research in structure optimisation

So far my project was more about the deformation process of the initial Piraeus Tower’s skin. Even though – while looking at the project’s representation – the structure seems to be a very important part, I somehow neglected it so far. This led to a rather weak structural solution which rises a lot of problems and questions.

As a next step I’ll try to find a more sophisticated engine driven solution for the tower’s structure. For that reason, I was researching in different structural optimisation processes. I haven’t yet chosen something for my project but it informs me what possibilities there are and what potential solutions I’d be able to create in the further development.
The following flowchart shows which way I’d like my project to go.

There are two main optimisation processes, which are the adaptive subdivision and the surface relaxation. The following extracts are only a small selection of what exists, but for me they seem to be quite illustrative and interesting:


Adaptive subdivision of mesh models
United States Patent 6356263

A computer-based system and method for refining of mesh model of a three-dimensional (3D) object or surface through adaptive subdivision that results in a smooth interpolation of the mesh surface. In one example, the system operates upon a triangulated mesh model and analyzes each edge of the triangle in question to determine whether that particular edge should be subdivided based on a predetermined subdivision criteria. After an analysis of each of the edges of that triangle (using the adaptive subdivision criteria) the system and method may make one of several different types of subdivisions – e.g. dividing the mesh triangle into two, three or four smaller triangles.

images taken from this PDF

There are also several analogies to the adaptive subdivision in many fields such as biology, chemistry, physics and art as well. The latter one seems very interesting to me because it could have paralellel ideas to my project as I could use gradients for instance to create subdivision depending on colours.


Dynamic Relaxation

Dynamic relaxation is a numerical method, which, among other things, can be used do “form-finding” for cable and fabric structures. The aim is to find a geometry where all forces are in equilibrium. In the past this was done by direct modelling, using hanging chains and weights (see Gaudi), or by using soap films, which have the property of adjusting to find a “minimal surface”.

The dynamic relaxation method is based on discretizing the continuum under consideration by lumping the mass at nodes and defining the relationship between nodes in terms of stiffness (see also the finite element method). The system oscillates about the equilibrium position under the influence of loads. An iterative process is followed by simulating a pseudo-dynamic process in time, with each iteration based on an update of the geometry.

Gaudi’s form-finding by hanging chains:

Soap film surfaces are examples of “minimal surfaces”, or surfaces with zero mean curvature. The surface that spans a given boundary set is the one that minimizes the surface area. That is, among all possible surfaces that could span the wire frame, the one that appears is the one with minimal surface area. This is due to the surface tension in the film.

Structural Relaxation

The object of a structural relaxation is to obtain the groundstate relaxed geometry of the system under consideration. This may include the equilibrium lattice constants. For a given ionic configuration, a self-consistent single-point calculation is carried out, and from this, the forces obtained. If these forces are greater than some minimum tolerance, then the ions are moved in the direction of the forces. This procedure is repeated until an equilibrium structure, with vanishing forces (within the numerical tolerance) is obtained. The lattice constants may also be varied by determining the stresses acting, and obtaining an equilibrium structure with vanishing stresses.

Surface Relaxation

This GH-forum entry has some quite interesting answers to the question “What is surface relaxation?” as for instance the following:
- a method of balancing forces in a dynamic manner
- dynamic relaxation should find surfaces of stable equilibrium

The two following projects are quite interesting and could give a better understanding of the structural optimisation process.

Santa Maria Del Pianto Station, Naples, Italy

The project brings up interesting question such as how to optimize a complex arrangement? The given answer shows an image and sais “Evolution of species: survival of the fittest”. It also shows very well that the optimisation is using an evolutionary algorithm to get to the final solution. A lot of infromation – included the present ones – can be found on a PDF on their site.

The British Museum Great Court Roof, London, England

The position of the nodes of the steelwork grid upon this surface was determined by a relaxation process applied to a ‘numerical grid’.The coarser structural grid is obtained by joining diagonal nodes of the numerical grid. The relaxation process involved moving each of the nodes on the numerical grid until it was the weighted average of the surrounding nodes. This process was repeated for the whole grid a large number of times, until the grid stopped moving. The weighting functions varied with position, mainly to try and limit the maximum size of glass panel. Figure 30 shows the grid before relaxation and figure 31 after relaxation.

As the structure has to be analysed to be improved there are several kinds to represent this. These stress analysis can give all information needed for a response as discribed with the flowchart in the beginning of this post. They could ask for a densification at certain parts using an adaptive subdivision or show the curvature analysis which can be used for a surface reflexation

from the book called Advances in Structural Optimisation published by Kluwer Academic Publishers

Some flow charts which show optimisation procedures:

So far, all these information are only research. They influence my thinking and further work might be using one or more of their ideas.


  1. Error: Unable to create directory /var/www/html/piraeus/wp-content/uploads/2019/07. Is its parent directory writable by the server? Nathaniel says:

    Well done making the survey of options very quickly. So what’s your intuition? In what direction do you head?

  2. Error: Unable to create directory /var/www/html/piraeus/wp-content/uploads/2019/07. Is its parent directory writable by the server? Nathaniel says:

    ps. There is a blockquote html tag that might be better for your large quoted passages.

  3. [...] my research on structural optimisation as a base for the further development of my project for the Piraeus Tower I’d like to show [...]

  4. Error: Unable to create directory /var/www/html/piraeus/wp-content/uploads/2019/07. Is its parent directory writable by the server? Martin says:

    So here you go. I put up some ideas, how I intend to go on. And thanks for the hint concerning the quotes. I also added some examplary flow charts at the end of this researche post.

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