Boundary integral methods for implicitly defined dynamic interfaces - PowerPoint PPT Presentation

Boundary integral methods for implicitly defined dynamic interfaces Celebrating Prof. Yoshikazu Giga’s 60th Birthday Richard Tsai with 
 C. Chen, C. Kublik, Y. Wu KTH Royal Institute of Technology, Sweden 
 and 
 The University of Texas at Austin, USA

Among the important things • The existence of a second stomach for sweets • The importance of “unagi” for survival of summer • The completion of a meal by soba-yu

The L-solutions of Giga-Sato Embed the solution graph as a level curve of a higher dimensional function, and look at convergence in Hausdorff distance of the subgraphs: y u(x) phi>0 phi<0 x x ⇒ φ t − φ y H ( t , x , y , − φ x u t + H ( t , x , u , u x ) = 0 ⇐ ) = 0 . φ y

Multivalued solutions 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 y 0.6 v(y) 0.4 0.4 0.2 0.2 0 0 − 0.2 − 0.2 − 0.4 − 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1 − 0.4 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x

Very singular diffusion of Giga H ( t , x , y , φ x , φ y ) = M | ∇φ | ∂ ∂ y ( φ y φ t + ˜ | φ y | ) . Numerical solution verifies the entropy condition. (Equal area rule). 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 − 0.2 − 0.2 − 0.4 − 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Bunching and anisotropy 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 − 0.5 − 0.5 − 1 − 1 − 1.5 − 1.5 − 2 − 2 − 2.5 − 2.5 − 2.5 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 2.5 − 2.5 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 2.5

Computation by T. Ohtsuka [Ohtsuka-Giga-T:2013]

Multivalued solutions for high frequency waves o t S þ jr x S j 2 þ V ð x Þ ¼ 0 ; 2 [Jin,Liu, Osher-T:2005]

Geometric motion of an interface Mullins-Sekerka dynamics: normal velocity of a moving interface  ∂ u � v n = ∂ n ∆ u = 0 x / ∈ ∂ Ω ( t ) Ω ( t ) u = κ , x ∈ ∂ Ω ( t ) + far field condition Ω ( t )

Solution of linear PDEs by boundary integrals ∆ u = 0 , x ∈ Ω u ( x ) = f ( x ) , x ∈ ∂ Ω Z f ( x ) = λγ ( y ) + K ( x, y ) γ ( y ) dS y • Solving for : γ ∂ Ω Z ˜ u ( x ) = K ( x, y ) γ ( y ) dS y • Evaluating the solution: ∂ Ω

Breaking of convexity

Volume preservation 4 3 2 1 0 − 1 − 2 − 3 − 4 − 4 − 3 − 2 − 1 0 1 2 3 4

Merging

Nontrivial surface area-volume relation Isothermal at infinity.

Closed simple curve Γ : γ ( s ) = ( X ( s ) , Y ( s )) , parametrized by arclength s . Z Z I = f ◦ γ ( s ) ds = R 2 f ( x , y ) δ ( Γ , x , y ) dxdy . φ >0 Explicit curves: [Peskin, Lai] R δ ( x − X ( s )) δ ( y − Y ( s )) ds . δ ( Γ , x , y ) = Implicit curves: φ <0 δ ( Γ , x , y ) = δ ( φ ( x , y )) | ∇φ ( x , y ) | . Surface integral: Z R d f ( x ) δ ( φ ) | ∇φ | dx .

� ∑ i , j δ ε ( Γ , x i , j ) f ( x i , j ) h 2 − � � E ε , h = R Γ f ( γ ( s )) ds � → 0 if ε = h α , 0 ≤ α < 1 . (wide support!) Typically E ε , h − Non-convergence if e.g. δ cos Ch

Relative error E = | S h − S | / S . Radii: r × > r ◦ > r � . 0.12 0.016 0.1 0.014 0.08 0.012 0.06 0.01 0 200 400 0 200 400 (b) δ L (c) δ L (a) Γ h 2 h − 2 10 − 1 10 − 3 10 − 4 10 − 2 10 − 5 10 2 3 2 3 10 10 10 10 (d) (e) δ cos δ cos 2 h ( d Γ ) 2 h ( phi )

Theorem ε ( d Γ ( x i , j )) h 2 exact, ∑ i , j ∈ I p , q δ L | p | + | q | if ( p , q ) = ∇ d Γ are relative prime and ε = ε ( p , q ) = p 2 + q 2 h. √ Polar plot of ε ( θ n ) ε (p,q) 1.5 ε ( θ n ) 1 0.5 θ n 0 − 0.5 − 1 − 1.5 − 1.5 − 1 − 0.5 0 0.5 1 1.5 Γ [Engquist-Tornberg-T:2005]


 
 
 Defining a “fat” integral equation Z f ( x ) = λρ ( y ) + Φ ( x, y ) ρ ( y ) dS y ∂ Ω • Proper definition of the equation for computation using implicit surfaces: 
 ∂ Ω = { x : φ ( x ) = 0 } Z f ext ( x ) ⇡ λρ ext ( y ) + R d Φ ( x, y ) ρ ext ( y ) δ ✏ ( φ ) | r φ | dy ? Need extension of and the unknown from to { − ✏ < � < ✏ } f ∂ Ω γ • Parameterization of the interface using nearby level sets • Averaging over different parameterizations

Advantages of implicit interface boundary integral method • Inherit certain advantages of boundary integral methods: 
 solution on exterior domain problem, easy for both Dirichlet, Neumann, Robin, transmission problems. • Inherit certain benefits from implicit interface formulation: 
 one grid for complicated, topologically changing dynamic interface • Flexible on the grid geometry used to embed the interface. Multi-resolution grid possible.

Inverse problem and shape optimization Ω Ω Ω

Analysis of cancellous bone from micro-CT scan Compute effective material properties. Segmented by an MBO type scheme [Esedoglu-T]

LIDARs and 3D scanners

New formulas for integration over implicit surfaces

Parametrizing by a nearby level set Z Z ρ ( γ ( s )) Φ ( x, γ ( s )) ds = ρ ( y ∗ ( s η )) Φ ( x, y ∗ ( s η )) J η ds η ∂ Ω ∂ Ω η z = y + ~ ds y z ∗ = y ∗ + J ~ y ∗ = y � η r d ( y ) y ∗ ds ∂ Ω η ∂ Ω

The Jacobian ds d τ A 0 = ( R 1 − η ) θ 1 · ( R 2 − η ) θ 2 ⌘~ n R 1 A η = ds d τ = R 1 θ 1 · R 2 θ 2 θ 1 R 2 A η R 1 R 2 = ( R 1 − η )( R 2 − η ) A 0 = 1 + ( κ 1 + κ 2 ) η + O ( η 2 ) θ 2 J η = 1 + 2 H η η + G η η 2 In fact, we have:

Average the identities Z Z I 0 = ρ ( γ ( s )) Φ ( x, γ ( s )) ds = ρ ( y ∗ ( s η )) Φ ( x, y ∗ ( s η )) J η ds η = I η ∂ Ω ∂ Ω η Z ✏ Z I 0 = ρ ( γ ( s ))Φ( x, γ ( s )) ds = δ ✏ ( η ) I ⌘ d η @ Ω − ✏ Z = R n ρ ( z ∗ ) Φ ( x, z ∗ ) δ ✏ ( d Ω ( z )) J ( z ; d Ω ) dz h z ∗ = z � d ( z ) r d ( z )

The singular values of D P Σ P Σ y = y ∗ φ ( y ) = min x ∈ Σ | y − x | = y � φ ( y ) r φ ( y ) Proposition 3. The following identity holds for su ffi ciently small ✏ : ˆ ˆ g ( x ) dS x = R 3 g ( P Σ ( y )) � 1 � 2 K ✏ ( � ( y )) dy, Σ where K ✏ is any symmetric kernel supported in [ − ✏ , ✏ ] having unit mass. λ 1 λ 2 = 0 at the boundary of Σ [Kublik-T:2015]

The Jacobian and singular values of Proposition 1. Let � 1 ( x ) be the largest singular value of D P Γ ( x ) . The following identity holds for ✏ <  ∞ , where  ∞ is the maximal unsigned curvature of Γ . gds = 1 R 3 g ( P Γ ( x )) � 1 ( x ) K ✏ ( � ) ˆ ˆ dx. 2 ⇡ � Γ δ ✏ ( x ; Γ ) λ 1 = 0 near the end points of Γ K(x)=(1 − cos(2 π x))/x 2.5 2 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 x

High order approximation Integration over a torus: Relative Error Order n 6 . 2030 ⇥ 10 − 3 32 � 1 . 8073 ⇥ 10 − 4 64 5 . 10 6 . 6838 ⇥ 10 − 6 128 4 . 76 4 . 1530 ⇥ 10 − 7 256 4 . 01 5 . 0379 ⇥ 10 − 8 512 3 . 04

Surfaces with boundaries 3/4 sphere and third-order one-sided finite differencing: Relative Error Order n 1 . 1726 ⇥ 10 − 2 32 � 1 . 1733 ⇥ 10 − 3 64 3 . 32 9 . 1325 ⇥ 10 − 4 128 0 . 36 3 . 8238 ⇥ 10 − 4 256 1 . 26 7 . 8308 ⇥ 10 − 5 512 2 . 29

Relative Error Order n 5 . 5078 ⇥ 10 − 3 60 � 1 . 1476 ⇥ 10 − 3 120 2 . 63 2 . 3409 ⇥ 10 − 4 240 2 . 29 3 . 7166 ⇥ 10 − 5 480 2 . 66

Summation over point clouds { d Γ N = η } has improved regularity Γ N ⊂ Σ 1 0.5 0 − 0.5 − 1 1 − 1 0 0 − 1 1 • 30 × 30 uniformly distributed point clouds sampling in spherical coordinate the quarter sphere patch. • 50 × 50 × 50 uniform Cartesian grid discretizing [ − 1 , 1] 3 . • Relative error using � = 0 . 05 = dx : − 0 . 56 . • Relative error using � = 0 . 2 = 4 dx : − 0 . 061 . P : x 2 { d Γ N = η } 7! x � η r d Γ N ( x ) “interpolates”

Implicit interface boundary integral equation x, y ∈ { − ✏ ≤ d Ω ≤ ✏ } Z f ( x ∗ ) = λρ ( x ∗ ) + R d Φ ( x ∗ , y ∗ ) ρ ( y ∗ ) δ ✏ ( y ; d Ω ) dy x ∗ = x � d Ω ( x ) r d Ω ( x ) δ ✏ ( x ; d Ω ) := δ ✏ ( d Ω ( x )) J ( x ; d Ω ) Discretized by simple Riemann sum over the grid nodes. Solve the resulting linear system by an iterative solver. [Kublik-Tanushev-T:2013]


 x Regularization y Ω ∂ Φ • Need regularization of near singularity 
 ∂ n ∂ Φ ∂ n ( x, y ) ' A ✏ ( x, y ) for || x � y || < ε • Approximate surfaces by paraboloids ∂ Φ • Approximate weakly in a neighborhood of x on the ∂ n paraboloid (the 3D case): ∂ Φ Z Z ( x, y ) α ( y ) dS ( y ) ' A ✏ ( x, y ) α ( y ) dS ( y ) ∂ n y ˜ U ( x ; " ) U ( x ; " )

Regularization P 0 O osculating paraboloid 1 Z ∂ Φ ( x, y ) dS ( y ) ' 8 π h ( κ x + κ y ) | U ( x ; h ) | ∂ n y U ( x ; h )

Condition numbers N = 50 3 h Cond. number with the tangent regularization Cond. number with the paraboloid regularization 4 dx 6 . 2113 7 . 8322 2 dx 7 . 3343 6 . 9390 dx 7 . 0467 7 . 4197 dx 6 . 8948 6 . 7572 2 N = 80 3 h Cond. number with the tangent regularization Cond. number with the paraboloid regularization 4 dx 8 . 0859 7 . 8024 2 dx 8 . 2791 7 . 7919 dx 7 . 4127 8 . 1231 dx 7 . 8830 8 . 1192 2