun Pure Appl Math 39(2):139–182CrossRefzbMATHMathSciNetGoogle Scholar
- Kohn R, Strang G (1986c) Optimal design and relaxation of variational problems (part III). Commun Pure Appl Math 39(3):353–377CrossRefzbMATHMathSciNetGoogle Scholar
- Liu Z, Korvink JG, Huang I (2005) Structure topology optimization: fully coupled level set method via FEMLAB. Struct Multidiscipl Optim 6(29):407–417CrossRefMathSciNetGoogle Scholar
- Mlejnek H (1992) Some aspects of the genesis of structures. Struct Optim 5(1–2):64–69CrossRefGoogle Scholar
- Nocedal J, Wright S (2006) Numerical optimization, 2nd edn. SpringerGoogle Scholar
- Schmit LA (1960) Structural design by systematic synthesis. In: 2nd ASCE conference of electrical compounds. Pittsburgh, pp 139–149Google Scholar
- Sigmund O (1994) Design of material structures using topology optimization, PhD thesis, Technical University of DenmarkGoogle Scholar
- Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):495–526CrossRefGoogle Scholar
- Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21(2):120–127CrossRefGoogle Scholar
- Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33:401–424CrossRefGoogle Scholar
- Sigmund O, Peterson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75CrossRefGoogle Scholar
- Sokół T (2011) A 99 line code for discretized michell truss optimization written in mathematica. Struct Multidiscip Optim 43(2):181–190CrossRefzbMATHGoogle Scholar
- Suresh K (2010) A 199-line matlab code for pareto-optimal tracing in topology optimization. Struct Multidiscip Optim 42:665–679CrossRefzbMATHMathSciNetGoogle Scholar
- Svanberg K (1987) The method of moving asymptotes-a new method for structural optimzation. Int J Numer Methods Eng 24:359–373CrossRefzbMATHMathSciNetGoogle Scholar
- Talischi C, Paulino G H, Pereira A, Menezes IFM (2012a) Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct Multidiscip Optim 45:309–328CrossRefzbMATHMathSciNetGoogle Scholar
- Talischi C, Paulino G H, Pereira A, Menezes IFM (2012b) Polytop: a matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45:329–357CrossRefzbMATHMathSciNetGoogle Scholar
- Wang MY, Chen S, Xia Q (2004) Structural topology optimization with the level set method. http://www2.acae.cuhk.edu.hk/cmdl/download.htm
- Wilson RB (1963) A simplicial method for convex programming, PhD thesis, Harvard UniversityGoogle Scholar
- Zhou M, Rozvany G (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comp Meth Appl Mech Eng 89:309–336CrossRefGoogle Scholar
- Zhou S, Wang MY (2005) 3d structural topology optimization with the simp method. http://www2.acae.cuhk.edu.hk/cmdl/download.htm