Discrete Material Optimization of Laminated Composites - SIMP vs. Global Optimization

Design of laminated composites structures is becoming increasingly important as the use of composite materials steadily increases. This development is driven by the aerospace, automotive and wind turbine industries who need still lighter and stiffer/stronger structures. This presents a very challenging design task that calls upon structural optimization tools for providing basic design ideas. However, existing methods for handling laminated composites suffer from problems with local optima when optimizing the fiber orientation, which is the key to efficient design with laminated composites. To counter this problem Discrete Material Optimization (DMO) was suggested in [

1

] where an alternative parametrization of the optimization problem is used, inspired by the procedures in topology optimization. The idea is to discretize the problem by using only a limited number of pre-defined candidate fiber orientations, each described by a constitutive matrix, C

i

. The optimization problem is then parameterized on the element level by expressing the constitutive matrix for lamina

j

as

$$ C_j = \sum _i x_{ij} C_i $$

where

$$ \forall x_{ij} \in \left\{ {0,1} \right\} $$

are the design variables for material i in lamina j. The objective of the optimization is then to choose one distinct material from the set of candidates, i.e.

$$ \sum _i x_{ij} = 1,\forall j $$

. The design variables, xij, may be associated with a specific lamina/element or a patch consisting of several laminae/elements, thereby significantly reducing the total number of design variables. The constitutive matrices,Ci, may represent any type of material, allowing for simultaneously optimization for fiber orientation and material choice.

The discrete problem stated above was solved successfully for minimum compliance of large-scale structures in [

1

] using relaxation and mathematical programming with a SIMP penalization strategy for obtaining 0/1 solutions. Inspired by this and the work in [

2

] the authors have recently solved smallerscale DMO problems to provable global optimum using the discrete variables directly in a nonlinear branch and bound framework. The relaxed problem is modeled as a convex program with a nonlinear objective function and linear constraints and solved using a Newton method. In this work we compare the results obtained using both the continuous SIMP relaxation and the global optimization strategy in order to evaluate the efficiency of the two methods. Preliminary results show that the SIMP method is ef- ficient at obtaining near-optimal 0/1 solutions for even large-scale problems but that the convergence to global optimum (if possible) is dependent on the chosen penalization strategy. The global optimization method converges successfully but is limited by computational requirements to smaller-scale problems. The global optimization method can provide valuable benchmark solutions and furthermore seems very promising for problems involving patches of design variables.