Rethinking Prototyping

2 Method: Topology Optimisation Using the Homogenization Method
2.1 Structural Optimisation

Topology optimisation deals with the optimum layout of material within a given design space. When performing structural optimisation, the general formulation is the minimisation of the objective function f(x)

minimise f(x) objective function

such that the constraints are fulfilled:

gj (x)≤ 0; j=1,mg Inequality constraints

hk=0; k=1,mh Equality constraints

xil ≤ xi ≤ xiu; i=1,n Side constraints, upper and lower bounds

More descriptively, the optimisation goal can be described for example as

 Develop a structure with minimum weight with given loadings and support conditions, with the constraint of the deflection not exceeding a given value (otherwise the optimisation algorithm would develop a structure with zero weight), or

 Develop a structure with minimum compliance (maximum stiffness) with given loadings and support conditions, with the constraint of only a given ratio of the design space, the so-called volume fraction, to be filled with material - otherwise the optimisation algorithm would fill up all of the design space with material.

The Homogenization Method is a gradient based optimisation method as an addendum to the Finite Element Method, with the basic idea of subdividing the design space into small domains (pixels or voxels). The initial process of dividing the design space into areas with material and areas without material (0-1 problem) is re-formulated into a continuum problem. The optimisation problem is then to find an optimal structure described by pixels with density 1 (full material) and pixels density 0(no material) plus intermediate values between 0 and 1(„porous“ material). The material layout generated by the optimisation algorithm is a design proposal, which can then be interpretated and futher developed by the designer.

Fig. 2 shows a high beam and an optimised structure under a point load (Ramm 1996). The basic idea is, the optimised structure shows a very homogeneous stress distribution and carries the same load with less material.

A MATLAB code for the application of the homogenization method to structural optimisation is presented in (Lochner, Schumacher 2014). The studies carried out in is contribution use the commercial software Altair OptiStruct, which also uses the homogenization method.

Fig. 2 Typical structure (left) and optimised structure (right) with homogenuous stress conditions

3 Shaping Structural Systems
3.1 Principles of Lightweight Structures: Descriptiveness, Deformations, Internal Forces

Frei Otto’s exclamation in 1977 „Stop building the way you build!“ is still resonating.His works are continuously published in manifold contributions.How can we apply Otto’s approaches to structural and architectural design using modern design tools?

The fundamental research of Frei Otto and his team on lightweight structures reveals basic principles about geometries in nature and building. Within given boundary conditions, structural shapes with an inherent logic are found. The approach of form-finding can be seen as a different kind of prototyping - with a clear differentiation from classical structural theory: It is not the thinking in elements but the thinking in conditions that sets up the design driver.

In structural teachings and design, this approach is very descriptive and aesthetic in a profound way. Approaching structures through thinking in deformations provides a very good understanding about how structures work and about how they can be designed. Deformations relate directly to spans; the distribution of internal forces, bending moments, principal moments or principal stresses, can be derived from a given design task in order to develop a structural geometry.

A design study based on these principles was carried out in a study at the Biberach University of Applied Sciences. The Neue Nationalgalerie in Berlin, an icon of classical modern architecture designed by Ludwig Mies van der Rohe and first opened in 1968, was taken as role model for the design task to develop a waffle slab with its geometry being derived from the support conditions of a square ground plan. The dimensions in ground plan of approximately 65m span were given as a design constraint; the layout of the waffle slab (with the built one measuring 1.80 m in height) was subject to the design. First studies show the interaction of support positions and the correspondent deformations of a continuous slab.

Fig. 3 Analysis models of a point supported slab with varying position of supports; deformation of the slab (with deformations superelevated)

Translating these deformations into an appropriate structure, the relation between span and deformation becomes clear. The support conditions can be transformed directly as a design driver for structural dimensions. They are of course only one singular aspect in design, since internal forces (here: predominantly bending moments and shear forces) are not a priori taken into account. However, a feasible approach to design is found.

The derivation of a structure from the deformation of a reduced model produces an aesthetic, feasable and understandable geometry, as shown in Fig. 4.

Fig. 4 Structural geometry of a point-supported waffle slab derived from its support conditions and correspondent deformations

Taking into account other aspects such as bending moments or principal moments produces different results for the same design task. Comparing the design results shows a great variety of proposals, while all of them are driven by structural constraints: With a very simple structural approach, a great variety of architectural designs can be developed.

3.2 Principles of Lightweight Structures: Efficiency, Structural Optimisation

Structural optimisation is a very useful tool for the design of cars and aircrafts, since the minimisation of dead load is an economical goal, as well as the complex geometries and loading conditions require modern computation tools. Architectural structures are not in motion and therefore it may not seem obvious to transfer these technologies into the building sector. However, when looking at the life cycle of buildings it becomes clear that from excavation, production, delivery, assembly until deconstruction, recycling or disposal, also here there is material in motion.This leads directly to the responsibility of all planners to limit material input sensibly. Furthermore, the principles of lightweight structures as design attitude open up a mind-set of design drivers that can be very useful in the development and understanding of structural geometries.

Fig. 5 Geometries derived from different structural aspects

The studies presented in this contribution use the commercial software Altair OptiStruct, which is based on the homogenization method and implies a wide range of special issues of structural optimisation. The authors would like to thank Altair Inc.for granting reduced and free licenses for academic studies.

3.3 Structural Parametric Patterns

Patterns, or ornaments, have a long history in building and can be found in all cultures. Traditionally, they are decoratively added elements. Adolf Loos’ essay „Ornament und Verbrechen“ (Ornament and Crime) marks a turning point in contemporary architecture: The elimination of any non-functional decorative elements sets the origin of classical modern architecture.

Studies of parametric geometries base on the systematic variation of geometrical patterns: the variation reveals characteristic elements, be it subjective aesthetical or objective structural qualities. Fig. 6 shows examples of parametric studies: Parametric geometrical studies by D‘rcy Wentworth Thompson dealing with the subject on growth and form (Thompson 1961); form-finding studies with systematic variation of boundary conditions using soap films within differently curved wires carried out by Frei Otto and his team (Bach 1988); and parametric optimisation studies carried out by the author developing optimal shapes within a given volume with varying orientation of the point loads applied to the structrure.

Fig. 6 Examples of parametric design studies

Today, the ornament seems to be back in architecture and parametric design is very fashionable in both architectural and structural design. New tools allow generating geometries adapting to variable boundary conditions - being structural demands, lighting conditions or formal conceptions. Fig. 7 shows examples of modern ornaments (left: exhibition stand Gasser Fassadentechnik, Swissbau 2012; right: Suedwestmetall facade, architects Allmann Wappner Sattler)with their design possibilities and manufacturing conditions directly related to new developments: repetition in patterns is other than some few decades or years ago redundant. Structural necessities as well as manufacturing constraints can be neglected. The focus is set on aesthetical logics and qualities.

Fig. 7 Examples of modern ornamental design

The missing link between pattern and structure can be achieved through use of structural optimisation methods. As an example, the procedure in this study shown in Fig. 8 is

 A given design space, here: a flat cuboid volume, with part of the space not being subject to the optimisation.

 A given structural system, here: evenly distributed load on the top surface; point supports.

 The optimisation formulation, here: minimum compliance as an objective, given volume fraction (15% of the design space to be filled with material) as a design constraint.

 Varying of parameters, here: systematic variation of the support points of the structure.

Fig. 8 The structural model as a plate with distributed loading + point supports, the optimisation result

The optimisation result shows the material distribution within the design space: ribs running directly between the point supports along the short distance and ribs connecting between the point supports reaching maximum height at mid-distance with the ribs merging and leaving a void in the middle.

The structural system modelled for this optimisation is a very broadly applied one: plates under distributed loadings can be inserted into many structural arrangements. The patterning is now produced through variation of the position of the points supports, which are located in the four corners of the initial model (Fig. 9).

Fig. 9 Position of point supports and bottom view of the corresponding optimisation results

Fig. 10 Possible arrangement of parametric pattern

All of the systems use the same amount of material, all of them are optimised according to the given support conditions. The result of this study is a patterning with an inherent structural logic, comprehensible to the viewer and derived from objective targets: the development of geometries relating to their structural system.

A possible application of parametrically optimised plates is shown in Fig. 10. The material of the plates could be any mouldable material adequate resistance, such as fibre reinforced concrete or plastic. The fixing points of the cladding plates are then positioned where they were located in the optimisation run.

The optimisation can basically be scaled within certain limits. Another possible application is the generation of plates at a larger scale is shown in Fig. 11. It resembles in its appearance the famous ceiling structure of the Gatti Wool Factory in Rome, designed by Pier Luigi Nervi, a pioneer in the design of aesthetic and efficient structures. To suit the support conditions determined by the optimisation procedure the columns are branched at varying heights, corresponding to the distance of the support points. Since the plates were optimised as individual structural elements, they are arranged at a distance.

Fig. 11 Pattern roof structure, referring to Nervi‘s Gatti ceiling

It was the merit of Pier Luigi Nervi to merge aesthetics, structural efficiency and construction - as he mentioned „good engineering seems to be a necessary, however not sufficient condition for good architecture“ (Nervi 1965). Modern manufacturing methods allow variations in geometry without rising costs. Nervi‘s idea of aesthetical engineering should be carried on using modern design and manufacturing tools.

Further parametric studies deal with the design of shells: the interaction of support geometries of a structure produces related shell geometries. The curvature of the shell is then directly linked to its structural system, a possibility to create structural parametric patterns in 3D space.

3.4 Re-Design of Natural Structures

Natural structures give impressive examples of structurally optimised geometries. Nature has developed a great variety of very lightweight structures, resulting from optimisation procedures running over billions of years. The methods of structural optimisation can be used to clarify the basic characteristic of their load-bearing behaviour and to develop structural geometries with the same efficiency, basically reproducing the process of optimisation in nature. The merging of studies of natural structures with the methods of structural optimisation can produce a new morphology of natural lightweight structures. Lightweight structures must and will play an important role in architecture and engineering when acting responsively in the field of material use.

A good example for an efficient natural structure is the skeleton of the columniform cactus, which reaches up to approximately 6m of height. Its structure can be described as a perforated tube. Developed by the SOM-affiliated engineer Fazlur Khan in the 1960s tube structures are very efficient structural systems for the design of tall slender buildings: The concentration of material along the outline of the structure allows for optimised structural efficiency in comparison to the classical core structure.

For this study, the structural model was set up as a vertical cast-in beam with a hollow tube section. The optimisation objective was to minimise the compliance, i.e. maximize stiffness of the structure with the design constraint of only 15% of the structural volume to be filled with material. The result is an organic geometry with thorough structural background. Fig. 12 shows the antetype, the structural system and the optimisation result.

The tube is now optimised for one dominant wind direction. In combination with an inner tube the tube-in-tube system, also initially developed by Fazlur Khan, the structure serves all wind directions. Wind loadings acting from varying wind directions are carried by two interacting structural systems: The wind loading at the lower part of the structure is assigned mainly to the outer tube, with the geometry not being

optimised for this load case, but working as a tube. The wind loading at the top part of the skyscraper is assigned to the inner tube, which acts as a (much more slender) cantilever, cast-in into the outer tube structure, with an appropriate slenderness of about 1:6 when considering only the top part actually working on its own.

Fig. 12 Skeleton of the columniform cactus; structural model and optimisation result

Fig. 13 Development of the structure into a tube-in-tube skyscraper structure

Another demonstrative example of efficient natural structures can be found at a much smaller scale: the diatoms. As a very good example of long-term optimisation processes in nature, there exists an evolutionary competition between crab and shell: the crab developing stronger pincers, while the diatom shell increasing in strength.

Ernst Haeckel is until today outstanding with his demonstration of the amazing variety of shapes in microscopic structures (Haeckel 1904) - with descriptive variations of structural shapes depending on the overall geometry as well as on the loading conditions of a structure. The division into Centrales with radial geometries and Pennales with bilaterally symmetric shapes leads to an overall classification of diatom geometries (Fig. 14).

Fig. 14 Diatoms, as documented by Ernst Haeckel

When carrying out studies of structural optimisation, geometrical influences have to be taken into account - as it can be seen from a comparative optimisation study of a circular geometry compared to an elongated geometry. The design proposals produced by the optimisation algorithm refer directly to the geometrical conditions of the design space. Fig. 15 shows optimisation studies, with the analysis model composed of one quarter of the circular shell and the interpretation of the design proposal into a structural geometry for a grid shell.

Fig. 15 Development of a diatom structure based on diatom‘s principles

Further development of optimisation studies of the circular shell, based on the design proposal shown in Fig. 15, produces in fact a structure resembling diatom structures. The optimisation study simulates the evolutionary process of nature resulting in a structure optimised for its loading (distributed loading is dominating) and support conditions (constant supports along the bottom edge) and of high aesthetic quality.

Fig. 16 Grid shell derived from the optimisation result (interior views)

Further structural studies including the spongiosa and sandwich-like systems in bone structures, the sea urchin, branching structures, seashells and dragonfly wings. They all conclude that nature is an excellent role model for the holistic design of structures, and that structural optimisation is a useful tool to grasp the inherent logic of natural structures.

4 Conclusions and Outlook

Structural optimisation is shown as a useful tool for a holistic approach to design with the profession of architects and engineers merging in a common task: the design of efficient and aesthetic structural geometries. Facing the challenges of the years to come, the requirement to provide liveable surroundings for an increasing population with limited resources; this can contribute to the beginning of a new team play of architects and engineers.

References

Bach, K., 1988: Seifenblasen - Foaming Bubbles: Eine Forschungsarbeit des Instituts für Leichte Flächentragwerke über Minimalflächen (IL 18). Stuttgart: Krämer.

Galilei, G., 1638: Discorsi e dimostrationi matematiche. Leiden.

Haeckel, E., 1904: Kunstformen der Natur - Kunstformen aus dem Meer. Re-edition 2013. Munich: Prestel.

Lochner, I.; Schumacher, A., 2014: Homogenization Method. Distribution of Material Densities. In: Adriaenssens, S. et al. (eds.): Shells for Architecture. Form Finding and Structural Optimisation. Abingdon: Routledge.

Michell, A.G.M., 1904: The Limits of Material in Frame Structures. Philos. Mag. Ser. 6, 8, pp. 589-597.

Ramm, E.; 1996: Force Follows Form oder Form Follows Force? Die Wechselwirkung von Form und Kraft bei Flächentragwerken. In: Wilke, J. et al. (eds.): Prozess und Form ‘Natürlicher Konstruktionen’. Berlin: Der SFB 230.

Thompson, D.W., 1961: On Growth and Form. Cambridge: Cambridge University Press.