Strength-based topology optimisation of plastic isotropic von Mises materials

Abstract

Conventionally, topology optimisation is formulated as a non-linear optimisation problem, where the material is distributed in a manner which maximises the stiffness of the structure. Due to the nature of non-linear, non-convex optimisation problems, a multitude of local optima will exist and the solution will depend on the starting point. Moreover, while stress is an essential consideration in topology optimisation, accounting for the stress locally requires a large number of constraints to be considered in the optimisation problem; therefore, global methods are often deployed to alleviate this with less control of the stress field as a consequence. In the present work, a strength-based formulation with stress-based elements is introduced for plastic isotropic von Mises materials. The formulation results in a convex optimisation problem which ensures that any local optimum is the global optimum, and the problems can be solved efficiently using interior point methods. Four plane stress elements are introduced and several examples illustrate the strength of the convex stress-based formulation including mesh independence, rapid convergence and near-linear time complexity.

Appendices

Appendix A: Equilibrium matrices for the plane stress elements

1.1 A.1 The Zouain element

$$ \hat{\mathbf{B}}^{T} = -\frac{1}{6}\left[\begin{array}{ccc} {\tilde{\mathbf{P}}}_{1} & 0 & 0 \\ 0 & {\tilde{\mathbf{P}}}_{2} & 0 \\ 0 & 0 & {\tilde{\mathbf{P}}}_{3} \\ -{\tilde{\mathbf{P}}}_{1} & {\tilde{\mathbf{P}}}_{3}-{\tilde{\mathbf{P}}}_{1} & {\tilde{\mathbf{P}}}_{2}-{\tilde{\mathbf{P}}}_{1} \\ {\tilde{\mathbf{P}}}_{3}-{\tilde{\mathbf{P}}}_{2} & -{\tilde{\mathbf{P}}}_{2} & {\tilde{\mathbf{P}}}_{1}-{\tilde{\mathbf{P}}}_{2} \\ {\tilde{\mathbf{P}}}_{2}-{\tilde{\mathbf{P}}}_{3} & {\tilde{\mathbf{P}}}_{1}-{\tilde{\mathbf{P}}}_{3} & -{\tilde{\mathbf{P}}}_{3} \end{array}\right] $$

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1.2 A.2 Standard six-node element

$$ \hat{\mathbf{B}}^{T} = -\frac{1}{18}\left[\begin{array}{ccc} 5{\tilde{\mathbf{P}}}_{1} & -{\tilde{\mathbf{P}}}_{1} & -{\tilde{\mathbf{P}}}_{1} \\ -{\tilde{\mathbf{P}}}_{2} & 5{\tilde{\mathbf{P}}}_{2} & -{\tilde{\mathbf{P}}}_{2} \\ -{\tilde{\mathbf{P}}}_{3} & -{\tilde{\mathbf{P}}}_{3} & 5{\tilde{\mathbf{P}}}_{3} \\ -2{\tilde{\mathbf{P}}}_{1} & 2{\tilde{\mathbf{P}}}_{2}+ 8{\tilde{\mathbf{P}}}_{3} & 2{\tilde{\mathbf{P}}}_{3}+ 8{\tilde{\mathbf{P}}}_{2} \\ 2{\tilde{\mathbf{P}}}_{1}+ 8{\tilde{\mathbf{P}}}_{3} & -2{\tilde{\mathbf{P}}}_{2} & 2{\tilde{\mathbf{P}}}_{3}+ 8{\tilde{\mathbf{P}}}_{1} \\ 2{\tilde{\mathbf{P}}}_{1}+ 8{\tilde{\mathbf{P}}}_{2} & 2{\tilde{\mathbf{P}}}_{2}+ 8{\tilde{\mathbf{P}}}_{1} & -2{\tilde{\mathbf{P}}}_{3} \end{array}\right] $$

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1.3 A.3 Upper bound element

$$ \hat{\mathbf{B}}^{T} = -\frac{1}{6}\left[\begin{array}{ccc} 3{\tilde{\mathbf{P}}}_{1} & -{\tilde{\mathbf{P}}}_{1} & -{\tilde{\mathbf{P}}}_{1} \\ -{\tilde{\mathbf{P}}}_{2} & 3{\tilde{\mathbf{P}}}_{2} & -{\tilde{\mathbf{P}}}_{2} \\ -{\tilde{\mathbf{P}}}_{3} & -{\tilde{\mathbf{P}}}_{3} & 3{\tilde{\mathbf{P}}}_{3} \\ 0 & 4{\tilde{\mathbf{P}}}_{3} & 4{\tilde{\mathbf{P}}}_{2} \\ 4{\tilde{\mathbf{P}}}_{3} & 0 & 4{\tilde{\mathbf{P}}}_{1} \\ 4{\tilde{\mathbf{P}}}_{2} & 4{\tilde{\mathbf{P}}}_{1} & 0 \end{array}\right] $$

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Appendix B: Tabulated results

Table 1 Minimum volume and computational time for the MBB-beam problem

Table 2 Minimum volume and computational time for the cantilever problem

Table 3 Minimum volume and computational time for the cantilever beam with circular hole problem

Table 4 Minimum volume and computational time for the portal structure problem with fixed supports