YOU Jun-feng，ZHONG Yan-yan

(1.The 41st Institute of the Forth Academy of CASC，National Key Laboratory of Combustion，F(xiàn)low and Thermo-Structure，Xi＇an 710025，China;2.Military Academy Engineering University of Capf.，Xi＇an 710038，China)

0 Introduction

Cabin(or the inter-stage section)is an important component of a rocket or missile，which delivers the thrust and bears all kinds of loads in flight.Nevertheless，because of some reasons such as design needs or processing errors，cabin(cascade)is often a complex structure with bossshaped bodies，brackets，grooves，holes，or windows，which can easily cause stress concentration［1］.Ideal results could be obtained for the stress concentration problem by mesh refinement and using the conventional finite element method(FEM).However，halving the element size results in four times the number of elements for plane problems and eight times the number for space problems.In other words，although a refined mesh could help obtain the ideal results，it requires more computing time and better hardware support.Sometimes，the simulation cannot be run due to the large number of elements and nodes.In order to solve this problem，the sub-model method was used to analyze a missile structure in this paper.

1 Sub-model method and steps

1.1 Sub-model method

Sub-model method is also called cut-boundary displacement method or specific boundarydisplacement method.The cut-boundary is just the cut from a coarse full FEM model，and the displacements of the nodes at the cut-boundary obtained/interpolated from the coarse full model are used as the displacement boundary conditions of the sub-models.The sub-model method is based on the Saint Venant＇s principle［2］，that is，using equivalent loads to replace the actually applied loads affects the stress and strain distributions only near the locations where the loads are applied.If the boundary of the sub-model is far away from the stress concentration area，accurate results could be obtained in the sub-model within the boundary.The analysis process of the sub-model method is shown in Fig.1.

Fig.1 Sub-model analysis process

1.2 Sub-model method analysis steps of ANSYS［3］

1.2.1 Generation and analysis of the coarse full model This step is to construct the model for the whole structure and to perform the simulation analysis，which has no difference with the analysis process of an ordinary model.The mesh is coarser than the sub-model and the simulation does not need to contain local details such as rounded corners.Some small structures could be omitted，however，

the finite element mesh must be detailed enough in order to obtain reasonably accurate displacements on cut-boundary，as demonstrated in Fig.2.

Fig.2 Coarse full model

1.2.2 Generation of the sub-model

To generate a sub-model model is actually to construct a new FEM model.The element types，real constants and material properties must be the same as those used in the coarse full model，and the position of the sub-models(relative to the global coordinate origin)should be the same as the corresponding part of the coarse full model，as shown in Fig.3.

Fig.3 Sub-model mesh

1.2.3 Cut-boundary

This step is key to the analysis of the sub-model method，which is to define the boundary location of the sub-model relative to the coarse full model，as shown in Fig.4.In ANSYS，the nodes on the boundary of the submodels can be arbitrarily chosen and do not have to be those of the coarse full model.Once a cut-boundary is defined，ANSYS software system can automatically get the displacement values of the nodes on the boundary of the sub-model by interpolating the displacements of the nodes in the coarse full model.

Fig.4 Refined mesh in cut-boundary

1.2.4 Analysis of the sub-model

In this step，the analysis type and analysis options of the sub-model must be chosen the same as those in coarse full model.In addition，all the corresponding load boundary conditions，constraints to the coarse full model must be copied to the sub-model.The sub-model can then be solved，as shown in Fig.5.

1.2.5 Verification of the rationality of the cut-boundary

The sub-model method requires that the boundary displacements of a sub-model be as far as possible consistent and continuous with the cut-boundary displacements of the coarse full model.This must be verified as shown in Fig.6.The verification method is to compare the sub-model boundary results with the corresponding boundary results of the coarse full model，as shown in Fig.7.If requirements are well satisfied，the cut-boundary selection is right;otherwise，a new cut-boundary far from the concerned area must be defined and a new sub-model should be established and solved.

Fig.5 Anlysis of sub-model

Fig.6 Continuous displacement contours

Fig.7 Comparison of results on the cut-boundary

2 Example

To verify the sub-model method in structure analysis，the stress in a stretched plate with a round hole is calculated，as shown in Fig.8.This problem is a typical example in continuum mechanics.The analytical solution in the polar coordinate is as follows［4］:

when ρ= α，the annular normal stress is σφ=q(1 －2cos2φ)along the edge of the hole.When φ=90°and ρ=a(i.e.point A，as shown in Fig.8)，σρ=0 ，σφ=3q and τρφ=τφρ=0.If q=1 MPa，σφ=3 MPa.

If using the coarse full model method，the finite element meshing is shown in Fig.9.The total number of nodes is 3 733 and the total number of elements is 1 359.The results are shown in Fig.10 and the error is(3 －2.832)/3=5.6%.

Fig.8 A stretched plate with a round hole

Fig.9 A coarse full model

Fig.10 Stress results obtained from the coarse full model

If using the sub-model method，the finite element model is shown in Fig.11，where the total number of nodes is 3 003 and the total number of elements is 1 209.The simulation results are shown in Fig.12，confirming that the stress contours are smoother within the cut-boundary.The calculation error are(3.042-3)/3=1.4%.This suggests that more accurate solutions could be obtained by applying the sub-model method and using less elements and nodes.

Fig.11 Sub-model mesh

Fig.12 Stress results of the sub-model

In order to compare the displacements on the cutboundary of the sub-model with those of the coarse full model，the displacements along the cut-boundary of the cutting path a-b-c-d-a，are drawn in Fig.13.

Fig.13 shows that the results from both the coarse model and the sub-model agree quite well.Thus，the cutboundary is right.

Fig.13 Comparison of displacement results on the cut-boundary

3 Structure analysis for a missile cabin

For example，a cabin，as shown in Fig.14，has two grooves with a depth of nearly the same as the wall thickness due to the manufacturing error.It undergoes combined axial compression and bending loads in applications.Stress analysis and strength assessment have to be done for the cabin because the defects are a kind of serious deviation from the design condition.If the conventional FEM method was adopted，a large amount of computing time and better hardware support would be required in order to get ideal results，and even so，sometimes，the calculation could not be executed under the current hardware conditions due to the large numbers of elements and nodes.So in this paper，the sub-model method was adopted to analyze the groove defects for the cabin.

Fig.14 A missile cabin structure

Fig.15 Sub-model for a missile cabin

The cabin is made of an aluminum alloy with a Young＇s modulus of 71 000 MPa，and a Poisson＇s ratio of 0.31.The Full FEM model and sub-model are shown in Fig.15 and the numbers of the elements and nodes are given in Table 1.

From Table 1，the sub-model is smaller compared to the coarse full model.By using the sub-model analysis technique，the model can be meshed coarsely for the whole structure and densely in the local region of interest，and hence the overall size of the problem and the demand for hardware can be significantly reduced.

Table1 Numbers of elements and nodes

Loads and boundary conditions:Front-end fixed，backend bears an axial compression load of 44 159 N and a bending load of 7.615 5×106N·mm，as shown in Fig.16.

Fig.16 Loads and boundary conditions

The displacement results are shown in Fig.17.It can be seen that under the application of the loads，the maximum deformation of the back-end face of the cabin relative to the front end face is 0.286 19 mm.

Fig.17 Displacement results

The displacement results of the cut-boundary and the area containing defects are shown in Fig.18.It can be seen that the displacement contours are continuous on and near cut-boundary，and smooth inside the sub-model.The selection of cut-boundary is therefore correct.As a result，more accurate results near the groove defects of the cabin have been obtained.

The stress results，as given in Fig.19，of the defect region show that themaximum Von-Mises stress is 139.88 MPa，smaller than the strength of aluminum alloy(310 MPa).Thus，the cabin containing the defects meets the strength requirement.

Fig.18 Displacement contours near groove defects of the cabin

Fig.19 Contours of Von-Mises stress near groove defects of the cabin

4 Conclusion

Using the sub-model method can effectively solve the stress concentration problem in missile structure analysis.More accurate numerical solutions can be obtained for the stress concentration parts by using the sub-model analysis technique.The ANSYS sub-model analysis technique has broad application prospects，and can be widely used to analyze complex engineering structures.This technique is helpful to reduce the size of the FEM model and the requirements for hardware，and to improve the work efficiency.

Reference:

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［3］PERA GLOBLE Ltd..ANSYS advanced finite element analysis technology［M］.Beijing:Astronautics press.1992.

［4］Xu Zhi-guan.Elastic mechanics［M］.Higher Education press.2006.