A Novel Method for Estimation of Intensity and

A Novel Method for Estimation of Intensity and Location of Multiple Point Heat Sources Based On Strain Measurements

N. Ghiasi 1, A. Khosravifard 1,* 1

Department of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran

Abstract A novel approach for estimation of the location and intensity of multiple point heat sources in 2D domains is proposed in this work. The proposed inverse method is based on the simultaneous application of an improved meshless method and artificial intelligence algorithms. The inverse problem is formulated as an optimization problem and is solved by an improved artificial bee colony algorithm. In this work, strain measurements, rather than the usual temperature measurements, are used in the estimation procedure. It is shown that by using the strains for estimation of heat sources, results with better accuracy and more reliability can be obtained. The estimation procedure includes several direct thermo-elasticity problems which are solved by the meshless radial point interpolation method (RPIM) with a modified integration technique. Some numerical results are presented to demonstrate the applicability and efficiency of the proposed method. The results of the proposed approach are compared with those obtained from temperature measurements, and it is shown that especially for materials with high thermal conductivity, the proposed approach is very promising. Keywords: Inverse problem; Meshless methods; Radial point interpolation method; Artificial bee colony algorithm; Background decomposition method (BDM) 1. Introduction Thermal analysis is of utmost importance in the structural design of high temperature components and devices. For instance, welding processes are employed in a large number of structures such as airplanes, power stations or oil pipes. Since these structures require high levels of safety, the manufacturing processes must be carefully considered and effective control over their performance must be reached. Also, within the nuclear industry, where strict safety criteria should be met whilst keeping human engagement to a minimum, remote monitoring of system performance can be highly beneficial. Inverse problems are encountered in many branches of engineering and science. In the past decades, the inverse heat conduction problem (IHCP) has been extensively investigated in different fields of science and engineering. It is well-known that the uniqueness, existence, and stability of an inverse problem’s solution are not usually guaranteed, and any random small error in the input data can lead to large errors in the solution. Consequently, inverse problems are most often classified as ill-posed problems. Due to this character of the inverse problems, one should usually employ a regularization algorithm like the Tikhonov regularization or the iterative regularization scheme [1], in the solution *

Corresponding Author: E-mail: [email protected]

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procedure. The use of conjugate gradient method along with adjoint equation is another common technique which encompasses regularization schemes [2]. This method is formulated as an optimization problem and utilizes iterative regularization. Minimization of the cost function leads to stabilized solutions in the iterative procedure. This technique can be used for solution of several types of inverse problems. [3]. The identification of point heat sources is a type of inverse source problem that has many practical applications in many areas of engineering, such as welding and casting processes. In such problems, the location and intensity of the point heat sources are determined from a set of temperature measurements at fixed locations of the domain. So far, several different approaches have been employed for the analysis of the inverse identification of heat source problems. The techniques for solving inverse problems are mainly categorized into two categories, gradient-based methods and stochastic methods. To name a few, the Gauss-Newton method, and the conjugate gradient method are gradient-based techniques. Huang and Özisik made use of the conjugate gradient method combined with the adjoint equation to obtain the intensity of heat sources in a plate [4]. The same procedure was employed by Silva and Özisik to solve inverse source problems [5]. Another approach of solving inverse problems is to use the stochastic methods, such as the particle swarm optimization (PSO), genetic algorithm, ant colony optimization (ACO) or artificial bee colony algorithm, which have also been successful. Liu estimated the planar heat source in an inverse heat transfer problem using a modified genetic algorithm [6]. Li et al. used the method of parallel ant colony optimization for estimation of the position of a point heat source in a 2-D domain [7]. One of the most efficient techniques to solve an inverse problem is to reformulate it as an optimization problem and find the best solution by minimization of a suitable cost function. Finding an effective method that is not affected by the initial guess values is very important in the solution of inverse problems. Artificial bee colony (ABC) is one of the most efficient stochastic methods which are used to solve optimization problems. The ABC algorithm imitates the natural behaviour of bees in searching for the nectar around the hive. The privilege of these modern optimization methods over the gradient-based methods is that they don’t need any gradient information and can search many possible solutions simultaneously. Hence, these stochastic methods have the potential to give more stable solutions. In the procedure of solving an inverse problem, it is required to solve several direct problems. In this regard, various numerical methods have been successfully utilized, e.g. the finite difference method (FDM), the finite element method (FEM), the boundary element method (BEM), method of fundamental solutions (MFS), and meshless techniques. Hon and Wei were among the first ones to use the MFS for analysis of inverse heat conduction problems [8]. They used the Tikhonov regularization technique along with the L-curve method to obtain stable numerical solutions. Ling et al. proposed a method for identification of point sources in Poisson equation under the assumption that the number and approximate position of the sources are known [9]. In their inverse algorithm, Dirichlet boundary data, i.e., temperature measurements were used. Marin et al. used the regularized MFS to estimate the missing mechanical and thermal boundary conditions in 2D and 3D steady state linear thermoelasticity problems [10, 11]. Mierzwiczak and Kołodziej applied the MFS for inverse transient heat source problems [12]. Amirfakhrian et al. proposed a new approximate method for an inverse time-dependent heat source problem using fundamental solutions and radial basis functions [13]. Karami and Hematiyan proposed a boundary element method to find either the intensity or the location of point or line distributed heat sources in a 2D non-linear problem [14]. Niliot and Lefèvre solved the inverse problem of identification of location and strength of multiplepoint heat sources in transient heat conduction using a method based on a boundary integral 2

formulation using space and time Green functions [15]. Yang proposed a numerical algorithm to determine two moving heat sources in a 2D inverse heat problem [16]. Zueco et al. investigate the inverse problem of unknown heat generation sources in two-dimensional homogeneous solids [17]. Lefèvre and Niliot proposed an inverse formulation to estimate multiple point heat sources in a 2D diffusive system using the BEM [18]. Cheng et al. determined the heat source term in an inverse heat source problem using the final temperature history of a cylinder [19]. In recent years, meshless methods, such as the meshless local Petrov–Galerkin (MLPG) method, the element free Galerkin (EFG) method, and the radial point interpolation method (RPIM), have been successfully used for the analysis of various types of problem in engineering and science. The reliance of the mesh-dependent methods, such as the FEM, on a properly generated mesh of elements makes their implementation for specific problems difficult and time consuming. Meshless methods can be a competitive alternative to obtain the discrete governing equations of physical phenomena. These methods have become increasingly popular because of their simplicity of implementation to problems of complicated geometries and they can be used for solving both direct and inverse problems. Zhang et al. proposed an improved element-free Galerkin method to solve three-dimensional transient heat conduction problems [20]. Reutskiy solved 2D steady-state heat conduction problems in anisotropic and inhomogeneous media using a meshless radial basis function method [21]. Hosseini et al. investigated two-dimensional transient analysis of coupled non-Fick diffusion–thermoelasticity, based on Green–Naghdi theory using the MLPG method [22]. Khosravifard et al. proposed an improved meshless RPIM to solve the nonlinear transient heat conduction problems in the presence of heat sources [23]. In another study, they considered the problem of nonlinear transient thermo-mechanical analysis of functionally graded materials using an improved meshless radial point interpolation method [24]. Yan et al. developed a meshless method to solve the inverse heat source problems and inverse spacewise-dependent heat source problems using the MFS [25-26]. Sladek et al. solved inverse heat conduction problems using meshless local Petrov–Galerkin method [27]. Shivanian and Khodabandehlo solved a 1D inverse heat conduction problem using the meshless local radial point interpolation (MLRPI) method [28]. In the present work, the improved RPIM introduced in [23, 24] is used for the required direct analyses. Since in the inverse algorithm, several direct problems with different positions of heat sources should be solved, the meshless methods are the best choice in order to get rid of the burdensome remeshing process. In the formulation of the weak-form meshless methods, there are several domain integrals which should be evaluated numerically. Hence, the overall efficiency and accuracy of the method used for computation of these domain integrations has a significant impact on the accuracy of the results obtained by the meshfree method. The conventional approach for computation of the domain integrals in the meshfree methods has been the use of a background mesh which is not quite accurate. To overcome this problem, Khosravifard et al. proposed the Cartesian transformation method (CTM) for the computation of domain integrals in meshfree techniques [29]. Although this method considers the geometry of the integration domain when the quadrature points are being generated, but the local distribution of the nodes are no taken into account. There are several types of problems in which the nodal distribution should be non-uniform, for instance, fracture mechanics problems or problems which contain point heat sources. Hematiyan et al. proposed a background decomposition method (BDM) for domain integrations which consider the nodal distribution while distributing the integration points [30]. In the present study, the domain integrals are computed using the BDM, since the nodal distribution needs to be non-uniform.

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So far, for identification of point heat sources, temperature measurements have been used. To the best knowledge of the authors, it is for the first time that the strain measurements are used for simultaneous identification of position and intensity of multiple point heat sources. It will be shown that the results of the inverse problem are more accurate when the strain measurements are used instead of the temperature measurements. The rest of the paper is organized as follows; the mathematical and numerical formulations of the direct and inverse heat conduction problem are explained in the Section 2. Then, in Section 3, the BDM is briefly explained for evaluation of the domain integrals. The procedure of ABC algorithm is described in the Section 4. Section 5 discusses the applicability of the proposed method by presenting the results of some numerical examples. Finally, the paper is concluded by discussing the findings of the work, in Section 6. 2. Problem description and formulation 2.1. The Mathematical formulation In this study, the magnitudes of strains/temperatures at some random locations of the domain are measured using several sensors. The solution procedure begins with the search algorithm. The intensity and location of the heat sources are guessed using the ABC algorithm and then the RPIM is used to calculate the strains/temperatures which are used in estimation of the cost function. These strains/temperatures are compared with the strain/temperature measurements at the locations of the sensors. To do so, several uncoupled steady-state thermo-elasticity problems should be solved. To do so, the heat conduction equation is solved first, and then the displacement field is obtained based on the computed temperature field. The equation governing the steady-state heat conduction in an isotropic and homogeneous domain Ω, including a point heat source G, located at (x0, y0) is written as follows:

k 2T (x , y )  G  (x  x 0 ) (y  y 0 )  0 in 

(1)

where k is the thermal conductivity, T is the temperature, G is the intensity of the point heat source and δ is the Dirac delta Function. Using the RPIM to solve Eq. (1) and its associated boundary conditions, one can obtain the temperature field [23]. The stress–strain relation in uncoupled thermo-elasticity problems is stated by the DuhamelNewmann equation as follows:

 ij   kk  ij  2  ij  (3  2  ) (T T 0 ) ij

(2)

In Eq. (2) λ is the so-called Lame's constant, μ is the modulus of rigidity, α is the coefficient of thermal expansion, and T0 is the reference temperature. Upon substitution of Eq. (2) and the strain– displacement relations into the equilibrium equation, the governing equation of the mechanical part of the problem is obtained as follows:

 ui , kk 

E E uk , ki  T,i  0 21    1

(3)

where υ is the Poisson's ratio and E is the Young's modulus. The displacement field can be obtained by solving Eq. (3). Having found the displacement and temperature fields, the strain field can now be obtained. 4

2.2. The numerical formulation In the present study, the meshless RPIM is used to solve the preceding equations of the uncoupled thermo-elasticity. Originally Kansa [31] proposed the use of radial basis functions (RBFs) for interpolation of data in hydrodynamics. Later, meshfree methods based on the use of the RBFs gained popularity. The approximation of any domain variable, for example the temperature, at any point x in the domain can be obtained using the RPIM as follows [23]: n

m

i 1

j 1

T h (x)   R i (x)ai   p j (x) b j  R T (x)a  p T (x)b

(4)

where Ri is a radial basis function (RBF), n is the number of nodes in the support domain of point x, Pj is a monomial in spatial coordinates, m is the number of polynomial basis functions, and ai and bj are unknown coefficients to be determined. Radial basis functions are defined in terms of the Euclidian distance between the point x and the nodes in its support domain. There are several specific RBFs used in the meshless RPIM. In the present study, the TPS functions are used for the construction of shape functions, i.e.: (5) where η is a constant, with the common values of 3.001, 4.001, or 5.001. To find the n+m unknown coefficients in Eq. (4), the Kronecker delta function property and some specific constraint equations are applied [32]. Afterwards, the approximation function for the temperature field can be written as: n

T h ( x)   i ( x) Ti  Φ T ( x)T

(6)

i 1

where T is a vector containing the nodal values of the temperature, and Φ is the vector of RPIM shape functions which contains the first n components of the following vector:

  [R T PT ]G -1 Φ

(7)

where

R G   T0  Pm

Pm  0 

(8)

1 1 ... 1  Pm   x 1 x 2 ... x n   y 1 y 2 ... y n 

(9)

 R1 (r1 ) R 2 (r1 ) ... R n (r1 )   R 0  R i (rk )       R1 (rn ) R 2 (rn ) ... R n (rn ) 

(10)

5

rk  (x k  x i )2  ( y k  y i )2

(11)

After substitution of Eq. (6) into the Galerkin weak form of Eq. (1), the discretized system of thermal equations for the meshless RPIM is obtained [24]: K th T  Fth

(12)

where

   j i  j   K ijth   k  i d  y y   x x 

(13)

Fi th   Gi d    q i d  

2

(14)

Where q is the applied heat flux on the boundary portion 2 . The temperature field can be obtained by solving Eq. (12). Then, using this temperature field, one can obtain the thermal strains, as follows:

 ijth   (T T 0 ) ij

(15)

Afterwards, the displacement field is approximated by the RPIM and is substituted in the Galerkin weak form of Eq. (3) to obtain the discretized system of elastic equations for the meshless RPIM: K el u  Fel

(16)

where

K elij   BTi DB j d 

(17)



Fiel 

 Φ td  +  Φ bd  +  B Dε i

4

i

i



th

d



(18)

Where t is the applied traction vector on the boundary portion 4 , b is the body force vector, D is the matrix of elastic constants, and Bi contains the derivatives of shape functions [24]. After the solution procedure for the direct problem is obtained, a suitable cost function in terms of the measured and computed strains should be defined. This cost function is subsequently minimized using the ABC algorithm in order to get the correct intensity and position of the unknown point heat sources. The cost function is defined as the summation of squares of differences of the computed strains using the RPIM and the measured strains by strain gauges, which is stated as follows: m

cos tfunction (G , x 0 , y 0 )   ( i  e i ) 2

(19)

i 1

where ei is the measured strains and εi is the computed strains using the RPIM and m is the number of sensors. 6

3. Numeriical computtation of th he domain integrals i byy the BDM M As can be seen from the t previouus section, th here are sevveral domaiin integrals to be evaluuated in thee m solution. Since in this study, thee problems containing point heat sources aree process of the problem ound decomp mposition meethod (BDM M) [30] is thhe best choice for the computationn analyzed, tthe backgro of these inntegrals. Thhe BDM is especially useful wheen several integrals i onn a domainn with non-uniform diistribution of o nodal pooints should d be calculaated. Hereinn, a brief review r of thhe BDM iss given. Suppose thhat the following domaain integral is i to be evalluated numeerically:

I   g (x)d 

(20)



In the BDM M the follow wing four stteps should be pursued [30]: In the firstt step, the average disstance of eaach node too its closesst nodes is computed and a termedd spacing vaalue of thatt node. For a two-dim mensional doomain, conssidering thee two closeest nodes iss sufficient tto evaluate the spacingg value of each e node. Based on the t computeed spacing values, thee nodal poinnts are categgorized into some differrent grades of nodal deensity. In the seccond step, the t domainn undergoess a quadtreee [33] parttitioning baased on thee evaluatedd spacing vaalues. For a 2D domainn, a square containing c thhe integratio on domain is selected. The squaree is divided recursivelyy into four quadrants, q until u either the nodes in each quaadrant are of o the samee n in thee quadrant. An example of quadtreee partitioning in a 2D D grade, or tthere are at most four nodes domain, with seven grrades of noddal density is illustrated in Fig. 1(aa).

o nodes; (bb) Integral Fig. 1. A 2D quadraant domain: (a) Quadtreee partitioniing with sevven grades of points Now that tthe nodal density d is uniform u in each e partition, in the third t step, the t integrall over eachh partition can c be com mputed by use u of any conventionnal quadratuure techniqque. In the BDM, thee Cartesian ttransformattion methodd (CTM) [2 29] is utilizzed for com mputation off the integrrals in eachh partition. A As the resultts, the integgral over eacch subdomaain  k can be expresseed as follow ws: Nk

I k  W i k (x i ) g (x i )

(21)

i 1

7

where xi is the coordinates of a CTM integration point, Wik is the corresponding integration weight, and Nk is the total number of integration points in  k . Finally, in the last step, by assuming the total number of partitions over the domain to be P, the global vectors can be generated to find the total value of the domain integral. Finally, BDM formulation for computation of the integral is written as follows: P

P

Nk

N

I   I k  W jk g (x kj )  W n g (x n )  WT g int k 1

k 1 j 1

(22)

n 1

where W is the global weight vector, the vector gint contains the values of the function g at the integration points, and N is the total number of integration points. Fig. 1(b) depicts the BDM integration points for the nodal arrangement shown in Fig 1(a). It can be observed that the distribution of the integration points is similar to that of the nodal points. 4. Artificial Bee colony Algorithm In this study, the minimization of the cost function is performed by the artificial bee colony algorithm. The ABC algorithm was inspired by the intelligent behavior of swarms of honey bees. It imitates the searching behavior of bees for food sources around the hive which includes a particular dance called the waggle dance to share the information about the quality and locations of the probable nectar sources. In this algorithm, the bees are classified into three different groups: employed bees, onlookers and scouts. Employed bees search the region to exploit the food sources and do the waggle dance to share the information. Onlooker bees are the bees waiting in the hive to receive the information about the food sources from the employed bees and select the best food source. Scouts are those which are randomly exploring new food sources around the hive. Hence, the region is investigated once generally to localize the best food sources of and then more precisely in vicinity of the best selected locations. In the following, the procedure of the ABC algorithm is defined briefly [34, 35]: Step 1: A distributed initial population (X1, …, XNB) is randomly generated, where NB is the number of food sources (solutions) and is equal to the number of onlooker or employed bees. Each solution, Xi = {xi1, xi2, …, xiD}, is a D-dimensional vector which is generated using: x ij  x min j  rand[0 1]  ( x max j - x min j ) (for j = 1, 2, …, D and i = 1, 2, …, NB)

(23)

where xmin j and xmax j denote respectively, the lower and upper bounds for the j-th dimension. After the initial population is generated, the fitness value, i.e. the value of the cost function, should be evaluated for each food source. Step 2: Investigating the vicinity of the current food source, each employed bee determines a new food source, using:

v ij  x ij  ij (x ij - x kj ) i  k , k  {1, 2,..., NB }, j {1, 2,..., D }, ij  [1,1]

(24)

where φij , k and j are chosen randomly. It is important to note that k should be different from i. Step 3: Evaluating the nectar amount (fitness value) of the new food source, a greedy selection will be performed to compare the new food source with the current position. If the new food source has

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an equal or better fitness compared to that of the previous source, the new food source replaces the previous one in the population. Step 4: The probability of the food source i to be selected by an onlooker bee, is based on its fitness value and is determined by: Pi 

fit i NB

 fit n 1

n

(25)

where fiti is the value of the cost function at food source Xi. When the onlooker selects a food source, steps 2 and 3 are repeated, e.g. a new food source will be obtained using eq. (24), then it will be evaluated and a greedy selection will be made. Step 5: In the ABC algorithm, a predetermined number of trials, called limit, is assigned in order to abandon a food source if its fitness value could not be further improved by this limit. In this case, the scout bee randomly chooses a new food source and the abandoned one is replaced by the new one. Step 6: The algorithm will stop and report the best food source (solution) when a termination condition is met; otherwise it returns to step 2. The termination condition is usually placed on the minimum value of the cost function found so far. 5. Results and Discussion In this Section, the developed techniques are applied to the inverse problem of estimation of intensity and location of point heat sources. Specifically, two example problems are analyzed; in the first one, only the position and intensity of a single heat source is sought. The effect of several factors, including the number of sensors, the location of sensors, the magnitude of measurement errors, and the material property on the accuracy of the results is studied in detail. In the second example, the same inverse problem is analyzed, but the intensities and locations of two point heat sources are sought simultaneously. In the present work, in accordance with many other works [36-39], the data measurements are numerically simulated. This means that instead of performing experiments, some numerical analyses are performed and a specific level of Gaussian error are added to the obtained results in order to represent the measurement errors. By this technique, a same level of measurement error can be considered for the cases of temperature and strain measurement, and therefore, a fair judgment can be obtained between the results of the inverse algorithm when the strain measurements are used instead of temperature measurements. Additionally, one can easily investigate the effect of level of measurement error on the performance of the inverse algorithm. It should be stated that the simulated experiments are performed by the well-known FEM package ANSYS, while the direct solver of the inverse algorithm is the meshfree RPIM code developed by the authors. As the results, the obtained results and the measurement data are not from the same numerical algorithm. This approach makes the simulated experiments as close as possible to the real experimental conditions.

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5.1. Exampple 1: Identiification of a single point source inn a quarterr circular doomain The probleem geomettry and bouundary connditions aree shown in Fig. 2. Baased on thee proposedd methods, tthe intensityy and locaation of a single s pointt source in the domaiin will be found. Thee following bboundary coonditions foor this probllem are connsidered: T (0, y )  T ( x , 0)  30 00 C ,

T |r 5  0 r

(26)

u ( x , 0)  v ( x , 0)  0

(27)

Fig. 2. A 2D D domain with w an unknnown point heat sourcee Three diffferent mateerials are considered c for the doomain to innvestigate the t effect of thermall conductivitty on the accuracy of o the resullts. The prroblem is solved s oncee using thee measuredd temperaturre data and once using the measurred strain daata, as the inputs i of thee ABC algoorithm. Thee ABC param meters are set s as follow ws: NB = 100; D = 3; lim mit = 6. The temperature and d strain measuremen m nts have been b simulated using g ANSYS. A set off me random positions inn the domaain is obtained, by connsidering a temperaturre/strain vallues at som point heat source G ass follows: G = 1000 at 1.5 3.6 t ABC alggorithm are the temperrature and strain valuess obtained by b ANSYS,, Hence, thee inputs of the after a randdom Gaussiian error with a maxim mum of 2%, is added to the results to t account for f inherentt measuremeent errors. The algoritthm has beeen executeed for diffeerent number of sensoors and thee stopping crriteria are defined d as foollows: min (cost ffunction) / number n of seensors < 10 0-3 (28) max (iterattion) = 300 (29) The sensorrs are distriibuted randdomly and scattered s alll over the domain. d Figg.3 shows four f typicall sensor disttributions. Inn this figuree, the sensors and the ppoint sourcee are represeented by cirrcles and ann asterisk, reespectively.

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F 3. Sensoors' distribuution over thhe domain of Fig. o Example 1 In order too accurately y model thee point heatt source, we have usedd an adaptiive concentrrated nodall distributionn in the seaarch algorithhm. In this method, inn the vicinitty of the pooint heat souurce a finerr distributionn of nodes is i used, whiile the nodees are distribbuted in a coarse mannner in other parts of thee domain. Thhis arrangem ment of thee nodal poinnts changess in each iteeration of thhe inverse problem, p ass the locatioon of the esstimated sources are changed c by the search algorithm (Fig. 4). Itt should bee pointed thaat only the nodes that are around the heat source need to t be movedd during thee iterations.. Since this procedure may repeaat hundreds and somettimes thoussands of tim mes, using a meshlesss a thereforre simplifiess the solution process efficiently.. method eliiminates thee remeshingg process and Additionallly, by using g the BDM M for compu utation of thhe domain integrals, thhe meshlesss algorithm m becomes vvery effectivve. By this approach, a acccurate resuults are obtaained very efficiently.

Fig. 4. Tyypical nodall distribution of the meeshless methhod, based on o the positiion of the heat h source

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5.1.1. First material – Zirconium Dioxide (ZrO2) In this case, the medium is made of zirconium dioxide (ZrO2) with the following mechanical and thermal properties: E=200 GPa, ν=0.29, k=2.04 w/m°C, α=7.11×10-6 1/°C The domain is represented using 105 distributed nodes with 36 concentrated dense nodes around the assumed heat source. The identification problem is solved first using temperature data. Table 1 lists the results obtained from the proposed algorithm. Table 1. Heat source identification in ZrO2 using temperature data with 2% input error, example 1 Number of thermometers

Number of iterations

Intensity of heat source (error)

Location of heat source (error)

300

min (cost function) / number of sensors 5.1

4

983.3 (1.7%)

5

300

22.9

986.7 (1.3)%

6

300

23.9

967.9 (3.2)%

7

300

13.1

955.3 (4.5)%

8

300

14.6

1016.2 (1.6)%

9

300

12.2

990.3 (1.0)%

10

300

18.0

1008.6 (0.9)%

1.52 3.65 1.3 1.5% 1.61 3.58 7.3 0.4% 1.45 3.39 3.2 5.7 % 1.60 3.74 6.7 3.9% 1.45 3.59 6.7 3.9% 1.56 3.70 3.8 2.7 % 1.53 3.58 2.1 0.5%

As can be seen, the ABC algorithm can precisely predict the location and intensity of the point heat source, and increasing the number of sensors can effectively decrease the relative error in the identification of the heat source. Although the algorithm is capable of finding a precise solution with small number of sensors, but the results obtained from more number of sensors are more reliable. Another point that should be noted is that the number of sensors should be at least one more than the number of the unknowns. The convergence graph of this algorithm is shown in Fig.5 for four different numbers of sensors.

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Fig. 5. The T cost fun nction conveergence graaph for ZrO2 and tempeerature data (N is the nuumber of sensors) The idea of o the preseent study is to investiggate the posssibility of finding f the point heat sources byy using the sstrains as thhe measuredd data. Now w the same problem is solved usinng the straiin data, andd the obtaineed results arre listed in Table T 2. Table 2. Heat source identificatio i on in ZrO2 using u strain data with 2% 2 input errror, examplee 1 Inttensity of heaat source ( (error)

Numbeer of straiin gaugess

Numbber of iteratioons

min (cost ( functiion) / numbber of senssors

4

40

8.2 e-4

11001.6 ( (0.2)%

5

73

8.0 e-4

11032.7 ( (3.3)%

6

79

9.8 e-4

944.2 ( (5.6)%

7

49

5.8 e-4

977.6 ((2.2)%

8

25

9.8 e-4

997.3 ( (0.3)%

9

32

8.9 e-4

11023.7 ( (2.4)%

10

1955

9.9 e-4

996.4 ((0.4)% 13

Location of heat sourrce (erro or)

1.47 3.50 1.6 2.7  % 1.50 3.53  0.1 1.9  % 3  1.55 3.59  3.3 0.2  % 3  1.55 3.59  3.3 0.2  % 3  1.48 3.68 1.2 2.2  % 1.47 3.58 0 %  2.0 0.4 1.51 3.61

 0.8

0.2 0 %

ned from thhe strain datta are consid derably bettter than thee It is observved that nott only the reesults obtain results obtaained from the temperaature data, but b also theiir stabilitiess have been improved. Also, A it cann be seen that by usingg the strain data, one can c find thee solution inn fewer num mber of iteerations andd considerabbly reduce thhe computaational time which is a very imporrtant factor in such prooblems. Thee convergencce graph off this algoriithm is show wn in Fig. 6 for four different d numbers of seensors. It iss observed tthat in casee of using strain s data, the graph converges more m rapidly and withhin a fewerr number off iterations, compared c too the graph of temperatture data.

u strain data (N is thhe number oof sensors) Fig. 6. Thee cost functtion converggence graphh for ZrO2 using The originnal ABC allgorithm chhanges onlyy one of the t unknow wn parameteers random mly in eachh iteration. B But in this study, s the allgorithm is modified by b changingg the values of all of thhe unknownn parameterss simultaneeously. Thee problem is now sollved with the modifieed algorithhm and thee obtained reesults are reeported in Table T 3. Table 3. H Heat source identificatioon in ZrO2 using the modified m AB BC algorithm m and strain data withh 2% input eerror, exampple 1 Numbeer of straiin gaugess

Numbber of iteratioons

4

17

min (cost ( functiion) / numbber of senssors 9.5 e-4

14

Inteensity of heaat source ( (error) 933.6 ((6.6)%

Location of heat sourcce (erroor)

1.53 10.8

3  3.58 0 % 0.5

5 6 7 8 9 10

29 16 1022 35 9 28

9.2 e-4

11062.2 ( (6.2)%

9.5 e-4

11013.9 ( (1.4)%

9.7 e-4

11000.3 ( (0.0)%

7.8 e-4

11003.6 ( (0.4)%

9.5 e-4

11043.5 ( (4.3)%

8.2 e-4

11067.6 ( (6.8)%

1.40  6.9 1.49  0.8 1.55  3.1 1.45  3.5 1.45  3.5 1.35 10.2

3  3.51 2.5  % 3.52 3  2 % 2.1

3.61 3  0.2  % 3  3.52

2.2  %

3  3.48 3.2  % 3  3.42 4 % 4.9

Comparingg the resultss of Tables 2 and 3, on ne can concllude that altthough this modificatioon does nott improve thhe accuracyy of the reesponse, buut it effectiively reduces the num mber of iterrations andd consequenttly, the com mputational time. From thesee tables it can be conncluded thaat the positiion and intensity of thhe heat souurce can bee obtained w with acceptaable accuracy, using thhe proposedd techniquee. The convvergence grraph of thiss algorithm iis shown in Fig.7 for foour differennt numbers of o sensors.

Fig. 7. Thee cost functtion converggence graphh for ZrO2 using u the moodified ABC C algorithm m and strain data (N is the t number of sensors)) 15

5.1.2. Second material – Stainless Steel AISI 304 Since the magnitude of the thermal conductivity has a significant effect on the distribution of the temperature and consequently the strain fields over the domain, a medium with higher thermal conductivity (stainless steel) is now considered to investigate the applicability of the proposed method to materials with larger conductivity. The mechanical and thermal properties of stainless steel are as follows: E=200 GPa, ν=0.29, k=16.2 w/m°C, α=1.73×10-5 1/°C At first, the domain is represented with the same number of nodes as the previous case. Table 4 reports the obtained results for the intensity and location of the heat source. Table 4. Heat source identification in Stainless Steel domain of Fig. 2 using ABC algorithm, with 105 distributed and 36 concentrated nodes and 2% input error, example 1 Location of heat Number data Number min (cost Intensity of of function) / of heat source sensors iterations number of source (error) sensors (error) 10 temperature 1000 92.3 1543.3 1.39 3.13 (54.3)%  7.3 13.0  % 10 strain 500 2.1 e-3 1052.3 1.58 3.58 (5.2)%  5.1 0.5 % It is observed that although the results are not as good as those of the previous material, nevertheless, the results obtained from the strain data are much better than the ones obtained from temperature data. Also, the identification results that are based on the strain data are found by less iteration. This is because in the presence of a point heat source, the distribution of temperature is more localized than the strains. In a domain subjected to a point heat source, temperature variations mostly occur in the vicinity of the source, whilst strains are distributed more uniformly all over the domain. This issue becomes more important in the domains with higher thermal conductivity. Since the thermal conductivity is higher than the previous material, the point heat source has more effect on the domain and the local variations of strains and temperatures increase over the domain. Hence we increase the number of nodes and represent the domain with 205 distributed and 36 concentrated dense nodes, to analyse the problem more precisely. The obtained results are listed in Table 5 for temperature input data and Table 6 for strain input data. Table 5. Heat source identification in Stainless Steel domain of Fig. 2 using ABC algorithm and temperature data, with 205 distributed and 36 concentrated nodes – 500 iterations, example 1 Number of min (cost thermometers function) / number of sensors 4 60.9 5

48.9



1476.8 (47.7)%

1.52 3.98 1.2 17.3 % 1.64 3.10  9.3 13.9  %

1282.2 (28.2)%

16

6 7 8 9 10

49.1 91.3 80.9 55.2 82.4

1213.6 (21.4)% 1105.9 (10.6)% 1120.6 (12.1)% 1148.7 (14.9)% 1227.4 (22.7)%

1.65 10.1 1.49  0.9 1.49  0.9 1.75 16.7 1.71 14.4

3.12 13.3 % 3.42 4.9  %

3.42 4.9  % 3.27  9.1 % 3.30 8.4  %

Table 6. Heat source identification in Stainless Steel domain of Fig. 2 using ABC algorithm and strain data with 205 distributed and 36 concentrated nodes – 300 iterations, example 1 Number of strain gauges

min (cost function) / number of sensors


4

1.4 e-3

1182.8 (18.3)%

5

2.5 e-3

1125.4 (12.5)%

6

5.8 e-3

1022.8 (2.3)%

7

3.1 e-3

1123.7 (12.4)%

8

3.7 e-3

1035.2 (3.5)%

9

2.8 e-3

983.1 (1.7)%

10

9.8 e-4

961.0 (3.9)%


1.55 2.92  3.3 18.9  % 1.56 2.59  4.1 13.9  % 1.50 3.16  0.0 12.3 % 1.40 2.91  6.3 19.2  % 1.56 3.57  4.2 0.8 % 1.48 3.74 1.4 17.3 % 1.34 3.31 10.4 7.9  %

It is concluded that the obtained results based on the temperature data have improved. However, the results obtained from strain data are still much better that those obtained from temperature data. This means that when the strain data are used for identification of point heat sources, fewer nodes can be used in the numerical algorithm.

17

5.1.3. Third material – Medium Carbon Steel In this case, a material with fairly high thermal conductivity is considered to make sure that the proposed method is accurate enough and also applicable to a wide range of material properties. Considering again the domain of Fig. 2, we assume that it is made of medium carbon steel with the following thermophysical properties: E=200 GPa, ν=0.29, k=54 w/m°C, α=11.5×10-6 1/°C The domain is defined using 377 distributed nodes with 139 concentrated dense nodes around the assumed heat source. The problem is solved based on the strain data as the input of the ABC algorithm. The results are listed in Table 7. Table 7. Heat source identification in carbon steel using strain data with 2% input error, example 1 Number of strain gauges



4

1.7 e-3

3374.1 (237.4)%

5

1.4 e-3

1323.2 (32.3)%

6

5 e-4

773.8 (22.6)%

7

6 e-4

1349.4 (34.9)%

8

1.5 e-3

1164.6 (16.5)%

9

1.5 e-3

1161.9 (16.2)%

10

1.4 e-3

1168.4 (16.8)%


1.05 2.05  29.9 42.9  % 1.47 3.52  2.3 2.3 % 1.23 4.53 17.9 25.8 %  2.69 2.46  79.2 31.6  % 1.52 3.50 1.5 2.7  % 1.47 3.50 1.7 2.6  % 1.47 3.51 1.7 2.6  %

Due to the high thermal conductivity of the material, and according to the results of Table 8, using sufficient number of sensors is essential to find an accurate answer. It is observed that the results obtained from fewer numbers of sensors are unacceptable and the minimum number of sensors required to obtain acceptable results is 8. In this case, since the thermal conductivity is relatively high, the point heat source causes the temperature variations to be more localized than the strains. Thus, it would be really difficult and time consuming if one wishes to identify the heat source by the temperature data. The convergence graph of this algorithm is shown in Fig. 8, for 9 and 10 sensors.

18

Fig. 8. Thee cost functiion converggence graph for carbon steel (N is the t number of sensors)) 5.1.4. Bounndary-only data measuurement Herein, another exam mple is solveed in order to investigaate the methhod's appliccability to situations inn which therre are restricctions for sttrain measurrement insidde the domaain and onlyy the bounddaries of thee domain aree accessiblee for data measurement m t. In this case, it is assuumed that the t medium m is made off zirconium dioxide (Z ZrO2), propperties of which werre given in n the precceding exam mples. Thee o the accesssible parts of the bounndary, i.e., those t parts that are nott distributionn of the straain gauges on mechanicaally constraiined, is shoown in Fig. 9. In this figure, f the circles, c andd the asterissk representt the sensorss and heat soource respeectively.

F 9. Sensoors' distribuution on the boundary of Fig. o the domaiin The numerrical domaiin is represeented by 1007 distributted nodes with w 36 con ncentrated dense d nodess around thee location off the heat soource. The identification problem m is solved based b on thhe boundaryy strain data as the input of the ABC algorithm m, and the obbtained resu ults are listeed in Table 8.

19

Table 8. H Heat source identificatio i on in ZrO2 using u bounddary strain data d with 2% % input erroor, examplee 1 Num mber of strainn gauges

m (cost min ffunction) / n number of sensors

8

1.4 e-3

Intensity of heat sourcce (error)

Loocation of heat source (error)

975.83 (2.4)%

1.55064 3.6016  0.44 0.04  %

It is observved that thee results obttained from m the mentiooned sensorr distributioon are accurrate and thee method is rreliable for identificatiion of heat sources, s whhen boundarry-only meaasurement is i available.. The converrgence grapph of the AB BC algorithm m for the caase of this example e is shown s in Figg. 10.

Fig. 10. Thhe cost funcction converrgence grapph for ZrO2 with 8 senssors located on the bounndary (N is the nu umber of seensors) 5.1.5. Effecct of measurrement erroor on the ouutput of the identificatio i on algorithm m The purposse of this seection is to investigate the effect of o measurem ment error on o the amouunt of errorr in the outpput of algorrithm. The problem is solved forr different amounts a off error, usinng the ABC C algorithm. In this probblem, the doomain is maade from staainless steell. The resultts are listed in Table 9. e of straains and tem mperatures on o the output of the iddentificationn Table 9. Effect of measurement error algorithm, using 10 seensors – 3000 iterations Type of input sensor

Percennt of input error e

minn (cost function) umber / nu of sensors

Inntensity of heeat source (error)

temperaturre

5% %

118.4

614.3 (38.6)%

20

Locatioon of heat souurce (errror)

1.96  30.8

3.07  14.6  %

1% %

3 3.1

1014.0 (1.4)%

5% %

2.99 e-2

1072.8 (7.3)%

1% %

6.55 e-4

1099.8 (10.0)%

strain

1.64  9.2 1.38  8.1 1.50  0.0

3.57  0.8  % 3.37  6.3 % 3.47  3.6  %

As can be observed, the t amount of error in the output generally g inncreases as the amountt of error inn the input inncreases. Buut the resultts of this tabble infer thaat, the results obtained from strainn inputs, aree much betteer and morre reliable than t those obtained o froom temperaature inputss, even in presence p off higher maggnitudes off error. Hennce it is recommended to use straain data insttead of tem mperature inn problems oof identificaation of poinnt heat sources. 5.2. Exampple 2: Identiification of two point heat h sourcess in a 2D doomain In this secttion, the ressults for idenntification of o the intensity and loccation of two point heatt sources inn a 2D simplly connected domain, made m of ZrO O2, shown inn Fig. 11, arre presentedd. The assum med boundarry conditionns for this prroblem are: T (0, y )  T (6, y )  T (x , 0)  1000C (30) (31) u x , 0  v  x , 0  0

Fig. 11. A 2D domainn with two unknown u heeat sources In this exaample, a sett of strain values v at som me positionns in the doomain is obtained, by considering c g two point hheat sourcess G1 and G2 as follows: G1 = 1000 at 1.5 2.5 , G2 = 18800 at 5.0 1.5 The strain measuremeents have been b simulaated using ANSYS. A Hence, the innputs of thee improvedd he strain vaalues obtainned by AN NSYS, after a random Gaussian eerror with a ABC algorrithm are th maximum of 2% is ad dded to the results to account a for inherent measurement m t errors. Sim milar to thee d nodal distrribution is used u to consider the efffects of thee previous exxample, an adaptive cooncentrated locality off the heat sources. Thee domain iss defined by b 103 distrributed noddes and 36 concentredd dense nodees around eaach heat souurce (Fig. 12).

21

Fiig. 12. A typpical nodal distributionn of domainn of examplee 2 The algoritthm has beeen executedd for differeent number of sensors distributed randomly all a over thee domain (Fiig. 13). Thee results of this t problem m are listed in Table 100.

Fig. 13. The position off sensors in the t domain of examplee 2 The stopping criteria are a defined as follows: n of seensors < 10 0-4 min (cost ffunction) / number max (iterattion) = 300

(32) (33)

Table 10. Multiple M heaat sources iddentification in ZrO2 and a coarse nodal n arranggement, exaample 2 Numbeer Numbeer of of strain iterationns gaugess

6

300

min (coost functionn) / numberr of sensorrs

Intennsity of heat source (eerror) 10029.6 (3.0)%

1.1 e--4

13304.6 (277.5)% 22

Locationn of heat sourrce (erro or)

1.53 2.54 1 %  2.1 1.5  4.84 1.52 1 %  3.1 1.1

G1 G2

7

144

8

252

9

300

10

300

972.1 (2.8)%

9 e-5

1521.3 (15.5)% 1022.8 (2.3)%

8 e-5

1425.0 (20.8)% 985.0 (1.5)%

8.3 e-4

1709.0 (5.0)% 1021.1 (2.1)%

6.8 e-4

1553.7 (13.7)%

1.52 2.60 1.2 4.1 %  4.94 1.55 1.1 3.5  % 1.59 2.53  6.3 1.3 %  4.94 1.48 1.3 1.2  % 1.49 2.38  0.6 4.9  % 5.02 1.63  0.3 8.8 % 1.50 2.38  0.3 4.5  %  4.95 1.58 1.0 5.1 %

G1 G2 G1 G2 G1 G2 G1 G2

From the table it is concluded that the positions and intensities of the heat sources can be obtained with acceptable accuracy, using the proposed technique. It is also seen that the accuracy of the results increases as the number of sensors is increased. In an attempt to acquire better results, we increase the number of nodes in the problem domain to 177 distributed and 36 concentrated nodes and solve the previous problem once again. The problem is solved for maximum of 500 iterations and with the stopping criteria defined in Eq. (28). The results of this problem are listed in Table 11. Table 11. Multiple heat sources identification in ZrO2 and fine nodal arrangement, example 2 Number Number of of strain iterations gauges

9

10

320

348


8.5 e-4

8.2 e-4

Intensity of heat source (error) 1006.8 (0.7)% 1635.1 (9.2)% 980.7 (1.9)%

23


1.57 2.51  4.5 0.3 %  4.82 1.27  3.7 15.6  % 1.52 2.42  0.8 3.3 %

G1 G2 G1

1904.6 (5.8)%

5.08 1.63 1.6 8.9  %

G2

It can be seen that with the increase of the domain nodes the results improve slightly. 6. Concluding Remarks This study proposed a method, based on the simultaneous application of an improved meshfree RPIM and ABC algorithm, to solve the inverse problem of estimating the location and intensity of unknown point heat sources in a 2D domain. Unlike other inverse studies which use temperature measurement, in this work, strain measurements are used in the estimation procedure, which results in more accurate and reliable results. The domain integrations, in the direct thermo-elasticity problem solved by the meshfree RPIM, were evaluated using the BDM, which has considerably improved the efficiency of the proposed method. The inverse problem is formulated as an optimization problem and is solved by a modified ABC algorithm. The main conclusions drawn from this study are listed below:    

  

The number of sensors in the domain has a great influence on the accuracy and efficiency of the algorithm, especially for media with high thermal conductivity. The identification results obtained from strain measurements are in general more precise than those obtained from temperature measurements. The number of iterations required for convergence of the algorithm can be decreased effectively, if the input data are strains, rather than temperatures. Using strain gauges instead of thermometers can be more efficient especially in the cases with high magnitude of thermal conductivity. Using an adequate number of nodes and an adaptive nodal distribution, can reduce the computational time and enhance the accuracy of the response. The proposed techniques of this study are applicable to cases in which the sensors are located inside the domain, as well as, to the situations in which only the boundary of the domain is accessible for data measurement. In the meshless RPIM, the computational accuracy of the domain integrations is very important. In the present study, the BDM is used which improves the accuracy of the results considerably. In the ABC algorithm, changing all of the unknown parameters simultaneously instead of changing only one of them can improve the convergence rate of the algorithm and reduce the computational time considerably.

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