Structural Displacement Requirement in a Topology Optimization Algorithm Based on Isogeometric Entities

Abstract

This work deals with the formulation of a general design requirement on the displacement of a continuum medium in the framework of a special density-based algorithm for topology optimization. The algorithm makes use of non-uniform rational basis spline hyper-surfaces to represent the pseudo-density field describing the part topology and of the well-known solid isotropic material with penalization approach. The proposed formulation efficiently exploits the properties of the isogeometric basis functions, which allow defining an implicit filter. In particular, the structural displacement requirement is formulated in the most general sense, by considering displacements on loaded and non-loaded regions. The gradient of the structural displacement is evaluated in closed form by using the principle of virtual work. Moreover, a sensitivity analysis of the optimized topology to the integer parameters, involved in the definition of the hyper-surface, is carried out. The effectiveness of the proposed approach is proven through meaningful 2D and 3D benchmarks.

Appendix

The details of the proof of Proposition 3.1 are given here below.

Proof

Inasmuch as body forces are identically null, the derivative of the right-hand side of Eq. (27) is

$$\begin{aligned} \frac{\partial \lbrace {\mathrm {f}}^{(i)} \rbrace }{\partial \varXi ^{(l)}_{\tau }}= \lbrace 0 \rbrace , \ i=R,V, \ l=1,2, \end{aligned}$$

(34)

which implies the following equality

$$\begin{aligned} \frac{\partial \left( [{\mathrm {K}}] \lbrace {\mathrm {d}}^{(i)} \rbrace \right) }{\partial \varXi ^{(l)}_{\tau }}= \lbrace 0 \rbrace , \ \Rightarrow \frac{\partial \lbrace {\mathrm {d}}^{(i)} \rbrace }{\partial \varXi ^{(l)}_{\tau }}= -[{\mathrm {K}}]^{-1} \frac{\partial [{\mathrm {K}}]}{\partial \varXi ^{(l)}_{\tau }}\lbrace {\mathrm {d}}^{(i)} \rbrace , \ i=R,V, \ l=1,2.\nonumber \\ \end{aligned}$$

(35)

By deriving both sides of Eq. (28) one obtains:

$$\begin{aligned} \frac{\partial \Vert {\mathbf {u}}_P^{(R)} \Vert }{\partial \varXi ^{(l)}_{\tau }}= \frac{\partial \left( \lbrace {\mathrm {d}}^{(V)} \rbrace ^{\mathrm {T}} [{\mathrm {K}}] \right) }{\partial \varXi ^{(l)}_{\tau }} \lbrace {\mathrm {d}}^{(R)} \rbrace + \lbrace {\mathrm {d}}^{(V)} \rbrace ^{\mathrm {T}} [{\mathrm {K}}]\frac{\partial \lbrace {\mathrm {d}}^{(R)} \rbrace }{\partial \varXi ^{(l)}_{\tau }}. \end{aligned}$$

(36)

By injecting Eq. (35) into Eq. (36), the following expression can be retrieved:

$$\begin{aligned} \frac{\partial \Vert {\mathbf {u}}_P^{(R)} \Vert }{\partial \varXi ^{(l)}_{\tau }}= - \lbrace {\mathrm {d}}^{(V)} \rbrace ^{\mathrm {T}} \frac{\partial [{\mathrm {K}}]}{\partial \varXi ^{(l)}_{\tau }} \lbrace {\mathrm {d}}^{(R)} \rbrace . \end{aligned}$$

(37)

Consider now the expression of the structure stiffness matrix of Eq. (10). Due to the local support property of Eq. (8) its derivative reads:

$$\begin{aligned} \frac{\partial [{\mathrm {K}}]}{\partial \varXi ^{(l)}_{\tau }}=\sum _{e=1}^{N_e} \alpha \rho _e^{\alpha -1} \frac{\partial \rho _e}{\partial \varXi ^{(l)}_{\tau }} [{\mathrm {K}}_e]=\sum _{e \in S_{\tau }} \dfrac{\alpha }{\rho _e} \frac{\partial \rho _e}{\partial \varXi ^{(l)}_{\tau }} \rho _e^{\alpha }[{\mathrm {K}}_e]. \end{aligned}$$

(38)

By multiplying both sides of the above expression by $\lbrace {\mathrm {d}}^{(V)} \rbrace ^{\mathrm {T}}$, one gets:

$$\begin{aligned} \lbrace {\mathrm {d}}^{(V)} \rbrace ^{\mathrm {T}} \frac{\partial [{\mathrm {K}}]}{\partial \varXi ^{(l)}_{\tau }}=\sum _{e \in S_{\tau }} \dfrac{\alpha }{\rho _e} \frac{\partial \rho _e}{\partial \varXi ^{(l)}_{\tau }} \lbrace \mathrm {f_e}^{(V)} \rbrace ^{\mathrm {T}}. \end{aligned}$$

(39)

Finally, Eq. (29) can be trivially obtained by injecting the above formula in Eq. (37). $\square$