DOI: 10.1002/adem.201200109 COMMUNICATION

Category Theory Based Solution for the Building Block Replacement Problem in Materials Design**

By Tristan Giesa, David I. Spivak and Markus J. Buehler*

An important objective in materials design is to develop a systematic methodology for replacing unavailable or expensive material building blocks by simpler and abundant ones, while maintaining or improving the functionality of the material. The mathematical field of category theory provides a formal specification language which lies at the heart of such a methodology. In this paper, we apply material ologs, category-theoretic descriptions of hierarchical materials, to rigorously define a process by which material building blocks can be replaced by others while maintaining large-scale properties, to the extent possible. We demonstrate the implementation of this approach by using algebraic techniques to predict concrete conditions needed for building block replacement. As an example, we specify structure–function relationships in two systems: a laminated composite and a structure–function analogue, a fruit salad. In both systems we illustrate how ologs provide us with a mathematical tool that allows us to replace one building block with others to achieve approximately the same functionality, and how to use them to model and design seemingly distinct physical systems with a consistent mathematical framework.

Thousands of labs across the world have been studying basic material building blocks such as molecules, mineral various biological and engineered synthetic materials from a particles, or chemical bonds. New research in the mathema- molecular perspective. This research provides a bottom-up tical field of category theory provides a precise framework to approach to materials which can be used to predict properties capture such relationships in an abstract space, as demon- from a fundamental molecular-level point of view,[1,2] strated in a recent paper.[5] When building a formal model of a establishing the field of materiomics.[3,4] Typically, for each material system using category theory, the result is a so-called material, one aims to identify how material function (e.g., material olog, which looks like a graph or network equipped strength and toughness) emerges based on the interplay of with additional information. The olog precisely expresses how structure and functionality can be composed, combined, and [*] Prof. M. J. Buehler, T. Giesa interconnected throughout the system. Laboratory for Atomistic and Molecular Mechanics, One of the most urgent questions for material scientists is Department of Civil and Environmental Engineering, how to design high performance materials by use of abundant [6] Massachusetts Institute of Technology, and cheap material constituents, similar to natural designs [7] 77 Mass. Ave. Room 1-235A&B, found in biological materials. In more abstract terms, the key Cambridge, MA 02139, USA problem is to determine the conditions necessary for replacing E-mail: [email protected] one or more of a system’s basic building blocks by other Prof. M. J. Buehler building blocks, while maintaining its overall functionality. Center for Computational Engineering, This is referred to as the ‘‘building block replacement Massachusetts Institute of Technology, problem.’’ For example, if we consider a polymer with 77 Mass. Ave. Room 1-235A&B, covalent crosslinks, it is of interest whether the covalent Cambridge, MA 02139, USA cross-links can be replaced by other molecular structures, such Dr. D. I. Spivak as weak van der Waals or hydrogen bonds, without Department of Mathematics, compromising the material’s properties. What types of Massachusetts Institute of Technology, structures, based on the new building blocks, need to be 77 Mass. Ave, Cambridge, MA 02139, USA introduced to achieve this? A simple example from linguistics E-mail: [email protected] illustrates this problem. Consider the sentence: [**] Support from AFOSR (FA9550-11-1-0199) and ONR (N00014-10-1-0562 and N00014-10-10841) is acknowledged. I drive my car from home to work. [A]

ADVANCED ENGINEERING MATERIALS 2012, DOI: 10.1002/adem.201200109 ß 2012 WILEY-VCH Verlag GmbH & Co. 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Fix functionality and then query the data store on the olog material and property vector, seeWe as visualized in then (Figure 2a). migration perform functors a regressive update, using the data replacement problems of the following two sorts: II. Fix one or more building blocks and explore the functor from the materialconstant (or within set a specified tolerance). while Finallyright-pushforward we perform of keeping a the result the and baseof change property with the the vector unit right-pushforward adjunctiondataset. applied In to the otherQuery-Result original words, our query is the fiber product within some tolerance interval of a collectionfunctionality. of other building blockscategory-theoretic while database query maintaining against theDat MaterialOlog/ and its application to databases, In other words, we hope to replace building block ) A ( Dat (or are , which Set) This X X , display all ! , H, each of p C ... 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ß , one given by arrow C ! 1 allow for aggregation A it is a functor , [5] [8] . (The notion of recording data 2012 Dat ), an olog is considered as a database followed by arrow 2, are equivalent (they [5] A C, one given by arrow 1 followed by arrow 3 and the other given by the implicit arrow A1 ! ! 1 A systems that have the same functionality as The most ubiquitous sort of numerical implementation Suppose we are given the following building block . Hierarchical categories DOI: 10.1002/adem.201200109 ADVANCED ENGINEERING MATERIALS blocks by more general objects. Incould our example, the be fruit salad substances, abstracted such as to a gravel a mixture in mixture concrete. of consisting ‘‘solving of the(as various olog’’ elsewhere is given by database query. Here Fig. 1. Material ologs used to describe the structure and function relationship for two functorially isomorphic materials, fruit salads, and lamina material mixture is composed of aeach collection material of has constituent a property building vector blocks (arrowtwo (fruits 3) and and paths laminate each A1 material layers). mixture Each aggregates constituent to is a made property out vector of (arrow 2). a The particula property vector is a five-dimensi T. Giesa et al./Building Block Replacement Problem schema, which in turn isC simply a finitely presented category refers to the tablesalads) of we layered have composites observeda (respectively, column and fruit for documented. the material Thisbuilding name, table a blocks, has column for andty-vector. Note its that constituent a the two paths column1 followed for by its arrowarrow 3 resulting and proper- the other given by the implicit olog has boxes (Figurewhich 1), labeled refers A, to A1, a B, table C, in the database. 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( 3, 4, FS ) ‘ovn h lg’masta ems n a find must we that means olog’’ the ‘‘Solving 1). oadt oegvnfutset fruit given some to add to 3 FS fruit 2) fruit opsdo,sy h ri e ( set fruit the say, of, composed 1 þ F [12] )adhvn oa at etr(ro 2) (arrow vector taste total a having and 4) c edmntaetesrrsn btato oe fologs: of power abstraction surprising the demonstrate We (c) . T ) uhta h at ilb the be will taste the that such 5), ( oyih c 92Srne Science Springer 1992 (c) Copyright FS ) oee,neither However, 3). DACDEGNEIGMATERIALS ENGINEERING ADVANCED FS T fruit FS eoe iertransforma- linear a denotes ,woetsew atto want we taste whose 1, R .Tecntan sthat is constraint The 2. 5 db iia ehius ept large despite techniques, similar by ed 1. osdrn concrete a Considering . O:10.1002/adem.201200109 DOI: unctor etrcntn,adapplying and constant, vector y a itr utb mimicked be must mixture ial F oa lgcnitn only consisting olog an to þ FS uiesMdaB.V. Media Business S nor 2 db pliga applying by ed h e of set the , FS FS ,a part as 1, FS ,i.e., 1, ,say 2, FS fruit 2012 3is FS 1, 2 , COMMUNICATION , , , k 1 5 2 ; 5 2, h 2, ¼ 1) ij k = R 2)). z 4 Þ Q MS [12,13] 1 MS 12 layer ( layer layer where ¼ 2 with MS Lay 1 A 8 k 4 with ( T = þ 3 with . Conse- T 5 with k 1, P Þ 2, so that 2 22 z 22 being the h ðÞ 3 layer 1), A k ; layer A . The over- ; = , Þ 1 and k ij MS layer = 1 layer k z layer z Þ h MS Q 12 ( 2 consists of ; ¼ . The elastic as the effective 11 1 02 MS 3 consisting of k A , 2 composed of . We assign to : T Þ ¼ and ij ( A Þ k 0 8 k yx 4 with . We need to solve ¼ MS h D H n MS Lay MS 03 Þ Þ = and P 005 : ð ; ¼ð k : xy 3 3 ; ¼ , 1 0 : : 3 0 ij n : 3 uses building block ¼ 0 0 layer xx xy 2 that yield an equal 0 3) A , n ij Þ E H ; h ; ð 4 ; ; MS A = ¼ , : ; MS 3 3 MS 15 H 2 mm with 0 : : k ( 33 : 005 xy : ; : 0 0 1, as part of the data base, ij 0 T 0 http://www.aem-journal.com A G 0 500 ; Q with respect to Þ¼ ; ; ¼ 1 composed of ( , ¼ ; , and h MS 2 nor ; ; k 8 ; yy , 700 2 k xy ij 1 and = ij E 15 200 900 MS z 900 : h G MS A , the set of laminates, to the in-plane 1, i.e., 0 ; A , 9000 , S MS ; ; ; ð½ xx h 1 ; = MS E k H þ ; Þ 2 k z z with distinct mechanical properties 12 10 000 3000 900 k 60 000 90 000 Þ¼ A ; 1 under the constraint that only 5 are available. Similar to the fruit salad, 1 ij 1 = S ; ; with distinct mechanical properties ; þ k ; ; Q 2 11 1.6 mm that are stacked symmetrically. All 130 000 5) MS z 3. S 1 denotes a linear transformation from the real jj¼ A MS layer ¼ ¼ ð 4) 8 k 3000 ¼ T h T ¼ð . For the in-plane composite properties we consider 3 we can add to the material layers layer k layer P 1 120 000 100 000 60 000 90 000 3 k 22 ; h 2 z layer A MS = 2). Again we wish to determine what material Lay 1 4, -calculus description of the aggregation function). ¼ð ¼ð ¼ð ¼ð is the linear transformation described above. 3, 4, and 1or l 1 ¼ð 2 4 3 5 ¼ As in Section 2.1, we now analyze a symmetric stack Again, þ MS 3 k ij yy ( 100 000 z layer could consist ofthickness eight equally thick laminates with total respective coordinate. Then T only the A-matrix such that Here, weso-called consider ABD-matrixenable to only the straightforwardsystem. simplify the connection the to in-plane calculation the part fruitof and eight salad of material layers the weighted summation ofmaterial the laminates ingredients,elasticity requires the theory a junctionconsequence (this procedure, of of e.g., isproblem the based not at particular on a hand,the physics limitation and that but which governs merely must the be a incorporated into determined system of equations can again be solved by Lay each laminatehomogenized). a Specifically, mechanical properties vector (already data iswith stored in appropriate units, i.e., MPa: single layer properties and H Similarly, thefour symmetric equally composite thick laminates: layers the mechanical propertiesmaterial will layer be stack the same as that of the and a( stack of four material layers layer uniquely given by quently, E B Lay Lay layer we can decouple the transformation of behavior to for the distributionbuilding blocks of from thicknesses in Lay ð behavior of thestiffness composite matrix is (ABD-matrix)nesses determined dependent on by its its ply laminate thick- due to the linearity of layer We require that neither vector space with basis mechanical properties space for a symmetrical laminate , ; Þ 0 (1) : [11] 0 Þ¼ Þ¼ 0.2, and 2 3in 1 2 ; ¼ 3 0.1001 ; FS FS FS a ð ð 9 ; ¼ : Tom T T 2 2 consists 2 f b ; FS þ ; 6 6 : ; 4 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 6 Pap 3. The best fit e ß . The total taste is a and tomatoes with ¼ð ¼ Þ FS Þ 1 0 2) Þ¼ð 0.1419 and . Similarly, 2 ; Pap FS ¼ Þ ; 5 ( 9 3 a FS : T ð 0 ; , apples with taste vector , where, e.g., T , ; 9 Þ 2 1 ; 6 Lem ; Þ : 2012 0.1. So we compute that ; 3 1 ; . We need to solve for 2 6 3 ¼ d Þ , where FS 5 ; d Þ ; : ð þ ; 0 1 C C C C C C C C C C A ; 1 5 0 T : 6 1 4 6 , papayas with 5 : : 4 9 0 : : : ; Þ ; 0 4 ¼ð Pap ; 0 2 2 3 1 0 ¼ð Þ¼ ; : 9 3 9 3 c ; 0.5. So we compute that : ; 0 B B B B B B B B B B @ 2 Pap 6 ; 5 FS ¼ þ Pap ; 3 ð f ; ¼ 5 ; T 7 : ; 0.1, and 4 3 2 : ; ¼ð App 2 7 ¼ b a ; c ; ; b . If, for instance, tomatoes become available and 0 ! 8 2 : Þ Ban 3 7 7 7 7 7 7 7 7 7 7 5 4 4 þ : Þ¼ð 0 1 ¼ð ; ¼ð ; 0.6, 4 0.5 and The only difference in the calculations lies in the The numerical solution yields 0 : Similar to the olog of the fruit salad, we can solve the Ban : FS 11 25 79 73 20 0 ð ¼ ¼ 7 2 6 6 6 6 6 6 6 6 6 6 4 DOI: 10.1002/adem.201200109 ADVANCED ENGINEERING MATERIALS (e.g., with a Householder reflection) predicts athe set of solutions, taste conditions,equation system for is given this by problem. In our setup, the T. Giesa et al./Building Block Replacement Problem vector App bananas becomecombination unavailable, of we fruit solve ingredients for in the linear (least-square) solution is over-determinedtaste-independent fruits for and under-determined less for than morefive. than In five many cases, matrix inversion via QR decomposition of papaya with building block replacement problemcomposites. A in functorial the isomorphism (uniquely caseup determined of to layered permutationsconverts of all arrows the and objects entries inin the the in laminated fruit composite salad olog the olog (Figure tocorresponds 1). property to those This a high isomorphism vector) degree of similarity insolving the calculation the for building blocksystems. replacement problem in the two aggregation function represented by arrowtical 2. model The – mathema- basedproperties of on the whole laminate laminate to theory thelayer properties – of the that is single connectsmathematical the more model complex. represented by Indeed, arrow while 2 is the a fruit simple salad 2.2. Laminated Composites (representing arespect volume to percentage theDepending in original on the the volume)additional context, mixture and constraints, such for with a example, a thatbe residuum calculation all positive. may coefficients of must require 5.28. and lemons with linear combinationa of the ingredients with Tom e ð T b the equation COMMUNICATION and qainyed h ehnclpoet etro h new the of vector property phase material mechanical the yields equation hc aeilfnto sraie,sc shierarchical as such by realized, patterns is similar follow function systems material analyzed which the of many Discussion 3. laminate. ply standard through a engineered of be rotation could laminate composite a Such nierdlyrcnb eemndeatyby exactly determined be the can of properties layer mechanical the 2.2, engineered Section in currently presented are processes. that manufacturing synthetic other genetic properties through or achievable engineering material are but engineers, in for unavailable result prediction The constraints. might various under functionality determined system’s be the can improve or maintain to conditions Ologs from Arising Predictions 2.3. constraints. likely other which has one, or blocks synthetic biological building a different a to includes from transferred be concepts This must bioinspired key system in the makeup. example, where physical design, for materials approach, their techniques, powerful similar in a by systems provides differences solved All be large can 2c). despite olog (Figure same reflection the having Householder and decomposi- QR examples tion simple category-theoretic these in similar context-specific tools, with mathematical very and design a the laminates guides to which (here, representation, back by systems trace salads) unalike (e.g., fruit ostensibly solved ologs: a the be of case desired, to In is has required. properties iteration). system fixed-point is plane equation thickness out-of nonlinear total of certain determination example, a for constraints, a that such additional context, require the on may Depending calculation 467. of residuum a and ply) 6 Q reflection Householder 2 , 4 6 6 6 6 6 6 6 2 ij ; fe tdigadvresto aeil n elzsthat realizes one materials of set diverse a studying After ncs h uligbokpoete r o xd the fixed, not are properties block building the case In osdrn h aiae opst example composite laminated the Considering power abstraction surprising the demonstrates above The h ueia ouinyields solution numerical The Q Q Q Q 4 6 6 6 6 6 6 2 h ¼ 33 12 22 11 24 5 2 0 0 / ; ; ; ; : : : . http://www.aem-journal.com 2 2 2 2 30 03 10 81 40 24 h : A 32 ¼ ij MS Q Q Q Q .5 rpeetn h hcns rcino each of fraction thickness the (representing 0.158 33 12 22 11 1 : : : : = ; ; ; ; 119 01 019 60 95 09 40 14 4 4 4 4 2 Lay Q Q Q Q Q 5 33 12 22 11 ; : ij 04 ¼ð 93 ; ; ; ; : : ; 5 5 5 5 4 8 8 h 5 7 7 7 7 7 7 7 3 5 7 7 7 7 7 7 3 4 @ B B 0 4146 64 @ B B 0 h h h Q 5 4 2 h h h A C C 1 ij 5 4 2 ; ; 5 A C C 1 h ¼ 1452 31 5 ¼ @ B B B B B B 0 h @ B B B B B B 0 A A A A h 2 / 11 MS 13 MS 12 MS 22 MS ; 3 13 1 0 h ouino this of solution The . h ß : : : 53 08 11 1 1 1 1 ¼ 125 : 02WLYVHVra mH&C.Ka,Weinheim KGaA, Co. & GmbH Verlag WILEY-VCH 2012 1 A C C C C C C 1 0.517, A C C C C C C 1 ; 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[14] hl uhsmlrte ewe silk, between similarities such While DACDEGNEIGMATERIALS ENGINEERING ADVANCED O:10.1002/adem.201200109 DOI: [15] ielinguistics, Like 2012 , COMMUNICATION 7 tional behavior similar to the ortar or carbon nanotubes with Possible replacements include a helical [9] http://www.aem-journal.com The olog, which is simultaneously [17–20] [10] Here, we introduced a set of strategies for solving certain need for a much weaker glue elementcarbon that nanotubes, can together connect pairs with of a stronglength lifeline (see with Figure hidden 3). Suchshown systems have to been be experimentally feasible. material ologs. Toas mixes transition and frominvolving layers to simple interaction more systems,would complex such materials, require terms e.g. moremore those or data. resources. A Of single hierarchicalcannot lab, particular produce even a the structures, necessity well-subsidized amountmake one, is of the simply kind structured of data discoveriesdata we necessary believe resources to are is possible. a Sharing keyproductivity. step toward increasing efficiency and computer-readable as a precisehuman-readable mathematical as construct, and a natural-languagecepts, is able presentation to precisely of describe and con- categorizeChallenges these may studies. arise ifthere the is given insufficient data olog in the is store.can too If be this simplistic is extended the or case, or if thedata additional olog migration data can collected, be applied andnew to functorial olog. move all the old data to the [16] -helix is abstracted to a brick–glue-lifeline system as in ref. b 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ß , 2012 Suppose we have a hierarchical protein material [9] In this paper we solved relatively simple ologs for fruit DOI: 10.1002/adem.201200109 ADVANCED ENGINEERING MATERIALS Fig. 3. The building replacement problem for a brick and glue system. (a) A salads and for composites of materialsimilar layers. In a manner, categorically itprotein is material possible ologs,literature. to such solve as the those more presented complex in recent One of the building blocks usedprotein in spider structure, silk is but the this beta-sheet choice may in not engineering. 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S. [2] 1 .Pne,M Mann, M. Pandey, A. [1] h rsn td rvdsafaeokfrteslto to solution the for framework a provides study present The ti la htsvrlo h mat itdaoe(.. the (e.g., above listed impacts the of several that clear is It n20 ilsmdls Ti medalist Fields 2009 In http://www.aem-journal.com [22–24] a.Biotechnol. Nat. iial,rcn rjcssc as such projects recent Similarly, netoa methods. onventional oh oesproposed Gowers mothy Nature rusi ossetmanner. consistent a in groups h fr ol eepoe osolve to employed be could ffort tn nsltosta a not may that solutions in lting eutl rsn n share and present dequately ae–eetterm This theorem. Hales–Jewett ia eso:My1,2012 15, May Version: Final 2003 2000 eevd ac 4 2012 14, March Received: ß , , 21 02WLYVHVra mH&C.Ka,Weinheim KGaA, Co. & GmbH Verlag WILEY-VCH 2012 405 enovo de 10. , 6788. , [25,26] materials. Asimilar [21] hta that 1]R .Christensen, M. R. [13] 1]O .Oha .N Reddy, N. J. Ochoa, O. O. [12] 2]S opr .Kai,A rule .Breo .Lee, J. Barbero, J. Treuille, A. Khatib, F. Cooper, S. [26] 1]L .A Adrianova, A. I. L. [11] 1]D .Spivak, I. D. [10] 1]M .Ocnel .Bu,L .Eisn .Huffman, C. Ericson, M. L. Boul, P. O’connell, J. M. [17] 1]J ytee .Gnh,S uiaa .H Ye, H. H. Sukigara, S. Gandhi, M. Ayutsede, J. Rubio, [19] A. Giannaris, C. Schadler, S. L. Ajayan, M. P. [18] 2]M hrs .S Pande, S. V. Shirts, M. Polymath, J. [25] H. D. Polymath, J. [24] H. D. Polymath, J. [23] H. D. [22] in: Gowers, T. [21] 1]M .Buehler, J. M. [16] Chomsky, N. Buehler, [15] J. M. [14] 2]N .Sho .Rn,J .Co .L,S .Chan, H. S. Li, L. Cho, W. J. Rana, S. Sahoo, G. N. [20] 9 .I pvk .Gea .Wo,M .Buehler, J. M. Wood, E. Giesa, T. Spivak, I. D. [9] 8 .Gea .I pvk .J Buehler, J. M. Spivak, I. D. Giesa, T. [8] Buehler, J. M. Cranford, W. S. [4] 7 .L,J v .K Niu, K. D. Lv, J. Li, N. Kirchhoff, [7] M. M. Anastas, T. Kent, E. P. R. Spivak, [6] I. D. [5] 3 .W rnod .J Buehler, J. M. Cranford, W. S. 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