3D simulation of laminated shell structures using the Proper Generalized Decomposition

TLDR: In this article, the authors explore an alternative to shell computation using the framework of the Proper Generalized Decomposition that is based on a separated representation of the solution and solve the full 3D solid problem separating the in-plane and the out-of-plane spaces.

About: This article is published in Composite Structures.The article was published on 2014-11-01 and is currently open access. It has received 13 citations till now. The article focuses on the topics: Shell (structure) & Finite element method.

Figures

Fig. 2. Laminated cylindrical shell.

Table 1 Results of the PGD compared with the exact solution for stresses and displacement.

Fig. 3. Stress for S ¼ 50: (a) rxx over the thickness for h ¼ p6. (b) rzz over the thickness for h ¼ p 6 . (c) rxz over the thickness for h ¼ 0.

Fig. 4. Displacement u over the thickness for h ¼ 0 and S ¼ 50.

Fig. 5. rxx field obtained with the PGD for S ¼ 10 (left) – Error in comparison with the exact solution for S ¼ 10(right).

Fig. 6. Curved geometry (left). Mid-plane surface and local basis at nodes (right).

Frequently asked questions

Q1. What is the alternative to shell elements for reducing the computational time?

An alternative to shell elements for reducing the computational time is to use model reduction strategies based on a separated representation of the solution. 

Q2. What is the purpose of this article?

The idea of this article is to use the PGDmethod to perform simulations of shell structures with complex shapes and curvatures without any a priori knowledge of the comportment in the thickness and using the PGD efficiency. 

Q3. What is the main interest of this approach?

The main interest of this approach is that it can treat complex phenomena like damaging in composites structures that are difficult to model with shell elements. 

Q4. What is the main advantage of the PGD?

shell models are generally more efficient in a computational point of view because they require only one 2D calculation when the PGD requires many 2D calculation (with less degrees of freedom though). 

Q5. What is the common way to extract the basis functions from data?

The most common way to extract the basis functions from data is to perform a Proper Orthogonal Decomposition (POD) which gives the most significant modes. 

Q6. What are the functions related to the thickness of the shell?

They represent a combination of modes related to bending, rigid displacements, compression of the normal fiber and also more complex modes related to boundary effects. 

Q7. What is the cost of shell modeling?

Shell models help reducing the computational cost required by full 3D finite element modeling but they require an approximation of the comportment in the thickness. 

Q8. What is the number of bases functions in the shell theory?

The number of basis functions is not related to the number of nodes like in the classic finite element method and is generally very restricted. 

Q9. What is the simplest way to solve the system?

This system can be solved using a finite elements discretization, the unknown being the nodal values of all the functions Fi for i ¼ f1;2; . . . ;Ng. 

Q10. What is the material properties of the GPa ET?

The materials properties are the ones used in [18]:EL ¼ 172 GPa ET ¼ 6:9 GPaGLT ¼ 3:4 GPa GTT ¼ 1:4 GPa mLT ¼ mTT ¼ 0:25ð23ÞL is the direction parallel to the fibers and T is the transverse direction. 

Q11. What is the mean relative error for the displacement?

The mean relative error (given as global indicator) is 0.08% for the displacement and around 3% for the different stress components. 

Q12. What is the common way to determine the basis functions?

The basis functions are determined ‘‘a posteriori’’ using for instance some results coming from a full computation or experimental data or just postulated from physic.