TY - JOUR

T1 - On the equivalence of optimality criterion and sequential approximate optimization methods in the classical layout problem

AU - Groenwold, A.A.

AU - Etman, L.F.P.

PY - 2008

Y1 - 2008

N2 - We study the classical topology optimization problem, in which minimum compliance is sought, subject to linear constraints. Using a dual statement, we propose two separable and strictly convex subproblems for use in sequential approximate optimization (SAO) algorithms.Respectively, the subproblems use reciprocal and exponential intermediate variables in approximating the non-linear compliance objective function. Any number of linear constraints (or linearly approximated constraints) are provided for. The relationships between the primal variables and the dual variables are found in analytical form.For the special case when only a single linear constraint on volume is present, we note that application of the ever-popular optimality criterion (OC) method to the topology optimization problem, combined with arbitrary values for the heuristic numerical damping factor proposed by Bends¿e, results in an updating scheme for the design variables that is identical to the application of a rudimentary dual SAO algorithm, in which the subproblems are based on exponential intermediate variables. What is more, we show that the popular choice for the damping factor =0.5 is identical to the use of SAO with reciprocal intervening variables.Finally, computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy, when compared to SAO subproblems based on reciprocal intermediate variables (and hence, the heuristic topology OC method hitherto used). This is attributed to the fact that a different exponent is computed for each design variable in the two-point exponential approximation we have used, using gradient information at the previously visited point.

AB - We study the classical topology optimization problem, in which minimum compliance is sought, subject to linear constraints. Using a dual statement, we propose two separable and strictly convex subproblems for use in sequential approximate optimization (SAO) algorithms.Respectively, the subproblems use reciprocal and exponential intermediate variables in approximating the non-linear compliance objective function. Any number of linear constraints (or linearly approximated constraints) are provided for. The relationships between the primal variables and the dual variables are found in analytical form.For the special case when only a single linear constraint on volume is present, we note that application of the ever-popular optimality criterion (OC) method to the topology optimization problem, combined with arbitrary values for the heuristic numerical damping factor proposed by Bends¿e, results in an updating scheme for the design variables that is identical to the application of a rudimentary dual SAO algorithm, in which the subproblems are based on exponential intermediate variables. What is more, we show that the popular choice for the damping factor =0.5 is identical to the use of SAO with reciprocal intervening variables.Finally, computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy, when compared to SAO subproblems based on reciprocal intermediate variables (and hence, the heuristic topology OC method hitherto used). This is attributed to the fact that a different exponent is computed for each design variable in the two-point exponential approximation we have used, using gradient information at the previously visited point.

U2 - 10.1002/nme.2071

DO - 10.1002/nme.2071

M3 - Article

SN - 0029-5981

VL - 73

SP - 297

EP - 316

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

IS - 3

ER -