A sequential linear programming (SLP) approach for uncertainty analysis-based data-driven computational mechanics

Abstract

In this article, an efficient sequential linear programming algorithm (SLP) for uncertainty analysis-based data-driven computational mechanics (UA-DDCM) is presented. By assuming that the uncertain constitutive relationship embedded behind the prescribed data set can be characterized through a convex combination of the local data points, the upper and lower bounds of structural responses pertaining to the given data set, which are more valuable for making decisions in engineering design, can be found by solving a sequential of linear programming problems very efficiently. Numerical examples demonstrate the effectiveness of the proposed approach on sparse data set and its robustness with respect to the existence of noise and outliers in the data set.

Appendix: The SLP-UADDCM algorithm for 3D elastic continuum

As illustrated in Table 11, the SLP-UADDCM algorithm for 3D elastic continuum is similar to its 1D counterpart in Table 1. It should be pointed out that, here, all the data points in $\mathcal{D}$ should be sorted according to the algebraic values of the inner product of $\langle {{\varvec{\varepsilon}}}_{j}^{\mathrm{d}},{\left(\mathrm{1,1},\mathrm{1,1},\mathrm{1,1}\right)}^{\boldsymbol{\top }}\rangle , j=1,\dots ,{N}_{\mathrm{d}}$. Moreover, in 3D cases, the third step of updating the data points for local convex hull construction is different from Table 1, which will be explained in detail as follows. Specifically, in order to enhance the feasibility of ${\mathcal{P}}^{(k)}$ in Eq. (6), the local convex hull in the next step should be able to contain the current stress and strain state as much as possible. However, only selecting a number of data points closest to the current state in the data set will easily cause that the current state is not covered by the local convex hull [3]. An effective treatment for this issue is to cover all possible directions in phase space when local convex hulls are constructed. To this end, we first determine a regular simplex [39] whose vertices are calculated as:

$$\left\{\begin{array}{c}{{\varvec{\varepsilon}}}_{1,e}^{\mathrm{s}}={{\varvec{\varepsilon}}}_{e}^{(k)}-{\left({\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1}\right)}^{\mathbf{\top }},\\ {{\varvec{\varepsilon}}}_{2,e}^{\mathrm{s}}={{\varvec{\varepsilon}}}_{1,e}^{\mathrm{s}}+{\left({p}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1}\right)}^{\mathbf{\top }},\\ {{\varvec{\varepsilon}}}_{3,e}^{\mathrm{s}}={{\varvec{\varepsilon}}}_{1,e}^{\mathrm{s}}+{\left({q}_{k+1},{p}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1}\right)}^{\mathbf{\top }},\\ \dots \\ {{\varvec{\varepsilon}}}_{7,e}^{\mathrm{s}}={{\varvec{\varepsilon}}}_{1,e}^{\mathrm{s}}+{\left({q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{p}_{k+1}\right)}^{\mathbf{\top }},\end{array}\right.$$

(10)

$$\left\{\begin{array}{c}{{\varvec{\sigma}}}_{1,e}^{\mathrm{s}}={{\varvec{\sigma}}}_{e}^{(k)}-{{\varvec{\mathcal{D}}}\left({\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1},{\gamma }_{k+1}\right)}^{\mathbf{\top }},\\ {{\varvec{\sigma}}}_{2,e}^{\mathrm{s}}={{\varvec{\sigma}}}_{1,e}^{\mathrm{s}}+{{\varvec{\mathcal{D}}}\left({p}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1}\right)}^{\mathbf{\top }},\\ {{\varvec{\sigma}}}_{3,e}^{\mathrm{s}}={{\varvec{\sigma}}}_{1,e}^{\mathrm{s}}+{{\varvec{\mathcal{D}}}\left({q}_{k+1},{p}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1}\right)}^{\mathbf{\top }},\\ \dots \\ {{\varvec{\sigma}}}_{7,e}^{\mathrm{s}}={{\varvec{\sigma}}}_{1,e}^{\mathrm{s}}+{{\varvec{\mathcal{D}}}\left({q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{q}_{k+1},{p}_{k+1}\right)}^{\mathbf{\top }},\end{array}\right.$$

(11)

where $\mathcal{D}$ is the scaling matrix,² and

$${p}_{k+1}=\frac{{L}^{(k+1)}}{6\sqrt{2}}\left(5+\sqrt{7}\right), {q}_{k+1}=\frac{{L}^{(k+1)}}{6\sqrt{2}}\left(\sqrt{7}-1\right), {\gamma }_{k+1}=\frac{1}{7}\left(5{q}_{k+1}+{p}_{k+1}\right),$$

(12)

respectively. In Eq. (10–12), we choose ${N}_{\mathrm{c}}=7$ and the value of ${L}^{(k+1)}$ represents the Euclidean distance between the regular simplex vertices. Since the vertices $\left({\left({{\varvec{\varepsilon}}}_{j,e}^{\mathrm{s}}\right)}^{\left(k+1\right)}, {\left({{\varvec{\sigma}}}_{j,e}^{\mathrm{s}}\right)}^{\left(k+1\right)}\right), j=1,\dots ,{N}_{\mathrm{c}}; e=1,\dots ,m$ may not coincide with data points, the data pairs $\left({\left({{\varvec{\varepsilon}}}_{j,e}^{\mathrm{d}}\right)}^{(k+1)}, {\left({{\varvec{\sigma}}}_{j,e}^{\mathrm{d}}\right)}^{(k+1)}\right), j=1,\dots ,{N}_{\mathrm{c}}; e=1,\dots ,m$ used to construct the local convex hulls in $(k+1)$-th iteration are determined as the data points closest to the vertices of the regular simplex in the data set $\mathcal{D}$ respectively, i.e.,

$$\Vert \left({\left({{\varvec{\varepsilon}}}_{j,e}^{\mathrm{d}}\right)}^{(k+1)}, {\left({{\varvec{\sigma}}}_{j,e}^{\mathrm{d}}\right)}^{(k+1)}\right)-\left({\left({{\varvec{\varepsilon}}}_{j,e}^{\mathrm{s}}\right)}^{\left(k+1\right)}, {\left({{\varvec{\sigma}}}_{j,e}^{\mathrm{s}}\right)}^{\left(k+1\right)}\right)\Vert =\underset{l=1,\dots ,{N}_{\mathrm{d}}}{\mathrm{min}}\Vert \left({{\varvec{\varepsilon}}}_{l}^{\mathrm{d}}{,\boldsymbol{ }\boldsymbol{ }\boldsymbol{ }{\varvec{\sigma}}}_{l}^{\mathrm{d}}\right)-\left({\left({{\varvec{\varepsilon}}}_{j,e}^{\mathrm{s}}\right)}^{\left(k+1\right)}, {\left({{\varvec{\sigma}}}_{j,e}^{\mathrm{s}}\right)}^{\left(k+1\right)}\right)\Vert , j=1,\dots ,{N}_{\mathrm{c}}; e=1,\dots ,m.$$

(14)

This updating strategy of data points for local convex hull construction is illustrated schematically in Fig. 

Fig. 17

A schematic illustration of the proposed adaptive local convexification scheme (2D case)

17. Similar to the Algorithm 1 in Table 1, if ${\mathcal{P}}^{(k)}$ in Eq. (6) is feasible, ${L}^{(k)}$ is reduced as ${L}^{(k+1)}=\mathrm{max}\left({L}^{(1)}/{\rho }^{k}, {L}_{\mathrm{min}}\right)$ with ${L}^{(1)}$, $\rho$, ${L}_{\mathrm{min}}$ denoting the initial length, scaling factor and the low bound of ${L}^{(k+1)}$. Otherwise, let ${L}^{(k+1)}$=${L}^{(k)}+0.1{L}^{(k)}$. In addition, the quantity ${\lambda }_{ej}, e=1,\dots ,m; j=1,\dots ,{N}_{\mathrm{c}}$ can also be relaxed to increase the feasible region of ${\mathcal{P}}^{(k+1)}$.