《纺织复合材料》课程参考文献（Textile Composites and Inflatable Structures）17 Form-Optimizing Processes in Biological Structures. Self-generating structures in nature based on pneumatics

Form-Optimizing Processes in Biological Structures.Self-generating structures in nature based on pneumatics Edgar Stachl University of Tennessee,College of Architecture and Design 1715 Volunteer Boulevard,Knoxville,TN 37996,USA stachOutk.edu "If architects designed a building like a body,it would have a system of bones and muscles and tendons and a brain that knows how to respond.If a building could change its posture,tighten its muscles and brace itself against the wind,its structural mass could literally be cut in half." Guy Nordenson,Ove Arup and Partners Summary.This case study is an investigation of self-generating forms in nature based on pneumatic structures and their use in architectural theory.It focuses on the concept of self organization as a defining principle in nature and in particular,on the mathematical,geometrical and physical properties of bubble clusters and shows eramples from nature,biology and engineering.Part of the research resulted in a se- ries of digital models and renderings of different bubble clusters and there polyhedral configuration.Advanced structural design methods are already using systems based upon self-generated models rooted in biological and genetic forms.Engineers are able to input a series of variables into a computer program which in turn,derive a struc- ture using a genetic algorithm resulting in the most efficient use of materials,etc. Numerous eramples of such procedures already erist in nature today,in particular, biology.Blueprints for these forms are stored in the genetic code of the DNA of all life forms.Until recent advances in computer technology,the ability to put such genetic algorithms to use has not been possible. 1 The DNA(deoxyribonucleic acid)is the carrier of our hereditary characteristics and that it is based on two strands twisted about one another forming a double he- lix.The strands consist of alternating carbohydrate and phosphate molecules.On each carbohydrate sits one of the four nitrogenous molecules Adenine,Cytosine, Guanine and Thymine.A DNA strand can thus be compared with a long sentence (sequence)of code words,where each word consists of three letters that can be combined in many different ways,e.g.CAG,ACT.Each code word can be read by components inside the cell and translated into one of the twenty amino acids that build proteins.The three-dimensional structure,and hence the function,of the proteins is determined by the order in which the different amino acids are linked together according to the genetic code.(www.nobel.se/chemistry). 285 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,285-303. 2005 Springer.Printed in the Netherlands

Form-Optimizing Processes in Biological Structures. Self-generating structures in nature based on pneumatics Edgar Stach1 University of Tennessee, College of Architecture and Design 1715 Volunteer Boulevard, Knoxville, TN 37996, USA stach@utk.edu “If architects designed a building like a body, it would have a system of bones and muscles and tendons and a brain that knows how to respond. If a building could change its posture, tighten its muscles and brace itself against the wind, its structural mass could literally be cut in half.” Guy Nordenson, Ove Arup and Partners Summary. This case study is an investigation of self-generating forms in nature based on pneumatic structures and their use in architectural theory. It focuses on the concept of self organization as a defining principle in nature and in particular, on the mathematical, geometrical and physical properties of bubble clusters and shows examples from nature, biology and engineering. Part of the research resulted in a se￾ries of digital models and renderings of different bubble clusters and there polyhedral configuration. Advanced structural design methods are already using systems based upon self-generated models rooted in biological and genetic forms. Engineers are able to input a series of variables into a computer program which in turn, derive a struc￾ture using a genetic algorithm resulting in the most efficient use of materials, etc. Numerous examples of such procedures already exist in nature today, in particular, biology. Blueprints for these forms are stored in the genetic code of the DNA1 of all life forms. Until recent advances in computer technology, the ability to put such genetic algorithms to use has not been possible. 1 The DNA (deoxyribonucleic acid) is the carrier of our hereditary characteristics and that it is based on two strands twisted about one another forming a double he￾lix. The strands consist of alternating carbohydrate and phosphate molecules. On each carbohydrate sits one of the four nitrogenous molecules Adenine, Cytosine, Guanine and Thymine. A DNA strand can thus be compared with a long sentence (sequence) of code words, where each word consists of three letters that can be combined in many different ways, e.g. CAG, ACT. Each code word can be read by components inside the cell and translated into one of the twenty amino acids that build proteins. The three-dimensional structure, and hence the function, of the proteins is determined by the order in which the different amino acids are linked together according to the genetic code. (www.nobel.se/chemistry). 285 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 285–303. © 2005 Springer. Printed in the Netherlands

286 Edgar Stach Key words:Form-optimization,lightweight membrane-construction,radiolarian, genetic algorithm polyhedra 1 Introduction The study of form-optimizing processes in biological structures has a long his- tory starting with Frei Otto,Werner Nachtigall and followed by many researchers (23][4][10].These researchers have outlined in a number of forms the mathematical relationships that control the overall geometry of polyhedral in biological structures [12].The research centers on an investigation how optimizing processes in biologi- cal structures are possible starting points to generate optimized architectural forms and structures.For this particular study the bubble cluster based on the pneus was selected.The pneu is a system of construction comprising of a non-rigid envelope having a certain tensile strength,and an internal filling,which is in most cases pres- surized.This system of construction can be translated into the architectural world in the form of pneumatic structures.This structural system,which can be found in many lightweight structures today,is based on the principals of those pneumatic structures found in nature. Fig.1.DNA double helix Fig.2.DNA,the genetic code 2 Pneumatic Structures in Nature One example of a pneumatic structure in nature is the soap bubble.In soap bubbles, growth is achieved through a system of division and inflation.This increased internal pressure encased in a reinforced membrane subject to tensile stress causes the bubble to grow in a process known as isomorphism or self-generation. Free-floating bubbles collect and form dense clusters known as foam.If three bubbles are placed on a glass surface and a fourth is added,the fourth bubble will relocate to the top of the three bubbles to form the simplest three-dimensional cluster consisting of four bubbles.If further bubbles are added they will automatically form a foam structure.If the bubbles are of equal size the liquid edges of the foam are straight and of equal length and the angles of incidence at the nodal points are equal.The total structure forms a net of equal mesh size called the "basic net"(Fig. 13)

286 Edgar Stach Key words: Form-optimization, lightweight membrane-construction, radiolarian, genetic algorithm polyhedra 1 Introduction The study of form-optimizing processes in biological structures has a long his￾tory starting with Frei Otto, Werner Nachtigall and followed by many researchers [2][3][4][10]. These researchers have outlined in a number of forms the mathematical relationships that control the overall geometry of polyhedral in biological structures [12]. The research centers on an investigation how optimizing processes in biologi￾cal structures are possible starting points to generate optimized architectural forms and structures. For this particular study the bubble cluster based on the pneus was selected. The pneu is a system of construction comprising of a non-rigid envelope having a certain tensile strength, and an internal filling, which is in most cases pres￾surized. This system of construction can be translated into the architectural world in the form of pneumatic structures. This structural system, which can be found in many lightweight structures today, is based on the principals of those pneumatic structures found in nature. Fig. 1. DNA double helix Fig. 2. DNA, the genetic code 2 Pneumatic Structures in Nature One example of a pneumatic structure in nature is the soap bubble. In soap bubbles, growth is achieved through a system of division and inflation. This increased internal pressure encased in a reinforced membrane subject to tensile stress causes the bubble to grow in a process known as isomorphism or self-generation. Free-floating bubbles collect and form dense clusters known as foam. If three bubbles are placed on a glass surface and a fourth is added, the fourth bubble will relocate to the top of the three bubbles to form the simplest three-dimensional cluster consisting of four bubbles. If further bubbles are added they will automatically form a foam structure. If the bubbles are of equal size the liquid edges of the foam are straight and of equal length and the angles of incidence at the nodal points are equal. The total structure forms a net of equal mesh size called the “basic net” (Fig. 13)

Form-Optimizing Processes in Biological Structures 287 Fig.3.Rendered computer model: Fig.4.Rendered computer model: Tetrahedron bubble structure Octahedron bubble structure Fig.5.Cube bubble structure Fig.6. Fig.7

Form-Optimizing Processes in Biological Structures 287 Fig. 3. Rendered computer model: Tetrahedron bubble structure Fig. 4. Rendered computer model: Octahedron bubble structure Fig. 5. Cube bubble structure Fig. 6. Fig. 7

288 Edgar Stach 3 2-D Bubble Clusters Net structures are formed through the solidification of a 2-D bubble cluster.Bubble clusters occur when bubbles are freely dispersed within a cell without touching each other.In the next phase,the bubbles are introduced to each other through points of contact and form patterns by agglomeration (Figs.5-13).These patterns are based on geometric forms such as cubes,tetrahedrons and octahedron (Figs.3-5). As solidification takes place,the membrane of the bubble dries out and the fiber net hardens(Fig.12).The bubble membrane then dissipates and the net structure is left (Fig.13). Fig.8. Fig.9. Fig.10. Fig.11

288 Edgar Stach 3 2-D Bubble Clusters Net structures are formed through the solidification of a 2-D bubble cluster. Bubble clusters occur when bubbles are freely dispersed within a cell without touching each other. In the next phase, the bubbles are introduced to each other through points of contact and form patterns by agglomeration (Figs. 5–13). These patterns are based on geometric forms such as cubes, tetrahedrons and octahedron (Figs. 3–5). As solidification takes place, the membrane of the bubble dries out and the fiber net hardens (Fig. 12). The bubble membrane then dissipates and the net structure is left (Fig. 13). Fig. 8. Fig. 9. Fig. 10. Fig. 11

Form-Optimizing Processes in Biological Structures 289 Fig.12. Fig.13. 4 Mathematics/Geometry Closest packing One of the fundamental geometric principles that drives the repetitive,self-generating forms in nature is the notion of closest packing [1]of spheres.It is this,which de- fines the curvature of an insect's compound eye or creates the formwork to mold a radiolarian's skeletal structure. As spheres are packed closely together,certain laws of physics cause geometric shapes to occur,such as hexagons.These polygons create repetitive surfaces among and around the spheres.In some cases,these surfaces find themselves useful for a number of functions,such as in an insect's eye.In other instances,these surfaces interlock together to create volumes,polygons create polyhedrons.These volumes may be used to serve a purpose. Often times,it is not the surface or volume that is put to use in these systems. Quite likely,it is the edges along which these spheres meet that are of use to the organism.Once again,the radiolarian and diatom gather silicate deposits along the edges where the spheres meet around the outer surface.It is along these edges that a skeletal system is formed. Fig.14.Closest packing of spheres

Form-Optimizing Processes in Biological Structures 289 Fig. 12. Fig. 13. 4 Mathematics/Geometry Closest packing One of the fundamental geometric principles that drives the repetitive, self-generating forms in nature is the notion of closest packing [1] of spheres. It is this, which de- fines the curvature of an insect’s compound eye or creates the formwork to mold a radiolarian’s skeletal structure. As spheres are packed closely together, certain laws of physics cause geometric shapes to occur, such as hexagons. These polygons create repetitive surfaces among and around the spheres. In some cases, these surfaces find themselves useful for a number of functions, such as in an insect’s eye. In other instances, these surfaces interlock together to create volumes, polygons create polyhedrons. These volumes may be used to serve a purpose. Often times, it is not the surface or volume that is put to use in these systems. Quite likely, it is the edges along which these spheres meet that are of use to the organism. Once again, the radiolarian and diatom gather silicate deposits along the edges where the spheres meet around the outer surface. It is along these edges that a skeletal system is formed. Fig. 14. Closest packing of spheres

290 Edgar Stach Cublo Cose Puckng Hegonl Coe Paoking fal lyor 3日图 Fig.15.Closest packing diagram Configuring and integrating form systems One of the greatest advantages to geometric systems based on the closest pack- ing model is the great variety of configurations from which to choose.Repetitive, self-generating form can be derived in the shape of hexagons,pentagons,and even triangles (Figs.3-5).These can be arranged independently or between various types [. After a particular form is created,it,too,can be arranged with other similar forms to create even more shapes or,in terms of architecture,spaces.Some examples of this can be seen here (Fig.16).Another important quality of these systems is the ability to obtain similar forms with varying degrees of complexity in terms of number of members,scale,etc.As you can see,very comparable forms can be achieved in different ways.The structural complexity of a geodesic dome is probably too complicated for that of a radiolarian's skeletal system.Yet,these two structures share obvious formal qualities with one another.At the same time,the mathematic and geometric basis from which both are derived are practically identical

290 Edgar Stach Fig. 15. Closest packing diagram Configuring and integrating form systems One of the greatest advantages to geometric systems based on the closest pack￾ing model is the great variety of configurations from which to choose. Repetitive, self-generating form can be derived in the shape of hexagons, pentagons, and even triangles (Figs. 3–5). These can be arranged independently or between various types [1]. After a particular form is created, it, too, can be arranged with other similar forms to create even more shapes or, in terms of architecture, spaces. Some examples of this can be seen here (Fig. 16). Another important quality of these systems is the ability to obtain similar forms with varying degrees of complexity in terms of number of members, scale, etc. As you can see, very comparable forms can be achieved in different ways. The structural complexity of a geodesic dome is probably too complicated for that of a radiolarian’s skeletal system. Yet, these two structures share obvious formal qualities with one another. At the same time, the mathematic and geometric basis from which both are derived are practically identical

Form-Optimizing Processes in Biological Structures 291 Table 6.1 sAum ans ct ed u品 pytednr 件Te和写 CierS 品 J.Ineaos G-a1小T3药 配 三CLb2 1 00g 3 14n4001 3 Hhy时 Telld eut 1 18 Cuboutaheer门 1230 1 6.an1OH上2 卷 n 号1 5. 寸：1mnrc 3 Fig.16.Closest packing chart,Spheres as Morphological Units [4 Pearce pp57] 5 Structural Optimization in Engineering Genetic algorithms In engineering fields,accomplishing an objective with a minimum of effort,either in terms of material,time or other expense,is a basic activity (Figs.18,19 and 20).For this reason it is easy to understand the interest designers have in different optimiza- tion techniques.Mathematical,as well as,model based tools have traditionally been

Form-Optimizing Processes in Biological Structures 291 Fig. 16. Closest packing chart, Spheres as Morphological Units [4 Pearce pp57] 5 Structural Optimization in Engineering Genetic algorithms In engineering fields, accomplishing an objective with a minimum of effort, either in terms of material, time or other expense, is a basic activity (Figs. 18, 19 and 20). For this reason it is easy to understand the interest designers have in different optimiza￾tion techniques. Mathematical, as well as, model based tools have traditionally been

292 Edgar Stach Fig.17.Various geometric configurations Here are a few varieties of geometric configurations that achieve similar overall results,spheres,while blending various geometric shapes together in a number of ways. employed for such optimization.In recent times,mathematical methods executed on computers have become predominant.Unfortunately,computer derived solutions often obscure the range of possible solutions from the designer by only exhibiting a final,'best'solution.Naturally,optimization methods can only respond to the objective parameters which are coded into the problem,and as a result,non-coded parameters,such as aesthetics,or context are left out of the optimization process, and ultimately left out of the final design solution. Structural optimization in engineering takes natural constructions as an exam- ple.Similar to nature itself,computer-generated genetic algorithms2 can be cal- culated using stated goals to achieve global optimization-the search strategy is, like in nature,goal-oriented.An evolutionary algorithm maintains a population of structures (usually randomly generated initially),that evolves according to rules of selection,recombination,mutation and survival,referred to as genetic operators.A shared 'environment'determines the fitness or performance of each individual in the population.The fittest individuals are more likely to be selected for reproduction (retention or duplication),while recombination and mutation modifies those indi- viduals,yielding potentially superior ones.Using algorithms,mechanical selection mutation and recombination improves generationally with a fixed parameter size and quality. 2 A genetic algorithm generates each individual from some encoded form known as a 'chromosome'and it is these which are combined or mutated to breed new individuals.The basis for the optimization is a vast array of possible solutions (population),where every solution (individual)is defined through a particular parameter(chromosome).The individuals within a generation are in competition with one another (selection),in other words,the value(fitness)of the individual is what allows the survival of the parameter(gene)until the next generation.The results of this computer-supported process are automatically generated and opti- mized.Evolutionary computation is useful for optimization when other techniques such as gradient descent or direct,analytical discovery are not possible.Combi- natory and real-valued function optimization in which the optimization surface or fitness landscape is 'rugged',possessing many locally optimal solutions,are well suited for evolutionary algorithms

292 Edgar Stach Fig. 17. Various geometric configurations Here are a few varieties of geometric configurations that achieve similar overall results, spheres, while blending various geometric shapes together in a number of ways. employed for such optimization. In recent times, mathematical methods executed on computers have become predominant. Unfortunately, computer derived solutions often obscure the range of possible solutions from the designer by only exhibiting a final, ’best’ solution. Naturally, optimization methods can only respond to the objective parameters which are coded into the problem, and as a result, non-coded parameters, such as aesthetics, or context are left out of the optimization process, and ultimately left out of the final design solution. Structural optimization in engineering takes natural constructions as an exam￾ple. Similar to nature itself, computer-generated genetic algorithms 2 can be cal￾culated using stated goals to achieve global optimization - the search strategy is, like in nature, goal-oriented. An evolutionary algorithm maintains a population of structures (usually randomly generated initially), that evolves according to rules of selection, recombination, mutation and survival, referred to as genetic operators. A shared ’environment’ determines the fitness or performance of each individual in the population. The fittest individuals are more likely to be selected for reproduction (retention or duplication), while recombination and mutation modifies those indi￾viduals, yielding potentially superior ones. Using algorithms, mechanical selection, mutation and recombination improves generationally with a fixed parameter size and quality. 2 A genetic algorithm generates each individual from some encoded form known as a ’chromosome’ and it is these which are combined or mutated to breed new individuals. The basis for the optimization is a vast array of possible solutions (population), where every solution (individual) is defined through a particular parameter (chromosome). The individuals within a generation are in competition with one another (selection), in other words, the value (fitness) of the individual is what allows the survival of the parameter (gene) until the next generation. The results of this computer-supported process are automatically generated and opti￾mized. Evolutionary computation is useful for optimization when other techniques such as gradient descent or direct, analytical discovery are not possible. Combi￾natory and real-valued function optimization in which the optimization surface or fitness landscape is ’rugged’, possessing many locally optimal solutions, are well suited for evolutionary algorithms

Form-Optimizing Processes in Biological Structures 293 Fig.18.Structure optimization in the shell structure of a sea urchin. Fig.19.Finite element analysis3 of sea urchin shell,color coded stress analysis [15 Process und Form,K.Teichmann]. Computer-compressed evolution Design space and finite elements Computer-compressed evolution follows the same construction principle that nature employs to promote for example the shell growth of a sea urchin (Figs.18/19)or the silica structure of radiolarian (Figs.23/25).Building material can be removed wherever there are no stresses,but additional material must be used where the stresses are greater.This is the simple principle that evolution has used for millions of years to produces weight optimized "components".Using computer programs based on computer-generated genetic algorithms like the SKO method4,scientists are now able to simulate this evolution and compress it into a short time span [9]. In order to simulate lightweight engineering strategy according to nature's guide- lines,scientists using the SKO method must first define a virtual design space,which represents the outermost parameters of the component being developed.To subdi- vide this design space into many small individual parts,the finite elements,a grid is applied.If now a virtually external load applied,the computer calculates the result- ing force exerted on every one of the finite elements.The FE model shows exactly where there is no load stress on a component and in turn shows where it is possible to make savings with regard to the materials used.On the other hand,for areas that bear heavy stress the simulation program indicates the need to reinforce the con- struction material.Like nature the computer let repeat this "finite element cycle" several times.As a result,they can refine a component repeatedly until the optimal form -one that evenly distributes the stresses within a component-is found. 3 The finite-element-method is a procedure used to solve structural-mechanical cal culations with precedence given to the three-dimensionality of the system.As a result,the construction is broken into discreet elements-Finite Elements (FE- such as columns,beams,plates,shells,etc.characterized by the individual con- nections (discreet points)where they are combined with one another. 4 The DaimlerChrysler Reaserch Center Ulm and Uni Karlsruhe,Prof.Claus Mattheck,in Germany developed the SKO method (Soft Kill Option).The method is based on the idea that is it only possible to achieve a combination of the lightweight and maximum strength in a design when the stresses are con- stant over the structure's entire surface area,ensuring that no area is under-or overstressed

Form-Optimizing Processes in Biological Structures 293 Fig. 18. Structure optimization in the shell structure of a sea urchin. Fig. 19. Finite element analysis3 of sea urchin shell, color coded stress analysis [15 Process und Form, K. Teichmann]. Computer-compressed evolution Design space and finite elements Computer-compressed evolution follows the same construction principle that nature employs to promote for example the shell growth of a sea urchin (Figs. 18/19) or the silica structure of radiolarian (Figs. 23/25). Building material can be removed wherever there are no stresses, but additional material must be used where the stresses are greater. This is the simple principle that evolution has used for millions of years to produces weight optimized “components”. Using computer programs based on computer-generated genetic algorithms like the SKO method4, scientists are now able to simulate this evolution and compress it into a short time span [9]. In order to simulate lightweight engineering strategy according to nature’s guide￾lines, scientists using the SKO method must first define a virtual design space, which represents the outermost parameters of the component being developed. To subdi￾vide this design space into many small individual parts, the finite elements, a grid is applied. If now a virtually external load applied, the computer calculates the result￾ing force exerted on every one of the finite elements. The FE model shows exactly where there is no load stress on a component and in turn shows where it is possible to make savings with regard to the materials used. On the other hand, for areas that bear heavy stress the simulation program indicates the need to reinforce the con￾struction material. Like nature the computer let repeat this “finite element cycle” several times. As a result, they can refine a component repeatedly until the optimal form –one that evenly distributes the stresses within a component– is found. 3 The finite-element-method is a procedure used to solve structural-mechanical cal￾culations with precedence given to the three-dimensionality of the system. As a result, the construction is broken into discreet elements - Finite Elements (FE – such as columns, beams, plates, shells, etc. characterized by the individual con￾nections (discreet points) where they are combined with one another. 4 The DaimlerChrysler Reaserch Center Ulm and Uni Karlsruhe, Prof. Claus Mattheck, in Germany developed the SKO method (Soft Kill Option). The method is based on the idea that is it only possible to achieve a combination of the lightweight and maximum strength in a design when the stresses are con￾stant over the structure’s entire surface area, ensuring that no area is under- or overstressed

294 Edgar Stach Force Design space Finite element cycle 20 Component approximation Refining components Virtual compon Fig.20.SKO method (Soft Kill Option).[9 HIGHTECH REPORT 1/2003,pp60- 63 6 Biological Models Radiolarians A number of self-generated,biological models based on the bubble cluster theory ex- ist.One of the best examples of this is the Radiolarian.Radiolarians are single-celled, marine organisms.These microscopic creatures extract silica from their environment to create a skeleton.Highly articulated geometric patterns define the usually spher- ically shaped structures.The resulting form resembles that of a dome. Fig.21.Computer generated spheri- Fig.22.Fossilized skeleton Radiolar- cal cluster skeleton based on the bub- ian [6] ble clusters theory (Figs.6-13)

294 Edgar Stach Fig. 20. SKO method (Soft Kill Option). [9 HIGHTECH REPORT 1/2003, pp60- 63] 6 Biological Models Radiolarians A number of self-generated, biological models based on the bubble cluster theory ex￾ist. One of the best examples of this is the Radiolarian. Radiolarians are single-celled, marine organisms. These microscopic creatures extract silica from their environment to create a skeleton. Highly articulated geometric patterns define the usually spher￾ically shaped structures. The resulting form resembles that of a dome. Fig. 21. Computer generated spheri￾cal cluster skeleton based on the bub￾ble clusters theory (Figs. 6–13) Fig. 22. Fossilized skeleton Radiolar￾ian [6]