- Research Article
- Open Access

# On the computation of plate assemblies using realistic 3D joint model: a non-intrusive approach

- Guillaume Guguin
^{1}Email author, - Olivier Allix
^{1}, - Pierre Gosselet
^{1}and - Stéphane Guinard
^{2}

**3**:16

https://doi.org/10.1186/s40323-016-0069-5

© Guguin et al 2016

**Received:**26 August 2015**Accepted:**26 April 2016**Published:**21 May 2016

## Abstract

Most large engineering structures are described as assemblies of plates and shells and they are computed as such using *adhoc* Finite Element packages. In fact their computation in 3D would be much too costly. In this framework, the connections between the parts are often modeled by means of simplified tying models. In order to improve the reliability of such simulations, we propose to apply a non-intrusive technique so as to virtually substitute the simplified connectors by a precise 3D nonlinear model, without modifying the global plate model. Moreover each computation can be conducted on independent optimized software. After a description of the method, examples are used to analyze its performance, and to draw some conclusions on the validity and limitation of both the modeling of junction by rigid connectors and the use of submodeling techniques for the estimation of the carrying capacity of bolted plates.

## Keywords

- Assembly
- Non-intrusive coupling
- Bolt

## Background

The simulation of large structures undergoing complex local nonlinear phenomena is still a major scientific and industrial challenge. One of the main difficulties originates from the difference of length scale between the global response of the structure and the localized phenomena. To address those problems a first type of computational approach is based on homogenization, as FE^{2} [1] but it works well as long as the scales are sufficiently separated. To overcome this limitation concurrent multiscale methods have been developed. They are often based on domain decomposition techniques like FETI [2], FETI-DP [3] or the LATIN multiscale method [4, 5] and its optimization for the multiscale treatment of nonlinear problems [6, 7].

Moreover most of large industrial structures are described as an assembly of plates and shells, whereas local phenomena often require 3D models to be properly analyzed. To deal with such problems, several methods have been applied or developed for the coupling of 2D and 3D models, like the Arlequin method [8, 9], transition elements [10, 11], MPCs approaches [12, 13] or Nitsche’s method [14].

Most of these methods are quite demanding in terms of software development and therefore they are seldom used in industrial packages. To overcome these drawbacks, non-intrusive approaches have recently been proposed [15]. They are nowadays the subject of extensions and developments: thermoelasticity with GFEM/FEM coupling [16], crack propagation in XFEM/FEM coupling [17], stochastic simulations [18] and dynamics [19, 20].

In [21] a non-intrusive coupling between plate and 3D models was proposed in the case of linear behaviors. The present paper concerns the extension of this approach to the simulation of bolted assemblies of plates where bolts are described with full 3D nonlinear models. Such structures are good candidates for the iterative global-local non-intrusive strategy for two main reasons. First, the detailed computation of a tightened bolt with frictional contact on all surfaces is a very hard and time-consuming task to perform using commercial software. Second, the construction of the reference model, which would correspond to the assembly of a plate model and a 3D model for the bolt, would be very complex. The non-intrusive framework provides answers to both these problems. First, it allows the use of dedicated software for the local computations (in our case, COFAST a parallel software based on the LATIN domain decomposition method [22]). Second, it allows the easy coupling of a general Finite Element software (here Code_Aster from EDF) for the plate computation, with COFAST, because the global model is unchanged during the iterative process. These properties were exploited in [23] for the simulation of damage in composite laminates at the meso and the micro scales using dedicated pieces of software.

The non-intrusive framework aims at solving the reference problem iteratively, by solving at each iteration both the global problem with prescribed residual traction at the interface and the local nonlinear problems submitted to prescribed displacement. Several techniques have been proposed to improve the convergence rate of the method by means of acceleration techniques [15, 17, 24] or improved interface conditions [25]. The method has therefore common points with so called nonlinear domain decomposition methods (or nonlinear relocalization techniques) [26, 27] which proved their efficiency and gain in robustness in the case of buckling [28], post-buckling [29] and damage analysis [30, 31]. Other proposals have been made to take into account the fact that the phenomena of interest are localized, aiming at a better representation of the target model [16, 32, 33].

The paper is organized as follows. In “The reference problem” section, a summary of the reference problem corresponding to the coupling of 2D and 3D models according to [21] is presented. The non-intrusive algorithm is presented in “The non-intrusive iterative algorithm” section. In “Analysis of the iterative corrections at the global and local levels” section, the results of the iterative coupling are analyzed in the case of a bolted joint. These results are compared to those corresponding to the plate solution and to the submodeling technique. In “Control of some parameters of the method and acceleration technique” section, the influence of some parameters is assessed regarding the rate of convergence of the iterative process and regarding the accuracy of the coupled model compared to a full 3D solution.

## The reference problem

### The plate model

The global plate model is the assembly of two plates, \(\omega ^{inf}\) and \(\omega ^{sup}\) respectively the lower plate and the top plate, and a rigid connector between them \(\omega ^{conn}\) Fig. 1 (left). The plates are 20 mm thick and 280 mm long. The lower plate is 160 mm wide whereas and the top plate is 80 mm wide. In order to simplify the presentation, the plates are assumed to be made out of homogeneous isotropic linear elastic material with Young modulus \(E =200\) GPa and Poisson ratio \(\nu = 0.3\); the handling of orthotropic composite plates is explained in [21].

### The 3D patch

The local patch is designed to replace the connector. Thus, a full 3D representation of the bolt is used with unilateral frictional contact interfaces between each part of the assembly. The dimensions of the bolt are given in Fig. 4a. The problem being symmetric, only half of the bolt is computed. The material used for the screw and the nut is linear elastic with Young’s modulus \(E=300\) GPa and Poisson’s ratio \(\nu =0.3\). Coulomb’s friction coefficient is equal to 0.3 on all interfaces.

Note that before coupling with the global problem, preload is applied by enforcing relative displacements between the nut and the screw. It is adjusted to match realistic values of tension in the screw (200 MPa).

### Connections between the models

The plates \(+\) connector model describes the entirety of the structure and occupies the (2D \(+\) 1D) domain \(\omega \). The 3D model of the bolt occupies what we call the zone of interest \(\Omega _I\). In the original method [15], the coarse modeling of \(\omega _I=\omega \cap \Omega _I\) was simply replaced by \(\Omega _I\) through iterations; in [21], it was shown that because of the edge effects which affect plate solutions, it is interesting to introduce a zone of transition between the two models.

This idea is sketched in Fig. 5: the 3D model is the only one taken into account in the inner zone of interest \(\Omega _{\widetilde{I}}\subset \subset \Omega _I\), the plate model is the only one taken into account in the outer complement zone \(\omega _{\widetilde{C}}\subset \subset \omega _C\), there exists an overlap \(\Omega _I\cap \omega _C\) also called buffer zone where the two models are equivalent in a certain sense.

*b*is represented on Fig. 6, together with dimension

*L*which characterizes the size of the 3D domain of interest \(\Omega _I\).

*L*corresponds to the size of the part of the 3D domain which is not strictly necessary to represent the bolt correctly but which was inserted as a way to keep the 3D/2D transition away from the zone dominated to by 3D effects.

## The non-intrusive iterative algorithm

The system (6, 7, 8, 9) can be interpreted as finding the traction \(\varvec{\delta }\) to be imposed to the global plate model on the inner interface \(\gamma _C\) in order to generate a reaction \(\lambda ^C\) in balance with the reaction of the inner zone of interest submitted to the recovery of the plate displacement on its boundary \(\Gamma _I\).

- (1)Run a global plate analysis with extra load \(\varvec{\delta }\):$$\begin{aligned} \mathbf {u}^G_n = {\mathbf {K}^G}^{-1}\left( \mathbf {f}_{ext}^G + \varvec{\delta }\right) \end{aligned}$$
- (2)Post-process the reaction on \(\gamma _C\):$$\begin{aligned} \varvec{\lambda }^{C}_n = \left( \mathbf {K}^{C}\mathbf {u}^G_{n|\omega _C} - \mathbf {f}^{C}_{ext}\right) _{|\gamma _C} \end{aligned}$$
- (3)Recover the 3D displacement on \(\Gamma _I\):$$\begin{aligned} \mathbf {u}^{L}_{n|\Gamma ^I} = \mathbf {R}\mathbf {u}^{G}_{n|\Gamma _I} +\mathbf {w}^G(\mathbf {u}^{G})_n \end{aligned}$$
- (4)Solve the local problem with imposed displacement on \(\Gamma _I\):$$\begin{aligned} \left\{ \begin{array}{ll} \mathbf {f}_{int}^L(\mathbf {u}^L_n) +\mathbf {f}_{ext}^L =0 \\ \mathbf {u}^{L}_{n|\Gamma _I} \text { given} \end{array}\right. \end{aligned}$$
- (5)Post-process the local reaction of \(\Gamma _C\):$$\begin{aligned} \varvec{\lambda }^{L}_n = -\left( \mathbf {f}_{int}^L(\mathbf {u}^L_n) +\mathbf {f}_{ext}^L\right) ^{\widetilde{I}}_{|\Gamma _C} \end{aligned}$$
- (6)Compute the residual on \(\gamma _C\):$$\begin{aligned} \mathbf {r}_n=\varvec{\lambda }^{C}_n + \mathbf {R}^{T}\varvec{\lambda }^{L}_n \end{aligned}$$
- (7)If residual is small enough then exit, else update the extra load:and go back to 1.$$\begin{aligned} \varvec{\delta }_{n+1|\gamma _C} = \varvec{\delta }_{n|\gamma _C} - \mathbf {r}_n \end{aligned}$$

*global step*1 and the

*local step*4 can be processed with different software. For the test-case developed here, Code_Aster is used for the global plate problem, and COFAST3D is used for the local 3D contact problem.

Note that running one global analysis (step 1) followed by a local reanalysis with given Dirichlet conditions (steps 3–4) without iterations corresponds to the industrialists’ practice called submodeling (or sometimes structural zoom). Such an approach is “purely descending” in the sense that there is no feedback from the local computation towards the global scale.

## Analysis of the iterative corrections at the global and local levels

*i*is defined as:

### Remark 1

With the proposed technique, for each global increment the first iteration corresponds to a classical submodeling approach: this enables us to easily measure the quality of a that approach, which is a question often raised by engineers. In this application, the level of the error of the submodeling approach is about \(25\,\%\).

### Global effect of the correction

#### Effect of the preload of the bolt

#### Additional global effects of the bolt on the plate solution

After the preload of the bolt, the plate is loaded in tension, in four global time steps. The analysis of the number of global increments is presented in “Control of some parameters of the method and acceleration technique” section. The global solution is modified along the iterations to match with the 3D model of the bolt, as can be seen on Fig. 9 which shows the values of the transverse displacement in the mid-section of the plate for the initial plate solution and the corrected one.

### Local effect of the correction

In this section the solution within the bolt is compared for three different approaches: a reference full 3D simulation, a submodeling approach and the mixed 2D–3D model with \(L=0\) (minimal size of the local 3D model) obtained at the convergence of the iterations (convergence threshold is \(10^{-6}\)).

To conclude, the coupling approach seems reliable contrarily to the submodeling approach for which the level of error is 20–25 % on global quantities and can be much more for local quantities of interest. The lack of conservatism of the submodeling extends similar results obtained in previous studies on localized plasticity or buckling for example.

## Control of some parameters of the method and acceleration technique

### Influence of the number of global time steps

### Influence of the position of the interface between the 2D and the 3D models

The question of the position of the interface raises in fact the question of the validity of the plate theory with respect to the 3D theory and has been largely discussed in [21]. From what is known on the validity of the plate and shell theories, in the case of isotropic materials, one expects the 2D–3D model to be a good approximation of the 3D reference, for an interface situated from the bolt at a distance superior to the thickness of the plate. It is therefore interesting to analyze the influence of the position of the interface and to compare the cases of \(L=0\) mm and \(L=15\) mm (see Fig. 6).

From a global point of view, the final deformed shapes of the corrected solutions in Fig. 15c are very close in the common plate domain. They only slightly differ in the local area of interest which are different for the two models.

From a local point of view, on Fig. 15b, both cases give close results. This is confirmed when analyzing the difference of the shear stress within the bolt \(\Delta \sigma _{xz} = \sigma _{xz}^{conv} - \sigma _{xz}^{0}\), as shown in the Fig. 16. As already analyzed, if for the first two steps, most of the differences are localized around the nut, during the sliding phase the correction mostly concerns the nut itself. Figure 17 shows the evolution of the local tangential jump with respect to the prescribed displacement. This is an interesting quantity in order to observe the initiation of the sliding. The 2D–3D models give close predictions for both values of *L*. They are much more closer to the reference than what is predicted by the simple submodeling.

### Quasi-Newton acceleration

Figure 18 presents convergence plots. The convergence is roughly three times faster with SR1. Moreover the rate of convergence is almost independent from the load step and the position of the interface. This property can be explained by the fact that the corrections induced by the SR1 acceleration technique are adapted to take into account the main differences between the 2D and 3D models.

## Conclusions

In this paper, a non-intrusive coupling between plate models and 3D models [21] has been extended to deal with the precise computations of bolted plates: the plate model with simplified connector was coupled with a full 3D nonlinear model of the bolt. The flexibility of the method was exploited to easily define the coupled model and to use of two dedicated pieces of software: Code_Aster for the plate computations, and COFAST3D for the nonlinear computation of the bolt (including many surfaces of friction).

The proposed technique enabled us to analyze the possibilities and limits of the use of plate connectors and submodeling approach in that case. It appears that, even when the bolt response is globally linear, the 3D effects induced by the bolt largely modify the plate solution itself. This shows that a rigid connector is a poor model to describe the connection between two plates. As expected, when important sliding occurs, such modeling becomes irrelevant. Moreover, the submodeling technique may lead to significant local errors and non-conservative results.

Another important feature for future applications concerning the treatment of multiple bolts in interaction, is that using an interface located at the limit of the bolt leads to acceptable results when compared to the reference solution. Works on that type of problem is in progress.

Another issue concerns the modeling of the bolt itself. In practice, some of the parameters of the bolted assembly are not precisely determined, as the preload of the nut or the friction coefficient. Such types of problems have been analyzed by dedicated techniques for the bolt computation [37, 38], including multiresolution [39]. The use of the proposed non-intrusive techniques allows to extend these studies to the case of 2D–3D structural analyses in a straightforward manner. In addition, the use of model reduction techniques for the global model itself, as proposed in [40] should lead to a very significant reduction of the computational time.

## Declarations

### Author's contributions

OA initiated the method with the help of PG. GG implemented the method and conducted the numerical experiments on the test cases provided by SG. OA and GG drafted the manuscript, PG finalized it. All authors read and approved the final manuscript.

### Acknowledgements

This work was partially funded by the French National Research Agency as part of project ICARE (ANR-12-MONU-0002-04).

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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