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Proceedings of the 6th International Conference on Mechanics and Materials in Design,

Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26

PAPER REF: 5358

A QUASI USER-INDEPENDENT MESHLESS ME

ANALYSIS OF PLATES

Carla Roque1(*)

, Pedro Martins1

1Institute of Mechanical Engineering (IDMEC), University of Porto, Porto, Portugal

(*)Email: [email protected]

ABSTRACT

A differential evolution optimization scheme

meshless numerical method. In the present paper, DE is used to optimize parameters in radial

basis function meshless method for simulation of isotropic plates in bending under uniform

load. Rectangular and triangular geometries are tested.

using a first order shear deformation theory (FSDT).

optimization is a good option to choose parameters

minimal intervention by the user

Keywords: differential evolution

optimization.

INTRODUCTION

Differential Evolution (DE) is a nature

and Price (1995) for global optimization. DE is a multipoint, derivative free optimization

method and is particularly suited to search for a global optimum or when

function is non-differentiable or noisy. DE has been used to solve many engineering problems

concerning intelligent material design (Loja et al. 2014; Zhang et al. 2015). In addition, DE

has been used to improve numerical methods for simulat

used DE as an alternative meshless approach to solve partial differential equations. Roque and

Martins (2015) used DE to improve the use of radial basis function meshless method, for the

study of composite plates in bendin

The use of radial basis functions for interpolation was proposed by Hardy (1971) and later

considered as one of the best methods in terms of accuracy for scatter data interpolation

(Franke 1982; Hardy 1971). The excellent behavior of radial basis funct

motivated their use for solving parabolic, hyperbolic or elliptical partial differential equations.

The RBF method proposed by Kansa is known as Kansa's collocation or unsymmetric

collocation since the collocation procedure produces

Kansa 1990b).

The meshless method uses radial basis functions (

assumed that any function,

differentiable basis functions,

where depends on a distance between grid points and may depend on a user defined

shape parameter . The quality of the solutions is greatly influenced by the choice of this


Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015

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INDEPENDENT MESHLESS METHOD FOR THE

ANALYSIS OF PLATES

Institute of Mechanical Engineering (IDMEC), University of Porto, Porto, Portugal

A differential evolution optimization scheme is used to optimize the radial basis function

In the present paper, DE is used to optimize parameters in radial

method for simulation of isotropic plates in bending under uniform

load. Rectangular and triangular geometries are tested. The deflection of the plate is studied

using a first order shear deformation theory (FSDT). Results show that differential evolutio

optimization is a good option to choose parameters for the numerical method requiring

minimal intervention by the user.

differential evolution, meshless method, grid optimization, shape parameter

Differential Evolution (DE) is a nature-inspired metaheuristics algorithm, proposed by Storn


method and is particularly suited to search for a global optimum or when

differentiable or noisy. DE has been used to solve many engineering problems


has been used to improve numerical methods for simulation. Panagant and Bureerat (2014)



study of composite plates in bending.



(Franke 1982; Hardy 1971). The excellent behavior of radial basis functions for interpolation



collocation since the collocation procedure produces asymmetric matrices (Kansa 1990a;

The meshless method uses radial basis functions ( ) to approximate a function

may be written as a linear combination

differentiable basis functions, :



THOD FOR THE

radial basis function

In the present paper, DE is used to optimize parameters in radial

method for simulation of isotropic plates in bending under uniform

The deflection of the plate is studied

Results show that differential evolution

for the numerical method requiring

, meshless method, grid optimization, shape parameter

inspired metaheuristics algorithm, proposed by Storn


method and is particularly suited to search for a global optimum or when the objective

differentiable or noisy. DE has been used to solve many engineering problems


ion. Panagant and Bureerat (2014)





ions for interpolation



asymmetric matrices (Kansa 1990a;

) to approximate a function . It is

a linear combination of continuously

(1)



Symposium_28

Advanced Discretization Techniques in Computational Mechanics

shape parameter, since it can improve the conditioning of

Usually this parameter is chosen by trial and error and depends on the user's experience,

which is not a desirable feature of the method. On the other hand, if the shape parameter is

well chosen, the method presents a fast

collocation points is also important for the problem's accuracy, since usually the condition

number of the system increases with increasing number of nodes. Adaptive methods can be

used to optimize point distribution in order to minimize the number of required collocation

points. Generally, adaptive methods focus points around areas with steeper gradients, by

adding/removing points or by moving an initial point distribution to a more appropriate

distribution (Barfeie et al. 2013; Driscoll and Heryudono 2007; Esmaeilbeigi and Hosseini

2012; Gonzalez-Casanova et al. 2009; Shanazari and Hosami 2012)

Regarding the shape parameter, Kansa suggests for the multiquadric

shape parameter for each grid point, with an exponential variation. This creates more distinct

coefficients for the multiquadric, improving the conditioning of the matrix. A simpler

approach uses a constant shape parameter for a

success by many authors, including in the analysis of plates and shells. The considered shape

parameter can for example be related to the number of grid points, (Fasshauer 2002; Franke

1982; Hardy 1971).

More complex optimization or analytical techniques have been proposed to choose a shape

parameter or point distribution. Rippa and Wang used a cross validation technique for shape

parameter optimization in multiquadric interpolation (Rippa 1999; Wang 2004). Ud

extended Rippa's algorithm for selecting a good value of shape parameter in solving time

dependent partial differential equations (Uddin 2014)

For plate bending problems, a cross validation technique developed by Rippa and Wang was

extended by Roque and Ferreira to Kansa's method for solving systems of PDEs (Roque and

Ferreira 2010). Using a cross validation technique it is possible to obtain good solutions for

the plate in bending problem, even with a reduced number of grid points, for regular and

irregular point distributions. Gherlone et al proposed an algorithm that finds an optimal value

of the shape parameter through a convergence analysis, inside a user

et al. 2012). In (Iurlaro et al. 2014) a principle of minimum of the t

used to choose a shape parameter inside a user

related to the physical problem to be solved, which sometimes can be difficult to obtain

Roque and Martins (2015) used differential evolu

in order to study the bending of composite plates. Differential evolution was used with

rectangular and L-shaped plates, with bounds on shape parameter

In the present paper, DE is used to optimize parameters in

method for simulation of isotropic plates in bending under uniform load. Rectangular and

triangular geometries are tested. Shape parameter is initially bounded in [0, 1] but bounds are

not used in subsequent generations, allo

bounds are not the best, the method is capable of finding a good shape parameter in interval

]0, +∞].

PLATE IN BENDING PROBLEM

A plate with thickness and area

of the plate. The deflection of the plate is studied according to

and for moderately thick isotropic plates a first order shear deformation theory can be used.


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shape parameter, since it can improve the conditioning of the problem if correctly chosen.



well chosen, the method presents a fast convergence and excellent accuracy. The number of



distribution in order to minimize the number of required collocation



on (Barfeie et al. 2013; Driscoll and Heryudono 2007; Esmaeilbeigi and Hosseini

Casanova et al. 2009; Shanazari and Hosami 2012).

Regarding the shape parameter, Kansa suggests for the multiquadric radial basis function, a



approach uses a constant shape parameter for all grid points. The methodology was used with



complex optimization or analytical techniques have been proposed to choose a shape


parameter optimization in multiquadric interpolation (Rippa 1999; Wang 2004). Ud

extended Rippa's algorithm for selecting a good value of shape parameter in solving time

dependent partial differential equations (Uddin 2014).


nd Ferreira to Kansa's method for solving systems of PDEs (Roque and



egular point distributions. Gherlone et al proposed an algorithm that finds an optimal value

of the shape parameter through a convergence analysis, inside a user-defined range (Gherlone

et al. 2012). In (Iurlaro et al. 2014) a principle of minimum of the total potential energy is

used to choose a shape parameter inside a user-define interval; the defined cost function is

related to the physical problem to be solved, which sometimes can be difficult to obtain

Roque and Martins (2015) used differential evolution with the radial basis meshless method


shaped plates, with bounds on shape parameter.

In the present paper, DE is used to optimize parameters in radial basis function meshless



not used in subsequent generations, allowing a quasi-user independent method. Even if initial


PLATE IN BENDING PROBLEM

and area is subjected to a load , applied on

of the plate. The deflection of the plate is studied according to a postulated displacement field,

or moderately thick isotropic plates a first order shear deformation theory can be used.

the problem if correctly chosen.



convergence and excellent accuracy. The number of



distribution in order to minimize the number of required collocation



on (Barfeie et al. 2013; Driscoll and Heryudono 2007; Esmaeilbeigi and Hosseini

radial basis function, a



ll grid points. The methodology was used with



complex optimization or analytical techniques have been proposed to choose a shape


parameter optimization in multiquadric interpolation (Rippa 1999; Wang 2004). Uddin

extended Rippa's algorithm for selecting a good value of shape parameter in solving time-


nd Ferreira to Kansa's method for solving systems of PDEs (Roque and



egular point distributions. Gherlone et al proposed an algorithm that finds an optimal value

defined range (Gherlone

otal potential energy is

define interval; the defined cost function is

related to the physical problem to be solved, which sometimes can be difficult to obtain.

tion with the radial basis meshless method


radial basis function meshless



user independent method. Even if initial


, applied on the top surface

a postulated displacement field,

or moderately thick isotropic plates a first order shear deformation theory can be used.



The FSDT uses five variables to describe the d

The theory can be simplified in the case of a symmetric plate, reducing the number of

variables to three, , and

coordinate of a point on the midplane (z =

axes, respectively.

The principle of virtual displacements can then be used to obtain a set of partial differential

equations, detailed in equations (2), (3) and (4) along with boundary condit

FSDT is well known and details can be found in (Reddy 1997)

Solving the system for ,

a load is applied.

with:

Parameters and are material properties (Young's modulus and Poisson ratio, respectively).

Parameter is a shear correction factor related to the first order shear deformation theory.

Boundary conditions for simply supported square plates of length

And for clamped plates,

With

And is the angle from the x-axis to the outward normal.



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The FSDT uses five variables to describe the displacement of a material point of the plate.


, where represents the displacement component along the z

midplane (z = 0) and and denote rotations about the


equations, detailed in equations (2), (3) and (4) along with boundary condit

FSDT is well known and details can be found in (Reddy 1997).

and gives information about the deflection of the plate when

are material properties (Young's modulus and Poisson ratio, respectively).

is a shear correction factor related to the first order shear deformation theory.

Boundary conditions for simply supported square plates of length are:

axis to the outward normal.

isplacement of a material point of the plate.


represents the displacement component along the z

denote rotations about the y and x


equations, detailed in equations (2), (3) and (4) along with boundary conditions (6). The

gives information about the deflection of the plate when

(2)

(3)

(4)

(5)

are material properties (Young's modulus and Poisson ratio, respectively).

is a shear correction factor related to the first order shear deformation theory.

(6)

(7)

(1)

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Analytical solutions

For simply supported rectangular plates, analytical solutions can be computed. For a plate

with lengths a, b, the solution for displacements

the displacements and applied transverse load

with

Substituting equations (9) in (2)

it is possible to compute an analytical solution for the problem

For triangular plates, a solution for vertical displacement can be found

MESHLESS NUMERICAL METHOD

Consider a boundary problem wi

by,

where and are differential operators in domain

Points and the domain respectively. The solution


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he solution for displacements is satisfied by the following expansions of

the displacements and applied transverse load (Reddy 1997),

(9) in (2)-(4) and solving the resulting equations for

it is possible to compute an analytical solution for the problem

For triangular plates, a solution for vertical displacement can be found in (Reddy 2007)

MESHLESS NUMERICAL METHOD

Consider a boundary problem with domain with an elliptic differential equation given

are differential operators in domain and in boundary

are distributed in the boundary and on

the domain respectively. The solution is approximated by :


satisfied by the following expansions of

(2)

and solving the resulting equations for

it is possible to compute an analytical solution for the problem.

in (Reddy 2007).

with an elliptic differential equation given

(10)

and in boundary , respectively.

are distributed in the boundary and on

(11)



Inserting equation (11) in (10) the following equations are obtained

where and are the prescribed values on boundary nodes

respectively.

In the present paper, the multiquadric

where can be the Euclidean distance between grid points and

The method is simple to implement but users must choose a grid distribution and shape

parameter in order to discretize

poorly chosen, results can be compromised. There are some thumb rules to choose the grid

and the shape parameter. For grids, regular or Chebyshev distributions are

the shape parameter, an expression related with the number of nodes in the grid is usually a

good choice.

However, different choices for shape parameter and grid distribution could provide good or

even better results. In this paper differential evolution optimizatio

grid distributions and shape parameters, with minimal user's input.

Differential evolution (DE) was chosen as an optimizer

particular, it has shown to be an adequate optimization technique for non

DIFFERENTIAL EVOLUTION

Being a population-based optimizer,

function at multiple initial points. Initial points can be randomly chosen or not, depending on

available information about the search space.

Classical differential evolution has four main stages: initialization, difference vector based

mutation, crossover/recombination and selection (Price et al. 2005). The algorithm is

controlled by 3 parameters:

1. F is the scaling factor typically between 0 and 1 that controls the differential mutation

process,

2. Cr is the crossover rate, which defines the probability of a trial

3. NP is the population size i. e., the number of competing solutions on any given

generation G.

The iterative process, whit each iteration being called a generation

defined stopping criteria is met.



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Inserting equation (11) in (10) the following equations are obtained,

are the prescribed values on boundary nodes and domain nodes,

In the present paper, the multiquadric radial basis function is used:

can be the Euclidean distance between grid points and is the shape parameter.


parameter in order to discretize equation (13). If a grid distribution or shape parameter are


and the shape parameter. For grids, regular or Chebyshev distributions are

arameter, an expression related with the number of nodes in the grid is usually a


even better results. In this paper differential evolution optimization is used to choose good

arameters, with minimal user's input.

was chosen as an optimizer for its simplicity and efficiency. In

particular, it has shown to be an adequate optimization technique for non-smooth functions.

DIFFERENTIAL EVOLUTION

based optimizer, DE starts solving the problem by sampling the objective

at multiple initial points. Initial points can be randomly chosen or not, depending on

available information about the search space.


ination and selection (Price et al. 2005). The algorithm is

is the scaling factor typically between 0 and 1 that controls the differential mutation

is the crossover rate, which defines the probability of a trial vector to survive

is the population size i. e., the number of competing solutions on any given

The iterative process, whit each iteration being called a generation G, stops when a user

defined stopping criteria is met.

(12)

and domain nodes,

(13)

is the shape parameter.


. If a grid distribution or shape parameter are


and the shape parameter. For grids, regular or Chebyshev distributions are often used. As for

arameter, an expression related with the number of nodes in the grid is usually a


n is used to choose good

for its simplicity and efficiency. In

smooth functions.

starts solving the problem by sampling the objective

at multiple initial points. Initial points can be randomly chosen or not, depending on


ination and selection (Price et al. 2005). The algorithm is

is the scaling factor typically between 0 and 1 that controls the differential mutation

vector to survive,

is the population size i. e., the number of competing solutions on any given

, stops when a user

Symposium_28


Initialization

DE initialization can be made by randomly generated candidate solutions (with or without

imposed restrictions) with NP D

current population G is

Initial population (G=0) can be defined

where is a random number,

length, ensures a distributed sampling of the parameter's domain interval

.

In the present case, far initialization is used since there is a prior knowledge of where a good

shape parameter could be located. This means that an interval is given for the search of shape

parameter for initial population G=0 but the constraint is not

generations, allowing the algorithm to search outside initial bounds.

Mutation

Differential mutation adds a scaled, randomly sampled, vector difference to a third vector.

Mutant vectors are obtained through differential

where F is a positive real number that controls the rate at which the population evolves.

Vectors , and are sampled

mutually exclusive integers chosen from interval

to look into the interval

interval . The search of an adequate value for F is a compromise between

exploration (large F) and exploitati

changing objective function

) (Das and Suganthan 2011; Price et al. 2005)

Crossover

Crossover enhances the potential diversity of a population. In

trial vectors are produced according to:


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E initialization can be made by randomly generated candidate solutions (with or without

imposed restrictions) with NP D-dimensional real valued parameter vectors. The

) can be defined as a uniform random initialization given by,

is a random number, which multiplied by the interval

ensures a distributed sampling of the parameter's domain interval



for initial population G=0 but the constraint is not maintained during subsequent

generations, allowing the algorithm to search outside initial bounds.


are obtained through differential mutation operation:


sampled randomly from the current population and

integers chosen from interval . A current approach to pick F is

although some authors restrict the parameter to the

The search of an adequate value for F is a compromise between

exploration (large F) and exploitation (small F), depending on the problem (slow/fast

changing objective function ) and the parameters' domain size (small/large interval

(Das and Suganthan 2011; Price et al. 2005).

Crossover enhances the potential diversity of a population. In the case of binomial crossover,

are produced according to:

E initialization can be made by randomly generated candidate solutions (with or without

dimensional real valued parameter vectors. The vector of

(14)

a uniform random initialization given by,

(15)

which multiplied by the interval

ensures a distributed sampling of the parameter's domain interval



maintained during subsequent


(16)


m the current population and , , are

. A current approach to pick F is

restrict the parameter to the

The search of an adequate value for F is a compromise between

on (small F), depending on the problem (slow/fast

) and the parameters' domain size (small/large interval

the case of binomial crossover,

(17)



Considering the classical DE scheme, DE/rand/1/bin

adequate values for the crossover coefficient

mutation rate F. In general, higher F values allow better results if accompanied by a higher

(Price et al. 2005). Condition

that has at least one component from

Selection

Selection may be understood as a form of competition, in line with many examples directly

observable in nature. Many evolutionary optimization schemes such as DE or GAs (Genetic

Algorithms) use some form of selection.

The selection criteria used in DE (

competes against one target vector, allowing to keep the population size

2008). The vector which minimizes the objectiv

generation, .

In the context of this work, the objective function can be understood as a measure of

proximity between the trial solutions and a reference solution. In equation (

prescribed values for boundary and domain nodes in the system of equations

is the right hand side computed by the numerical method.

Stopping criteria

The processes of mutation, recombination and selection are repeated until a

stopping criterion is met. There are many stopping criteria, often used together in a DE

optimization scheme. The most common of these criteria are,

1. a user defined value to reach (VTR) of the objective function is obtained,

2. a maximum number of generations is reached,

3. optimization running time exceeds a defined

4. the user stops the optimization.

Stopping criteria 1 and 2 were used simultaneously in all optimizations carried out during this

study. This approach allowed to contain the co

boundaries and to pursue the main optimization goal



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g the classical DE scheme, DE/rand/1/bin, there are some guidelines to choose

adequate values for the crossover coefficient (a probability

mutation rate F. In general, higher F values allow better results if accompanied by a higher

. Condition (where is a random number

has at least one component from (Das and Suganthan 2011).



Algorithms) use some form of selection.

selection criteria used in DE (18) is a one-to-one survivor selection where one trial vector

competes against one target vector, allowing to keep the population size

. The vector which minimizes the objective function , survives into the next


proximity between the trial solutions and a reference solution. In equation (

boundary and domain nodes in the system of equations

is the right hand side computed by the numerical method.

The processes of mutation, recombination and selection are repeated until a


optimization scheme. The most common of these criteria are,

a user defined value to reach (VTR) of the objective function is obtained,

.

r of generations is reached, .

optimization running time exceeds a defined .

the user stops the optimization.

were used simultaneously in all optimizations carried out during this

study. This approach allowed to contain the computational efforts within reasonable

boundaries and to pursue the main optimization goal.

, there are some guidelines to choose

), for a given

mutation rate F. In general, higher F values allow better results if accompanied by a higher Cr

), ensures



(18)

one survivor selection where one trial vector

competes against one target vector, allowing to keep the population size constant (Storn

, survives into the next

(19)


proximity between the trial solutions and a reference solution. In equation (19) are the

boundary and domain nodes in the system of equations (2)-(4), whereas

The processes of mutation, recombination and selection are repeated until a established


a user defined value to reach (VTR) of the objective function is obtained,

were used simultaneously in all optimizations carried out during this

mputational efforts within reasonable

Symposium_28


Population size

The population size , is the number of competing solutions on a given generation

general

parameters to be determined by the optimization process. There are some guidelines to find an

ideal population size based on the number of parameters (

Price, the creators of DE recommend

the population size, whereas Das et al.

between 3 and 8.

NUMERICAL EXAMPLES

Isotropic plates under uniform load are considered. Young modulus and

and respectivel

correction factor is . Geometries for rectangular and equilateral triangular plates are

illustrated in Figure 1. To ensure the enforcement of boundary conditions, points are

uniformly distributed at boundaries. Interior points with coordinates

differential evolution algorithm.

DE, Out of bounds parameters for coordinates

reinitialization.

Optimization can be defined as,

Previous experience by the authors shows that all pairs of parameters

results, although some pairs produce good results with fewer generations

2015). For the present case, parameter

results.

Cost function is computed using a refined regular grid. For all examples, the value to reach

was chosen as

.

Computed errors are computed using:

And relative errors, in percentage are computed using:


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, is the number of competing solutions on a given generation

. is the dimension of the problem i.e., the numbe


ideal population size based on the number of parameters ( ) of a given problem. Storn and

Price, the creators of DE recommend (Price et al. 2005) as a sensible choice for

the population size, whereas Das et al. (Das and Suganthan 2011) point to a value of

NUMERICAL EXAMPLES

Isotropic plates under uniform load are considered. Young modulus and

respectively. Ratio length/thickness is and shear deformation

. Geometries for rectangular and equilateral triangular plates are


ndaries. Interior points with coordinates are optimized using

differential evolution algorithm. Initial constraints for shape parameter are added to initiate

Out of bounds parameters for coordinates are handle by r

Optimization can be defined as,

Previous experience by the authors shows that all pairs of parameters and

results, although some pairs produce good results with fewer generations (Roque and Martins

For the present case, parameters and were used in all presented


. For rectangular plates, and for triangular plates,

Computed errors are computed using:

And relative errors, in percentage are computed using:

, is the number of competing solutions on a given generation . In

is the dimension of the problem i.e., the number of


) of a given problem. Storn and

as a sensible choice for

point to a value of

Isotropic plates under uniform load are considered. Young modulus and Poisson ratio are

and shear deformation

. Geometries for rectangular and equilateral triangular plates are


are optimized using

are added to initiate

are handle by random

and produce good

(Roque and Martins

were used in all presented


and for triangular plates,

(20)

(21)



Fig. 1 - Geometries of rectangular and equilateral triangular plates

Rectangular plate

Rectangular plates of lengths

rotations are compared with Navier analytical solutions, at every interpolated point. All

examples consider 25 interior points with coordinates

As a first example, a square plate (

the evolution of the average cost with the number of generations, for 25 independent runs. It

can be seen that the cost function decreases as the number of generations increases.

Fig. 2 - Square plate, average cost per generation <cost> over 25 runs, for

0 x

y



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Geometries of rectangular and equilateral triangular plates

s are considered. Solutions for vertical displacement

are compared with Navier analytical solutions, at every interpolated point. All

examples consider 25 interior points with coordinates to be optimized.

As a first example, a square plate ( ) is considered, with

tion of the average cost with the number of generations, for 25 independent runs. It


Square plate, average cost per generation <cost> over 25 runs, for F=0.9

0 x0

y

are considered. Solutions for vertical displacement and

are compared with Navier analytical solutions, at every interpolated point. All

to be optimized.

. Figure 2 shows

tion of the average cost with the number of generations, for 25 independent runs. It


F=0.9 and Cr=0.1.

Symposium_28


VTR was not reached in 500 generations. However, solutions are excellent, as can be seen in

Figure 3. Figure 3 shows one of the 25 runs for this optimization problem (run nº 25).

Analytical and numerical RBF solutions are not graphically distinguishable. S

vertical displacement and rotations are computed in Figure 3 in the refined grid used for cost

computation. In addition, optimal grid distribution is plotted in the lower right corner of

Figure 3.

For the same run, the cost function is depict

with approximately 100 generations. Shape parameter evolution with generations is shown in

Figure 5. About 100 generations were enough to find a stable value for the shape parameter.

Fig. 3 - Square plate, a=b=1; N=7; F=0.9; Cr=0.1;

error

Fig. 4 - Square plate, Cost evolution with generations, for run nº25.

vertical deformation

00

0.5

y

6

4

2

0

-21

rotation

00

0.5

y

-10

0

10

20

-201

50 100

0.035

0.04

0.045

0.05

0.055


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Analytical and numerical RBF solutions are not graphically distinguishable. S



For the same run, the cost function is depicted in Figure 4. A stable solution was achieved



=b=1; N=7; F=0.9; Cr=0.1; =500; Cost=0.0106; error

error ( )=0.0588 ; error( )=0.0477, run nº25.

Square plate, Cost evolution with generations, for run nº25.

10.8

0.6

x

vertical deformation =0.66603

0.40.2

Analitical solution

RBF numerical solution

0.40.2

rotationx

00

0.5

y

10

20

-20

-10

0

1

10.8

0.6

x

0.40.2

y

x

00

1optimized grid

Generation150 200 250 300 350 400



Analytical and numerical RBF solutions are not graphically distinguishable. Solutions for



ed in Figure 4. A stable solution was achieved



=500; Cost=0.0106; error (w)=0.0103;

Square plate, Cost evolution with generations, for run nº25.

10.8

0.6

x

0.4

1

450



Fig. 5 - Square plate, shape parameter evolution with generations, for

The influence of ratio is shown in Table 1. For this optimization problem,

used. Table 1 shows the mean and standard deviation of maximum values for

considering 25 runs. Relative error, in percentage is computed using Navier

solutions. For vertical displacements

relative errors are higher, especially for higher

Table 1 - Mean < > and standard deviation

lengths a, b. Optimization parameters are

b/a

Relative

error, (%)

0.5 0.0007 9.8x10-6

1.4

1 0.0042 6.0x10-5

2.3

1.5 0.0078 2.6x10-4

2.5

2 0.0102 3.3x10-4

2.9

2.5 0.0116 3.5x10-4

1.7

3 0.0122 4.9x10-4

3.2

Triangular plate

Triangular equilateral plates with

displacement are compared with analytical solutions. The number of interior points is 21.

Figure 6 shows the evolution of the average cost with the

independent runs, considering a maximum of 500 generations. As expected, the average cost

decreases with increasing generations.

50 1000.65

0.7

0.75

0.8

0.85

0.9

0.95



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Square plate, shape parameter evolution with generations, for trial nº25.

is shown in Table 1. For this optimization problem,

used. Table 1 shows the mean and standard deviation of maximum values for

considering 25 runs. Relative error, in percentage is computed using Navier

solutions. For vertical displacements and rotation errors are bellow 4%. For rotation

relative errors are higher, especially for higher ratios.

Mean < > and standard deviation of maximum values over 25 runs for rectangular

. Optimization parameters are Gmax=100, F=0.9, Cr=0.1

Relative

error, (%)

Relative

error (%)

1.4 00024 9.1x10-5

4.0 0.0040

2.3 0.0131 2.1x10-4

3.0 0.0131

2.5 0.0242 7.0x10-4

3.2 0.0172

2.9 0.0316 0.0011 3.1 0.0181

1.7 0.0362 0.0012 1.9 0.0180

3.2 0.0379 0.0016 3.3 0.0183

Triangular equilateral plates with are considered. RBF numerical solutions for vertical

are compared with analytical solutions. The number of interior points is 21.

the evolution of the average cost with the number of generations, for 25


decreases with increasing generations.

Generation150 200 250 300 350 400

trial nº25.

is shown in Table 1. For this optimization problem, is

used. Table 1 shows the mean and standard deviation of maximum values for ,

considering 25 runs. Relative error, in percentage is computed using Navier analytical

errors are bellow 4%. For rotation

of maximum values over 25 runs for rectangular plates with

=100, F=0.9, Cr=0.1.

Relative

error (%)

7.7x10-5

2.4

2.3x10-4

3.0

7.4x10-4

5.5

0.0010 8.1

0.0017 10.9

0.0018 9.9

are considered. RBF numerical solutions for vertical

are compared with analytical solutions. The number of interior points is 21.

number of generations, for 25


450

Symposium_28


Fig. 6 - Triangular plate, average cost per generation (<cost>) over 25 runs, for

Figure 7 shows the vertical displacement and optimized grid for run nº 25. Relative error for

maximum vertical displacement is

Fig. 7 - Triangular plate, F=0.9; Cr=0.1;

vertical displacement

-0.60.8

0.6

0.4

0.2

x

0

-0.2

-0.4

0

2

4

6

8

-2

12

10

10-4


-2298-

Triangular plate, average cost per generation (<cost>) over 25 runs, for F=0.9


maximum vertical displacement is 3.7%.

F=0.9; Cr=0.1; =500; Cost=0.039; error(w)=2.22x10

0.6

0.4

0.2

y

0

vertical displacement =1.0715

-0.2

-0.4

-0.6

Optimized Solution

x

-0.4 -0.2 0 0.2 0.4-0.6

-0.4

-0.2

0

0.2

0.4

0.6Optimized grid

Analytical solution

RBF numerical solution

F=0.9 and Cr=0.1.


=500; Cost=0.039; error(w)=2.22x10-5

; run nº25.

0.4 0.6 0.8



-2299-

The correspondent cost and shape parameter evolution with each generation is shown in

Figure 8 and Figure 9, respectively. This is an example where the best shape parameter is not

contained within initial bounds. Despite an initial poor choice of shape parameter bounds, the

optimization method is robust enough to find a better value providing an improved solution.

This important for unexperienced users, or when dealing with complex geometries. Figures 8

and 9 are an example of the typically fast convergence of DE based methods, even using a

random initialization technique for dealing with out of bounds parameters.

Fig. 8 - Triangular plate, Cost evolution with generation, for run nº25.

Figure 1: Triangular plate, shape parameter evolution with generation, for run nº25.

Generation

50 100 150 200 250 300 350 400 450

0.04

0.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

Generation50 100 150 200 250 300 350 400 450

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

Symposium_28


FINAL REMARKS

Differential evolution is used to optimize shape parameter and node distribution in the

analysis of isotropic rectangular and triangular plates in bending under uniform load. The

optimization method is capable of choosing a good shape parameter for radial basis functions,

with minimal intervention by the user. The method was also capable of choosing a good grid

distribution for different plate geometries. For rectangular plates, different ratios

used. The optimization method and the meshless method were able to find excellent solutions

for vertical displacements and rotations, even for large ratios of

Results show that differential evolution associated with the RBF meshless method is an

excellent technique to solve the proposed problems. This combined strategy provides a highly

versatile quasi-user independent meshless method for the analysis of systems of partial

differential equations.

ACKNOWLWDGEMENT

The support of Ministério da Ci

Europeu (MCTES and FSE) under programs

SFRH/BPD/71080/2010 is gratefully acknowledged.

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is used to optimize shape parameter and node distribution in the


d is capable of choosing a good shape parameter for radial basis functions,


distribution for different plate geometries. For rectangular plates, different ratios


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