Design complexity control in truss optimization

Abstract

Truss optimization based on the ground structure approach often leads to designs that are too complex for practical purposes. In this paper we present an approach for design complexity control in truss optimization. The approach is based on design complexity measures related to the number of bars (similar to Asadpoure et al. Struct Multidisc Optim 51(2):385–396 2015) and a novel complexity measure related to the number of nodes of the structure. Both complexity measures are continuously differentiable and thus can be used together with gradient based optimization algorithms. The numerical examples show that the proposed approach is able to reduce design complexity, leading to solutions that are more fit for engineering practice. Besides, the examples also indicate that in some cases it is possible to significantly reduce design complexity with little impact on structural performance. Since the complexity measures are non convex, a global gradient based optimization algorithm is employed. Finally, a detailed comparison to a classical approach is presented.

Appendices

Appendix A Derivatives of complexity measures

The derivative of C _b according to a given design variable is given by

$$ \frac{\partial C_{b}}{\partial x_{i}} = \frac{\partial c(x_{i})}{\partial t}, $$

(10)

where

$$ \frac{\partial c}{\partial t} = \frac{a}{(t-\varepsilon+1)^{1+a}} , \quad t \geq \varepsilon. $$

(11)

The derivative of C _n according to a given design variable is given by

$$ \frac{\partial C_{n}}{\partial x_{i}} = {\sum}_{j=1}^{n} \frac{\partial c}{\partial b_{j}} \frac{\partial b_{j}}{\partial x_{i}}. $$

(12)

The derivative ∂ c/∂ b _j is as presented in (11). The derivative ∂ b _j /∂ x _i , on the other hand, is simply

$$ \frac{\partial b_{j}}{\partial x_{i}} = \left\lbrace \begin{array}{ll} 1, & \textrm{if } i \in N_{j}\\ 0, & \textrm{if } i \notin N_{j}\\ \end{array}\right. $$

(13)

Note that both derivatives are very simple to evaluate and should not lead to significant increases in the computational effort required for sensitivity analysis.

Appendix B Global optimization algorithm

The global optimization algorithm used in this paper is based on starting local searches from different initial design vectors. The algorithm is very similar to the restart procedure first proposed by Luersen et al. (2004) and already applied for truss optimization by Torii et al. (2011).

We first start the local search with an initial design vector x ₀ to obtain an optimum design $\mathbf {x}_{0}^{\ast }$. Both x ₀ and $\mathbf {x}_{0}^{\ast }$ are then included in a set of design vectors

$$ S_{0} = \{\mathbf{x}_{0}, \mathbf{x}_{0}^{\ast}\}, $$

(14)

that will store design vectors where we already started/ended local searches. The next step is to randomly generate a set of M trial design vectors y _i

$$ T = \{\mathbf{y}_{1}, \mathbf{y}_{2}, ..., \mathbf{y}_{M}\} $$

(15)

satisfying the bounds on the design variables, but not necessarily satisfying the nonlinear constraints (check of nonlinear constraints are avoided because of the computational costs involved). The average distance from all trial vectors y _i ∈T to all vectors in S ₀ is then evaluated, i.e.,

$$ d\left( \mathbf{y}_{i}, S_{0} \right) = \frac{1}{ \text{card}(S_{0})}\sum\limits_{\mathbf{x} \in S_{0}} \Vert \mathbf{y}_{i} - \mathbf{x} \Vert, \quad i = 1, 2, ..., M, $$

(16)

where card (S ₀) stands for cardinality of S ₀ (i.e. number of elements in S ₀) and ∥.∥ is some vector norm (in this work Euclidean norm is used). We then take the next starting point as the one in T with maximum average distance from the vectors in S ₀, i.e.,

$$ \mathbf{x}_{1} = \textrm{arg max }_{\mathbf{y} \in T} \ d(\mathbf{y},S_{0}). $$

(17)

In this way we choose a starting point that is, in the average sense, the most distant from the ones in S ₀. A new local search is then started with x ₁ to obtain an optimum design $\mathbf {x}_{1}^{\ast }$ and both vectors are included in a new set S ₁ (together with x ₀ and $\mathbf {x}_{0}^{\ast }$). A new set of trial vectors T is randomly generated and the process is repeated. After k restarts (k+1 local searches) we have the set of starting/found design vectors

$$ S_{k} = \{\mathbf{x}_{0}, \mathbf{x}_{0}^{\ast}, \mathbf{x}_{1}, \mathbf{x}_{1}^{\ast}, ..., \mathbf{x}_{k}, \mathbf{x}_{k}^{\ast}\}, $$

(18)

we randomly generate a set of trial vectors T and choose the new starting design vector as the one that maximizes

$$ \mathbf{x}_{k+1} = \textrm{arg max }_{\mathbf{y} \in T} d(\mathbf{y},S_{k}). $$

(19)

The parameters required to use this algorithm are the number of local searches k to be made and the number of trial vectors M generated before each restart. The local searches can be made using any optimization algorithm. In this paper an interior point algorithm is used (Nocedal and Wright, 1999; Luenberger and Ye, 2008).

The only difference between this algorithm and the one proposed by Luersen et al. (2004) is that here we take the trial vector y _i ∈T that maximizes the average distance from the ones in S _k . In the original approach the new starting design vector was chosen as the one that minimizes a normal multidimensional probability density function, representing the probability of arriving at some point from S _k when starting from some trial vector y _i ∈T.

Appendix C Prager-Parkes’ approach

The approach presented by Prager (1974) and Parkes (1975) was originally conceived for problems based on volume minimization. The idea is to replace the real volume of the structure, given by $V = {\sum }_{i=1}^{m} l_{i} x_{i}$ (where l _i are the bars lengths), by the modified volume

$$ V_{p} = \sum\limits_{i=1}^{m} (l_{i} + s) x_{i}, $$

(20)

where s>0 is a parameter named by Parkes (1975) as joint radius. In this way, shorter bars are penalized in comparison to longer ones. In order to see this fact, it is interesting to rewrite V _p as

$$ V_{p} = \sum\limits_{i=1}^{m} \phi(l_{i}) l_{i} x_{i}, $$

(21)

with

$$ \phi(l_{i}) = \left( 1 + \frac{s}{l_{i}} \right). $$

(22)

This expression puts in evidence that the approach is the same as multiplying the volume of each bar l _i x _i by a unitary cost per volume ϕ(l _i ). This cost per volume, however, is smaller the longer the bar is (i.e. $\lim _{l_{i} \rightarrow \infty } \phi (l_{i}) = 1$). On the other hand, for shorter bars we have $\lim _{l_{i} \rightarrow 0} \phi (l_{i}) = \infty$, i.e. shorter bars have increased unitary costs. In this way, the optimization algorithm will try to avoid shorter bars, that are comparatively more expensive than longer ones. It is important to point out that this approach was originally developed in order to tackle analytical solutions.