Mathematics Of 3D Printing

Alexander Hulpke Department of Mathematics Colorado State University

Fort Collins, CO, 80523, USA http://www.math.colostate.edu/~hulpke

3D Printing For Mathematics

Feb/6/15

Thank YouCollege of Natural Sciences

What Is 3D Printing• Additive (not subtractive)

manufacturing.

• A hot glue gun on a robot arm.

• Build object in thin slices (layers).

• Material: thermoplastic (PLA, ABS, PET). Temperature 215-270°C

What Is 3D Printing• Additive (not subtractive)

manufacturing.

• A hot glue gun on a robot arm.

• Build object in thin slices (layers).

• Material: thermoplastic (PLA, ABS, PET). Temperature 215-270°C

What Is 3D Printing• Additive (not subtractive)

manufacturing.

• A hot glue gun on a robot arm.

• Build object in thin slices (layers).

• Material: thermoplastic (PLA, ABS, PET). Temperature 215-270°C

What Is 3D Printing• Additive (not subtractive)

manufacturing.

• A hot glue gun on a robot arm.

• Build object in thin slices (layers).

• Material: thermoplastic (PLA, ABS, PET). Temperature 215-270°C

Other Technologies

• Other meltable material.

• Sliced Paper.

• Sintering (Sugar, Plastic, Metal Powder)

• Photosensitive liquid (Stereolithography).

Other Technologies

• Other meltable material.

• Sliced Paper.

• Sintering (Sugar, Plastic, Metal Powder)

• Photosensitive liquid (Stereolithography).

Other Technologies

• Other meltable material.

• Sliced Paper.

• Sintering (Sugar, Plastic, Metal Powder)

• Photosensitive liquid (Stereolithography).

Other Technologies

• Other meltable material.

• Sliced Paper.

• Sintering (Sugar, Plastic, Metal Powder)

• Photosensitive liquid (Stereolithography).

Other Technologies

• Other meltable material.

• Sliced Paper.

• Sintering (Sugar, Plastic, Metal Powder)

• Photosensitive liquid (Stereolithography).

New Features• Individualized, adaptable

dimensions.

• Replacement parts.

• Complicated craft structures.

• Impossible structures.

• Replace mechanical skills with mathematics.

New Features• Individualized, adaptable

dimensions.

• Replacement parts.

• Complicated craft structures.

• Impossible structures.

• Replace mechanical skills with mathematics.

New Features• Individualized, adaptable

dimensions.

• Replacement parts.

• Complicated craft structures.

• Impossible structures.

• Replace mechanical skills with mathematics.

New Features• Individualized, adaptable

dimensions.

• Replacement parts.

• Complicated craft structures.

• Impossible structures.

• Replace mechanical skills with mathematics.

New Features• Individualized, adaptable

dimensions.

• Replacement parts.

• Complicated craft structures.

• Impossible structures.

• Replace mechanical skills with mathematics.

Object Features

• Outer shell

• Honeycomb structure inside, save on material

• Complete Rubik's Cube would be ~100g.

• Sharing Websites (thingiverse.com)

Extruder

New

Some Use

Extruder, Back And Inside

Back Inside (old model)

Cost• Printer: $500-$5000 (Fuse Deposit Modeling)

• $5000-$30000 (Stereolithography)

• $100000 and +++ (Metal Sintering)

• Plastic: $20- $50 per kg, special material (mixed with carbon fiber, metal, wood) more.

• Replacement Extruder: $180

Use For Mathematics

• Models of structures in R3, both research and teaching

• Being able to manipulate with hands is genuine benefit.

• Easier than perspective drawings etc.

• Sufficiently cheap and stable for class use.

SimplicesFor k+1 points p0,…,pk⊆Rn, affinely independent (that

is p1-p0,p2-p0,…,pk-p0 lin. independent), define the

simplex as the set {Σi≧0αipi | αi ≧0, Σi≧0αi =1}.

A simplicial complex is a set of simplices, closed under

faces (simplices defined by subsets of corners) with any two simplices intersection being a face of both. As a subset of Rn it is also called a polyhedron.

Geometric Structures• Given only the outside faces, we can reconstruct

an equivalent simplicial complex.

• File format: STL (surface triangulization language). Describe the outer triangles (as tuples of vertices). (There are other file formats.)

• Floating point coordinates of vertices, rounding issues can produce holes that require fixing.

The print builds up the object in many layers.

For each layer the slicer determines the polygon of object intersecting the plane:

• Edges intersecting plane (special case if triangle in plane)

• Edges on same triangle connect

• Determine inside by having triangles oriented (clockwise, when viewed from outside).

• Many heuristics in optimizing travel time.

It Slices, It Dices

Print Process

Print Process

Practical Issues• Gravity might imply need

for nonstandard orientation.

• Support overhangs (>45°) with extra material.

• Software can add support, not well.

• Object must sit firmly on build-surface — add raft. (Flat bottom advantageous.)

Practical Issues• Gravity might imply need

for nonstandard orientation.

• Support overhangs (>45°) with extra material.

• Software can add support, not well.

• Object must sit firmly on build-surface — add raft. (Flat bottom advantageous.)

Practical Issues• Gravity might imply need

for nonstandard orientation.

• Support overhangs (>45°) with extra material.

• Software can add support, not well.

• Object must sit firmly on build-surface — add raft. (Flat bottom advantageous.)

Practical Issues• Gravity might imply need

for nonstandard orientation.

• Support overhangs (>45°) with extra material.

• Software can add support, not well.

• Object must sit firmly on build-surface — add raft. (Flat bottom advantageous.)

Less Than Perfect

• Object might not hold well on build platform (and move/tip over while building).

• Stringing.

• Removing support can be difficult.

Less Than Perfect

• Object might not hold well on build platform (and move/tip over while building).

• Stringing.

• Removing support can be difficult.

Less Than Perfect

• Object might not hold well on build platform (and move/tip over while building).

• Stringing.

• Removing support can be difficult.

Real-World Aspects• Extruder may clog. (Oil helps)

• Material may not stick (or stick too much) on build surface (blue painter tape).

• Brittle material can break when feeding.

• Vertical resolution .1, .2 (default), or .3 mm. Printing time @default is about 45 minutes for matchbox-sized object.

• Infill can be increased for stability.

• Post-Processing: Sanding, Painting, Casting

Mathematical Problems

• How to orient an object for optimal printing (firm hold on platform, little overhangs).

• If not, decomposition? Good support?

• Thin connections, Singularities, are hard

Software

Construction: MapleUse plot3d to form parametric surfaces.

Grid needs to be chosen sufficiently fine.

Export in Collada (.dae) format — Need conversion.

Output often is badly formed (orientation, holes).

Intersecting surfaces problematic (Surface vs. volume.)

Conversion, FixingConvert to .STL format: MeshLab (Open Source)

Add thickness: Blender (Open Source)

Fix holes, orientation: netfabb (Commercial, free use), online service (requires signup) at netfabb.azurewebsites.net

Additional Fixes, Add Support: meshmixer (Commercial, free use)

Construction: OpenSCADProgramming language (limited). Open-Source.

Geometric primitives (spheres, boxes, cylinders/cones).

Affine transformations, Boolean operations.

Limited calculation functionality — produce code with Maple program etc.

Construction: 123D Design

Freemium Software (Autodesk).

Mouse-oriented.

Geometric Primitives, Boolean Operations.

STL input and output.

Geometric Calculations• Linear algebra for coordinate change, affine

transformations.

• Parameterization of surfaces.

• Need to solve polynomial equations (e.g. circle through three points): Gröbner-basis, Resultant, Numerical Alg.Geom. techniques

• Symmetric structures: Map by coset representatives for stabilizer.