A 2D topology optimisation algorithm in NURBS framework with geometric constraints

Abstract

In this paper, the solid isotropic material with penalisation (SIMP) method for topology optimisation of 2D problems is reformulated in the non-uniform rational BSpline (NURBS) framework. This choice implies several advantages, such as the definition of an implicit filter zone and the possibility for the designer to get a geometric entity at the end of the optimisation process. Therefore, important facilities are provided in CAD postprocessing phases in order to retrieve a consistent and well connected final topology. The effect of the main NURBS parameters (degrees, control points, weights and knot-vector components) on the final optimum topology is investigated. Classic geometric constraints, as the minimum and maximum member size, have been integrated and reformulated according to the NURBS formalism. Furthermore, a new constraint on the local curvature radius has been developed thanks to the NURBS formalism and properties. The effectiveness and the robustness of the proposed method are tested and proven through some benchmarks taken from literature and the results are compared with those provided by the classical SIMP approach.

Appendices

Appendix 1: Sensitivity analysis of compliance and volume fraction

Let G be a generic scalar quantity whose gradient with respect to the mesh elements is known (i.e. $\frac{\partial G}{\partial \rho _e}$). Now, the derivatives $\frac{\partial G}{\partial \overline{\rho _{s,t}}}$ and $\frac{\partial G}{\partial w_{s,t}}$ need to be computed, where $\overline{\rho _{s,t}}$ is the generic control point of the NURBS scalar function and $w_{s,t}$ the respective weight. Let $I_{s,t}$ be the local support of the blending function associated to the control point $\overline{\rho _{s,t}}$: it is evident that only those elements lying in the support will contribute to the sensitivity analysis. Therefore, the following general expressions can be inferred by the chaining rule for derivatives:

$$\begin{aligned} \frac{\partial G}{\partial \overline{\rho _{s,t}}}& {} = \sum _{e \in I_{s,t}} \frac{\partial G}{\partial \rho _e} \frac{\partial \rho _e}{\partial \overline{\rho _{s,t}}}, \end{aligned}$$

(67)

$$\begin{aligned} \frac{\partial G}{\partial w_{s,t}}& {} = \sum _{e \in I_{s,t}} \frac{\partial G}{\partial \rho _e} \frac{\partial \rho _e}{\partial w_{s,t}}. \end{aligned}$$

(68)

The derivative $\frac{\partial \rho _e}{\partial \overline{\rho _{s,t}}}$ can be easily computed from Eq. (20):

$$\begin{aligned} \frac{\partial \rho _e}{\partial \overline{\rho _{s,t}}}=R_{s,t}(u_e,v_e). \end{aligned}$$

(69)

The derivative $\frac{\partial \rho _e}{\partial w_{s,t}}$ is evaluated by explicitly inserting Eq. (2) in Eq. (20). The final expression can be retrieved after few computations:

$$\begin{aligned} \frac{\partial \rho _e}{\partial w_{s,t}}=\frac{R_{s,t}(u_e,v_e)}{w_{s,t}}(\overline{\rho _{s,t}}-\rho _e). \end{aligned}$$

(70)

Consequently, the sensitivity analysis for the compliance and the volume fraction can be deduced combining Eqs. (69) and (70) with Eqs. (18) and (19).

Appendix 2: Sensitivity analysis of constraints

1.1 Sensitivity analysis of the minimum member size constraint

Sensitivity of the monotonicity integral $M_{\gamma _i}(\rho )$ with respect to the control points:

$$\begin{aligned} \begin{aligned}\dfrac{\partial M_{\gamma _i}(\rho )}{\partial \overline{\rho _{s,t}}}&= \sum _{j=1}^{N_{\varvec{{\mathrm {\gamma }}}_i}-1} \dfrac{\partial }{\partial \overline{\rho _{s,t}}} \left( \sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2} \right) - \left( \dfrac{\partial }{\partial \overline{\rho _{s,t}}} \sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2} \right) \\& =\sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1}\dfrac{(\rho _{j+1}-\rho _j) \left( \dfrac{\partial \rho _{j+1} }{\partial \overline{\rho _{s,t}}}- \dfrac{\partial \rho _{j} }{\partial \overline{\rho _{s,t}}}\right) }{\sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2}}-\dfrac{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)\left( \dfrac{\partial \rho _{N_{{\varvec{\gamma }}_i}} }{\partial \overline{\rho _{s,t}}} - \dfrac{\partial \rho _1}{\partial \overline{\rho _{s,t}}}\right) }{\sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2}}\\& =\sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1}\dfrac{(\rho _{j+1}-\rho _j) \left( R_{s,t}(u_{j+1},v_{j+1})- R_{s,t}(u_{j},v_{j})\right) }{\sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2}}+\\&\quad -\,\dfrac{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)\left( R_{s,t}(u_{N_{{\varvec{\gamma }}_i}},v_{N_{{\varvec{\gamma }}_i}})- R_{s,t}(u_{1},v_{1})\right) }{\sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2}} . \end{aligned} \end{aligned}$$

(71)

Sensitivity of the monotonicity integral $M_{\gamma _i}(\rho )$ with respect to the NURBS weights:

$$\begin{aligned} \begin{aligned} \dfrac{\partial M_{\gamma _i}(\rho )}{\partial w_{s,t}} & = \sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1} \dfrac{\partial }{\partial w_{s,t}} \left( \sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2} \right) - \left( \dfrac{\partial }{\partial w_{s,t}} \sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2} \right) \\& =\sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1}\dfrac{(\rho _{j+1}-\rho _j) \left( \dfrac{\partial \rho _{j+1} }{\partial w_{s,t}}- \dfrac{\partial \rho _{j} }{\partial w_{s,t}}\right) }{\sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2}}-\dfrac{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)\left( \dfrac{\partial \rho _{N_{{\varvec{\gamma }}_i}} }{\partial w_{s,t}} - \dfrac{\partial \rho _1}{\partial w_{s,t}}\right) }{\sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2}}\\& =\dfrac{1}{w_{s,t}}\sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1}\dfrac{(\rho _{j+1}-\rho _j) \left( \dfrac{\partial \rho _{j+1} }{\partial \overline{\rho _{s,t}}}(\overline{\rho _{s,t}}-\rho _{j+1})- \dfrac{\partial \rho _{j} }{\partial \overline{\rho _{s,t}}}(\overline{\rho _{s,t}}-\rho _{j}) \right) }{\sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2}}+\\&\quad -\dfrac{1}{w_{s,t}}\dfrac{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)\left( \dfrac{\partial \rho _{N_{{\varvec{\gamma }}_i}} }{\partial \overline{\rho _{s,t}}}(\overline{\rho _{s,t}}-\rho _{N_{{\varvec{\gamma }}_i}}) - \dfrac{\partial \rho _1}{\partial \overline{\rho _{s,t}}}(\overline{\rho _{s,t}}-\rho _1)\right) }{\sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2}}\\& =\dfrac{\overline{\rho _{s,t}}}{w_{s,t}} \left[ \sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1}\dfrac{(\rho _{j+1}-\rho _j) \left( \dfrac{\partial \rho _{j+1} }{\partial \overline{\rho _{s,t}}}- \dfrac{\partial \rho _{j} }{\partial \overline{\rho _{s,t}}}\right) }{\sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2}}-\dfrac{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)\left( \dfrac{\partial \rho _{N_{{\varvec{\gamma }}_i}} }{\partial \overline{\rho _{s,t}}} - \dfrac{\partial \rho _1}{\partial \overline{\rho _{s,t}}}\right) }{\sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2}} \right]\\&\quad +\dfrac{1}{w_{s,t}} \Bigg [ \sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1}\dfrac{(\rho _{j+1}-\rho _j) \left( \rho _j \dfrac{\partial \rho _{j} }{\partial \overline{\rho _{s,t}}}- \rho _{j+1} \dfrac{\partial \rho _{j+1} }{\partial \overline{\rho _{s,t}}} \right) }{\sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2}} +\\&\quad - \left.\dfrac{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)\left( \rho _1 \dfrac{\partial \rho _{1} }{\partial \overline{\rho _{s,t}}} -\rho _{N_{{\varvec{\gamma }}_i}} \dfrac{\partial \rho _{N_{{\varvec{\gamma }}_i}} }{\partial \overline{\rho _{s,t}}} \right) }{\sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2}} \right]\\& = \dfrac{\overline{\rho _{s,t}}}{w_{s,t}} \dfrac{\partial M_{\gamma _i}(\rho )}{\partial \overline{\rho _{s,t}}}+\dfrac{1}{w_{s,t}} \Bigg [ \sum _{j=1}^{N_{{\varvec{\gamma }}_i}-1}\dfrac{(\rho _{j+1}-\rho _j) \left( \rho _j R_{s,t}(u_{j},v_{j})- \rho _{j+1} R_{s,t}(u_{j+1},v_{j+1}) \right) }{\sqrt{(\rho _{j+1}-\rho _j)^2+\epsilon ^2}} +\\&\quad \left.- \dfrac{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)\left( \rho _1 R_{s,t}(u_{1},v_{1}) -\rho _{N_{{\varvec{\gamma }}_i}} R_{s,t}(u_{N_{{\varvec{\gamma }}_i}},v_{N_{{\varvec{\gamma }}_i}}) \right) }{\sqrt{(\rho _{N_{{\varvec{\gamma }}_i}}-\rho _1)^2+\epsilon ^2}} \right]. \end{aligned} \end{aligned}$$

(72)

1.2 Sensitivity analysis of the maximum member size constraint

The gradient of the maximum member size constraint is computed here with respect to the NURBS control points and weights:

$$\begin{aligned}&\dfrac{\partial g_{d_{max}}}{\partial \overline{\rho _{s,t}}}=\dfrac{1}{\dfrac{\pi d_{max}^2}{4}(1-\psi )} \dfrac{\partial }{\partial \overline{\rho _{s,t}}} \left( \sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \hat{\rho _i} A_i \right) ^\chi \right) ^{\dfrac{1}{\chi }}\nonumber \\&\quad =\dfrac{\left( \sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi \right) ^{\dfrac{1}{\chi }-1}}{\chi \dfrac{\pi d_{max}^2}{4}(1-\psi )} \left( \sum _{e=1}^{N_e} \dfrac{\partial }{\partial \overline{\rho _{s,t}}} \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^\chi \right) \nonumber \\&\quad =\dfrac{\left( \sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi \right) ^{\dfrac{1}{\chi }}}{\dfrac{\pi d_{max}^2}{4}(1-\psi )} \dfrac{\left( \sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^{\chi -1} \left( \sum _{i \in \Omega _e} \alpha \rho _i^{\alpha -1} \dfrac{\partial \rho _i}{\partial \overline{\rho _{s,t}}} A_i \right) \right) }{\sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi } \nonumber \\&\quad =\alpha (g_{d_{max}}+1)\dfrac{ \sum _{e=1}^{N_e} \left( \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^{\chi -1} \left( \sum _{i \in \Omega _e} \rho _i^{\alpha -1} R_{s,t}(u_i,v_i) A_i \right) \right) }{\sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi }. \end{aligned}$$

(73)

$$\begin{aligned}\dfrac{\partial g_{d_{max}}}{\partial w_{s,t}}&=\alpha (g_{d_{max}}+1)\dfrac{ \sum _{e=1}^{N_e} \left( \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^{\chi -1} \left( \sum _{i \in \Omega _e} \rho _i^{\alpha -1} \dfrac{\partial \rho _i}{\partial w_{s,t}} A_i \right) \right) }{\sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi }\nonumber \\& =\alpha (g_{d_{max}}+1)\dfrac{ \sum _{e=1}^{N_e} \left( \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^{\chi -1} \left( \sum _{i \in \Omega _e} \rho _i^{\alpha -1} \dfrac{\partial \rho _i}{\partial \overline{\rho _{s,t}}}\dfrac{\overline{\rho _{s,t}}-\rho _i}{w_{s,t}} A_i \right) \right) }{\sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi }\nonumber \\& = \dfrac{\overline{\rho _{s,t}}}{w_{s,t}}\alpha (g_{d_{max}}+1)\dfrac{ \sum _{e=1}^{N_e} \left( \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^{\chi -1} \left( \sum _{i \in \Omega _e} \rho _i^{\alpha -1} \dfrac{\partial \rho _i}{\partial \overline{\rho _{s,t}}} A_i \right) \right) }{\sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi }+\nonumber \\&\quad -\dfrac{1}{w_{s,t}}\alpha (g_{d_{max}}+1)\dfrac{ \sum _{e=1}^{N_e} \left( \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^{\chi -1} \left( \sum _{i \in \Omega _e} \rho _i^{\alpha } \dfrac{\partial \rho _i}{\partial \overline{\rho _{s,t}}} A_i \right) \right) }{\sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi }\nonumber \\& =\dfrac{\overline{\rho _{s,t}}}{w_{s,t}}\dfrac{\partial g_{d_{max}}}{\partial \overline{\rho _{s,t}}}-\dfrac{\alpha (g_{d_{max}}+1)}{w_{s,t}} \dfrac{ \sum _{e=1}^{N_e} \left( \left( \sum _{i \in \Omega _e} \rho _i^\alpha A_i \right) ^{\chi -1} \left( \sum _{i \in \Omega _e} \rho _i^{\alpha } R_{s,t}(u_i,v_i) A_i \right) \right) }{\sum _{e=1}^{N_e} \left( \sum _{i \in \Omega _e} \overline{\rho _i}A_i \right) ^\chi }. \end{aligned}$$

(74)

1.3 Sensitivity analysis of the local curvature radius constraint

The gradient of the local curvature radius is computed with respect to the control points and to the weights. Let us write again Eq. (50) in a more convenient form:

$$\begin{aligned} r=-\,\dfrac{1}{WH} \dfrac{r_N}{r_D}. \end{aligned}$$

(75)

Thus, the derivatives write

$$\begin{aligned}&\dfrac{\partial r}{\partial \overline{\rho _{s,t}}}=r \left( \dfrac{1}{r_N} \dfrac{\partial r_N}{\partial \overline{\rho _{s,t}}} - \dfrac{1}{r_D} \dfrac{\partial r_D}{\partial \overline{\rho _{s,t}}} \right) , \end{aligned}$$

(76)

$$\begin{aligned}&\dfrac{\partial r}{\partial w_{s,t}}=r \left( \dfrac{1}{r_N} \dfrac{\partial r_N}{\partial w_{s,t}} - \dfrac{1}{r_D} \dfrac{\partial r_D}{\partial w_{s,t}} \right) . \end{aligned}$$

(77)

The complete expression of Eqs. (76) and (77) are not provided here for sake of brevity; anyway the reader can easily deduce them by using the following formulae to compute the derivatives of each term. They are deduced from the results of “Appendix 1”:

$$\begin{aligned}&\frac{\partial }{\partial \overline{\rho _{s,t}}} \left( \dfrac{\partial \rho }{\partial u} \right) =\dfrac{\partial R_{s,t}}{\partial u}, \end{aligned}$$

(78)

$$\begin{aligned}&\frac{\partial }{\partial \overline{\rho _{s,t}}} \left( \dfrac{\partial \rho }{\partial v} \right) =\dfrac{\partial R_{s,t}}{\partial v}, \end{aligned}$$

(79)

$$\begin{aligned}&\frac{\partial }{\partial \overline{\rho _{s,t}}} \left( \dfrac{\partial ^2 \rho }{\partial u^2} \right) =\dfrac{\partial ^2 R_{s,t}}{\partial u^2}, \end{aligned}$$

(80)

$$\begin{aligned}&\frac{\partial }{\partial \overline{\rho _{s,t}}} \left( \dfrac{\partial ^2 \rho }{\partial u \partial v} \right) =\dfrac{\partial ^2 R_{s,t}}{\partial u \partial v}, \end{aligned}$$

(81)

$$\begin{aligned}&\frac{\partial }{\partial \overline{\rho _{s,t}}} \left( \dfrac{\partial ^2 \rho }{\partial v^2} \right) =\dfrac{\partial ^2 R_{s,t}}{\partial v^2}, \end{aligned}$$

(82)

$$\begin{aligned}&\frac{\partial }{\partial w_{s,t}} \left( \dfrac{\partial \rho }{\partial u} \right) =\dfrac{\overline{\rho _{s,t}}-\rho }{w_{s,t}}\dfrac{\partial R_{s,t}}{\partial u}-\dfrac{R_{s,t}}{w_{s,t}} \dfrac{\partial \rho }{\partial u}, \end{aligned}$$

(83)

$$\begin{aligned}&\frac{\partial }{\partial w_{s,t}} \left( \dfrac{\partial \rho }{\partial v} \right) =\dfrac{\overline{\rho _{s,t}}-\rho }{w_{s,t}}\dfrac{\partial R_{s,t}}{\partial v}-\dfrac{R_{s,t}}{w_{s,t}} \dfrac{\partial \rho }{\partial v}, \end{aligned}$$

(84)

$$\begin{aligned}&\frac{\partial }{\partial w_{s,t}} \left( \dfrac{\partial ^2 \rho }{\partial u^2} \right) =\dfrac{1}{w_{s,t}} \left( (\overline{\rho _{s,t}}-\rho ) \dfrac{\partial ^2 R_{s,t}}{\partial u^2} -2\dfrac{\partial R_{s,t}}{\partial u}\dfrac{\partial \rho }{\partial u} -R_{s,t}\dfrac{\partial ^2 \rho }{\partial u^2} \right) , \end{aligned}$$

(85)

$$\begin{aligned}&\frac{\partial }{\partial w_{s,t}} \left( \dfrac{\partial ^2 \rho }{\partial u \partial v} \right) =\dfrac{1}{w_{s,t}} \left( (\overline{\rho _{s,t}}-\rho ) \dfrac{\partial ^2 R_{s,t}}{\partial u \partial v} -\dfrac{\partial R_{s,t}}{\partial u}\dfrac{\partial \rho }{\partial v}- \dfrac{\partial R_{s,t}}{\partial v}\dfrac{\partial \rho }{\partial u} -R_{s,t}\dfrac{\partial ^2 \rho }{\partial u \partial v} \right) , \end{aligned}$$

(86)

$$\begin{aligned}&\frac{\partial }{\partial w_{s,t}} \left( \dfrac{\partial ^2 \rho }{\partial v^2} \right) =\dfrac{1}{w_{s,t}} \left( (\overline{\rho _{s,t}}-\rho ) \dfrac{\partial ^2 R_{s,t}}{\partial v^2} -2\dfrac{\partial R_{s,t}}{\partial v}\dfrac{\partial \rho }{\partial v} -R_{s,t}\dfrac{\partial ^2 \rho }{\partial v^2} \right) . \end{aligned}$$

(87)