# Fast assembly of Galerkin matrices for 3D solid laminated composites using finite element and isogeometric discretizations

## TL;DR

This paper introduces a new method to significantly accelerate the assembly of stiffness matrices for 3D laminated composites by decomposing the problem into lower-dimensional integrals, reducing computational complexity.

## Contribution

The novel approach reduces assembly time for laminated composite matrices by splitting terms into in-plane and out-of-plane parts, independent of the number of layers.

## Key findings

- Reduces computational time by orders of magnitude for high-layer composites.
- Matrices assembled are identical to standard methods up to machine precision.
- Performance validated through numerical experiments.

## Abstract

This work presents a novel methodology for speeding up the assembly of stiffness matrices for laminate composite 3D structures in the context of isogeometric and finite element discretizations. By splitting the involved terms into their in-plane and out-of-plane contributions, this method computes the problems's 3D stiffness matrix as a combination of 2D (in-plane) and 1D (out-of-plane) integrals. Therefore, the assembly's computational complexity is reduced to the one of a 2D problem. Additionally, the number of 2D integrals to be computed becomes independent of the number of material layers that constitute the laminated composite, it only depends on the number of different materials used (or different orientations of the same anisotropic material). Hence, when a high number of layers is present, the proposed technique reduces by orders of magnitude the computational time required to create the stiffness matrix with standard methods, being the resulting matrices identical up to machine precision. The predicted performance is illustrated through numerical experiments.