# Truss layout optimization within a continuum

- 731 Downloads
- 9 Citations

## Abstract

The present work extends truss layout optimization by considering the case when it is embedded in a continuum. Structural models often combine discrete and continuum members and current requirements for efficiency and extreme structures push research in the field of optimization. Examples of varied complexity and dimensional space are analyzed and compared, highlighting the advantages of the proposed method. The goal of this work is to provide a simple formulation for the discrete component of the structure, more specifically the truss, to be optimized in presence of a continuum.

## Keywords

Truss layout optimization Topology optimization Michell truss Truss geometry optimization Discrete-continuum optimization Embedded formulation## 1 Introduction

Structural optimization research is rapidly moving forward with a constant push for more efficient, lighter, cheaper and extreme structures (Hemp 1973). Structural optimization is commonly carried out by optimizing the material distribution (Bendsoe and Sigmund 2004), optimizing a truss (Felix and Vanderplaats 1987; Hansen and Vanderplaats 1988; Lipson and Gwin 1977; Ohsaki 2010), and optimizing the continuum shape (Haslinger and Mäkinen 2003) to name a few. Optimal truss layout has greatly evolved with the *ground structure* method (Dorn et al. 1964; Sokół 2010) and proves to be a reliable and stable method for truss structures. Optimizing material distribution with an overlaying discrete element structure connected has been previously studied (Allahdadian et al. 2012; Liang et al. 2000; Liang 2007; Mijar et al. 1998), and recent refinements make it suitable for real applications (Stromberg et al. 2012). Previously, a formulation for embedding reinforcement (discrete elements) in the context of reinforced concrete was developed (Elwi and Hrudey 1989), and later extended to three-dimensions (Barzegar and Maddipudi 1994). Optimization of reinforced concrete using this embedded formulation was also explored (Kato and Ramm 2010). The ground-structure method for optimization combined with discrete elements embedded in a continuum has also proven to be feasible (Amir and Sigmund 2012). The present work attempts to solve the problem where discrete structures, linked to a continuum (or embedded), are optimized with the discrete nodes not directly matching over continuum nodes using a convolution-based coupling to embed the discrete onto the continuum. Some examples of structures typically modeled in a discrete-continuum fashion are: reinforced concrete, cable supported bridges, column supporting a slab and beam-wall connections to name a few.

This works aims to develop a simple technique that allows for truss layout optimization (nodal locations and cross-sectional area) to be optimized, in presence of a continuum, with linkage between both. If the continuum is modeled using traditional \(C^{0}\) elements, the first derivatives are discontinuous, thus making the embedded formulation difficult to optimize using traditional gradient based optimizers. The discontinuity problem could potentially be solved using \(C^{1}\) elements, however, the formulations for these are complex, especially for higher dimensions. An alternative procedure is presented here which is easy to implement and shows agreement with analytical results or demonstrates stability of the optimized solution, regardless of numerical variations in the model.

This formulation is based on small deformation theory, and because nodes are treated as a cloud, any type or order of finite elements can be used (i.e. the element connectivity is not used). The examples in the present work deal with compliance optimization. Nevertheless, the technique can be applied to any objective function based on stiffness for which an expression for the gradient can be obtained.

The article is organized as follows: The formulation is derived and described in detail in Section 2. In Section 3, the method is verified and the stability tested against a problem for which the solution is known. Several demonstrative examples are optimized in Section 4. Finally, conclusions and remarks of the method are discussed in Section 5. The nomenclature and symbols used are listed in the Appendix.

## 2 Formulation

Truss layout optimization has been explored previously with good results (Felix and Vanderplaats 1987; Hansen and Vanderplaats 1988; Lipson and Gwin 1977; Ohsaki 2010). The formulation for truss layout optimization presented here is analogous to the one presented in (Hansen and Vanderplaats 1988), but better suited for any-dimensional (1D, 2D, 3D) problems and extended by combining it with a continuum.

*A*,

*E*and

*L*being the element’s cross-sectional area, Young modulus, and length respectively. The stiffness matrix in global coordinates \(\mathbf {K}_{e}\) for truss element

*e*is defined in terms of the stiffness matrix in the element’s local coordinates \(\mathbf {K}^{\star }_{e}\) and the transformation matrix \(\mathbf {T}_{e}\)

*n*of node

*j*of the truss member is

*L*representing the truss element’s length, \(n=\left \{x,y,z\right \}\) and \(j=\left \{1,2\right \}\). The derivatives of the element’s length

*L*, with respect to the coordinate

*n*, are

### 2.1 Mapping discrete to continuum representation

Consider the stiffness matrix of a continuum \(\mathbf {K}_{c}\) obtained by means of a finite element method (FEM), and the stiffness matrix from a single truss element \(\mathbf {K}_{e}\). The challenge is to add the contribution of \(\mathbf {K}_{e}\) onto \(\mathbf {K}_{c}\) in a coherent fashion (energy conservation), and with a smooth derivative field. An approach based on energy conservation and FEM shape functions meets the first requirement, but because the FEM shape functions are discontinuous across elements, it does not have a smooth derivative field.

The choice of the shape functions \(\mathbf {N}\) used in the mapping to \(\mathbf {K}^+_{e}\) is of critical importance to obtain an embedded formulation with a smooth gradient field. Besides the inter-element discontinuity of the derivative field in traditional FEM shape functions, the truss node position needs to be mapped into the parent element coordinates if an isoparametric formulation is used, as in previous embedded formulations (Elwi and Hrudey 1989; Barzegar and Maddipudi 1994). The alternative proposed in the present work is to use shape functions based on a convolution operator. These can be arbitrary smooth up to any derivative depending on the convolution function (although we are only interested in the first derivative), and do not need to be mapped to parent coordinates since they operate in the actual node coordinates.

### 2.2 Convolution operator

*R*defined as the convolution operator radius, and

*r*the distance between the truss member and continuum node. In addition, the shape functions \(\tilde {\mathbf {N}}\) must preserve partition of unity

*a*is

*a*with respect to coordinate

*n*is

*r*). The sum in the denominator is through all the nodes in the continuum, but because the convolution function is zero for \(r>R\), the sum only encompasses a few of the total nodes. The continuum nodes that fall within the convolution operator are found using a tree data structure (quadtree and octtree in two and three dimensions respectively), making the search for different truss nodes linking to continuum efficient.

The convolution shape functions lack desirable properties like the Kronecker delta property (\(\delta _{ii}=1\) and \(\delta _{ij}=0\) for nodes \(i\neq j\)), because these shape functions are not associated to a specific node as with FEM shape functions, but to a cloud of nodes instead. However, it does comply with partition of unity (13) and has no negative values. These convolution shape functions possess continuous first derivative field, a desirable property and required for the present work.

### 2.3 Optimization issues

This coupling to the continuum works by smearing the displacement field around the truss member node. Provided that the convolution radius is not too big, the error introduced by this method is controllable and more importantly, it provides a smooth derivative field throughout the continuum. The smearing error will have a higher impact when closer to a rapid variation of the field (i.e. sharp edges, single node loads and boundary conditions).

*m*enforces small variations from one iteration to the next. This results in a more cautious progression towards the optimum, and with the step size controlled by the move limit

*m*, as follows:

## 3 Verification

*E*and

*A*are the same but for the anchor cable. The design variable is the anchoring distance \(\beta L\). This problem is of particular interest because an analytical solution can be obtained. The compliance of a single bar problem of length

*L*, subjected to body force

*b*and an end force

*P*as in Fig. 3 is:

*P*taken by the bar segment of length \(\beta L\) is

Optimal anchor location with varied mesh refinement

\(\beta _{c}\) | \(C\left (\beta _{c} L\right )\) | |
---|---|---|

Exact | 0.7434 | 1.5252 |

\(N_E=10\) | 0.7492 | 1.5144 |

\(N_E=20\) | 0.7346 | 1.5169 |

\(N_E=40\) | 0.7431 | 1.5203 |

Optimal anchor location with varied convolution radius

\(\beta _{c}\) | \(C\left (\beta _{c} L\right )\) | |
---|---|---|

Exact | 0.7434 | 1.5252 |

\(R=0.1L\) | 0.7333 | 1.5187 |

\(R=0.2L\) | 0.7396 | 1.5177 |

\(R=0.4L\) | 0.7393 | 1.5258 |

The optimization problem for 50 iterations, with a starting point \(\beta _{0}=0.5\) is performed for \(N_E=20\) (element mesh), with randomly spaced elements of size \(0.7L/N_E\leq \Delta x \leq 1.3L/N_E\). The only constraint or technique used is the move limit as detailed in (20) with \(m=0.1\). The optimizer is the *Method of Moving Asymptotes* (MMA) (Svanberg 1987). The convergence towards the optimal point \(\beta _{c} L\) is shown in Fig. 9a and the compliance plot in Fig. 9b. There is an oscillatory behavior between iterations 17 and 30 due to the *adventurous* behavior of the optimizer close to the optimum. The oscillations can be eliminated by taking a smaller move limit, or decreasing it with each iteration.

## 4 Examples

The examples explored here aim to verify the method, and portray some applications that can be tackled with the method. The optimizer is the *Method of Moving Asymptotes* (MMA) (Svanberg 1987), and the convolution function used is \(h_{2}\left (\cdot \right )\) from (14). Similarly to the previous 1D example, 2D and 3D problems are optimized for compliance (\(J=\mathbf {u}^T \mathbf {K}_{e} \mathbf {u}\)) of the coupled structure. For the specific case of two-dimensional problems, unit thickness and plane stress is assumed.

### 4.1 Beam with cable supports

*b*, the domain is regularly partitioned in \(N_{x} \times N_{y}\) four node quadrilateral elements (Q4). The design variables of the problem are the anchor location coordinates \(x_{1}\) and \(x_{2}\), with the only constraint or technique being the move limit as in (20) with \(m=0.05\).

The problem data is \(L_{x}=2\) (\(2L_{x}=4\)), \(L_{y}=0.8\), \(b=-2\), \(E_{c}=100\), \(\nu =0.3\), \(EA=300\) and \(R=0.3\). The objective function (compliance) for a \(N_{x}=20\) and \(N_{y}=8\) mesh is plotted in Fig. 11a, and the gradient fields in Fig. 11b and c. The gradient fields are smooth enough that a gradient-based optimizer should converge to the optimum (it could be a local optimum). Analysis of Fig. 11d for \(\nabla {}C=\mathbf {0}\) gives \(x_{1}=0.8165\) and \(x_{2}=0.3699\) as the global optimum, but also hints of a few potholes that could trap the optimizer. The global optimum location does change with the mesh refinement, and together with other numerical optimization artifacts cause the solution to the problem to experience small changes if the problem parameters change.

Optimal anchor location and compliance with varied mesh size

\(x_{1}\) | \(x_{2}\) | \(C\left (x_{1},x_{2}\right )\) | |
---|---|---|---|

\(10\times 04\) | 0.9999 | 0.5100 | 0.1956 |

\(20\times 08\) | 0.8541 | 0.4016 | 0.1910 |

\(40\times 16\) | 0.8094 | 0.3712 | 0.1971 |

\(80\times 32\) | 0.8635 | 0.3711 | 0.2025 |

Optimal anchor location and compliance for a Q9 mesh with varied mesh size

\(x_{1}\) | \(x_{2}\) | \(C\left (x_{1},x_{2}\right )\) | |
---|---|---|---|

\(10\times 04\) | 0.8497 | 0.4030 | 0.1952 |

\(20\times 08\) | 0.8688 | 0.3736 | 0.2006 |

\(40\times 16\) | 0.9004 | 0.3833 | 0.2056 |

\(80\times 32\) | 0.9272 | 0.3721 | 0.2101 |

Optimal anchor location and compliance for a Q9 mesh with varied convolution radius

\(x_{1}\) | \(x_{2}\) | \(C\left (x_{1},x_{2}\right )\) | |
---|---|---|---|

\(R=0.2L\) | 0.8763 | 0.4025 | 0.2154 |

\(R=0.3L\) | 0.8688 | 0.3736 | 0.2006 |

\(R=0.4L\) | 0.8747 | 0.3752 | 0.1900 |

\(R=0.5L\) | 0.8769 | 0.3748 | 0.1816 |

The anchor path throughout the iterations for this problem is shown in Fig. 14. This path exhibits a steady and consistent approach towards the optimal solution, where the cable efficiently supports the continuum.

### 4.2 Tapered building with truss superstructure

The continuum is meshed with \(N_E=1520\) Q8 elements, with dimensions and material properties: \(L_{x1}=1.0\), \(L_{x2}=0.6\), \(L_{y}=2\), \(E_{c}=10\), \(\nu =0.3\). The truss consists of 4 spans with equal properties for all bars \(EA=300\) and convolution radius \(R=0.075\). The structure is loaded by self-weight of the continuum \(b=-10\). The design variables are the nodal positions of the truss (cross-sectional areas are not being optimized). The problem is optimized for 50 iterations with a move limit as in (20) with \(m=0.015\), and a truss volume constraint as in (22) with \(V_{\max }=32\) (note that initially the truss has a volume \(V_{0}=34.76\)).

Final nodal locations for the symmetry constrained and free problems with node numbering in accordance with Fig. 15b

Symm | Free | Symm | Free | ||
---|---|---|---|---|---|

\(x_{1}\) | \(-0.3958\) | \(-0.3959\) | \(y_{1}\) | 0.0000 | 0.0000 |

\(x_{2}\) | \(-0.3522\) | \(-0.3433\) | \(y_{2}\) | 0.5376 | 0.5406 |

\(x_{3}\) | \(-0.2770\) | \(-0.2733\) | \(y_{3}\) | 0.9426 | 0.9546 |

\(x_{4}\) | \(-0.2379\) | \(-0.2491\) | \(y_{4}\) | 1.3449 | 1.4098 |

\(x_{5}\) | \(-0.2188\) | \(-0.2385\) | \(y_{5}\) | 1.7725 | 1.8019 |

\(x_{6}\) | \(\phantom {-}0.3958\) | \(\phantom {-}0.4167\) | \(y_{6}\) | 0.0000 | 0.0000 |

\(x_{7}\) | \(\phantom {-}0.3522\) | \(\phantom {-}0.3547\) | \(y_{7}\) | 0.5376 | 0.5027 |

\(x_{8}\) | \(\phantom {-}0.2770\) | \(\phantom {-}0.3063\) | \(y_{8}\) | 0.9426 | 0.9485 |

\(x_{9}\) | \(\phantom {-}0.2379\) | \(\phantom {-}0.2275\) | \(y_{9}\) | 1.3449 | 1.3711 |

\(x_{10}\) | \(\phantom {-}0.2188\) | \(\phantom {-}0.2086\) | \(y_{10}\) | 1.7725 | 1.8003 |

\(x_{11}\) | \(\phantom {-}0.0000\) | \(-0.0285\) | \(y_{11}\) | 0.5544 | 0.5494 |

\(x_{12}\) | \(\phantom {-}0.0000\) | \(\phantom {-}0.0535\) | \(y_{12}\) | 0.9420 | 0.9122 |

\(x_{13}\) | \(\phantom {-}0.0000\) | \(-0.0201\) | \(y_{13}\) | 1.3124 | 1.3115 |

\(x_{14}\) | \(\phantom {-}0.0000\) | \(-0.0318\) | \(y_{14}\) | 1.7901 | 1.7828 |

The compliance plot in Fig. 16a has an initial increase while the optimizer is fulfilling the truss volume constraint, as shown in Fig. 16b. Once the constraint is satisfied, the optimizer is free to search for the optimal truss geometry (using the node locations only). The final compliance for the symmetry imposed and free cases are \(C_{\mathrm {symm}}=1.1215\) and \(C_{\mathrm {free}}=1.1296\). The optimized compliance for the symmetric case is surprisingly lower. However, if iterations continue, the less-constrained unsymmetric case will have a lower final value. The unsymmetric case has more than twice the number of design variables compared to the symmetric case, resulting in a (slightly) lower rate of convergence.

### 4.3 Full truss layout optimization for tapered building

This is an extension of the previous problem, adding the truss member’s cross-sectional areas as design variables for the optimization of the symmetric case. The simultaneous optimization of both sizing and geometry of the truss translated into a full layout optimization of the building’s truss superstructure. Previously, the final volume of the truss does not match \(V_{\max }\) because the design variables are the node locations only (Fig. 16b). The gradient of the cross-sectional areas follow (23). The constraints in (24) are also used with \(A_{\min }=0.015\) and \(m_{a}=0.015\).

Final cross-sectional areas for truss members in accordance with Fig. 15b

\(A_{1}\) | 3.6269 | \(A_{9}\) | 3.1658 | \(A_{17}\) | 3.0460 |

\(A_{2}\) | 3.6165 | \(A_{10}\) | 3.1296 | \(A_{18}\) | 2.7400 |

\(A_{3}\) | 3.5832 | \(A_{11}\) | 3.0016 | \(A_{19}\) | 2.7346 |

\(A_{4}\) | 3.3270 | \(A_{12}\) | 2.8983 | \(A_{20}\) | 2.6999 |

\(A_{5}\) | 3.6269 | \(A_{13}\) | 3.1658 | \(A_{21}\) | 3.0460 |

\(A_{6}\) | 3.6165 | \(A_{14}\) | 3.1296 | \(A_{22}\) | 2.7400 |

\(A_{7}\) | 3.5832 | \(A_{15}\) | 3.0016 | \(A_{23}\) | 2.7346 |

\(A_{8}\) | 3.3270 | \(A_{16}\) | 2.8983 | \(A_{24}\) | 2.6999 |

### 4.4 Three-dimensional beam with truss reinforcements

Initial truss nodal locations within the three-dimensional beam

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

1 | 0.5000 | 1.2000 | 1.6000 |

2 | 3.0000 | 1.2000 | 0.4000 |

3 | 7.0000 | 1.2000 | 0.4000 |

4 | 9.5000 | 1.2000 | 1.6000 |

5 | 0.5000 | 1.8000 | 1.6000 |

6 | 3.0000 | 1.8000 | 0.4000 |

7 | 7.0000 | 1.8000 | 0.4000 |

8 | 9.5000 | 1.8000 | 1.6000 |

Final truss nodal locations within the three-dimensional beam

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

1 | 0.4972 | 0.6366 | 1.2482 |

2 | 2.2785 | 0.7085 | 0.0000 |

3 | 7.7148 | 0.7021 | 0.0000 |

4 | 9.5002 | 0.6281 | 1.2542 |

5 | 0.5005 | 2.3713 | 1.2562 |

6 | 2.2828 | 2.2935 | 0.0000 |

7 | 7.7181 | 2.2898 | 0.0000 |

8 | 9.5012 | 2.3635 | 1.2485 |

### 4.5 Reinforced double corbel

The loads coming from the upper column and the beam are distributed over the column cross-sectional area and plate respectively. Analysis will be carried out on a \(t=1\:in\) thick model, with *plane stress*. Given the depth dimension of the corbel, a three-dimensional analysis would be more appropriate but the simplicity of a *plane stress* analysis is more appropriate to showcase the method in an application setting. The steel in compression has cross-sectional area \(A_{sc}=0.1\:in^{2}\) (not to be designed), and the steel in traction has initially \(A_{st}=0.1\:in^{2}\). The elastic modulus of steel is \(E_{s}=29000\:kips\), and for the concrete \(E_{c}=3600\:kips\) and \(\nu =0.2\). The model with the loads, boundary conditions and initial steel placement (for a \(1\:in\) thick model) is presented in Fig. 20b.

The concrete is modeled using 23312 \(T6\) elements, and 47065 nodes. The steel rebars are modeled as several pin-jointed bars \(1\:in\) apart to allow for linkage with the continuum throughout the length of the bar. The convolution radius is \(R=0.25\:in\). The optimization is done for compliance subject to constant volume, and the design variables are steel cross-sectional areas of the bars and the vertical (*y* direction) node positions of the bar in traction (layout optimization). The node movement is limited to \(1\:in\) away from the concrete edges to allow for steel cover. The constraints or restrictions included are a move limit \(m=0.1\) as in (20) for the node locations, and in the cross-sectional areas \(m_{a}=\:0.005in^{2}\) and \(A_{\min }=0.001\:in^{2}\) as in (24). The optimization is run for 200 iterations for a symmetric mesh, with symmetry not enforced.

*moustache shape*. The optimized cross-sectional areas vary as in Fig. 22a, but most importantly the bar assumes a constant stress behavior as in Fig. 22b in accordance with Michell’s fully stressed requirements (Hemp 1973; Michell 1904; Rozvany 1996, 1997). In the final configuration there is no shear in the bar, that along with the constant stress (smaller than the previous maximum stress), makes a more efficient use of the steel available and thus a better design. The final position and cross-sectional areas for the (half) bar are in Tables 10 and 11 respectively.

Final node locations for steel in traction (*in*)

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

1 | 0 | 16.2521 | 11 | 10 | 16.9086 |

2 | 1 | 16.2555 | 12 | 11 | 16.7158 |

3 | 2 | 16.2695 | 13 | 12 | 16.5538 |

4 | 3 | 16.2921 | 14 | 13 | 16.4070 |

5 | 4 | 16.3344 | 15 | 14 | 16.2939 |

6 | 5 | 16.3995 | 16 | 15 | 16.2710 |

7 | 6 | 16.5203 | 17 | 16 | 16.3699 |

8 | 7 | 16.7265 | 18 | 17 | 16.4352 |

9 | 8 | 16.9813 | 19 | 18 | 16.4119 |

10 | 9 | 17.0000 |

Final cross-sectional areas for steel in traction for bars between nodes *i* and *j* (\(in^{2}\))

\(node_{i}\) | \(node_{j}\) | \(A_{s}\) | \(node_{i}\) | \(node_{j}\) | \(A_{s}\) |
---|---|---|---|---|---|

1 | 2 | 2.5389 | 10 | 11 | 1.3553 |

2 | 3 | 2.5596 | 11 | 12 | 1.0042 |

3 | 4 | 2.5937 | 12 | 13 | 0.3947 |

4 | 5 | 2.6694 | 13 | 14 | 0.0148 |

5 | 6 | 2.6540 | 14 | 15 | 0.0144 |

6 | 7 | 2.7265 | 15 | 16 | 0.0143 |

7 | 8 | 2.6469 | 16 | 17 | 0.0143 |

8 | 9 | 2.1364 | 17 | 18 | 0.0142 |

9 | 10 | 1.6514 | 18 | 19 | 0.0142 |

Corbel reinforcement steel in traction

Rebar | Horizontal Position |
---|---|

\(3\#5\) | \(-12.5\:in \text { to } 12.5\:in\) |

\(2\#5\) | \(-9.0\:in \text { to } 9.0\:in\) |

\(2\#5\) | \(-7.5\:in \text { to } 7.5\:in\) |

## 5 Conclusions

The method presented here extends truss layout optimization to combine a continuum with the discrete elements, allowing for mixed-element type optimization problems to be solved. This is possible because the derivative field remains continuous and sufficiently smooth even if the convolution radius is small.

Convolution coupling to the continuum does violate the energy principle of the problem, but when used with a reasonable sized convolution radius, the results are shown to agree up to some level with an exact solution when available. In cases where an analytical solution cannot be easily found, the method exhibits stable results (i.e. converging to an almost equivalent state regardless of changing some parameters). The optimum location has a small variation that can be attributed to the difference in the FEM solutions with refinement, and numerical inaccuracies.

There is no restriction over the objective function provided that the derivation procedure for the stiffness follows (18). Restrictions to the optimization are easily implemented and examples with volume constraint, minimum cross-sectional areas and member lengths are given. The method requires however a small step size (move limit) between iterations due to the highly nonlinear behavior of the problem. The situation worsens with an increasing number of truss nodes or the inclusion of member sizing, and thus the optimization can easily diverge.

The method is shown to effectively reach optimal configurations, however, an *acceptable* initial guess must be given because of the large number of local minima in these problems. Note that a truss can have an infinite number of spatial configurations, thus relying on the *engineer’s common sense* to provide a starting point for the optimization.

## References

- ACI Committee (2002) SP-208: Examples for the Design of Structural Concrete with Strut-and-Tie ModelsGoogle Scholar
- Allahdadian S, Boroomand B, Barekatein A (2012) Towards optimal design of bracing system of multi-story structures under harmonic base excitation through a topology optimization scheme. Finite Elem Anal Des 61:60–74MathSciNetCrossRefGoogle Scholar
- Amir O, Sigmund O (2013) Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization. Struct Multidiscipl Optim 47(2):157–174Google Scholar
- Barzegar F, Maddipudi S (1994) Generating reinforcement in FE modeling of concrete structures. J Struct Eng 120(5):1656–1662CrossRefGoogle Scholar
- Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods and applications, 2nd edn. In: Engineering online library. Springer, Berlin, GermanyGoogle Scholar
- Dorn W, Gomory R, Greenberg H (1964) Atomatic design of optimal structures. J Mech 3:25–52Google Scholar
- Elwi A, Hrudey T (1989) Finite element model for curved embedded reinforcement. J Eng Mech 115(4):740–754CrossRefGoogle Scholar
- Felix J, Vanderplaats GN (1987) Configuration optimization of trusses subject to strength, displacement and frequency constraints. J Mech Des 109:233–241Google Scholar
- Hansen SR, Vanderplaats GN (1988) An approximation method for configuration optimization of trusses. AAIA J 28(1):161–168CrossRefGoogle Scholar
- Haslinger J, Mäkinen RAE (2003) Introduction to shape optimization: theory, approximation, and computation. In: Advances in design and control. Society for Industrial and Applied Mathematics, Philadelphia, PA, USAGoogle Scholar
- Hemp WS (1973) Optimum structures. In: Oxford engineering science series. Clarendon Press, Oxford, UKGoogle Scholar
- Imran I, Pantazopoulou SJ (1996) Experimental study of plain concrete under triaxial stress. ACI Mater J 93(6):589–601Google Scholar
- Kato J, Ramm E (2010) Optimization of fiber geometry for fiber reinforced composites considering damage. Finite Elem Anal Des 46(5):401–415CrossRefGoogle Scholar
- Liang Q (2007) Performance-based optimization of structures: theory and Applications. Wiley, Ltd, Chichester, UKGoogle Scholar
- Liang Q, Xie Y, Steven G (2000) Optimal topology design of bracing systems for multistory steel frames. J Struct Eng 127(7):823–829CrossRefGoogle Scholar
- Lipson S, Gwin L (1977) The complex method applied to optimal truss configuration. Comput Struct 7(6):461–468CrossRefGoogle Scholar
- Michell AGM (1904) The limits of economy of material in frame-structures. Phil Mag Ser 6 8(47):589–597MATHCrossRefGoogle Scholar
- Mijar AR, Swan CC, Arora JS, Kosaka I (1998) Continuum topology optimization for concept design of frame bracing systems. J Struct Eng 124(5):541–550CrossRefGoogle Scholar
- Ohsaki M (2010) Optimization of finite dimensional structures. Taylor & Francis, Boca Raton, FL, USA CrossRefGoogle Scholar
- Rozvany G (1996) Some shortcomings in Michell’s truss theory. Struct Multidisc Optim 12(4):244–250CrossRefGoogle Scholar
- Rozvany G (1997) Some shortcomings in Michell’s truss theory. Struct Multidisc Optim 13(2–3):203–204Google Scholar
- Sokół T (2010) A 99 line code for discretized Michell truss optimization written in Mathematica. Struct Multidisc Optim 43(2):181–190Google Scholar
- Stromberg L, Beghini A, Baker W, Paulino G (2012) Topology optimization for braced frames: combining continuum and beam/column elements. Eng Struct 37:106–124CrossRefGoogle Scholar
- Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetMATHCrossRefGoogle Scholar