Some studies on vorticity Dynamics for Two Dimensional flows on fixed smooth surfaces XiLin XIE, Qian SHI and Yu CHEN Department of Mechanics Engineering Science, Fudan University, Shanghai, CHINA Configurations and Coordinates Casel: Wakes of a Circular Cylinder on Fixed Smooth Surfaces Current Parametric Cowfigarwon Top View Incompressible Flow on Twin-Bells Surface Fixed Smooth Surface E=Ef(5,n) Re=100 .·, x=x(2,) Initial Parametric Configuration Re=500 2(5) Deformation Gradient FA2(Es, t)g (as, t)8G(zg)ET() b Surface Gradient Operator ) 垂≡()。-(y)g。-(9,89) Distribution Vo-9)8g+时hg。-n)g+时go-9)n b Levi-Civita Gradient Operator vo-4到gV当)0-(189)会o-出时189) =VA,g。-g)8g Intrinsic Generalized Stokes Formulas of pressure To-更d= n x Momentum Conservation Curve integrals transferred to surface integral c f ar (, )do+ m- fov)v d=AA b Surface Tension Frem:=f 7T x nd=ys H ndo dx/dal()=(Dr)dx/dx b Inner Pressure Fet: =fx(pn)dl=JE - da sity Fris: fu(r x n).(V &V)d=uf V(Vi ase2: Flows on Fixed Undulated Helical Surface b Novel NSEs Curvatures Roles Distribution of Mean Curvature pa=-2(x,)+yH+可1()+Ka+fm pan=B(-P)+p2V吲+ Distribution of Vorticity(Re=100) b Mass Conservation 0=/需=如+…0)=m=如+/on)如 0+(M)=(+小+p=+== vorticity Stream-function for Incompressible flows NSEs Curvature Role 4=E[VV+2V(KGV)]+-6vAfaurd b Pressure Equation for Incompressible flows NSEs Curvature Role △p=(v⑧V):(VV)+Kd|V-2V·(VKa)-V·fs Governing Equation for General 2D Compressible flows b Stokes-Helmholtz Decomposition V=Vo+Vx(n), with(Ao=VV=e △=-(VxV)·n=-3 Corollary( Casewell Formula) On any fixed solid boundary where the general viscous boundary condition is 6=-[(V 8v):(VaV)+KaIVF]+VpVp satisfied by the fluid, namely V=0 on the boundary, the strain tensor can be represented as following vp(△V+V+KV)+21△+·(kv)+fm D=n②n+(u×n)8n+n②(axn) D-+kd2+vp·vp--△ oundary, the directions corresponding to the maximum or mi ge of element material 共v+vxu+2T+++v,V)+Jm ngth with the same absolute value 1-1/2 is a/4 or 3/4 with respect to the tangent direction of the boundary. b Vorticity Equation 3-Vp,-Vp+叫-Vx如+2(Ve+KcV)n △+2×(K2V)]n-vfm,n+(xfm)n Casel: Wakes of a Circular Cylinder on Fixed Smooth Surfaces Streamline Distribution(3D View) Vorticity Distribution(3D View Reference 1. Xie X L, Chen Y, Shi Q. Some studies on mechanics of continuous mediums viewed as differential manifolds. Sci China-Phys Mech Astron, 2013, 56: 432-4 2. Xie X L. A theoretical framework of vorticity dynamics for two dimensional flows on fixed smooth surfaces. arXiv: 1304.-5145v1 [physics. flu-dynJ 18 Apr 2013 http://jpkc.fudaneducn/s/353/main.htm xiexilinofudan. edu. cn
Some Studies on Vorticity Dynamics for Two Dimensional Flows on Fixed Smooth Surfaces XiLin XIE, Qian SHI and Yu CHEN Department of Mechanics & Engineering Science, Fudan University, Shanghai, CHINA Configurations and Coordinates I Deformation Gradient F , ∂xi Σ ∂ξA Σ (ξΣ, t)gi (xΣ, t) ⊗ GA (xΣ) ∈ T 2 (R 3 ) I Surface Gradient Operator Σ ∇ ◦ −Φ ≡ g l ∂ ∂xl ◦ − Φ i ·jgi ⊗ g j , g l ◦ − ∂ ∂xl Φ i ·jgi ⊗ g j =∇lΦ i ·j (g l ◦ −gi ) ⊗ g j + Φi ·j bli(g l ◦ −n) ⊗ g j + Φi ·j b j l (g l ◦ −gi ) ⊗ n I Levi-Civita Gradient Operator ∇ ◦ −Φ ≡(g l∇ ∂ ∂xl ) ◦ −(Φi ·jgi ⊗ g j ) , g l ◦ −∇ ∂ ∂xl (Φi ·jgi ⊗ g j ) =∇lΦ i ·j (g l ◦ −gi ) ⊗ g j Intrinsic Generalized Stokes Formulas I C τ ◦ −Φ dl = Z Σ n × Σ ∇ ◦ −Φ dσ I C Φ ◦ −τ dl = Z Σ Φ ◦ − n × Σ ∇ dσ I Momentum Conservation:Curve integrals transferred to surface integral Z t Σ ∂(ρV ) ∂t (x, t) dσ + I ∂ t Σ n · (ρV )V dl = Z t Σ ρ a dσ = Ften + F int pre + Fvis + Fsur I Surface Tension Ften := H c γτ × ndl = γ R Σ H ndσ I Inner Pressure F int pre := − H c τ × (p n)dl = R Σ − Σ ∇p − pHn dσ I Inner Viscosity Fvis := H c µ(τ × n) · (V ⊗ Σ ∇)dl = µ R Σ Σ ∇ · ( Σ ∇ ⊗ V )dσ I Novel NSEs Curvatures Roles ρal = − ∂p ∂xl (x, t) + µ g ij∇i∇jVl + ∇l (∇sVs) + KGVl + fsur,l ρan = H(γ − p) + µ 2b ij∇iVj + fsur,n I Mass Conservation 0 = Z t Σ ∂ρ ∂t(x, t) dσ + I ∂ t Σ n · (ρV ) dl = Z t Σ ∂ρ ∂t(x, t) dσ + Z t Σ ∇ · (ρV ) dσ ∂ρ ∂t(x, t) + ∇ · (ρV ) = ∂ρ ∂t(x, t) + V i ∂ρ ∂xi (x, t) + ρ∇iV i = ρ˙ + ρ θ = 0, θ := ∇iV i I Vorticity & Stream-function for Incompressible flows NSEs Curvature Role 4ψ , g ij ∂ 2ψ ∂xi∂xj (x, t) − Γ k ij ∂ψ ∂xk (x, t) = −ω 3 ω˙ 3 = µ ρ ∇s∇sω 3 + 2 kl3∇k(KGVl) + 1 ρ kl3∇kfsur,l I Pressure Equation for Incompressible flows NSEs Curvature Role −∆p = ρ[(V ⊗ ∇) : (∇ ⊗ V ) + KG|V | 2 ] − 2µV · (∇KG) − ∇ · f Σ, Governing Equation for General 2D Compressible flows I Stokes-Helmholtz Decomposition V = ∇φ + ∇ × (ψn), with ∆φ = ∇ · V =: θ ∆ψ = −(∇ × V ) · n = −ω 3 I Dilation Equation ˙θ = − (V ⊗ ∇) : (∇ ⊗ V ) + KG|V | 2 + 1 ρ 2∇ρ · ∇p − 1 ρ ∆p − µ ρ 2∇ρ · (∆V + ∇θ + KGV ) + 2µ ρ [∆θ + ∇ · (KGV )] + ∇ · fsur = − D : D − |ω| 2 2 + KG|V | 2 + 1 ρ 2∇ρ · ∇p − 1 ρ ∆p − µ ρ 2∇ρ · [−∇ × ω + 2(∇θ + KGV )] + 2µ ρ [∆θ + ∇ · (KGV )] + ∇ · fsur I Vorticity Equation ω˙ 3 = −θω3 − 1 ρ 2 [∇ρ, −∇p + µ[−∇ × ω + 2 (∇θ + KGV )], n] + µ ρ [∆ω + 2∇ × (KGV )] · n − 1 ρ 2 [∇ρ, fsur, n] + 1 ρ (∇ × fsur) · n Case1:Wakes of a Circular Cylinder on Fixed Smooth Surfaces Streamline Distribution (3D View) Vorticity Distribution (3D View) Introduction I Most face recognition approaches are sensitive to registration errors . rely on a very good initial alignment and illumination I We propose/analyze: . grid-based and dense extraction of local features . block-based matching accounting for different viewpoints and registration errors Feature Extraction I Interest point based feature extraction . SIFT or SURF interest point detector . leads to a very sparse description I Grid-based feature extraction . overlaid regular grid . leads to a dense description Orig. IP Grid Feature Description I Scale Invariant Feature Transform (SIFT) . 128-dimensional descriptor, histogram of gradients, scale invariant I Speeded Up Robust Features (SURF) . 64-dimensional descriptor, histogram of gradients, scale invariant I face recognition: invariance w.r.t. rotation is often not necessary . rotation dependent upright-versions U-SIFT, U-SURF-64, U-SURF-128 Feature Matching I Recognition by Matching . nearest neighbor matching strategy . descriptor vectors extracted at keypoints in a test image X are compared to all descriptor vectors extracted at keypoints from the reference images Yn, n = 1, · · · , N by the Euclidean distance . decision rule: X → r(X) = arg max c n max n X xi∈X δ(xi , Yn,c) o . additionally, a ratio constraint is applied in δ(xi , Yn,c) I Viewpoint Matching Constraints . maximum matching: unconstrained . grid-based matching: absolute box constraints . grid-based best matching: absolute box constraints, overlapping I Postprocessing . RANSAC-based outlier removal . RANSAC-based system combination Matching Examples for the AR-Face and CMU-PIE Database Feature Maximum Grid Grid-Best Maximum Grid Grid-Best Feature SIFT SURF U-SIFT U-SURF I Matching results for the AR-Face (left) and the CMU-PIE database (right) . maximum matching show false classification examples . grid matchings show correct classification examples . upright descriptor versions reduce the number of false matches Case1:Wakes of a Circular Cylinder on Fixed Smooth Surfaces Top View Incompressible Flow on Twin-Bells Surface Plane Re=100 Re=500 Distribution of vorticity Distribution of pressure Distribution of first eigenvalue of the strain tensor ˙ d t X/dλ R3 (λ) = (τ · D · τ ) d t X/dλ R3 Case2: Flows on Fixed Undulated Helical Surface Distribution of Vorticity (Re=100) Distribution of Mean Curvature Distribution of Gaussian Curvature Corollary (Caswell Formula) Corollary (Casewell Formula) On any fixed solid boundary where the general viscous boundary condition is satisfied by the fluid, namely V = 0 on the boundary, the strain tensor can be represented as following D = θn ⊗ n + 1 2 (ω × n) ⊗ n + 1 2 n ⊗ (ω × n) Corollary For any two dimensional incompressible viscous flow on any fixed smooth surface, on any fixed solid boundary, the directions corresponding to the maximum or minimum rate of change of element material arc length with the same absolute value |ω 3 |/2 is π/4 or 3π/4 with respect to the tangent direction of the boundary. Reference 1. Xie X L, Chen Y, Shi Q. Some studies on mechanics of continuous mediums viewed as differential manifolds. Sci China-Phys Mech Astron, 2013, 56:432-456. 2. Xie X L. A theoretical framework of vorticity dynamics for two dimensional flows on fixed smooth surfaces. arXiv:1304.5145v1 [physics.flu-dyn] 18 Apr 2013 Databases I AR-Face . variations in illumination . many different facial expressions I CMU-PIE . variations in illumination (frontal images from the illumination subset) Results: Manually Aligned Faces I AR-Face: 110 classes, 770 train, 770 test Descriptor Extraction # Features Error Rates [%] Maximum Grid Grid-Best SURF-64 IPs 164 × 5.6 (avg.) 80.64 84.15 84.15 SIFT IPs 128 × 633.78 (avg.) 1.03 95.84 95.84 SURF-64 64x64-2 grid 164 × 1024 0.90 0.51 0.90 SURF-128 64x64-2 grid 128 × 1024 0.90 0.51 0.38 SIFT 64x64-2 grid 128 × 1024 11.03 0.90 0.64 U-SURF-64 64x64-2 grid 164 × 1024 0.90 1.03 0.64 U-SURF-128 64x64-2 grid 128 × 1024 1.55 1.29 1.03 U-SIFT 64x64-2 grid 128 × 1024 0.25 0.25 0.25 I CMU-PIE: 68 classes, 68 train (“one-shot” training), 1360 test Descriptor Extraction # Features Error Rates [%] Maximum Grid Grid-Best SURF-64 IPs 164 × 6.80 (avg.) 93.95 95.21 95.21 SIFT IPs 128 × 723.17 (avg.) 43.47 99.33 99.33 SURF-64 64x64-2 grid 164 × 1024 13.41 4.12 7.82 SURF-128 64x64-2 grid 128 × 1024 12.45 3.68 3.24 SIFT 64x64-2 grid 128 × 1024 27.92 7.00 9.80 U-SURF-64 64x64-2 grid 164 × 1024 3.83 0.51 0.66 U-SURF-128 64x64-2 grid 128 × 1024 5.67 0.95 0.88 U-SIFT 64x64-2 grid 128 × 1024 16.28 1.40 6.41 Results: Unaligned Faces I Automatically aligned by Viola & Jones Descriptor Error Rates [%] AR-Face CMU-PIE SURF-64 5.97 15.32 SURF-128 5.71 11.42 SIFT 5.45 8.32 U-SURF-64 5.32 5.52 U-SURF-128 5.71 4.86 U-SIFT 4.15 8.99 I Manually aligned faces I Unaligned faces Results: Partially Occluded Faces I AR-Face: 110 classes, 110 train (“one-shot” training), 550 test Descriptor Error Rates [%] AR1scarf AR1sun ARneutral AR2scarf AR2sun Avg. SURF-64 2.72 30.00 0.00 4.54 47.27 16.90 SURF-128 1.81 23.63 0.00 3.63 40.90 13.99 SIFT 1.81 24.54 0.00 2.72 44.54 14.72 U-SURF-64 4.54 23.63 0.00 4.54 47.27 15.99 U-SURF-128 1.81 20.00 0.00 3.63 41.81 13.45 U-SIFT 1.81 20.90 0.00 1.81 38.18 12.54 U-SURF-128+R 1.81 19.09 0.00 3.63 43.63 13.63 U-SIFT+R 2.72 14.54 0.00 0.90 35.45 10.72 U-SURF-128+U-SIFT+R 0.90 16.36 0.00 2.72 32.72 10.54 Conclusions I Grid-based local feature extraction instead of interest points I Local descriptors: . upright descriptor versions achieved better results . SURF-128 better than SURF-64 I System robustness: manually aligned/unaligned/partially occluded faces . SURF more robust to illumination . SIFT more robust to changes in viewing conditions I RANSAC-based system combination and outlier removal Created with LATEXbeamerposter http://jpkc.fudan.edu.cn/s/353/main.htm xiexilin@fudan.edu.cn