An optimality criteria method hybridized with dual programming for topology optimization under multiple constraints by moving asymptotes approximation

Abstract

The moving asymptotes function is widely applied as sequential explicit convex approximation in topology optimization, due to the controllable conservatism and convergence. In virtue of these advantages, it is supposed that the efficiency of optimality criteria method will become higher if constraint functions are approximated by moving asymptotes function instead of linear function. This work presents an optimality criteria method for topology optimization under multiple constraints where the constraint functions are approximated by moving asymptotes function. The dual feasibility condition is customarily adopted to establish explicit update scheme of topological variable, where hypothesis of active topological variable set is avoided, in the case that gradient of objective function is positive. The complementary slackness condition and primal feasibility condition are combined into a simplified dual programming to solve for the Lagrange multipliers, where hypothesis of active constraint set is avoided. Three benchmark examples under multiple displacement, stress, compliance or eigenfrequency constraints are solved by the presented optimality criteria method, the results are compared to the method of moving asymptotes and the optimality criteria method with linear constraint approximation.

Appendix A: The OCLCA method

Through taking the update scheme of topological variable into linear approximation of constraint, the following equation is obtained:

$$\begin{aligned} \begin{aligned} {{\overline{g}} _j}( {{\varvec{\lambda }}} ) = d_j^{(k)} + \sum \limits _{t = 1}^m {{\lambda _t}D_{j,t}^{(k)}} \end{aligned} \end{aligned}$$

(64)

where parameters $d_j^{(k)}$ and $D_{j,t}^{(k)}$ are as follows:

$$\begin{aligned}&\begin{aligned} d_j^{(k)}&= {g_j}( {{{{\varvec{x}}}^{(k)}}} ) + \sum \limits _{i \notin {{\mathscr {A}}^{(k + 1)}}} {\frac{{\partial {g_j}( {{{{\varvec{x}}}^{(k)}}} )}}{{\partial {x_i}}}( {x_{m ,i}^{(k + 1)} - x_i^{(k)}} )}\\&+ ( {\beta - 1} )\sum \limits _{i \in {{\mathscr {A}}^{(k + 1)}}} {\frac{{\partial {g_j}( {{{{\varvec{x}}}^{(k)}}} )}}{{\partial {x_i}}}x_i^{(k)}} \end{aligned} \end{aligned}$$

(65)

$$\begin{aligned}&\begin{aligned} D_{j,t}^{(k)} = ( {\beta - 1} )\sum \limits _{i \in {{\mathscr {A}}^{(k + 1)}}} {\frac{\displaystyle {x_i^{(k)}\frac{{\partial {g_j}( {{{{\varvec{x}}}^{(k)}}} )}}{{\partial {x_i}}}\frac{{\partial {g_t}( {{{{\varvec{x}}}^{(k)}}} )}}{{\partial {x_i}}}}}{\displaystyle {\frac{{\partial f( {{{{\varvec{x}}}^{(k)}}} )}}{{\partial {x_i}}}}}} \end{aligned} \end{aligned}$$

(66)

$$\begin{aligned}&\begin{aligned} x_{{\mathop { m}\nolimits } ,i}^{(k + 1)} = \left\{ {\begin{array}{*{20}{l}} {x_{\min ,i}^{(k + 1)}}&{}{x_i^{(k + 1)} = x_{\min ,i}^{(k + 1)}}\\ {x_{\max ,i}^{(k + 1)}}&{}{x_i^{(k + 1)} = x_{\max ,i}^{(k + 1)}} \end{array}} \right. \end{aligned} \end{aligned}$$

(67)

The corresponding simplified dual programming is established as follows:

$$\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{l}} {\mathrm{{Find}}}&{}{{{\varvec{\lambda }}} = {{( {{\lambda _1},{\lambda _2}, \ldots ,{\lambda _m}} )}^\mathrm{{T}}}}\\ {\mathrm{{Min}}}&{}{ - {{{\varvec{\lambda }}} ^\mathrm{{T}}}{{{\varvec{D}}}^{( k )}}{{\varvec{\lambda }}} - {{{\varvec{d}}}^{( k ),\mathrm{{T}}}}{{\varvec{\lambda }}} }\\ {\mathrm{{S}}\mathrm{{.t}}\mathrm{{.}}}&{}{{{{\varvec{D}}}^{( k )}}{{\varvec{\lambda }}} + {{{\varvec{d}}}^{( k )}} \le {{\varvec{0}}}}\\ {}&{}{{\lambda _j} \ge 0,j = 1,2, \ldots ,m} \end{array}} \right. \end{aligned} \end{aligned}$$

(68)

where the element at the j-th row and t-th column of matrix ${{\varvec{D}}}^{( k )}$ is $D_{j,t}^{(k)}$, the element at the j-th row of vector ${{\varvec{d}}}^{( k )}$ is $d_j^{(k)}$.

The KKT optimality condition of the above quadratic programming is as follows:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{l} - 2{{{\varvec{D}}}^{( k )}}{{\varvec{\lambda }}} - {{{\varvec{d}}}^{( k )}} + {{{\varvec{D}}}^{( k ),\mathrm{{T}}}}{{\varvec{y}}} = {{\varvec{0}}}\\ {y_j}{z_j} = 0,j = 1,2, \ldots ,m\\ {{{\varvec{D}}}^{( k )}}{{\varvec{\lambda }}} + {{{\varvec{d}}}^{( k )}} + {{\varvec{z}}} = {{\varvec{0}}}\\ {\lambda _j},{y_j},{z_j} \ge 0,j = 1,2, \ldots ,m \end{array} \right. \end{aligned} \end{aligned}$$

(69)

where ${\varvec{y}}$ is the Lagrange multiplier vector, $y_j$ is the Lagrange multiplier of the j-th constraint, ${\varvec{z}}$ is the vector of slack variables, $z_j$ is the slack variable of the j-th constraint.

Because the optimum value of simplified dual function is zero, the third equation of Eq. (69) transfers to ${{{\varvec{\lambda }}} ^\mathrm{{T}}}{{\varvec{z}}} = 0$ and then Eq. (69) is simplified as follows:

$$\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{{\varvec{z}}} + {{{\varvec{D}}}^{( k )}}{{\varvec{\lambda }}} = - {{{\varvec{d}}}^{( k )}}}\\ {{z_j},{\lambda _j} \ge 0,j = 1,2, \ldots ,m}\\ {{\lambda _j}{z_j} = 0,j = 1,2, \ldots ,m} \end{array}} \right. \end{aligned} \end{aligned}$$

(70)

The above linear complementary problem is adopted to solve for the Lagrange multipliers. It is consistent with the guide-weight method and verifies the simplified dual programming.