Optimization of Mix Proportion of Concrete under Various Severe Conditions by Applying the Genetic Algorithm

Transcription

1 Optimization of Mix Proportion of Concrete under Various Severe Conditions by Applying the Genetic Algorithm Ippei Maruyama, Manabu Kanematsu, Takafumi Noguchi and Fuminori Tomosawa University of Tokyo, Japan Abstract This paper presents a method for optimizing concrete mixture proportions according to the required performances and two case studies showing how to solve proportioning problems under severe conditions. Since various qualities are usually required of concrete, we categorize the proportioning problems as multicriteria optimization problems. As the concept of optimality is not clear in multicriteria problems and there is no way to calculate the optimum solution to such a problem, we dealt with the notion of Pareto Optimality to derive the optimum solution and applied it to a genetic algorithm. In this paper, we solved two proportioning problems for concrete used in severe conditions. One is exposed to chlorides together with freezing and thawing attack and the other is exposed to carbonation attack. 1. Introduction Concrete is required to exhibit high performance under severe environments involving penetration of chloride ions, sulfate attack, and freezing-and-thawing action. Besides high performance, concrete should conventionally have workability, strength and durability at all times. Thanks to technological progress, we can make concrete that meets those requirements. However, there has been no established method whereby the mixture proportions of concrete can be optimized according to the required performance. Only a few attempts[1][2] have so far been made at that problem. The main reason is that a wide variety of mixture proportions are possible and there is no way to optimize the problem under many criteria (represented by objective functions) mathematically. In this report we are concerned with the method of optimizing mixture proportions of concrete involving severe environmental conditions. Optimization MARUYAMA 1/8

2 We cannot solve proportioning problems by the usual methods which consist in searching for the best solution with a single objective function, such as linear programming problems and nonlinear programming problems. The reason is that various qualities are required of concrete corresponding to the environments in which the concrete is used, and we have to consider those qualities to solve the problem. It follows from what has been said that a proportioning problem is classified as a multicriteria optimization problem and we have to formulate a way to solve the multicriteria optimization problem. In multicriteria optimization, the notion of optimality is not obvious at all. For example, it is difficult to compare strength with fluidity. But the concept of Pareto optimality helps us to do this in a rational way. We define the Pareto optimal set as stated below. When a vector x is partially less than y, the mathematical expression of which is: ( x< py) ( i) ( x i y i ) ( i) ( x i < y i ) (The lower the better) 1 Under these conditions, we define that point x dominates point y. If a point is not dominated by any other, we define that it is noninferior. The set of these noninferior points is what we call a Pareto optimal set. According to this definition, if there is a point which is not less than any other points by all criteria, only the best point will get a good evaluation. If there is no such a point, a set of noninferior points, which trade off one of the set of points for another, will be evaluated as good. We use this definition to solve the proportioning problem with genetic algorithms applied to multicritera problems. 2. How to Apply Genetic Algorithm to Multicriteria Problems 2.1 Genetic Algorithms Genetic Algorithms (GA) are optimizing, learning, and searching algorithms based on the mechanics of natural selection and natural genetics. What is important in GA is the unnecessity of the developed method to solve the target problem. Because GA can solve the problem with the relative evaluation in the noninferior set and is more efficient than any other existing method, GA is widely applied to the engineering field, especially to combination problems. Since proportioning is a type of combination problem, there is a good reason to apply GA to proportioning problems. GA deal with genotype, which is the natural parameter set of the optimization problem coded as a finite-length string with binary digits. The simple string is called individual. The algorithms make a phenotype (characteristic form and quality in the designed system) from each genotype and calculate fitness value from the phenotype in the designed system. According to that fitness, an individual in the population (or set of individuals) Optimization MARUYAMA 2/8

3 will be reproduced with crossover and mutation from generation to generation. 2.2 Genetic Algorithms Applied to Multicriteria Problem To apply genetic algorithms to a multicriteria problem, we developed an algorithm that is able to gain the Pareto optimal set. Goldberg[3] has designed similar algorithms. We have improved on those algorithms and applied GA to the problem as follows: We assume that there are P criteria and N individuals (1) Make n individuals randomly. (2) Select criterion No. 1 and determine the fitness of each individual s genotype. Then choose parents A and B with the roulette method in which the probability of its selection is in proportion to fitness. (3) Crossover A and B and reproduce child A and child B. (4) Repeat steps (2) and (3) from criteria No. 1 to No. P. Now we have N individuals in the new generation (child generation) and N individuals in the old generation (parent generation). (5) Produce a temporary generation with the new and parent generations. (6) Mutate genes by reversing the number at certain loci arbitrarily with a constant probability of 1%. (Loci is the plural word of locus. Locus measn the position of the gene.) (7) Select Pareto individuals from the temporary generation and make the next generation that consists of N individuals. (8) (I) If the number of Pareto individuals is less than N, then preserve all the Pareto individuals. Until the number of all individuals becomes N, select individuals one by one from the rest according to criteria No. 1 to No. P in proportion to their fitness. This method means that good genes in the remainder should be carried on in the next generation. (II) On the other hand, if the number of Pareto individuals is N or more, select individuals from the Pareto individuals according to criteria No. 1 to No. P in proportion to their fitness in order. (9) Repeat steps (2) to (8) until the convergent condition is satisfied. In this algorithm the convergent condition is repeating this flow 300 times. 2.3 Individual Design [Genotype] To apply GA to a proportioning problem, we used genotypes to represent various components and various mixture proportions. We designed the genotypes as given in Fig. 1. The genotypes consist of two parts. One is a coded binary string and the other is a database containing components such as cement, aggregate, admixture, and so on. As described in Fig. 1, the string has linkage parts and volumetric ratio parts. The linkage parts have a connection to the database, which is used when the fitness is calculated with volumetric rates. The volumetric ratio parts show the volumes of components in concrete Optimization MARUYAMA 3/8

4 as component/water ratios. In reality, we designed the binary string as 256 bytes whereas it is not shown in Fig. 1. Component Database Cement type, specific-gravity, strength, price Fine aggregate factory, sort, type, specific-gravity, absorption-rate, maximum size, solid-content, F.M,. price Coarse aggregate factory, sort, type, specific-gravity, absorption-rate, maximum size, solid-content, F.M., price Additives type, specific-gravity, sepecific-surface-area, price Admixtures type, effectiveness, recommended-additive-rate, water-reducing-rate, supposed-air-volume, price Genotype Cement sort volume Fine aggregate Coarse aggregate Additives Admixture sort volume sort volume sort volume sort volume Phenotype W/C Water Air Mix Proportion Volume/Mass per unit volume of concrete C F.A C.A. Add Adm Strength specific-gravity Young's modulus Property/Performance of concrete Creep per load Carbonation speed coefficient Chlorides diffusion coefficient Drying shrinkage Slump Plastic viscosity Yield value Setting time Durability Factor* Figure 1 Genotype, phenotype and database of mix proportion [Phenotype] Each designed genotype mentioned above has its phenotype. In our system, the term phenotype means the concrete properties and performance that are estimated Price * Relative dynamic modulus of elasticity after 300 cycles of freezing and thawing action Optimization MARUYAMA 4/8

5 from mixture proportions coded in the genotype. The kinds of concrete properties and performances are also shown in Fig. 1. By studying the literature, we formulated prediction formulas for each property. [Fitness function] In the genetic algorithm, the fitness of phenotypes is evaluated and determinds the individuals ability to reproduce the next generation like individuals surviving natural selection according to its fitness for the environment. To derive a Pareto optimal set by our system, we designed the following suitable functions for fitness. The appearances of these functions are shown in Fig. 2 together with the equations. Fitness Fitness y = property value Young s modulus, Durability Factor ( x u) T 1 y = ( x u) T property value Carbonation speed coefficient, Chlorides diffusion coefficient Fitness Fitness y = ( x u) n T Tu y = x Strength, Slump, property value Specific gravity Setting time Price, Creep per load, Drying-shrinkage property value Figure 2 Fitness functions The method described above is characterized by the evolving population having evaluated genes and Pareto conditions. We should notice, however, that a Pareto optimum set is a set of noninferior points. This definition implies that the evolving population has individuals that have an outstanding performance by a certain performance criterion. The Pareto set conditions do not provide a single exact solution but help to search for an optimal set. Since there still remain some undesirable elements, we have to select individuals that satisfy the required performances from among the final population by GA in the final stage. Optimization MARUYAMA 5/8

6 3. Trial & Results Following are two case studies optimizing the mixture proportion of concrete under severe conditions by this Genetic Algorithms system. 3.1 Case 1 (Severe conditions under freezing and thawing attack and chloride attack) In case 1, we consider concrete that is used in the structure of a two-story reinforced concrete building with a cover depth of 4cm. We assume that the building is located near the seashore, which is the so-called splash and spray zone ranked as XS3 in pren2[4], and attacked by freeze/thaw cycles at a level of XF4 in pren2[4]. The client wants to use this building for 100 years. According to this assumption, the concrete is required to have the qualities given in Table 1. We also assume that the chloride limit condition of the structure is a 4-cm depth of chloride diffusion after 100 years. 3.2 Case 2 (Severe conditions under carbonation attack and intense drying shrinkage) In case 2, we assume that a 10-story steel-framed reinforced concrete building is located in a tropical rain region where carbonation is as hard as in the region ranked XC4 in pren2. This assumption leads to concrete performance requirements shown Table 1. Similarly to case 1, the client wants to use this building for 100 years and the cover depth is 4cm. Table 1 Requested performances under severe condition * Property or Performance Unit Case 1 Case 2 Slump[SL] cm Initial Setting[Si] h 5 5 Final Setting[Sf] h 7 7 Specific Gravity[Gr] t/m Compressive Strength[Fc] MPa Young s Modulus[E] GPa Drying Shrinkage[DS] 8E-04 6E-04 Creep per load[cr] 1/MPa 2.0E- 2.0E- Carbonation Speed Coefficient[D_c] Chlorides Diffussion Coefficient[D_Cl] Durability Factor at 300 cycles[df] cm/ year cm 2 /year *. The circle " " is the sign of main requested performance Price[P] YEN/m Optimization MARUYAMA 6/8

7 Table 2 Samples of the obtained Phenotype *B means Binder i.e. blast furnace, fly ash, silica fume **high early strength cement ***moderate heat cement ****low heat cement *****high range water reducer ******water reducer 3.3 Results SL Si Sf Gr Fc E DS Cr 2.58E- 4.81E- 2.56E- 2.07E- 5.63E- 3.27E- 6.70E- D_c D_Cl 8.32E E E E E E E- 01 DF E- 3.47E- 01 P Table 3 Mixture proportions corresponding to Table 2 Case 1 Case 2 a b c d a b c d W/(C+B) Water (kg) Cement (kg) Fine aggregate (kg) Coarse aggregate (kg) Additive(kg) Admixture (Cwt%) Air (%) Cement (type) HS ** MH *** HS HS MH MH LH **** OPC Additive (type) Admixture (type) fly-ash fly-ash fly-ash fly-ash fly-ash silica fume SP ***** non SP SP non SP SP WR ****** non WR Optimization MARUYAMA 7/8

8 After running the system with 100 individuals and 300 generations, we obtain Tables 2 and 3. The former is a list of phenotypes of individuals and the latter is a list of the corresponding mixture proportions. These lists are selected from 100 individuals in the final stage. Both a and b satisfy the target performances (circled in Table.1), whereas c and d are samples of the gene that can not satisfy the target performances. In case 1, gene a and b satisfy almost all the required performances. Gene c can not satisfy the slump property because of its a little higher addition of superplasticizer and gene d has no fluidity with too little water. In case 2, similar to case 1, genes a and b satisfy the target performances perfectly and almost other required performances. On the other hand, Genes c and d do not exhibit good performance in Drying shrinkage. This can be explained by their high water content and the great paste mass. This is reflected by the tendency of prediction formulas which is based on the many literatures[5]. 4. Conclusions The main results from this study can be summarized as follows; 1 Genetic algorithm system using the concept of Pareto optimality was developed for solving the multicriteria optimization problem. 2 As shown by the exapmles presented in this study, the GA system can derive the appropriate mix proportions from the variety of mixture proportions and sorts of content. This system is maintained by suitable fitness evaluation, reasonable reproduction and correct prediction formulas. 3 An improvement in prediction formulas is still needed to derive more trustworthy results. 5. References [1] Marks W. and Potrzebowski J. Multicriteria optimization of structual concretemixes, Arch.Civil Engineering 38(4), pp.77-01, 1992 [2] Piasta Z. and Czarneski L. Analysis of material efficiency of resin concrete. IN Brittle Matrix Composite, Elsevier Apllied Science London and New York, pp [3] Avid E. Goldberg Genetic Algorithms in Search Optimization & Machine Learning Addison Wesley, pp , 1989 [4] pren (1 st draft), Eurocode 2: Design of concrete structures - Part 1: General rules and rules for buildings, p.52, December, 1999 [5] e.g. G.E.Troxell, Proper Methods of Design and Construcion of Concrete Structures to Prevent Damage from Volumetric Changes of Concrete,ACI Vol.30, Jan-Feb, pp , 1934 Optimization MARUYAMA 8/8