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(A bit more) Finite Element Method
Stelian Coros
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Goal: find equilibrium configuration i.e., 𝐟𝑖 = 0 ∀𝑖
Given 𝐱 with 𝐟 𝐱 ≠ 0, find Δ𝐱 such that 𝐟 𝐱 + Δ𝐱 = 𝟎
𝐟 𝐱 + Δ𝐱 = 𝐟 𝐱 + 𝐊Δ𝐱 + 𝑂(Δ𝐱2)
Solve 𝐊Δ𝐱 = −𝐟 𝐱 for Δ𝐱
From last time…
Stiffness matrix
𝐊 =𝜕𝐟
𝜕𝐱
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• Discretize into triangles/tetraderons
• For each element
– Compute deformation gradient 𝐅 = 𝒆𝑬-1
– Use material model to define energy density Ψ(𝐅)
– Integrate over element to compute energy: W
– Compute nodal forces as:
FEM recipe
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FEM recipe
𝒇 = −𝜕𝑊
𝜕𝒙= −𝑉
𝜕Ψ
𝜕𝑭
𝜕𝑭
𝜕𝒙
First Piola-Kirchhoff stress tensor P
Area/volume of element
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FEM recipe
𝒇 = −𝜕𝑊
𝜕𝒙= −𝑉
𝜕Ψ
𝜕𝑭
𝜕𝑭
𝜕𝒙
First Piola-Kirchhoff stress tensor P
Area/volume of element
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FEM recipe
𝒇𝟏 𝒇𝟐
𝒇𝟑 = −𝑉𝑷𝑬−𝑻; 𝒇𝟒 = −𝒇𝟏 − 𝒇𝟐 − 𝒇𝟑
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FEM recipe
• For each element
– Compute deformation gradient 𝐅 = 𝒆𝑬-1
– Use material model to define energy density Ψ(𝐅)
– Integrate over element to compute energy: W
– Compute nodal forces as:
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Assignment 3
• Implement FEM Simulation
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Inverse Design Example
• Design an object that can deform like this:
• Need model that predicts the way in which
objects deform
– Design parameters: material parameters, points of
application for forces or constraints
“Computational Design of Actuated Deformable Characters”, Skouras et al, 2013
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Inverse Design Example
• Design an object that can deform like this:
• FEM Model
– Design parameters: material parameters, points of
application for forces or constraints
“Computational Design of Actuated Deformable Characters”, Skouras et al, 2013
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Pipeline
Actuator Location
Optimization
Material
Optimization
Fabricated
Deformable Model
Input and Target
Shapes
+
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Mathematical Formulation
𝐸 𝐱, 𝐩 = 𝐸𝑚𝑎𝑡𝑐ℎ 𝐱, 𝐱𝑡𝑎𝑟𝑔𝑒𝑡
Design
Parameters
𝐟𝑡𝑜𝑡𝑎𝑙 𝐱, 𝐩 = 0 subject
to
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Actuator Location Optimization
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Material Distribution Optimization
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Results
Input
Animation
Fabricated Prototype
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Results
Target Pose
Rest Pose
Stiff Soft
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Project Proposals and some Project Ideas
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Project Proposals
• 1 page of text/4 paragraphs
– Problem and motivation
– Background (include previous work)
– Proposed technical method
– Expected results (low and high bar)
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Project Proposal Presentations
• In-class on March 5th
• 5 minutes/project + discussion/questions
• Slides
– Problem and motivation (1 slide)
– Background and Tools (1 slide)
– Proposed technical method (1-2 slides)
– Expected results (1 slide)
• Send me slides at least 1 hour before the
lecture
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Parametric music box
http://www.thingiverse.com/thing:53235
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Parameterized flute
http://www.thingiverse.com/thing:12301
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Puzzle Creator
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3D Printing animations (zoetrope)
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Robot Hands
http://www.3ders.org/articles/20150219-build-an-air-powered-prosthetic-hand-on-a-regular-fdm-3d-printer.html