Advanced approximations for sequential optimization with discrete material interpolations

Abstract

The DMO (discrete material optimization) technique is employed in structural synthesis dealing with the selection of a material belonging to a group of candidate materials. It has mainly been employed in the optimization for orientation of layers of composite laminates. The DMO is based on an interpolation in the form of a weighted sum of candidate materials. The weights are nonlinear functions of penalized design variables that are solved by continuous optimization, leading to proper material selection. The preferred way of solution has been by sequential approximate optimization (SAO), based on Taylor series approximations (TSA) as surrogate functions of the structural responses. However, due to the complexity of the DMO formulation the classical local surrogate techniques become of questionable efficiency in adequately capturing structural response behavior. To improve the quality of the surrogate models, it is here proposed the use of the weighting functions to form intermediate design variables in terms of which a higher quality TSA is created. Improvements in the convergence characteristics of the SAO is observed, opening new perspectives to the efficient application of the DMO concept.

Appendix A: Common material weighting functions

The DMO uses material interpolations like in (1). Popular and efficient weighting functions w _i (x) are the so-called DMO4, DMO5 ( Stegmann 2004; Stegmann and Lund 2005) and the SFP ( Bruyneel 2011).

1.1 A.1 DMO4

The DMO4 scheme uses the following weights w _i (x) to define the material interpolation (1):

$$ w_{i}(\mathbf{x})=(x_{i})^{p} \prod\limits_{j=1;j\neq i}^{n}\left[1-(x_{j})^{p}\right] $$

(22)

In the (22) above, the weights w _i are functions of the x _i design variables, bounded in such a way that 0≤x _i ≤1. This way, the w _i are also bounded between 0 and 1. The parameter p is a penalization power which aims to penalize intermediate combinations of materials, by reducing its relative stiffness when p>1, as found in the well-known SIMP (Solid Isotropic Material with Penalization) ( Bendsøe and Sigmund 2004). The number n of candidates can be any.

This material weighting works specially well in problems of compliance minimization ( Stegmann 2004) but the weights w _i do not necessarily add up to unity for intermediate values of x _i . This cause ill-characterized material physical representation during intermediate optimization stages, where the weights possibly present intermediate values between 0 and 1. This is critical when interpolating mass properties in general (Lund and Stegmann 2006: Niu et al. 2010 and optimizing for buckling loads ( Lund 2009). Trying to deal with this issue, the DMO5 formulation was developed.

1.2 A.2 DMO5

The DMO5 weighting is basically a normalization of the DMO4:

$$ w_{i}(\mathbf{x})=\frac{\hat{w}_{i}}{{\sum}_{i=1}^{n}\hat{w}_{i}} \quad\text{and}\quad \hat{w}_{i}(\mathbf{x})=(x_{i})^{p} \prod\limits_{j=1;j\neq i}^{n}\left[1-(x_{j})^{p}\right] $$

(23)

It always ensures ${\sum }_{i=1}^{n}w_{i}=1$, which provides a much better material representation during intermediate iterations of an optimization process. However, it is reported that this normalization adds some difficulties in the optimization convergence, facilitating the appearance of intermediate values of the w _i in the final optimized solutions ( Stegmann 2004). The reason for this behavior seems to be the more complicated functions w _i (x) now used, and because the weights’ normalization also affect the intermediate materials penalization ( Niu et al. 2010).

1.3 A.3 SFP

In Bruyneel et al. (2010) and Bruyneel (2011) it was proposed a new DMO-like scheme called SFP (shape function with penalization), which consists in interpolating materials in (1) using functions commonly employed in the FE method. Its most common form uses the bi-linear interpolation functions of a four-node quadrilateral element to select materials among four candidates.

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} &w_{1}(\mathbf{x})=\left[\frac{1}{4}(1-x_{1})(1-x_{2})\right]^{p} &w_{2}(\mathbf{x})=\left[\frac{1}{4}(1+x_{1})(1-x_{2})\right]^{p}\\ &w_{3}(\mathbf{x})=\left[\frac{1}{4}(1+x_{1})(1+x_{2})\right]^{p} &w_{4}(\mathbf{x})=\left[\frac{1}{4}(1-x_{1})(1+x_{2})\right]^{p} \end{array} \end{array} $$

(24)

These functions are defined in terms of the variables x ₁,x ₂, that could be seen as elemental natural coordinates, and now bounded to permit −1≤x _i ≤1 ( Reddy 1996). In (24), the penalty p has the same functionality as in SIMP and DMO4/5. However, the use of p≠1 does not renders the characteristic of weights adding up to unity. When four materials are interpolated the SFP reduces in a half the number of design variables needed in comparison to the DMOs, since it uses x ₁,x ₂ when the DMOs would use x ₁,...,x ₄. However, it is clear that the set of weights shown in (24) does not permit working with any number of candidates in (1). Nevertheless, in composite laminates practical applications, it is common the problem of distributing laminae at 0^∘/±45^∘/90^∘ orientations, and SFP is clearly effective for that. Variations and extensions of this weighting can be found in Bruyneel et al. (2011) and Gao et al. (2012), but (24) is the form of interest here.