- Email: [email protected]
Shape Optimization of Soil-steel Structure by Simulated Annealing
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 91 (2014) 304 – 309
XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) (TFoCE 2014)
Shape Optimization of Soil-Steel Structure by Simulated Annealing Maciej Sobótkaa*, Dariusz àydĪbaa a
Institute of Geotechnics and Hydrotechnics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland
Abstract The paper presents results of shape optimization of underground structure. First, the objective function is proposed based on the energetic equivalence principle. The excavation is replaced, within the approach proposed, by an artificial very compliant material and the optimal shape of the excavation is postulated as the one that minimizes the energy of volumetric strain cumulated in it. The solution is looked for using the stochastic optimization approach, namely the simulated annealing (SA) procedure. A soil-steel bridge structure with different loads and boundary conditions is investigated. The results representing the optimized shapes of the structures examined are presented. © 2014 The Authors. Published by Elsevier Ltd. © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil (http://creativecommons.org/licenses/by-nc-nd/3.0/). Engineering (23RSP). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) Keywords: shape optimization; simulated annealing; underground structure; soil-steel structure.
1. Introduction A considerable amount of literature has been published on size, shape and topology optimization of bridge structures in the last decades [1, 2, 3, 4]. The researchers use a wide spectrum of optimization methods, inter alia, structural evolutionary optimization (ESO) [5, 6], bi-directional ESO (BESO) [7], genetic algorithm (GA) [4], and others, e.g. [2]. In most of these works, the bridge structure is treated as an “independent” system, attached rigidly to the ground at specifically established fixed points. However, in the case of the soil-shell type structure the interaction between the backfill and the shell is of crucial importance. Shape optimization procedure of that kind of bridge has been reported in [2]. In the proposed approach both, the concrete arc and the soil engineered backfill are considered. The shape corresponding to minimum tensile stress within the arc has been found for a given load and geometric constraints. This paper concerns soil-steel structure with flexible steel shell. It has been reported that most of the load is carried by the soil backfill, whereas the shell acts as an excavation support [8]. Thus, from the constructional point of view, such structure is similar to shallow tunnel. The paper takes into account stated above facts. In particular, it
* Corresponding author. Tel.: +48-320-41-27. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) doi:10.1016/j.proeng.2014.12.065
305
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
focuses on application of the energetic optimality condition formulated for tunnels [9] and the simulated annealing procedure [10, 11] to shape optimization of a soil steel-bridge. The paper is organised as follows. First, the formulation of the energetic condition is presented. Subsequent section is concerned with the simulated annealing optimization procedure. Next, the results of the optimization for different load types and their combinations are presented. Finally, the outcomes are discussed and the implications of the results to future research are sketched. 2. Energetic optimality condition In 2009 the authors of present work formulated an energetic optimality condition for tunnel excavation with a structural support [12]. A crucial assumption taken is as follows: the less the convergence calculated with no support involved, the better shape from the view of the support effort. For the case of tunnels at great depths, where the dominant load is deformation pressure, a primary task of a support is to ensure the stability of excavation. This is performed by retaining the convergence. Thus, the objective function should be a measure of the convergence. In the original formulation [9] two forms of the cost function have been postulated. The first one has the following form:
'V
&&
³ u n dS ,
(1)
S
where the designation of quantities used in the formula above is presented in Fig. 1.
Fig. 1. Adopted designation: ī, ȍ – rock mass and excavation area, respectively; & & S – excavation contour, u – displacement vector, n unit normal vector.
The physical meaning of the formula (1) is the convergence calculated as a change in the area of the void region (tunnel cross-section) under the load. Even so, it does not satisfy the typical conditions imposed on a cost function. It is not positive definite. Thus, the second measure has been introduced. In the proposed approach, a void region is replaced by a very compliant material. Then, the measure of convergence can be expressed as an energy of elastic volumetric strain cumulated in the excavation region ȍ (tunnel core). Assuming plain strain, the following form of a cost function is postulated:
E0
2 1 K ³ H x H y d: , 2 :
(2)
where K denotes the bulk modulus, İx, İy are the values of normal strain components, horizontal and vertical, respectively. The quantity (2) is positive definite and, thus, easier to implement in numerical calculations. In addition, no change in the geometry of a problem domain is required during the optimization process because of the manner the void region is modelled. Both conditions (1) and (2) have been examined in the problem of shape optimization of a tunnel in elastic rock mass. The detailed results are presented in [9]. The main finding from this work is as follows: incorporation of the
306
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
conditions (1) and (2) gives the same result: the optimal shape of a tunnel in the view of support effort is the ellipse with sizes ratio equal to the square root of in situ principal stresses ratio [9,11]. 3. Simulated annealing Simulated annealing [10] is a heuristic, stochastic method of optimization. The basis of the procedure derives from physical process in metallurgy, called annealing. It involves heating of the material to above its critical temperature, maintaining at suitable temperature for some period and then cooling with prescribed, enough slow rate. At higher temperature, the rearrangement of particles is a continuous, chaotic process, described in terms of statistical mechanics. During the cooling, the movements of the particles become less chaotic and the microstructure is gradually refined. If the cooling rate is enough slow, the particles eventually set at low energy levels and the system reaches a stable state, optimal in sense of the value of the internal energy. A simple algorithm for a simulation of a collection of atoms at given temperature is proposed in [13]. A system is brought to equilibrium by an iterative procedure in which, first, an atom is given a random displacement and then if the energy of the system in the new configuration is less than in previous one (ǻE<0), the change is accepted; otherwise, the change is accepted with the probability:
P('E ) e
'E k BT
,
(3)
where kB is the Boltzmann constant and T is the actual temperature. An analogous procedure is adopted in the simulated annealing. The internal energy is replaced by a suitable cost function and the temperature is an artificial parameter controlling the randomness of the process. When the temperature decreases, the system becomes less chaotic. Finally, the system is brought to the stable, optimal state. 4. A procedure of shape optimization This section gives an example of simulated annealing procedure for shape optimization of a soil-steel culvert. It is assumed that the height of the structure is 4.2 m, and the value of “void” area under the bridge is equal to 12.7 m2. The values correspond to the bridge on the rout DK8 near Niemcza, Poland [14,15]. The sequence of boundary value problems has been solved using commercially available code Itasca Flac 7.00 [16] that is based on finite difference method. Taking the advantage of so-called FISH programming language in Flac, entire procedure has been developed in a single piece of software. A plain boundary value problem being a simplified model of a soil-steel bridge is considered. Only a half of the structure is analysed due to the symmetry of the system. The model at initial state of shape is presented in the Fig. 2 below. Colour filled areas indicate the search grid. The void region (the lighter one) cannot protrude the area of the regular grid of 30 x 30 zones.
Fig. 2. A boundary value problem under consideration – finite difference grid, fixed gridpoints and initial shape of the structure.
307
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
Oedometric boundary conditions are assumed, i.e. the displacements perpendicular to the truncation lines are set to 0. The top boundary segment has prescribed proper load condition. In this particular case the structure is subjected to uniformly distributed load. The intensity of the load is q=4.0 kN/m. The dead load is not taken into account in this example. Linear elastic model of the material is adopted. The parameters of engineered backfill correspond to the coarse sand in dense state of compaction (density index ID=0.8). According to the polish standard PN-81/B 03020 such soil is characterized by the Young modulus E=150 MPa, the Poisson’s ratio Ȟ=0.25 and density ȡ=2.0 t/m3. The parameters of the void region are negligible in relation to those of backfill, i.e. the stiffness and density are assumed as 106 times smaller than those of the backfill. The cost function adopted in the problem of shape optimization is the energy E0 defined by the equation (2). The permissible change in shape in a single iteration is a swap of two zones belonging to the different regions, i.e. one to the void region and the other to the soil region. It follows that void region area is preserved. The other geometrical constraints imposed on the shape are as follows. The area of the void must remain simply connected space. In other words, the region under the bridge must never divide into two or more separate regions and must not have the “islands” of the soil region. Such assumption restricts the topology optimization problem to the case of shape optimization but it is not the limitation of the algorithm. The last constraint is that top row of zones is protected, i.e. the bridge deck is continuous. The procedure starts from a certain initial shape. In the case considered it is a rectangle of 18 x 18 zones. The artificial temperature parameter at start must be specified too. The manner of determining the initial temperature is given later on. At given temperature a hundred of iteration steps are executed. Iteration consists of: x a random change in shape, x calculation of a change in the cost function ǻE0, x an acceptance or rejection of a change. A change is always accepted if causes a decrease of function E0, i.e. ǻE0<0 (the shape becomes better). Otherwise, the change is accepted with the probability P(ǻE0) given by Eq. (3). If ǻE0>0, a random number x between 0.0 and 1.0 is drawn from a uniform distribution. Then, the change is accepted if x< P(ǻE0), otherwise it is not. A temperature T0 at the start of the optimization procedure is determined in a manner that implies the probability of acceptance of a change in case of ǻE>0 equal to approximately 0.5. To estimate the starting temperature T0 the average ǻE0*of all positive ǻE0 values for a thousand of trials is determined for starting shape. Then the temperature T0 is derived from inequality (3) by following substitutions: P(ǻE0)=0.5, ǻE0= ǻE0*, T =T0. Finally the starting temperature is:
T0
'E0* . ln 0.5
(4)
The cooling kinetics is realised as a geometric sequence, i.e. a new temperature is 0.9 times of the previous one. For each value of temperature a hundred iterations are performed, the temperature is decreased, etc. The optimization procedure ends when further decreasing of the temperature does not make any difference in the shape. The evolution of shape during the optimization procedure is presented in Fig. 3. Subsequent shapes starting from the initial one and then after 1000, 2000, 5000 and 10000 iterations are shown.
308
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
a)
b)
d)
c)
e)
Fig. 3. Evolution of the shape during simulated annealing: a) the initial shape; b) c) d) and e) – shapes after 1000, 2000, 5000 and 10000 iterations, respectively.
5. Numerical examples The procedure described in the previous section has been utilised to find optimal shapes for different load combinations and for different support conditions. Two different load types have been analysed. Dead weight of a structure (backfill) and uniformly distributed load acting on the surface are considered. The intensity of the load is q=4.0 kN/m. Density of the soil backfill is ȡ=2.0 t/m3. Three load combinations have been tested, namely: uniform load on top (combination I), dead weight of the structure (combination II) and both loads acting on the structure (combination III). The procedure has been performed for two different displacement conditions at the bottom boundary segment, i.e. fixed vertical displacement (type I) and fully fixed displacement in both, vertical and horizontal directions (type II). The optimal shapes obtained are presented in Figures below:
a)
b)
c)
Fig 4. Optimal shapes for boundary condition type I; a) load combination I, b) load combination II, c) load combination III.
a)
b)
c)
Fig 5. Optimal shapes for boundary condition type II; a) load combination I, b) load combination II, c) load combination III.
6. Discussion of the results and conclusions In all considered cases of different load and support conditions the shape stabilizes after approximately 70009000 iterations. Nevertheless, the procedure has been continued up to 10000 to make sure that the process is
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
309
completed. For the uniform load acting on the structure, the resulting optimal shape does not depend on boundary condition assumed for bottom boundary segment. On the other hand, in the cases in which the own weight of the soil is considered, the support condition at bottom has an influence on the shape. For fully fixed displacement, the resulting optimal shape is more oval than in the case with only the vertical displacement fixed. Furthermore, for load on the top, the soil layer above the shell is thicker than in the case of taking the only the dead weight into account. Obtained shapes suggest that the methodology could give reasonable results. However, the practical application of the procedure requires accounting reliable geometrical constraints like fixed clearance gauge or fixed span length and proper load conditions as well. It will be a topic of further study of the authors. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Y.M. Xie, Z.H. Zuo, X. Huang, T. Black, P. Felicetti, Application of topological optimisation technology to bridge design, Struct. Eng. Int. 2 (2014) 185–191. V. Houšt, J. Eliáš, L. Miþa, Shape optimization of concrete buried arches, Eng. Struct. 48 (2013) 716–726. J. Norato, et al., A geometry projection method for shape optimization, Int. J. Numer. Meth. Eng. 60.14 (2004) 2289–2312. S.D. Rajan, Sizing, shape, and topology design optimization of trusses using genetic algorithm, J. Struct. Eng. 121.10 (1995) 1480–1487. Y.M. Xie, G.P. Steven, A simple evolutionary procedure for structural optimization, Comput. Struct. 49.5 (1993) 885–96. Y.M. Xie, G.P. Steven, G.P., Evolutionary structural optimization, Springer, Berlin, 1997. V. Young, et al., 3D and multiple load case bi-directional evolutionary structural optimization (BESO), Structural optimization 18.2–3 (1999) 183–192. L. Janusz, A. Madaj, Engineering structures made form corrugated steel plates (in Polish), WKiL, Warszawa, 2007. A. RóĪaĔski, M. Sobótka, A procedure of underground excavation shape optimization (in Polish), Górnictwo i GeoinĪynieria 1 (2009) 519–529. S. Kirkpatrick, C. Gelatt, M. Vecchi, Optimization by simulated annealing, Science 220.4598 (1983) 671–680. M. Sobótka, D. àydĪba, A. RóĪaĔski, Shape optimization of underground excavation by simulated annealing, Studia Geotechnica et Mechanica 35.1 (2013) 209–218. M. Kawa, A. RóĪaĔski, M. Sobótka, A verification of shape optimization procedures of tunnel underground excavations (in Polish), Górnictwo i GeoinĪynieria 2 (2011) 535–541. N. Metropolis, et.al., Equation of state calculation by fast computing machines, J. Chem. Phys. 21. 6 (1953) 1087–1092. C. Machelski, G. Antoniszyn, B. Michalski, Live load effects on a soil-steel bridge founded on elastic supports, Studia Geotechnica et Mechanica 28, 2–4 (2006) 65–82. M. Sobótka, Numerical simulation of hysteretic live load effect in a soil-steel bridge, Studia Geotechnica et Mechanica 36.1 (2014) 103– 109. FLAC Fast Lagrangian Analysis of Continua. User’s Guide, Itasca Consulting Group Inc., Minneapolis, 2011.
ScienceDirect Procedia Engineering 91 (2014) 304 – 309
XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) (TFoCE 2014)
Shape Optimization of Soil-Steel Structure by Simulated Annealing Maciej Sobótkaa*, Dariusz àydĪbaa a
Institute of Geotechnics and Hydrotechnics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland
Abstract The paper presents results of shape optimization of underground structure. First, the objective function is proposed based on the energetic equivalence principle. The excavation is replaced, within the approach proposed, by an artificial very compliant material and the optimal shape of the excavation is postulated as the one that minimizes the energy of volumetric strain cumulated in it. The solution is looked for using the stochastic optimization approach, namely the simulated annealing (SA) procedure. A soil-steel bridge structure with different loads and boundary conditions is investigated. The results representing the optimized shapes of the structures examined are presented. © 2014 The Authors. Published by Elsevier Ltd. © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil (http://creativecommons.org/licenses/by-nc-nd/3.0/). Engineering (23RSP). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) Keywords: shape optimization; simulated annealing; underground structure; soil-steel structure.
1. Introduction A considerable amount of literature has been published on size, shape and topology optimization of bridge structures in the last decades [1, 2, 3, 4]. The researchers use a wide spectrum of optimization methods, inter alia, structural evolutionary optimization (ESO) [5, 6], bi-directional ESO (BESO) [7], genetic algorithm (GA) [4], and others, e.g. [2]. In most of these works, the bridge structure is treated as an “independent” system, attached rigidly to the ground at specifically established fixed points. However, in the case of the soil-shell type structure the interaction between the backfill and the shell is of crucial importance. Shape optimization procedure of that kind of bridge has been reported in [2]. In the proposed approach both, the concrete arc and the soil engineered backfill are considered. The shape corresponding to minimum tensile stress within the arc has been found for a given load and geometric constraints. This paper concerns soil-steel structure with flexible steel shell. It has been reported that most of the load is carried by the soil backfill, whereas the shell acts as an excavation support [8]. Thus, from the constructional point of view, such structure is similar to shallow tunnel. The paper takes into account stated above facts. In particular, it
* Corresponding author. Tel.: +48-320-41-27. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) doi:10.1016/j.proeng.2014.12.065
305
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
focuses on application of the energetic optimality condition formulated for tunnels [9] and the simulated annealing procedure [10, 11] to shape optimization of a soil steel-bridge. The paper is organised as follows. First, the formulation of the energetic condition is presented. Subsequent section is concerned with the simulated annealing optimization procedure. Next, the results of the optimization for different load types and their combinations are presented. Finally, the outcomes are discussed and the implications of the results to future research are sketched. 2. Energetic optimality condition In 2009 the authors of present work formulated an energetic optimality condition for tunnel excavation with a structural support [12]. A crucial assumption taken is as follows: the less the convergence calculated with no support involved, the better shape from the view of the support effort. For the case of tunnels at great depths, where the dominant load is deformation pressure, a primary task of a support is to ensure the stability of excavation. This is performed by retaining the convergence. Thus, the objective function should be a measure of the convergence. In the original formulation [9] two forms of the cost function have been postulated. The first one has the following form:
'V
&&
³ u n dS ,
(1)
S
where the designation of quantities used in the formula above is presented in Fig. 1.
Fig. 1. Adopted designation: ī, ȍ – rock mass and excavation area, respectively; & & S – excavation contour, u – displacement vector, n unit normal vector.
The physical meaning of the formula (1) is the convergence calculated as a change in the area of the void region (tunnel cross-section) under the load. Even so, it does not satisfy the typical conditions imposed on a cost function. It is not positive definite. Thus, the second measure has been introduced. In the proposed approach, a void region is replaced by a very compliant material. Then, the measure of convergence can be expressed as an energy of elastic volumetric strain cumulated in the excavation region ȍ (tunnel core). Assuming plain strain, the following form of a cost function is postulated:
E0
2 1 K ³ H x H y d: , 2 :
(2)
where K denotes the bulk modulus, İx, İy are the values of normal strain components, horizontal and vertical, respectively. The quantity (2) is positive definite and, thus, easier to implement in numerical calculations. In addition, no change in the geometry of a problem domain is required during the optimization process because of the manner the void region is modelled. Both conditions (1) and (2) have been examined in the problem of shape optimization of a tunnel in elastic rock mass. The detailed results are presented in [9]. The main finding from this work is as follows: incorporation of the
306
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
conditions (1) and (2) gives the same result: the optimal shape of a tunnel in the view of support effort is the ellipse with sizes ratio equal to the square root of in situ principal stresses ratio [9,11]. 3. Simulated annealing Simulated annealing [10] is a heuristic, stochastic method of optimization. The basis of the procedure derives from physical process in metallurgy, called annealing. It involves heating of the material to above its critical temperature, maintaining at suitable temperature for some period and then cooling with prescribed, enough slow rate. At higher temperature, the rearrangement of particles is a continuous, chaotic process, described in terms of statistical mechanics. During the cooling, the movements of the particles become less chaotic and the microstructure is gradually refined. If the cooling rate is enough slow, the particles eventually set at low energy levels and the system reaches a stable state, optimal in sense of the value of the internal energy. A simple algorithm for a simulation of a collection of atoms at given temperature is proposed in [13]. A system is brought to equilibrium by an iterative procedure in which, first, an atom is given a random displacement and then if the energy of the system in the new configuration is less than in previous one (ǻE<0), the change is accepted; otherwise, the change is accepted with the probability:
P('E ) e
'E k BT
,
(3)
where kB is the Boltzmann constant and T is the actual temperature. An analogous procedure is adopted in the simulated annealing. The internal energy is replaced by a suitable cost function and the temperature is an artificial parameter controlling the randomness of the process. When the temperature decreases, the system becomes less chaotic. Finally, the system is brought to the stable, optimal state. 4. A procedure of shape optimization This section gives an example of simulated annealing procedure for shape optimization of a soil-steel culvert. It is assumed that the height of the structure is 4.2 m, and the value of “void” area under the bridge is equal to 12.7 m2. The values correspond to the bridge on the rout DK8 near Niemcza, Poland [14,15]. The sequence of boundary value problems has been solved using commercially available code Itasca Flac 7.00 [16] that is based on finite difference method. Taking the advantage of so-called FISH programming language in Flac, entire procedure has been developed in a single piece of software. A plain boundary value problem being a simplified model of a soil-steel bridge is considered. Only a half of the structure is analysed due to the symmetry of the system. The model at initial state of shape is presented in the Fig. 2 below. Colour filled areas indicate the search grid. The void region (the lighter one) cannot protrude the area of the regular grid of 30 x 30 zones.
Fig. 2. A boundary value problem under consideration – finite difference grid, fixed gridpoints and initial shape of the structure.
307
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
Oedometric boundary conditions are assumed, i.e. the displacements perpendicular to the truncation lines are set to 0. The top boundary segment has prescribed proper load condition. In this particular case the structure is subjected to uniformly distributed load. The intensity of the load is q=4.0 kN/m. The dead load is not taken into account in this example. Linear elastic model of the material is adopted. The parameters of engineered backfill correspond to the coarse sand in dense state of compaction (density index ID=0.8). According to the polish standard PN-81/B 03020 such soil is characterized by the Young modulus E=150 MPa, the Poisson’s ratio Ȟ=0.25 and density ȡ=2.0 t/m3. The parameters of the void region are negligible in relation to those of backfill, i.e. the stiffness and density are assumed as 106 times smaller than those of the backfill. The cost function adopted in the problem of shape optimization is the energy E0 defined by the equation (2). The permissible change in shape in a single iteration is a swap of two zones belonging to the different regions, i.e. one to the void region and the other to the soil region. It follows that void region area is preserved. The other geometrical constraints imposed on the shape are as follows. The area of the void must remain simply connected space. In other words, the region under the bridge must never divide into two or more separate regions and must not have the “islands” of the soil region. Such assumption restricts the topology optimization problem to the case of shape optimization but it is not the limitation of the algorithm. The last constraint is that top row of zones is protected, i.e. the bridge deck is continuous. The procedure starts from a certain initial shape. In the case considered it is a rectangle of 18 x 18 zones. The artificial temperature parameter at start must be specified too. The manner of determining the initial temperature is given later on. At given temperature a hundred of iteration steps are executed. Iteration consists of: x a random change in shape, x calculation of a change in the cost function ǻE0, x an acceptance or rejection of a change. A change is always accepted if causes a decrease of function E0, i.e. ǻE0<0 (the shape becomes better). Otherwise, the change is accepted with the probability P(ǻE0) given by Eq. (3). If ǻE0>0, a random number x between 0.0 and 1.0 is drawn from a uniform distribution. Then, the change is accepted if x< P(ǻE0), otherwise it is not. A temperature T0 at the start of the optimization procedure is determined in a manner that implies the probability of acceptance of a change in case of ǻE>0 equal to approximately 0.5. To estimate the starting temperature T0 the average ǻE0*of all positive ǻE0 values for a thousand of trials is determined for starting shape. Then the temperature T0 is derived from inequality (3) by following substitutions: P(ǻE0)=0.5, ǻE0= ǻE0*, T =T0. Finally the starting temperature is:
T0
'E0* . ln 0.5
(4)
The cooling kinetics is realised as a geometric sequence, i.e. a new temperature is 0.9 times of the previous one. For each value of temperature a hundred iterations are performed, the temperature is decreased, etc. The optimization procedure ends when further decreasing of the temperature does not make any difference in the shape. The evolution of shape during the optimization procedure is presented in Fig. 3. Subsequent shapes starting from the initial one and then after 1000, 2000, 5000 and 10000 iterations are shown.
308
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
a)
b)
d)
c)
e)
Fig. 3. Evolution of the shape during simulated annealing: a) the initial shape; b) c) d) and e) – shapes after 1000, 2000, 5000 and 10000 iterations, respectively.
5. Numerical examples The procedure described in the previous section has been utilised to find optimal shapes for different load combinations and for different support conditions. Two different load types have been analysed. Dead weight of a structure (backfill) and uniformly distributed load acting on the surface are considered. The intensity of the load is q=4.0 kN/m. Density of the soil backfill is ȡ=2.0 t/m3. Three load combinations have been tested, namely: uniform load on top (combination I), dead weight of the structure (combination II) and both loads acting on the structure (combination III). The procedure has been performed for two different displacement conditions at the bottom boundary segment, i.e. fixed vertical displacement (type I) and fully fixed displacement in both, vertical and horizontal directions (type II). The optimal shapes obtained are presented in Figures below:
a)
b)
c)
Fig 4. Optimal shapes for boundary condition type I; a) load combination I, b) load combination II, c) load combination III.
a)
b)
c)
Fig 5. Optimal shapes for boundary condition type II; a) load combination I, b) load combination II, c) load combination III.
6. Discussion of the results and conclusions In all considered cases of different load and support conditions the shape stabilizes after approximately 70009000 iterations. Nevertheless, the procedure has been continued up to 10000 to make sure that the process is
Maciej Sobótka and Dariusz Łydżba / Procedia Engineering 91 (2014) 304 – 309
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completed. For the uniform load acting on the structure, the resulting optimal shape does not depend on boundary condition assumed for bottom boundary segment. On the other hand, in the cases in which the own weight of the soil is considered, the support condition at bottom has an influence on the shape. For fully fixed displacement, the resulting optimal shape is more oval than in the case with only the vertical displacement fixed. Furthermore, for load on the top, the soil layer above the shell is thicker than in the case of taking the only the dead weight into account. Obtained shapes suggest that the methodology could give reasonable results. However, the practical application of the procedure requires accounting reliable geometrical constraints like fixed clearance gauge or fixed span length and proper load conditions as well. It will be a topic of further study of the authors. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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