Data-driven topology optimization (DDTO) for three-dimensional continuum structures

Abstract

Developing appropriate analytic-function-based constitutive models for new materials with nonlinear mechanical behavior is demanding. For such kinds of materials, it is more challenging to realize the integrated design from the collection of the material experiment under the classical topology optimization framework based on constitutive models. The present work proposes a mechanistic-based data-driven topology optimization (DDTO) framework for three-dimensional continuum structures under finite deformation. In the DDTO framework, with the help of neural networks and explicit topology optimization method, the optimal design of the three-dimensional continuum structures under finite deformation is implemented only using the uniaxial and equi-biaxial experimental data. Numerical examples illustrate the effectiveness of the data-driven topology optimization approach, which paves the way for the optimal design of continuum structures composed of novel materials without available constitutive relations.

Appendices

Appendix A

$$r\left(\theta ,\varphi \right)=\Vert {\varvec{S}}\left(u\left(\theta \right), v\left(\varphi \right)\right)\Vert =\Vert \sum_{k=0}^{K}\sum_{l=0}^{L}{N}_{k,p}\left(u\left(\theta \right)\right){N}_{l,q}\left(v\left(\varphi \right)\right){{\varvec{P}}}^{k,l}\Vert ,$$

(A.1)

where ${N}_{k,p}\left(u\right)$,${N}_{l,q}\left(v\right)$ are the B-spline basis functions of $p$ th and $q$ th orders, $K=14$ and $L=6$ are the total number of control points in the $\theta$ and $\varphi$ directions, respectively. The symbols ${{\varvec{P}}}^{k,l}={\left({P}_{x}^{k,l},{P}_{y}^{k,l},{P}_{z}^{k,l}\right)}^{\top },\boldsymbol{ }k=0,\dots ,K, l=0,\dots ,L$ are the coordinates of the corresponding control points. In order to avoid self-intersection of the surfaces, the control points are constructed as follows:

$${P}_{x}^{k,l}={r}^{k,l}\mathrm{sin}\left({\varphi }_{l}\right)\mathrm{cos}\left({\theta }_{k}\right),$$

(A.2a)

$${P}_{y}^{k,l}={r}^{k,l}\mathrm{sin}\left({\varphi }_{l}\right)\mathrm{sin}\left({\theta }_{k}\right),$$

(A.2b)

$${P}_{z}^{k,l}={r}^{k,l}\mathrm{cos}\left({\varphi }_{l}\right),$$

(A.2c)

where ${r}^{k,l},\boldsymbol{ }k=0,\dots ,K, l=0,\dots ,L$ is the distance from the control point to the center of the void, ${\theta }_{k}=\frac{2k\pi }{K}, k=0,\dots ,K, {\varphi }_{l}=\frac{l\pi }{L}, l=0,\dots ,L$. Since the surface is a closed region, it is also required that

$${r}^{0,l}={r}^{K,l}, l=0,\dots ,L,$$

(A.3a)

$${r}^{k,0}={r}^{\mathrm{0,0}}, k=1,\dots ,K,$$

(A.3b)

$${r}^{k,L}={r}^{0,L}, k=1,\dots ,K.$$

(A.3c)

Appendix B

$$\frac{\partial {\chi }^{\mathrm{s}}}{\partial {\varvec{D}}}=\sum_{i=1}^{nv}\frac{\partial {\chi }^{\mathrm{s}}}{\partial {\chi }^{i}\left({{\varvec{D}}}^{i};\,{\varvec{x}}\right)}\frac{\partial {\chi }^{i}\left({{\varvec{D}}}^{i};\,{\varvec{x}}\right)}{\partial {\varvec{D}}}=\sum_{i=1}^{nv}\frac{{e}^{{-\lambda \chi }^{i}\left({{\varvec{D}}}^{i};\,{\varvec{x}}\right)}}{\sum_{i=1}^{nv}{e}^{{-\lambda \chi }^{i}\left({{\varvec{D}}}^{i};{\varvec{x}}\right)}}\frac{\partial {\chi }^{i}\left({{\varvec{D}}}^{i};\,{\varvec{x}}\right)}{\partial {\varvec{D}}},$$

(B.1)

In Eq. (2.11), the derivative of ${\chi }^{i}\left({{\varvec{D}}}^{i};{\varvec{x}}\right)$ with respect to each design variable can be calculated as follows:

$$\frac{\partial {\chi }^{i}\left({{\varvec{D}}}^{i};{\varvec{x}}\right)}{\partial {C}_{x}^{i}}=\frac{{C}_{x}^{i}}{\sqrt{{\left(x-{C}_{x}^{i}\right)}^{2}+{\left(y-{C}_{y}^{i}\right)}^{2}+{\left(z-{C}_{z}^{i}\right)}^{2}}} ,$$

(B.2)

$$\frac{\partial {\chi }^{i}\left({{\varvec{D}}}^{i};{\varvec{x}}\right)}{\partial {C}_{y}^{i}}=\frac{{C}_{y}^{i}}{\sqrt{{\left(x-{C}_{x}^{i}\right)}^{2}+{\left(y-{C}_{y}^{i}\right)}^{2}+{\left(z-{C}_{z}^{i}\right)}^{2}}} ,$$

(B.3)

$$\frac{\partial {\chi }^{i}\left({{\varvec{D}}}^{i};{\varvec{x}}\right)}{\partial {C}_{z}^{i}}=\frac{{C}_{z}^{i}}{\sqrt{{\left(x-{C}_{x}^{i}\right)}^{2}+{\left(y-{C}_{y}^{i}\right)}^{2}+{\left(z-{C}_{z}^{i}\right)}^{2}}} ,$$

(B.4)

$$\frac{\partial {\chi }^{i}\left({{\varvec{D}}}^{i};{\varvec{x}}\right)}{\partial {r}^{k,l}}=\frac{1}{r\left(\theta ,\varphi \right)}\left[{S}_{x}\left(\theta ,\varphi \right)\frac{\partial {S}_{x}\left(\theta ,\varphi \right)}{\partial {r}^{k,l}}+{S}_{y}\left(\theta ,\varphi \right)\frac{\partial {S}_{y}\left(\theta ,\varphi \right)}{\partial {r}^{k,l}}+{S}_{z}\left(\theta ,\varphi \right)\frac{\partial {S}_{z}\left(\theta ,\varphi \right)}{{\partial r}^{k,l}}\right],$$

$$k=0,\dots ,K, l=0,\dots ,L,$$

(B.5)

where

$$\frac{\partial {S}_{x}\left(\theta ,\varphi \right)}{{\partial r}^{k,l}}=\sum_{n=1}^{K}\sum_{m=1}^{L}{N}_{n,p}\left(u\left(\theta \right)\right){N}_{m,q}\left(v\left(\varphi \right)\right)\frac{\partial {P}_{x}^{n,m}}{{\partial r}^{k,l}}, n=0,\dots ,K, m=0,\dots ,L, (\mathrm{B}.6)$$

$$\frac{\partial {S}_{y}\left(\theta ,\varphi \right)}{{\partial r}^{k,l}}=\sum_{n=1}^{K}\sum_{m=1}^{L}{N}_{n,p}\left(u\left(\theta \right)\right){N}_{m,q}\left(v\left(\varphi \right)\right)\frac{\partial {P}_{y}^{n,m}}{{\partial r}^{k,l}}, n=0,\dots ,K, m=0,\dots ,L, (\mathrm{B}.7)$$

$$\frac{\partial {S}_{z}\left(\theta ,\varphi \right)}{{\partial r}^{k,l}}=\sum_{n=1}^{K}\sum_{m=1}^{L}{N}_{n,p}\left(u\left(\theta \right)\right){N}_{m,q}\left(v\left(\varphi \right)\right)\frac{\partial {P}_{z}^{n,m}}{{\partial r}^{k,l}}, n=0,\dots ,K, m=0,\dots ,L. (\mathrm{B}.8)$$

If $n=k$ and $m=l$, $\partial {P}_{x}^{k,l}/\partial {r}^{k,l}$, $\partial {P}_{y}^{k,l}/\partial {r}^{k,l}$, $\partial {P}_{z}^{k,l}/\partial {r}^{k,l}$ are be expressed as

$$\frac{\partial {P}_{x}^{k,l}}{\partial {r}^{k,l}}=\mathrm{sin}\left({\varphi }_{l}\right)\mathrm{cos}\left({\theta }_{k}\right), \frac{\partial {P}_{y}^{k,l}}{\partial {r}^{k,l}}=\mathrm{sin}\left({\varphi }_{k}\right)\mathrm{sin}\left({\theta }_{l}\right), \frac{\partial {P}_{z}^{k,l}}{\partial {r}^{k,l}}=\mathrm{cos}\left({\varphi }_{l}\right), (\mathrm{B}.9)$$

otherwise, $\partial {P}_{x}^{n,m}/\partial {r}^{k,l}=0$, $\partial {P}_{y}^{n,m}/\partial {r}^{k,l}=0$, $\partial {P}_{z}^{n,m}/\partial {r}^{k,l}=0$.

Appendix C

See Tables 7 and 8.

Table 7 The mean square errors of ANN and polynomials in fitting different numbers of ideal data of the uniaxial tension experiment

Table 8 The mean square errors of ANN and polynomials in fitting different numbers of noisy data of the uniaxial tension experiment

As shown in the above two tables, the ANN performs better on both ideal and noisy data sets, and ANN is adopted as the fitting tool in this work.