An efficient stress-constrained sizing design optimization method with <i>O</i> ( <i>n</i> ) computational complexity

The bottleneck of solving a large-scale stress-constrained structural optimization model is the expensive computational cost. Since stress is a local strength measurement, the number of constraints introduced due to this factor is huge in a lightweight design model for complex engineering structures in order to ensure structural integrity. In a gradient-based optimization framework, two points contribute significantly to the computational expense, i.e., the sensitivity analysis and solving the Schur complement. In order to reduce such computational cost of stress-constrained sizing optimization, three numerical improvements are proposed in this work. The first improvement stems from enriching the fully stressed design approach with complementary terms to achieve a more accurate stress approximation. In particular, a convex, separable, and scalable stress approximation is developed, which splits the approximation into a local fully stressed part and a global load redistribution part. Secondly, an implicit sensitivity analysis, which is based on the adjoint and reanalysis methods, is proposed for stress constraints to avoid the heavy computational effort required for determining their gradient matrix within an interior-point method. Finally, a diagonal preconditioner is proposed for the resolution of the Schur complement, which is derived from the optimality condition, with the conjugate gradient method. The enhanced algorithm reduces the computational complexity effectively from its original <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , with n the number of the FEM nodes and b the bandwidth of the stiffness matrix. The proposed algorithm is tested with four test cases. The computational complexity is validated to increase only linearly with respect to the problem size n in two-dimensional (2D) cases. This makes the method computationally attractive for engineering problems of large size.