Analytical formfinding

Abstract

In this article, formfinding theories for different structures are considered. The beginning of analytical formfinding theories coincides with the analysis of the Olympic Roof in Munich (1972). Therefore, this article starts with the derivation of the force-density method with respect to cable net calculations as those procedures were used for the cable net calculation of the Munich stadium. Later on, the formfinding theories were adapted to textile membranes and foils, so the constitutive equations are extended by crimp- and shear-stiffness. The crimp-stiffness produces a correlation between warp- and weft-stiffness in textile membranes. The shear-stiffness causes shear-stresses because of angle-deformations. Pneumatic structures such as air-halls, air-domes, and ethylene tetrafluoroethylene cushions are widely used. The formfinding calculation of those pneumatically stressed membranes is also demonstrated. An important formfinding input value for those structures is the size of the volume itself. Therefore, an additional constraint in the volume formfinding strategy is the so-called volume equation, which is related to the internal pressure. Minimal surfaces such as soap films (and pneumatically stressed bubbles) have a very specific material behavior which is extensively described. The analytical formfinding theories were always developed due to physical modeling procedures. Mixed formfinding tools combine stiff elements with soft surface materials, as is often done in physical models when, for example, a stiff timber bar is used together with a soft nylon mesh.

Keywords

anticlastic cable net double curvature ethylene tetrafluoroethylene cushions force-density formfinding pneumatic structures soap bubbles soap film synclastic

Introduction

Lightweight structures such as cable nets or textile membranes cannot be designed as conventional structures: conventional design means, in this context, the architect fixes the real geometry on a drawing board. This is not possible with respect to pre-stressed lightweight structures because internal forces or stresses and the surface geometry are not independent of each other. Therefore, when the usual design procedure is not possible, a formfinding process is needed.

For the formfinding procedure, we have in general two choices: to create either physical models (e.g. using nylons, meshes, and soap films) or analytical models. We developed analytical computer models for the formfinding and analysis of the Olympic Stadium Roofs in Munich (1972).

In fact, these analytical methods were only partially used for the calculation of these roofs. At this time, a procedure was applied which can be seen as a mixture of physical and analytical formfinding. A 1:125 scale model was built and this model was measured by means of photogrammetry. The measured coordinates of all points were input as starting values in the computer program and the—by pre-stress balanced—geometry was calculated.

We should mention that nonlinear processes need starting values for the determination of the unknowns, whereas linear processes can be executed without having starting values for the unknowns.

This really complex way (building a model, measuring it photogrammetrically, and importing the measurements as starting values for a nonlinear cable net program) was later substituted by analytical formfinding tools. In the following section, we will develop the formfinding theory by adapting the nonlinear equations for the cable net calculations. Then, we extend the theory for textile membranes and pneumatic constructions (Figure 1).

Figure 1.

Model of the Munich Olympic Roof.

In the physical modeling, soap films play an important role. Therefore, we also present an analytical soap film calculation theory. Finally, we demonstrate “Mixed Formfinding,” where soft and stiff elements are combined.

Cable nets

As in general mechanics, the calculations of pre-stressed cable net structures are possible using three types of equations: equations for the equilibrium, equations for the geometrical compatibility, and equations based on the material law (constitutive equations).

We are going to describe the situation with a simple example. We have an internal free point, $i$ , which is connected by four cables— $a$ , $b$ , $c$ , and $d$ —to its fixed neighbor points $m$ , $n$ , $k$ , and $l$ . The point $i$ carries the external load $p$ (Figure 2).

Figure 2.

Single point with neighbors.

The equation of equilibrium reveals that the sum of all forces in the point $i$ is equal to the external load $p$ . We write this equation in its components $x$ , $y$ , $z$ .

The cable forces are named $s_{a}$ , $s_{b}$ , $s_{c}$ , and $s_{d}$ . We get the x-component of the cable force $s_{a}$ , for example, by multiplying the force $s_{a}$ with the direction cosine in x-direction $s_{a} \cos (a, x)$ . Hence, we receive

\begin{array}{l} s_{a} \cos (a, x) + s_{b} \cos (b, x) + \\ s_{c} \cos (c, x) + s_{d} \cos (d, x) = p_{x} \\ s_{a} \cos (a, y) + s_{b} \cos (b, y) + \\ s_{c} \cos (c, y) + s_{d} \cos (d, y) = p_{y} \\ s_{a} \cos (a, z) + s_{b} \cos (b, z) + \\ s_{c} \cos (c, z) + s_{d} \cos (d, z) = p_{z} \end{array}

(1)

Now, the direction cosine in x-direction is substituted by

\cos (a, x) = \frac{x_{m} - x_{i}}{l_{a}}

(2)

where $l_{a}$ (stressed length) is the length between point $i$ and point $m$ .

We get

\begin{array}{l} \frac{s_{a}}{l_{a}} (x_{m} - x_{i}) + \frac{s_{b}}{l_{b}} (x_{j} - x_{i}) + \\ \frac{s_{c}}{l_{c}} (x_{k} - x_{i}) + \frac{s_{d}}{l_{d}} (x_{l} - x_{i}) = p_{x} \\ \frac{s_{a}}{l_{a}} (y_{m} - y_{i}) + \frac{s_{b}}{l_{b}} (y_{j} - y_{i}) + \\ \frac{s_{c}}{l_{c}} (y_{k} - y_{i}) + \frac{s_{d}}{l_{d}} (y_{l} - y_{i}) = p_{y} \\ \frac{s_{a}}{l_{a}} (z_{m} - z_{i}) + \frac{s_{b}}{l_{b}} (z_{j} - z_{i}) + \\ \frac{s_{c}}{l_{c}} (z_{k} - z_{i}) + \frac{s_{d}}{l_{d}} (z_{l} - z_{i}) = p_{z} \end{array}

(3)

However, these equations for the equilibrium are not enough to solve the system. We have to add the equations for the geometrical compatibility

l_{a} = \sqrt{{(x_{m} - x_{i})}^{2} + {(y_{m} - y_{i})}^{2} + {(z_{m} - z_{i})}^{2}}

and the constitutive equations

s_{a} = E A \frac{l_{a}}{l_{a 0} - l_{a}}

(4)

analogue for $l_{b}$ , $l_{c}$ , and $l_{d}$ and $s_{b}$ , $s_{c}$ , and $s_{d}$ .

The above equation results from Hooke’s law $σ = E ϵ$ , where $σ = s / A$ and $ϵ = (l - l_{0}) / l_{0}$ . Here, $l_{0}$ is the so-called unstressed length. In order to clarify the situation, we assume the unstressed length $l_{0}$ and the stiffness $E A$ to be known. We can now count the unknowns: three coordinates $x_{i}$ , $y_{i}$ , and $z_{i}$ ; four stressed lengths $l_{a}$ , $l_{b}$ , $l_{c}$ , and $l_{d}$ ; and four cable forces $s_{a}$ , $s_{b}$ , $s_{c}$ , and $s_{d}$ . In total, we have 11 unknowns as well as 11 equations.

The nonlinear system can be solved if starting values are available. This extensive method was used for the calculation of the Olympic Roof in Munich (1972), where the physical measurement model delivered the starting values for all points.

Force-density method

The so-called “force-density method” was developed by Schek, Linkwitz, and Gründig using equations (2), (3), (4), (5) and (6).

The ratio between force $s$ and stressed length $l$ was defined as force-density $q$ and assumed to be known. For example, for the cable $a$

\frac{s_{a}}{l_{a}} = \frac{s_{a, x}}{l_{a, x}} = c o n s t = q_{a}

(5)

So, we receive from equation (3) the following system

\begin{array}{l} q_{a} (x_{m} - x_{i}) + q_{b} (x_{j} - x_{i}) + \\ q_{c} (x_{k} - x_{i}) + q_{d} (x_{l} - x_{i}) = p_{x} \\ q_{a} (y_{m} - y_{i}) + q_{b} (y_{j} - y_{i}) + \\ q_{c} (y_{k} - y_{i}) + q_{d} (y_{l} - y_{i}) = p_{y} \\ q_{a} (z_{m} - z_{i}) + q_{b} (z_{j} - z_{i}) + \\ q_{c} (z_{k} - z_{i}) + q_{d} (z_{l} - z_{i}) = p_{z} \end{array}

(6)

Equation (6) shows three linear equations for the three unknowns $x_{i}$ , $y_{i}$ , and $z_{i}$ . It means the solution of the system is the pre-stressed geometry of the net.

After having calculated the coordinates, the stressed lengths can be calculated by Pythagoras’ theorem followed by the forces, for example, in the cable a by

s_{a} = q_{a} l_{a}

(7)

The force-density approach is completely defined by the last three equations (6).

The only difficulty is in getting reasonable values for the force-densities which are already solved in our software package.

As we need it later, we describe the theory more generally and introduce as potential $π$ the total energy.⁷ The derivations of this potential $π$ to the unknowns, under consideration of the force-densities, lead to the following equations^1,2

\begin{array}{l} Π = \frac{1}{2} v^{t} P v - p_{x}^{t} (x - x_{0}) - \\ p_{y}^{t} (y - y_{0}) - p_{z}^{t} (z - z_{0}) \Rightarrow s t a t \end{array}

(8)

\begin{matrix} \sum_{i = 1}^{m} (x_{j} - x_{k}) q_{i} - p_{x} & = 0 \\ \sum_{i = 1}^{m} (y_{j} - y_{k}) q_{i} - p_{y} & = 0 \\ \sum_{i = 1}^{m} (z_{j} - z_{k}) q_{i} - p_{z} & = 0 \end{matrix}

(9)

and the diagonal matrix P with the elements q_i

\begin{matrix} v_{x i} = x_{j} - x_{k} \\ v_{y i} = y_{j} - y_{k} \\ v_{z i} = z_{j} - z_{k} \end{matrix}

(10)

Membranes and foils

The extension of the force-density method to membranes and foils seemed to pose a challenge since the theory was initially developed for cable nets. In the case of pre-stressed membranes, we can see the similarity between the warp and weft directions of the fabric and an orthogonal cable net. These warp and weft lines can also be used in the formfinding procedure. The only difference is the calculation of the force-densities itself. For membranes, the desired pre-stress leads to the specific force-densities in the material lines.

The statics is more complex as the constant stiffness values from the cable net calculation must be substituted by width-dependent stiffness values for the membranes.

First, the adaption for the textile membranes was made: here, we assumed that Young’s moduli exist for warp and weft directions (equation (11)). Then, the material law was extended for transverse extension effects. The so-called crimp-stiffness was introduced; this means that the warp-stress depends not only on the warp-strain but also on the weft-strain, and vice versa (equation (12)). Finally, the shear-stiffness completes the constitutive equations; for the shear-stress, an un-deformed angle $α_{0}$ was introduced; the difference $Δ α$ between the actual angle $α$ and $α_{0}$ leads, by using the third line of equation (13), to the shear-stress $τ$ (Figure 3).

Figure 3.

Angle between warp and weft directions.

Orthotropic material

σ = [\begin{matrix} σ_{u} \\ σ_{v} \end{matrix}] [\begin{matrix} m_{11} & 0 \\ 0 & m_{22} \end{matrix}] [\begin{matrix} ε_{u} \\ ε_{v} \end{matrix}] = M ε

(11)

Orthotropic material with crimp

σ = [\begin{matrix} σ_{u} \\ σ_{v} \end{matrix}] [\begin{matrix} m_{11} & m_{12} \\ s y m . & m_{22} \end{matrix}] [\begin{matrix} ε_{u} \\ ε_{v} \end{matrix}] = M ε

(12)

Anisotropic with crimp- and shear-stiffness

σ = [\begin{matrix} σ_{u} \\ σ_{v} \\ τ \end{matrix}] = [\begin{matrix} m_{11} & m_{12} & 0 \\ m_{22} & 0 \\ s y m . & m_{33} \end{matrix}] [\begin{matrix} ε_{u} \\ ε_{v} \\ Δ α \end{matrix}] = M ε

(13)

In the special case of an isotropic material, equation (13) becomes

Isotropic material

[\begin{matrix} σ_{u} \\ σ_{v} \\ τ \end{matrix}] = \frac{E}{1 - υ^{2}} [\begin{matrix} 1 & υ & 0 \\ 1 & 0 \\ s y m . & (1 - υ) / 2 \end{matrix}] [\begin{matrix} ε_{u} \\ ε_{v} \\ Δ α \end{matrix}]

(14)

Pneumatic structures

Pre-stressed membrane structures can be of two different types:

(a) Mechanically pre-stressed;

(b) Pneumatically pre-stressed.

The surface of mechanically pre-stressed membranes is anticlastic doubly curved; the pneumatically ones are synclastic. (Here, pressure differences are used to introduce the pre-stress.) In case (a), an initial stiffness exists; the cutting point in Figure 4 can carry downward forces (snow) as well as upward forces (wind suction). The pneumatically stressed surface (Figure 5) carries snow-loads by reducing the membrane stress and increasing the internal pressure, while we find the opposite situation for wind suction.

Figure 4.

Mechanically pre-stressed.

Figure 5.

Pneumatically pre-stressed.

The total energy⁶ can be written as

\begin{array}{l} Π = \frac{1}{2} v^{t} P v - p_{x}^{t} (x - x_{0}) - p_{y}^{t} (y - y_{0}) - \\ p_{z}^{t} (z - z_{0}) - p_{i} (V - V_{0}) \Rightarrow s t a t \end{array}

(15)

The derivation of the potential $π$ to the unknowns leads to the following equations

\begin{matrix} \frac{\partial Π}{\partial x} = & \sum_{i = 1}^{m} (x_{j} - x_{k}) q_{i} - p_{x} - p_{i} \frac{\partial V}{\partial x} & = 0 \\ \frac{\partial Π}{\partial y} = & \sum_{i = 1}^{m} (y_{j} - y_{k}) q_{i} - p_{y} - p_{i} \frac{\partial V}{\partial y} & = 0 \\ \frac{\partial Π}{\partial z} = & \sum_{i = 1}^{m} (z_{j} - z_{k}) q_{i} - p_{z} - p_{i} \frac{\partial V}{\partial z} & = 0 \\ \frac{\partial Π}{\partial p_{i}} = & V - V_{0} & = 0 \end{matrix}

(16)

In system (16), the internal pressure $p_{i}$ can be seen as so-called Lagrange multiplier. The fourth column in equation (16) shows that our boundary condition $V - V_{0}$ is obtained by the derivation of energy to this Lagrange multiplier. The vector $[\begin{matrix} \frac{\partial V}{\partial x_{i}} & \frac{\partial V}{\partial y_{i}} & \frac{\partial V}{\partial z_{i}} \end{matrix}]$ describes the normal direction in the point $P_{i} (x_{i}, y_{i}, z_{i})$ , and the vector size is equal to the loaded area. By a set of given force-densities for all elements as well as a given volume $V_{0}$ , we end up with a pre-stressed, and balanced, pneumatic system with an internal pressure $p_{i}$ .

The force-densities $q$ and the internal pressure $p_{i}$ are not independent of each other.

In this article, formfinding theories are generally shown. Here, we want to put the focus on the situation for pneumatics. In our theory, three cases are possible:

Given internal pressure p (snow);

Given volume V (water);

Given product p × V (wind).

These possible cases could be, for example, (a) an air-hall under snow-loading (a specific internal pressure is set to resist the snow-loads), (b) a membrane filled with an incompressible fluid (water-bag), and (c) a pneumatic cushion loaded by a fast wind-gust; here, the gas law (p × V = const) is valid.

In order to illustrate cases (a) and (c), we introduce an ethylene tetrafluoroethylene (ETFE) cushion with a fixed rectangle (2 m × 4 m) as boundary and a volume $V = 4.000 m^{3}$ and an internal pressure $p_{i} = 0.300 k N m^{- 2} .$ The maximum membrane stress is $σ_{m a x} = 0.35 k N m^{- 1}$ (Figure 6).

Figure 6.

Pneumatic ETFE-cushion.

A 200-µm ETFE foil is an isotropic material with $E = 180.0 k N m^{- 1}$ and $ν = 1 / 3$

\begin{array}{l} \frac{180.0}{1 - {(\frac{1}{3})}^{2}} [\begin{matrix} 1 & 1 / 3 & 0 \\ 1 & 0 \\ s y m . & (1 - \frac{1}{3}) / 2 \end{matrix}] = \\ [\begin{matrix} 202.5 & 67.5 & 0 \\ 202.5 & 0 \\ s y m . & 67.5 \end{matrix}] \end{array}

A wind suction load of $0.5 k N m^{- 2}$ is applied. Using (c), we receive an internal pressure $p_{i} = - 0.077 k N m^{- 2}$ and a volume $V = 4.015 m^{3}$ . The stresses remain moderate with $σ_{m a x} = 0.50 k N m^{- 1}$ .

In our opinion, it is wrong to use (a) for fast wind-gusts, but we know it is often done. This leads to very high stresses; in our case, we receive $σ_{m a x} = 1.0 k N m^{- 1}$ .

Minimal surfaces

Soap films (Figure 7) play an important role in physical modeling. In fact, Frei Otto has built a soap film machine. To use, you create the physical boundary conditions by arcs, wires, and so on; submerge the form into the soap sud; and take photos of the soap film surface. Because it is an isotropic material, the “soap film” always generates a so-called “minimal surface.” The stress situation is identical in all points of the surface, and when we transfer this model into reality, we obtain an optimal stress situation. This optimization is what made the soap film material for analytical formfinding theories so useful. Today, we can use this “soap film material” for mechanically and pneumatically stressed structures. In the following sections, we demonstrate how those minimal surfaces can be used in formfinding strategies, first for films and then for bubbles.

Figure 7.

Soap film model.

Soap films

Minimal surfaces are the smallest areas in a given closed boundary. Here, we describe minimal surfaces by nets which are discretized in $n$ triangles and $m$ points. The total area of the surface (the total potential) is the sum of the areas of all single triangles (single potential). The potential is given by

π_{N} = \sum_{i = 1}^{n} π_{Δ i} \to \min

(17)

We receive the equation system if we differentiate the potential $π_{N}$ to the unknown coordinates $x_{N}^{t} = [\begin{matrix} \begin{matrix} x_{1} & y_{1} & z_{1} \end{matrix} & \dots & \begin{matrix} x_{m} & y_{m} & z_{m} \end{matrix} \end{matrix}]$ of the total net

\frac{\partial Π}{\partial x_{N}} = g (x) = 0

(18)

For solving nonlinear problems, we use Newton’s method

g (x) = g (x_{0}) + \frac{\partial g (x)}{\partial x} Δ x = 0

(19)

where we need starting values $x_{0}$ for this iterative process.

Hereafter, we describe only one triangle (Figure 8), representing all triangles of the net. The triangle in three dimensions is given by three corner points, for example, $P_{1} (x_{1}, y_{1}, z_{1})$ , $P_{2} (x_{2}, y_{2}, z_{2})$ , and $P_{3} (x_{3}, y_{3}, z_{3})$ .

Figure 8.

Single triangle.

The lengths of the triangle $l_{1}$ , $l_{2}$ , and $l_{3}$ are calculated with Pythagorean theorem. With the vector of coordinate differences

u_{12 (3, 1)} = [\begin{matrix} x_{1} - x_{2} \\ y_{1} - y_{2} \\ z_{1} - z_{2} \end{matrix}]

We receive for the first length

l_{1} = ∥ u_{12}^{t} u_{12} ∥

(20)

With Heron’s formula, we are able to calculate the area of a single triangle

F_{Δ} = \sqrt{s (s - l_{1}) (s - l_{2}) (s - l_{3})}

(21)

where $s = (l_{1} + l_{2} + l_{3}) / 2$ .

If we insert the variable $s$ and summarize the multiplied terms, we receive

16 F_{Δ}^{2} = - l_{1}^{4} - l_{2}^{4} - l_{3}^{4} + 2 l_{1}^{2} l_{2}^{2} + 2 l_{2}^{2} l_{3}^{2} + 2 l_{3}^{2} l_{1}^{2}

(22)

This leads to a quadratic form, which we can write using matrix notation

\begin{array}{l} 16 F_{Δ}^{2} = [\begin{matrix} l_{1}^{2} & l_{2}^{2} & l_{3}^{2} \end{matrix}] \\ [\begin{matrix} - 1 & 1 & 1 \\ - 1 & 1 \\ s y m . & - 1 \end{matrix}] [\begin{matrix} l_{1}^{2} \\ l_{2}^{2} \\ l_{3}^{2} \end{matrix}] = l^{t} L M L l \end{array}

(23)

where $l_{(1, 3)}^{t} = [\begin{matrix} l_{1} & l_{2} & l_{3} \end{matrix}]$ and $L_{(3, 3)} = [\begin{matrix} l_{1} & 0 & 0 \\ 0 & l_{2} & 0 \\ 0 & 0 & l_{3} \end{matrix}]$

For the differentiation of the lengths to the unknown coordinates

x_{(1, 9)}^{t} = [\begin{matrix} x_{1}^{t} & x_{2}^{t} & x_{3}^{t} \end{matrix}]

with

x_{1 (1, 3)}^{t} = [\begin{matrix} x_{1} & y_{1} & z_{1} \end{matrix}] and so on

With the identity matrix $E$ of the dimension 3

E_{(3, 3)} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

We can define the branch-node-matrix $C$ for a triangle

C_{(9, 9)} = [\begin{matrix} E & - E & 0 \\ 0 & E & - E \\ - E & 0 & E \end{matrix}]

For the matrix of coordinate differences $U$ , we receive

U_{(9, 3)} = [\begin{matrix} u_{12} & 0 & 0 \\ 0 & u_{23} & 0 \\ 0 & 0 & u_{31} \end{matrix}]

So, we are able to write the differentiation of the three lengths to the nine coordinates in a short form with the notation of matrices as

{(\frac{\partial l}{\partial x})}_{(3, 9)} = L^{- 1} U^{t} C

As in adjustment theory, where the quadratic weighted sum of the residuals $v$ are minimized, we try to minimize the area of one triangle, and therefore, we define with $f_{Δ}$ the square root of the area of the triangle $F_{Δ}$

Π_{Δ} = v^{t} P v = f_{Δ}^{t} f_{Δ} = F_{Δ}

(24)

The differentiation to the unknown coordinates $x$ leads to

\frac{\partial Π_{Δ}}{\partial x} = \frac{\partial F_{Δ}}{\partial x} = \frac{\partial F_{Δ}}{\partial l} \frac{\partial l}{\partial x}

(25)

The single partial derivatives with the notation of matrices below are

\frac{\partial F_{Δ}}{\partial l} = \frac{1}{8 F_{Δ}} l^{t} L M L

and

\frac{\partial l}{\partial x} = L^{- 1} U^{t} C

This leads to our equation system which we have to solve

g {(x)}_{(9, 1)} = C^{t} U \frac{1}{8 F_{Δ}} M L l = C^{t} U q = C^{t} Q C x = 0

(26)

where we summarize the terms which do not belong to the branch node matrix $C$ C and the matrix of the coordinate differences $U$ to a vector

q_{(3, 1)} : = \frac{1}{8 F_{Δ}} M L l = \frac{1}{8 F_{Δ}} [\begin{array}{l} - l_{1}^{2} + l_{2}^{2} + l_{3}^{2} \\ l_{1}^{2} - l_{2}^{2} + l_{3}^{2} \\ l_{1}^{2} + l_{2}^{2} - l_{3}^{2} \end{array}] = [\begin{matrix} q_{1} \\ q_{2} \\ q_{3} \end{matrix}]

(27)

We define this vector as the vector of force-densities.

It was shown in Singer⁵ that the derivations of the area to the coordinates of the corner points are vectors perpendicular to the opposite side with half of the length (Figure 9).

Figure 9.

Derivations of the area to the coordinates of the corner points from one triangle.

For example, $s_{1}^{t} u_{23} = 0$ and $‖ s_{1} ‖ = (1 / 2) l_{2}$ .

Now we have all explanations for one triangle and we can go back to the total net. For solving equation system (26) by Newton’s method, we need starting values $x_{N 0}$

{(C^{t} Q C)}_{N} Δ x_{N} = - {(C^{t} Q C)}_{N} x_{N 0}

(28)

After each iteration, we have to improve the coordinates, and using the new coordinates $x_{i + 1} = x_{i} + Δ x$ , we have to update the force-densities and repeat the calculation.

Case study A. The net is discretized by $m = 2880$ points and $n = 5472$ triangles, where the points of the upper and lower rings are fixed. The force-densities of all links in the left computer model are constant. For example, with $q_{i} = 1$ , we receive the figure of equilibrium in a linear step and the area of the surface is $F = 674.2 m^{2}$ (Figure 10).

Figure 10.

Formfinding result–minimal surface.

For the right computer model, we used the same net as in the left picture. After some iterations, the described algorithm leads to this figure of equilibrium. The result is a minimal surface and the area of the surface is $599.0 m^{2}$ .

Soap bubbles

At first, we have to describe minimal surfaces around a closed chamber. Every single triangle of the net creates with the origin a tetrahedron. If the chamber is closed, the sum of the tetrahedrons gives the correct volume

V (x_{N}) = \sum_{i = 1}^{m} V_{Δ i} (x)

(29)

The potential with the unit $(m^{2})$ changes

Π_{N} = \sum_{i = 1}^{n} F_{Δ i} - k (V (x_{N}) - V_{N}) \to s t a t

(30)

The variable $k$ is named as Lagrange multiplier. Between the unknown coordinates and one chamber, we have exactly one condition

\begin{matrix} \frac{\partial Π_{N}}{\partial x_{N}} = & {(\frac{\partial Π_{N}}{\partial x_{N}})}^{t} - {(\frac{\partial V}{\partial x_{N}})}^{t} k = 0 \\ \frac{\partial Π_{N}}{\partial k} = & - (V (x_{N}) - V_{N}) = 0 \end{matrix}

(31)

And we receive for one triangle

\begin{matrix} g_{1} (x, k) = & C^{t} U \frac{1}{8 F_{Δ}} M L l - b k \\ g_{2} (x, k) = & - (V (x_{N}) - V_{N}) \end{matrix}

The differentiation with $H_{(9, 9)} = \partial b k / \partial x$ leads to

\begin{matrix} \frac{\partial g_{1} (x, k)}{\partial x} = & C^{t} Q C - H \\ \frac{\partial g_{2} (x, k)}{\partial x} = & - b^{t} \end{matrix}

and

\begin{matrix} \frac{\partial g_{1} (x, k)}{\partial k} = & - b \\ \frac{\partial g_{2} (x, k)}{\partial k} = & 0 \end{matrix}

to following equation system

\begin{array}{l} [\begin{matrix} {(C^{t} Q C - H)}_{N} & - b_{N} \\ s y m . & 0 \end{matrix}] [\begin{matrix} Δ x_{N} \\ Δ k \end{matrix}] = \\ [\begin{matrix} - {(C^{t} Q C)}_{N} x_{N 0} + b_{N} k_{0} \\ V_{0} (x_{N}) - V_{N} \end{matrix}] \end{array}

(32)

From potential (30), we know that the Lagrange multiplier $k$ has the unit $(m^{- 1})$ and its geometrical expression is known as curvature.

Case study B. At first, we create six meshes on a cuboid with the length $a = 3 m$ , width $b = 2 m$ , and height $h = 1 m$ . The volume of the cuboid is, therefore, $V = a b c = 6 m^{3}$ . The net is discretized by $m = 786$ points and $n = 1568$ triangles (Figure 11).

Figure 11.

Mesh on a cuboid–minimal surface.

In the next step, we set all points free and ask the algorithm, which we described below, to search the minimal surface under this given volume. The unknowns are the $x,$ $y$ , $z$ coordinates of each point and the Lagrange multiplier $k$ K for one chamber—so the number of unknowns is $3 \times 786 + 1 = 2359$ . The starting values $x_{0}$ are the coordinates of the net on the surface of the cuboid. As a result, we receive, after some iterations, a figure of equilibrium corresponding to a sphere and the Lagrange multiplier comes to be $1.7751 m^{- 1}$ . From the curvature $k$ , we find the radius $r = 2 / k = 1.1267 m$ . The theoretical formula of a sphere $V_{m a t h} = (4 / 3) π r^{3}$ leads, with our given volume $V_{m a t h} = 6 m^{3}$ , to the radius $r_{m a t h} = 1.1272 m$ .

From this pure geometrical approach, we described minimal surfaces and minimal surfaces under conditions; we have to introduce the stress of the surface $σ (N m^{- 1})$ as a constant factor. The potential which we have to minimize now has the unit $(N m)$

π_{N} = \sum_{i = 1}^{n} σ F_{Δ i} - k (V (x_{N}) - V_{N}) \to s t a t

Then, the Lagrange multiplier gets the unit $(N m^{- 2})$ and represents exactly the inner pressure $p$ .

Between the stress on the surface $σ$ , the inner pressure $p$ , and the radius $r$ , we know from a sphere the following relation

2 σ = p r

(33)

Now we have a look at a soap bubble with two chambers and we receive the relation

\frac{p_{1}}{p_{2}} = \frac{2 σ / r_{1}}{2 σ / r_{2}} = \frac{r_{2}}{r_{1}}

(34)

If the membrane between the two spheres is in equilibrium, its inner pressure $p_{3}$ is the difference of both the pressures

p_{3} = \frac{2 σ}{r_{3}} = 2 σ (\frac{1}{r_{2}} - \frac{1}{r_{1}}) = 2 σ (\frac{r_{1} - r_{2}}{r_{1} r_{2}})

(35)

This leads to the fact that the membrane between the spheres must have the curvature

r_{3} = \frac{r_{1} - r_{2}}{r_{1} r_{2}}

(36)

With both radius $r_{1}$ and $r_{2}$ , and the distance $d$ between the center points of both spheres, we are able to determine the angle

γ = \arccos \frac{r_{1}^{2} + r_{1}^{2} - d^{2}}{2 r_{1} r_{2}}

(37)

In our next example C, we start again with our cuboid below and insert a dividing wall. We define two chambers with the volume $V_{1} = 4 m^{3}$ and $V_{2} = 2 m^{3}$ (Figure 12).

Figure 12.

Meshes created on a cuboid–Minimal surface under two conditions.

As a result, our algorithm calculates the inner pressures $p_{1} = 1.9639 N m^{- 2}$ and $p_{2} = 2.3596 N m^{- 2}$ . Hence, we get with the stress of the surface $σ = 1 N m^{- 1}$ the radius $r_{1} = 2 σ / p_{1} = 1.0184 m$ and $r_{2} = 2 σ / p_{2} = 0.8476 m$ .

From our computer model, we can determine between the two midpoints $M_{1}$ and $M_{2}$ the distance $d = 0.9467 m .$ This leads to the angle $γ = 60.15 °$ and shows us the validity of our calculation (Figure 13).

Figure 13.

Computer model–CAD construction.

The angle between the two soap bubbles is always 120°.¹

Mixed formfinding

In a “mixed form-finding-procedure,” soft (force-density controlled) elements are mixed with stiff (elastically controlled) elements. This method simulates the physical modeling, when, for example, stiff timber elements as struts are connected to a soft mesh (nylon).

From the theoretical point of view, we must say that the linear pure formfinding theory is combined with the nonlinear static analysis calculation. As this part is nonlinear, the whole procedure for mixed formfinding is nonlinear. The “linear” elements are steered by their force-densities and the nonlinear ones by their unstressed lengths and stiffness values. In Figure 14, we can see an example. All struts and suspension cables, two fish columns, and the top of the cones are stiff, and the membrane is soft in the formfinding procedure.

Figure 14.

Mixed formfinding computer model.

Conclusion

In this article, it is shown that today we have for all kind of structures formfinding theories. The formfinding result is the precondition of statics. So, we can simulate structures with respect to their geometry and load carrying behavior by computer modeling.

The calculations become more and more realistic by further development of theories and computer methods. The validity of our procedures is shown with simple examples from nature.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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