A Sequential Element Rejection and Admission (SERA) method for compliant mechanisms design

Abstract

A Sequential Element Rejection and Admission (SERA) method to design compliant mechanisms with topology optimization techniques is presented in this work. This procedure allows material to flow between two different material models: ‘real’ and ‘virtual’. The method works with two separate criteria for the rejection and admission of elements to efficiently achieve the optimum design. The SERA method overcomes the problems encountered by the ESO method when used to design compliant mechanisms. Three benchmark problems are presented to show the validity and robustness of the SERA method to design complaint mechanisms, regardless of the design parameters chosen.

Appendix: Sensitivity analysis

This section shows the derivation of the sensitivity analysis of the objective function with respect to the design variable.

The derivative of the MPE with respect to the element density is given in (21).

$$ \begin{array}{rll} \frac{\partial \mathit{MPE}}{\partial \rho_e} & =&\frac{\partial} {\partial \rho_e} \left( {\boldsymbol{U}_2^T \cdot \boldsymbol{K}\cdot \boldsymbol{U}_1} \right)\\ & =&\left({\frac{\partial \boldsymbol{U}_2^T}{\partial \rho_e} \cdot \boldsymbol{K}\cdot \boldsymbol{U}_1 +\boldsymbol{U}_2^T \cdot \frac{\partial \boldsymbol{K}}{\partial \rho_e} \cdot \boldsymbol{U}_1 +\boldsymbol{U}_2^T \cdot \boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_1} {\partial \rho_e} } \right) \end{array} $$

(21)

Since the stiffness matrix is symmetric, the first derivative is then given in (22).

$$ \begin{array}{rll} \frac{\partial \boldsymbol{U}_2^T} {\partial \rho_e} \cdot \boldsymbol{K}\cdot \boldsymbol{U}_1 &=&\left( {\boldsymbol{K}\cdot \boldsymbol{U}_1} \right)^T\cdot \frac{\partial \boldsymbol{U}_2} {\partial \rho_e} \\ &=&\boldsymbol{U}_1^T \cdot \boldsymbol{K}^T\cdot \frac{\partial \boldsymbol{U}_2} {\partial \rho_e} =\boldsymbol{U}_1^T \cdot \boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_2} {\partial \rho_e} \end{array} $$

(22)

Giving the derivative of the MPE to be (23).

$$ \begin{array}{rll} \frac{\partial \mathit{MPE}}{\partial \rho_e} &={}&\left( {\boldsymbol{U}_1^T \cdot \boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_2} {\partial \rho_e} +\boldsymbol{U}_2^T \cdot \frac{\partial \boldsymbol{K}}{\partial \rho_e} }\right.\\ &&\quad\cdot\left. {\boldsymbol{U}_1 +\boldsymbol{U}_2^T \cdot \boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_1} {\partial \rho_e} } \right) \end{array} $$

(23)

The two equilibrium (4) and (5) are differentiated with respect to the density and are given in (24) and (25). The input load is independent from the design variables and its derivative is zero.

$$ \frac{\partial \boldsymbol{K}}{\partial \rho_e} \cdot \boldsymbol{U}_1 +\boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_1} {\partial \rho_e} =0\to \frac{\partial \boldsymbol{K}}{\partial \rho_e} \cdot \boldsymbol{U}_1 =-\boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_1} {\partial \rho_e}$$

(24)

$$ \frac{\partial \boldsymbol{K}}{\partial \rho_e} \cdot \boldsymbol{U}_2 +\boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_2} {\partial \rho_e} =0\to \frac{\partial \boldsymbol{K}}{\partial \rho_e} \cdot \boldsymbol{U}_2 =-\boldsymbol{K}\cdot \frac{\partial \boldsymbol{U}_2} {\partial \rho_e}$$

(25)

The equivalences obtained in (24) and (25) are introduced to (23), giving the derivative of the MPE to be (26).

$$ \begin{array}{rll} \frac{\partial \mathit{MPE}}{\partial \rho_e} &=& \left(-\boldsymbol{U}_1^T \cdot \frac{\partial \boldsymbol{K}}{\partial \rho_e} \cdot \boldsymbol{U}_2 - \boldsymbol{U}_2^T \cdot \boldsymbol{K} \cdot \frac{\partial \boldsymbol{U}_1}{\partial \rho_e} + \boldsymbol{U}_2^T \cdot \boldsymbol{K} \cdot \frac{\partial \boldsymbol{U}_1}{\partial \rho_e}\right) \\ &=& -\boldsymbol{U}_1^T \cdot \frac{\partial \boldsymbol{K}}{\partial \rho_e} \cdot \boldsymbol{U}_2 \end{array}$$

(26)

As each density variable corresponds to a unique mesh element, only the displacements and stiffness of that element needs to be considered in the calculation. The sensitivity number for an element $e$, $\alpha _{e}$ can be calculated using (27).

$$ \alpha_e = - \boldsymbol{U}_{1e}^T \cdot \frac{\partial \boldsymbol{K}_e} {\partial \rho_e} \cdot \boldsymbol{U}_{2e} $$

(27)

where: U _1e is the displacement vector of element e due to load case 1; U _2e is the displacement vector of element $e$ due to load case 2; and $\frac {\partial \mathbf {K}_e} {\partial \rho _e}$ is the derivative of the elemental stiffness matrix with respect to the density.

The derivative of the stiffness matrix with respect to the density can only be approximated to the variation of the elemental stiffness (28). This is because the design variables are discrete (density can only be zero or the unit) and as a consequence, the elemental stiffness can only be the value of a ‘real’ material, K _e or a negligible value equivalent to zero.

$$ \frac{\partial \boldsymbol{K}_e} {\partial \rho_e} \approx \Delta \boldsymbol{K}_e $$

(28)

When the approximation to the variation of the elemental stiffness in (28) is substituted to the expression of the elemental sensitivity number (27) and the relative volume of the FE is factored, (6) is obtained.