Multi-scale approach for reliability-based design optimization with metamodel upscaling

Abstract

For multi-scale materials, the interplay of material and design uncertainties and reliability-based design optimization is complex and very dependent on the chosen modeling scale. Uncertainty quantification and management are often introduced at lower scales of the material, while a more macroscopic scale is the preferred design space at which optimization is performed. How the coupling between the different scales is handled strongly affects the efficiency of the overall model and optimization. This work proposes a new iterative methodology that combines a low-dimensional macroscopic design space with gradient information to perform accurate optimization and a high-dimensional lower-scale space where design variables uncertainties are modeled and upscaled. An inverse problem is solved at each iteration of the optimization process to identify the lower-scale configuration that meets the macroscopic properties in terms of some statistical description. This is only achievable thanks to efficient metamodel upscaling. The proposed approach is tested on the optimization of a composite plate subjected to buckling with uncertain ply angles. A particular orthonormal basis is constructed with Fourier chaos expansion for the metamodel upscaling, which provides a very efficient closed-form expression of the lamination parameters statistics. The results demonstrate a drastic improvement in the reliability compared to the deterministic optimized design and a significant computational gain compared to the approach of directly optimizing ply angles via a genetic algorithm.

Appendix 1: Trigonometric polynomials: FCE construction

1.1 Useful trigonometric formula

For the sake of simplicity, we will next consider a single random variable X, that is normally distributed, $X\sim f_X = {\mathcal {N}}(\mu ,\sigma ^2)$, the expectation of $\sin (aX)$ and $\cos (aX)$ can be explicitly computed:

$$\begin{aligned}&{\mathbb{E}}\left[ \cos (aX) \right] =\cos ( a\mu ) \times w, \end{aligned}$$

(27)

$$\begin{aligned}&{\mathbb{E}}\left[ \sin (aX) \right] =\sin ( a\mu ) \times w, \end{aligned}$$

(28)

with $a \in {\mathbb{R}}$ and $w = \text{exp}(-0.5a^2\sigma ^2)$.

The product of the trigonometric functions $\cos (kx)$ and $\sin (lx)$ can be expressed as:

$$\begin{aligned} \cos (kx)\sin (lk)=\frac{1}{4}\left( ie^{i(k-l)x}+ie^{-i(k+l)x} \right) \\ \left( -ie^{i(k+l)x}-ie^{i(l-k)x}\right) \end{aligned}$$

(29)

Since only on the real part is of interest, Eq. (29) can be written, and its expected value as Eq. (31) using Eqs. (27, 28).

$$\begin{aligned}\cos (kx)\sin (lx) &=-0.5\ \sin ((k-l)x) \\&\quad +0.5 \ \sin ((k+l)x), \end{aligned}$$

(30)

$$\begin{aligned} {\mathbb{E}}[\cos (kX)\sin (lX)] &= -0.5\ {\mathbb{E}}[\sin ((k-l)X)] \\&\quad +0.5\ {\mathbb{E}}[\sin ((k+l)X)]. \end{aligned}$$

(31)

Similarly:

$$\begin{aligned}{\mathbb{E}}[\cos (kX)\cos (lX)] &= 0.5\ {\mathbb{E}}[\cos ((l-k)X)] \\&\quad + 0.5\ {\mathbb{E}}[\cos ((k+l)X)], \end{aligned}$$

(32)

$$\begin{aligned}&{\mathbb{E}}[\sin (kX)\sin (lX)] = 0.5\ {\mathbb{E}}[\cos ((l-k)X)] \\&- 0.5\ {\mathbb{E}}[\cos ((k+l)X)]. \end{aligned}$$

(33)

Given $\sin ^2(x) = \frac{1-\cos (2x)}{2}$, $\cos ^2(x) = \frac{1+\cos (2x)}{2}$, we can write:

$$\begin{aligned}&{\mathbb{E}} \left[ \sin ^2(aX)\right] = 0.5 - 0.5 \ {\mathbb{E}} \left[ \cos (2aX) \right] , \end{aligned}$$

(34)

$$\begin{aligned}&{\mathbb{E}} \left[ \cos ^2(aX)\right] = 0.5 + 0.5 \ {\mathbb{E}} \left[ \cos (2aX)\right] . \end{aligned}$$

(35)

1.2 Gram-Schmidt algorithm

For the construction of the orthonormal basis, we rely on the Gram-Schmidt algorithm. It calculates the coefficients of the polynomials using the inner product to ensure each polynomial is orthonormal to all of its predecessors:

$$\begin{aligned} \psi _0({\hat{X}})&= 1 \\ \psi _i({\hat{X}})&= u_i({\hat{X}}) - \sum _{k=0}^{i-1} C_{ik} \psi _k({\hat{X}}), \end{aligned}$$

(36)

where ${\hat{X}}=\sigma _{\Theta }X$, $u_i$ are the set of Fourier polynomials ($u_0 = 1$, $u_1 = \sin ({\hat{X}})$, $u_2 = \cos ({\hat{X}})$, $u_3 = \sin (2{\hat{X}})$, $u_4 = \cos (2{\hat{X}})$, $u_5 = \sin (3{\hat{X}})$, $u_6 = \cos (3{\hat{X}}),\cdots$) to be orthogonalized and the coefficients $C_{ik}$ are computed as:

$$\begin{aligned} C_{ik} = \frac{{\mathbb{E}}\left[ u_i({\hat{X}}) \psi _k({\hat{X}}) \right] }{{\mathbb{E}}\left[ \psi _k({\hat{X}}) \psi _k({\hat{X}}) \right] } \end{aligned}$$

(37)

Here an example of the first polynomials construction:

• $\psi _0 = u_0 = 1$,

• $\psi _1$:

  $$\begin{aligned} \psi _1({\hat{X}})&= u_1 - \frac{{\mathbb{E}}\left[ u_1 \psi _0\right] }{{\mathbb{E}}\left[ \psi _0^2 \right] } \psi _0 \\&= \sin ({\hat{X}}) - {\mathbb{E}}\left[ \sin ({\hat{X}}) \right] \end{aligned}$$

  The expected value can be computed with Eq. (28) and is equal to 0. Then:

  $$\begin{aligned} \psi _1&= \sin ({\hat{X}}) \end{aligned}$$

  and can be normalized as:

  $$\begin{aligned} \psi _1^{\text{n}}&= Z_{11}\sin ({\hat{X}}) \\ \text{with } \quad Z_{11}&= \frac{1}{\sqrt{{\mathbb{E}}\left[ \sin ({\hat{X}})^2 \right] }} \end{aligned}$$

  which can be computed with the Eq. (34).

• $\psi _2$:

  $$\begin{aligned} \psi _2&= \cos ({\hat{X}}) - \frac{{\mathbb{E}}\left[ u_2\psi _0\right] }{{\mathbb{E}}\left[ \psi _0^2 \right] } \psi _0 - \frac{{\mathbb{E}}\left[ u_2\psi _1\right] }{{\mathbb{E}}\left[ \psi _1^2 \right] }\psi _1 \\&= \cos ({\hat{X}}) - {\mathbb{E}}\left[ \cos ({\hat{X}})\right] - \frac{{\mathbb{E}}\left[ \cos ({\hat{X}}) \sin ({\hat{X}}) \right] }{{\mathbb{E}}\left[ \sin ({\hat{X}})^2 \right] } \psi _1 \\&= \cos ({\hat{X}}) - C_{20} - C_{21}\psi _1 \end{aligned}$$

  The expected values can be computed with Eq. (27), Eqs. (31) and (28). Here $C_{21}$ is equal to 0, then:

  $$\begin{aligned} \psi _2&= \cos ({\hat{X}}) - C_{20} \end{aligned}$$

  and can be normalized:

  $$\begin{aligned} \psi _2^{\text{n}}= & {} Z_{21}(\cos ({\hat{X}}) - C_{20}) \\ \text{with}\; Z_{21}= & {} \frac{1}{\sqrt{{\mathbb{E}}\left[ (\cos ({\hat{X}}) - C_{20})^2 \right] }} \end{aligned}$$

  which can be computed thanks to Eqs. (35, 27). First orthonormal polynomials are represented in Fig. 12 between $-\pi$ and $\pi$.

Fig. 12

Orthonormal Fourier basis

1.3 Uncertainty quantification of lamination parameters

In this work, accurate statistics of the lamination parameters are important, especially for the inverse problem resolution in Eq. (8). The expected values of the lamination parameters of the bending matrix stiffness are expressed as:

$$\begin{aligned} \mu _{H(\varvec{\Theta })}&= {\mathbb{E}}[\mathbf {v^D}] \\&= \frac{12}{t^3} \sum _k^N y [{\mathbb{E}}\left( \cos (2\Theta _k)\right) , {\mathbb{E}}\left( \sin (2\Theta _k)\right) , \\&\quad {\mathbb{E}}\left( \cos (4\Theta _k)\right) , {\mathbb{E}}\left( \sin (4\Theta _k)\right) ] \end{aligned}$$

(38)

with $y = \frac{(z_k^3-z_{k-1}^3)}{3}$, t the thickness of the laminate, N the total number of plies and $z_k$ is the coordinate of the $k^{th}$ ply. The trigonometric functions can be written as a combination of Fourier polynomial functions:

$$\begin{aligned}&\cos \left( a\left( \Theta _k(X)\right) \right) \approx \sum _{i=0}^p e_{i}^{\text{cos},a}\psi _i(\sigma _{\Theta _k}X), \\&\sin \left( a\left( \Theta _k(X)\right) \right) \approx \sum _{i=0}^p e_{i}^{\text{sin},a}\psi _i(\sigma _{\Theta _k}X), \end{aligned}$$

(39)

with $(p-1)$ the total number of terms in the expansion and where a can take the value of 2 or 4. Using the Fourier chaos expansion, the expected values of lamination parameters can be simply expressed as:

$$\begin{aligned}{} & {} E[\mathbf {v^D}] = \frac{12}{h^3} \sum _k^N y [e_{0c2}^k(\mu _{\Theta _k}), e_{0s2}^k(\mu _{\Theta _k}), \\{} & {} e_{0c4}^k(\mu _{\Theta _k}), e_{0s4}^k(\mu _{\Theta _k})]\ \end{aligned}$$

(40)

with $e_{0ca}$ and $e_{0sa}$ the first coefficient of Eq. (39) who have to be computed. In the same manner, with independent random variables, the variances are expressed as:

$$\begin{aligned}&\text{Var }[\mathbf {v^D}] = \left( \frac{12}{h^3}\right) ^2 \sum _k^N y^2 [\text{Var }\left( \cos (2\Theta _k)\right) , \\&\text{Var }\left( \sin (2\Theta _k)\right) , \text{Var }\left( \cos (4\Theta _k)\right) , \text{Var }\left( \sin (4\Theta _k)\right) ] \\&\text{Var }[\mathbf {v^D}] = \left( \frac{12}{h^3}\right) ^2 \sum _k^N y^2 \left[ \sum _{i=1}^p e_{ic2}^k(\mu _{\Theta _k})^2, \right] \\&\left[ \sum _{i=1}^p e_{is2}^k(\mu _{\Theta _k})^2, \sum _{i=1}^p e_{ic4}^k(\mu _{\Theta _k})^2, \sum _{i=1}^p e_{is4}^k(\mu _{\Theta _k})^2\right] \ \end{aligned}$$

(41)

In the similar manner, the covariance between the lamination parameters can be computed with coefficients products.

1.3.1 FCE coefficients computation

The coefficients in Eq. (39) can be computed analytically with the Fourier basis. An example can be shown using $\cos \left( a\left( \Theta _k(X)\right) \right) =\sum _i^p e_{ica}\psi _i(\sigma _{\Theta _k}X)$ with a taking the value of 2 or 4. We can write the function:

$$\begin{aligned} e&= \cos \left( a(\mu _{\Theta }+\sigma _{\Theta }X)\right) \\&= c_1\cos (a{\hat{X}}) - s_1\sin (a{\hat{X}}) \end{aligned}$$

with $c_a = \cos (a\mu _{\Theta })$ and $s_a = \sin (a\mu _{\Theta })$.

The coefficients $e_{ica}$ are obtained as projections of the functional of interest (e.g. e) onto each member of the Fourier basis. For example, the first two coefficients can be written as:

$$\begin{aligned} e_{0ca}&= {\mathbb{E}}[e\times \phi _0] = {\mathbb{E}}[e\times 1] \\&= c_a{\mathbb{E}}\left[ \cos (a{\hat{X}})\right] - s_a{\mathbb{E}}\left[ \sin (a{\hat{X}})\right] \end{aligned}$$

(42)

$$\begin{aligned} e_{1ca}&= {\mathbb{E}}[e\times \phi _1] = {\mathbb{E}}[e\times Z_{11}\sin ({\hat{X}})] \\&= Z_{11} \left( c_a {\mathbb{E}}\left[ \cos (a{\hat{X}})\sin ({\hat{X}})\right] \right) \\&\left( -s_a{\mathbb{E}}\left[ \sin (a{\hat{X}})\sin ({\hat{X}}) \right] \right) , \end{aligned}$$

(43)

where the expected values are computed with Eqs. (31, 33, 28, 27) and (28). The coefficients are obtained until the order p, in the same manner, using the equations in Sect. A.1.

The procedure is the same with the function $\sin \left( a\left( \Theta (X)\right) \right) =\sum _i^p e_{isa}\psi _i(X)$. Then a database is created for every orientations $\mu _{\Theta }$ possible (in this case [${-75}^\circ , {-60}^\circ , {-45}^\circ , {-30}^\circ , {-15}^\circ , {0}^\circ , {15}^\circ , {30}^\circ , {45}^\circ , {60}^\circ , {75}^\circ , {90}^\circ$]). Once this database is available the means and covariances of lamination parameters of any stacking sequence are directly obtained from Eqs. (40) and (41).

1.3.2 Validation of the representation

The FCE approach is numerically validated by computing the statistics associated to a simple case of lamination parameters. Increasing the size of the approximation basis, the variances of the bending lamination parameters $\mathbf{v^D}$ are compared to the variances computed with a numerical quadrature applied to the first equation of Eq. (41). For each ply, the variance of the $\cos (2\Theta _k)$ function, for example, is written as

$$\begin{aligned} \text{Var }\left( \cos (2\Theta _k)\right)&= {\mathbb{E}}[\cos (2(\mu _{\Theta } + \sigma _{\Theta }X))^2] \\&\quad -{\mathbb{E}}[\cos (2(\mu _{\Theta } + \sigma _{\Theta }X))] \end{aligned}$$

(44)

and the reference expected values can be computed with a numerical integration tool of SciPy. The metamodel is validated for a stacking sequence of 16 plies ([${45}^\circ , {30}^\circ , {0}^\circ , {-45}^\circ , {90}^\circ$,-30,${-15}^\circ , {15}^\circ$]$_s$) and the relative error is plotted in Fig. 13.

Fig. 13

Relative error of the variances obtained from Fourier chaos expansion of $\mathbf {v^D}$ lamination parameters of a stacking sequence

We notice, as expected in this case, the spectral convergence of the error to very small values for 4-term FCE.