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A generalized Bayesian regularization network approach on characterization of geometric defects in lattice structures for topology optimization in preliminary design of 3D printing

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Abstract

In this work, we developed a Generalized Bayesian Regularization Network (GBRN) approach that can quantitatively identify the defect shapes and locations by mapping the distorted lattice structure to its original designed configuration, making registration between manufactured parts with defects and the perfect design models in the preliminary design stage of 3D printing.. On the one hand, it shows the proposed GBRN method has quantitatively comparable accuracy to the Coherent Point Drift (CPD) method in 2D boundary points registration problems. On the other hand, we have shown that the proposed GBRN method can find the possible geometric defects in the 3D printed lattice structure model and identify inherent defect-prone lattice structure parameters with obvious advantages over those two-dimensional point registration methods, i.e., coherent point drift (CPD) method, in registration of interior points of 3D lattice structures.

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Notes

  1. In designing and 3D printed samples, there are always points mismatch, i.e., points of design models \(\varvec{X}_i, i=1,2,\ldots ,N\) and manufactured points \(\varvec{y}_j, j=1,2,\ldots ,M\) [26].

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Author information

Authors and Affiliations

  1. University of California, 738 Davis Hall, Berkeley, CA, 94720, USA

    Yuxi Xie, Shaofan Li & Chao Wang

  2. ANSYS Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, CA, 94551, USA

    Yuxi Xie, C. T. Wu & Dandan Lyu

  3. Ford Motor Corporation, Detroit, MI, 48226, USA

    Danielle Zeng

Authors
  1. Yuxi Xie
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  2. Shaofan Li
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  3. C. T. Wu
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  4. Dandan Lyu
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  5. Chao Wang
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  6. Danielle Zeng
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Correspondence to Shaofan Li.

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Appendices

Appendix A: Linear system with approximate radial basis function

The variational functional is shown as

$$\begin{aligned} H \left[ \varvec{\varPsi } \right] = \sum _{i=1}^N \left\| \varvec{x}_i - \varvec{\varPsi }\left( \varvec{X}_i \right) \right\| ^2 + \lambda \phi \left( \varvec{\varPsi } \right) \end{aligned}$$
(20)

and the solution to Eq. (20) with approximate radial basis function is

$$\begin{aligned} \varvec{\varPsi }^{*} \left( \varvec{X} \right) = \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{X}-\varvec{t}_{\alpha } \right) \end{aligned}$$
(21)

The Eq. (21) is substituted back Eq. (20), one is to have

$$\begin{aligned} H\left[ \varvec{\varPsi }^{*} \right] = \sum _{i=1}^N \left\| \varvec{x}_i - \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{X}_i-\varvec{t}_{\alpha } \right) \right\| ^2 + \lambda \phi \left( \varvec{\varPsi }^{*} \left( \varvec{X} \right) \right) \end{aligned}$$
(22)

The smoothness constraint (second term) has the formula as

$$\begin{aligned} \phi \left( \varvec{\varPsi }^{*} \left( \varvec{X} \right) \right) = \int _{{\mathbb {R}}^d} \frac{ \left\| \tilde{\varvec{\varPsi }}^{*}\left( \varvec{s} \right) \right\| ^2}{{\tilde{G}}\left( \varvec{s} \right) } d\varvec{s} = \int _{{\mathbb {R}}^d} \frac{ \tilde{\varvec{\varPsi }}^{*}\left( \varvec{s} \right) \tilde{\varvec{\varPsi }}^{*}\left( \varvec{-s} \right) }{{\tilde{G}}\left( \varvec{s} \right) } d\varvec{s} \end{aligned}$$

since \(\tilde{\varvec{\varPsi }}^{*}\left( \varvec{s} \right) = \tilde{\varvec{\varPsi }}^{*}\left( \varvec{-s} \right) \) for \(\varvec{\varPsi }^{*}\) is real.

in which,

$$\begin{aligned} \tilde{\varvec{\varPsi }}^{*} \left( \varvec{s} \right)= & {} \int _{{\mathbb {R}}^d} \varvec{\varPsi }^{*} \left( \varvec{t} \right) e^{-i\varvec{t} \cdot \varvec{s}} d\varvec{t} \\= & {} \frac{\int _{{\mathbb {R}}^d} \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{t}-\varvec{t}_{\alpha } \right) e^{-i \varvec{t}\cdot \varvec{s}} d\varvec{t} \cdot e^{i\varvec{t}_{\alpha }\cdot \varvec{s}} }{e^{i\varvec{t}_{\alpha }\cdot \varvec{s}}} \\= & {} \sum _{\alpha =1}^n \varvec{c}_{\alpha } \int _{{\mathbb {R}}^d} G \left( \varvec{t}-\varvec{t}_{\alpha } \right) e^{-i \left( \varvec{t}-\varvec{t}_{\alpha } \right) \cdot \varvec{s}} d \left( \varvec{t}-\varvec{t}_{\alpha } \right) \cdot e^{-i \varvec{t}_{\alpha }\cdot \varvec{s}} \\= & {} \sum _{\alpha =1}^n \varvec{c}_{\alpha } {\tilde{G}} \left( \varvec{s} \right) e^{-i\varvec{t}_{\alpha }\cdot \varvec{s}}~. \end{aligned}$$

and

$$\begin{aligned} \tilde{\varvec{\varPsi }}^{*} \left( -\varvec{s} \right)= & {} \int _{{\mathbb {R}}^d} \varvec{\varPsi }^{*} \left( \varvec{t} \right) e^{-i\varvec{t} \cdot \left( -\varvec{s} \right) } d\varvec{t} \\= & {} \frac{\int _{{\mathbb {R}}^d} \sum _{\beta =1}^n \varvec{c}_{\beta } G \left( \varvec{t}-\varvec{t}_{\beta } \right) e^{-i\varvec{t} \cdot \left( -\varvec{s} \right) } d\varvec{t} \cdot e^{i\varvec{t}_{\beta }\cdot \left( -\varvec{s} \right) } }{e^{i\varvec{t}_{\beta }\cdot \left( -\varvec{s} \right) }} \\= & {} \sum _{\beta =1}^n \varvec{c}_{\beta } \int _{{\mathbb {R}}^d} G \left( \varvec{t}-\varvec{t}_{\beta } \right) e^{-i\left( \varvec{t}-\varvec{t}_{\beta } \right) \cdot \left( -\varvec{s} \right) } d\left( \varvec{t}-\varvec{t}_{\beta } \right) \cdot e^{i\varvec{t}_{\beta }\cdot \varvec{s}} \\= & {} \sum _{\beta =1}^n \varvec{c}_{\beta } {\tilde{G}} \left( -\varvec{s} \right) e^{i\varvec{t}_{\beta }\cdot \varvec{s}} ~. \end{aligned}$$

Therefore, the regularization term can be written as

$$\begin{aligned}&\int _{{\mathbb {R}}^d} \frac{\left\| \tilde{\varvec{\varPsi }}^{*} \left( \varvec{s} \right) \right\| ^2}{{\tilde{G}}\left( \varvec{s} \right) } d\varvec{s} \nonumber \\&\quad = \int _{{\mathbb {R}}^d} \frac{ \left[ \sum _{\alpha =1}^n \varvec{c}_{\alpha } {\tilde{G}}\left( \varvec{s} \right) e^{-i\varvec{t}_{\alpha }\cdot \varvec{s}} \right] \cdot \left[ \sum _{\beta =1}^n \varvec{c}_{\beta } {\tilde{G}}\left( -\varvec{s} \right) e^{i\varvec{t}_{\beta }\cdot \varvec{s}} \right] }{{\tilde{G}}\left( \varvec{s} \right) } d\varvec{s} \nonumber \\&\quad = \int _{{\mathbb {R}}^d} \frac{\sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \cdot \varvec{c}_{\beta } {\tilde{G}}\left( \varvec{s} \right) {\tilde{G}}\left( -\varvec{s} \right) \cdot e^{i\varvec{s}\left( \varvec{t}_{\beta }-\varvec{t}_{\alpha } \right) } }{{\tilde{G}}\left( \varvec{s} \right) } d\varvec{s} \nonumber \\&\quad = \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \cdot \varvec{c}_{\beta } \int _{{\mathbb {R}}^d} d\varvec{s} \cdot e^{i\varvec{s}\cdot \left( \varvec{t}_{\beta }-\varvec{t}_{\alpha } \right) } {\tilde{G}}\left( -\varvec{s} \right) \nonumber \\&\quad = \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \cdot \varvec{c}_{\beta } G \left( \varvec{t}_{\alpha } - \varvec{t}_{\beta } \right) ~. \end{aligned}$$
(23)

Then, Eq. (22) can be converted as

$$\begin{aligned} H\left[ \varvec{\varPsi }^{*} \right]&= \sum _{i=1}^N \left\| \varvec{x}_i - \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{X}-\varvec{t}_{\alpha } \right) \right\| ^2 \nonumber \\&\quad + \lambda \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha }\cdot \varvec{c}_{\beta } G \left( \varvec{t}_{\alpha }-\varvec{t}_{\beta } \right) \end{aligned}$$
(24)

Then taking derivative of Eq. (24) with respect to \(\varvec{c}_{\gamma }\) yields,

$$\begin{aligned} \frac{\partial H\left[ \varvec{\varPsi }^{*} \right] }{\partial \varvec{c}_{\gamma }}= & {} \sum _{i=1}^N 2 \left[ \varvec{x}_i - \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{X}_i-\varvec{t}_{\alpha } \right) \right] \\&\quad \cdot \bigg [-G \left( \varvec{X}_i -\varvec{t}_{\gamma } \right) \bigg ] + 2 \lambda \sum _{\alpha =1}^n \varvec{c}_{\alpha } \cdot G \left( \varvec{t}_{\alpha }-\varvec{t}_{\beta } \right) \\= & {} -\sum _{i=1}^N \varvec{x}_i \cdot G \left( \varvec{X}_i-\varvec{t}_{\gamma } \right) + \sum _{i=1}^N \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{X}_i-\varvec{t}_{\alpha } \right) \\&\quad \cdot G \left( \varvec{X}_i-\varvec{t}_{\gamma } \right) + \lambda \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{t}_{\alpha }-\varvec{t}_{\gamma } \right) = \varvec{0} \end{aligned}$$

It can be written in a matrix form as follows,

$$\begin{aligned} \varvec{G}^T \varvec{G} \varvec{c} + \lambda \varvec{g} \varvec{c} = \varvec{G}^T \varvec{x} \end{aligned}$$
(25)

in which,

$$\begin{aligned} \varvec{c}= & {} \left[ \varvec{c}_1, \varvec{c}_2, \ldots , \varvec{c}_n \right] ^T, \quad \varvec{x} =\left[ \varvec{x}_1, \varvec{x}_2, \ldots , \varvec{x}_N \right] ^T\\ \varvec{G}_{N\times n}= & {} \begin{bmatrix} G\left( \varvec{X}_1-\varvec{t}_1 \right) &{} G\left( \varvec{X}_1-\varvec{t}_2 \right) &{} \cdots &{} G\left( \varvec{X}_1-\varvec{t}_n \right) \\ G\left( \varvec{X}_2-\varvec{t}_1 \right) &{} G\left( \varvec{X}_2-\varvec{t}_2 \right) &{} \cdots &{} G\left( \varvec{X}_2-\varvec{t}_n \right) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ G\left( \varvec{X}_N-\varvec{t}_1 \right) &{} G\left( \varvec{X}_N-\varvec{t}_2 \right) &{} \cdots &{} G\left( \varvec{X}_N-\varvec{t}_n \right) \end{bmatrix}\\ \varvec{g}_{n \times n}= & {} \begin{bmatrix} G\left( \varvec{t}_1-\varvec{t}_1 \right) &{} G\left( \varvec{t}_1-\varvec{t}_2 \right) &{} \cdots &{} G\left( \varvec{t}_1-\varvec{t}_n \right) \\ G\left( \varvec{t}_2-\varvec{t}_1 \right) &{} G\left( \varvec{t}_2-\varvec{t}_2 \right) &{} \cdots &{} G\left( \varvec{t}_2-\varvec{t}_n \right) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ G\left( \varvec{t}_n-\varvec{t}_1 \right) &{} G\left( \varvec{t}_n-\varvec{t}_2 \right) &{} \cdots &{} G\left( \varvec{t}_n-\varvec{t}_n \right) \end{bmatrix}~. \end{aligned}$$

Appendix B. The derivative of \(H_{{\mathbf {w}}}\left[ \varvec{\varPsi }^{*} \right] \) with \(\varvec{W}\)

The weighted variational functional is given as follows,

$$\begin{aligned} H_{{\mathbf {w}}} \left[ \varvec{\varPsi }^{*} \right] = \sum _{i=1}^N \left\| \sum _{\alpha =1}^n \varvec{c}_{\alpha } G\left( \varvec{W}\varvec{X}-\varvec{W}\varvec{t}_{\alpha } \right) - \varvec{x}_i \right\| ^2 + \lambda \phi \left[ \varvec{\varPsi }^{*} \left( \varvec{W}\varvec{X} \right) \right] \end{aligned}$$
(26)

The regularization term can be converted as

$$\begin{aligned}&\phi \left[ \varvec{\varPhi }^{*} \left( \varvec{W}\varvec{X} \right) \right] = \int _{{\mathbb {R}}^d} \frac{\left\| \tilde{\varvec{\varPhi }}^{*} \left( \varvec{W}\varvec{s} \right) \right\| ^2}{{\tilde{G}}\left( \varvec{W}\varvec{s} \right) } d\varvec{W}\varvec{s} \\&\quad = \int _{{\mathbb {R}}^d} \frac{ \tilde{\varvec{\varPhi }}^{*} \left( \varvec{W}\varvec{s} \right) \cdot \tilde{\varvec{\varPhi }}^{*} \left( -\varvec{W}\varvec{s} \right) }{{\tilde{G}}\left( \varvec{W}\varvec{s} \right) } d\varvec{W}\varvec{s} \end{aligned}$$

in which,

$$\begin{aligned}&\tilde{\varvec{\varPhi }}^{*} \left( \varvec{W}\varvec{s} \right) = \int _{{\mathbb {R}}^d} \varvec{\varPhi }^{*} \left( \varvec{W}\varvec{t} \right) e^{-i\left( \varvec{W}\varvec{t} \right) \cdot \left( \varvec{W}\varvec{s} \right) } d\varvec{W}\varvec{t} \\&\quad = \frac{ \int _{{\mathbb {R}}^d} \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{W}\varvec{t}-\varvec{W}\varvec{t}_{\alpha } \right) e^{-i\left( \varvec{W}\varvec{t} \right) \cdot \left( \varvec{W}\varvec{s} \right) } d\varvec{W}\varvec{t} \cdot e^{i\left( \varvec{W}\varvec{t}_{\alpha } \right) \cdot \left( \varvec{W}\varvec{s} \right) } }{e^{i\left( \varvec{W}\varvec{t}_{\alpha } \right) \cdot \left( \varvec{W}\varvec{s} \right) }} \\&\quad = \sum \nolimits _{\alpha =1}^n \varvec{c}_{\alpha } \int _{{\mathbb {R}}^d} G\left( \varvec{W}\varvec{t}-\varvec{W}\varvec{t}_{\alpha } \right) e^{-i \left( \varvec{W}\varvec{t}-\varvec{W}\varvec{t}_{\alpha } \right) \cdot \varvec{W}\varvec{s}} d\left( \varvec{W}\varvec{t}-\varvec{W}\varvec{t}_{\alpha } \right) \\&\quad \cdot e^{-i\varvec{W}\varvec{t}_{\alpha }\cdot \varvec{W}\varvec{s}} \\&\quad = \sum _{\alpha =1}^n \varvec{c}_{\alpha } {\tilde{G}} \left( \varvec{W}\varvec{s} \right) e^{-i\varvec{W}\varvec{t}_{\alpha } \cdot \varvec{W}\varvec{s}} \end{aligned}$$

and

$$\begin{aligned}&\tilde{\varvec{\varPhi }}^{*} \left( -\varvec{W}\varvec{s} \right) = \int _{{\mathbb {R}}^d} \varvec{\varPhi }^{*} \left( \varvec{W}\varvec{t} \right) e^{-i\left( \varvec{W}\varvec{t} \right) \cdot \left( -\varvec{W}\varvec{s} \right) } d\left( \varvec{W}\varvec{t} \right) \\&\quad = \frac{ \int _{{\mathbb {R}}^d} \sum _{\beta =1}^n \varvec{c}_{\beta } G\left( \varvec{W}\varvec{t}-\varvec{W}\varvec{t}_{\beta } \right) e^{-i \varvec{W}\varvec{t}\cdot \left( -\varvec{W}\varvec{s} \right) }d\varvec{W}\varvec{t} \cdot e^{i\varvec{W}\varvec{t}_{\beta }\cdot \left( -\varvec{W}\varvec{s} \right) } }{e^{i\varvec{W}\varvec{t}_{\beta }\cdot \left( -\varvec{W}\varvec{s} \right) }} \\&\quad = \sum \nolimits _{\beta =1}^n \varvec{c}_{\beta } \int _{{\mathbb {R}}^d} G \left( \varvec{W}\varvec{t}-\varvec{W}\varvec{t}_{\beta } \right) e^{-i\varvec{W}\left( \varvec{t}-\varvec{t}_{\beta } \right) \cdot \left( -\varvec{W}\varvec{s} \right) } d\varvec{W}\left( \varvec{t}-\varvec{t}_{\beta } \right) \\&\qquad \cdot e^{i\left( \varvec{W}\varvec{t}_{\beta } \right) \cdot \left( \varvec{W}\varvec{s} \right) } \\&\quad = \sum \nolimits _{\beta =1}^n \varvec{c}_{\beta } {\tilde{G}}\left( -\varvec{W}\varvec{s} \right) e^{i\varvec{W}\varvec{t}_{\beta }\cdot \varvec{W}\varvec{s}} \end{aligned}$$

Therefore, the weighted smoothness constraint term can be written as

$$\begin{aligned}&\int _{{\mathbb {R}}^d} \frac{\left\| \tilde{\varvec{\varPsi }}^{*} \left( \varvec{s} \right) \right\| ^2}{{\tilde{G}}\left( \varvec{W}\varvec{s} \right) } d\varvec{W}\varvec{s} \nonumber \\&\quad = \int _{{\mathbb {R}}^d} \frac{ \left[ \sum _{\alpha =1}^n \varvec{c}_{\alpha } {\tilde{G}}\left( \varvec{W}\varvec{s} \right) e^{-i\varvec{W}\varvec{t}_{\alpha }\cdot \varvec{W}\varvec{s}} \right] \cdot \left[ \sum _{\beta =1}^n \varvec{c}_{\beta } {\tilde{G}}\left( -\varvec{W}\varvec{s} \right) e^{i\varvec{W}\varvec{t}_{\beta }\cdot \varvec{W}\varvec{s}} \right] }{{\tilde{G}}\left( \varvec{W}\varvec{s} \right) } d\varvec{W}\varvec{s} \nonumber \\&\quad = \int _{{\mathbb {R}}^d} \frac{\sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \cdot \varvec{c}_{\beta } {\tilde{G}} \left( \varvec{W}\varvec{s} \right) {\tilde{G}}\left( -\varvec{W}\varvec{s} \right) e^{i\varvec{W}\varvec{s}\cdot \varvec{W}\left( \varvec{t}_{\beta }-\varvec{t}_{\alpha } \right) } d\varvec{W}\varvec{s} }{ {\tilde{G}}\left( \varvec{W}\varvec{s} \right) } \nonumber \\&\quad = \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha }\cdot \varvec{c}_{\beta } \int _{{\mathbb {R}}^d} d\varvec{W}\varvec{s} \cdot e^{i\varvec{W}\varvec{s}\cdot \varvec{W}\left( \varvec{t}_{\beta }-\varvec{t}_{\alpha } \right) } {\tilde{G}}\left( -\varvec{W}\varvec{s} \right) \nonumber \\&\quad = \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \cdot \varvec{c}_{\beta } G\left( \varvec{W}\varvec{t}_{\alpha }-\varvec{W}\varvec{t}_{\beta } \right) \end{aligned}$$
(27)

Subsequently, Eq. (26) can be converted as

$$\begin{aligned} H_{{\mathbf {w}}} \left[ \varvec{\varPsi }^{*} \right]= & {} \sum _{i=1}^N \left\| \varvec{x}_i - \sum _{\alpha =1}^n \varvec{c}_{\alpha } G\left( \varvec{W}\varvec{X}-\varvec{W}\varvec{t}_{\alpha } \right) \right\| ^2 \nonumber \\&\quad + \lambda \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \cdot \varvec{c}_{\beta } G\left( \varvec{W}\varvec{t}_{\alpha }-\varvec{W}\varvec{t}_{\beta } \right) \end{aligned}$$
(28)

Then taking derivative of Eq. (28) with respect to \(\varvec{W}\) yields,

$$\begin{aligned} \frac{\partial H_{{\mathbf {w}}}\left[ \varvec{\varPsi }^{*} \right] }{\partial \varvec{W}}= & {} \sum _{i=1}^N 2 \left[ \varvec{x}_i - \sum _{\alpha =1}^n \varvec{c}_{\alpha } G\left( \varvec{W}\varvec{X}-\varvec{W}\varvec{t}_{\alpha } \right) \right] \nonumber \\&\cdot \left[ -\sum _{\alpha =1}^n \varvec{c}_{\alpha } G^{\prime }\left( \varvec{W}\varvec{X}-\varvec{W}\varvec{t}_{\alpha } \right) \right. \nonumber \\&\quad \left. \cdot \left( \varvec{W}\varvec{X}-\varvec{W}\varvec{t}_{\alpha } \right) \cdot \left( \varvec{X}-\varvec{t}_{\alpha } \right) \right] \nonumber \\&+ \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \cdot \varvec{c}_{\beta } G^{\prime } \left( \varvec{W}\varvec{t}_{\alpha }-\varvec{W}\varvec{t}_{\beta } \right) \nonumber \\&\quad \cdot \varvec{W}\left( \varvec{t}_{\alpha }-\varvec{t}_{\beta } \right) \cdot \left( \varvec{t}_{\alpha }-\varvec{t}_{\beta } \right) = \varvec{0} \end{aligned}$$
(29)

So that we can have

$$\begin{aligned} \frac{\partial H_{{\mathbf {w}}} \left[ \varvec{\varPsi }^{*} \right] }{\partial \varvec{W}} =&\sum _{i=1}^N 2 \left[ \varvec{x}_i - \sum _{\alpha =1}^n \varvec{c}_{\alpha } G \left( \varvec{W}\varvec{X}_i - \varvec{W}\varvec{t}_{\alpha } \right) \right] \nonumber \\&\cdot \left[ - \sum _{\alpha =1}^n \varvec{c}_{\alpha } G^{\prime }\left( \varvec{W}\varvec{X}_i-\varvec{W}\varvec{t}_{\alpha } \right) \right. \nonumber \\&\quad \left. \cdot \varvec{W}:\left( \varvec{X}_i-\varvec{t}_{\alpha } \right) \otimes \left( \varvec{X}_i-\varvec{t}_{\alpha } \right) \right] \nonumber \\&+ \sum _{\alpha ,\beta =1}^n \varvec{c}_{\alpha } \varvec{c}_{\beta } G^{\prime } \left( \varvec{W}\varvec{t}_{\alpha }-\varvec{W}\varvec{t}_{\beta } \right) \nonumber \\&\quad \cdot \varvec{W}:\left( \varvec{t}_{\alpha }-\varvec{t}_{\beta } \right) \otimes \left( \varvec{t}_{\alpha }-\varvec{t}_{\beta } \right) = \varvec{0} \end{aligned}$$
(30)

Appendix C. Derivative of \({\tilde{Q}}\) with respect to \(\varvec{C}\)

First, the derivative of \({\tilde{Q}}\) with respect to \(\varvec{C}\) can be shown as,

$$\begin{aligned} \frac{\partial {\tilde{Q}}}{\partial \varvec{C}}= & {} \frac{\partial }{\partial \varvec{C}} \sum _{i=1}^N \sum _{j=1}^M \phi _j \left( \varvec{y}_{old} | \varvec{X}_i \right) \\&\quad \frac{\left\| \varvec{X}_i - \sum _{\alpha =1}^q \varvec{c}_{\alpha } G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0\alpha } \right) \right\| ^2}{2\sigma _j^2} \\&\quad + \lambda \frac{\partial }{\partial \varvec{C}} \mathbf{Tr} \left( \varvec{C}^T \varvec{g} \varvec{C} \right) \end{aligned}$$

For the first term, we shall consider the derivative with one element of matrix \(\varvec{C}\), i.e., \(\varvec{c}_r\),

$$\begin{aligned}&\frac{\partial }{\partial \varvec{c}_r} \sum _{i=1}^N \sum _{j=1}^M \phi _j \left( \varvec{y}_{old}|\varvec{X}_i \right) \frac{\left\| \varvec{X}_i-\sum _{\alpha =1}^q \varvec{c}_{\alpha } G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0\alpha } \right) \right\| ^2}{2\sigma _j^2} \\= & {} \sum _{i=1}^N \sum _{j=1}^M \phi _j \left( \varvec{y}_{old}|\varvec{X}_i \right) \frac{2\left( \varvec{X}_i-\sum _{\alpha =1}^q \varvec{c}_{\alpha } G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0\alpha } \right) \right) }{2\sigma _j^2} \\&\quad \cdot \left( -G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0r} \right) \right) \\= & {} \sum _{i=1}^N \sum _{j=1}^M \phi _j \left( \varvec{y}_{old}|\varvec{X}_i \right) \frac{-\varvec{X}_i+\sum _{\alpha =1}^q \varvec{c}_{\alpha }G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0\alpha } \right) }{\sigma _j^2}\\&\quad \cdot G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0r} \right) \\= & {} \sum _{i=1}^N \sum _{j=1}^M \frac{\phi _j \left( \varvec{y}_{old}|\varvec{X}_i \right) \sum _{\alpha =1}^q \varvec{c}_{\alpha } G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0\alpha } \right) }{\sigma _j^2} \\&\quad G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0r} \right) - \sum _{i=1}^N \sum _{j=1}^M \frac{\phi _j \left( \varvec{y}_{old}|\varvec{X}_i \right) }{\sigma _j^2} \\&\quad \varvec{X}_i G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0r} \right) \\= & {} \frac{1}{\sigma ^2} \sum _{j=1}^M \left( \sum _{i=1}^N \frac{\phi _j \left( \varvec{y}_{old}|\varvec{X}_i \right) }{\left( \sigma _j/\sigma \right) ^2} \right) \sum _{\alpha =1}^q \varvec{c}_{\alpha } G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0\alpha } \right) \\&\quad \cdot G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0r} \right) \\&-\frac{1}{\sigma ^2} \sum _{j=1}^M \frac{\sum _{i=1}^N \phi _j\left( \varvec{y}_{old}|\varvec{X}_i \right) \varvec{X}_i}{\left( \sigma _j/\sigma \right) ^2} G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0r} \right) \\= & {} \frac{1}{\sigma ^2} \sum _{j=1}^M G\left( \hat{\varvec{y}}_{0r}-\varvec{y}_{0j} \right) diag \left( \tilde{\varvec{\varPhi }}\varvec{1} \right) \varvec{G}\left( j,: \right) \varvec{C}\\&\quad - \frac{1}{\sigma ^2} \sum _{j=1}^M G\left( \hat{\varvec{y}}_{0r}-\varvec{y}_{0j} \right) \left( \tilde{\varvec{\varPhi }} \varvec{X} \right) \\= & {} \frac{1}{\sigma ^2} \varvec{G}^T \left( r,: \right) diag\left( \tilde{\varvec{\varPhi }}\varvec{1} \right) \varvec{G}\varvec{C}\\&\quad - \frac{1}{\sigma ^2} \varvec{G}^T\left( r,: \right) \left( \tilde{\varvec{\varPhi }}\varvec{X} \right) \end{aligned}$$

Therefore, the derivative of the first term with respect to \(\varvec{C}\) matrix is,

$$\begin{aligned}&\frac{\partial }{\partial \varvec{C}} \sum _{i=1}^N \sum _{j=1}^M \phi _j \left( \varvec{y}_{old}|\varvec{X}_i \right) \frac{\left\| \varvec{X}_i -\sum _{\alpha =1}^q \varvec{c}_{\alpha } G\left( \varvec{y}_{0j}-\hat{\varvec{y}}_{0\alpha } \right) \right\| ^2}{2\sigma _j^2} \nonumber \\&\frac{1}{\sigma ^2} \left[ \varvec{G}^T diag \left( \tilde{\varvec{\varPhi }}\varvec{1} \right) \varvec{G}\varvec{C} - \varvec{G}^T \left( \tilde{\varvec{\varPhi }}\varvec{X} \right) \right] \end{aligned}$$
(31)

The derivative of second term with respect to \(\varvec{C}\) can be shown as indices form,

$$\begin{aligned}&\frac{\partial }{\partial \varvec{C}} \mathbf{Tr} \left( \varvec{C}^T \varvec{g} \varvec{C} \right) \\&\quad = \frac{\partial }{\partial C_{\gamma k}} \left( C_{\alpha j} g_{\alpha \beta } C_{\beta j} \right) \\&\quad = \delta _{\alpha \gamma } \delta _{jk} g_{\alpha \beta } C_{\beta j} + C_{\alpha j} g_{\alpha \beta } \delta _{\gamma \beta } \delta _{jk}\\&\quad = g_{\gamma \beta } C_{\beta k} + C_{\alpha k} g_{\alpha \gamma } = \varvec{g} \varvec{C} + \varvec{C}^T \varvec{g} = 2\varvec{g} \varvec{C} \end{aligned}$$

Sum the two terms above together to have,

$$\begin{aligned} \frac{{\tilde{Q}}}{\varvec{C}}&= \varvec{G}^T diag \left( \tilde{\varvec{\varPhi }}\varvec{1} \right) \varvec{G}\varvec{C} \\&\quad - \varvec{G}^T \left( \tilde{\varvec{\varPhi }}\varvec{X} \right) + 2\sigma ^2 \lambda \varvec{g} \varvec{C} = \varvec{0} \end{aligned}$$

so that the matrix \(\varvec{C}\) can be obtained

$$\begin{aligned} \varvec{C} = \left[ \varvec{G}^T diag\left( \tilde{\varvec{\varPhi }}\varvec{1} \right) \varvec{G} + 2\sigma ^2 \lambda \varvec{g} \right] ^{-1} \varvec{G}^T \left( \tilde{\varvec{\varPhi }} \varvec{X} \right) \end{aligned}$$
(32)

in which, \(\varvec{G}\) is a \(M\times q\) matrix, \(\tilde{\varvec{\varPhi }}\) is a \(M \times N\) matrix, \(\varvec{1}\) is a \(N \times 1\) vector, \(\varvec{C}\) is a \(q \times d\) matrix, \(\varvec{X}\) is a \(N \times d\) matrix and \(\varvec{g}\) is a \(q \times q\) matrix.

Appendix D. The brief outline to Coherent Point Drift algorithm

First of all, for convenience of the presentation, we introduce the following notations:

  • N - the number of target point sets

  • M - the number of source point sets

  • D - the dimension of the space where a point exists

  • \({\mathbf {X}} = \left( x_1^1, x_1^2, \cdots , x_1^D, \cdots , x_N^1, x_N^2, \cdots , x_N^D \right) ^T\) - target point set

  • \({\mathbf {Y}} = \left( y_1^1, y_1^2, \cdots , y_1^D, \cdots , y_M^1, y_M^2, \cdots , y_M^D \right) ^T\) - source point set

  • \({\mathcal {T}}\) - mapping function transforming the shape represented by \({\mathbf {Y}}\) to the shape represented by \({\mathbf {X}}\)

  • \(\sigma ^2 I_D\) - covariance matrix with D-dimension

Then a component of Gaussian mixture model, as basis of the CPD method, is defined as follows,

$$\begin{aligned} p\left( {\mathbf {x}}_n|{\mathbf {y}}_m, \sigma ^2 \right) = |2\pi \sigma ^2 I_D|^{-1/2} \exp \left\{ -\frac{1}{2\sigma ^2} \left\| {\mathbf {x}}_n - {\mathcal {T}}\left( {\mathbf {y}}_m \right) \right\| ^2 \right\} \end{aligned}$$

in which, \({\mathcal {T}}\left( \varvec{Y} \right) \) is assumed as expectation of the Gaussian mixture model and a target point \({\mathbf {x}}_n\) is the data point produced from the Gaussian mixture model.

Therefore, after considering outlier points distribution and all components, a complete Gaussian mixture model is defined as follows,

$$\begin{aligned} p\left( {\mathbf {x}}_n; \sigma ^2 \right) = wp_{out}\left( {\mathbf {x}}_n \right) + \left( 1-w \right) \sum _{m=1}^M \frac{1}{M} p\left( {\mathbf {x}}_n | {\mathbf {y}}_m, \sigma ^2 \right) \end{aligned}$$
(33)

in which, w is the probability of an outlier appearance.

Then the EM algorithm is applied to find out the local minimum of the negative likelihood of the Guassian mixture model. Two iterative steps are conducted:

  • Computation of the probability of \(p_{mn}\) between \({\mathbf {x}}_n\) and \({\mathbf {y}}_m\)

  • Minimization of an objective function Q in terms of \(p_{mn}\)

The Q-function is defined as

$$\begin{aligned} Q = \frac{{\hat{N}}D}{2} \ln \sigma ^2 + \frac{1}{2\sigma ^2} \sum _{n=1}^N \sum _{m=1}^M p_{mn} \left\| {\mathbf {x}}_n - {\mathcal {T}}\left( {\mathbf {y}}_m \right) \right\| ^2 \end{aligned}$$
(34)

in which \({\hat{N}} = \sum _{n=1}^N \sum _{m=1}^M p_{mn}\) and

$$\begin{aligned} p_{mn} = \frac{\exp \left( -\frac{1}{2\sigma ^2}\left\| {\mathbf {x}}_n - {\mathcal {T}}\left( {\mathbf {y}}_m \right) \right\| ^2 \right) }{\frac{w}{1-w} \frac{M}{N} |2\pi \sigma ^2 I_D|^{1/2} + \sum _{k=1}^K \exp \left( -\frac{1}{2\sigma ^2} \left\| {\mathbf {x}}_n - {\mathbf {y}}_k \right\| ^2 \right) } \end{aligned}$$

The complete derivation and some intermediate formulas can be found in Ref. [32].

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Xie, Y., Li, S., Wu, C.T. et al. A generalized Bayesian regularization network approach on characterization of geometric defects in lattice structures for topology optimization in preliminary design of 3D printing. Comput Mech 69, 1191–1212 (2022). https://doi.org/10.1007/s00466-021-02137-8

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