Additive . . . It Is Important to . . . How Complexity Is . . . Open Problem Which Material Design Is Commonsense . . . Possible Under Additive Commonsense . . . How Accurate Is Any . . . Manufacturing: How to Combine . . . Our Result A Fuzzy Approach Home Page Title Page Francisco Zapata 1 , Olga Kosheleva 2 , and Vladik Kreinovich 2 ◭◭ ◮◮ 1 Department of Industrial, Manufacturing, and Systems Engineering ◭ ◮ 2 University of Texas at El Paso, El Paso, Texas 79968, USA faxg74@gmail.com, olgak@utep.edu, vladik@utep.edu Page 1 of 15 Go Back Full Screen Close Quit
Additive . . . It Is Important to . . . 1. Additive Manufacturing: Successes and Limi- How Complexity Is . . . tations Open Problem • Additive manufacturing – aka 3-D printing – is a Commonsense . . . promising way to generate complex material designs. Commonsense . . . How Accurate Is Any . . . • It allows us to generate objects layer-by-layer, and How to Combine . . . thus, come up with very complex objects. Our Result • While additive manufacturing has many successes, it Home Page is not a panacea. Title Page • Current equipment for additive manufacturing only re- ◭◭ ◮◮ produces the desired design with a certain accuracy. ◭ ◮ – For simpler shapes, small deviations from the de- Page 2 of 15 sired configuration do not affect their functionality. – However, for objects, with small elements, a small Go Back change in the configuration can ruin the result. Full Screen • Example: design of tiny blood vessels. Close Quit
Additive . . . It Is Important to . . . 2. It Is Important to Estimate the Complexity of How Complexity Is . . . a Given Design Open Problem • An ideal blood vessel should have uniform width. Commonsense . . . Commonsense . . . • A vessel with widely varying width impedes the blood How Accurate Is Any . . . flow. How to Combine . . . • For such complex objects, we need more accurate Our Result equipment, whose use is very expensive. Home Page • It is therefore desirable to be able to estimate the com- Title Page plexity of the design before the manufacturing, so that: ◭◭ ◮◮ – we would be able to see whether a given equipment ◭ ◮ can implement this design – and Page 3 of 15 – if it can, whether this same design can be imple- Go Back mented by a cheaper equipment. Full Screen Close Quit
Additive . . . It Is Important to . . . 3. How Complexity Is Estimated Now: an Empir- How Complexity Is . . . ical Formula Open Problem • The state-of-the-art empirical formula is based on the Commonsense . . . division of the design into several sections i . Commonsense . . . How Accurate Is Any . . . • If we make sections sufficiently small, then in each sec- How to Combine . . . tion, we have at most two different materials. Our Result • So there is no need to consider sections with three or Home Page more materials. Title Page • Let us denote the fraction of this section which is filled ◭◭ ◮◮ with one of these materials by v i ∈ [0 , 1]. ◭ ◮ • Then, the other material fills the fraction 1 − v i . Page 4 of 15 • In this case, the empirical formula for the complexity C is C = � Go Back v η i · (1 − v i ) η , for some η . i Full Screen Close Quit
Additive . . . It Is Important to . . . 4. Open Problem How Complexity Is . . . • When a formula does not have a theoretical justifica- Open Problem tion, it is less reliable, because it is not clear Commonsense . . . Commonsense . . . – whether this dependence indeed follows from first How Accurate Is Any . . . principles — and thus, can be safely applied, How to Combine . . . – or is somewhat accidental and probably will not Our Result hold in other cases. Home Page • In this paper, we provide a theoretical justification for Title Page this formula. ◭◭ ◮◮ • In this justification, we use fuzzy logic ideas. ◭ ◮ Page 5 of 15 Go Back Full Screen Close Quit
Additive . . . It Is Important to . . . 5. Commonsense Analysis of the Problem How Complexity Is . . . • When we have only one material, i.e., when v i = 0 or Open Problem v i = 1, there is no complexity: c (0) = c (1) = 0 . Commonsense . . . Commonsense . . . • When both materials are present, there is complexity, How Accurate Is Any . . . so we should have c ( v i ) > 0 for v i ∈ (0 , 1). How to Combine . . . • Many physical ideas are based on the fact that: Our Result – every analytical function can be expanded in Taylor Home Page series, and, Title Page – as a good approximation, we can take the sum of ◭◭ ◮◮ the first few terms in this expansion. ◭ ◮ • Let us consider c ( v i ) = c 0 + c 1 · v i + c 2 · v 2 i + c 3 · v 3 i + . . . Page 6 of 15 • 0-th order c ( v i ) = c 0 is not enough: c (0) = 0 implies Go Back c 0 = 0 and c ( v i ) ≡ 0, but c ( v i ) > 0 for v i ∈ (0 , 1). Full Screen • Linear approximation c ( v i ) = c 0 + c 1 · v i is also not good: c (0) = c (1) = 0 implies c ( v i ) = 0 for all v i . Close Quit
Additive . . . It Is Important to . . . 6. Commonsense Analysis and Fuzzy Logic How Complexity Is . . . • So, we need quadratic terms: c ( v i ) = c 0 + c 1 · v i + c 2 · v 2 i . Open Problem Commonsense . . . • The conditions c (0) = c (1) = 0 imply that c ( v i ) = Commonsense . . . c 1 · ( v i − v 2 i ) , with c 1 > 0. How Accurate Is Any . . . • We are interested in is relative complexity of designs. How to Combine . . . • So, nothing will change if wedivide all the complexity Our Result values by c 1 and take c ( v i ) = v i · (1 − v i ) . Home Page • The section is complex if the first material is present Title Page and the second material is present. ◭◭ ◮◮ • It is reasonable to take the proportion v i as the degree ◭ ◮ to which this material is present in this section. Page 7 of 15 • Similarly, 1 − v i can be taken as the degree to which Go Back the other material is present. Full Screen • If we use one of the simplest and widely used “and”- operations f & ( a, b ) = a · b , we get c = v i · (1 − v i ). Close Quit
Additive . . . It Is Important to . . . 7. How Accurate Is Any Taylor Approximation? How Complexity Is . . . • If addition of one more term drastically changes the Open Problem situation, then: Commonsense . . . Commonsense . . . – our original approximation is rather crude, and How Accurate Is Any . . . – we should not trust the results of using this approx- How to Combine . . . imation too much. Our Result • If the addition of one more term does not change the Home Page result, the original approximation was accurate. Title Page • From this viewpoint, let us see what happens if we add ◭◭ ◮◮ a cubic term c ( v i ) = c 0 + c 1 · v i + c 2 · v 2 i + c 3 · v 3 i . ◭ ◮ • Values v i and 1 − v i describe the same situation modulo Page 8 of 15 re-naming, so, c ( v i ) = c (1 − v i ). Go Back • This implies c 3 = 0; so, an extra term doesn’t change Full Screen much; thus, the quadratic approximate is accurate. Close Quit
Additive . . . It Is Important to . . . 8. How to Combine Complexity of Sections into a How Complexity Is . . . Single Complexity Value? Open Problem • Let f ( C 1 , C 2 ) be an overall complexity of a 2-section Commonsense . . . design with section complexities C 1 and C 2 . Commonsense . . . How Accurate Is Any . . . • The complexity does not depend on the order of the How to Combine . . . sections: f ( C 1 , C 2 ) = f ( C 2 , C 1 ). Our Result • The complexity of a 3-section design can be computed Home Page in two ways: Title Page – as combination of 1-2 and 3, then the complexity ◭◭ ◮◮ is f ( f ( C 1 , C 2 ) , C 3 ), ◭ ◮ – or as combination as 1 and 2-3: f ( C 1 , f ( C 2 , C 3 )) . Page 9 of 15 • These values should coincide, so f ( C 1 , C 2 ) must be as- sociative. Go Back • If we increase the complexity of one of the sections, the Full Screen overall complexity increases, so f is increasing. Close Quit
Additive . . . It Is Important to . . . 9. Combining Complexities: Scale-Invariance How Complexity Is . . . • Small changes in complexities C i should lead to small Open Problem changes in the overall complexity; so f is continuous. Commonsense . . . Commonsense . . . • These are properties of “or”-operation (t-conorm) in How Accurate Is Any . . . fuzzy logic: namely, of Archimedean t-conorms. How to Combine . . . • Thus, we can use the known classification of such t- Our Result conorms and conclude that Home Page f ( C 1 , C 2 ) = g − 1 ( g ( C 1 ) + g ( C 2 )) for some g ( C ) . Title Page • This is equivalent to g ( C ) = g ( C 1 ) + g ( C 2 ) . ◭◭ ◮◮ • As we have mentioned earlier, complexity is defined ◭ ◮ modulo a measuring unit C → λ · C . Page 10 of 15 • It is reasonable to require that the combination oper- Go Back ation should not depend on this re-scaling, i.e., that Full Screen g ( C ) = g ( C 1 )+ g ( C 2 ) implies g ( λ · C ) = g ( λ · C 1 )+ g ( λ · C 2 ) . Close Quit
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