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Lecture 24: Heat Flow COMPSCI/MATH 290-04 Chris Tralie, Duke University 4/12/2016 COMPSCI/MATH 290-04 Lecture 24: Heat Flow Announcements Group Assignment 3 Out: First Deadline Monday 4/18. Final Deadline Wednesday 4/27 (Sakai says 4/26

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Lecture 24: Heat Flow

COMPSCI/MATH 290-04

Chris Tralie, Duke University

4/12/2016

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 2

Announcements

⊲ Group Assignment 3 Out: First Deadline Monday 4/18. Final Deadline Wednesday 4/27 (Sakai says 4/26 but that’s wrong...e-mail me solution if go until 4/27) ⊲ Final Project Final Deadline 5/3 5:00 PM

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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◮ Group Assignment 3 Preview ⊲ Scalar Fields / Laplacian Review ⊲ Heat Flow

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Group Assignment 3 Preview

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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⊲ Group Assignment 3 Preview ◮ Scalar Fields / Laplacian Review ⊲ Heat Flow

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Mesh Scalar Fields (Functions)

⊲ A real number for every point on the surface

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Coordinates As Functions

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Laplacian Eigenfunctions (Homer Modes)

Lf = λf

1 2

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 9

Discrete Circle Laplacian Eigenvectors

https://github.com/COMPSCI290-S2016/NumpyDemos/blob/master/1DLaplacian.py COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Discrete Circle Laplacian Eigenvectors

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Curvature Vector

δx = Lx, δy = Ly, δz = Lz δ =

j∈N(i)

(xi, yi, zi) − (xj, yj, zj)

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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⊲ Group Assignment 3 Preview ⊲ Scalar Fields / Laplacian Review ◮ Heat Flow

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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1D Heat Equation

Let f(x, t) be the distribution of heat over a 1D bar of uniform material parameterized by x at time t. Then heat flow is governed by ∂2f(x, t) ∂x2 = ∂f(x, t) ∂t

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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1D Heat Equation

Let f(x, t) be the distribution of heat over a 1D bar of uniform material parameterized by x at time t. Then heat flow is governed by ∂2f(x, t) ∂x2 = ∂f(x, t) ∂t ⊲ The higher the curvature of the heat distribution, the faster it dissipates

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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1D Heat Flow Example

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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1D Heat Flow Example

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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1D Heat Flow Example

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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1D Heat Flow Example

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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1D Heat Flow Example

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Generalizing To Surfaces

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t Let f be the discrete values of a function on a mesh. What is the heat equation?

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 21

Generalizing To Surfaces

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t Let f be the discrete values of a function on a mesh. What is the heat equation? Lf = −∂f ∂t

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 22

Generalizing To Surfaces

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t Let f be the discrete values of a function on a mesh. What is the heat equation? Lf = −∂f ∂t Video example...

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t fω = cos(ωx)e−ω2t

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t fω = cos(ωx)e−ω2t Lf = −∂f ∂t

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t fω = cos(ωx)e−ω2t Lf = −∂f ∂t Let φk be the kth eigenvector of L and λk be the associated

eigenvalue. Then solutions are

fk = φke−λt

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

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Initial Conditions

What if initial condition f0 doesn’t happen to be an eigenvector

f L?

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 28

Initial Conditions

What if initial condition f0 doesn’t happen to be an eigenvector

f L?

⊲ Project f0 onto eigen basis, sum the solutions of each eigenvector individually f(t) =

(f T

0 φk)e−λktφk

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 29

Initial Conditions

What if initial condition f0 doesn’t happen to be an eigenvector

f L?

⊲ Project f0 onto eigen basis, sum the solutions of each eigenvector individually f(t) =

(f T

0 φk)e−λktφk

Demo...

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 30

Heat Kernel Signature

For every point, the fraction of heat that stays at that point after a certain amount of time t = 20 t = 500 What does this look like?

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 31

Heat Kernel Signature

For every point, the fraction of heat that stays at that point after a certain amount of time t = 20 t = 500 What does this look like? Multiscale curvature

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 32

Heat Kernel Signature: Computations

f(t)[i] =

(f T

0 φk)[i]e−λktφk[i]

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 33

Heat Kernel Signature: Computations

f(t)[i] =

(f T

0 φk)[i]e−λktφk[i]

By definition, starting with a unit amount of heat at every point f0[a] = 1 a = i

therwise
COMPSCI/MATH 290-04

Lecture 24: Heat Flow

SLIDE 34

Heat Kernel Signature: Computations

f(t)[i] =

(f T

0 φk)[i]e−λktφk[i]

By definition, starting with a unit amount of heat at every point f0[a] = 1 a = i

therwise
f(t)[i] =
k

φk[i]e−λktφk[i] f(t)[i] =

e−λktφk[i]2

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

SLIDE 35

Another Example

t = 20 t = 500

COMPSCI/MATH 290-04 Lecture 24: Heat Flow

Lecture 24: Heat Flow

COMPSCI/MATH 290-04

Chris Tralie, Duke University

4/12/2016

Announcements

⊲ Group Assignment 3 Out: First Deadline Monday 4/18. Final Deadline Wednesday 4/27 (Sakai says 4/26 but that’s wrong...e-mail me solution if go until 4/27) ⊲ Final Project Final Deadline 5/3 5:00 PM

Table of Contents

◮ Group Assignment 3 Preview ⊲ Scalar Fields / Laplacian Review ⊲ Heat Flow

Group Assignment 3 Preview

Table of Contents

⊲ Group Assignment 3 Preview ◮ Scalar Fields / Laplacian Review ⊲ Heat Flow

Mesh Scalar Fields (Functions)

⊲ A real number for every point on the surface

Coordinates As Functions

Laplacian Eigenfunctions (Homer Modes)

Lf = λf

1 2

Discrete Circle Laplacian Eigenvectors

Discrete Circle Laplacian Eigenvectors

Curvature Vector

δx = Lx, δy = Ly, δz = Lz δ =

(xi, yi, zi) − (xj, yj, zj)

Table of Contents

⊲ Group Assignment 3 Preview ⊲ Scalar Fields / Laplacian Review ◮ Heat Flow

1D Heat Equation

Let f(x, t) be the distribution of heat over a 1D bar of uniform material parameterized by x at time t. Then heat flow is governed by ∂2f(x, t) ∂x2 = ∂f(x, t) ∂t

1D Heat Equation

Let f(x, t) be the distribution of heat over a 1D bar of uniform material parameterized by x at time t. Then heat flow is governed by ∂2f(x, t) ∂x2 = ∂f(x, t) ∂t ⊲ The higher the curvature of the heat distribution, the faster it dissipates

1D Heat Flow Example

1D Heat Flow Example

1D Heat Flow Example

1D Heat Flow Example

1D Heat Flow Example

Generalizing To Surfaces

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t Let f be the discrete values of a function on a mesh. What is the heat equation?

Generalizing To Surfaces

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t Let f be the discrete values of a function on a mesh. What is the heat equation? Lf = −∂f ∂t

Generalizing To Surfaces

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t Let f be the discrete values of a function on a mesh. What is the heat equation? Lf = −∂f ∂t Video example...

Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t

Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t fω = cos(ωx)e−ω2t

Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t fω = cos(ωx)e−ω2t Lf = −∂f ∂t

Solutions To The Heat Equation

∂2f(x, t) ∂x2 = ∂f(x, t) ∂t fω = cos(ωx)e−ω2t Lf = −∂f ∂t Let φk be the kth eigenvector of L and λk be the associated

fk = φke−λt

Initial Conditions

What if initial condition f0 doesn’t happen to be an eigenvector

Initial Conditions

What if initial condition f0 doesn’t happen to be an eigenvector

⊲ Project f0 onto eigen basis, sum the solutions of each eigenvector individually f(t) =

(f T

0 φk)e−λktφk

Initial Conditions

What if initial condition f0 doesn’t happen to be an eigenvector

⊲ Project f0 onto eigen basis, sum the solutions of each eigenvector individually f(t) =

(f T

0 φk)e−λktφk

Demo...

Heat Kernel Signature

For every point, the fraction of heat that stays at that point after a certain amount of time t = 20 t = 500 What does this look like?

Heat Kernel Signature

For every point, the fraction of heat that stays at that point after a certain amount of time t = 20 t = 500 What does this look like? Multiscale curvature

Heat Kernel Signature: Computations

f(t)[i] =

(f T

0 φk)[i]e−λktφk[i]

Heat Kernel Signature: Computations

f(t)[i] =

(f T

0 φk)[i]e−λktφk[i]

By definition, starting with a unit amount of heat at every point f0[a] = 1 a = i

Heat Kernel Signature: Computations

f(t)[i] =

(f T

0 φk)[i]e−λktφk[i]

By definition, starting with a unit amount of heat at every point f0[a] = 1 a = i

φk[i]e−λktφk[i] f(t)[i] =