Design of multi-phase composites

In comparison with periodic cellular materials, with microstructures composing of one material phase and a void phase, composites with two or more material phases are more favourable and attractive for practical application. One of the reasons for such advantageous properties is that, by combining different materials, a wider range of properties could be achieved which are not attainable by individual phases. In fact, it is mathematically shown that increasing the number of constituent phases of the composite will widen the G-closure (Zhou and Li, 2008a). On the other hand, multi-functional materials are inevitably composites of two or more constituent phases (Gibson, 2010). Therefore the development of multi-phase composites provides the basis for development of materials with combined functional properties.

The SIMP method is used for the design of 2D periodic microstructures for composites with two material phases and a void phase (Gibiansky and Sigmund, 2000). The objective functions were the extremization of bulk modulus and thermal conductivity (Gibiansky and Sigmund 2000) or the thermal expansion (Sigmund and Torquato, 1997). The key point in these studies was the introduction of three design variables , and , for each element i that corresponds to three constituent phases. The design variables are defined in the interval of ]0, 1],with the following condition (superscript in parentheses denotes the phase numbers):

(2.49) ) 1 ( i x x_i(2) x(_i3) 1 ) 3 ( ) 2 ( ) 1 ( + + = i i i x x x

The above assumption makes it possible to define a simple artificial mixture function to correlate the local stiffness or thermal strain tensors of the element i with the design variables. For instance, in the case of designing microstructures with two material phases (phases 2 and 3) and one void phase (phase 1), the relationships between design variables xi, and the

elements of local stiffness tensor are defined as:

(2.50)

where ; the penalty exponents of p and q are introduced to prevent intermediate densities. The mixing rule for two materials and void mixture proposed by (Sigmund and Torquato 1997) is a combination of the classical Voigt’s mixing rule for two solid materials and the power-law interpolation between the void phase and other (i.e. q=1 and p>1). However, as it is mentioned in Swan and Kosaka (1997), by using the Voigt’s mixing rule, the phase separation does not happen appropriately in the final result. A formulation that uses hybrid combinations of the classical Reuss’ and Voigt’s mixing rules was proposed by Swan and Kosaka (1997), but it may not provide an accurate constitutive model of mixtures. Gibiansky and Torquato (1996b) used the power-law mixing rules for all phases in the multi-phase material design. The penalty exponents are taken equal to 3 (p=q=3) (Gibiansky and Sigmund, 2000) at the beginning of the procedure to prevent the algorithm to trap in local optima. The penalty exponents are then gradually increased, to make intermediate densities uneconomical in a process known as “continuation method” (Bendsøe and Sigmund, 2003). ] ) 1 3 2 3 3 23 ( ) ijkl q ) ( i ) ( ijkl q ) ( i p ) ( i ijkl(x) (x ) [( (x ) E (x ) E E = − + ) 3 ( ) 2 ( ) 23 ( i i i x x x = +

Nevertheless, as an inherent problem with the SIMP, such density-based method leads to intermediate (grey) densities in the final topology. In comparison with microstructures designed with one materials and a void phase, in multi-phase materials with closer physical properties, the SIMP usually causes more ambiguity in interpretation and identification of the boundaries and increasing the penalty values not only cannot solve the problem completely, but it may also result in numerical instability (Kohn and Strang, 1986, Swan and Kosaka, 1997, Zhou and Wang, 2007).

On the other hand, the application of optimality criteria (OC) causes numerical instability in multiphase topology optimization scenarios (Yin and Yang, 2001, Zhou and Li, 2008a). Zhou and Li (2008a) applied the Sequential Linear Programming (SLP) for the design of multiphase microstructures of materials. As mentioned before, the SLP solves the minimization problem sequentially, by approximating the objective and constraint functions using the first-order Taylor series. Although the application of SLP ensures a stable linearized procedure, numerical experience has shown that the move-limit for the design variables should be kept fairly small (Fujii et al., 2001) and that there may be some difficulties in convergence (Zhou and Li, 2008a). In the above studies, the procedure typically needed around 8000 iterations for a 60×60 discretization elements model, including some interactions by the user (Sigmund and Torquato, 1997). Therefore the design process is not fully automatic and cost efficient.

In another study, Zhou and Wang (2007) applied a phase field model for compliance minimization of multi-phase structures based on Cahn-Hillard theory (Cahn and Hillard, 1958). The introduced phase field is a model that enables interpretation of interface between

material constituent phases based on Cahn–Hilliard partial differential equations which deal with the dynamics of phase changes. By using a variational method, the topology optimization problem is transformed into a problem of solving a set of partial differential equations. In addition to mathematical complexity, the main disadvantage of the procedure is its time consumption. For example the topology design of a 2D cantilever beams may need 200,000 iterations (Zhou and Wang, 2007).

Zhou and Li (2008a) developed a method for the design of multi-phased periodic microstructures of materials for extremal conductivity. Although a similar density-approach has been applied, several modifications have been made to improve the abovementioned SIMP procedure. To make the design problem self-weighted, instead of using the SIMP interpolation scheme, the HS bound has been used for interpolation. Thereby the need for choosing the penalty factor is alleviated. In addition, due to the above mentioned numerical issues in applying the optimality criteria (OC) and SLP, an approach based on the Methods of Moving Asymptotes (MMA) has been applied. It has shown that it would result in faster convergence and more numerical stability. To reduce the blurring effects in the boundaries of constituent phases, a method based on non-linear diffusion (discussed in Chapter 6) has been adopted. However, the method is still unable to completely remove the grey areas that form the boundaries of the constituent phases.

A generalized new BESO has been developed which utilizes a material interpolation scheme, with penalization and which is capable of optimization of continuum structures with multiple material constituents (Huang and Xie, 2009b, Huang and Xie, 2010b). One of the advantages of the method is that the procedure is independent of the selection of the penalization factor.

Better convergence of the procedure together with high computational efficiency and more importantly the capability of the new BESO in separating the constituent phases, have made BESO a promising tool for topology optimization of structures with multiple materials (Cadman et al., 2013).