Post on 22-May-2020
Geometric Partial Differential Equations Methods inGeometric Design and Modeling
Reporter: Qin Zhang1
Collaborator: Guoliang Xu,2 C. L Bajaj1, Dan Liu2
1CVC, University of Texas at Austin, TX, USA
2LSEC, AMSS of CAS, Beijing, China
09/03/2008
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 1 / 40
Outline
1 Intoduction
2 The Construction of GPDE
3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods
4 Conclusion
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 2 / 40
Outline
1 Intoduction
2 The Construction of GPDE
3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods
4 Conclusion
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 2 / 40
Outline
1 Intoduction
2 The Construction of GPDE
3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods
4 Conclusion
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 2 / 40
Outline
1 Intoduction
2 The Construction of GPDE
3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods
4 Conclusion
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 2 / 40
1. Introduction
What is a GPDE?
A PDE which controls the motion of curves orsurfaces and is merely formulated by the geometricentries is known as Geometric PDE.
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 3 / 40
1. Introduction
What is a GPDE?
A PDE which controls the motion of curves orsurfaces and is merely formulated by the geometricentries is known as Geometric PDE.
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 3 / 40
Examples
1 Mean Curvature Flow (MCF) (1956, Mullins)
∂x∂t
= 2H
2 Willmore Flow (WF) (1923, Thomsen)
∂x∂t
= −[∆sH + 2H(H2 − K )]n
3 Minimal Mean-Curvature-Variation Flow (MMCVF) (2006, Xu &Zhang)
∂x∂t
= (∆2sH + 2(2H2 − K )∆sH + 2〈∇sH,♦H〉 − 2H‖∇sH‖2)n
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 4 / 40
Several Differential Geometric Operators of First Order
Assume
S = x(u, v) ∈ R3 : (u, v) ∈ Ω ⊂ R2,f : S → R
Tangential Gradient Operator of Surface S
∇sf = [xu, xv ][ gαβ ][fu, fv ]T ∈ R3,
The Second Tangential Operator
♦f = [xu, xv ][ K bαβ ][fu, fv ]T ∈ R3.
Tangential Divergence Operator
div(v) =1√
g
[∂
∂u,
∂
∂v
] [√g [ gαβ ] [xu, xv ]T v
].
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 5 / 40
Several Differential Geometric Operators of SecondOrder
Laplace-Beltrami Operator
∆sf = div(∇sf ).
Giaquinta-Hildebrandt Operator
f = div(♦f ).
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 6 / 40
Why GPDE is important?
Theory aspect: Relate intimately togeometric analysismanifold theorytopologycomplex analysisPDEcalculus of variationgeometric measure theorycritical point theory
Application aspect: Relate intimately tophysics,chemistry,biology,computational geometry,computer graphics,image processing
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 7 / 40
Why GPDE is important?
Theory aspect: Relate intimately togeometric analysismanifold theorytopologycomplex analysisPDEcalculus of variationgeometric measure theorycritical point theory
Application aspect: Relate intimately tophysics,chemistry,biology,computational geometry,computer graphics,image processing
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 7 / 40
What can GPDE do in geometric design andmodeling?
Surface ProcessingSurface ReconstructionSurface RestorationSurface BlendingFree-Form DesignBiomolecular DesignImage ProcessingInterface Simulation· · · · · · · · ·
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 8 / 40
What’s about the result of GPDE?
Some optimal property, e.g.,minimize area
∫S dA = min
minimize the sum of square of principal curvatures∫S(κ2
1 + κ22)dA = min
minimize the variation of mean curvature∫S ‖∇sH‖2dA = min
etc.∫S f (H, K )dA = min
The surfaces designed by GPDEsClear geometric sense
Boundary condition with prescribed smoothness
Fairness
Aesthetic demanding
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 9 / 40
What’s about the result of GPDE?
Some optimal property, e.g.,minimize area
∫S dA = min
minimize the sum of square of principal curvatures∫S(κ2
1 + κ22)dA = min
minimize the variation of mean curvature∫S ‖∇sH‖2dA = min
etc.∫S f (H, K )dA = min
The surfaces designed by GPDEsClear geometric sense
Boundary condition with prescribed smoothness
Fairness
Aesthetic demanding
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 9 / 40
Surface Processing
input Mean curvature flow
(From Bajaj, Xu 2004)
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 10 / 40
Surface Reconstruction
scattered data reconstructed surface(Level set method)
(From Bajaj, Xu, Zhang 2006)
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 11 / 40
Surface Blending
initial input MCF (2nd order)
WF (4th order) MMCVF (6th order)
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 12 / 40
Free-Form Surface Design I
input initial surface surface diffusion flow
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 13 / 40
Free-Form Surface Design II
input skeleton PDE surface of 4th order
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 14 / 40
Surface Restoration I
input head without jaws MCF (2nd order)
WF (4th order) MMCVF (6th order)
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 15 / 40
Surface Restoration II
MCF SDF 6th order flow
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 16 / 40
Biomolecular Design
van der Waals surface SES
(from Bajaj, Xu, Zhang 2006)
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 17 / 40
Image Processing
initial input denoising process with MCF
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 18 / 40
Comparison Results Among 2,4,6-th Order Flows
MCF — G0 WF — G1 MMCVF — G2
2nd order flows usually used4th order flows often used6th order flows occasionally used
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 19 / 40
GPDE vs. Classical PDE
The domains of classical PDEs are usually fixed, but GPDE’s arealways changeable.
GPDE is geometrically intrinsic, independent of the parameterrepresentation of surface.
GPDE is highly nonlinear, which makes theoretical analysischallenging.
Classical methods can not be used directly and classical theoriesare not effective.
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 20 / 40
2. The Construction of GPDE
How to construct GPDE?
Manual approach
∂, ∇s, divs, ∆s, , ♦, · · · ,
⇓E(x) = 0, x ∈ S
For example, ∆2sx = 0, ∆2
sH = 0, · · · .
⇓∂x∂t = ±E(x)n, x ∈ S(t),
S(0) = S0.
Energy based variational approach
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 21 / 40
2. The Construction of GPDE
How to construct GPDE?
Manual approach
∂, ∇s, divs, ∆s, , ♦, · · · ,
⇓E(x) = 0, x ∈ S
For example, ∆2sx = 0, ∆2
sH = 0, · · · .
⇓∂x∂t = ±E(x)n, x ∈ S(t),
S(0) = S0.
Energy based variational approach
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 21 / 40
Energy Based Variational Approach to ConstructGPDE
Energy E (S) - Variational CalculusE ′(S)
- Construct Flow∂x∂t = −E ′
Energy∫S
dA - −2Hn - ∂x∂t = 2Hn
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 22 / 40
More Details on the Construction (Example)
Define inner product space:
S = x(u, v) ∈ R3 : (u, v) ∈ Ω is smooth.
f, g ∈ C2(S, R3)
define inner product
(f, g) =
∫S〈f, g〉dA
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 23 / 40
Three Steps to Construct GPDE
1 Define energy functional
E (S) =
∫S
dA
2 Compute first order variation
δ(E (S),Θ) =
∫S〈E ′
c(S),Θ〉dA,
here, E ′c(S) is named as the L2 gradient of E (S).
3 Construct GPDE of weak form and general form.∫S〈∂x∂t
,Θ〉dA = −∫S〈E ′
c(S),Θ〉dA, ∀Θ ∈ C∞0 (S, R3).
This is the starting point of finite element method. Then GPDE is∂x∂t
= −E ′c(S) ∈ R3.
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 24 / 40
Three Steps to Construct GPDE
1 Define energy functional
E (S) =
∫S
dA
2 Compute first order variation
δ(E (S),Θ) =
∫S〈E ′
c(S),Θ〉dA,
here, E ′c(S) is named as the L2 gradient of E (S).
3 Construct GPDE of weak form and general form.∫S〈∂x∂t
,Θ〉dA = −∫S〈E ′
c(S),Θ〉dA, ∀Θ ∈ C∞0 (S, R3).
This is the starting point of finite element method. Then GPDE is∂x∂t
= −E ′c(S) ∈ R3.
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 24 / 40
Three Steps to Construct GPDE
1 Define energy functional
E (S) =
∫S
dA
2 Compute first order variation
δ(E (S),Θ) =
∫S〈E ′
c(S),Θ〉dA,
here, E ′c(S) is named as the L2 gradient of E (S).
3 Construct GPDE of weak form and general form.∫S〈∂x∂t
,Θ〉dA = −∫S〈E ′
c(S),Θ〉dA, ∀Θ ∈ C∞0 (S, R3).
This is the starting point of finite element method. Then GPDE is∂x∂t
= −E ′c(S) ∈ R3.
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 24 / 40
A General Approach
First-order energy (second-order equation):
E1(S) =
∫S
h(x, n)dA
Second-order energy (fourth-order equation):
E2(S) =
∫S
f (H, K)dA
Third-order energy (sixth-order equation):
E3(S) =
∫S‖∇sg(H, K)‖2dA
and their combinations:
E (S) = αE1(S) + βE2(S) + γE3(S)
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 25 / 40
GPDEs of General Functional — 1st Order Energy
For first-order energy
E1(S) =
∫S
h(x, n)dA,
we obtain second-order equation∂x∂t = −∇xh − divs(∇nh)n−∇sn∇nh + divs(h∇sx),S(0) = S0.
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 26 / 40
GPDEs of General Functional — 2nd Order Energy
For the second-order energy
E2(S) =
∫S
f (H, K )dA,
we obtain fourth-order equation∂x∂t = −(fK n)− 1
2∆s(fHn) + divs(fH∇sn) + divs((f − 2KfK )∇sx),S(0) = S0,
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 27 / 40
GPDEs of General Functional — 3rd Order Energy
For the third-order energy
E3(S) =
∫S‖∇sg(H, K )‖2dA,
we obtain sixth-order equation
∂x∂t
=[∆s(gH ∆sgn) + 2(gK ∆sgn)− 2divs[gH ∆sg∇sn
− 2KgK ∆sg∇sx] + 2divs[R∇sg(R∇sg)T ]− divs[‖∇sg‖2∇sx]],
where
R =1√
g[−xv , xu] [xu, xv ]T .
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 28 / 40
Special Examples
1. Mean Curvature Flow [Mullins, 1956], (Minimal surface, Euler,1744).
f (H, K ) = 1
E ′n(S) = −2H
2. Surface Diffusion Flow [Mullins, 1957] .f (H, K ) = 1
E ′n(S) = 2∆sH
—- in the sense (·, ·)H−1 .3. Sixth-order Flow [Xu, Pan and Bajaj, 2006].
f (H, K ) = 1
E ′n(S) = −2∆2
sH
—- in the sense (·, ·)H−2 .
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 29 / 40
Special Examples (Cont.)
4. Weighted Mean Curvature Flow [cf. Zhang and Xu, 2005].h(x, n) = γ(n)
E ′n(S) = ∇sn:(∇2
nnγ)
5. Gaussian Curvature Flow [Firey, 1974].f (H, K ) = H
E ′n(S) = K
6. Willmore Flow. f (H, K ) = H2
E ′n(S) = ∆sH + 2H(H2 − K )
Willmore functional is initially introduced by Thomsen in 1923........
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 30 / 40
3. Numerical Solutions of GPDEs
Explicit solutions of the GPDEs are hard to obtain. Numerical solutionsare necessary and freasible.
Generalized Finite Difference Methods
Finite Element Methods
Level-Set Methods
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 31 / 40
Outline
1 Intoduction
2 The Construction of GPDE
3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods
4 Conclusion
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 32 / 40
Generalized Finite Difference Methods
Two Scenarios:
Normal motion equation
nT ∂x∂t
= 2H,
All directions motion equation
∂x∂t
= ∆sx,
Two Pivot Problems:
Discretization of differential geometry operatorsBoundary treatment
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 33 / 40
Outline
1 Intoduction
2 The Construction of GPDE
3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods
4 Conclusion
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 34 / 40
Mixed Finite Element Methods
Two Scenarios:
Total variation form
Normal variation form
Two Pivot Problems:
Construction of the function spaces of finite element
Boundary treatment
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 35 / 40
Outline
1 Intoduction
2 The Construction of GPDE
3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods
4 Conclusion
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 36 / 40
Level-Set Methods
Key Techniques:
Narrow band technique
Runga-Kutta method with adaptive step size
Fast computation of higher-order level-set function
Fast reinitialization
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 37 / 40
Finite Element Methods vs. Finite Difference Methods
Finite Difference MethodsEasy to implementLow costDepends on the discretization of differential geometric operators
Finite Element MethodsMore stableSolid mathematical foundationDepends on the construction of the space of finite element
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 38 / 40
4. ConclusionInterdisciplinary, GPDEs Methods relate to
differential geometry, manifold, equationsvariational calculusfunction approximationcomputational mathematicscomputer graphics
Challenging, GPDEs deep intogeometry analysis, manifold theorytopologygeometric measure theorycritical theory
Practical, GPDEs closely relate toPhysics and chemistry settings, the motion of interfaces problems,e.g., dissolution, combustion, erosionBiology field, biomembrane vesicle problem, the construction ofprotein surfaceImage processing, edge detection, noise removal, imagerestoration......
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 39 / 40
4. ConclusionInterdisciplinary, GPDEs Methods relate to
differential geometry, manifold, equationsvariational calculusfunction approximationcomputational mathematicscomputer graphics
Challenging, GPDEs deep intogeometry analysis, manifold theorytopologygeometric measure theorycritical theory
Practical, GPDEs closely relate toPhysics and chemistry settings, the motion of interfaces problems,e.g., dissolution, combustion, erosionBiology field, biomembrane vesicle problem, the construction ofprotein surfaceImage processing, edge detection, noise removal, imagerestoration......
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 39 / 40
4. ConclusionInterdisciplinary, GPDEs Methods relate to
differential geometry, manifold, equationsvariational calculusfunction approximationcomputational mathematicscomputer graphics
Challenging, GPDEs deep intogeometry analysis, manifold theorytopologygeometric measure theorycritical theory
Practical, GPDEs closely relate toPhysics and chemistry settings, the motion of interfaces problems,e.g., dissolution, combustion, erosionBiology field, biomembrane vesicle problem, the construction ofprotein surfaceImage processing, edge detection, noise removal, imagerestoration......
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 39 / 40
Thank you!
Email:zqyork@lsec.cc.ac.cn, zqyork@ices.utexas.edu
Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 40 / 40