Optimisation of Recycled Thermoplastic Plate (Tile)

Document Type: Regular Article

Authors

Department of Civil Engineering, University of Ilorin, Ilorin, Nigeria

Abstract

The purpose of this paper is to perform a structural optimisation of a flat thermoplastic plate (tile). This task is developed computationally through the interface between an optimisation algorithm and the finite element method with the goal of minimising the equivalent stress with a specified target stress of 2 MPa when applied with a load intensity of 1000N. A 300 x 300 x 20 mm thermoplastic plate was selected for the optimisation, which was performed with a tool in MATLAB R2012b known as genetic algorithm accompanied with a static analysis in ANSYS 15. The results produced the optimum equivalent stress (δopt) of 2.136 MPa with the optimum dimensions of 305 x 302 x 20 mm. Also, the dimensions of the plate with the optimum value of the equivalent stress were discovered to be within the lower and upper bound dimensions of the plate. The thermoplastic plate object of the optimization was a square plate of 300 x 300mm and 20 mm thick with isotropic properties and a particular load and boundary conditions were applied on the entire plate.

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1. Introduction

The amount of solid waste is ever increasing due to increase in population, developmental activities, changes in life style, and socio-economic conditions. Plastics waste is a significant portion of the total municipal solid waste [1] consumption of plastic products has increased dramatically over the past few decades, they account for more than 70% of total plastics market [2]. This trend results in the generation of a vast waste stream that needs to be properly managed to avoid environmental damage [3]. Many countries around the world are continuously faced with the problem of generation and disposal of plastic wastes. Governments around the world have funded hundreds of research projects to find efficient waste treatment technology [4]. Water sachet polyethylene are the most widely used polyethylene in developing countries especially Nigeria. Sachet water making factories both licenced and unlicensed are found in virtually every street in cities, towns and villages of Nigeria. During the dry season about 70% Nigerian adults drink at least one sachet of pure water daily indicating that about 50-60 million used-water sachet are thrown into the streets of Nigeria on a daily basis [7]. The packaging of this sachet water is made of non-biodegradable synthetic polyethylene (polythene), which does not decay, decompose or corrode, and which when burnt, produces oxides of carbon, nitrogen and sulphur
which can harm man and the environment [9] and [10]. Waste recycling is often seen as an important aspect of an efficient and effective solid waste management system, plastic materials can be formed into shapes by different processes: extrusion, moulding and casting or spinning. Mechanic properties particularly stiffness, strength and toughness of polymeric materials are decisive properties in industrial, technological and household applications [9]. Several studies have been carried out on the recycling of polyethylene [13], [15], [16] and [17]. Since these water sachets still possess some properties of polyethylene, they can be used in blend with pure polyethylene in making different polyethylene products like baskets, hats, ropes, bags, mats and for fabric sewing [18].

The amount of solid waste is ever increasing due to increase in population, developmental activities, changes in life style, and socio-economic conditions. Plastics waste is a significant portion of the total municipal solid waste [1], consumption of plastic products has increased dramatically over the past few decades, they account for more than 70% of total plastics market [2]. This trend results in the generation of a vast waste stream that needs to be properly managed to avoid environmental damage [3]. Many countries around the world are continuously faced with the problem of generation and disposal of plastic wastes. Governments around the world have funded hundreds of research projects to find efficient waste treatment technology [4].Water sachet polyethylene are the most widely used polyethylene in developing countries especially Nigeria. Sachet water making factories both licenced and unlicensed are found in virtually every street in cities, towns and villages of Nigeria. During the dry season about 70% Nigerian adults drink at least one sachet of pure water daily indicating that about 50-60 million used-water sachet are thrown into the streets of Nigeria on a daily basis [7]. The packaging of this sachet water is made of non-biodegradable synthetic polyethylene (polythene), which does not decay, decompose or corrode, and which when burnt, produces oxides of carbon, nitrogen and sulphur
which can harm man and the environment [9] and [10]. Waste recycling is often seen as an important aspect of an efficient and effective solid waste management system, plastic materials can be formed into shapes by different processes: extrusion, moulding and casting or spinning. Mechanic properties particularly stiffness, strength and toughness of polymeric materials are decisive properties in industrial, technological and household applications [11]. Several studies have been carried out on the recycling of polyethylene [13], [15], [16] & [17]. Since these water sachets still possess some properties of polyethylene, they can be used in blend with pure polyethylene in making different polyethylene products like baskets, hats, ropes, bags, mats and for fabric sewing [18]. Andrei carried out a work on recycling of waste polyethylene into thread but was not blended with pure polyethylene and melt drawing was not carried out. This study is aimed at recycling polyethylene waste by converting waste water sachets into threads for different uses thereby combating the problem of waste management.

The utilization of Genetic Algorithms (GA) in tackling engineering problems has been a major issue arousing the curiosity of researchers and practitioners in the area of systems and engineering research, operations research and management sciences in the past decades. The vast areas of applications of GA optimization techniques in tackling problems that cannot be handled using the conventional methods and stochastic search are the focal areas of keen interest for consideration in this paper. GA is a type of evolutionary algorithm (EA) that is found useful in so many engineering applications which includes numerical and combinatorial optimization problems, filter design as in the field of Signal processing, designing of communication networks, semiconductor layout, spacecraft [12] and [14]. It is founded on the bases of natural biological evolution process which is used to mimic nature in searching for optimal solution of a specific problem [5]. In the description of GA, the definition of chromosome and fitness functions is of paramount importance. Chromosomes are abstract representation of candidate solutions. The fitness function is used in quantifying the desirability of a solution, which is closely correlated with the objective of the algorithm or optimization process. The fitness level is used in evaluating candidate solutions, that is, the values being generated characterize the performance of candidate solutions [6].

In GA, the most promising search space areas are being explored through the utilization of probabilistic rule, hence minimizing the risk of convergence to local minima. This is achieved by simultaneously considering many points in the search space and favouring the mating of the fitter individuals [5] and [6]. GA is a robust search algorithm that enables the quick location of high quality solution areas in a complex and large search space. The fundamental principle of GA includes selection, reproduction, population solution, encoding and decoding, fitness function evaluation and convergence [6] and [8]. This paper presents the optimisation of recycled thermoplastic plate (tile).

2. Materials and Methods

Optimization process

The optimization is performed through an algorithm which exploits the capabilities of the Optimization toolbox included in Matlab R2012b. It contains different functions among which Genetic Algorithmhas been chosen as explained as follows.

The procedure is the following:

  1. The Matlab R2012b algorithm is run and the optimization starts;
  2. An initial value is assigned to the variables and txt file;
  3. ANSYS15 reads the parameters present in the .txt file and uses them as input;
  4. The static analysis is performed and the total equivalent stress value is exported to another .txt file;
  5. Matlab R2012b reads this value and continues to iterate in this way until a minimum (at least local) of the equivalent stress is reached.

3. Static analysis

The goal of the static analysis is to find the defects whether or not they may cause failure using the Finite Element (FE) method to perform a static analysis of the plate (tile) through the engineering tool ANSYS15. ANSYS15 is launched by Matlab R2012b at each iteration. In this way, the optimization process needs only to be started by the user in Matlab R2012b, and it continues to iterate until either an optimal solution is reached or a stopping criterion is satisfied.

Properties of material and Parameters used in the FE analysis

The property of the materials used for the FE analysis of the material is shown in Figure 1.

 

Fig. 1. Properties of thermoplastic tile

The design variables of the plate are length (l)and width (w)which appear explicitly in this phase, and the thickness of the plate is specified but was not included as a design variable.

On the other hand, it is necessary to choose between triangular or quadrilateral meshes, which are the two possible choices when the mesh has to be defined over an area. In the optimization performed a quadrilateral mesh has been adopted because, besides a better quality of the solution it speeds up the analysis time.

4. Boundary conditions and loads

In the optimization performed on the tile, the boundary condition adopted was an elastic support of foundation stiffness 18N/mm3 with all the edges simply supported as shown in Figure 2. Also, aforceof intensity 1000 N was applied on the plate before beginning the optimisation as shown in Figure 3.

 

Fig. 2. Boundary conditions of material

 

Fig. 3. Load intensity on the material

Once the boundary conditions and the loads are defined in the ANSYS15, the static analysis is performed and the total equivalent stress and deformation of the structure is calculated by summing the single equivalent stress and deformation values of each element that forms the panel. For this, the ANSYS15 element tableof the post processingstep is used to obtain the equivalent stress and deformation of each element and then each component of the table defined is summed in order to obtain the total value.

In the end, ANSYS15 writes the total equivalent stress and the total deformation value on a .txt file, and this represents the last step of the interface between Matlab R2012b and ANSYS15.

 

5. Results and discussion

Optimisations with l and w as design variables

Table 1 shows the optimizations done for different initial points, presenting in particular the minimum value of the equivalent stress reached, the optimal values of the design variables lopt and wopt are also presented in the table.

Table 1. Various design points (iterations)

Design Points

Length(lopt)mm

Width(wopt)mm

Equivalent stress (δopt)MPa

1

305

302

2.1381

2

300

302

2.1563

3

310

302

2.1251

4

305

300

2.1412

5

305

304

2.1300

6

300

300

2.1683

7

310

300

2.1402

8

300

304

2.1788

9

310

304

2.1344

 

In this case the initial value of the thickness (t)is kept constant, and only the length and the width of the plateis changed because it turned out to be the most problematic in the convergence of the solution. The upper and lower bounds of the variables in these optimizations are presented in Table 2.

Table 2. Upper and lower bounds for optimisation

Bounds

Length (mm)

Width (mm)

Upper bound

310

304

Lower bound

300

300

 

From the optimisation performed, a targeted stress 2 Mpa was sought and running the optimisation program again, it yielded three candidate points of which if a more refined optimisation is performed, these three points will yield the targeted stress. These points are presented in Table 3.

Table 3. Candidate points for the sought target

 

Candidate points 1

Candidate points 2

Candidate points 3

Length(mm)

308.0189

308.0150

308.0250

Width(mm)

300.1763

302.0762

301.0762

Stress(MPa)

2.1282

2.1282

2.1282

From the FE Analysis performed in ANSYS15, the deformations and equivalent stress were determined as shown in Figure 4 and Figure 5 respectively. The result also shows that their maximum values were obtained to be 2.1683MPa and 0.0025mm respectively.

 

Fig. 4. Deflections of plastic plate when analysed

 

Fig. 5. Equivalent stress of plastic plate when analysed

However, from the optimisation performed, the equivalent stress determined at each design point (iteration) was plotted as shown in Figure 6. Also, the maximum and minimum equivalent stress was determined at the design point 8 and design point 3 respectively.

 

Fig. 6. Design points (iteration) against parameters

Moreover, the dimensions determined at various design points were plotted against the equivalent stress (length and width) and the feasible points of the plate were found with a length of 305mm, width of 302mm and the equivalent stress of this dimensions was 2.1364MPa. The graph of these feasible points is presented in Figure 7 for 2D and Figure 8 for 3D.

 

Fig. 7. Dimensions of the plate (mm) against Equivalent stress (MPa) in 2D

 

Fig 8. Dimensions of the plate (mm) against Equivalent stress (MPa) in 3D

The equivalent stress predicted from the response surface before performing the optimisation was also plotted against the equivalent stress observed at various design points (iteration) and a goodness of fit line was drawn for the best possible straight line which was found not passing through the origin as shown in Figure 9.

 

Fig. 9. Goodness to fit for predicted against observed values of the stress.

Moreover, the equivalent stress determined at various design points were plotted against the dimensions (length and width) which shows a cubic curve and which when interpolated with a length of 305mm gives the optimum value of the equivalent stress of 2.1364MPa as shown in Figure 10 for 3D and Figure 11 for 2D.

 

Fig. 10. Equivalent stress (MPa) against Dimensions of the plate (mm) in 3D

 

Fig. 11. Equivalent stress (MPa) against Dimensions of the plate (mm) in 2D

The local sensitivity and the local sensitivity curves (LSC) were also plotted and indicated the same optimum value as in the previous cases. The equivalent stress is plotted against the LSC as shown in Figure 9, this allows not only to assess the overall accuracy but also to select the best threshold. With a very low threshold, everything will be detected but without specificity. With a very high threshold, there will be perfect specificity but will never classify anything as positive. It is possible to plot the specificity on the x-axis and just reverse the direction so it goes from 1 to 0 instead of 0 to 1. It’s more intuitive with a specificity of 1, which is the rate of more false positives among all cases that should be negative. As we move along the curve, we get more true positive but also more false negative. This description is best shown in Figure 12 and Figure 13. Note that, the point of intersection between the horizontal line and the curve marks the optimum point.

 

Fig. 12. Local sensitivity curves for Equivalent stress (MPa)

 

Fig. 13. Local sensitivity for Equivalent stress (MPa)

The parameters parallel chart was also plotted which shows the updated design points performed in the optimisation. This chart also displays the lower and the upper bound of both the length and the width and also the minimum and maximum value of the equivalent stress performed during the optimisation process. This chart is presented in Figure 14.

 

Fig. 14. Parameters parallel chart for the optimisation.

As it is visible from the graphs plotted, all the simulations converged to same values of length 305mm, width 302mm and with the equivalent stress of 2.1364MPa.

6. Conclusions

The goal of this paper was to build a computational model able to find the optimum value of a recycled thermoplastic plate (tile).

The thermoplastic plate object of the optimization was a square plate of (300x300x10mm) with isotropic properties and a particular load and boundary conditions were applied on the entire plate.

In particular, the total equivalent stress of the plate was chosen as objective function to minimize, and the design variables were the length (l) and the width (w) of the plate for the first set of optimizations (the most relevant), only two design variables at once were considered in each optimization, because otherwise the computational effort would have been too big and the time required for each simulation as well.

An interface between the engineering tools Matlab R2012b and ANSYS15 was created to perform such optimization that adopted the Genetic algorithm from Matlab R2012b optimization toolbox, and a FE analysis to calculate the equivalent stress of the panel at each iteration, which the optimum point was reached and was found to be 2.1364MPa with the optimum dimensions of 305mm and 302mm for the length and the width of the plate respectively and a thickness of 10mm (fixed) with a load intensity of 1000N. It can therefore be concluded that, for a tile of this type, the proposed dimensions of 300x300x20mm should not be used for the load intensity of 1000N but the optimum dimensions of 305 x 302 mm and 20 mm thick will be suitable for that load.

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