Material Interpolation

Recall that the original topology optimization problem is formulated as an integer problem (see e.g. Bendsøe and Sigmund (2003)),

$p_{e} = 1$ or $p_{e} \approx 0$

meaning that each element should be filled up with material (hard elements with the relative material volume equal to 1) or contain no material (void elements with the relative material volume close to 0 but always positive). Thus, the relative material volume of each element is defined as a design variable. Numerically, the above optimization problem is an integer problem and therefore difficult to solve for a large number of design variables. Therefore, the above optimization problem is relaxed so the relative densities can have values in between 0 and 1,

$p_{e} =] 0, 1]$

The material interpolation schemes penalizing the Young’s modulus E for the intermediate densities are introduced for obtaining the solution of the relaxed problem resulting in all elements being either hard or void elements. SIMULIA Tosca Structure contains two material interpolation schemes described in the following. Both material interpolation schemes contain a penalization factor p that defines to what extent the solution is forced toward a solution having with hard or void elements only.

The SIMP (solid isotropic material with penalization) material interpolation scheme is defined by

$E = E_{0} ρ^{p}$

and is very popular in the literature, e.g., see Mlejnek (1992), Bendsøe and Sigmund (1999) and Bendsøe and Sigmund (2003) and references therein. The Young’s modulus $E_{0}$ is the Young’s modulus when the element is filled up with material, thus being the Young’s modulus defined by the user. The penalty factor p should be greater than 1. Numerical experiments indicate that p=3 is a good value.

The RAMP (rational approximation of material properties) material interpolation scheme is defined by

$E = E_{0} \frac{ρ}{1 + p (1 - ρ)}$

The RAMP interpolation scheme is desirable for dynamic problems because the interpolation scheme is concave, see Hansen (2005) and Stolpe and Svanberg (2001). For the RAMP interpolation scheme, the interpolation factor p should be greater than 0. Numerical experiments indicate that p=3 is a good value.

The following figure shows the interpolation of the Young’s modulus for the SIMP and RAMP material scheme, respectively.

For p=1 in SIMP and p=0 in RAMP, the material interpolation is the so-called thickness optimization problem leads to many elements having intermediate densities. By increasing the penalization factor p, the intermediate densities turn out to be computationally more expensive because elements having intermediate densities require a higher density for having the same stiffness as in the thickness optimization problem. Thus, when the penalty factor is increased, fewer elements contain have an intermediate density. On the other side, one should not choose too large penalty because then the likelihood for ending up in a local minimum during the optimization is significantly higher, and, if the penalty factor is extremely high, one will always end up in a local minimum.

Consequently, a compromise for the penalty factor will be to choose a value in the interval between 2 and 4 that also agrees with the numerical experiments.

By default, the optimization algorithm chooses the SIMP interpolation scheme with p=3 if all load case are static. However, if at least one dynamic load case is used, the RAMP interpolation scheme with p=3 is chosen by default.

The 4_TH (4th order polynomial) material interpolation is defined by

$E = E_{0} (a_{4} ρ^{4} + a_{3} ρ^{3} + a_{2} ρ^{2} + a_{1} ρ + α_{0})$

This interpolation scheme should only be used by experienced users. The coefficients have to be defined in a way that the material interpolation is positive in the given range of the relative density. It is up to the user that physical bounds are full-filled e.g. the Hashin-Shtrikman bounds. By default there is no penalization build in meaning that one would not yield solid-void designs.

The user has the possibility to choose the interpolation scheme and the value of the penalty factor using the commands MAT_INTERPOLATION and MAT_PENALTY, respectively. The valid values for MAT_INTERPOLATION parameter are SIMP, RAMP and 4_TH.

E.g., the SIMP material interpolation scheme and a penalty of 3.5 are used in the following example:

OPT_PARAM
 ID_NAME = opt_params
 OPTIMIZE = id_of_optimize

 MAT_INTERPOLATION = SIMP
 MAT_PENALTY = 3.5
END_

The RAMP material interpolation scheme and a penalty of 2.5 are used in the following example:

OPT_PARAM
 ID_NAME = opt_params
 OPTIMIZE = id_of_optimize

 MAT_INTERPOLATION = RAMP
 MAT_PENALTY = 2.5
END_

The 4_TH material interpolation scheme and the corresponding coefficients are defined in OPT_PARAM command as follows:

OPT_PARAM
 ID_NAME = opt_params
 OPTIMIZE = id_of_optimize

 MAT_INTERPOLATION = 4_TH
 MAT_0_TH_FACTOR = 0.0
 MAT_1_TH_FACTOR = 0.0
 MAT_2_TH_FACTOR = 0.0
 MAT_3_TH_FACTOR = 3.0
 MAT_4_TH_FACTOR = 0.0
END_

For the 4_TH material interpolation scheme the user should also set the required lower and upper bounds for the relative density using for example the following commands:

OPT_PARAM
 ...
 DENSITY_LOWER = 0.4
 DENSITY_UPPER = 0.9
 ...
END_