FEA Information Engineering Journal - FEAIEJ · FEA Information Engineering Journal ... David Dubois, Ph. ... presented during last LS-DYNA Conference,

ISSN 2167-1273 Volume 2, Issue12, December 2013

FEA Information Engineering Journal

Optimization


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Cover: Fig. 1: Model of two aircraft fuselage barrels during assembly Stochastic Simulation of Aircraft Fuselage Assembly Considering Manufacturing Uncertainties

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TABLE OF CONTENTS

Volume 2, Issue No. 12 December 2013 Publications are © to LS-DYNA 2013 9th European Users‘ Conference LS-OPT Parameters Identification on Concrete Sample Tests for an Impact Simulation on Concrete Slab

Nicolas VAN DORSSELAER- DynaS+, Ivry-sur-Seine, France Vincent LAPOUJADE - DynaS+, Ivry-sur-Seine, France Georges NAHAS, Bertrand CIREE, François TARALLO, Jean-Mathieu RAMBACH Institut de Radioprotection et de Sûreté Nucléaire, Fontenay-aux-roses, France

Using LS-OPT for Parameter Identification and MAT_FABRIC with FORM=-14 David Dubois, Ph. – Autoliv France Jimmy Forsberg, Ph. D - DYNAmore Nordic AB

Multi-disciplinary Topology Optimization for Vehicle Bonnet Design David Salway - GRM Consulting Ltd Dr Tayeb Zeguer - Jaguar Land Rover Ltd

Stochastic Simulation of Aircraft Fuselage Assembly Considering Manufacturing Uncertainties

Dietmar C. Vogt, Sönke Klostermann EADS Innovation Works, Germany

All contents are copyright © to the publishing company, author or respective company. All rights reserved.

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© 2013 Copyright by Arup

LS-OPT Parameters Identification on Concrete Sample Tests for an Impact Simulation on Concrete

Slab

Nicolas VAN DORSSELAER(1), Vincent LAPOUJADE(1), Georges NAHAS(2), Bertrand CIREE(2), François TARALLO(2), Jean-Mathieu RAMBACH(2)

(1) DynaS+, Ivry-sur-Seine, France (2) Institut de Radioprotection et de Sûreté Nucléaire, Fontenay-aux-roses, France

[email protected]

1 Introduction The dynamic behavior of Concrete is one of the most common and difficult problem of simulation in Nuclear, Defense and Civil fields. In most cases, the data available for modeling problems is much reduced; engineers are obliged to predict the behavior with non sufficient information. Due to this lack of experimental sample based input parameters, the result of simulation becomes “engineer dependent”, leading to much different results than people doing the same modeling problem. In previous paper ([5], [6]) presented during last LS-DYNA Conferences, we showed that a probabilistic approach for concrete modeling can be used to reduce these differences due to the modeling choices. But one of the main conclusions of these papers was that all these modeling techniques never replace experimental concrete sample tests to obtain the right material behavior before simulation.

This paper is based on a work realized for an international OECD benchmark initiated by IRSN and CNSC. The main goal of IRIS_2012 Benchmark was to evaluate the ability of simulation to reproduce experimental tests of impacts on concrete slabs. Contrary to the earlier benchmark (IRIS_2010), experimental results of concrete sample tests was this time available in order to calibrate numerical constitutive laws before simulations on real tests. This paper, as the rest of our previous papers about IRIS_2010, will present the use of LSTC products capabilities in this kind of approach.

In a first time, a complete LS-DYNA concrete model based on compressive strength will be created using automatic parameters generation capabilities of LS-DYNA. Then this model will be compared to experimental sample results of several cylindrical sample tests (simple compression and confined compressions at several confinement pressures). After sensitivity analysis to identify which parameters of the concrete model can be used to fit experimental results, LS-OPT parameters identification will be performed simultaneously on all cases.

Based on the VTT Punching test simulation of IRIS_2012, we will compare the results between simulation with parameters automatically generated, simulation with fitted parameters and experiment. This comparison will be focused on missile velocity after impact and slab concrete damage.

We precise that all the calculations presented here are performed with LS-DYNA solver, coupled with LS-OPT software for the probabilistic part of the studies (DoE studies, Monte Carlo Analysis, Robustness and Optimizations).

9th European LS-DYNA Conference 2013 _________________________________________________________________________________


2 Fitting of Concrete Sample Tests The concrete sample tests used for this study were realized in 2011 by 3S-R Laboratory of Grenoble University. They have been performed on cylindrical samples (7 cm diameter and 14 cm length) using a high pressure testing machine named „Giga“. The following figure shows the experimental sample test.

Fig. 1: Experimental sample test

For IRIS_2012 Benchmark, five curves of stress versus strain were given corresponding to:

- A simple compression test, - Four compression tests with confinement at 15.5 MPa, 26 MPa, 47 MPa and 100 MPa.

LS-DYNA finite elements models have been created for each test case.

Fig. 2: LS-DYNA model for samples

LS-DYNA® software has several advanced constitutive models developed to simulate concrete material behavior, the most usual ones are currently *MAT_PSEUDO_TENSOR (*MAT_16), *MAT_CONCRETE_DAMAGE_Release3 (*MAT_72r3), *MAT_WINFRITH_CONCRETE (*MAT_84) and *MAT_CSCM (*MAT_159). Most of them have automatic generation capability of concrete law parameters. Indeed, LS-DYNA is able to provide, starting from a first set of physical parameters (unconfined compressive strength Fc, unconfined tension strength Ft,…) a second set of parameters by internally fitting experimental reference results.

Starting from the Fc, we chose to use *MAT_72r3 material law with automatic parameter generation as a starting point to simulate sample tests. Then parameters previously generated will be optimized to better fit experimental results.



In a first time, a simulation has been performed using automatic parameter generation using Fc = 70 MPa. The results obtained have been compared to the experimental ones. On the following figure, there is the comparison between simulation and experiment for simple compression test. We can see that *MAT_72r3 automatic parameter generation gives acceptable results, excepted for a small numerical problem at high strain.

Fig. 3: Comparison between simulation and experiment for simple compression test

Regarding to this result, some options are identified to optimize the behavior of this material. In fact, the Young modulus of the curve and the softening are not optimized for this test. Starting from this first set of parameters, a lot of tests and calculations have been performed with direct LS-DYNA simulation and LS-OPT sensitivity analysis in order to identify which parameters can be used in optimization to fit the experimental result. On the following figure, there is an example of LS-OPT sensitivity analysis performed.

Fig. 4: Example of LS-OPT sensitivity analysis

After this sensitivity part, an optimization with LS-OPT has been performed on identified parameters. The following picture shows the result of this optimization.



Fig. 5: Results of LS-OPT Optimization

After this optimization phase, we can show the difference between simulation and experiment for automatic parameter generation and after optimization. On the following curves, “A” curve represents the fitted parameters, “B” curve represents the automatic generation and “C” curve represents the experiment.

Fig. 6: Results for simple compression test

Fig. 7: Results for compression test with 15.5 MPa confinement



Fig. 8: Results for compression test with 26 MPa confinement





3 Results on VTT Punching Test All the results presented before show that the fitting approach has improved the results on sample tests. It is now interesting to see the effect of this parameters fitting on a real test case of IRIS Benchmark: VTT Punching test.

The VTT Punching test is composed on two parts:

A missile with a steel dome and a concrete cylinder with a steel skin, with a total mass (about 50 kg). This missile impacts the slab at 135 m/s.

A concrete slab of 200 x 200 x 25 cm hold by a UPN Steel part, reinforced by a square mesh of longitudinal rebars on each side of the slab.

This test is modeled by a 3D half model; the goal is to use one symmetry plane to limit the number of elements without forcing a distortion mode.

Concrete is modeled by under integrated constant stress solid element (one integration point per volume). Reinforcement is modeled by Hugues-Liu with cross section integration beam elements. The ratio between slab and missile element size guarantees a good behavior during the contact.

The UPN Steel part, surrounding the concrete slab, is explicitly modeled by Belystchko fully integrated shell element and is merged into the concrete part.

The missile for the VTT Punching test is explicitly modeled. Light-weight concrete and steel dome are modeled using under integrated constant stress solid elements (one integration point per volume). Steel pipe and steel plate are modeled with Belytschko fully integrated shell elements merged with the concrete solid.

Fig. 11: View of VTT Punching LS-DYNA model

The constitutive law of steel elements is a *MAT_PIECEWISE_LINEAR_PLASTICITY able to model the behavior of steel with a complex plasticity curve and to include strain rate effects. Engineer values are changed into true values up to striction and then interpolated using a swift law. Without stress-strain curves for different strain rates, a simple way to take into account strain rate effects is to add a Cowper-Symonds law.

Rebars are not merged to the concrete elements; the interaction is modeled by a coupling method based on a constrained approach. Junctions between two longitudinal rebars are merged.

Two types of contact are used to model the interaction between missile and slab:

*CONTACT_ERODING_SINGLE_SURFACE deals with the contact between missile and solid concrete and the auto contact of the missile on itself. This contact is based on penalty method



with a segment based option for contact detection (instead of node based) to avoid penetration.

*CONTACT_ERODING_NODES_TO_SURFACE deals with a possible contact between reinforcement nodes and missile segments (if erosion leads to such a possibility).

Using firstly the automatic generation of parameters, and secondly the optimized ones, we can compare the results with experiment for final missile velocity after penetration and concrete slab damage.

The residual missile velocity in the simulation can be compared to the corresponding experimental value. The following table shows a comparison of these values. We can see that the automatic generation of simulation parameters underestimates the residual velocity. However, with the optimized parameters, the results show a residual speed exactly in the experimental range.

Experimental speed range

Automatic generation parameters Optimized parameters

35-40 m/s 23.2 m/s 36.2 m/s

Fig. 12: Table of speed comparison We can also compare the concrete damage of the reinforced concrete slab for the two calculations. In the following figure, we can see that comparison with a fringe of concrete damage (internal variable law of *MAT_72r3) for both simulations.

Fig. 13: Concrete damage comparison

In the previous figure, we can see that the slab damage with automatic generation is too important on the front of the slab. Indeed, the upper half of the slab is totally damaged, which is not consistent with the experimental results. For the optimized parameters calculation, we notice a more physical damage, with a damage cone located in the impact area and damages near boundary conditions, which is more consistent with the observation of cracks of the experimental results (see following figures).

Fig. 14: Experimental slab damages



4 Conclusion In a first time, a fitting approach on sample compression tests has been performed to optimize the concrete material law using *MAT_72r3 automatic parameter generation capabilities and experimental results. We showed that with LS-OPT DoE studies, sensitivity analysis and optimizations, it is possible to fit experimental stress/strain curves.

Based on the VTT Punching test simulation of IRIS_2012, we have also compared the results between simulation with parameters automatically generated, simulation with fitted parameters and experiment on a real test case. This comparison showed an improvement of quality results for the missile velocity after impact and slab concrete damage.

5 Summary This paper is based on a work realized for an international OECD benchmark initiated by IRSN and CNSC. The main goal of IRIS_2012 Benchmark was to evaluate the ability of simulation to reproduce experimental tests of impacts on concrete slabs. Contrary to the earlier benchmark (IRIS_2010), experimental results of concrete sample tests was this time available in order to calibrate numerical constitutive laws before simulations on real tests. This paper, along with the two corresponding papers related to IRIS_2010 ([5], [6]), present the use of LSTC products capabilities in this kind of approach.

For the first phase of IRIS benchmark (IRIS_2010), the data available for modeling was reduced to minimum. As a consequence; engineers were forced to predict the behavior with non sufficient information and consequently to rely mainly on the parameters generated automatically by concrete material law and / or to determine relatively arbitrary missing parameters or their variation from the values generated automatically. As a consequence, the result of simulation becomes greatly “engineer dependent”, leading to much different results than people doing the same modeling problem. In previous paper ([5], [6]) presented during last LS-DYNA Conferences, we demonstrate that a probabilistic approach for concrete modeling can be used to reduce these differences due to the modeling choices or at least to assess the dispersion of results based on possible variation of input parameters. Yet, one of the main conclusions of these papers was that all these modeling techniques would never replace experimental concrete sample tests to obtain a proper material behavior.

The second phase of IRIS benchmark (IRIS_2012), for which sufficient experimental data were available, allowed us to supplement the previous papers highlighting the joint capabilities of LS-DYNA and LS-OPT for predictively assess the consequences of an impact on a slab reinforced concrete. In a preliminary phase, from data automatically generated by the constitutive law, a massive use of LS-OPT has enabled an extremely precise calibration of material parameters to fit with great accuracy the experimental data. Simply based on these recalibrated parameters, the results for the impact on the concrete slab were very much improved. The exit velocity of the projectile, initially undervalued, has been heavily modified and is found in the range observed during the tests. Meanwhile, the deformation and damage modes observed in the tests are predicted in a much more realistic manner.

This paper, and previous one presented by DynaS+ related to the IRIS benchmark, highlights the interest to minimize the uncertainties of a joint and widespread use of LS-OPT software along with LS-DYNA for this kind of applications.

6 References [1] CEB-FIP Model Code 1990 – Comité Euro-International du Béton – 1990 – Thomas Telford

House.

[2] Malvar, Crawford, Morill : “K&C Concrete Material Model Release III” – Karagozian & Case – 2000.

[3] LS-DYNA Keyword Users’ Manual and Theory Manual – LSTC.

[4] Van Dorsselaer N., Lapoujade V., Nahas G. , Tarallo F., Rambach J.M. : “General Approach for Concrete Modelling : Impact on reinforced concrete” - 12th International LS-DYNA Conference, June 2012, Detroit, USA.

[5] Van Dorsselaer N., Lapoujade V., Nahas G. , Tarallo F., Rambach J.M. : “Impact Simulations on Concrete Slabs: LS-OPT® fitting approach” - 8th European LS-DYNA Users Conference, May 2011, Strasbourg, France



Using LS-OPT for Parameter Identification and MAT_FABRIC with FORM=-14

David Dubois, Ph. D

Autoliv Avenue de l’Europe

76220 Gournay en Bray France

Jimmy Forsberg, Ph. D

DYNAmore Nordic AB Brigadgatan 14

SE-587 58 Linköping Sweden

1 Abstract This work was carried out as a methodology development project in a joint venture between Autoliv and DYNAmore Nordic AB. The outset of the project was to obtain a better component behavior due to a more realistic material behavior in the simulation of airbag models. The observation underlying the project was that the current fabric model used in most airbag models is *MAT_FABRIC and FORM=14. In FORM=14 there is no consideration taken to a stiffened response due to a bi-axial stress state in the fabric. To consider the bi-axial stress state, FORM=-14 was implemented some years ago but has, until now, not been used. The objective with this implementation is to increase the stiffness in the fabric when subjected to a bi-axial stress state. This paper presents a resume of the features found in *MAT_FABRIC, a methodology to fit the simulation model to material test data using LS-OPT and finally a comparison between the behavior of the different FORM options.

2 Introduction The material formulation -14 was implemented some time ago in order to capture the behavior of fabric materials under a bi-axial stress state. This paper describes the work flow when going from material testing to material data input into LS-DYNA when using MAT_FABRIC with FORM=-14. The material parameters are described as values or load curves in LS-DYNA. In many material models in LS-DYNA these parameters will have to be found using finite element, (FE), simulations models of the material test procedure, i.e. parameter identification optimization problems. Hence, the need for an optimizer such as LS-OPT is obvious. When using FORM=14 for MAT_FABRIC in LS-DYNA, the user will supply load curves describing the material response for uni-axial loading in terms of a 2’nd Piola-Kirchoff/Green-Lagrange curve. This curve can be converted from a uni-axial test analytically and directly input to LS-DYNA. The LS-DYNA simulation model is then able to exactly reproduce this material test. When using FORM=-14, the user will also supply load curves which describes the material response for a bi-axial stress state. Typically this will increase the stiffness of the material response. In contrary to the uni-axial test, the results from material tests cannot be converted to input data to LS-DYNA



analytically. Instead the load curves to be used in LS-DYNA will have to be found by use of reversed engineering. Reversed engineering for parameter identification is a well-known area in the field of optimization. In this work we used LS-OPT and Sequential Response Surface Methodology, (SRSM) as optimization technique. The main difficulty with this optimization procedure is to keep the number of design variables as low as possible. Apart from FORM=14/-14 a number of new features have been added to the material model lately. Therefore, a small description of some of the different features in MAT_FABRIC will be discussed as well in the initial part of the paper.

3 *MAT_FABRIC The features of *MAT_FABRIC that will be discussed is the liner, coating and form selection and their influence on the material response under compression.

3.1 Liner

The liner in *MAT_FABRIC is a way to deal with compressive loads. The liner is defined by three parameters: EL, PRL and LRATIO which are the Youngs modulus, Poisson’s ration and thickness of the liner, respectively. If the elastic liner is used it will also affect the material response under tension. The material response for the liner is a pure elastic one, including the Poisson’s effect.

3.2 Fabric Coating

For -14 there is a new feature called the coating which is an elasto-plastic material model for representation of the coating of the fabric. Three extra input parameters are read which are the ECOAT, SCOAT and TCOAT which stands for Youngs modulus, yields strength and thickness of the coating respectively. The thickness is applied on both surfaces of the membrane, see Figure 1.. The intention of is to be able to capture the increased bending stiffness due to a coating. Observe, the coating will add membrane stiffness in both tension and compression, unless it is turned off.

Figure 1: Fabric coating

3.3 FORM 14/-14 under compressive loads

During compression loading there has been an update for both FORM 14 and -14. Earlier the two implementations were intended to define the material response under tension loads using load curves. The compression behavior was treated using the liner characteristics and in many cases the material response from the FORM option was turned off in compression. Now, it is also possible to define the compression behavior using negative stress/strain values in the behavior curves. Still, if a liner is defined, the total behavior during compression will be the cumulative of the liner and the defined behavior curves.

4 Uni-axial behavior: FORM=14 When using FORM=14, the 2’nd Piola-Kirchoff stress versus the Green-Lagrange strain is given using load curves in each fiber direction. To establish these curves, material tests are performed. The results from the test are transformed into load curve data which is entered into the simulation model (input deck) directly.



4.1 Uni-axial test

The uni-axial test is carried out using a strip of fabric material with the initial length, l0, and cross-section area, A0, see Figure 2. The measurements taken from the test are the grip handle displacement d and the force f(d) as a function of the displacement.

Figure 2 : Uni-axial test, Bias test, Picture frame test configurations

The stress-strain curve to be used in the material model is then obtained by the transformations:

2

11

2

12

0

l

dE and (1)

)(

)(

0

0

0 dl

l

A

dfS

. (2)

4.2 Shear test

Two types of shear test are presented in Figure 2. A Bias test [1], where the fabric is oriented at 45° compared to the loading direction. A Picture Frame test [2], where the the fabric is clamped in a rigid frame. and a hinge is modeled at each corner of the rigid frame. Two opposite corners are pulled apart. The measurements from the test is the force –displacement curve f(d). The stress-strain curve to be used in the material model is then obtained by the transformations:

112

2

0

l

dEE YXXY (3)

and

)2(

)(

0 dlt

dfS XY

. (4)

5 Bi-axial test – FORM=-14 The aim of FORM=-14 is to capture the stiffening effect found when the fabric is subjected to a bi-axial stress state. Another aspect of FORM=-14 is that in an uni-axial loadcase it should give the exact same result as FORM=14. The test configuration is given in Figure 3. The force – displacement curves fbx(d) and fbY(d) in the two directions are measured.



Figure 3 : Bi-axial test configuration.

Two new curves, f‘bX and f‘b

Y are introduced to model the tensile stress-strain curves in the fiber directions when EXX=EYY, i.e., in bi-axial strain. Assuming that EXX>0 and EYY>0, the stress in the fiber directions are computed according to

)/,1min()(')/1,0max()( XXYYXX

X

bXXYYXX

X

lXX EEEfEEEfS (5)

)/,1min()(')/1,0max()(´ YYXXYY

Y

bYYXXYY

Y

lYY EEEfEEEfS (6)

This means that the stress will follow the uni-axial loading curve for uni-axial strain and the bi-axial loading curve for bi-axial strain. The stress is linearly interpolated in between these two curves using the normalized parameters min(1,EXX/EYY) and min(1,EYY/EXX), respectively.

This means that f’ is a stress/strain relationship that is used in the material definition in LS-DYNA and f’ cannot be found through a neat equation as for the uni-axial load case. To find f’ we need to optimize the appearance of f’ to capture the test configuration material response f.

6 Optimization From the bi-axial test the force displacement curves are known. To determine the optimal curves to use in LS-DYNA in order to capture the material test results the curves to be used must be found through some sort of parameter identification. In this study the SRSM approach in LS-OPT was used.

6.1 Simulation model

As a balanced rate of bi-axiality was used (same prescribed displacements at all four ends), the center of the sample remains in the center of the device. A simplification of the finite element model was then applied. A quarter of the test piece was modeled, using symmetry conditions and prescribed velocities along the other boundaries, see Figure 4. The element size is chosen to be of the same order as the element size used for the restraint system simulation.



Figure 4: Simulation model for the bi-axial loading case.

6.2 Parameterization

Load curves need to be parameterized. A Hermetic Cubic Spline formulation is used in order to generate continuous load curves form the optimization. Each load curve is divided into 3 segments with a cubic polynomial interpolation for each respective segment, see equations below. C1 continuity criterion was used between the segments.

with and

Shown in Figure 5 is the final parameter definition input needed to LS-OPT. There are 18 independent parameters. Since the simulation model is small the number of parameters is not discarding.

Figure 5 : Parameter window in LS-OPT. There are 2 dependent variables, 3 constants and 9

parameters for each yarn direction, hence a total of 22 design variables.



6.3 Responses, Constraints and Objective

The objective is to minimize the difference of the force-displacement curve between the test and the simulation. The measure used in this study is the Root Mean Squared (RMS) error evaluated at each point in the force-displacement curve from the test.

6.4 Results from the optimization

The optimization process runs for 7 iterations using 33 simulations in each iteration, see Figure 6. As it can be seen, the optimization process rapidly reaches the objective. Several different starting positions have been tested but all of them converge to the same solution.

Figure 6 : The optimization history. The optimization history shows that the optimization process converges very fast towards an optimal value.

Shown in Figure 7 are the obtained force-displacement curves from the test as well as the initial and final force-displacement curves from the simulation. The appearance of the load curves used in the material model, LCAA and LCBB in respective yarn direction, in the simulation model are shown in Figure 8.

Figure 7: The force-displacement results from test, initial simulation and the optimized simulation results, respectively.



Figure 8: The shape of the used load curves in the material model for the two yarn directions.

7 Component simulations To check the numerical stability and the stiffening of the material response in case of a distributed loading, a component simulation model was needed. Autoliv’s leakage device (GES), developed by Manfred Schlenger [3], was chosen to perform this analysis. This device is regularly used to define the fabric leakage parameters needed to simulate the behavior of uncoated airbags. It consists of a pressurized metal cone. At one end, a known quantity of gas is released and at the other end, a circular fabric sample is fixed. The pressure history in the cone and fabric bulge amplitude are measured and recorded. These physical measurements are the parameters according to which the Saint Venant-Wantzel leakage coefficients are tuned. To analyze the local variation of strain due to fabric scrimp interchange, two semicircles with different warp/weft orientations (90°/45°) were used instead of the classical circular fabric sample. To join the two pieces of fabric, a straight line of seam was horizontally sewn.

7.1 Numerical model and sample fiber orientation

The aim of these simulations was to compare MAT34 Form = 14 and Form = -14. The pressure measured during GES tests was applied to the back side of the two semicircular meshes, see Figure 9, right.

Figure 9 : Numerical model used for the comparison of Form = 14 vs Form = -14

Fixed at their outer diameters, two fabric panels were numerically sewn using a simple line of common nodes. The different fabric orientations (90°/45°) of the two semicircles were taken into by different local warp/weft orientation of the MAT 34 finite elements, see Figure 9, left.



7.2 Strain distribution over the surface

To assess the validity of the new formulation and the set of behavior laws obtained by optimization, the strain distributions over the surface between Mat 34 Form = 14 and Form = -14 were compared as shown Figure 10.

FORM = 14 / GL. Strain along Weft FORM = -14 / GL. Strain along Weft

FORM = 14 / GL. Strain along Warp FORM = -14 / GL. Strain along Warp

Figure 10 : Comparison of strain distribution over the surface at the maximum pressure.

The comparison of the strain distribution over a fabric sample subjected to a pressure field, successfully shows a stiffening of the material properties. Hence the new formulation of the MAT34 enables to simulate the variation of fabric behavior due to the scrimp interchange. In both warp / weft directions, a reduction of a maximum of 20% of the strain values is measured in the center of the circular sample of fabric. This value corresponds to the variance between uni-axial and bi-axial tensile behavior laws.



7.3 Analysis of Bulge amplitudes

To confirm the stiffening of the fabric material using FORM = -14, a global measurement was also assessed. The pressure generated inside the GES, induces a curvature of the circular fabric sample. The amplitude of this bulge experimentally measured by a laser, was compared between the two FORM 14 / -14.

Measurement of the max bulge point Variation of the bulge amplitude vs. time

Figure 11 : Comparison of Bulge amplitude

Initially equivalent, a stiffening of the fabric property of FORM = -14 compared to FORM = 14 is progressively observed on Figure 11. At the maximum of pressure, a significant reduction of 10% of the bulge amplitude is obtained. This modification of behavior is in accordance with the expected result and validates the whole methodology. As a consequence, an update of the Saint Venant Wantzel leakage coefficients is needed for un-coated airbag fabric when it is modeled with MAT 34 Form =-14.

8 Summary This paper describes a methodology to determine material behavior using reversed engineering. The use of automated procedures enables to obtain reliable numerical behavior law and to match test data. The comparison of the strain distribution over a fabric sample submitted to a pressure field shows that the new formulation MAT 34 Form = -14 enables to reproduce the effect on the fabric scrimp interchange. The material becomes stiffen and a global observable, the bulge amplitude, is modified by 10%.

9 Perspectives The current Form = -14 is a linear interpolation between uni-axial behaviour laws and balanced bi-axial behaviour laws. During the airbag deployment and the restraint phase, various cases of bi-axial loadings are expected. The next development aims at extrapolating this new formulation to the general cases of bi-axiality.

10 Acknowledgement The work in this study was made with the contribution of Thomas Borrvall, DYNAmore Nordic AB, Pontus Bergman, Autoliv Sweden and Marc Marchand, Autoliv France.

11 References [1] Mohammed, U.; Lekakou, C.; Dong, L. and Bader, M.G.: "Shear deformation and

micromechanics of woven fabrics", Composites: Part A, Elsevier, 2000, pp. 299-308. [2] McGuinness GB.; Bradaigh CMO. : “Development of rheological models for forming flows and

picture-frame shear testing of fabric reinforced thermoplastic sheets”. Journal of Non-Newtonian Fluid Mechanics 1997;73:1–28. [3] Schlenger M. : "A New Model for Simulation of Fabric Leakage in LS-DYNA”, LS-DYNA Forum,

Bamberg, 2010, pp. D-III-7-22



Multi-disciplinary Topology Optimization for Vehicle Bonnet Design

David Salway1, Dr Tayeb Zeguer2

1GRM Consulting Ltd, 2Jaguar Land Rover Ltd

1 Abstract

Bonnet Pedestrian Head Impact and Structural Stiffness and Strength targets have conflicting design requirements which currently result in design compromises, and the current CAE methods use different models and solvers. This paper highlights a new CAE capability to provide Multi-Disciplinary Optimization of bonnet geometry to achieve the conflicting Pedestrian Head Impact and structural stiffness/strength targets at lowest weight and cost. The aim has been to combine all bonnet load cases using one code “LS-DYNA®” and carry out trade-offs and optimize weight using LS-OPT®. A new developed topology process employing VR&D Genesis® for HIC optimization is presented and compared with LS-TASC® tools for a generic bonnet design.

2 Objective

The objective of this study is to establish a robust MDO process for bonnet design. Currently bonnet design is done in an empirical manner, by carrying over existing design features that are known to work in other vehicle programs. Using Topology optimization for Head Impact applications has not been perfected before. A simple hypothesis was developed; that it was hoped would allow the problem to be solved:

‘The Head Impact properties of a bonnet are linked to the linear characteristics of the bonnet structure when no secondary

impacts are considered’ The motivation behind developing this process is to reduce the development cycle time of a 5* car, producing the lowest cost and mass design, whilst achieving the maximum category in consumer and legal testing for pedestrian head impacts. To achieve the best results when using optimisation it is necessary to apply it to every stage of the development cycle. In the case of the bonnet it became apparent that this wasn’t occurring. The current design route is shown in Figure 1

Figure 1 Current Bonnet Design Process



By applying a more optimisation/CAE led approach it should be possible to reduce development time and improve design efficiency, the proposed work plan is as set out in Figure 2

Figure 2 Proposed Bonnet Design Process

It is anticipated that the ‘Concept Optimisation’ phase will be carried out using a Topology optimiser, specifically VR&D Genesis®. An automated loop was proposed, such that the optimisation could be carried out, and then verification could be carried out using LS-DYNA for the head impact loads. The results of this verification would be used to modify the Topology constraints and hence guide the design. Following this it is proposed that the analyst would create geometry based on the optimisation result. This would require the analyst to create a feasible solution, but this would be carried out in an FE pre-processor such as ANSA as opposed to full CAD at this stage of the process. This design could then be further developed through size and shape optimisation using LS-OPT. By coupling LS-OPT to ANSA it is possible to create complex morphing variables and then use LS-OPT to drive ANSA in batch mode and create the models for each design. Currently the bonnet component analysis is carried out using LS-DYNA for the safety analysis (Head and Upper Leg) and Abaqus for the static load cases. It is proposed to use LS-DYNA implicit to replace Abaqus in the static analysis. This would then allow a common model approach to be employed, making the optimization set-up simpler.

3 Process Development

The whole process relies on being able to produce the concept design from the Topology optimisation. The later phases of the process are currently possible with the commercially available tools and as such do not need any significant research effort. The bulk of the effort was therefore put into developing a method to generate the concept design.

4 Model creation

Through discussion with the relevant CAE teams it became apparent which of the load cases were the most important to consider during the method development. Although in the application of the method it would be necessary to consider all of the requirements. The most important load cases considered in the method development were:

Torsional Stiffness Rear Beam Stiffness Latch Over Bend

Analysis models of these load cases were gathered along with the requirements for each load. At this stage a baseline assessment was carried out to see how the current model compared to the



requirements. The results of this analysis showed that for most of the requirements the bonnet exceeded the target, but did not comply with a small minority. Currently the head impact models are run in LS-DYNA and feature the complete front-end of the vehicle, with powertrain and ancillaries. The run time of this model on 16 CPUs is approximately 8 hours, because of this a simplified model was produced. As stated in the hypothesis at this stage the effect of secondary impacts will not be considered, in light of this the under bonnet components were not required. The model was setup with the hinges, latches and bump stops in position with the top half of the fenders and the front end carrier. The hinges and latches feature the kinematics of the full LS-DYNA model. Two models were created, the first a Nastran linear model to be used to optimise in GENESIS and second an LS-DYNA model to be used for the Head Impacts and also the LS-TASC investigation. These can be seen in Figure 4. To establish the constraints for the optimisation the models were run with the existing production bonnet. The static values were recorded, and used as constraint for the optimisation. Further simple models were used during the process development to improve the turnaround time, an example of one of these models can be seen in Figure 3.

Figure 3 Quick Test Model

Figure 4 LS-DYNA and GENESIS Models

5 Concept Optimisation Method Development

The initial phase of the optimisation workflow is to complete a linear optimisation to determine the material placement considering both the static load cases and also an approximation of head impacts spread across the bonnet. The total number of assessment points for this bonnet was approximately 180, it was decided that it wasn’t practical to consider so many points. For the optimisation 20 points were considered spread across one side of the bonnet, and symmetry constraints were applied. The purpose of this was to try and minimise the number of active constraint that the optimiser would encounter, thereby improving solve time and convergence. Two core methods were identified for optimising the reinforcement. Further variations on each method were considered. The first core method explored was a solid topology approach. By using sheet manufacturing constraints with this approach it was anticipated that the optimised part would require minimum interpretation to produce a manufacturable pressing.



An alternative approach was free solid topology without any manufacturing constraints, it was anticipated that this would provide a more straight forward problem for the optimiser than the constrained problem, but would require more interpretation to get to a design. The second core method was to use shell Sizing and Shape optimisation. Various combinations of Sizing, Topometry, Topography, and Shape optimisation were tried. It was intended that shape optimisation would be used to provide the form to the inner, and then Sizing optimisation would be used to design the thickness. When using the topology approach the connection between the bonnet skin and the inner is taken care of by the nature of the topology optimisation. For the shell based approach the connection between the Skin and inner needs to be controlled based upon the distance of the inner from the skin. In addition to a variety of different optimisation design methods to consider, a number of different constraint/objective setups were considered as shown in Figure 5

Response Type Minimise Target Constrain

Linear Head Impact Response

Mass Fraction

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Figure 5 Possible Constraint System Combinations

As the GENESIS models were being run it became apparent, that the method we would ultimately end up using had to achieve various requirements, it had to produce a converged solution, and it had to produce a coherent structure. Many times the optimisation would not converge, either by not meeting constraints, or in the case of Topology by not forcing elements to be 1 or 0.

6 Solid Topology Investigation

The solid Topology baseline model featured a large lump of solid material on the underside of the bonnet skin. The dimensions of this package space were based on the general depth of a current production bonnet. Due to the simple shape of this generic model the solids were created by offsetting the bonnet skin to create a hexa/penta mesh. The main sections of the particular production bonnet that was employed as a guide are all aluminium; as such this material choice was carried to the new mesh. In other solid topology applications where a performance constraint or target is being employed we have found benefits to scaling the solid package space material properties. The logic to this is very simple; the solid topology result is never going to be as thin as the production panel, therefore by scaling the material properties, the mass and stiffness properties of the solid result will more closely match the eventual thin pressed sheet. This approach has been used with both the sheet metal and free Topology method. Experience of using Genesis and advice from the developers suggest that the most effective way to use solid Topology is to constrain mass fraction and maximise stiffness. Unfortunately for this particular example that is not really feasible as we want to tune the stiffness to be stiff enough to achieve the static requirements, but compliant enough to give us the 5* HIC scores. From this it was apparent the less favoured approach of minimising mass fraction with either constrained or targeted performance as the best route forward.

7 Sheet Topology Method

This method is intended to give you a topology result that can be pressed from a single sheet. As the bonnet is made up of pressed parts, it was anticipated that this would provide an answer that would require the minimum interpretation into a feasible design. An example can be seen in Figure 6



Figure 6 Example of Sheet Topology

The method is unique to VR&D Genesis and was developed with significant input from GRM. Because of past experience with this we were able to employ non-default parameters in a bid to improve convergence. From running with the sheet Topology we encountered problems with convergence; generally the Topology should converge towards elements having a value of either 1 or 0. An example of this is shown in Figure 7 both models are identical apart from the one on the left having the sheet constraint activated.

Figure 7 Effect of Sheet Constraint on Result

In addition to the convergence issue we found a problem with connections. Where the rear beam load was being applied into the structure via an RBE3(interpolation element), we discovered a feature of the method where the sensitivities and approximations were allowing the material to move away from the connection and then not being able to reconnect. An example of the disconnect is shown in Figure 8

Figure 8 Loss of Connection on Rear Beam

By including the manufacturing constraints in the optimisation we are making life more difficult for the optimiser, as such this combination was sufficient to justify dropping the Sheet Constraints. It is, however, anticipated that the method will be further investigated in future.



8 Topology Method

Following the decision to drop the sheet constraints from the Topology further research was carried out using the solid topology method. We explored the best problem formulation for the optimisation, trying various approaches. As previously discussed the most efficient method for solid Topology in Genesis is to use a fixed mass fraction. However, due to the nature of this problem this is not really appropriate as the objective is to achieve a specific stiffness and this will require material to be added and removed. The next approach considered, was to minimise the mass fraction whilst constraining the static and head impact cases. The optimiser prioritises the constraints, so it will increase the mass fraction to meet the constraints. Due to the large number of constraints this makes the optimisation quite slow, and also makes it uncertain if the optimiser will meet the constraints. It can also result in the optimiser going down a blind alley from which it cannot return. The option that is currently favoured for the problem formulation is to constrain the static targets (torsion, bending, and rear beam stiffness) and then target the desired performance for the head impacts. This was achieved by subtracting the performance from the target, and then constraining the absolute result of this. At the target this gives a value of 0, to allow the optimiser some freedom this value was constrained to have an upper bound of 2.0 (due to the magnitude of the values we were using this equates to approximately 10%. The target value has no direct relationship to the HIC, as such experimentation was used to establish this value. The criteria were that the value had to provide sensible HIC scores, when the results were assessed in LS-DYNA, and that the value had to be such that the constraints in the linear solver were achievable. This target value was the value that would then be updated as the process loops and gets feedback from the LS-DYNA runs. Various techniques have been discovered to encourage convergence. Through a number of tests it was concluded that control of the optimisation move limits and convergence parametrs should be changed from the default setting chosen for traditional linear analysis problems. Another effect that was observed is that when an element gets to a density value of 1 it becomes trapped. In the same way as steadily reducing the move limits, trapping elements at 1 is done to encourage convergence in linear cases. To allow freedom in this case the value was reset from the deafult to cycle 100. Following the first runs it was apparent the optimum mass fraction would be at a value 0.1 or lower, this is considered to be a low mass fraction, as such some non-default move limits were used to compensate. These non-default values alter the minimum move limit and also the fraction by which the move limit decreases each cycle.

9 Shell Shape and Gauge Method

As an alternative to the Topology method, a shell approach was also pursued. The inner was made as an offset shell with the intention of then using either Topography or free Shape optimisation to form the bonnet. As previously discussed this raises problems with controlling the connectivity of the inner and outer parts of the bonnet. As the inner moves away from the outer the connection between the two should detach, this effect is captured in the Topology method as the material is removed. To achieve the same using shape variables requires the connection element to be linked to the design variable. Due to the nature of Topography where the individual design variables are generated internally it is not possible to create this link. Because of this a switch was made to using Shape Optimisation where each design variable is created by the user at the start. To speed up this process a script was created, which also generated the connections and created the design variable links. To achieve this, the connection was modelled using a CBAR element, with the stiffness of the CBAR linked to the shape change at the connection point. This approach was used right across the bonnet, with a separate CBAR property for each node to node connection. As the shape variable moved the inner away from the outer the connection stiffness would be reduced, using a power function, forcing it to decide between big sections or many connections. An example of the model setup is shown in Figure 9



Figure 9 CBAR Connections to Link with Shape Variable

Various studies were carried out using this approach, to try and achieve a convincing shape. One of the advantages of Topography is that the method includes smoothing functions. By using the Shape approach a high level of noise was seen in the resulting structures. To try and smooth out the spikes a constraint was placed on the volume of the sheet, a spiky sheet has a much higher volume than the same basic shape with a smooth finish. The other issue that was identified was that the optimiser seemed to be struggling to converge on a definite shape. To overcome this various starting positions were considered; fully against the outer, fully away from the outer and at 50% of the potential move. We achieved some convincing results using this approach, but experienced convergence issue. It was therefore concluded that the Topology method provided a better solution and had more scope. However, there is definitely scope within this method and it warrants further investigation.

10 LS-TaSC comparison

Using LS-DYNA for the head impact analysis allowed us to test LS-TaSC with the problem. All of the models have been constructed to run in both GENESIS and LS-DYNA and as such it was very straight forward to setup the LS-TaSC optimisation. In LS-TaSC the individual load cases were run separately, and then combined. Individual results are shown in Figure 10

Figure 10 LS-TaSC Static Results

Something that was observed when comparing the results to the Genesis results in Figure 11 is that the LS-TaSC results have much more material. Investigation shows that whilst LS-TaSC had the constraints set for upper bounds on displacement, the optimised result exceeds the constraints by considerable margin. Results from the optimisation can be seen in Figure 12

Figure 11 Genesis Static Results

LS-TaSC GENESIS

Torsion Latch Beam Torsion Latch Beam

Constraint 1 1 1 1 1 1 Result 0.446 0.524 0.348 0.994 1.000 0.999

Figure 12 Comparison of Static Constraint



The comparable Genesis results exhibit displacements that are significantly larger than the size of the displacements in the LS-TaSC models, whilst still meeting the constraints. Discussing with LSTC about this has suggested that the constraints in LS-TaSC are still being developed. The constraint method selected for the dynamic impacts was contact force, both peak and average were considered. The result of an average run is shown below in Figure 13 and the results are very encouraging. The next phase of the LS-TaSC study would be to incorporate a user defined HIC constraint and see how that effects the results.

Figure 13 LS-TaSC Head Impact Result

11 Detail Design using LS-OPT and morphing

The constraint values from the Topology optimisation are provided in Figure 14

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Figure 14 Normalised Topology Results

Following the topology optimisation the penultimate stage of the process is interpreting the Topology density result into a pressed design. As previously discussed by removing the manufacturing constraints from the Topology optimisation a degree of interpretation was required to achieve a pressed design. Some features of the design are quite clear such as the deep drawn section at the rear, and the general layout of the material. The main fore/aft members in the result are not laid out in a classic pressed design; a section plot is shown in Figure 15

Figure 15 Section plot of Fore/Aft Members, showing Topology Result and Interpreted Pressed Design

In the pressed sheet some consideration has been given to both the pressing and assembly requirements and so extra material has been added around the outside to improve the feasibility. This is particularly evident around the hinges, latches and over the fender. The Topology result and interpreted design are shown in Figure 16



Figure 16 Topology Result and Interpreted Design

The outside of the whole pressing was attached to the outer using the same clinching approach as the original design. This is an area that has been identified as having potential for further optimisation. The connections between the inner and outer have long been considered to be important. The initial design has been developed so that there is a flange suitable to support an adhesive connection; the locations of these are based on where the Topology result has a connection between the inner and outer. A baseline run was carried out to verify the performance was close to the desired values. Some small detailed changes were made to the baseline model to improve the performance of the latch pull. The final stage of optimisation was to carry out a shape and gauge optimisation by coupling ANSA morphing to LS-OPT. The morphing was setup using direct morphing approach as recommended by Beta CAE. Variables were setup to vary the depth and width of the major sections and the thickness of the pressed inner. The connections were included in the morphing and then the adhesive was regenerated for each shape change and the mesh smoothed to improve the quality. When creating the shape variables special attention was paid to how they would combine and to ensure that the mesh remained viable. The design variables and morphing parameters are shown in Fehler! Verweisquelle konnte nicht gefunden werden..

Figure 17 Design variables and Morphing Parameters

The linking of LS-OPT to ANSA was carried out following the principles defined in the Beta Tutorial ‘Optimisation with LS-OPT’ [7]. A DOE study was carried out using a space filling approach to define the start points with a radial basis metamodel. The meta model was then used to carryout virtual optimisation to achieve different design objectives.

12 MDO Results

The DOE results have been used to explore the performance that can be achieved at different mass values. The first check was to verify the acuracy of the meta model, the accuracy plots, for each of the responses was checked. In this example a small number of points was used for each loadcase to keep the computation cost to a minimum for non-essential cost. To verify the results the chosen metamodel design should be verified using an LS-DYNA analysis. The metamodel accuracy was checked using LS-OPT, by plotting the predicted results against the analysis results. The following optimisation schemes were tried:

Minimise Mass, Static Constraints Minimise Mass and HIC for All Points, Static Constraints 4.0Kg Mass Constraint, minimise HIC, Static Constraint 4.5Kg Mass Constraint, minimise HIC, Static Constraint



5.0Kg Mass Constraint, minimise HIC, Static Constraint In the multi-objective cases the mass is dominated by the HIC scores due to the orders of magnitude involved, i.e the mass is in Kg and the HICs can be expected to be in a region from ~400-1700. By adjusting the weighting it was possible to overcome this, but this had the effect of causing the thickness variable to flip-flop between the upper and lower bounds. The results of the virtual optimisation and the thcikness variable dominating the mass can be clearly seen in Figure 18

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Minimise Mass & HIC, Static Constraints 0.00 0.80 0.15 0.50 0.45 0.30 0.30 0.10 0.50 0.55 0.00 0.00 1.00 1.40 1.00 1.00 1.00 Thickness at lower Bound

Minimise Mass & HIC, Static Constraints 1.00 0.00 1.00 0.10 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 7.00 0.50 0.50 1.00 Thickness at Upper Bound (Mass Objective Dominating)

4.0Kg Mass Constaint, Min HIC, Static Constraint 0.50 0.05 0.85 0.65 1.00 0.00 0.10 0.00 1.00 0.25 0.15 0.00 1.00 4.00 0.95 0.60 1.00



Design Variables

Figure 18 Virtual Optimisation Results

The bonnet pedestrian scores have been postporcessed for the 5.0Kg reinforcement scheme and the scores are presented in Figure 20

13 Summary

As a method for improving the development time and costs of the bonnet reinforcement structure this process shows good potential. The correct use of Optimisation should reduce development time and produce a more efficient design. The post Topology phase is widely being used and available in commercial software application, some further exploration and adaptation to better suit the JLR HPC could be carried out, but really only to make it more user friendly. The use of Topology to determine the material placement is unique and as such has presented a pretty big challenge. The development done in the Topology phase so far has shown that there is definitely some truth in the hypothesis. The work to develop this will continue. The ultimate goal is to make the Topology phase automated, and once the optimisation strategy has been fully refined, this will be the next goal.

14 References

[1] Vanderplaats Research and Development: GENESIS Design Manual, Version 12.2, September 2012 [2] Vanderplaats Research and Development: GENESIS Analysis Manual, Version 12.2, September 2012 [3] LSTC: LS-DYNA Keyword User’s Manual, Version 971, May 2007 [4] LSTC: LS-TaSC Topolgy and Shape Computations for LS-DYNA, User’s Manual, Version 2.0 April 2.0 [5] LSTC: LS-OPT User’s Manual, Version 4.2, February 2012 [6] Beta CAE: ANSA V14.0.1 User’s Guide, February 2013 [7] Beta CAE: Optimisation with LS-OPT, v 14.0.1, February 2013

Figure 19 4.0Kg and 5.0Kg Optimisation Scheme Pedestrian Scores

Thickness/Mass Flip Flop Effect

4.0Kg 5.0Kg



Stochastic Simulation of Aircraft Fuselage Assembly Considering Manufacturing Uncertainties

Dietmar C. Vogt, Sönke Klostermann

EADS Innovation Works, Germany

1 Introduction

In aircraft production the use of rivets as permanent mechanical fastener to assemble lightweight sheet metal structures is very common. At assembly the rivet is placed in a through boring and the buck-tail is plastically deformed to create a second head. Thus rivets are positive locking and can carry axial tension loads. However, rivets are mainly used to transfer shear loads via the seating stress of their cylindrical shaft. The remaining pre-stress in the rivet and its local area after the riveting process is subject to immanent manufacturing scatter. When assembling the fuselage of commercial aircrafts additional inherent uncertainties are impacting the riveting process. The cylindrical barrels of a fuselage are typically manufactured from large thin walled shell structures underlying geometric tolerances and variations of the boundary conditions. Managing these uncertainties has a significant impact on the geometrical and structural product quality. In this paper the resulting variations of the three-dimensional residual stress condition will be analysed by simulation. To simulate the fuselage assembly process the model must be able to predict the influence of manufacturing uncertainties appropriately. Therefore these uncertainties will be considered already during the modelling by stochastic parameters and random fields. While most application examples in the literature are quite simple [1], the present paper aims to apply random fields in an industrial application using LS-DYNA®. By utilizing non-invasive methods the approach can be adapted to different FE-Solvers without too much effort.

2 Modelling of the deterministic fuselage structure

The stochastic simulation is based upon a finite element model of two aircraft fuselage barrels during assembly process (Figure 1). The fuselage barrels consist mainly of three types of parts, the circular ribs, the stringers and the sheet metal skin. In the LS-DYNA model these are represented by shell elements. In addition the basic floor structure including Samer Rods and bilge structures are modelled. The floor is also represented by shell elements while the remaining structure is modelled from beam elements. Metallic parts are modelled with a linear-elastic material model of a typical aluminium-copper alloy. The floor made of carbon fibre sandwich material with orthotropic properties a suitable material model is chosen. The two fuselage barrels are assembled by riveting their flanges on one shared rib. While in reality the barrels are joined by pushing them onto the shared rib from both sides the simulation is modelled to start in-situ directly after moving the barrels together. A Surface-to-Surface-Interference contact is defined to avoid initial penetrations and consider potential pre-stresses resulting from geometrical imperfections of the barrels. This penalty based contact includes the shared rib as well as the sheet metal skin of the barrels in the flange area.



Fig. 1: Model of two aircraft fuselage barrels during assembly The penalty based interference contact allows controlling the contact stiffness and damping individually. In modelling state the manufacturing uncertainties of both barrels are represented by geometric imperfections (Figure 2). Before simulation start the contact is initialised with high damping and progressively increasing contact stiffness up to material stiffness. Potential penetrations are transformed into pre-stresses. After the static equilibrium is reached the damping is reduced to normal values and the actual simulation starts.

Fig. 2: Contact initialisation between fuselage cross-sections with imperfections The rivets are modelled from beam elements that are mesh independently fixed with a tied contact to the shell elements of the fuselage sheet metal skin and the shared rib. By introducing initial local pressures on the regarding shell elements during initialisation of the tied contact the specific pre-stressing of the rivets is controlled. Figure 3 shows a close-up view of the model where the barrels and the shared rib (indicated in blue) are riveted together. The rivets are grouped by sectors whose borders are defined by the stringers for each fuselage skin area. The simulation model is setup to execute the riveting process sector by sector.

Model with geometricimperfections

Static equilibriumContact initialisation



Fig. 3: Detailed model view of the shared rib and flange areas to be riveted

3 Modelling of the uncertainties

Two major manufacturing uncertainties will be considered. The remaining pre-stress in the rivets and the geometrical imperfections of the fuselage barrels deviating from an ideal cylinder. In separate studies the variation of the riveting pre-stress was analysed. It is assumed the riveting force follows a normal distribution function with a mean of 4450 N and a standard deviation of 360 N. To compare the influence of the riveting sequence the stochastic study is performed two times with different setups (Figure 4).

Fig. 4: Analysed riveting setups (4-quadrant sequence, left; random sequence, right) In manufacturing it is very common to perform the riveting parallel in four quadrants with clockwise sequence (Figure 4, left). For comparison a random riveting sequence is chosen (Figure 4, right). Each rivet is assigned with an individual stochastic parameter controlling the variation of the riveting force. Unlike the pre-stressing of rivets only little is known about the detailed geometrical imperfections of the fuselage barrels resulting from manufacturing. Detailed analysis and experimental measurements about spatial imperfections of thin walled cylindrical shell structures were performed in [2]. These measured sample database is available as a set of Fourier coefficients for a two-dimensional Fourier series from [3]. It is assumed the geometrical imperfections of the fuselage barrels have a similar order of magnitude in relation to the size of the cylindrical shell structure. To obtain suitable sample data the imperfections are scaled to the aircraft fuselage size and used to model the random fields. A random field represents the variation of a property of a given parameter as a function of space or of time and space. Applying random fields to FEA the grain size of the field is only limited by the element discretization. Several approaches for the implementation of random fields into simple FEA problems

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are described in literature [4, 5, 6]. For implementing the random field a non-invasive approach is chosen, since the LS-DYNA solver will remain unmodified. In this paper the Karhunen-Loève transform is used to model the random field based on the sample database. Similar to a Fourier series the stochastic property is modelled by a series of orthogonal base functions. For a one-dimensional random field the Karhunen-Loève series around the mean value is defined as follows:

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In this case the characteristics of the random field can be derived from the covariance matrix of the direct measurements of the geometric uncertainty (e.g. distortions). Metaphorically the Eigenvectors create the geometric basic shapes which assemble the stochastic field by random superposition (Figure 5).

Fig. 5: Geometrical representation of 6 Eigenvectors from the covariance function To match the spatial resolution of the fuselage barrels FE mesh with the random field resulting from the discrete Karhunen-Loève transform and interpolation based on radial basis functions is implemented. For the interpolation the thin plate spline base function described in [7] is used, for best approximation of the thin walled fuselage barrel. This interpolation method allows a very good continuity of the mesh and therefore high quality representation of the random field.

4 Simulation results

The stochastic simulation is executed by means of the Monte Carlo method using a Latin-Hypercube strategy with 100 samples each. To solve the simulations of both comparative studies a high performance compute cluster is used. The whole computational time takes about 8300 hours to solve all simulation samples. Figure 6 shows a single simulation sample directly after the interference contact initialisation before the riveting process starts. Some areas exhibit internal stress due to geometric imperfections of the fuselage barrels while others remain nearly free of stress. A detailed analysis shows small gaps between the parts in these stress free regions. Of course the stress distribution and intensity is different for all stochastic simulation samples.



Fig. 6: Stress after contact initialisation caused by geometrical imperfections (exploded view) Taking a look at the global stochastic simulation results gives further insight. Figure 7 shows the matrix of Spearman's rank correlation coefficients for the global parameters. The correlation matrix of both studies is nearly identical. Like supposed there is a noticeable dependency rSP = 0.34 between the riveting force and the remaining pre-stress in the rivets while these parameters are independent from all others. A very high correlation of rSP = 0.9 can be observed between the global distances before and after joining of both barrels that are caused by their geometrical imperfections. A negative correlation of rSP = -0.68 indicates that larger distance between the barrels before joining will decrease the stress before the riveting process. The opposite counts for the distances between the barrels after joining them and the remaining stress after the riveting process, rSP = 0.73.

Fig. 7: Correlation matrix of global stochastic simulation results Comparing the probability density functions of both stochastic simulation studies (Figure 8) exhibit some remarkable details. Both densities have a similar shape with one distinct accumulation point and a positive skew. But the variation of the 4-quadrant sequence is higher and thus the process is less robust. There is a visible higher amount of results with an average residual stress after riveting between 40 N/mm2 and 50 N/mm2.

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Fig. 8: Density estimate of average residual stress after riveting These findings are supported by the arithmetic mean and standard deviation of the residual stress after riveting. While the random sequence has a mean of 19.9 N/mm2 and a standard deviation of 4.4 N/mm2 the 4-quadrant sequence has a mean of 20.6 N/mm2 and a standard deviation of 7.3 N/mm2. Hence the riveting process in four quadrants with clockwise sequence is less robust against manufacturing uncertainties. A detailed look at the local stress distribution around the fuselage barrels after riveting shows why. Figure 9 shows the arithmetic mean of the residual stress after riveting plotted over sectors around the barrel. The 4-quadrant sequence started riveting at sectors 1, 23, 45 and 67. Precisely at the end of each quadrant the peak stresses occur. Following the riveting sequence over simulation time one can observe the accumulation of initial gaps in front of the current riveting sector. At the end of each quadrant the accumulated gap is closed with the last riveted sector which results in stress up to 4 times higher than the average. Contrary to the 4-quadrant sequence no particular accumulation of gaps can be observed. In general the variation of the residual stress after riveting correlates well with the distance of the barrels due to geometric imperfections, for each sector.

Fig. 9: Arithmetic mean of residual stress plotted over riveting sectors

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ua

l st

ress

[N

/mm

²]

Sector

random sequence

4-quadrant sequence



5 Conclusion

In the present paper manufacturing uncertainties of an aircraft fuselage assembly process have been modelled by stochastic parameters and random fields. Two stochastic simulation studies with different riveting sequences were performed for comparison. A strong dependency between the geometric imperfections of the fuselage barrels and residual stresses in the structure was identified while the variation of the riveting force showed no significant impact. Surprisingly the common manufacturing practise of parallel riveting in four quadrants with clockwise sequence was very sensitive to these uncertainties and caused high local stress concentrations. In comparison the random riveting sequence exhibited a more robust behaviour against geometric imperfections of the fuselage barrels. The analysis shows the importance of considering manufacturing processes and regarding uncertainties in combination. Based on this study, future work aims to analyse the influence of additional uncertainties, material combinations and different riveting sequences. The final goal is to optimise the riveting strategy, to design a process with low residual stresses and robustness against inherent manufacturing uncertainties.

6 Literature

[1] Bayer, V. ; Roos, D.: “Non-Parametric Structural Reliability Analysis using Random Fields and Robustness Evaluation”, Weimarer Op-timierungs- und Stochastiktage 3.0, Dynardo GmbH, 23 & 24.11.2006

[2] Arbocz, J. ; Babcock, C. D.: “Experimental Investigation of the Effect of General Imperfections on the Buckling of Cylindrical Shells”, Pasadena : California Institute of Technology, 1968

[3] Arbocz, J. ; Abramovich, H.: “The Initial Imperfection Data Bank at the Delft University of Technology Part I”, Delft : TU Delft, 1979

[4] Bucher, C.: “Introduction to Stochastic Structural Analysis”, Weimar : Bauhaus-Universität, Institute of Structural Mechanics, 2006

[5] Marczyk, J.: “Principles of Simulation-Based Computer-Aided Engineering”, Barcelona : FIM Publications, 1999

[6] Vanmarcke, E.: “Random Fields : Analysis and Synthesis”, Cambridge : The MIT Press, 1983 [7] Wright, G. B.: “Radial Basis Function Interpolation: Numerical and Analytical Developments”,

Colorado : University of Colorado (PhD Thesis), 2003


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FEA Information Engineering Journal - FEAIEJ · FEA Information Engineering Journal ... David Dubois, Ph. ... presented during last LS-DYNA Conference,