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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
http://www.math.colostate.edu/~hulpke
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Thank YouCollege of Natural Sciences
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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
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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
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Other Technologies
• Other meltable material.
• Sliced Paper.
• Sintering (Sugar, Plastic, Metal Powder)
• Photosensitive liquid (Stereolithography).
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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).
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New Features• Individualized, adaptable
dimensions.
• Replacement parts.
• Complicated craft structures.
• Impossible structures.
• Replace mechanical skills with mathematics.
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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.
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Object Features
• Outer shell
• Honeycomb structure inside, save on material
• Complete Rubik's Cube would be ~100g.
• Sharing Websites (thingiverse.com)
http://thingiverse.com
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Extruder
New
Some Use
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Extruder, Back And Inside
Back Inside (old model)
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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
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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.
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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.
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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.
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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
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Print Process
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Print Process
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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.)
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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.)
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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.)
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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.)
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Less Than Perfect
• Object might not hold well on build platform (and move/tip over while building).
• Stringing.
• Removing support can be difficult.
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Less Than Perfect
• Object might not hold well on build platform (and move/tip over while building).
• Stringing.
• Removing support can be difficult.
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Less Than Perfect
• Object might not hold well on build platform (and move/tip over while building).
• Stringing.
• Removing support can be difficult.
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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
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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
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Software
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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.)
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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)
http://netfabb.azurewebsites.net
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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.
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Construction: 123D Design
Freemium Software (Autodesk).
Mouse-oriented.
Geometric Primitives, Boolean Operations.
STL input and output.
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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.