Fast assembly of Galerkin matrices for 3D solid laminated composites using finite element and isogeometric discretizations

Abstract

This work presents a novel methodology for speeding up the assembly of stiffness matrices for laminate composite 3D structures in the context of isogeometric and finite element discretizations. By splitting the involved terms into their in-plane and out-of-plane contributions, this method computes the problems’s 3D stiffness matrix as a combination of 2D (in-plane) and 1D (out-of-plane) integrals. Therefore, the assembly’s computational complexity is reduced to the one of a 2D problem. Additionally, the number of 2D integrals to be computed becomes independent of the number of material layers that constitute the laminated composite, it only depends on the number of different materials used (or different orientations of the same anisotropic material). Hence, when a high number of layers is present, the proposed technique reduces by orders of magnitude the computational time required to create the stiffness matrix with standard methods, being the resulting matrices identical up to machine precision. The predicted performance is illustrated through numerical experiments.

Appendix: A fast assembly of stiffness matrices for laminated composites without using Voigt’s notation

Planas et al. introduced in [53] an alternative methodology for the assembly of stiffness matrices in the context of solid mechanics, denoted as “$\varvec{\mathsf {B}}$ free”, that overcomes the use of Voigt’s notation. Based on that approach, we detail in this Appendix the construction of the operators $\varvec{\mathsf {P}}^{{ij_s}}_{l,\alpha \beta }$ defined in (27) avoiding the use of Voigt’s notation, i.e, without building the material matrix $\varvec{\mathsf {D}}$ (11) and the strain-displacement operators ${\hat{\varvec{\mathsf {B}}}}^i$ (24).

Using the bracket operator $\bullet \left\{ \bullet ,\bullet \right\} :\mathbb { R}^{3\times 3\times 3\times 3}\times \mathbb {R}^3\times \mathbb {R}^3\rightarrow \mathbb {R}^{3\times 3}$ defined in [53], the stiffness matrix (9) can be computed as

$$\begin{aligned} \varvec{K}^{ij} = \int _\varOmega \mathbb {C}(\varvec{x})\left\{ \nabla B^i(\varvec{x}), \nabla B^j(\varvec{x})\right\} \text {d}\varvec{x}\,, \end{aligned}$$

(30)

where the contraction $\mathbb {C}(\varvec{x})\left\{ \nabla B^i(\varvec{x}), \nabla B^j(\varvec{x})\right\} \in \mathbb {R}^{3\times 3}$ is such that the vector $\nabla B^i$ is contracted with the second component of $\mathbb {C}$, and $\nabla B^j$ with the fourth one, i.e.:

$$\begin{aligned} \varvec{a}\cdot \left( \mathbb {C}\{\varvec{b},\varvec{d}\}\,\varvec{c}\right) = \left( \varvec{a}\otimes \varvec{b}\right) :\mathbb {C}:\left( \varvec{c}\otimes \varvec{d}\right) \,. \end{aligned}$$

(31)

Thus, being $\{\varvec{e}_1,\varvec{e}_2,\varvec{e}_3\}$ the Cartesian orthonormal basis, the contraction $\mathbb {C}\left\{ \varvec{e}_i,\varvec{e}_j\right\} \in \mathbb {R}^{3\times 3}$ for small strain linear isotropic materials can be computed as [53]:

$$\begin{aligned} \mathbb {C}\left\{ \varvec{e}_i,\varvec{e}_j\right\} = \lambda \,\varvec{e}_i\otimes \varvec{e}_j + \mu \left( \varvec{e}_i\cdot \varvec{e}_j\,\varvec{I}+ \varvec{e}_j\otimes \varvec{e}_i\right) \,, \end{aligned}$$

(32)

where $\lambda$ and $\mu$ are the Lamé coefficients and $\varvec{I}\in \mathbb {R}^{3\times 3}$ is the identity tensor. For orthotropic materials the contraction reads:

(33)

with $\bar{\varvec{a}}_{\beta ,i} = \left( \varvec{a}_\beta \cdot \varvec{e}_i\right) \,\varvec{a}_\beta$, where $\varvec{a}_1$ and $\varvec{a}_2$ are the main orthonormal in-plane material directions, and $\lambda$, $\mu$, $\alpha _k$, with $k=1,\dots ,7$, are the nine material coefficients (see e.g., [60, Section 3.3] for further details).

Let us now write the map gradient ${\hat{D}} \varvec{F}$ as a function of the covariant basis $\{\varvec{g}_1,\varvec{g}_2,\varvec{g}_3\}$:

$$\begin{aligned} {\hat{D}} \varvec{F}(\bar{\varvec{\xi }}) = \sum _{i=1}^{3}\varvec{e}_i\otimes \varvec{g}_i(\bar{\varvec{\xi }})\,, \end{aligned}$$

(34)

where the Assumption 2 was considered. In the same way, its inverse ${\hat{D}} \varvec{F}^{-\top }$ can be expressed as

$$\begin{aligned} {\hat{D}} \varvec{F}^{-\top }(\bar{\varvec{\xi }}) = \sum _{i=1}^{3}\varvec{e}_i\otimes \varvec{g}^i(\bar{\varvec{\xi }})\,, \end{aligned}$$

(35)

where $\{\varvec{g}^1,\varvec{g}^2,\varvec{g}^3\}$ is the contravariant basis, such that $\varvec{g}^i\cdot \varvec{g}_i=1$ and $\varvec{g}^i\cdot \varvec{g}_j=0$ if $i\ne j$, for $i,j=\lbrace 1,2,3\rbrace$.

Pulling-back the computation of $\varvec{K}^{ij}$ to the parametric domain, as in (13), we obtain:

(36)

that can be rewritten as

(37)

where ${\tilde{\mathbb {C}}}$ is the pull-back of ${\hat{\mathbb {C}}}$ with ${\hat{D}}\varvec{F}^{-\top }$ for the second and fourth components.

On the other hand, ${\hat{\nabla }}{\hat{B}}^i$ can be split according to its Cartesian components as (see Assumption 1):

$$\begin{aligned} \begin{aligned}&{\hat{\nabla }}{\hat{B}}^i(\xi ^1,\xi ^2,\xi ^3) = \dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^1}\,\hat{T}^{i_t} (\xi ^3)\,\varvec{e}_1\\&\quad +\, \dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^2}\,\hat{T}^{i_t} (\xi ^3)\,\varvec{e}_2+\hat{S}^{i_s} (\xi ^1,\xi ^2)\,\dfrac{\partial \hat{T}^{i_t} (\xi ^3)}{\partial \xi ^3}\,\varvec{e}_3\,, \end{aligned} \end{aligned}$$

(38)

and the contraction ${\tilde{\mathbb {C}}}(\varvec{\varvec{\xi }})\left\{ {\hat{\nabla }} {\hat{B}}^i(\varvec{\xi }), {\hat{\nabla }} {\hat{B}}^j(\varvec{\xi })\right\} \in \mathbb {R}^{3\times 3}$ reads:

$$\begin{aligned} \begin{aligned}&{\tilde{\mathbb {C}}}(\varvec{\varvec{\xi }})\left\{ {\hat{\nabla }} {\hat{B}}^i(\varvec{\xi }), {\hat{\nabla }} {\hat{B}}^j(\varvec{\xi })\right\} {=} \left[ \tilde{\varvec{C}}_{11}(\varvec{\xi }) \dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^1}\,\dfrac{\partial \hat{S}^{j_s} (\xi ^1,\xi ^2)}{\partial \xi ^1}\right. \\&\quad \left. +\,\tilde{\varvec{C}}_{12}(\varvec{\xi })\dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^1}\,\dfrac{\partial \hat{S}^{j_s} (\xi ^1,\xi ^2)}{\partial \xi ^2}\right. \\&\quad \left. +\,\tilde{\varvec{C}}_{21}(\varvec{\xi })\dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^2}\,\dfrac{\partial \hat{S}^{j_s} (\xi ^1,\xi ^2)}{\partial \xi ^1}\right. \\&\quad \left. +\,\tilde{\varvec{C}}_{22}(\varvec{\xi })\dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^2}\,\dfrac{\partial \hat{S}^{j_s} (\xi ^1,\xi ^2)}{\partial \xi ^2}\right] \hat{T}^{i_t} (\xi ^3)\hat{T}^{j_t} (\xi ^3) \\&\quad +\, \left[ \tilde{\varvec{C}}_{13}(\varvec{\xi }) \dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^1}\right. \left. +\,\tilde{\varvec{C}}_{23}(\varvec{\xi }) \dfrac{\partial \hat{S}^{i_s} (\xi ^1,\xi ^2)}{\partial \xi ^2}\right] \\&\quad \times \hat{S}^{j_s} (\xi ^1,\xi ^2) \hat{T}^{i_t} (\xi ^3)\dfrac{\partial \hat{T}^{j_t} (\xi ^3)}{\partial \xi ^3}\\&\quad +\, \left[ \tilde{\varvec{C}}_{31}(\varvec{\xi }) \dfrac{\partial \hat{S}^{j_s} (\xi ^1,\xi ^2)}{\partial \xi ^1}\right. \left. +\,\tilde{\varvec{C}}_{32}(\varvec{\xi }) \dfrac{\partial \hat{S}^{j_s} (\xi ^1,\xi ^2)}{\partial \xi ^2}\right] \\&\quad \times \hat{S}^{i_s} (\xi ^1,\xi ^2) \dfrac{\partial \hat{T}^{i_t} (\xi ^3)}{\partial \xi ^3}\hat{T}^{j_t} (\xi ^3)\\&\quad +\,\tilde{\varvec{C}}_{33}(\varvec{\xi }) \hat{S}^{i_s} (\xi ^1,\xi ^2)\hat{S}^{j_s} (\xi ^1,\xi ^2) \dfrac{\partial \hat{T}^{i_t} (\xi ^3)}{\partial \xi ^3}\dfrac{\partial \hat{T}^{j_t} (\xi ^3)}{\partial \xi ^3} \end{aligned} \end{aligned}$$

(39)

where $\tilde{\varvec{C}}_{\alpha \beta }\in \mathbb {R}^{3\times 3}$ is

$$\begin{aligned} \tilde{\varvec{C}}_{\alpha \beta }(\varvec{\xi }) = \sum _{i,j=1}^{3} {\hat{\mathbb {C}}}(\varvec{\xi })\left\{ \varvec{e}_i, \varvec{e}_j\right\} \, \left( \varvec{g}^i(\bar{\varvec{\xi }})\cdot \varvec{e}_\alpha \right) \, \left( \varvec{g}^j(\bar{\varvec{\xi }})\cdot \varvec{e}_\beta \right) \,. \end{aligned}$$

(40)

Finally, substituting (39) into (37), and including the Assumptions 2 and 3, the stiffness matrix terms can be rearranged in the same way as we did for (26), and the $\varvec{\mathsf {P}}^{{ij_s}}_{l,\alpha \beta }$ matrices (27) become

(41a)

(41b)

(41c)

(41d)

The terms $\tilde{\varvec{C}}_{\alpha \beta ,l}$ are computed following (40) and particularizing ${\hat{\mathbb {C}}}$ for every material layer l, with $l=1,\dots ,m$. Therefore, the matrices $\varvec{\mathsf {P}}^{{ij_s}}_{l,\alpha \beta }$ can be computed, avoiding the use of Voigt’s notation, by means of in-plane terms only.