Erratum to: Struct Multidisc Optim (2016)

DOI:10.1007/s00158-016-1574-3

The original version of the above article unfortunately contained some typographical errors on page 12. These typos should be corrected as follows:

1. Page 12, left hand column, first line after Equation (1c*) : (1b**) should read (1b*)

2. Page 12, left hand column, second line after Equation (1c*) : (1c**) should read (1c*)

3. Page 12, left hand column, fifth line after Equation (1c*) : (1b**) should read (1b*)

4. Page 12, left hand column, sixth line after Equation (1c*) : (1c**) should read (1c*)

5. Page 12, right hand column, seventh line from top: (1c**) should read (1c*).

Thus, the two paragraphs after Equation (1c*) should read:

Here, (1b*) expresses the constraint for the static compliance C s with \( {\overline{C}}_s \) as a given upper bound, and (1c*) is the static equilibrium equation, where P denotes the static loading, and the corresponding static displacement vector U* is defined as U* = U(ω = 0). Based on Eqs. (1b*) and (1c*), the sensitivity C ′ S of the static compliance is obtained as

$$ {C}_s^{\prime }=\mathbf{P}{\prime}^{\mathbf{T}}{\mathbf{U}}^{*}+{\mathbf{P}}^{\mathbf{T}}\mathbf{U}{\prime}^{*}=2{\mathbf{U}}^{*\mathbf{T}}\mathbf{P}^{\prime }-{\mathbf{U}}^{*\mathbf{T}}\mathbf{K}^{\prime }{\mathbf{U}}^{*}, $$
(14)

where prime denotes partial derivative with respect to the design variable ρ e , and the sensitivity P ′ of the load vector vanishes if P is design-independent. Note that the gradient C ′ s in (14) is reduced analogously to the gradient of the objective function F’ for the squared dynamic compliance in (10) to facilitate treatment by adjoint sensitivity analysis.

With the inclusion of the upper bound on the static compliance of a structure, the dynamic and static equilibrium equations (1c) and (1c*) are solved by Gauss elimination, and the constrained topology design problem (1a-e, b*,c*) is solved iteratively to convergence by means of the gradients C ′ d and C ′ s of the dynamic and static compliances, by usage of the MMA optimizer (Svanberg, 1987).

The original article was corrected.