Content uploaded by Ahmed H. Abdulaziz

Author content

All content in this area was uploaded by Ahmed H. Abdulaziz on Apr 13, 2019

Content may be subject to copyright.

2019 UKACM Conference City, University of London

1

PARAMETRIC STUDY OF HONEYCOMB COMPOSITE STRUCTURE

USING OPEN SOURCE FINITE ELEMENT SOFTWARE

*Ahmed H. Abdulaziz1,2, Mohammed Hedaya1, John P. McCrory2, Karen M. Holford2,

Adel Elsabbagh1

1Design and Production Engineering Department, Faculty of Engineering,

Ain Shams University, Abbaseya, Cairo, Egypt, 11517

2School of Engineering, Cardiff University, Wales, United Kingdom, CF24 3AA

* ahesham@eng.asu.edu.eg

Summary

This paper deals with the parametric optimisation of a simply supported sandwich panel made

of honeycomb composite structure using sequential quadratic programming SQP. The panel

consists of aluminum honeycomb sandwiched between two orthotropic fiberglass faces. The

parameters studied are fibreglass thickness, tf, honeycomb height, h, and honeycomb wall

thickness, tc. The objective was to minimise weight to bending stiffness ratio by using the

nonlinear MATLAB function fmincon, considering the maximum central displacement and

intercellular buckling as the constraints. Following this, a static structural analysis was

conducted on the optimised structure using the open source finite element solver CalculiX and

Salome Platform software for preprocessing. The utmost displacement of the honeycomb

panel was desirably less than the serviceability limit. Preliminary results show that composite

honeycomb structures can be optimised yielding low weight to bending stiffness ratio using

SQP method and CalculiX for design evaluation.

Keywords: Parametric optimisation; Sequential quadratic programming; CalculiX; Salome

Platform; Honeycomb composites

Introduction

In the wind turbines industry, blade materials must have a low weight to bending stiffness ratio

for optimal aerodynamics. Aluminum honeycomb can play a vital role in manufacturing longer

wind turbine blades with fibre glass as outer skin due to the cost benefit when compared with

using carbon fibre composites alone. This study proposes a parametric optimisation using

Sequential Quadratic Programming algorithm SQP, which can be used in MATLAB’s fmincon

function to minimise weight to bending stiffness. This function deals with nonlinear constrained

convex objective functions with linear/nonlinear equalities and inequalities. To evaluate the

optimum values, structural analysis using finite element method was carried out using the open

source finite element solver CalculiX. CalculiX has many interesting features such as wide

compatibility with open source CAD modelling and mesh generating softwares such as Salome

platform, FreeCad and GMSH. Further, it is extremely versatile as is it can be modified using

Python programming on Linux platform. For instance, recently Genao et al [1] have proposed

a framework to merge Calculix FE solver with NASA’s Micromechanics Analysis Code MAC to

promote multiscale analysis of the composite materials effectively. Galehdari et al [2] have

optimised honeycomb structural parameters using SQP and genetic algorithm for minimising

the weight to absorbed energy ratio to increase the crashworthiness. Park et al [3], have

conducted structural linear analysis on a cantilever model and sliding contact analysis using

CalculiX and Code_Aster comparing the results favourably with ANSYS commercial FE

software. Therefore, in this paper, Salome platform was used for meshing the honeycomb

composite structure and CalculiX FE solver was used for conducting structural analysis.

Methodology

The common failure modes of sandwich structures may happen due to severe shear force,

intercellular buckling, core crushing, delamination in case of orthotropic composite facets,

2019 UKACM Conference City, University of London

2

shear crimping and punching shear [4]. The sandwich panel dimensions width, b, and length,

l, are 0.2m× 0.2m respectively, and the honeycomb top/bottom faces are fibre glass with

thickness, tf, honeycomb height, h, and with wall thickness, tc. The sandwich panel is treated

as a shell structure considering the length/width are significantly larger than the height [5]. The

top/bottom faces consist of three laminates with a combined thickness of tf mm, and a

[0o/90o/0o] layup meaning that the in-plane/bending deformations are decoupled. The fibre

glass composite micromechanics properties are computed using the Halpin-Tsai empirical

approach. Moreover, the research methodology workflow is presented in Figure 1.

Figure 1: Block diagram of the research methodology

According to Bitzer [6] the equivalent bending stiffness of honeycomb sandwich panel can

be computed using Eq. (1).

where; E1 is longitudinal Young’s modulus of faces and Ec is the honeycomb Young’s modulus,

is equal to (1-and are Poisson’s ratio in longitudinal and transverse directions of

composite layer, is (1-

is Poisson’s ratio of aluminum. The weight is

where, g is gravity acceleration, b is breadth, l is length and is

honeycomb density and is top/bottom face material density. Fmincon function in MATLAB

ustilises sequential quadratic programming SQP algorithm to obtain the optimum minimum

value. Therefore, the objective function is to minimise weight to bending stiffness which is

formulated as in Eq. (2)

(2)

The nonlinear constraints of the design are displacement and intercellular buckling. The

displacement at the panel centre should not exceed span/100. The displacement of the panel

at the centre can be computed by Lèvy’s single series as shown in Eq. (3)

(3)

Since the panel is square of side , Eq. (3) can be re-written as given in Eq. (4) [7], and the

intercellular buckling load must be less than critical value as computed in Eq. (5).

≤ (span/100 = 2mm)

(4)

(5)

where k is 5.75, Es is the honeycomb Young’s modulus, The upper/lower limits of the design

variables are tabulated in Table 1. SQP function tolerance is 10-6.

Table 1: Optimisation design variables with upper/lower bounds

(1)

2019 UKACM Conference City, University of London

3

Design Variable

x (1)

x (2)

x (3)

Geometric Parameter

Upper bound

0.002

0.010

0.001

Lower bound

0.0015

0.001

0.0001

As a side note, the positive definite Hessian matrix is a measure of function convexity over the

domain [8]. Therefore, the eigenvalues of the Hessian have been computed and they are

positive. After computations, a local minimum that satisfies the nonlinear constrains has been

detected. Further, the variations of weight to bending stiffness ratio according to different

honeycomb height and faces thickness are plotted in Figure 2. The eigen values of the Hessian

matrix are computed. The iterations stopped as the objective function is non-decreasing in the

feasible region. The buckling load factor (BLF)has been computed (i.e. /) and it

is larger than 1 which indicates safety of the honeycomb cell wall under buckling.

Figure 2: Weight to bending stiffness ratio for different height and face thickness

The optimum values are tabulated in Table 2. Furthermore, the MATLAB script used to obtain

this result is provided at the end of this paper.

Table 2: Honeycomb optimum values

tf (m)

H(m)

tc(m)

W (N)

Deq(N.m)

W/ Deq

(N/N.m)

Intercellular

Critical

Buckling (N)

Buckling

Load

Factor

0.0020

0.0100

0.0003

3.31564

5.8898e+03

5.6295e-04

1599.4

1.6

Finite Element Model

To evaluate the optimisation results, a honeycomb composite panel of zero thickness is

processed in Salome Platform and meshed with “S6” and “S8R” shell elements [9] using

Netgen 1D-2D option with maximum length 3 mm and minimum length 1.5 mm. Further, for

better accuracy in solution, second order approximation for the meshing process is followed.

However, care must be taken in meshing process as unlike commercial softwares, node-to-

node connectivity is not assured for multiple surfaces automatically. Therefore, the sandwich

panel must be partitioned into multiple shells and edges to assure the nodal connectivity. Yet,

after partitioning it, the honeycomb core and top/bottom faces must be grouped as well the two

edges at the bottom face to form the elemental and nodal groups which will be used later for

2019 UKACM Conference City, University of London

4

materials definitions and boundary conditions in CalculiX. The mesh is saved as “.unv” file to

obtain a Python code of elements data. Afterwards, unical mesh converter in CalculiX is used

to convert the (.unv) mesh file into input deck for further finite element analysis. Figure 3a)

presents a block diagram of the FE process and 3b) shows the meshed honeycomb panel. It

consists of 18,309 quadrangle elements “S8R” and 2,863 triangular elements “S6”. These

elements expand to 3D quadratic brick elements and 3D wedge elements in modelling the

top/bottom composite faces .

Figure 3: a) Finite element steps, b) Honeycomb Meshing in Salome Platform

The lateral concentrated force 5,000 N is positioned at the centre of the panel and structural

static analysis is conducted. The maximum central deflection is 1.31 mm as shown in Figure

4a. Compared to serviceability limit (i.e. span/100) which has been utilised within the

optimisation, the utmost deflection obtained by CalculiX for the panel is desirably less. It is

noteworthy to mention that in CalculiX section definition, the shell elements after expanding to

build the required thickness may intersect at the corners as shown in Figure 4b. This

intersection is dependent on the shell offset value and its normal direction whether negative or

positive.

Figure 4: a) Maximum central displacement of the simply supported panel, b) the shell

elements corner intersection after expanding

Conclusions

In conclusion, the honeycomb composite panel has been optimised using sequential quadratic

programming. The ratio of weight to bending stiffness is minimised considering the intercellular

buckling and lateral deflection as the main constraints functions. The optimum geometric

parameters are the faces thickness, core height and core thickness. After the optimisation, a

a) b)

a) b)

2019 UKACM Conference City, University of London

5

numerical model is processed and meshed with S8R/S6 shell elements in Salome platform

then a structural static analysis has been carried out in the open source finite element solver

CalculiX. Overall, it is demonstrated that coding with CalculiX is flexible nevertheless care must

be taken in the section definition. Mainly, the shell element offset and normal direction because

the results are dependent on them. The maximum displacement retrieved from CalculiX was

1.31 mm which is less than the limiting value specified in SQP optimisation. Future research

should be devoted to couple the SQP optimisation code within Salome Python code of the

geometry/mesh to be processed after that in Calculix input deck. In addition, optimisation of

honeycomb composite structure might prove an important area for future optimisation research

so it is recommended that another optimisation technique such as Method of Moving

Asymptotes MMA or Genetic algorithms is used and all optimisation results are examined.

Acknowledgements

This research is carried out within the PhD project entitled “Investigation of Honeycomb

Composite Structure for Wind Turbine Blades with Acoustics Emissions Damage Assessment”

funded by Newton-Mosharafa Fund in Egypt, I.D: (NMJ 3/18). Many thanks to Prof Otto Ernst

Bernhardi in Karlsruhe University of Applied Sciences, Germany, Hossem Elnachmie and

Haoluan Li in Cardiff University for recommendations and discussions about some technical

issues encountered in finite element modelling and optimisation.

References

1

F. A. Yapor Genao, E. J. Pineda, B. A. Bednarcyk and P. A. Gustafson, "Integration of

MAC/GMC into CalculiX, an open source finite element code," in AIAA SciTech Forum,

San Diego, California, 7-11 January 2019.

2

S. Galehdari, M. Kadkhodayan and S. Hadidi-Moud, "Analytical, experimental and

numerical study of a graded honeycomb structure under in-plane impact load with low

velocity," International Journal of Crashworthiness, vol. 20, no. 4, pp. 1754-2111, 2015.

3

S. K. Park, D.W. Seo, H. Jeong and M. Kim, "Performance evaluation of open-source

structural analysis solver, CalculiX and Code_Aster, for linear static and contact

problems," ICIC Express Letters , vol. 12 , no. 7, pp. 655-662, 2018.

4

G. Lubin, Handbook of Composites, Springer Science & Business Media , 2013.

5

T. Kubiak, Static and Dynamic Buckling of Thin-Walled Plate Structures, Lodz, Poland:

Springer, 2013.

6

T. N. Bitzer, Honeycomb Technology: Materials, Design, Manufacturing, Applications

and Testing, Springer Science & Business Media, 2012.

7

S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, Singapore:

Mcgraw-Hill Inc., 1959.

8

A. Messac, Optimization in Practice with MATLAB® for Engineering Students and

Professionals, Cambridge: Cambridge University Press, 2015.

9

E. J. Barbero, Finite Element Analysis of Composite Materials using Abaqus, Taylor and

Francisn Group, 2013.

MATLAB M-script

%This code is written to perform

parametric optimisation using SQP

algorithm-All dimensions are in SI-

units~~After reading left hand

column to its end, continue reading

ub= [0.002,0.010,0.001];

nonlcon = @Constrains;

x0 = [0.0018,0.002,0.0002] ;

%Initialisation point

2019 UKACM Conference City, University of London

6

from top of right column to its

end. ~~File 1 consists of 3

sections. File 2 presents

constraints functions

% File 1-Section 1: Halpin-Tsai

empirical approach for

micromechanics computations &

parameters definition

clc

rhof = 1800 ; % Faces density

rhos = 2700; % Aluminum density

% Material is Al honeycomb

L=0.2; % length

w=0.2; % breadth

g=9.81; % Gravity acceleration

E=60e9;

Ef = 73.1e9; %Fibre glass Young's

modulus

Em = 3.45e9;

vf=0.55; %volume fibre fraction

vm=0.45 ; % volume matrix fraction

uf = 0.22; % poisson ratio

fiberglass

um = 0.33; % poisson ratio Epoxy

Gf = 30e9; %Shear rigidity of fibre

Gm = 1.25e9 ; %Epoxy shear rigidity

v12 = uf*vf + um*vm ;

v21 = v12;

k = 0.9;%k:fibre misalignment

factor

E1 =k*(Ef*vf+Em*vm);%Longitudinal

Young's modulus

zeta = 2;

etae = ((Ef/Em)-1)/((Ef/Em)+zeta);

E2 = Em*(1+zeta*etae*vf)/(1-

etae*vf);%Longitudinal Young's

modulus

etaG= ((Gf/Gm)-1)/((Gf/Gm)+zeta);

G12 = Gm*(1+zeta*etaG*vf)/(1-

etaG*vf);

lambda = 1-v12*v21 ;

v_Al = 0.3 ; %Poisson ratio Al

lambdac = 1-v_Al;

s=0.0064;%Side length of cell

%File 1-Section 2:Optimisation

Formulation

f=@(x)2*g*w*L*(rhof*x(1)+(x(2)*x(3)

*rhos/(s*(3^0.5))))/(((E1*x(1)*(x(2

)+x(1))^2/(2*lambda))

+(E1*x(1)^3/(6*lambda))+(E*x(3)^3/(

12*lambdac))));

A =[];

b = [];

Aeq = [];

beq = [];

lb =[0.0015,0.001,0.0001];

options =

optimoptions('fmincon','Algorithm','

sqp','Display','iter') ;

[x,fval,exitflag,output,lambda,grad,

hessian]=fmincon(f,x0,A,b,Aeq,beq,lb

,ub,nonlcon,options);

disp (hessian);

e = eig(hessian);

[~,r] = chol(hessian);

disp (e)

disp (x)

M=2*g*w*L*(rhof*x(1)+(x(2)*x(3)*rhos

/(s*(3^0.5))));

K = M/fval ;% stiffness at optimum

point

% File 1-Section 3: 3D plot of the

variables and corresponding

objective function

tf1 = linspace(0.001,0.009,10) ;

h1 = linspace(0.005,0.05,10) ;

tc1 = linspace(0.0001,0.009,10);

[XX,YY] = meshgrid(tf1,h1);

[VV] = meshgrid(tc1);

WW =

2*g*w*L.*(rhof*XX+(YY.*VV*rhos./(s*(

3^0.5)))); %Weight at optimum point

; %Weight

DD =(E1*XX.*(YY+VV).^2/ 2*0.9274)

+(E1*XX.^3/(6*0.9274))+(E*VV.^3/(12*

0.7));

Func = WW./DD ;

[FF] = meshgrid(Func);

%plot objective function vs design

variables core height&face thickness

figure

set(gcf, 'PaperPosition', [0 0 4

4]);

C = contourf(XX,YY,Func);

clabel(C,'FontSize',12)

xlabel('Faces Thickness in

m','FontSize',12,'Color','k');

ylabel('Honeycomb Height in

m','FontSize',12,'Color','k' );

%File-2:Constraints.m File

function [c,ceq] = Constrains(x)

%Displacement at centre due to 5000

N concentrated force

c(1)=0.00406*5000*0.2^4/((3.758175e1

0*x(1)*(x(2)+x(1))^2/(2*0.9274))

+(3.758175e10*x(1)^3/(6*0.9274))+(6e

10*x(3)^3/(12*0.7)))-

0.002;%deflection

c(2) = 1000 -(5.75*60e9*x(3)^3/((1-

0.3^2)*0.0064)); %intercellular

buckling acting force on the side

ceq = [];

end