INTRODUCION FEATURES

This project develops a custom finite element solver integrated with NaxToPy as a pre-processor, SfePy[1] as a solver and custom code as post-processor, in order to demonstrate the capabilities of generalization of NaxToPy in a pre-processing environment. The project is based on the resolution of general three-dimensional isotropic shell elements with isotropic and composite materials, and also isotropic and composite three-dimensional elements. The results obtained via several test models have been checked to ensure its validity by comparing the results with a certified commercial solver: Altair Optistruct[2].

The finite element software industry has experienced a considerable increase in licensing prices in the last two decades, becoming less accessible to the general public in a sector that is continuously growing. It is for this reason that this project arises in which we have developed a custom finite element solver, integrated with NaxToPy as preprocessor, SfePy as solver, custom code as postprocessor, with the use of the open-source packages NumPy and SciPy.

SMARTM IMPLEMENTATION

The framework of linear elasticity resolution using NaxToPy in the pre-processing and post-processing stage, and SfePy as a solver has been implemented in the Research & Development project at Idaero Solutions, SMARTM.

The SMARTM project seeks to improve the competitiveness of Resin Transfer Molding (RTM) processes through the development of digital predictive, monitoring and process control technologies. These technologies include digital twin visualization, advanced monitoring systems and real-time process simulation to ensure final product quality. With a focus on defect reduction and resin flow optimization, SMARTM aims to increase efficiency and sustainability in out-of-autoclave composite manufacturing in the aerospace industry.

The developed framework is used as a structural twin that computes the behavior of a composite model that has undergone the manufacturing processes of Automatic Fiber Placement (AFP) and Resin Transfer Molding (RTM).

These processes can cause defects in the production chain that negatively affect the properties and the performance of the model, which can result it unusable structures. The defects studied within the possible failure modes in the AFP process are: gaps, overlaps and misorientation of the plies. The defects within the RTM process are related to the expansion and behavior of the resin over time. The structural twin will not study this effect directly, but rather will receive its effect in the set of defects and associated porosity.

The structural twin in the SMARTM environment will model the set of defects according to existing experimental models and previous bibliography: [1], [2], [3]

[1]	H. Naceur, S. Shiri, D. Coutellier and J. L. Batoz, "On the modeling and design of composite multilayered structures using solid-shell finite element model," Finite Elements in Analysis and Design, Vols. 70-71, pp. 1-14, 2013.
[2]	X. Li, S. Hallett and M. Wisnom, "Modelling the effect of Gaps and Overlaps in Automated Fibre Placement (AFP) manufactured laminates," Science and Engineering of Composite Materials, vol. 22, pp. 115-129, 2015.
[3]	Y. Torres, J. Pavon Palacio and J. A. Rodriguez-Ortiz, "Influence of sintering conditions on the microstructural and mechanical properties of porous Ti c.p. for biomedical applications," Anales de la Mecánica de Fractura, vol. 1, pp. 63-68, 2010.

THEORETICAL BACKGROUND

The framework between NaxToPy and SfePy, will consider static analysis and linear elasticity. The systems under a static behaviour (static equilibrium and static loading) can be represented by the following equation:

Which is a linear system of equations. In this system, K represents the stiffness matrix of the system, x represents the static response of the system, and F represents the vector containing the forces applied to the system.

The linear elasticity problem focuses on the linear relationship between stresses and strains. The equations defining the problem derive from the Cauchy momentum equation in convective form. The boundary value problem (BVP) of linear elasticity is defined by the equilibrium equations, the strain-displacement equations and the constitutive equations:

The materials are represented in the constitutive matrix C. In the case of isotropic materials, that is, materials whose properties are uniform independently of the direction; the components of the constitutive matrix can be written as[5]:

Therefore, the relationship between stresses and strains can be simplified to:

Additionally, the previous expression can be simplified using plane stress. For composite materials, the laminate theory is used to obtain the constitutive matrix between stresses and strains. The stiffness of a ply under plane stress assumption can be obtained as[6]:

This matrix is transformed according to the orientation of the ply, , and then all the ply matrices contribute to the global stiffness matrix of the material as follows:

The displacements are solved using SciPy’s SuperLU solver, which is a fully-pivoted LU decomposition of the problem matrix. SciPy also provides iterative methods such as CG and GMRES.

SfePy does not support the computation of stresses and strains for 2D elements but it does for 3D elements.

For 2D elements, the post-processing of stresses and strains was obtained through the degenerated isoparametric shell element transformation of displacement to strains.

The definition of the shape functions as well as its derivatives (matrix B) and the Jacobian matrix of transformation (matrix J) was done according to El-Zeiny[7].

Several models have been tested, and the results can be observed below:

LIMITATIONS

The limitations and requirements of the tool are depicted below:

The program works with the following inputs:

Binary file of the geometry mesh (.vtk, .hmascii (Hypermesh), .gmsh, .cdb (Nastran), .hdf5, .XYZ, .mesh3d)
Input file with the load case: In .fem, .dat format with Nastran/Optistruct cards
Microsoft .csv defects file with the format indicated in the User Guide (empty file with the headers if not working with defects in composite materials for 3D elements)

Only first order elements have been tested.

The framework is focused on solving the linear elasticity problem. Other analysis involving dynamic loading, modal analysis, etc. are not considered within this framework

[1] Cimrman, R., Lukeš, V., Rohan, E., 2019. “Multiscale finite element calculations in Python using SfePy.” Advances in Computational Mathematics 45, 1897-1921. https://doi.org/10.1007/s10444-019-09666-0

[2] https://altair.com/optistruct

[3] Boon, Y., Joshi, S., Bhudolia, S., & Gohel, G. "Recent Advances on the Design Automation for Performance-Optimized Fiber Reinforced Polymer Composite Components," in Journal of Composites Science, vol. 4, pp. 61, 2020.

[4] Bhat, P., Merotte, J., Simacek, P., & Advani, S.G. "Process analysis of compression resin transfer molding," in Composites Part A: Applied Science and Manufacturing, vol. 40, no. 4, pp. 431-441, 2009.

[5] R. Radovitzky, «Constituitive Equations,» 2013. [En línea]. Available: https://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive.html. [Último acceso: 2024 April 19].

[6] Nettles, Alan. (1994). Basic Mechanics of Laminated composite plates.

[7] A. A.-W. El-Zeiny, «Nonlinear Tilne-Dependent Seismic Response of Unanchored Liquid Storage Tanks,» PhD Thesis. Department of Civil and Environmental Engineering. University of California, Irvine, Irvine, CA, 1995.