# Coupling grid topology generation and form-finding for the design of architectural meshes

Romane Boutillier, Cyril Douthe, Laurent Hauswirth, Olivier Baverel

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Romane Boutillier, Cyril Douthe, Laurent Hauswirth, Olivier Baverel. Coupling grid topology generation and form-finding for the design of architectural meshes. Structures, 2024, 68, pp.107119. 10.1016/j.istruc.2024.107119. hal-05414118

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# Coupling grid topology generation and form-finding for the design of architectural meshes

Romane Boutillier<sup>a,b,*</sup>, Cyril Douthe<sup>a</sup>, Laurent Hauswirth<sup>b</sup>, Olivier Baverel<sup>a,c</sup>

<sup>a</sup>Laboratoire Navier UMR8205, Ecole des Ponts, Univ Eiffel, CNRS, Champs-sur-Marne, 77455, France

<sup>b</sup>Université Gustave Eiffel, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR8050, Champs-sur-Marne, 77455, France

<sup>c</sup>GSA / ENS Architecture Grenoble, Grenoble, France

## Abstract

The performances of gridshells depend on the grid topology but also the geometry of the underlying surface. In optimal design, such as funicular shapes, topology and geometry are linked. When lifting a grid, the resulting shape depends on the provided topology. Topology generation is thus a tool for structural design. This paper shows its practical application for engineers. The topology generation method described in (Boutillier et al., 2024b) is employed and provides grids adapted to the support conditions of the project. Grids are then lifted and evaluated in two case studies: the spatial forms result from an optimisation process balancing surface funicularity with panel planarity. In the first case, the mechanical performances are then evaluated according to different criteria. A non-intuitive result shows that a large variety of topologies is situated on the Pareto front, legitimating a posteriori the topology generation method. In the second case, the method's applicability is evaluated through the fabrication of a wooden pavilion with a funicular structure and flat quadrilateral panels, making it possible to reconcile topology with manufacturing and project constraints.

*Keywords:* Topology finding, Gridshell, Form-finding, Funicular Pavilion.

*romane.boutillier@enpc.fr

Preprint submitted to *Structures*

December 12, 2025
<figure><img src="image_2.png" /><figcaption>Figure 1: Milan Trade Fair (Fuksas/SBP)</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Isler’s hanging model Chilton and Isler (2000)</figcaption></figure>

# 1. Introduction

## 1.1. Necessity of topology generation

Meshes are everywhere in free-form architecture, especially when constructing surfaces. To be built, surfaces must be subdivided, i.e. discretised, into manufacturable elements (beams and panels). This discretisation of the surface leads naturally to a two-dimensional mesh (i.e. a collection of triangles or quadrilaterals) where panels are faces and beams are edges. The grid is thus a visible structural layout that determines the discretisation of the surface (regular panels and a uniform member distribution may be preferred), the manufacturing and implementing simplicity (flat panels are for example less expensive and easier to implement), the mechanical efficiency (limiting deflection and reducing bending), etc.

The grid topology is therefore crucial: it is often seen as a project constraint that is dictated by the strong architectural and formal intentions a designer may have.

For example, in the case of the Milan Trade Fair (Figure 1), the abrupt change in curvature makes it necessary to introduce higher valency nodes (i.e. nodes that connect more elements, coloured on the image) to limit the distortion of the panels (Schlaich et al., 2005).

But when formal intentions are not paramount and the surface can be modified, the grid topology can be used to control the resulting shape. This form-finding process is illustrated by the image of a Heinz Isler model (Figure 2). This shape was obtained by suspending a net, whose topology (the textile orientation) influences the deformation under gravity loading. The influence of the topology is also illustrated in Chilton and Chuang (2017) where different fabric patterns are lifted, and in Figure 5 where a quad mesh
with isolated singularities is submitted to gravity with a constraint on panel planarity.

## 1.2. *Existing methods of topology generation*

Topological generation has not been widely studied for architectural structures but is a subject of research in the computer graphics community (see surveys by (Bommes et al., 2013; Campen, 2017) and (Dong et al., 2005; Jezdimirovic et al., 2021) which are closer to the method described hereafter).

Topological generation is not to be confused with topological optimization, which optimises the distribution of material in an element (Deaton and Grandhi, 2014).

On given surfaces, there are methods for defining a high-performance grid by integrating curvature lines (Liu et al., 2006; Pellis and Pottmann, 2018; Tellier et al., 2019) or stress lines (Pellis and Pottmann, 2018; Schiftner and Balzer, 2011). Panozzo et al. (2013) generate form diagrams by aligning a mesh with a cross-field constrained by heuristics adapted to masonry to fit a target surface. However, these methods do not allow transformations to be performed on the resulting grids.

Given the numerous criteria guiding the choice of mesh, methods for generating and exploring topologies are relevant and constitute a tool for engineers. Shea and Cagan (1997) introduce a dome grammar and use shape annealing method for exploring designing solutions and creating domes. Other methods don't rely on the input of a target surface but offer pure topological exploration: Oval et al. (2019a,b) developed a method of topology generation based on the boundary of the surface to mesh, as well as features such as points or curves chosen by the designer. The exploration of the different grids is then controlled by a grammar of elementary operations on strips. The method presented in Boutillier et al. (2024b) is a continuation of this approach called "topology finding". It proposes a new paradigm where the exploration of the grids is controlled by continuous parameters defined at the boundary. It is therefore intuitive and starts from the support conditions which are a crucial element in the behaviour of the structure but also for its integration in an architectural context.

These topological generation methods produce grids which are then lifted using various form-finding tools and optimised to satisfy multiple constraints.
### 1.3. Form-finding methods with an input grid

Form-finding methods require a grid to be supplied and lift it to meet different criteria. Some methods focus on geometric constraints, such as the Marionette method (Mesnil et al., 2017), which produces a grid with flat panels projecting onto the initial flat grid.

Others focus more on mechanical performance. Historical methods developed by Frei Otto at the Institute for Lightweight Structures (Otto et al., 1974), Gaudí or Isler (Isler, 1961) used models and nets transformed by natural laws to get shapes in equilibrium. The force density method (Schek, 1974) can be used to determine all equilibrium shapes of a cable network. The updated reference strategy (Bletzinger and Ramm, 1999) generalises this method to structures modelled by finite elements. Dynamic relaxation, introduced in particular by Day (1965), determines an equilibrium state of a structure by increments and with damping. Cited by Barnes (1988) for finding the shape of cable networks and membrane roofs, this method can be coupled with iso-geometric analysis (Hughes et al., 2005) for finding the shape of suspended models (Alic and Persson, 2016). Particle-spring systems (Kilian and Ochsendorf, 2005) is another interactive form-finding method, that computes the equilibrium of a system of particles with a mass and connected by springs.

Finally, graphic statics is a visual method for controlling both the shape and the forces in the structure. Block and Ochsendorf (2007) have developed the thrust network analysis which produces compression shapes under gravity loading, a method implemented in RhinoVAULT (Rippmann et al., 2012). Ohlbrock and Schwartz (2016) propose a method (combinatorial equilibrium modelling) that, in the upstream phase, can generate a variety of structural typologies without being limited to pure compression structures.

### 1.4. Summary of contributions

This paper builds upon the conference paper (Boutillier et al., 2024a). It uses a method described in (Boutillier et al., 2024b) that generates topologies by varying continuous parameters. The generation method guarantees that the obtained grids are adapted to the support conditions of the considered surface: elements converge towards the supports and arrive perpendicular to the edges otherwise. The paper here shows how engineers and architects can deploy the topology generation method as a design tool, by looking at two case studies which are more or less subject to project and site constraints.
This parallel application to two different projects gives a broader view of the
tool's benefits:

* the first is more conceptual, more upstream, with no a priori struc-
tural typology. Performance is used as an indicator relative to other
topologies and structural types.

* the second, on the other hand, is heavily influenced by manufacturing,
with performance constrained by precise critical values.

All grids are evaluated according to a reduced choice of metrics, which are
discussed in the following sections (the reader is invited to consult Bruno
et al. (2023); Raffaele et al. (2024) which propose a general framework for
the assessment of gridshell performances).

After a brief review in Section 2 of the topological generation method and
a description of the form-finding method that is used, two applications are
studied. In the first case (Section 3), a variety of grids are generated on a
given boundary and evaluated according to different criteria using a Pareto
front. The second case (Section 4) illustrates the use of this tool in a real
project, namely the construction of a pavilion.

## 2. Topology and geometry generation method

2.1. *Topology generation*

The topology finding process is detailed in Boutillier et al. (2024b). Figure 3 outlines the method and introduces the vocabulary. The method generates topologies (and more precisely topological skeletons) and grids based on parameters defined along the edge or contour. From the projection of the architectural boundary and scalar data at the boundary, topological skeletons are constructed by integrating the height gradient over a minimal surface. The boundary of the minimal surface is chosen according to the supports of the real architectural surface, in order to find a topology and a grid where beams converge at the supports and arrive perpendicular to the edge elsewhere. They are therefore optimal from a mechanical point of view. Moreover, beams intersect at almost 90 degrees throughout the mesh.

The boundary of the minimal surface, defined by the designer through the
height of anchoring points, can be modified, thus leading to the generation
of various topological skeletons. The latter are then subdivided into dense
mesh and lifted to define the structure's geometry.
<figure><img src="image_4.png" /></figure>

Figure 3: Topology and grid generation process. A crenellated boundary (in red) is built over the real supports of the surface. This artificial boundary is controlled by height parameters. A minimal surface (in blue) is computed on this boundary. A topological skeleton is then retrieved (centre top). It is made of singularities (red dots) of the height gradient field. Streamlines lines of the gradient are shown in blue, the level set is shown in orange. The skeleton defines patches that can be discretised regularly. The field is finally integrated to get patch subdivisions (centre bottom). The grid can then be lifted, using various methods. Two examples are shown using the Marionette method (right).

## 2.2. Determination of the geometry

### 2.2.1. Direct method for determining the strip subdivisions

The subdivision is the same for all patches of a strip. Strips and patches are illustrated in Figure 3: a patch is the topological entity, a zone that can be regularly subdivided with quads (or triangles around poles), and a strip is a series of patches that are connected along their opposite edges. The subdivision is computed to get a target average edge length (of about 1.2 metres in the case study of Section 3 with a hexagonal boundary) or can be distributed by hand (in the case of the pavilion shown in Section 4).

The resulting grid is the discretisation of level and gradient lines, which intersect at right angles. It is made of quads (except at poles) that are overall regularly distributed in the mesh. However, the concentration of lines at poles, and large faces at singularities can be noted. This could easily be remeshed locally (as will be seen in Section 4.3 for the pavilion). The former grids are thus kept for evaluation in the first study, despite this constructive remark.
### 2.2.2. Lifting method to achieve flat-panelled funicular structures

In the next two studies, grids are lifted and optimised to get compression-only structures with flat panels. Panel planarity is a frequent criterion, for both economic and manufacturing reasons. This is the objective of many form-finding methods (Pottmann et al., 2015). Kangaroo for Grasshopper ® (Piker, 2013) is an optimisation algorithm that adapts dynamic relaxation with the combination of projection operators. Different goals (geometric constraints, forces, etc.) are translated into energies and the combination of these is minimised. Other methods are available for obtaining plane-faced funicular gridshells. For example, Adriaenssens et al. (2012) use Maxwell reciprocal diagram to achieve planarity in a funicular grid obtained with dynamic relaxation. Millar et al. (2023) interpret Airy functions for funicular flat panel structures under vertical loading.

Kangaroo is used for both studies. It is a fast, deterministic tool that can be used to obtain comparable structures and thus evaluate the different topologies generated in Section 3. In the second study (Section 4), the lifting parameters are modified in order to obtain the best three-dimensional grid from the chosen topology. Ideally, a two-stage optimisation would be carried out: upstream on the topology, then on the parameters of the lifting method, these two layers being linked in the search for a high-performance grid.

## 3. First case study: generation of grids on a hexagonal boundary

The method is illustrated on a regular shape, that can be used to cover an interior space or as a building (like the restaurant of Felix Candela in Xochimilco (Moreyra Garlock and Billington, 2008)). We consider a point-supported structure, whose boundary projection is a hexagon, where supports are located at the discontinuities of the boundary. As the topological variety depends on the number of supports, considering a hexagon with 6 supports serves to illustrate the diversity of grids. The crenellated contour is thus made of six slots, and five parameter heights are considered (after removing the rigid body motion). Several topological skeletons are calculated and evaluated according to different criteria. They are represented on a Pareto front, which can be used as a design tool enabling the engineer to navigate through the space of topologies. Some topologies that are found are shown in Figure 4.
### 3.1. Topologies that can be reached with this method

Even if this contour is very symmetrical, many topologies can appear. Topologies depend on the valency (i.e. the number of edges that are incident to the vertex) of singularities and poles, their number, and the strip distribution. The types of topologies that can occur are controlled by the Poincaré Hopf theorem which relates the point valency (a local feature) and the topology of the surface (a global information). The sum of the indices of the vertices must be equal to the Euler characteristic of the surface $\chi$, where $\chi = V - E + F$ and $V$ is the number of vertices, $E$ the number of edges and $F$ the number of faces of the graph. As we consider disks, the Euler characteristic equals 1. (See Boutillier et al. (2024b) for details).

<figure><img src="image_5.png" /><figcaption>Figure 4: Selection of topologies that can be achieved with the presented method. Various combinations of vertex indices can be created.</figcaption></figure>

Several distributions of singularities can be obtained. Some of them are shown in Figure 4, and the reader is invited to visit the website (Boutillier et al.) to explore a broader variety of topologies. For every topological skeleton of Figure 4, the combination of singularities and poles is given in Table 1. For example, for the topological skeleton a., there is one singularity of valency 12, that has an index -2, and 6 poles of index 1/2.

### 3.2. Lifting procedure

In this study, in order to compare topologies, only one shape is computed for each topology. This shape depends on the topology but is not the only one that is adapted to this topology. It is the result of an optimisation under two major constraints: panel planarity and grid funicularity. The funicular load is proportional to the tributary area in the plane (with a factor 2), therefore all grids, that have the same planar projection, are submitted to the same loading. Other goals are applied for achieving a smooth shape: mesh edge lengths are controlled (lengths are kept with a strength 1000), stiffness is provided at the ridge beams to smooth curvature (with a bending strength of 10), all points are anchored in the horizontal directions (with a strength of 1000), support points are anchored (with a strength of 100 000), mesh faces
<table><caption>Table 1: List of the available singularities with their indices, and examples of possible combinations of singularities and poles so that the sum of the indices is 1. The configurations correspond, in order, to the images in Figure 4. Other singularity combinations are available online (Boutillier et al.).</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Index</th><th colspan="6">Possible configurations</th></tr><tr><th>a.</th><th>b.</th><th>c.</th><th>d.</th><th>e.</th><th>f.</th></tr></thead><tbody><tr><td>Topological skeleton</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Singularity of valency 12</td><td>-2</td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Singularity of valency 8</td><td>-1</td><td></td><td></td><td>2</td><td>1</td><td></td><td>1</td></tr><tr><td>Singularity on the boundary of valency 5</td><td>-1/2</td><td></td><td></td><td></td><td>2</td><td>4</td><td></td></tr><tr><td>Singularity in a corner of valency 5</td><td>-1/2</td><td></td><td></td><td></td><td></td><td></td><td>1</td></tr><tr><td>Singularity in a corner of valency 3</td><td>0</td><td></td><td>4</td><td></td><td></td><td></td><td></td></tr><tr><td>Pole</td><td>1/2</td><td>6</td><td>2</td><td>6</td><td>6</td><td>6</td><td>5</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Figure 5: Two grid examples that are lifted with a gravity load and constrained to have planar panels. The two lifted shapes differ because grid topologies are different.</figcaption></figure>

are asked to be planar (with a strength of 2000). The optimised shape is rescaled to get a maximum target height that is the same for all grids (and corresponds to a ratio height/span of 0.2). This lifting process is illustrated on two grids with different topologies in Figure 5: because the shape depends on the topology, the resulting geometries vary.

As mentioned in the introduction, this method for generating topologies can also be used as a form-finding method. Indeed, every topology can be regarded as a new fabric weft and giving rise to new forms in Heinz Isler's experimental process illustrated in Figure 2. The resulting geometry is directly related to the topology of the grid. Therefore, each geometry is also evaluated as an isotropic shell, in order to compare the performance of the shape (ignoring the structure).
<table><caption>Table 2: Correspondence between the properties of the braced grid and the isotropic shell.</caption><thead><tr><th></th><th>Axial</th><th>Bending</th><th>In-plane shear</th><th>Out-of-plane shear</th></tr></thead><tbody><tr><td>Braced gridshell</td><td>E<sub>A</sub>/L</td><td>E<sub>I</sub>/L</td><td>2E<sub>A<sub>d</sub></sub>/L</td><td>G<sub>A<sub>r</sub></sub>/L</td></tr><tr><td>Shell</td><td>E<sub>s</sub>t/(1-ν<sup>2</sup>)</td><td>E<sub>s</sub>t<sup>3</sup>/(12(1-ν<sup>2</sup>))</td><td>G<sub>12</sub>t = E<sub>s</sub>/(2(1+ν))t</td><td>G<sub>3</sub>t</td></tr></tbody></table>

### 3.3. Mechanical model

#### 3.3.1. Structural behaviour

Three structural types are tested: a shell (isotropic), a braced gridshell (anisotropic but with a membrane shear stiffness), and a gridshell without diagonal bracing but fully clamped (anisotropic). A gridshell is a grid of beams approaching a double-curved surface, and whose elements are mainly axially loaded, so they can be thin. The braced gridshell serves as a reference for determining equivalent mechanical properties. The bracing section for the diagonals of the braced gridshell is taken as a tenth of the section of the regular beam. The shell properties ($E_s$, $t$ and $\nu$) are chosen to get a similar rigidity to the braced gridshell. Correspondences are shown in Table 2 and determined according to Tsuboi et al. (1984). Notations in the table are as follows: for the braced gridshell (made of beams and diagonals), $E$ is the Young's modulus of the beams, $A$ is the section of the beams, $I$ the section modulus of the beams, $L$ is the average length of the beams (a measure of the width of a cell), $A_d$ is the section of the diagonals (the bracing beams), $A_r$ is the reduced section of the beams, and $G$ the shear modulus of the beams. For the shell, $E_s$ is the Young's modulus, $t$ is the thickness, $\nu$ is the Poisson's ratio, $G_{12}$ and $G_3$ are the shear modulus in the resp. tangent and normal directions.

Equivalent shell properties are thus:

$$t^2 = 12 \frac{EI}{EA} \qquad (1)$$

$$1 - \nu = 4 \frac{EA_d}{EA} \qquad (2)$$

When comparing bending over axial rigidity, one gets that the equivalent shell thickness is related to the ratio of bending and axial stiffness (Equation 1). When comparing axial and shear rigidity, one gets that the Poisson's ratio is related to the ratio of shear over axial stiffness (Equation 2).
To ensure that the grids are comparable, the cross-section of the beams (hollow circular tubes with the same cross-section on the same grid) is calculated so that the total mass of the structure and the inertia of the cross-section are the same for all the grids (even if they do not have the same cumulative length of beams).

### 3.3.2. Applied loads and boundary conditions

Symmetric gravity load (the mesh weight) is applied to mesh points: this load is the same for all grids, as the beam section is computed to keep the structure mass constant. The load distribution depends on the grid but varies relatively little as there is a criterion on the edge length. A vertical surface load of 0.37 kN/m², corresponding to a typical panel/cover weight, is applied on the whole grid, and distributed to mesh edges according to the spatial tributary area. These two loads are gathered in the "symmetrical loading" case.

Another surface load of 2 kN/m² is applied to a third of the structure. The six possible load distributions (by rotation of the loaded zone around the central point) are tested and the worst one is kept for the study. The load is applied to mesh points and proportional to the tributary area in projection (it corresponds to a snow load). This distribution in the plane is therefore uniform and identical for all meshes. The total applied load is also the same for all configurations. This load refers to the "asymmetrical loading" case.

Hinged supports are set at the six corners of all grids.

Mechanical analysis is performed with Karamba3D (Preisinger, 2013), a finite element analysis tool for parametric design.

### 3.3.3. Initialisation of the model

Some parameters are chosen at the first iteration and kept constant for the rest of the generation process (same structure mass and section inertia in all grids). The only variables of the generation are those changing the topology, i.e. the slot heights of the boundary of the minimal surface.

### 3.3.4. Evaluation of the grids

The grid geometry is thus given, and its properties can be evaluated. Among all the criteria that can be chosen, metrics are constructed (and calculated for each load case):

* the horizontal thrust divided by the vertical reaction, noted *R*, a dimensionless ratio, measured as the average of horizontal reaction forces
divided by the average of vertical reactions at supports. It can be a
design criterion for structures on top of existing buildings for example,

* the ratio between axial and bending strain energies, noted *E*, computed as the total axial deformation energies divided by the sum of bending energies in all elements. It characterises deformations and forces in the structure,

* the deflection $\delta$ which is a design factor for relatively low structures. It is taken as the maximal displacement at midpoints and endpoints of elements. It is not normalised as the structure mass and projection of the grid are the same for all structures (within a structural topology). In order to compare models obtained with the same topology, as done in Table 3, deflection is normalised by the vertical reaction $P$.

Since several criteria are involved, the Pareto front of the generated grids is evaluated. These three metrics make sense for structural design but are not an exhaustive list. They have been chosen to show how the topological generation method and the Pareto front can be used as design tools.

## 3.4. Results

Altogether, three structural typologies are tested: a gridshell with and without bracing, and an isotropic shell. More than a hundred grids are generated and are available on the website Design Explorer (Boutillier et al.), but one cannot prove that it constitutes an exhaustive scan of the space of grids achievable with this method. Nevertheless, this sample allows to see the diversity of possible designs and to compare grids between them.

### 3.4.1. Pareto Front

The Pareto front is calculated for the gridshell without bracing. Over the 115 tested grids, 23 of them belong to the Pareto front for the symmetrical loading. They are shown in Figure 6 and are grouped into topological families, identified by coloured boxes. Indeed, the generation method controls both the topology of the grid and the geometry of the members in the plane. Some topologies are also equivalent (with one rotation) and are grouped. Finally, ten different topologies are present on the Pareto front.

These grids are plotted in the objective space, a three-dimensional space whose axes are the reaction forces at the supports, the ratio of strain energies, and deflection. The grids belonging to the Pareto front for the symmetrical
<figure><img src="image_7.png" /><figcaption>Figure 6: Grids that belong to the Pareto front when evaluated for a symmetric loading. Different topologies appear, but also geometric variations of the same topology, which are framed with colours.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Projection of the grids without bracing, in the plane of the support reactions (abscissa) and deflection (ordinate) criteria. The grids belonging to the Pareto front are shown in colours. The colours correspond to the topological families identified in Figure 6, except for the black dots which represent unique topologies on the Pareto Front. One representative of each topology family of the Pareto front is displayed on the side.</figcaption></figure>

loading are represented with coloured disks among all other generated grids represented with grey circles in Figure 7. The results are represented as
a projection in the plane of two criteria (in this case the deflection on the
ordinate and the ratio of horizontal to vertical reactions on the abscissa). A
representative of each family is also displayed. This visualisation tool shows
how the grids are positioned in relation to each other. It also allows the
designer to make choices according to the criteria to be met. In projection
in the plane of the energy ratio and the deflection, one sees that these two
criteria are correlated (see Figure 8).

<figure><img src="image_9.png" /><figcaption>Figure 8: Projection of the grids without bracing, in the plane of the energy ratio *E* (abscissa) and deflection *δ* (ordinate) criteria. The grids belonging to the Pareto front are shown in colour. The colours correspond to the topological families identified in Figure 6, except for the black dots which represent unique topologies on the Pareto Front. One sees that the two criteria are correlated, which corresponds to our intuition.</figcaption></figure>

For the asymmetrical loading, 20 grids belong to the Pareto front and are
on the Pareto front for the symmetrical loading too. As the loads applied
are close to real loads, the dominant load remains symmetrical and tests
under asymmetrical loads do not show any major differences. Furthermore,
as the structure topologies are asymmetrical (except for the topology with
a central singularity), the symmetry of the loading has little influence on
the behaviour. Non-symmetrical structures are on the Pareto front, and
the symmetrical structure is not necessarily the best-performing one, under
either symmetrical or asymmetrical loading.
<table><caption>Table 3: Comparison of the performance of two grids, for three structural types (R for reactions, E for strain energy, and δ/P for deflection normalized by the load), under symmetrical loading.</caption><thead><tr><th rowspan="2"></th><th colspan="3">3-symmetrical topology</th><th colspan="3">asymmetrical topology</th></tr><tr><th>R</th><th>E</th><th>δ/P (mm/kN)</th><th>R</th><th>E</th><th>δ/P (mm/kN)</th></tr></thead><tbody><tr><td>Gridshell</td><td>1.84</td><td>4.70</td><td>81.4</td><td>2.13</td><td>5.16</td><td>139</td></tr><tr><td>Braced gridshell</td><td>1.79</td><td>1.44</td><td>59.8</td><td>2.11</td><td>1.29</td><td>95.5</td></tr><tr><td>Shell</td><td>1.83</td><td>0.47</td><td>42.6</td><td>2.19</td><td>0.59</td><td>163</td></tr></tbody></table>

### 3.4.2. *Comparison of the structure typologies*

To go more into detail, two topologies are picked up: one 3-symmetrical topology that is on the Pareto front, with one singularity in the centre, and another topology that is not on the Pareto front, with a singularity in the surface and two in corners. These topologies are shown in Figure 10 and the resulting 3D shapes are shown in Figure 9. The behaviour of these two shapes is then evaluated for three different structural typologies: an unbraced gridshell, a braced gridshell, and an isotropic shell (with equivalent properties given in Table 2). As the loads (in this case gravity loads) change according to the structural type, the deflection is normalised by the applied load in order to compare the types. Results are displayed in Table 3 for the symmetrical loading.

<figure><img src="image_10.png" /><figcaption>Figure 9: Perspective views of the grid whose performances are compared for the different structural typologies: a symmetric topology with a central singularity (left) and an asymmetric topology with one singularity in the surface and two at corners (right).</figcaption></figure>

As expected, adding bracing drastically improves the behaviour of the grid, in particular with regard to strain energies. The relative performance between the symmetrical and the asymmetrical topologies depends on the type of structure considered, and are not significantly different: with bracing, the asymmetrical topology works slightly better than the symmetrical topology for the energy criterion, whereas the opposite is observed when con-
sidering the deflection criterion. As with the Pareto front, similar conclusions
can be drawn for the study of the two topologies under asymmetric loading.

Furthermore, the shell model can be used as an evaluation of the geometry obtained with the grid topology. Stress lines can be computed on this shell, and compared to the grid that was used to create the geometry. Two examples are shown in Figure 10. The stress lines of the two funicular forms are completely different. However, the shape of the lines is close to the underlying grid topology. The deviations may be due to the added planarity objective, or to the method of determining these lines on imperfect meshes. This topic deserves to be explored in more detail but visually confirms that the selected grids are well chosen to transfer load toward the supports.

<figure><img src="image_11.png" /><figcaption>Figure 10: Stress lines under gravity loading computed on the symmetric grid (left) and on a grid with no symmetries and singularities located in the surface and on the boundary (right). Stresses are superimposed with the topological skeleton used for grid generation.</figcaption></figure>

4. Second case study: construction of a wooden gridshell

The topology generation method presented in Boutillier et al. (2024b) and
applied in this paper to a parametric study was also implemented on-site for
the construction of a 12-square-metre pavilion in wood. The main objective
of this project is to illustrate topology finding and form-finding for covering
a space. The technical solution has been developed in order for the pavilion
to be buildable by students with a simple CNC machine during a one-week
workshop. It could be improved for more realistic applications similar to
Robeller et al. (2022). The design and construction were guided by stringent
fabrication constraints, influenced by the tools and materials available on
site.
### 4.1. Design constraints

The main constraints are linked to manufacturing and have guided the design of the pavilion. The panels are laser-cut, which limits their size (in this case 600 mm x 900 mm). With the panels being assembled by hand, we wanted to limit the number of panels to around fifty. This therefore determined the scale of the project.

As the structure is small, it was also important to limit the number of special cases that occur at the supports (poles) and at the boundary in general.

At columns, the number of panels must be even to allow triangular faces to be converted into quadrangular faces and then only quadrangular faces to be manufactured.

The structure has a boundary segment contained in a vertical plane and supported by an existing building. There are two reasons for this: being supported by a building, the structure benefits from its horizontal stiffness and there are fewer point supports (that come with distorted quads) to carry the structure. The final structure has thus 4 supports, located at the corners of the contour shown in Figure 11, but a hexagonal contour was considered for the topological generation phase to allow a greater grid richness.

### 4.2. Topology and form preliminary studies

The idea here is to find a shape variety, which entails exploring the topological variety. The topological variety (number and position of singularities) is related to the boundary variety. For a four-height crenellated boundary, it is possible to have one singularity in the grid or two at the boundary. By adding two extra heights, we can have up to two singularities in the grid, or one singularity in the grid and one or two at the edge, or four at the edge (see Figure 11).

As in Section 3, the space of crenellated boundaries was explored and allowed the generation of topological skeletons and grids. However, in this study, a Pareto front was not used: given the many fabrication constraints, in order to limit the number of special cases and to show the topology (i.e. show the singularities), a skeleton with two central singularities was arbitrarily chosen.

### 4.3. Remeshing

Once the topology has been chosen, the height of the slots can be slightly varied to move the singularities while remaining in the same topological fam-
<figure><img src="image_12.png" /><figcaption>Figure 11: Selection of topological skeletons generated over a non-symmetric hexagonal boundary. The chosen topology is framed in red.</figcaption></figure>

This allows the singularities to be moved away from the boundaries to make room for a panel. The density of the mesh is chosen so as to have a suitable number of panels (around fifty), a good distribution of subdivisions, and an even number of panels at the poles. It is anticipated that the singularities will be remeshed, areas of low density are thus left close to the singularities. The dense mesh, discretising the smooth gradient and level set lines, is shown in Figure 12.1.

<figure><img src="image_13.png" /><figcaption>Figure 12: Remeshing procedure: 1. Topological skeleton is densified, 2. Valency 5 points are added around the singularities, with the introduction of the two red squares, 3. Diagonals are removed and the valency 8 singularities disappear, 4. Triangles are merged into quads at poles, 5. The mesh is smoothed to prevent flat T-junctions at poles.</figcaption></figure>

Before being lifted, the mesh is altered in plan around the singularities (to avoid panel distortion) and the poles (to eliminate the triangles and keep only quads over the whole structure, and therefore a unique construction method). Valency 8 singularities are exploded into four valency 5 singularities each. To do so, red squares are added (Figure 12.2), and diagonals are removed (Figure 12.3). Triangles are then removed all over the structure. This consists in removing diagonals and merging triangles at poles (Figure 12.4). This is part of the remeshing procedure proposed in Boutillier et al. (2024b). To
prevent flat T-junctions, that are mechanically weak - especially at poles i.e. close to the supports, a (tangential) smoothing is performed and the resulting mesh is shown in Figure 12.5. The final mesh is well suited for fabrication and mechanical behaviour. It has four poles at supports and one pole on the boundary along one edge. It has four inner singularities of valency 5 and twelve nodes of valency 3 around poles.

### 4.4. Form-finding

The grid is then lifted using the same method as in Section 3. The structure is optimised to get flat panels and being funicular: the planarity criterion is satisfied as the ratio (distance between diagonals / diameter of a circle approximating the four corners of a face) is below 1/200 (which is a value commonly used in design offices for glass panels, the reader is invited to read (Pietroni et al., 2015) for other criteria and value). After the lifting, a rounding of the mesh is introduced to distribute the curvature across the mesh. Indeed, curvature is stiffness: by modifying the shape, the structure moves away from the funicular shape of its own weight, but a priori, its behaviour is improved under the other loads. We thus proceed as follows: the dihedral angle at each edge, which relates to the discrete mean curvature of the mesh, is calculated and shows flat areas in the structure. Point loads are then arbitrarily applied to four mesh vertices and added to the optimisation in order to increase the curvature at the identified points, thereby improving the structure's stiffness. These loads do not reflect actual loading, but a targeted modification by the designer: in real conditions, particularly with lightweight structures, loads are not uniformly distributed. The final shape is shown in Figure 13: two valleys appear and join the tip to the supports and to the pole on the boundary. The dihedral angle is 13 degrees on average, ranging from 3 to 24 degrees.

### 4.5. Offset

Once the form has been found, it needs to be given thickness. The question of the direction of the offset is a recurring problem (Liu et al., 2006; Pottmann et al., 2007). Given the manufacturing constraints in this project (laser cutting of flat boards, bolted assembly with a bolt of constant length), the offset was optimised to minimise beam torsion and have a constant face offset and flat faces. To this end, the method proposed by Tellier et al. (2024) to explore the space of parallel transformations related to fabrication constraints was adapted to fit within a global optimisation problem where it is
<figure><img src="image_14.png" /></figure>

Figure 13: Shape obtained after the form-finding procedure. The structure is globally funicular and has flat panels. Regular arches join the supports. Two valleys appear in the structure and connect the supports and the pole on the boundary. Dimensions are given in metres.

combined with other conditions. An additional goal was added on the edge segment that would be supported on a building, ensuring its vertical offset.

<figure><img src="image_15.png" /></figure>

Figure 14: Angle deviation of the beams (angle between the beam axis and a normal to the mesh edge). The angle deviation is 8 degrees on average, ranging from 0.12 to 31 degrees.

<figure><img src="image_16.png" /></figure>

Figure 15: Angle deviation of the nodes (angle between the node axis and a normal to the mesh vertex). The angle deviation is 12 degrees on average, ranging from 1.6 to 34 degrees.

The deviation between the directions obtained and the mesh normals is calculated (aligning the beams and nodes with the normal maximises the structural height and therefore the bending stiffness):

* the direction of the beams (i.e. the normal to the edges) is compared
with the normal calculated as the average of the normals to the adjacent faces, weighted by the area of the faces (as illustrated in Figure 16). The angle deviation is 8 degrees on average, ranging from 0.12 to 31 degrees. The angle distribution is shown in Figure 14.

* a normal to the nodes, calculated as the sum of the normals to the faces of the star of the node, weighted by the area of the faces, is compared to the axis obtained for each node by joining the points of the original mesh and its offset (as illustrated in Figure 17). The angle deviation is 12 degrees on average, ranging from 1.6 to 34 degrees. The angle distribution is shown in Figure 15.

<figure><img src="image_17.png" /><figcaption>Figure 16: Deviation of the beam axis (blue line) from the normal to the mesh edge, computed as a weighted average of the faces normals (orange arrows)</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 17: Deviation of the node axis (blue line) from the normal to the mesh vertex, computed as a weighted average of the faces normals (orange arrows)</figcaption></figure>

Apart from the deviations that may appear at the boundary, the shell has ribs that are fairly aligned with the mesh normals, which suggests good structural behaviour. In the end, beam and face panels are almost flat (the criterion described in Section 4.4 is still below 1/200), which allows their fabrication from flat plywood boards.

## 4.6. Geometrical properties and mechanical behaviour

This shape enables the structure to be mainly in compression under self-weight, as illustrated in Figure 18 where arches in compression connect the supports. The average values of axial loads, out-of-plane moments, in-plane moments, and axial and bending strain energies are shown in Table 4 and show that axial loads are the dominant stresses. The grid is modelled with
<figure><img src="image_19.png" /><figcaption>Figure 18: Axial load under self weight [-1.3 kN; 0.34 kN].</figcaption></figure>

diagonals (that represent the panels) that can only take tension, and whose stresses are not shown in the picture.

Supports are finally added and extend the structure tangentially (in projection in the plane of the building for the two supports located against a wall), as shown in Figure 23.

The structure is modelled as a grid of braced beams to take account of the panels in the behaviour. The beams are connected via the boxes, resulting in a relatively flexible connection.

<figure><img src="image_20.png" /><figcaption>Figure 19: Out-of-plane moments under self weight [-9.8 Nm; 7.3 Nm]</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 20: In-plane moments under self-weight [-2.9 Nm; 1.4 Nm]</figcaption></figure>
<table><caption>Table 4: Mechanical performances of the grid under self-weight. The beam section is a wood rectangle of 70 mm by 55 mm.</caption><tr><td>Average axial load (absolute values)</td><td>0.17 kN</td></tr><tr><td>Average out-of-plane moment (absolute values)</td><td>1.1 Nm</td></tr><tr><td>Average in-plane moment (absolute values)</td><td>0.17 Nm</td></tr><tr><td>Total axial strain energy</td><td>0.017 Nm</td></tr><tr><td>Total bending strain energy</td><td>0.013 Nm</td></tr></table>

### 4.7. Construction

It was chosen to build and assemble wood beams, connected by wood boxes, for several reasons:

* we wanted to construct a covered structure and not only the structural skeleton, including the cover in the structural element is one solution.

* with laser cutting, angled cuts are not possible, which complicates the manufacture of nodes where the beams do not intersect at 90 degrees.

* the more geometry is included in the components to be assembled, the simpler the assembly (we build without formwork, with students who are inexperienced workers).

Ultimately, the structure is a wooden gridshell connected and braced by plywood boxes.

The composition of the unit cell (the box) is always the same and is illustrated in Figure 21. It is made of four panels of 5 mm plywood. Notches at both ends provide an (unbraced) frame for the box. Slots on the top and bottom fibres allow assembly with the top and bottom panels. The top panel and holed bottom panel brace and lock the geometry of the box. The angle between the top / bottom panels and the lateral panels gives curvature to the structure once these components are assembled (remember that in differential geometry, the mean curvature is proportional to the dihedral angle). Wooden beams, with a cross-section of 45 mm by 70 mm, are bolted. The assembly of boxes is shown in Figure 22. The whole structure was built in two days with 22 students.

All boxes are unique and there is only one way to assemble them. The finished structure is shown in Figure 23. At each support, a beam between
<figure><img src="image_22.png" /><figcaption>Figure 21: Exploded component: top and bottom panels dark blue, lateral panels with slots in light blue, and spacers in grey.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 22: Assembly of two components. The beams (panels + spacers) are assembled to form the frame, bolts are placed across the beams to join the boxes together, and the top and bottom panels are positioned to brace the structure.</figcaption></figure>

two boxes is extended and enclosed by two beams to make a composite column.

### 4.8. *Experimentation on the built structure*

This structure is not traditional: it is made of plywood boxes assembled with beams, but the nodes are not materialised. The technology chosen for the connections stems from a compromise between the tools available, ease of assembly by students and reduced costs. The beams are connected by the plywood panels, via slots distributed along the length of the beams. Rather
<figure><img src="image_24.png" /><figcaption>Figure 23: Photo of the pavilion built on the site of the Ecole des Ponts (by courtesy C. Boutemy)</figcaption></figure>

than faithfully modelling the structure (e.g. with plates and connectors along their edges), the question is what stiffness of the connections should be taken into account in the gridshell model used for the mechanical calculations, to represent the overall behaviour of the structure. The structure is therefore modelled as a braced gridshell, and the aim is to determine the equivalent stiffness of the connections. It is loaded with different sets of loads (6 loads cases are tested where sandbags of 7 kg are suspended to the structure nodes), and the displacements are measured. The whole instrumentation, measurements, and analysis of raw data is carried out by a team from ENSG. A scan of the structure is shown in Figure 24.

The structure is modelled in Grasshopper with wood beams, and 5 cm wide diagonals are considered (a portion of the top panel). Supports are pinned at wooden column ends. The end of the metal column (see Figure 23 on the right) is fixed. The last support against the wall is an elastic hinge. Elastic hinges are also introduced at every beam connection in the mesh. Hence two different spring stiffnesses are considered for the connections in the mesh and at the support. Their values in the 6 directions (i.e. a total of 12 parameters) can be adapted to align the modelled and measured displacements.

Figure 25 shows the modelled displacements (with the coloured squares) and the measured displacements (with the coloured disks) of the structure under the worst load case, where 20 sandbags are regularly distributed over the structure. The colour distribution indicates a similar behaviour and a
<figure><img src="image_25.png" /><figcaption>Figure 24: Picture of the structure during experimentation. The structure is loaded with a few bags and is fitted with black and white marks. A scanner is visible under the structure (ENSG picture).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure 25: Displacement comparison between the measures and the model. The coloured squares in the middle of the beam represent the modelled displacement. The coloured disks in the faces represent the measured displacement. The colour scale is the same for both symbols and is given in metres at the bottom of the figures. The amplitude of horizontal displacement is shown on the left. Vertical displacement is shown on the right (note that there is an uplift).</figcaption></figure>

satisfactory calibration of the node stiffness in the numerical model.

The displacements of Figure 25 are obtained with the following stiffness for the beam joints: $C_t = 480 \times (1 \ 1 \ 1)$ kN/m in translation and $C_r = 400 \times (1 \ 0.004 \ 1)$ kNm/rad in rotation for the mesh nodes, and $C_t = 300 \times (0.1 \ 0.1 \ 1)$ kN/m in translation and $C_r = 30 \times (1 \ 0.1 \ 1)$ kNm/rad in rotation at the support. All factors were normalised, and the behaviour of the structure depends on the translation in the axial and vertical
tical local directions ($C_{t,x}$ and $C_{t,z}$) and mainly on the out-of-plane bending
stiffness $C_{r,y}$. This study shows that the bending stiffness of the structure is
very poor. The structure had been optimised to be funicular, so the connec-
tions were not designed to withstand bending. However, given the equivalent
stiffnesses measured, and the discrepancy with the structural elements, stiffer
connections more typical of timber frames should be envisaged (considering
an average length of 52 cm, the bending stiffness of the beams is 32 kN.m,
i.e. 20 times greater than the stiffness of the connection). As the measured
and modelled behaviours globally match, this model can be regarded as an
equivalent gridshell and used to provide an overall representation of the built
structure.

A few comments can be made on the realisation of the structure and the
stiffness study.

* 12 parameters were considered, which is a lot, but still not enough for representing the joints, as every beam has a different number of slots.

* there are non-linearities in the model, particularly as it is made of wood. During loading, there may be sliding within connections and the structure is changing. The loading history is unknown. To get a better calibration, one should consider all relative displacements between all load cases, i.e. look at the displacement under load case *m* and load case *n* for every *n* ≠ *m* between 0 (self-weight) and 5.

* the diagonal stiffness is of the same order of magnitude as the translational stiffness because they are made of the same plywood and use the same connections with slots. Moreover, diagonals are deleted when working in compression, which introduces non-linearity in the model.

Overall, while the study warrants further exploration, it clearly shows
that the pavilion has poor bending stiffness as a result of the chosen fabrica-
tion process. The latter was quite simple to design and implement during a
student workshop but corresponds to the scale of a model rather than a real
wood structure.

**5. Conclusion and perspectives**

This study implements a method for generating a variety of grid topolo-
gies, all adapted to the support conditions, by varying parameters defined
along the edge. This variety is reflected in the Pareto front of the first case
study: a real choice is offered to the designer according to the chosen criteria.
The topology generation tool can also be used upstream of many form-finding
methods, for which the topology is given. Navigating the topological space,
or editing topologies, can require a certain amount of intuition. In addi-
tion, not all valencies and topologies are accessible (compared to Oval et al.
(2019b,a) for example). However, this constitutes a catalogue of grids, and a
first draft which can then be edited and re-meshed. The second case study,
involving the construction of a funicular wooden pavilion, demonstrates this
process. This practical case provides an opportunity to compare a theoret-
ical research tool with the inerrant constraints of a real project. The grid
is confronted with a lifting method that makes it possible to satisfy certain
manufacturing constraints, in this case, to have flat panels. It could be used
to meet other constraints, for example relating to the nodes: the grid does
indeed have beams that intersect at almost 90 degrees, but this property is
not currently being exploited. Finally, the underlying topological skeleton
can serve to generate other 2-colourable grids. We thus demonstrate the
value of the method for the practical design of gridshells and the feasibility
of such funicular panelled gridshells.

**6. Acknowledgements**

The authors gratefully acknowledge funding from the Labex MMCD pro-
vided by the national program Investments for the Future of the French
National Research Agency (ANR-11-LABX-022), from the I-Site FUTURE
(16-IDEX-0003) and Université Gustave Eiffel. The authors would also like
to thank Krittika Walia and Maryse Maugenest for their involvement in the
design and construction of the pavilion, civil engineering students from the
Ecole des Ponts for the assembly of the pavilion, and the ENSG team (es-
pecially Franck Tertre and Jean-Pierre Papelard) for the instrumentation of
the pavilion.

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