# Name: 		_README.txt fr Ordner "\03_Multiscale\01"

# Content:		This folder contains all data for the first multiscale example.
				
				The first multiscale example is a clamped beam subject to bending. Different mesostructures are to be 
				compared by using three different RVEs at mesoscopic scale
					1) a homogeneous RVE with linear elastic material (E = 100, nu = 0.3)
					2) an RVE with cylindrical inclusion (radius r = 0.106) parallel to the beam axis
					   linear elastic material is used (E_M = 27000, nu_M = 0.2, E_F = 142000, nu_F=0.35)
					3) an RVE with cylindrical inclusion (radius r = 0.106) perpendicular to the beam axis
					   linear elastic material is used (E_M = 27000, nu_M = 0.2, E_F = 142000, nu_F=0.35)
				
				Two different beam length are investigated to examine the influence of shear: L=6 and L=20.
				
				The RVE has dimensions lx x ly x h, where he in-plane RVE size of the RVE is chosen to be equal in both 
				directions (i.e. lx=ly=LRVE) and is consecutively increased, LRVE=1,2,4,8,16,32,64. The RVE height is
				kept constant and is equal to the macroscopic shell thickness (h=1).
				
				The vertical tip displacement of the beam is compared. 
				
				Case 1) 
					In Fig. 7.13 the relative error (compared to the analytical solution) is plotted vs. the size of the 
					RVE LRVE. All three types of boundary conditions (tbc, sbc, pbc) are presented. The results for the
					short beam L=6 and the longer beam L=20 are presented in subfigures (a) and (b), respectively.
								
				Case 2) 
					In Fig. 7.15 the absolute tip displacement is plotted vs. the size of the RVE (LRVE) for different 
					boundary conditions. As a reference the solutions from a full-scale volumetric model are given as 
					dashed lines.
				
				Case 3) 
					In Fig. 7.16 the absolute tip displacement is plotted vs. the size of the RVE (LRVE) for different 
					boundary conditions. As a reference the solutions from a full-scale volumetric model are given as 
					dashed lines.
				
				Three different reference solutions are constructed for the full-scale volumetric model, which differ 
				mainly in their treatment of the displacement uz at the boundary, compare Fig. 7.14 [1].
				They form an upper, lower and intermediate value for the multiscale example. Standard quadrilateral 
				elements have been used for the reference solution.
				
				Abbreviation of the boundary conditions:
				pbc - periodic boundary conditions
				sbc - shell boundary conditions
				tbc - traction boundary conditions
				
				Detailed description may be found in [1].

# Datum:		28.08.2024

# Author:		Leonie Mester

# Folder:		01_homogeneous	- Data for clamped beam example using homogeneous mesostructures
				02_longitudinal - Data for clamped beam example using RVE with cylindrical inclusion parallel to beam axis
				03_transversal	- Data for clamped beam example using RVE with cylindrical inclusion perpendicular to beam axis

# Files:		Fig_7-12.pdf	- Figure 7.12 from [1], illustration of clamped beam subjected to line load p and the
								  three different mesostructures
				Fig_7-13.pdf 	- Figure 7.13 from [1], relative error of the vertical tip displacement for different
								  RVE sizes (LRVE), boundary conditions (pbc,sbc,tbc), and macroscopic beam length (L)
				Fig_7-14.pdf	- Figure 7.14 from [1], illustration of boundary conditions for full-scale volumetric model
				Fig_7-15.pdf	- Figure 7.15 from [1], comparison of absolute vertical tip displacement for longitudinally
								  reinforced beam using different RVE sizes (LRVE), boundary conditions (pbc,sbc,tbc), 
								  and macroscopic beam length (L)
				Fig_7-16.pdf	- Figure 7.16 from [1], comparison of absolute vertical tip displacement for transversally
								  reinforced beam using different RVE sizes (LRVE), boundary conditions (pbc,sbc,tbc), 
								  and macroscopic beam length (L)

				
# References:	[1]  	Leonie Mester, Computational Homogenisation and Multiscale Modelling Employing an Image-based
						Approach for the Structural Analysis of Shells, RWTH Aachen, PhD Thesis (2024)