1.2.2 SIMP

To relax the binary problem, a classical solution is to use the continuous density-property model. The mechanical property is determined by a power-law interpolation function of relative density. The power law penalizes intermediate densities, driving the variables towards purely 0 and 1. This is the Solid Isotropic Material with Penalization (SIMP) method[8].

In SIMP, element Young's modulus E_i is described by element density x_i by,

where E₀ is Young's modulus of the solid material and p is the penalization power (p＞1).

To avoid zero E_i, a modified SIMP approach is given by,

where E_min denotes the Young's modulus of the void.

The optimization problem (1-2) can be solved by the gradient-based optimality criteria (OC) method. According to the classical education paper of SIMP[9], the updating scheme of design variable is as follows:

where m is a positive move limit, η (set as 1/2) is a numerical damping coefficient, and B_e is obtained from the optimality condition as:

where the Lagrangian multiplier λ should be chosen to satisfy the volume constraint and the appropriate value can be found by means of a bisection algorithm.

The sensitivities of the objective function c and the material volume Vol with respect to the element densities x_eare given by:

The optimized problem (1-2) will be solved using the updating scheme mentioned above. However, the solution of topology optimization may have unsatisfied features such as the formation of checkerboard patterns, which is unable to fabricate in practice. To ensure the existence of solutions, some restrictions on the design must be imposed, for example, the application of filters. A whole range of filtering methods is thoroughly described by Sigmund[10]. The commonly used sensitivity filter and density filter are introduced hereby. The sensitivity filter modifies the sensitivities as follows:

where N_e is the set of elements i for which the center-to-center distanceΔ(e, i) to element e is smaller than the filter radius r_min, and H_ei is a weight factor defined by:

The term γ (set as 10-3) in Eq.(1-9) is a small positive number introduced in order to avoid division by zero.

The density filter transforms the original densities x_e as follows:

Applying the density filter, the sensitivities with respect to the design variable x_j are obtained by means of the chain rule:

where the function ψ represents either the objective function c or the material volume Vol.

Several classical Matlab implementations of SIMP-based optimization are list blow.

● A 99 line topology optimization code written in MATLAB[11].

● Efficient topology optimization in MATLAB using 88 lines of code[9].