DBD_Management of Compex Geometries

École Nationale Des Ponts et ChaussÊes MS Design by Data

Assignment #2 Conceptual Structural Design | Management and Design of Complex Geometries O. Baverel | R. Mesnil

Mario Di Sibio 11/11/2018

École Nationale Des Ponts et Chaussées

Mario Di Sibio

Select one out of three provided surfaces, perform a curvature analysis and, according to the result, design a structure using some remarkable lines.

0. SURFACES SELECTION The three surfaces [Fig. 1] are all different: the first one is a synclastic surface since has double positive curvature; the second one is an anticlastic double ruled surface and presents a negative double curvature; the third one presents zones with positive and negative double curvature. According to this elementary analysis it is possible to define which are the best lines to use in each surface: it is possible to compute principal curvature lines, isocurves and geodesics in every surface, as well as any projected curve. Asymptotic lines can be computed just in the second surface and in the negative curved area of the third one since this kind of lines exists only when the darboux frame generates shapes that allows for asymptotes (eg. hyperboloid).

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Fig. 1 The provided surfaces; Isocurves highlighted

These assumptions give us an idea of how to proceed: • 1st Srf: Geodesic lines | Principal Curvature lines: mixing these two lines or use one of them.

• 2ndSrf: Asymtotic lines | Principal Curvature | Geodesics: mixing these two lines or use one of them.

• 3rdSrf: Geodesic lines | Principal Curvature | Geodesics |Asymptotic lines:

divide the surface along the 0 curvature line and treat these zones according to the curvature sign. Study the transition between the zones.

For this assignment I selected the first one.

1. CURVATURE ANALYSIS This a dome-like surface, trimmed from a vault in correspondance with the key point of each arc.

The dome spans 15m x 10m along the principal axis.

The choosen surface is synclastic [Fig. 2, left] since the Dupin’s indicatrix is > 0 in every point and the Darboux frame generates closed, planar ellipses (elliptic points).

This surface presents symmetry in respect to X and Y axis: this property is easy to read from the gaussian and mean curvature diagrams [Fig. 3].

The Gaussian curvature is calculated as K1 x K2 where K1 and K2 are the inverse of the radius of the osculating circle calculated in the direction of the principal axis; The max and min values of the Gaussian curvature ranges from 0.00418 to 0.0329 [Fig.3, left]; the curvature is not 0 but it is quite close.

The Mean curvature is also an interesting property to evaluate; is calculated as (K1 + K2)\2. The max and min values are ranging from 0.06473 to 0.19652 [Fig.3, right]; since these values are not constant we understand this is not a minimal surface.

Fig. 2 left: Dupin’s Indicatrix and Darboux Frame | right: Principal Curvature lines

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École Nationale Des Ponts et Chaussées

Mario Di Sibio

2. STRUCTURE STRATEGY AND OPTIMIZATION Given that we want every element maximum span to be between 0.8m and 2.5m, the principal directions are divided in order to equalize a lenght of ~1.08m, a lenght that is convenient for each axis. The calculation results in 16 division points in Y axis and 12 along X axis [Fig. 4]. It is also remarkable that the difference of the gaussian curvature between subsequent sample points along these lines is on average 1.17 x 10-12 with a maximum of 0.0031, so we can assume is nearly zero. This is important due to the the development properties of zero Gaussian surfaces.

Fig. 3 left: Gaussian Curvature diagram | right: Mean Curvature diagram

The structure chosen is a timber frame following geodesic lines in order to obtain the shortest path between the base points and consequently optimize the material usage and the overall weight of the structure. The covering is composed by developable stripes of perforated fabric, the boudaries of which are anchored to the corresponding half of each beam.

Choosing the right base edges is crucial as this is going to shape the beam pattern over the surface. The first trial was to cut the base curve of the dome with the principal directions lines but that yelded to a “diagonal” pattern with very curved geodesics. The aim, in this case, was to obtain the most rational pattern possible so a second trial was made. In order to find the original 4 edges, the surface has been untrimmed. The intersection between the dome base curve and the diagonals gives the vertices of the 4 base curves [Fig. 5].

Fig. 4 Sample Points along principal directions | 16 Yellow dot along Y and 12 blue dots along X

According to the sample points count, each base edge is being subdivided, in order to obtain a geodesic for each sample point. If the edges are subdivided equally, the geodesics tend to pass further from the center [Fig. 6.1] in a non linear way. In order to optimize the distrubution of the geodesics and let them pass throw the sample points a local optimization logic has been setup with Goat (BOBYQA): the distribution of the endpoints parameters of the geodesic curves is constrained to a parabola formula x2+x (which is nearly the behaviour of the geodesics in relation to the center) in order to compute a non linear distribution over the base curve [Fig. 6-7]. After less than 1 minute the alorithm converged: the maximum distance between a geodesic curve and its respective sample point is 0.55m and the minimum is 0.00048m; the max is a big gap but it is located on the first (and last) curve and is dued to another project constraint that will keep the endpoints of the “boundary” geodesic lines coincident so actually not giving enough space for optimizing them.

The optimization process yelded to elements ranging from ~0.78m to ~1.46m so it seems to be an acceptable result.

Fig. 5 First Step: determining points to cut the base curve

Fig. 6 Second Step: before (1) and after (2) geodesic endpoints optimization

Fig. 7 Parabolic graphic for generating the base points parameters

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École Nationale Des Ponts et Chaussées

Mario Di Sibio

2. STRUCTURE ORGANIZATION After the optimization process a principal and a secondary structure has been identified. The two principal lines are intended as arcs, composing the principal frame with the four boundary geodesic curves [Fig. 8] joined with steel or concrete supports [Fig.10]. The surfaces of the beams are created sweeping a straight line (the lenght of wich is equal to the beam thickness) along the geodesic lines following the normal vector to the surface. This results in Monge surfaces which are nearly developable surfaces and can be unrolled.

Fig. 8 Principal structure

The secondary structure is composed by smaller beams [Fig. 9] with little bending capabilities on the weak axis in order to accomodate the very small curvature. This secondary structure is intended to sustain the roof covering and it stops on the boundary beams. As the principal ones, these are also developable surfaces and must be unrolled in order to prototype them.

The roofing consists in 14 strips generated, in the short direction, from one geodesic and the following one [Fig.11].

Due to the very small difference in the Gaussian curvature between the sample points, these strips are nearly developable. They successfully unrolled within a 0.005% area tolerance.

Another covering system can be made with a panelization system: calculating the determinant of the panels it shows that along the principal lines the quads tend to be planar, but near the support corners they become curved. The values of the height of the tetrahedron ranges from 0m to 0.081m with an average of 0.018m according to the diagram [Fig. 12]. The corners values are too high and out of normal tolerances for flat panels but due to the double simmetry of the shape and rounding the determinant result to 3 decimal places (because errors are introduced with the geodesics integration) we can assume that there are 21 different panels types for a total of 168.

Fig. 9 Secondary structure (the “waffle” system is for prototyping purposes)

Fig. 10 Boundary Joins

Fig. 11 Roof Skin and unrolled stripes

Fig. 12 Planarization diagram (Blue is planar | Red is curved)

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École Nationale Des Ponts et Chaussées

Mario Di Sibio

3. ISSUES • The geodesic component in Grasshopper peforms some kind of approximation (Euler, Runge-Kutta,...) to compute the curves. There is not much documentation about the default settings of the integration process, like the step size.

This is why many of the geodesic lines does not intersect properly to each others with an absolute tolerance set at 0.001m. The right result is achieved setting the tolerance at 0.08m that is absolutely too high. The problem has been solved using a component written by Jon Mirtschin that can compute curve intersections changing the tolerance just for this operation.

• While testing the fabrication of the prototype a mistake in the waffle system occurred caused by the region difference component. It planarizes the beams on a given plane and consequently doesnt allows for the joins to be perfectly aligned. Since the choosen material was 6mm MDF it was impossible to properly assemble the structure. A solution was to subtract solid boxes from the beam surfaces and unroll them. Using a thinner and less strong material for secondary beams, like cardboard, will permit to perform minimal bending allowing the beams to join.

Fig. 13 Render - The dome with coloured stripes

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École Nationale Des Ponts et Chaussées

Mario Di Sibio

Fig. 14 Render - Top View of the structure

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École Nationale Des Ponts et Chaussées

Mario Di Sibio

Fig. 15 Render - Right View of the structure

Fig. 16 Render - Front View of the structure

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