Numerical issues in material distribution methods

As discussed earlier, in general, the structural topology optimization thorough material distribution involves with discretizing a predefined domain into finite elements and devising a numerical procedure to find the element-wise design parameters. Such a procedure frequently encounters numerical instabilities that are categorized into “mesh dependency”, “checkerboard pattern” and “local minima” (Sigmund and Petersson, 1998). In the following subsections, these problems will be briefly addressed.

p i s x E E(xi) () = ) (s E p

2.1.7.1 Checkerboard pattern

Checkerboard pattern refers to the situation where regions of alternating solid-void (strong- weak) elements are formed in some areas of the structure (Díaz and Sigmund, 1995, Petersson and Sigmund, 1998). Diaz and Sigmund (1995) and Jog and Haber (1996) have shown that the main cause of the checkerboard patterns lies in poor numerical modeling of the structure specifically when the low-order finite elements are used.In fact arranging 4-node elements in a checkerboard fashion would maximize the calculated strain energy density in elasticity problems (Díaz and Sigmund 1995); therefore, the checkerboard areas have larger stiffness due to numerical errors which prevents the algorithm to converge to the optimum solution (Sigmund and Petersson, 1998). The topology optimization approaches such as SIMP, ESO and BESO, based on material distribution are prone to such numerical instability.

It has been shown in 2D continuum structures’ elasticity problems that the checkerboard pattern can be prevented by using 8 or 9-node finite elements (Jog and Haber, 1996, Díaz and Sigmund, 1995). This is attributed to the fact that the higher order elements have many more degrees of freedoms per design variable. This makes the numerical calculations more accurate. However, increasing the degrees of freedom makes the topology optimization procedure costly.In addition, when a large value of penalty exponent is used with the SIMP method, applying elements of higher order often does not yield a checkerboard-free result (Sigmund and Petersson, 1998). In addition, the technique may not be helpful with the ESO and BESO methods.

Inspired by the filtering scheme from image processing, Sigmund (1994b) implied a technique in combination with the SIMP topology optimization to overcome the checkerboard pattern. The filtering scheme modifies the design sensitivity of each element based on the weighted average of the element itself, as well as its neighboring elements (Sigmund and Petersson, 1998). Although the filtering scheme is a computationally efficient method, it usually weakens the effects of the SIMP method to eliminate the intermediate densities and results in blur boundaries (Zhou and Li, 2008b). Several other techniques have been purposed for solving this issue in the SIMP topology optimization. Heaviside projections algorithm (see next section) (Guest et al., 2004), nonlinear diffusion techniques (Wang et al., 2004a) and the phase field approach based on Cahn-Hilliard model (Zhou and Wang, 2007) are some of the these methods.

Recently an adaptive refinement approach was applied which uses an “analysis-mesh separated density field” for tackling the numerical instabilities and achieving an improved boundary description with SIMP (Wang et. al. 2013). In this method, the design variables are defined on some points in the finite element model which are known as density points. By defining an interpolation scheme the density of these points are evaluated and restricted to be either 0 or 1. The approach uses an additional refinement procedure to identify and rectify the densities on the boundary regions. It is shown that the method enables the reduction in the number of design variables in the model, hence reduces the computational time of optimization.

As mentioned before, in relation to the sensitivity based ESO, Li et al. (2001) introduced a checkerboard suppression algorithm, based on the smoothing the sensitivities through a

filtering scheme. In the modified soft-kill BESO a similar filtering scheme has been applied which has been shown that efficiently suppresses the numerical instability of checkerboard pattern and mesh dependency. The details of this scheme will be discussed in the next chapters.

2.1.7.2 Mesh dependency

By using finer mesh sizes in topology optimization, it is expected that the same structure with better description of boundaries is attained. However the “mesh dependency” using finer mesh results in qualitatively different topologies with more members of smaller sizes (Bendsøe and Sigmund, 1999, Huang and Xie, 2010b). One of the reasons for such numerical problems is the so called non-uniqueness of the solution which means that there might be several optimum solutions with the same performance and structural weight or volume (Sigmund and Petersson 1998). Another source of this instability is the non-existence of the solution. In a solid-void topology optimization, it is known that the introduction of more void spaces into the structure provides higher stiffness and no closeness in the possible sets of solution could be found (Sigmund and Petersson 1998; Jog and Haber, 1996).

The non-uniqueness of the solution can be controlled, to some extent, by introducing the manufacturing performance constraint (Ambrosio and Buttazzo, 1993). On the other hand one way for solving the problem of the non-existence of solution is by relaxation as used in the Homogenization and the SIMP topology optimization approaches. However, because of the existence of composite regions (grey areas) in the final solution, the result of a relaxed formulation cannot be interpreted easily for manufacturing purposes (Sigmund and Petersson

1998). In many studies, the imposing of restrictions on the variation of design variables has been demonstrated as an efficient method for alleviating the mesh-dependency problem.

Restriction on variation of design variables can be imposed through different techniques. Many of these restriction techniques can also address the problem of checkerboard patterns as were discussed previously. As other examples of the restriction methods, Haber et al. (1996) proposed a method based on imposing constraints on the circumferences of the void regions and outer boundaries of the structure. In the SIMP approach, Bourdin (2001) used a technique in which the densities are filtered and regulated using a convolution operator. Wang and Wang (2005a) applied the bilateral filtering technique to solve the numerical instabilities in topology optimization. The bilateral filtering is a type of non-linear filtering scheme and has already been used in the image processing. Also the nonlinear diffusion technique has been developed by Wang et al. (2004a) based on a similar method for the image processing (Aubert and Kornprobst, 2006).

Recently, Heaviside filtering method has widely applied for suppressing the intermediate densities together with the SIMP approach. It has been developed by Guest et al (2004) in order to obtain 0/1 solutions. The original formulation of the method modifies the densities by a filtering scheme in the form of:

∑

∑

in which x is the density of element and _j w is a weighting function. The next step is to ij apply a smoothing function for instance in the form of:

τ τ − − ₊ − = e xe x_{i 1} xˆi _{ˆi (2.25b)}

When τ =0 the formulations act as an isotropic filtering which diffuses the mass linearly among surrounding elements. When τ approaches infinity the element’s density becomes

1

=

i

x if x)_i >0 and will be equal to 0 otherwise. Therefore if an element is surrounded by other elements with a density higher than zero, its density will switches to 1 too. It can be seen that the method suppresses the checkerboard pattern and mesh dependency concurrently. For practical application however, τ needs to be properly determined by the user experimentally; otherwise a stable convergence may not be attainable (Sigmund 2007).

2.1.7.3 Local optima

The occurrence of local optima in structural topology optimization refers to the situation where different solutions are obtained by changing the design parameters, such as move limits or evolutionary rate, geometry of the initial design, number of finite elements or filtering parameters and so forth. Mathematically, the function has a local minimum (maximum) at the point if (or ) for all in a small neighbourhood of and has a global minimum (or maximum) if the statement holds for all feasible points of . Structural topology optimization problems usually have many local optima and essentially are not convex. The flatness of the objective function in most topology optimization problems causes the algorithms to be unable to avoid a nearby solution and trap in local optima.

) (x f * x f(x*)≤ f(x) f(x*)≥ f(x) x _x* x

To solve this problem in procedures that use intermediate densities, the idea of the continuation method was introduced. Different procedures for the continuation methods have been proposed. For instance Allaire and Kohn (1993) proposed an approach in which in the first step of the optimization procedure, the intermediate densities are allowed in the structure by applying a low penalization factor and after convergence of the procedure they are penalized to earn a 0-1 solution. In a different approach Sigmund (1997) and Sigmund and Torquato (1997) used a filtering scheme with a large filtering size at the initial steps and gradually reducing filtering size. It should be mentioned that the concept of the continuation method is not mathematically coherent. Although the experimental application of the procedure may lead to better solutions, it still does not guarantee the earning of a global optimum (Sigmund and Petersson 1998).