Integration of PGD-virtual charts into an engineering design process

Abstract

This article deals with the efficient construction of approximations of fields and quantities of interest used in geometric optimisation of complex shapes that can be encountered in engineering structures. The strategy, which is developed herein, is based on the construction of virtual charts that allow, once computed offline, to optimise the structure for a negligible online CPU cost. These virtual charts can be used as a powerful numerical decision support tool during the design of industrial structures. They are built using the proper generalized decomposition (PGD) that offers a very convenient framework to solve parametrised problems. In this paper, particular attention has been paid to the integration of the procedure into a genuine engineering design process. In particular, a dedicated methodology is proposed to interface the PGD approach with commercial software.

Appendices

Appendix 1

The Proper Generalized Decomposition is illustrated through a linear elastic problem where the constitutive law is parametrised by a set of Young’s moduli $\varvec{\alpha }=\left( \alpha _1,\ldots ,\alpha _m\right) \in \varvec{\mathcal {A}}={\mathcal {A}}_1\times \cdots \times {\mathcal {A}}_m$. The problem is defined by the following governing equations and conditions:

• Equilibrium equations

  $$\begin{aligned} {\left\{ \begin{array}{ll} \varvec{\nabla } \cdot \varvec{\sigma }=\varvec{0} &{}\quad \text { in }\quad {\varOmega } \times \varvec{\mathcal {A}}\\ \varvec{\sigma } \cdot \varvec{n} = \varvec{f} &{} \quad \text { on } \quad \partial _d {\varOmega } \times \varvec{\mathcal {A}} \end{array}\right. } \end{aligned}$$

  (44)

• Constitutive law

  $$\begin{aligned} \varvec{\sigma }= & {} \mathbb {C}\left( \varvec{\alpha }\right) \varvec{\varepsilon } \quad \text { in } \quad {\varOmega } \times \varvec{\mathcal {A}} \end{aligned}$$

  (45)
  $$\begin{aligned}= & {} \mathbb {C}\left( \varvec{\alpha }\right) \varvec{\nabla }_s \left( \varvec{u}\right) \quad \text { in } \quad {\varOmega } \times \varvec{\mathcal {A}} \end{aligned}$$

  (46)

  where $\varvec{\nabla }_s$ stands for the symmetrical part of $\varvec{\nabla }$ and $\mathbb {C}$ is the Hooke’s operator.

• Homogeneous boundary condition

  $$\begin{aligned} \varvec{u}=\varvec{0} \quad \text { on } \quad \partial _u {\varOmega } \times \varvec{\mathcal {A}} \end{aligned}$$

  (47)

One writes the full variational formulation of the problem, i.e. the problem is not only integrated on space but also on parameters. For that purpose, we introduce the following functional spaces $\mathcal {V}=H_0^1\left( {\varOmega }\right) =\left\{ \varvec{v} \in H^1\left( {\varOmega }\right) \left| \varvec{v}_{\left| \partial _u{\varOmega }\right. }=\varvec{0} \right. \right\}$, $\varvec{\mathcal {I}}=L^2\left( \varvec{\mathcal {A}}\right)$ and $\mathcal {I}_j=L^2\left( {\mathcal {A}}_j\right)$ for $j=1,\ldots ,m$. The problem therefore writes:

Find $\varvec{u} \in \varvec{\mathcal {I}}\otimes \mathcal {V}$ such that

$$\begin{aligned} \forall \,\varvec{v} \in \varvec{\mathcal {I}}\otimes \mathcal {V},\quad \int \limits _{\varvec{\mathcal {A}}\,\,\,}\int \limits _{{\varOmega }}{\varvec{\sigma }\left( \varvec{u}\right) :\varvec{\varepsilon }\left( \varvec{v}\right) } \,\mathrm d {\varOmega }\,\mathrm d \varvec{\mathcal {A}}= \int \limits _{\varvec{\mathcal {A}}\,\,\,}\int \limits _{\partial _d {\varOmega }}{\varvec{f} \cdot \varvec{v}}\,\mathrm d S \,\mathrm d \varvec{\mathcal {A}}\nonumber \\ \end{aligned}$$

(48)

where $\otimes$ stands for the tensor product.

One seeks a PGD approximation of the displacement.

$$\begin{aligned} \varvec{u}(\varvec{\alpha }, \varvec{X}) \approx \varvec{u}_n ( \varvec{\alpha }, \varvec{X})= \sum _{i=1}^n \lambda _i(\varvec{\alpha })\varvec{\Lambda }_i(\varvec{X})= \sum _{i=1}^n \prod _{j=1}^m\lambda _i^j(\alpha _j)\varvec{\Lambda }_i(\varvec{X})\nonumber \\ \end{aligned}$$

(49)

The different modes are computed through an iterative algorithm. At the enrichment step n, we suppose the separated variables representation $\varvec{u}_n$ known. The new $(m+1)$-uplet $\left( \lambda ^1,\ldots ,\lambda ^m,\varvec{\Lambda }\right) \in \mathcal {I}_{1}\times \cdots \times \mathcal {I}_{m}\times \mathcal {V}$ is then computed at enrichment step $n+1$:

$$\begin{aligned} \varvec{u}_{n+1}=\varvec{u}_n+\prod _{r=1}^m\lambda ^r\varvec{\Lambda } \end{aligned}$$

(50)

To do so, the following expression of the test function $\varvec{v}$ is chosen:

$$\begin{aligned} \varvec{v}=\prod _{r=1}^m\lambda ^r\varvec{\Lambda }^*+\sum _{j=1}^m\lambda ^{j*}\prod _{\begin{array}{c} r=1\\ r \ne j \end{array}}^m\lambda ^r\varvec{\Lambda } \end{aligned}$$

(51)

with $\varvec{\Lambda }^*\in \mathcal {V}$ and $\lambda ^{j*}=\mathcal {I}_{j}$ for $j=1,\ldots ,m$. (5) becomes:

Find $\left( \lambda ^1,\ldots ,\lambda ^m,\varvec{\Lambda }\right) \in \mathcal {I}_{1}\times \cdots \times \mathcal {I}_{m}\times \mathcal {V}$ such that

$$\begin{aligned}&\forall \left( \lambda ^{1*},\ldots ,\lambda ^{m*},\varvec{\Lambda }^*\right) \in \mathcal {I}_{1}\times \cdots \times \mathcal {I}_{m}\times \mathcal {V},\nonumber \\&\int \limits _{\varvec{\mathcal {A}}\,\,\,}\int \limits _{{\varOmega }}\mathbb {C}\left( \varvec{\alpha }\right) \,\varvec{\nabla }_s \left( \varvec{u}_n+\prod _{r=1}^m\lambda ^r\varvec{\Lambda }\right) :\varvec{\nabla }_s \left( \prod _{r=1}^m\lambda ^r\varvec{\Lambda }^*\right. \nonumber \\&\left. +\sum _{j=1}^m\lambda ^{j*}\prod _{\begin{array}{c} r=1 \\ r \ne j \end{array}}^m\lambda ^r\varvec{\Lambda }\right) \,\mathrm d {\varOmega }\,\mathrm d \varvec{\mathcal {A}}\nonumber \\&\quad =\int \limits _{\varvec{\mathcal {A}}\,\,\,}\int \limits _{\partial _d {\varOmega }}{\varvec{f} \cdot \left( \prod _{r=1}^m\lambda ^r\varvec{\Lambda }^*+\sum _{j=1}^m\lambda ^{j*}\prod _{\begin{array}{c} r=1 \\ r \ne j \end{array}}^m\lambda ^r\varvec{\Lambda }\right) }\,\mathrm d S \,\mathrm d \varvec{\mathcal {A}}\nonumber \\ \end{aligned}$$

(52)

The computations of the different integrals are highly facilitated by the separated variables representation of the integrand since they can be done independently from one another. However, the problem is not linear any more with respect to the test function given in (51). The $(m+1)$-uplet $\left( \lambda ^1,\ldots ,\lambda ^m,\varvec{\Lambda }\right)$ is consequently computed thanks to a fixed-point algorithm. The parametric and space problems are solved alternatively. In practice, the fixed-point algorithm is stopped before reaching convergence (only 2 or 3 iterations are performed). The space basis $\left( \varvec{\Lambda }_i\right)$ is, then, conserved and the functions $\left( \lambda _i^1,\ldots ,\lambda _i^m\right)$ are computed, once again, during the so-called update step [25].

Appendix 2

Fig. 9

The test structure is partitioned into three sub-domains, the Young’s modulus of each domain being considered as a parameter

Fig. 10

First modes associated to each Young’s modulus. Modes associated to the Young’s modulus $E_1$ (a), modes associated to the Young’s modulus $E_2$ (b) and msodes associated to the Young’s modulus $E_3$ (c) considered as parameters

Let us illustrate the procedure described in Appendix 1 through a simple numerical example. The considered geometry is a $\Gamma$-shaped structure (see Fig. 9) which is clamped at one side and subjected to an upward load of 1000 N at the other side. The structure is partitioned into three different sub-domains and the Young’s modulus of each domain, being considered as a parameter, can vary from 50 to 1000 GPa. The Poisson’s ratio is homogeneous and equal to 0.3. A PGD approximation of the generalised displacement field $\varvec{u}$ is sought:

$$\begin{aligned}&\varvec{u}\left( \varvec{\alpha },\varvec{X}\right) =\varvec{u}\left( E_1,E_2,E_3,\varvec{X}\right) \nonumber \\&=\sum _{i=1}^m \lambda _i^1\left( E_1\right) \lambda _i^2\left( E_2\right) \lambda _i^3\left( E_3\right) \tilde{\varvec{u}}_i\left( \varvec{X}\right) \end{aligned}$$

(53)

which gives after space discretisation:

$$\begin{aligned} \mathbf U\left( E_1,E_2,E_3\right) =\sum _{i=1}^m \lambda _i^1\left( E_1\right) \lambda _i^2\left( E_2\right) \lambda _i^3\left( E_3\right) \tilde{\mathbf U}_i \end{aligned}$$

(54)

The modes, i.e. the functions $\lambda _i^1$, $\lambda _i^2$, $\lambda _i^3$ and $\tilde{\mathbf U}_i$ are computed thanks to Algorithm 1. The four first modes associated to the Young’s moduli $E_1$, $E_2$ and $E_3$ are plotted in Fig. 10 and the four first spatial modes are represented in Fig. 11.

Fig. 11

First spatial modes. a First spatial mode $\tilde{\mathbf U}_1$, b second spatial mode $\tilde{\mathbf U}_2$, c third spatial mode $\tilde{\mathbf U}_3$, d fourth spatial mode $\tilde{\mathbf U}_4$