28-Jul-16

Computational Tools for 3D Printing
Ariel Shamir
Bernd Bickel
Wojciech Matusik

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Part 4: Modeling and Analysis
Ariel Shamir

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1

28-Jul-16

Modeling 3D Objects• A difficult problem on its own:
• 3D world on 2D displays• 3D manipulation using 2D (or 2.5D) devices• Complex: mostly done by experts• Issues relating to the digital representation

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Challenges Specifically for
Fabrication & Printing (not just Graphics)•

Need to actually be constructed or printed:
•
•
•
•

•

Fitting parts
Finding intersections
Defining connectors
Checking printability

Geometry

Need to be physically plausible:••••

Appearance
Materials
WeightForcesPhysics

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28-Jul-16

Two Approaches
1. Inverse ModelingSpecification

Optimization

Model

Fabrication

2. Interactive modeling:
Interaction

Model

Fabrication

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Examples of Inverse Design• Appearance fabrication• Deformable objects fabrication• Basic idea:
Input
A given shape+
High level specification

OutputA recipe for fabricating
the object (shape,material, structure…)

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28-Jul-16

More Inverse Modeling Examples
Chopper: Partitioning Models into 3D-Printable Parts
Linjie Luo, Ilya Baran, Szymon Rusinkiewicz, Wojciech Matusik
ACM Transactions on Graphics, 31(6), (SIGGRAPH Asia), 2012

Make It Stand: Balancing Shapes for 3D Fabrication
Romain Prévost, Emily Whiting, Sylvain Lefebvre, Olga
Sorkine-Hornung, ACM Trans. Graph. 32, 4, Article 81(July 2013)

Build-to-last: strength to weight 3D printed objects.
Lin Lu, Andrei Sharf, Haisen Zhao, Yuan Wei, Qingnan Fan,
Xuelin Chen, Yann Savoye, Changhe Tu, Daniel Cohen-Or, andBaoquan Chen. 2014 ACM Trans. Graph. 33, 4, Article 97 (July2014)

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More Inverse Modeling Examples
Specify: shape and size

Specify: balance

Specify: strength to a given weight

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28-Jul-16

Strati: First 3D Printed Car• Made by Local Motors for theInternational Manufacturing
Technology Show (IMTS)• Strati = “layers” in Italian• 18 months of design• 44 hours of printing

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Usually Print Volume is Limited• Typically 10cm x 10cm x 10cm up to 50cm x 40cm x 30cm
• Printing large objects requires chopping and assembly

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28-Jul-16

Chopper: Two Challenges• How to segment?
• How to connect?

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Chopper: Two Challenges• How to segment?
• Use planes to define parts• Cut top down recursively• How to connect?

BSP• Use male/female connectors:

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28-Jul-16

Overview
Pick a bunch of
potential cuts
Evaluate them andfind the best ones
Recurse on pieces
that are still to big
Place connectors on
cross-sections

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Optimization Objective
•
•
•
•
•
•

Connector feasibility
Part fragility
Structural soundness when assembled
Number of parts
Printing volume utilization
Seam aesthetics

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28-Jul-16

Optimization: Beam Search [Lowerre 1976]
0.59
0.59
0.67

chop

evaluateselect
0.67
0.71
0.82
0.82

chop

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Make It Stand Challenge• Designing things that can really stand:

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28-Jul-16

Overview

input model
inner carving

shape deformation

balancedmodel

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Optimization

does it stand?

similarity tooriginal shape

inner carving

shape deformation

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28-Jul-16

User Assistance• Changing the shape using handles

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Build-to-Last Challenge• Reduce the object weight while providing a durable 3D printout
sustaining given forces.
• Inspiration from porous structure– Lightweight– Strong– Ability to absorb energy, vibration

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28-Jul-16

Basic Idea
Introducing the honeycomb-cell structure, which is of minimal material cost
while providing strength in tension.

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Overview

Strength-to-Weight
Optimization

Structural
Analysis

ResultInput(model &forces)Voronoi
Tessellation

Pore
Extraction

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28-Jul-16

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

𝛽 (Hollowing
amount)20𝛽 = 𝛽1 ,∙∙∙, 𝛽𝛼

50
100
200
400
600
800

𝛼 (# of cells)

Optimization
Shape• Objective:

argmin 𝑊𝑠 𝛼, 𝛽Forces𝑠. 𝑡. 𝑆𝑀 𝑆, 𝐹 < 𝜒𝛼,𝛽Weight (volume)

Yield point = stressat which amaterial begins to
deform plasticallyStrength (stress map)

• Two loops for optimizing 𝛼 and 𝛽• Outer loop: optimize 𝛼 for compactness• Inner loop: optimize 𝛽 for sustaining stress
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28-Jul-16

Results

Initial setting &Stress map

Iterations

Optimal result

3D pores50.79 𝑐𝑚3 (40.6%)

125.07 𝑐𝑚3

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More Examples of Inverse Design Objectives[Umetani et al., 2014]

Flight ability[Baecher et al., 2014]

Spin ability

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28-Jul-16

Inverse Design Methodology• Given a 3D shape (usually boundary surface),
optimize some objective (size, balance,
strength) by changing shape parameters:– Inner shape – does not change outside appearance– Outer part – usually want to constrain not to differ
too much from original shape

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Modeling vs. Modifying a Given Input Shape• Shape and size: finds segmentation• Balance: modify internal and external shape• Strength to weight: modify internal shape mostly

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28-Jul-16

Creating a Whole New Model?

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Makers Make Connectors

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28-Jul-16

Objective• Input — Two geometries– Position / orientation– Weights– Auxiliary parameters (e.g., free directions)• Output — A customized connector– Automatically generated– Fabricable (physics, geometry)• No initial shape to begin withFOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE(at least one is hand-held) (acquisition is out of scope)

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28-Jul-16

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Example 2

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28-Jul-16

Connectors Definition• Two holder parts

Bridge part

– Each holder connects/holds
one of the target objects• Bridge part
– Simple bar connecting thetwo holders
Holder parts

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Two Holder Types
1. Holder for standard shapes
Must have sufficient grip strength
Physics

2. Holders for freeform shapes
Object must not fall off (hold-ability)
Geometry

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28-Jul-16

Holders Database

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Parametrized Holders
Every holder is parametrized by a few parameters

e.g. heighte.g. closeness

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This parametrization is done by one of the authors who has mechanical engineering background19

28-Jul-16

Considering Grip Strength

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Freeform Holders by Area Expansion

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28-Jul-16

Two Types of Optimizations• Optimizing holder parameters for Grip Strength• Optimizing surface for Holdability
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28-Jul-16

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28-Jul-16

Physical Considerations•: avoid cutting at weak points in the mesh•: balancing•: avoid large pores at weak points•: avoid slipping or falling

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Structural Analysis• In many cases there is a need to analyze the “strength” or“weakness” of the shape

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23

28-Jul-16

Forces & Load

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Yield• A yield strength or yield point is the material
property defined as the stress at which a material
begins to deform plastically.
• Prior to the yield point the material will deform
elastically and will return to its original shape when
the applied stress is removed. Once the yield point
is passed, some fraction of the deformation will bepermanent and non-reversible.
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28-Jul-16

Strain vs. Stress

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Continuum Mechanics: Structural Analysis• Standard method in engineering uses Finite Elements Methods

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28-Jul-16

FEM• Define a variational formulation• Discretize the space• Define basis functions• Solve (iterative)• Post processing

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In Graphics
Stress Relief: Improving Structural Strength of 3D
Printable Objects[Stava et al. SIG 2012]

Worst-case Structural Analysis[Zhou et al. SIG 2013]

Cross-sectional Structural Analysis for 3D Printing
Optimization[Umetani & Schmidt SIGA 2013]

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26

28-Jul-16

Three Key Questions for Fabrication
ﬁngers. The value of each conﬁguration ⇤G is then computed asY⇤G =n A p · l G , p OG (p),(1)

1. What is the applied load?
2. What are failure locations?
3. How to fix failures?
Figure 3: The convex hull and the center of mass are used to ﬁnd
the most probable rest orientations of the object.

following concepts detailed in [Fu et al. 2008]. We compute theconvex hull of the object along with its center of mass (see Fig. 3).
Each face of the convex hull represents a candidate orientation for
placing the object on a ﬂat surface. We test each face by orthogonally projecting the center of mass onto the face plane. If the projected center fallsoutside theface, theconﬁguration isunstable. We
sort the stable faces by their decreasing area and then move the default orientation to the head of the list. The area criterion provides
a simple heuristic to prioritize the more likely orientations if theobject were to be randomly placed down. The prioritized ordering
allows test cases to be truncated into a smaller set when computing
resources are limited. We note that the set of cases may be reduced
further by exploiting symmetries.
An object can be manipulated in many different ways. Because of the physical size of objects that are usually
printed, we restrict ourselves to the analysis of loads generated by apinch grip, where the object is grasped by two ﬁngers. Two-ﬁnger
grips represent only a limited sub-set of possible handling conﬁgurations, but they are especially important for printability analysis
because the forces applied to the model are higher when only two
ﬁngers are used.
Imposed Loads

p 2 { P1 ,P2 }where we compare the grip axis direction l G , p to the average normal direction n A p on the grasped surface A p . The size of the surface A p is deﬁned by the average size of a human ﬁnger and its
center is the position p of a given grasp site P . This ensures that
we preferably choose grip conﬁgurations where the grasped surface
is ﬂat and aligned with the grip axis. The accessibility of each siteis then estimated using ambient occlusion OG (p).
We select N G grip conﬁgurations with highest ⇤ while keeping
them signiﬁcantly different from each other. Equation (1) would
lead to similar ⇤values for closely located grip conﬁgurations. For
each selected grip conﬁguration G s , we multiply thep value of ⇤forall remaining conﬁgurations with a penalty term dG i , G s , wheredG i , G s is the average geodesic distance between grip sites.
The force F gr i p of the grip is evaluated using an empirical modelof [Kinoshita et al. 1997] that predicts the grip force based on thetangential force F t and the tangential torque Tz on the ﬁngers holding the object:
F gr i p =

Ft
|Tz |
0.011
+
−
F t |Tz |,µiµ r ot
µi

(2)

where
µ i INTERNAL
and µ r ot
thecoefﬁcients
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USEareONLY,
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DISTRIBUTE of linear and rotational friction, respectively, and their values are determined by the propertiesof the material. The tangential force is F t = F g (|l G ⇥ g|), where
g is the direction of gravity and F g is the gravitational force. The
value of tangential torque depends on the relative positions of thegrasp sites
P p 1 and p 2 to the center of mass of the object co as
Tz = F g i = 1, 2 l G · ((p i − co ) ⇥ g).

Stress Relief – Pinching Load

• To further
Compute“Pinchlocations:
reduce the search
space Grip”
of grip conﬁgurations,
we ﬁrstcreate a set of plausible grip positions that are then evaluated andfrom
the best ones
are selected.
determine plausible• whichLabeltriangle
onToconvex
hullgripaspositions, we ﬁrst mark each face center of the input mesh as apossible grasp
siteP for thegrip
ﬁrst ﬁnger.
Forforeach site
P , wedeﬁnepotential
site
first
fingera set of grip directions that determine the axis of a single pinch
grip. The set of directions should represent the most natural ways
• Cast ray in predefined direction toby which a person would grasp the given object.
determine second triangle
1••

Note that if the grip axis l G is parallel to g, both F t and Tz are zeroand the given grip requires no force. However, it is still important totest these conﬁgurations because the effects of gravity alone might
be signiﬁcant, especially when l G is far away from the center ofmass of the object. An example of the ﬁrst ﬁve detected pinch grips
for a model of a kitten is shown in Fig. 4.

1

As shown in [Balasubramanian et al. 2010], people tend to instinctively hold objectsin such waysthat thecenter of themassislocatedeither in the grip axis itself or somewhere on the plane deﬁned by
the grip axis and the gravity vector. To ﬁnd such conﬁgurations forevery site P1 , we deﬁne one grip direction that connects the site to
the center of the object’s mass and one direction that points to thecenter of mass projected to a plane that is parallel to the groundlevel. Another common case is holding the object by some local
feature, such as the handle of a cup. We capture these cases by creating a grip direction that points to the closest point on the medial
axis to P1 , and we create one ray for each of these directions. We
then ﬁnd all intersections of the ray with the object, and for each
one we create a second grasp site P2 that, together with the givensite P1 , forms a grip conﬁguration GP 1, P 2 .

Filter grips based on biomechanicalcriteria
Compute load using model frombiomechanics

The above process results in a large number of possible grip conﬁgurations that is lessened by analyzing geometric properties of thegrip conﬁgurations and checking how accessible they are for human

Figur e 4: Pinch grip positions. Each grip pair has the same color.

3.2

Structural Analysis

The structural stability of a printed model depends on the computed load cases and on the printing technology and material thatare used. To make our approach applicable to a wide variety of 3D
printing
settings, we compute the stability from the solution of the
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linear elasticity problem considering a homogenous material. The
accuracy of the structural analysis can be improved by using moreadvanced methods targeted at speciﬁc 3D printers and materials.

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Stress Relief• Finite Element Analysis for allorientations and grip locations• Fixing failures• Automatically thicken thin parts of
the object• Add struts to support parts of theobject• Hollow parts of the object to reduceweightFOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE

Worst-Case Structural Analysis• Find a load that maximizes stress in the object

StressDeformed Shapeof Object

Pressure Applied

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28-Jul-16

Worst-Case Structural Analysis• Compute failure locations by finding stresses that exceed a
Results
threshold• No fixing discussed

66

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How does a structure break ?
Structure breaks at slender parts

Structure breaks by bending

How a beam bends?

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28-Jul-16

Cross-sectional Structural Analysis• Requires just the surface mesh to perform
analysis of the stress• Treats objects as if they are beams

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Compute Cross Sections

Fixedboundary

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28-Jul-16

Divide object into two pieces using a cutting plane

Cutting plane

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Fix boundary on one piece

Cutting plane

Fixedboundary

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28-Jul-16

Apply Forces
Force
Force

Cutting plane

Fixedboundary

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Compute maximum stress on cross sections
Force
Force

Cutting plane

Fixedboundary

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28-Jul-16

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

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Application• Not fixing but choosing the printing direction
max load: 0.41kg
weak

printing
direction

max load: 3.91kg

strongprinting
direction
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28-Jul-16

Structural Analysis Using Finite Elements• Choose loading conditions• FEM Solve for each loading condition

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Worst-Case Analysis• Solving for multiple forces is too slow, so find most destructive force• Finding this single force sample is slow and it may be unrealistic

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28-Jul-16

New: Stochastic Finite Elements

Probability• Succinctly describe many forces as distributions• Perform Finite Element Analysis on Force Distributions
InsteadStochastic Structural Analysis for Context-Aware Design and Fabrication
Timothy Langlois, David I.W. Levin, Daniel Dror, Ariel Shamir, Wojciech Matusik Conditionally Accepted SIG Asia 2016

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Probability

Probability

Generate Distributions via Sampling

Linear Velocity

Angular Velocity

1.5m

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28-Jul-16

Scene – Drop Test

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Operate in Reduced Sample SpaceFOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE

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28-Jul-16

Fast Stochastic Analysis using Reduced Space
1. Unlike previous methods,
encodes context efficiently
2. Handles complicated contact
scenarios
3. Expresses reliability
compactly as a single failure
percentage
Probability of Failure: 71%
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Stochastic Topology Optimization• Because distributions are easy to compute via
sampling, the method can be used for topology
optimization

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28-Jul-16

Back to Modeling
1. Inverse ModelingSpecification

Optimization

Model

Fabrication

2. Interactive modeling:
Interaction

Model

Fabrication

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Modeling from Scratch is a Challenge

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28-Jul-16

Interactive Modeling• We can still allow the user some control and design
intelligent tools• Three attempts to allow more interactive tools:– Modeling from Photographs– Modeling by (Part) Examples– Customization of Models

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28-Jul-16

3Sweep Motivations• Modeling from an image is challenging for automatic computer
algorithms• It is a challenge for humans• Can we combine forces?

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3-Sweep: Image Based Object Modeling• A simple intuitive gesture to define 3D primitives using 3 mouse
sweeps:FOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE

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28-Jul-16

Modeling One Primitive

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3Sweep Overview

Pre-Processing

Input image

3-Sweep:
Modeling One Primitive

Extracted edges

Sweep to form theDrawing two strokes toform the profile of a primitive 3D model of a primitive

Optimization:
Modeling Composite

Applying geo-semantic
Editing and pasting
constraint to achieve final model
the object

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28-Jul-16

Behind the Scene: Outline Snapping Rules• Prefer the outline that is parallel to the 3rd stroke• Stick to the same outline, jump to the one with minimal angle change ifnecessary• Use symmetry and continuity to deal with missing outline• Fit to uniform or linearly changing diameters and use diameters smoothing(assumption: the profile can only be scaled uniformly).

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Behind the Scene: Geo-Semantic Constraints

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28-Jul-16

Expert Knowledge: Geo-Semantic Constraints• Defined in terms of major axes of the primitives.
• Support six constraint types:
•
•
•
•
•
•

parallelism
orthogonality
collinear anchors
overlapping anchors
coplanar anchors
coplanar axes

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Modeling Session

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Virtual vs. RealFOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE

Many Parts (~300)

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The Design for Fabrication Process

Designer

Engineer

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Fabrication by Example Motivation• Designing objects that can be really fabricated:• Requires many small details• Includes how to connect parts• Involves materials, physics…

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28-Jul-16

Key Idea: Database of Example
hierarchical parametricobjects (templates)

Modeling System

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Modeling Session

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28-Jul-16

Approach

Input

Expert Data

Composition Tool

Fabrication

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ExamplesFOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE

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Customizable Models:Parameters & Constraints

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Part Based Modeling

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Snapping•

User interaction•

Preserve alignment constraints

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Connectors

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28-Jul-16

ExamplesFOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE

ExamplesFOR TWDC INTERNAL USE ONLY, DO NOT DISTRIBUTE

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28-Jul-16

Customization• Instead of the user involved in modeling –design customizable objects and allow the user
to modify them before fabrication

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Difficulties• Which parameters to expose?
• It takes time to compute after change ofparameters• Changing parameters may break thedesign in various ways:– Geometrical (self intersections, invalid
boundary, holes)– Physical (too thin, unstable, un-printable)
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28-Jul-16

Customizable
Objects for
Fab Forms
Fab Form
1. Customizable✓

OK

2. Valid
3. Interactive

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Overview

Our Method
Parametric
DesignFabFormDesigner

User

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28-Jul-16

Fab Form Implementation
Fab Form
Implementation

design space

Fab Form
Requirements

Parametric Design✓ 1. Customizable

Valid Regions of
the Design Space✓ 2. Valid

Geometry Cache✓ 3. Interactive
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Inputs and Outputs
1. Parametric Design

Fab Form
Implementation
Parametric
Designdesign space

2. Automatic Tests• Printability analysis• Design-specific testsf[Doubilet et al. ’84, Wu ’94,
Stava et al. ’12, Zhou et al. ’13]

Valid Regions of
the Design Space
design space

Geometry
Cache

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28-Jul-16

Run-time
Precomputation
Design Space Sampling
Fab Form
Implementation
Parametric
Designdesign space

Valid Regions of
the Design Space

Cache Optimization
design space

Geometry
Geometry
Cache
Cache

Geometry
Cache

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Design Space Sampling
Sampling objectives:• approximate the valid region
design space• populate geometry cache
Intuition:• sample more where geometry changes• sample more where validity changes

Geometry
Cache

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28-Jul-16

Using the Valid Region
Non-convex, noncontiguous regionsdifficult to navigate
with sliders alone.

current design point during
run-time customization

design space[Marks et al. ’97, Talton et al. ’09, Shapira et al. ’09, Koyama et al. ‘14]

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Design Exploration Points
graph ofvalid samples

design space

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28-Jul-16

Using Design Exploration Points

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Summary• 3D Modeling for the physical world involves additional
challenges : Geometric and Physical Analysis• Three points of views need to work together for effectivemodeling:

Designer

Engineer

User

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28-Jul-16

Summary• 3D Modeling for the physical world involves additional
challenges: Geometric and Physical Analysis• Three points of views need to work together for effectivemodeling: Designer,
Engineer,
User• Inverse modeling: mostly changes existing models to fit
certain specification• Interactive modeling: still very difficult for novices – need forintelligent tools to assist
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57