PERFORMANCE BASED DESIGN SYSTEM FOR CONCRETE MIXTURE WITH MULTI-OPTIMIZING GENETIC ALGORITHM

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1 PERFORMANCE BASED DESIGN SYSTEM FOR CONCRETE MIXTURE WITH MULTI-OPTIMIZING GENETIC ALGORITHM Takafumi Noguchi 1, Iei Maruyama 1 and Manabu Kanematsu 1 1 Deartment of Architecture, University of Tokyo, Tokyo, Jaan. noguchi@bme.arch.t.u-tokyo.ac.j ABSTRACT This aer resents a method for otimizing concrete mixture roortions according to the required erformance and also resents several case studies. Since various qualities are usually required of concrete, we characterize the roortioning roblems as multi-criteria otimization roblems. We dealt with the notion of Pareto otimality to derive the otimum solution and alied it to a genetic algorithm. In this contribution, two roortioning roblems are solved by the genetic algorithm: a request of cost-erformance at 60 MPa in strength as a simle roblem, and a concrete mix for mass concrete in cold weather at the coast. 1. INTRODUCTION Concrete is required to exhibit erformance corresonding with alied environment. And nowadays, thanks to technological rogress, it is ossible to make the concrete meet those requirements. However, there has been no established method. Only a few attemts [1][2] have so far been made in relation to these roblems. The main reason for few outcomes is that a wide variety of mixture roortions are ossible and there is no way to otimize mathematically the roblem under many criteria, which are reresented by objective functions. This contribution describes the method of otimizing mixture roortions of concrete by alication of the Genetic Algorithm (GA) and several engineering models reresenting the relationshi between the roerties of concrete and mix roortions. Proortioning roblems cannot be solved by the usual methods that consist of searching for the best solution with a single objective function, such as linear rogramming roblems and nonlinear rogramming roblems. The reason for this imossibility is that various erformances are required of concrete corresonding to the environments in which the concrete is used, and it is imossible to exress the lural requests in a single objective function. It follows from the features of the roortioning roblem mentioned above that a roortioning roblem is considered to be a multicriteria otimization roblem. 2. HOW TO APPLY GENETIC ALGORITHM TO MULTI-CRITERIA PROBLEMS 2.1 Genetic Algorithms Genetic algorithms are otimizing, learning and searching algorithms based on the mechanism of natural selection and natural genetics. What is imortant in GA is that it is unnecessary to develo a method to solve the target roblem. Because GA can solve the roblem with the relative evaluation of the non-inferior set and is easier to formulate rules of evaluation than any other existing method, GA is widely alied to the engineering field, esecially to combination roblems. Since roortioning is a tye of combination roblem, there is a good reason to aly GA. Proceedings of the 11 th International Congress on the Chemistry of Cement (ICCC) May 2003, Durban, South Africa Cement s Contribution to the Develoment in the 21 st Century ISBN Number: Hosted by: The Cement and Concrete Institute of South Africa Editors: Dr G. Grieve and G. Owens CD-ROM roduced by: Document Transformation Technologies Congress Organised by: Event Dynamics

2 In GA genotye is the arameter set, which means a vector in the set of otimization roblem, coded as a finite-length string with binary digits. The simle string is called individual. To aly GA to a roortioning roblem, the genotye is designed to reresent various comonents and various mixture roortions (see Fig.1). The genotye consists of two arts. One is a coded binary string and the other is a database containing comonents such as cement, aggregate, admixtures, and so on. As demonstrated in Figure 1, the string has linkage arts and volumetric ratio arts. The linkage arts have a connection to the database, which is used when the fitness is calculated with volumetric ratios. The volumetric ratio arts show the volumes of comonents in concrete as comonent/water ratios. In reality, the binary string is fixed as 256 bytes whereas it is not shown in Figure 1. The algorithms make a henotye, which demonstrates characteristic form and quality in the designed system, from each individual according to its genotye. Each designed genotye has its henotye. In our system, the term henotye means the concrete roerties and erformance that are estimated from mixture roortions coded in the genotye. The kinds of concrete roerties and erformance are also shown in Figure 1. Prediction formulas are formulated statistically for each roerty. Figure 1. Schematic of GA alied for roortioning roblem Fitness value gives the numerical evaluation of each individual and each henotye in the designed system. According to the fitness values, which can be lural in Pareto otimality roblem, an individual in the oulation reresenting the set of individuals will be reroduced with crossover and mutation from generation to generation. Thus the individuals in the set become fitted under alied environment. In the develoed algorithm, suitable functions are designed to meet the way of erformance requesting and those are shown in Figure 2 together with the equations. U is the arameter reresenting the required erformance of concrete mixture and T is the arameters

3 that determine shae of function. These two arameters exress the allowable range of requirements for erformance. Figure 2. Fitness function and its shaes 2.2 Genetic Algorithms Alied to Multi-criteria Problem In multi-criteria otimization, the notion of otimality is not obvious at all. For examle, it is difficult to comare strength with fluidity. There does not exist the way to evaluate and comare different things. But the concet of Pareto otimality hels the evaluation of multile criteria in a rational way. The Pareto otimal set is defined as stated below. When a vector x is artially less than y and all criteria are minimizing criteria, the mathematical exression of the vector x in the Pareto otimal set is: ( x< y) = ( i)( x < y ) ( i)( x < y ) i i i i (1) Under the condition of equation (1), oint x dominates oint y. If the oint is not dominated by any other, that oint is non-inferior. The set of these non-inferior oints is what we call a Pareto otimal set. According to this definition, if there is a oint, which is not less than any other oints by all criteria, only the best oint will get a good evaluation. If there is no such a oint, a set of non-inferior oints, which trade off one of the set of oints for another, will be evaluated as good. To aly genetic algorithms to a multi-criteria roblem, an algorithm that can derive the Pareto otimal set is develoed. Goldberg [3] had designed similar algorithms. Imrovement on Goldberg s algorithms and alication to the roortioning roblem are conducted. The develoed rogram alied to concrete mixture roblem is named MixGA. The rogram-flow of MixGA is detailed as follows. 1. Assume that there exist P criteria and N individuals 2. Make N individuals randomly. 3. Select criterion No. 1 and determine the fitness value of each individual's genotye. Choose arents A and B with the roulette method in which the robability of its selection is in roortion to fitness value. Crossover A and B, and reroduce child C and child D. 4. Reeat stes (2) and (3) from criteria No. 1 to No. P.

4 5. After getting N individuals in the new oulation (child generation) and N individuals in the old oulation (arent generation). 6. Produce a temorary generation with the child and arent generations. 7. Mutate genes by reversing the number at certain loci arbitrarily with a constant robability of 1%. (Loci is the lural word of locus. Locus means the osition of the gene.) 8. Select Pareto individuals from the temorary generation and make the next generation that consists of N individuals. Reduction rule of oulation is dominated by two roles below. (i) If the number of Pareto individuals is less than N, then reserve all the Pareto individuals. U to the number of all individuals becomes N, select individuals one by one from the rest with the criteria No. 1 to No. P. The robability of selection is in roortion to their fitness value. This method intends that the 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. The robability of selection is in roortion to their fitness in order. 9. Iterate stes (2) to (8) until the given number of times. 10. Conduct final selection. The algorithm detailed above is characterized by the evolution of oulation with highly evaluated genes and Pareto otimal conditions. The highly evaluated individual has a good feature in a art of the binary string. And these arts of a binary string in individuals are inherited to next generation exlicitly. An individual in the next generation inherits several good arts of a binary string that can be highly evaluated through henotye. This rocess of evolution makes it ossible to find out the otimal concrete mixture among an almost infinite number of combinations of materials and roortions; otherwise the rocess can be a random search and that will fail. But it should be noted, however, that a Pareto otimum set is a set of non-inferior oints. This definition imlies it is ossible that the evolved oulation has individuals, which have an outstanding erformance by a certain erformance criterion in exchange for bad fitness values of the others. The Pareto set conditions do not rovide a single exact solution but hel to search for an otimal set widely. Because of this undesirable asect of Pareto otimality, the concet of metaroerty is used. In MixGA the average of fitness values of target roerties is used as one of roerties and this kind of roerty of individual is called meta-roerty. With the meta-roerty concet it is ossible to inherit another ossibility in gene that shows balancing in target roerties of concrete to the next generation. Besides a final rocess of evaluation, a final selection is conducted using the fitness values and eliminating the individuals that do not meet the required erformance. 3. TRIAL OF MIXGA The followings are two case studies, in which the mixture roortion of concrete is otimized under required conditions by using MixGA. 3.1 Case study 1 In case study 1, simle cost erformance roblem of strength roerty is examined. In detail, the concrete is required to have 60 MPa, Slum 18 cm and lower cost. Other roerties are less significant Prediction of Strength of Concrete As is mentioned above, redicting functions that enable us to calculate the roerties of concrete with any mix roortion are formulated. Regarding the strength roerties of concrete, the

5 formulating is artially based on the theoretical aroach and artially based on the statistical aroach. The formulas used are shown below. Strength of mortar: F = ( a( B/ W) + b) K m where F m denotes the strength of mortar, a, b and K denote material arameter deending on cement, W denotes the water content er unit weight [kg/m 3 ], and B denotes the summation of cement and admix content er unit weight [kg/m 3 ]. In this contribution 4 kinds of cement are formulated; namely Ordinary Portland Cement (OPC), Low Heat Cement (LHC), Moderate Heat Cement (MHC) and High Early Strength Cement (HESC). (2) Effect of coarse aggregate on strength: 1 V rg = 1 d 1 e B / W + f LogX LogA W / B ( ( ) ( )) (3) where r g denotes the effect of coarse aggregate on strength, d, e and f denote material arameter deending on aggregate, V denotes the volume of coarse aggregate er unit weight, X denotes the maximum size of coarse aggregate and A denotes the minimum size of aggregate that is able to affect the loss of strength. Effect of tye of coarse aggregate focused on the aste interface: rg 2 = k (4) where r g2 denotes the effect of coarse aggregate tye and k denotes the material arameter of aggregate tye deending on crushed aggregate (CR) or gravel (GR). Effect of air content: r = 1 k V air 2 air (5) where r air denotes the effect of air content on strength, k 2 denotes the arameter deending on water to binder ratio and tye of coarse aggregate, and V air denotes the volume of air er unit volume of concrete. Effect of addition of Ground Granulated Blast furnace Slag (GGBS): X < 0.3: 3/2 1 X rmix = 1 + ( g( X2 h) ) + ( lx2 + m) X X 0.3: X 0.3 r0.3 ( a1 X2 + b1 ) 0.4 (6) (7) where r mix denotes the effect of admixture on strength, g, h, l, m, a 1, b 1 denote the material arameter, X denotes the relacement ratio, X 1 denotes the water to binder ratio, X 2 denotes secific surface area of slag and r 0.3 denotes the value calculated by equation (6) with X of 0.3.

6 Effect of addition of Fly-Ash (FA): ( ) X < 0.45: r = 1 a X b X mix X 0.45 r mix = (8) (9) where r mix denotes the effect of admix on strength, a 2 and b 2 denote the material arameter, X denotes the relacement ratio and X 1 denotes the water to binder ratio. Effect of addition of Silica Fume (SF): X > 0.2 r = 1+ X mix X 0.2 r mix = 1.2 (10) (11) where r mix denotes the effect of admix on strength, X denotes the relacement ratio. Strength of Concrete: F = F r r r r c m g g2 air mix (12) where F c denotes the strength of concrete Prediction of cone slum Function for redicting of the value of cone slum is develoed statistically with rheological aroach. Kikukawa [4] roosed equation (13), which exresses the rheological arameters of cement aste at 20 o C that are lastic viscosity and yield value, as a function of solid content of cement and volumetric density of cement aste. This exression was on the basis of study of Roscoe [5]. Plastic viscosity of cement aste: η V = ηw 1 c ( KV 1 + K2) (13) where η denotes lastic viscosity of cement aste [Pa s], ηw denotes lastic viscosity of water, V denotes volumetric density of cement aste, c denotes solid content of cement, K 1,K 2 denote constants affected by the roerties of the cement aste and 12.0 are used resectively. Yield value : τ = aη + b (14) where τ denotes yield value of cement aste [Pa], a and b denote constant arameters and are used resectively

7 Temerature effect on the rheological arameters: η = η T + η T τ = 3.03τ T + τ T (15) (16) where η 20 denotes lastic viscosity of cement aste at 20 o C, τ 20 denotes yield value of cement aste at 20 o C, T denotes temerature, η T denotes lastic viscosity of cement aste at T o C and τ T denotes yield value of cement aste at T o C. Effect of admixtures and additions: Collecting the result of exeriments, the effects of the contents on the rheological arameters are integrated into the formulas statistically. radmix, fi ( X) r η =, Gi ( X) admix τ = (17) (18) where radmix, η and radmix, τ are the coefficient of viscosity and yield value deending on the admixtures and X is the admixture to cement ratio. In this contribution, 4 tye of admixtures are formulated; namely High Range Water Reducer (HRWR), AE agent (AE), Accerelator (AC) and Retarder (RE) radd, f j ( X) η = (19) radd, τ = Gj ( X) (20) where radd, η and radd, τ are the coefficient of viscosity and yield value deending on the addition and X denotes the relacement ratio. Solid content of aggregate: Ooi [6] investigated the comlex nature of acking articles with random shae and wide grading, and develoed mathematical model redicting the comacted bulk density of aggregate. This model is exressed as a second order olynomial of the sieve residuals as follows: Z n n = Aij Xi X j (21) i j where Z denotes solid content of combined fine aggregate and coarse aggregate, A ij is coefficient reresenting the acking erformance of combined aggregates of two different diameter, X i and X j exress volumetric ratios of the aggregates at the reresentative sieve sizes. Rheological arameters of fresh concrete: Recently Oh [7] found a henomenon that the rheological arameters of fresh concrete have good correlation to a relative thickness of excess aste on the basis of excess aste theory develoed by Kennedy [8]. The relation between the relative thickness of excess aste and the rheological arameters of fresh concrete is formulated as follows:

8 P = 1 V / Z e Γ= n i a P e nsd i i i (22) (23) where P e denotes volume of excess aste, V a is volume of aggregate, Γ denotes relative thickness of excess aste, ni is number of aggregate of size i, s i denotes surface area of each aggregate of size i, D i is diameter of aggregate of size i. Using relative thickness of excess aste and rheological arameters of cement aste, the viscosity and yield value of concrete can be calculated. c 1.69 ( ) η = η + Γ τ = τ + Γ c 2.22 ( ) (23) (24) where η c denotes lastic viscosity of fresh concrete and τ c denotes yield value of fresh concrete Cone slum: Using the exerimental data of slum redicted rheological values, the regression curve is formulated as the function of yield value and arameter of mix roortion as below: 1 S = a Ln( τ ) + b 1 + W / B 3 c 3 (25) where S denotes the value of slum, a 3 and b 3 are coefficients Costs In this roblem, the function of cost is simly the summation of material cost. 3.2 Case study 2 In case 2, assuming that construction site is in cool temerate zone and average temerature is 10 o C, concrete is required 15 cm of slum, 40 MPa of strength, 6.0x10-4 of drying shrinkage, 35 o C of adiabatic temerature rise at 7 day for mass concrete. Additionally 100 of durability factor, which means relative dynamic modulus of elasticity after 300 cycles of freezing and thawing actions, and 5.0x10-8 of Chlorides-diffusion-coefficient are required strictly. The other arameters are less imortant Adiabatic temerature rise Adiabatic temerature rise is modelled as follows: Q= Q (1 Ex( γ t)) Q = a4b + b4 (26) (27) γ = ab+ b 5 5 where Q denotes the adiabatic temerature rise at t day, a 4, b 4, a 5 and b 5 denote the material arameters and B denotes the weight of binder er unit volume. (28)

9 3.3 Results The required arameters of concrete in both cases are summarised in the same Table 1. Durability factor Chlorides diffusion coefficient Carbonation seed coefficient Table 1. Required erformances Strength Young's modulus Adiabatic Drying temeratur shrinkage e rise Initial setting Final setting Unit cm 2 /year cm/year 0.5 MPa GPa 10-6 K hour hour cm Yen Case E Case E Slum Cost Case 1: After running the MixGA with 120 individuals and 100 generations, mix roortions shown in Tables 2 are derived regarding with case 1. As final selection, elimination of the individuals, which do not meet required erformance of strength and slum simultaneously, is conducted. Additionally in the final selection, as an index of cost erformance, a value of strength divided by cost is used. Three mix roortions listed in Table 2 are to-to-third in strength-cost index. In left of Figure 3 shows the degree of conformity of three mix roortions listed in Table 2. Among three roortions, mix A has the best strength-cost index. It should be noted that, as is shown in left grah of Figure 3, mix C is much cheaer than mix A. The value of strength of mix A is higher than that of mix C. But in MixGA both mix A and Mix C is evaluated equally and they survive selections from generation to generation. Table 2. Pareto otimal mixture roortions derived by MixGA W/B Water Cement (tye) Addition (tye) Fine aggregate (tye) Coarse aggregate (tye) Admixture 1 (tye) Admixture 2 (tye) Case 1 A (OPC) 52 (GGBS) 936 (CR) 987 (GR) B (HESC) 76 (GGBS) 967 (CR) 925 (GR) (AC) (AE) C (MHC) 52 (GGBS) 962 (CR) 921 (CR) (AE) D (LHC) (CR) 898 (CR) (HRWR) (AE) Case 2 E (LHC) (CR) 893 (CR) (HRWR) (AE) F (LHC) (CR) 906 (CR) (HRWR) (AE) G (MHC) (CR) 898 (CR) (HRWR) (AE) Figure 3. Ratio between henotye and required erformance of concrete mixtures derived by MixGA

10 Case 2: After running the MixGA with 140 individuals and 200 generations with meta-roerty concet, mix roortions shown in Tables 2 are derived regarding with case 2. As final selection, elimination of the individuals, which do not meet required erformance of strength, slum, drying shrinkage, chlorides diffusion coefficient, durability factor and adiabatic temerature simultaneously, is conducted. In right of Figure 3 shows the degree of conformity of four mix roortions listed in Table 2. As shown in Figure 3, almost all the mixtures meet the required erformance. It should be noted here that mix E has low value in slum and this mix is evaluated equally from the Pareto otimal oint of view. If only the Pareto otimization is used, it is ossible that such undesirable individual that do not have good value in all several roerties is able to survive the selection. The genetic algorithm with the notion of Pareto otimality has a good otential in initial convergence tendency but it has a risk of undesirable acceleration in a few roerties as well. With meta roerty concet, which disareciate the rominence and unbalancing it is ossible to derive the sufficient set of mixture roortions. Using the suitable fitness function and evaluation method is imortant to get sufficient set. 4. CONCLUSIONS The results of this contribution are summarized as follows: 1. A genetic algorithm system integrating the concet of Pareto otimality, which is named MixGA, was develoed for solving multicriteria otimization roblems in concrete mix roortioning. 2. As shown in the examles resented in this study, MixGA can derive the aroriate mix roortions from the vast combinations of sorts of content and roortions of mixture to exlore. This system is maintained by suitable fitness evaluation, reasonable reroduction and correct rediction formulas. 3. The genetic algorithm with the notion of Pareto otimality has a good otential in initial convergence tendency but it has a risk of undesirable acceleration in a few roerties as well. Using meta roerty concet, it is ossible to comensate for this risk of the notion of Pareto otimality. REFERENCES [1] Marks, W. and Potrzebowski, J., Multicriteria otimization of structual concretemixes, Architecture and Civil Engineering 38(4), 1992, [2] Piasta, Z. and Czarneski, L., Analysis of material efficiency of resin concrete. in Brittle Matrix Comosite, Elsevier Allied Science London and New York, 1989, [3] Goldberg, A. E., Genetic Algorithms in Search Otimization & Machine Learning, Addison Wesley, 1989, [4] Kikukawa, H., Studies on viscosity equation of Portland cement aste, Journal of Material, Concrete Structure and Pavements, V-2, 354, 1985, (In Jaanese) [5] Roscoe, R., The Viscosity of Susension of Rigid Sheres, British Jouranl of Alied Physics, Vol.3, 1952, [6] Ooi, T., Comarted bulk density of aggregate with random shae and widely ranged article size distribution, Journal of Structure and Construction Engineering. AIJ, No.423, 1991, (In Jaanese) [7] Oh, S. G.., Noguchi, T., and Tomosawa, F., Toward Mix Design for Rheology of Self-Comacting Concrete, 1st International RILEM Symosium on Self-Comacting Concrete, Stockholm, 1999, [8] Kennedy, C.T., The Design of Concrete Mixes, Proceedings of the American Concrete Institute, Vol.36, 1940,