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The interpolation schemes described above are, in essence, computational approximations to the black-and-white 0-1 problem. As the problems are different in form, the results obtained with the various methods are, as expected, not the same. Conceptually, there are strong sim- ilarities, but the differences in detail can be quite significant. This is not a major problem when employing the techniques in a design context, as long as these differences are understood and acknowledged.

Figure 4 The interpolation schemes described above are, in essence, computational approximations to the black-and-white 0-1 problem. As the problems are different in form, the results obtained with the various methods are, as expected, not the same. Conceptually, there are strong sim- ilarities, but the differences in detail can be quite significant. This is not a major problem when employing the techniques in a design context, as long as these differences are understood and acknowledged.

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Fig. 1. The generalized shape design problem of finding the optimal material distribution to a situation where the volume fraction is allowed any value between zero and one. It may also involve finding an appropriate method for limiting geometric complexity, for example, ex- changing the total variation of a density for the perimeter of a domain. Problem (1) is a discrete optimization problem, and for many applications it is useful to consider reformulations in terms of continuous variables, with the goal of using derivative based mathematical programming algorithms. This means that one changes the model for material properties, i.e., the relations defined in (1) as
Fig. 1. The generalized shape design problem of finding the optimal material distribution to a situation where the volume fraction is allowed any value between zero and one. It may also involve finding an appropriate method for limiting geometric complexity, for example, ex- changing the total variation of a density for the perimeter of a domain. Problem (1) is a discrete optimization problem, and for many applications it is useful to consider reformulations in terms of continuous variables, with the goal of using derivative based mathematical programming algorithms. This means that one changes the model for material properties, i.e., the relations defined in (1) as
The discussion above holds for planar problems. In 3-D, there is, in a sense, more geometric freedom to construct microstructures, and here the Hashin-Shtrikman bounds lead to the condition on the power p in the SIMP model. This condition can be derived as outlined above, but as the algebra is rather lengthy this is omitted here. Example bounds are here so some lower values of p are possible in dimension three. Note, however, that for v = 1/3 we have the same bounds in 2-D and in 3-D.
The discussion above holds for planar problems. In 3-D, there is, in a sense, more geometric freedom to construct microstructures, and here the Hashin-Shtrikman bounds lead to the condition on the power p in the SIMP model. This condition can be derived as outlined above, but as the algebra is rather lengthy this is omitted here. Example bounds are here so some lower values of p are possible in dimension three. Note, however, that for v = 1/3 we have the same bounds in 2-D and in 3-D.
Fig. 4. Microstructures of material and void realizing the material properties of the SIMP model with p = 4; Eq. (11), for a base material with Poisson’s ratio v = 0 and v = 0.5, respectively. As in Fig. 3, nonstructural areas are seen at the centers of the cells
Fig. 4. Microstructures of material and void realizing the material properties of the SIMP model with p = 4; Eq. (11), for a base material with Poisson’s ratio v = 0 and v = 0.5, respectively. As in Fig. 3, nonstructural areas are seen at the centers of the cells
Fig. 6. Optimal design results for material and void, using various powers p in the SIMP interpolation scheme, and using the Hashin-Shtrikman upper bound. Problem definition as in Fig. 1 Figure 6 shows examplary optimal designs for a simple, planar, minimum-compliance de- sign problem using the Voigt upper-bound interpolation, the Hashin-Shtrikman upper-bound interpolation and SIMP for various powers of p. For the latter cases, the power is maintained fixed in the iterative optimization scheme, except in one situation. Note that the Voigt upper- bound interpolation does not satisfy our goal of finding a black-and-white design. The com-
Fig. 6. Optimal design results for material and void, using various powers p in the SIMP interpolation scheme, and using the Hashin-Shtrikman upper bound. Problem definition as in Fig. 1 Figure 6 shows examplary optimal designs for a simple, planar, minimum-compliance de- sign problem using the Voigt upper-bound interpolation, the Hashin-Shtrikman upper-bound interpolation and SIMP for various powers of p. For the latter cases, the power is maintained fixed in the iterative optimization scheme, except in one situation. Note that the Voigt upper- bound interpolation does not satisfy our goal of finding a black-and-white design. The com-
Fig. 7. The shape of single inclusions of void in a cell of a homogenized, periodic medium minimizing complementary energy (Vigdergauz-like structures for v = 1/3 and a density p = 0.5). Results for a range of principal stress ratios of a macroscopic stress field The plots of the complementary energy explain many features of computational experience with various interpolation schemes. For compliance optimization, the complementary energy should be minimized. As ranked laminates are efficient also at intermediate densities, optimal
Fig. 7. The shape of single inclusions of void in a cell of a homogenized, periodic medium minimizing complementary energy (Vigdergauz-like structures for v = 1/3 and a density p = 0.5). Results for a range of principal stress ratios of a macroscopic stress field The plots of the complementary energy explain many features of computational experience with various interpolation schemes. For compliance optimization, the complementary energy should be minimized. As ranked laminates are efficient also at intermediate densities, optimal
For a SIMP-like interpolation model, it is most convenient to interpolate first between the two nonzero phases and then between this ‘material’ and void. The resulting model is
For a SIMP-like interpolation model, it is most convenient to interpolate first between the two nonzero phases and then between this ‘material’ and void. The resulting model is
Fig. 10. Optimal design results for two-materials design (for E; = 1, Ey = 0.1, and v, = v. = 1/3), usin; various interpolation schemes. The geometry and loading of the problem as in Fig. 1, comp. Fig. 6. The compliances of the designs lie within a few percent of one another
Fig. 10. Optimal design results for two-materials design (for E; = 1, Ey = 0.1, and v, = v. = 1/3), usin; various interpolation schemes. The geometry and loading of the problem as in Fig. 1, comp. Fig. 6. The compliances of the designs lie within a few percent of one another
Fig. 11. Optimal design results for three-materials design (two materials with v) = v. = 1/3 and with stiffness E; = 1, E, = 0.1, and void), using various powers p in the interpolation scheme (28). The ge- ometry and loading of the problem as in Fig. 1. Compare with Figs. 6 and 10
Fig. 11. Optimal design results for three-materials design (two materials with v) = v. = 1/3 and with stiffness E; = 1, E, = 0.1, and void), using various powers p in the interpolation scheme (28). The ge- ometry and loading of the problem as in Fig. 1. Compare with Figs. 6 and 10
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