Hot Wire Cuttings for the Building Industries

David Brander, Andreas Bærentzen, Anton Evgrafov, Jens Gravesen, Steen Markvorsen, Toke Nørbjerg∗, Peter Nørtoft, and Kasper Steenstrup∗

∗Ph.D. students, i.e., doing the work.

Department of Applied Mathematics and Computer Science Technical University of Denmark

MaDiFa Weierstrass Institute, Berlin, 2014 BladeRunner Project BladeRunner Project 2013–2016

Three year project 2013–2016 supported with 1.5 million EURO by The Danish National Advanced Technology Foundation

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 2 / 42 BladeRunner Project Partners:

I Odico: Project manager HotWire and HotBlade technology

I CONFAC: Concrete elements

I 3XN: Architectural design

I Danish Technological Institute: Robot technology

I The Technical University of Denmark: DTU Compute: Surfaces and rationalization (2 Ph.D. students) DTU Mechanical Engineering: Thermal and mechanical modeling (1 Ph.D. student)

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 3 / 42 Outline

1 BladeRunner Project

2 Outline

3 Is the building industry an industry? Architectural examples The present procedure Examples

4 Concepts and ideas

5 Odico I : Hot Wire Technology

6 Odico II : Hot Blade Technology

7 The mathematics

8 Conclusion Is the building industry an industry? Is the building industry an industry?

I Labor takes ∼50% of the cost in the Danish building industry

I The corresponding number is 10–20% for other industries in Denmark

I Architecture with non-trivial geometry is even more labor intensive

The architecture of “standard” buildings is mostly very boring.

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 5 / 42 Is the building industry an industry? Architectural examples 3XN architects

Lighthouse, Aarhus, Denmark.

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 6 / 42 Is the building industry an industry? Architectural examples 3XN architects

Blue Planet, Copenhagen, Denmark. Design for mixed-use high rise building.

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 7 / 42 Is the building industry an industry? The present procedure Handmade construction of a mold (wooden formwork).

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 8 / 42 Is the building industry an industry? Examples CONFAC: Handmade unique plywood-mold for a balcony

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 9 / 42 Is the building industry an industry? Examples CONFAC: EPS-form work directly on casting-bed (table)

The price is reduced to the half.

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 10 / 42 Is the building industry an industry? Examples Snøhetta: San Fransisco Museum of Modern Art

The EPS-mold for the facade requires one year of non stop NC milling. → estimated 2–3 weeks of HotWire cutting

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 11 / 42 Concepts and ideas Concepts and ideas : An example

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 12 / 42 Concepts and ideas Concepts and ideas : An example

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 13 / 42 Concepts and ideas Concepts and ideas : An example

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 14 / 42 Concepts and ideas Concepts and ideas : An example

It is : 1. Ad hoc

2. Imprecise

3. Messy, dirty

4. Timeconsuming

5. and very Costly

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 15 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 16 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 17 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 18 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

It is : 1. Automatic

2. Precise

3. Clean (?)

4. Fast

5. and relatively Cheap

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 19 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 20 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 21 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 22 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 23 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 24 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 25 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 26 / 42 Odico I : Hot Wire Technology RoboticHotWireCutting

But it can only produce surfaces with non-positive Gaussian curvature:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 27 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Positive Gaussian curvature can be obtained by sweeping ELASTIC CURVES:

γ(s) = (−2k cn(s) , s − 2 E(am(s, k), k)) (1)

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 28 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Positive Gaussian curvature can be obtained by sweeping ELASTIC CURVES:

γ(s) = (−2k cn(s) , s − 2 E(am(s, k), k)) (2)

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 29 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Positive Gaussian curvature can be obtained by sweeping ELASTIC CURVES:

 2  s   2  2  s   γ(s) = − dn , s − 1 − E(am , k , k (3) k k k2 k k

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 30 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Positive Gaussian curvature can be obtained by sweeping ELASTIC CURVES:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 31 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Positive Gaussian curvature can be obtained by sweeping ELASTIC CURVES:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 32 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Positive Gaussian curvature can be obtained by sweeping ELASTIC CURVES:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 33 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Positive Gaussian curvature can be obtained by sweeping ELASTIC CURVES:

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 34 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 35 / 42 Odico II : Hot Blade Technology RoboticFlexibleBladeCutting a´la Euler

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 36 / 42 Odico II : Hot Blade Technology The HotWire and HotBlade technology

I Cutting with a wire → ruled surface (or is it?)

I Cutting with an elastic blade → surface swept by elastica (or is it?)

The “or is it” question will be answered by DTU Mechanical Engineering and by experiments.

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 37 / 42 The mathematics The mathematical problem: Surface rationalization

I Given a CAD model of the architecture or more precisely a spline surface modeling some (“interesting/expensive”) piece of the building.

I Approximate it by a collection of surfaces that can be produced by the HotWire or the HotBlade technology.

I That is, approximate it by a collection of ruled surface and surfaces swept by elastica.

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 38 / 42 The mathematics Surfaces swept by elastica

I Not studied before (to our knowledge)

I Can produce surfaces with arbitrary curvature 2 I Can be pieced together with C continuity

I An elasticum can be expressed in closed form using elliptic functions

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 39 / 42 The mathematics

Approximation by lines. Approximation by elastica.

Approximation. One sided approximation.

The one sided approximation can be the starting point for NC-milling. Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 40 / 42 The mathematics Other problems

I Smoothness conditions/tolerances between patches

I Distance measures

I Segmentation adapted to the block structure

I Production constraints

I Elastic and thermal deformations

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 41 / 42 Conclusion Conclusion

I There is a need for industrialization of the building industry

I The BladeRunner project is a contribution to this

I Mathematics, including differential geometry, has a key role to play in this endeavor

Thank you for your attention!

Steen Markvorsen (DTU Compute) Hot Wire Cuttings for the Building Industries MaDiFa 42 / 42