Cross-sectional Structural Analysis
for 3D Printing Optimization

Nobuyuki Umetani

Ryan Schmidt

Shape Design for 3D Printing is Difficult
Beginners cannot design structurally sound shape

3D printing	(1-2 hours)	

(1-3 weeks)	

[source: DBrager14@Thingiverse]	

Goal: Real-time Weakness Detection During Design
Design with interactive trial & error

3D printing	The user thicken leg	This	  part	  is	  Here	  we	  go	  !	  
too	  fragile	  Related Works

Related Works: Simulation during Interactive Design
Structural analysis for solid 3D shape had been extremely difficult[Umetani et al. NIME 2010]	

[Umetani et al. Siggraph 2011]	

[Furuta et al. Eurographics 2010]	

[Umetani et al. Siggraph 2012]	

Related Works: Structural Weakness Detection
All previous works are based on Finite Element Method[Stava et al. Siggraph 2012]	

[Luo et al. Siggraph Asia 2012]	

[Zhou et al. Siggraph 2013]	

Drawbacks of Finite Element Method•Slow•Mesh construction•Linear solver•FEM requires fair mesh	•Without hole & intersection	(only less than 25% meshes are closed manifold, 90% are self-intersecting [Gilbert 2013])	

•Well-shaped triangle	Solution: Cross Sectional Structural Analysis
Key idea: bending momentum conservation	

Stress	  ?	  	  Stress!	  
Moment arm	Force	

Bending moment	

Fixed 	boundary	

Bending moment	

Cross-section	

Problem is simplified very much
We don’t care	• Location of fixed boundaries	

• Object’s in-between 3D shape	

Stress	  Fixed boundary	

Force	

Fixed 	

boundary	

Fixed boundary	How the structure break ?
Structure breaks at slender parts	

Structure breaks by bending	How	  a	  beam	  bends?	  History of the Beam Theory[Galileo 1638]	

[Euler 1750]	

2nd industrial revolution	

[source: wikipedia]	

[source:The Stillman Drake Collection]	

[ source:wikipedia]	

Euler-Bernoulli Assumption"

Neutral axis

centroid	  Deformation	

"

Cross sections do not deform"Neutral axis does not stretch"

Cross sections remain orthogonal to neutral axis

Stress on a Cross-Section
cross-section	

neutral axis	

Deformation	

Second moment	of area	

Fracture condition	

How we can extend beam theory into 3D shape?

Extension to 3D Shape"

Topology analysis of cross sections
Force	

Force	

Cutting plane	

Fixed 	boundary	Neutral Axis Detection"

Stress computation from arbitrary cross section
cross section	

to compute stress	neutral axis	

centroid	Comparison with FEM
Linear solid material analysis	

with 2nd–order finite element method	

FEM	Ours	

External 	Force	Structural Weakness is Both Local and Global
Weak	  

Strong	  Strong	  Strong	  Shorter	

Smaller	Wider	Weakness of a Cross-Section

Minimum 	breaking 	

force	

Fixed 	

boundary	

Bending moment	Acceleration of Weakness Computation

Weak Cross Sections Performance"

13 directions of cross section, 32 slices in each direction|T| = 17k	n-sample=60k	

time = 0.18sec	

|T| = 34k	n-sample=200k	

time = 0.31sec	

|T| = 600k	

n-sample=414k	

time = 3.8sec	

Live Demo with

Please download and check it out !	

http://www.meshmixer.com/	

Application: 3D printing optimization

Printed Objects Have Large An-isotropy

printing 	

direction	

strong	weak	

Printing Direction Determines Strength
max	  load:	  0.22kg	  weak	

printing 	

direction	

max	  load:	  3.51kg	  strong	

printing 	

direction	

Printing Optimization"

Filament should goes through weak cross section"

Printing direction should be perpendicular to weak cross sections

Results

Experiment #1max	  load:	  0.21kg	  weak	

printing 	

direction	

max	  load:	  1.43kg	  strong	

printing 	

direction	

Experiment #2
max	  load:	  0.51kg	  weak	

printing 	

direction	

max	  load:	  1.5kg	  strong	

printing 	

direction	

Experiment #2
max	  load:	  0.41kg	  weak	

printing 	

direction	

max	  load:	  3.91kg	  strong	

printing 	

direction	

Discussion

Analysis on Skinny Triangle Mesh"

Desirable for industrial designAnalysis on Holey Mesh"

Skip the cross sections on the holeComparison with a Typical Procedure of FEM

FEM	  Input	  mesh	  Ours	  Input	  mesh	  manual intervention+Mesh	  simpliﬁcaIon	  +Hole	  ﬁlling	  +Embedded	  mesh	  not robust
Tetrahedral	  mesh	  generaIon	  slow & memory intensive
Linear	  system	  construcIon	  Solving	  linear	  system	  Cross-­‐secIon	  generaIon	  Beam	  equaIon	  (closed	  form)	  Limitations"Complex fracture is difficult(e.g. Buckling)

Round object is difficult"F"

F

Some boundary conditions are hardFuture Work: Direct Stress Analysis on CSG"

Real-time stress analysis for interactive CAD system

Cross section[source: IRIT user’s manual]Cross-sectional Structural Analysis
for 3D Printing Optimization
Nobuyuki Umetani

Ryan Schmidt