Topology optimization of continuum structures subjected to the variance constraint of reaction forces

Abstract

In this paper, the attainment of uniform reaction forces at the specific fixed boundary is investigated for topology optimization of continuum structures. The variance of the reaction forces at the boundary between the elastic solid and its foundation is firstly introduced as the evaluation criterion of the uniformity of the reaction forces. Then, the standard formulation of optimal topology design is improved by introducing the variance constraint of the reaction forces. Sensitivity analysis of the latter is carried out based on the adjoint method. Numerical examples are dealt with to reveal the effect of the variance constraint in comparison with solutions of standard topology optimization.

Appendix: Sensitivity analysis

The sensitivity analysis of the global compliance and the variance of reaction forces with respect to design variables is detailed as follow.

1.1 Sensitivity analysis of the global compliance

Based on the definition of the global compliance, the sensitivity of the latter then corresponds to

$$ \frac{\partial C}{\partial {x}_i}=2{\mathbf{U}}^{\mathrm{T}}\frac{\partial \mathbf{F}}{\partial {x}_i}-{\mathbf{U}}^{\mathrm{T}}\frac{\partial \mathbf{K}}{\partial {x}_i}\mathbf{U}=2{\mathbf{U}}_{\mathrm{c}}^{\mathrm{T}}\frac{\partial {\mathbf{F}}^{\mathrm{a}}}{\partial {x}_i}-{\mathbf{U}}_i^{\mathrm{T}}\frac{\partial {\mathbf{K}}_i}{\partial {x}_i}{\mathbf{U}}_i $$

(A.1)

In this work, the applied forces are supposed to be design-independent. This implies that

$$ \frac{\partial {\mathbf{F}}^{\mathrm{a}}}{\partial {x}_i}=0 $$

(A.2)

The sensitivity of the global compliance is then rewritten as

$$ \frac{\partial C}{\partial {x}_i}=-{\mathbf{U}}_i^{\mathrm{T}}\frac{\partial {\mathbf{K}}_i}{\partial {x}_i}{\mathbf{U}}_i $$

(A.3)

1.2 Sensitivity analysis of the variance of reaction forces

Based on the definition in Section 3, the sensitivity of the variance of the specific reaction forces corresponds to

$$ \frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}=\frac{1}{n_{\mathrm{r}}}{\sum}_j2\left({R}_j-\overset{-}{R}\right)\left(\frac{\partial {R}_j}{\partial {x}_i}-\frac{\partial \overset{-}{R}}{\partial {x}_i}\right) $$

(A.4)

where the partial derivative of the mean reaction force is expressed as

$$ \frac{\partial \overset{-}{R}}{\partial {x}_i}=\frac{1}{n_{\mathrm{r}}}{\sum}_{\zeta}\frac{\partial {R}_{\zeta }}{\partial {x}_i} $$

(A.5)

Thus, we have

$$ \begin{array}{l}\frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}=\frac{1}{n_{\mathrm{r}}}\sum_j2\left({R}_j-\overline{R}\right)\left(\frac{\partial {R}_j}{\partial {x}_i}-\frac{1}{n_{\mathrm{r}}}\sum_{\zeta}\frac{\partial {R}_{\zeta }}{\partial {x}_i}\right)\\ {}=\frac{2}{n_{\mathrm{r}}}{\sum}_j\left({R}_j-\overline{R}\right)\left(\left(1-\frac{1}{n_{\mathrm{r}}}\right)\frac{\partial {R}_j}{\partial {x}_i}-\frac{1}{n_{\mathrm{r}}}{\sum}_{{}_{\zeta }{,}_{\zeta}\ne j}\frac{\partial {R}_{\zeta }}{\partial {x}_i}\right)\end{array} $$

(A.6)

This expression can be rewritten as

$$ \frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}={\boldsymbol{\Lambda}}^{\mathrm{T}}\frac{\partial \mathbf{R}}{\partial {x}_i} $$

(A.7)

in which Λ is a vector

$$ \boldsymbol{\Lambda} =\left\{\begin{array}{c}{\varLambda}_1\\ {}\begin{array}{c}\vdots \\ {}{\varLambda}_j\\ {}\vdots \end{array}\\ {}{\varLambda}_{n_{\mathrm{s}}}\end{array}\right\} $$

(A.8)

with each term being calculated by

$$ {\Lambda}_j=\left\{\begin{array}{cc}\frac{2}{n_{\mathrm{r}}}\left(\left({R}_{\boldsymbol{j}}-\overset{-}{R}\right)\left(1-\frac{1}{n_{\mathrm{r}}}\right)-\frac{1}{n_{\mathrm{r}}}{\sum}_{\boldsymbol{\zeta}, \boldsymbol{\zeta} \ne \boldsymbol{j}}\left({R}_{\boldsymbol{\zeta}}-\overset{-}{R}\right)\right)& j\in {\Gamma}_{\mathrm{RV}}\\ {}0& j\notin {\Gamma}_{\mathrm{RV}}\end{array}\right. $$

(A.9)

To calculate $\frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}$, the partial derivative of the reaction force vector R is necessary. According to (6), we have

$$ \frac{\partial \mathbf{R}}{\partial {x}_i}=\frac{\partial {\mathbf{K}}_{cs}^{\mathrm{T}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}}+{\mathbf{K}}_{cs}^{\mathrm{T}}\frac{\partial {\mathbf{U}}_{\mathrm{c}}}{\partial {x}_i} $$

(A.10)

and

$$ \frac{\partial {\mathbf{U}}_{\mathrm{c}}}{\partial {x}_i}={\mathbf{K}}_{\mathrm{c}\mathrm{c}}^{-1}\left(\frac{\partial {\mathbf{F}}^{\mathrm{a}}}{\partial {x}_i}-\frac{\partial {\mathbf{K}}_{\mathrm{c}\mathrm{c}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}}\right) $$

(A.11)

Under the assumption of design-independent force (i.e., $\frac{\partial {\mathbf{F}}^{\mathrm{a}}}{\partial {x}_i}=0$), the above relation can be simplified as

$$ \frac{\partial {\mathbf{U}}_{\mathrm{c}}}{\partial {x}_i}=-{\mathbf{K}}_{\mathrm{c}\mathrm{c}}^{-1}\frac{\partial {\mathbf{K}}_{\mathrm{c}\mathrm{c}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}} $$

(A.12)

The substitution of (A.12) into (A.10) yields

$$ \frac{\partial \mathbf{R}}{\partial {x}_i}=\frac{\partial {\mathbf{K}}_{cs}^{\mathrm{T}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}}-{\mathbf{K}}_{cs}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{c}\mathrm{c}}^{-1}\frac{\partial {\mathbf{K}}_{\mathrm{c}\mathrm{c}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}} $$

(A.13)

Now, the sensitivity of the variance of the considered reaction forces can be derived as

$$ \frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}={\boldsymbol{\Lambda}}^{\mathrm{T}}\frac{\partial {\mathbf{K}}_{cs}^{\mathrm{T}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}}-{\boldsymbol{\Lambda}}^{\mathrm{T}}{\mathbf{K}}_{cs}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{c}\mathrm{c}}^{-1}\frac{\partial {\mathbf{K}}_{\mathrm{c}\mathrm{c}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}} $$

(A.14)

in which the adjoint method is applied to calculate the second term.

Suppose

$$ {\boldsymbol{\Lambda}}^{\mathrm{T}}{\mathbf{K}}_{cs}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{cc}}^{-1}={\boldsymbol{\uplambda}}^{\mathrm{T}} $$

(A.15)

The adjoint vector λ can then be obtained by

$$ {\mathbf{K}}_{\mathrm{cc}}\boldsymbol{\uplambda} ={\mathbf{K}}_{cs}\boldsymbol{\Lambda} $$

(A.16)

or

$$ \left[\begin{array}{cc}{\mathbf{K}}_{\mathrm{cc}}& {\mathbf{K}}_{cs}\\ {}{\mathbf{K}}_{cs}^{\mathrm{T}}& {\mathbf{K}}_{ss}\end{array}\right]\left\{\begin{array}{c}\boldsymbol{\uplambda} \\ {}\mathbf{0}\end{array}\right\}=\left\{\begin{array}{c}{\mathbf{K}}_{cs}\boldsymbol{\Lambda} \\ {}{\mathbf{R}}_{\uplambda}\end{array}\right\} $$

(A.17)

Here, ${\mathbf{R}}_{\uplambda}$ is the reaction forces in the additional analysis related to (A.17). Obviously, only one additional finite element analysis needs to be carried out under the same boundary conditions as the original structural analysis in (2) no matter what the adjoint load is.

Thus, by virtue of the adjoint method, $\frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}$ is established as

$$ \frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}={\boldsymbol{\Lambda}}^{\mathrm{T}}\frac{\partial {\mathbf{K}}_{cs}^{\mathrm{T}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}}-{\boldsymbol{\uplambda}}^{\mathrm{T}}\frac{\partial {\mathbf{K}}_{\mathrm{c}\mathrm{c}}}{\partial {x}_i}{\mathbf{U}}_{\mathrm{c}} $$

(A.18)

In this expression, $\frac{\partial {\mathbf{K}}_{cs}^{\mathrm{T}}}{\partial {x}_i}$ and $\frac{\partial {\mathbf{K}}_{\mathrm{cc}}}{\partial {x}_i}$ can be easily derived at the element level.

A particular case should be mentioned herein. If the applied force is constant during optimization and the variance constraint is applied to all fixed nodes, the mean reaction force $\overset{-}{R}$ is constant as well. As a result, the derivative of the mean reaction force with respect to all design variables in (A.5) is zero and the sensitivity of the variance of specific reaction forces in (A.4) could be simplified as

$$ \frac{\partial D\left(\mathbf{R}\right)}{\partial {x}_i}=\frac{1}{n_{\mathrm{r}}}\sum_j2\left({R}_j-\overset{-}{R}\right)\frac{\partial {R}_j}{\partial {x}_i} $$

(A.19)

Thus, the artificial vector Λ is simplified as

$$ {\boldsymbol{\Lambda}}_j=\left\{\begin{array}{cc}\frac{1}{n_{\mathrm{r}}}{\sum}_j2\left({R}_j-\overset{-}{R}\right)& j\in {\Gamma}_{\mathrm{RV}}\\ {}0& j\notin {\Gamma}_{\mathrm{RV}}\end{array}\right. $$

(A.20)