B- Spline Constrained Deformations

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B-Spline Constrained Deformations Submitted by :- Course Instructor :- Avinash Kumar (10105017) Prof. Bhaskar Dasgupta Piyush Rai (10105070) (ME 751)

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B- Spline Constrained Deformations. Submitted by : - Course Instructor :- Avinash Kumar (10105017) Prof. Bhaskar Dasgupta Piyush Rai (10105070)(ME 751). Objective. To develop a deformable model and application of loads subjected to geometric constraints (point, boundary ,etc.) - PowerPoint PPT Presentation

Transcript of B- Spline Constrained Deformations

Page 1: B- Spline Constrained Deformations

B-Spline Constrained Deformations

Submitted by:- Course Instructor:-Avinash Kumar (10105017) Prof. Bhaskar DasguptaPiyush Rai (10105070) (ME 751)

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Objective

To develop a deformable model and application of loads subjected to geometric constraints (point, boundary ,etc.)

Defining the deformation energy functional to solve for deformed shape of curves and surfaces

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Direct manipulations of B-spline curves

A B-spline curve is defined by the following equation:

where are B-spline basis functions

Pi(u) are control point vectors

p = order of curve

Another way of representing B-spline curve is :C(u) = F1,p(u) F2,p(u) ………..Fn,p(u) P1 P2 ……….. PnT = [F][P]

where [F] = Blending matrix , [P] = matrix of control points of order (n x 3)

n

ipii uNPuC

0, )()(

)(, uN pi

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Deformable modelsThe extent of a curve’s deformation depends on two factors:

1. The external forces and constraints. Point constraint Boundary constraint

2. The physical properties of the curve, e.g. α and β terms, where α represents resistance to stretching, and β represents resistance to bending.

Fig. 1. - - - Initial B-spline curve. ___ Modified curve.

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Deformation Energy functional Finite Element approach

First, a B-spline curve is meshed into small curve segments, and each curve segment is regarded as an element, such that adjacent knot vectors are taken as an element, e.g. ti , ti+1 is an element .

For a curve, the energy functional is given by –

Uc = (1/2) ∫c [α (∂C(u)/ ∂u)2 + β(∂2C(u)/ ∂u2)2 ] du ………….. (i)

where, Uc is the deformation energy of the curve

C(u) is the arbitrary point on the curve α = Stretching stiffness , β = bending stiffness

By minimizing the energy functional Uc , the shape of a deformable model can be obtained.So, putting C(u) in eq. (i) , we get

Uc = (1/2) ∫c [α [P]T [∂F/ ∂u]T [∂F/ ∂u] [P] + β [P]T [∂2F/ ∂u2]T [∂2F/ ∂u2] [P] ] du

= (1/2) [P]T [∫c [α [∂F/ ∂u]T [∂F/ ∂u] + β[∂2F/ ∂u2]T [∂2F/ ∂u2] ] du].[P]

This equation resembles with the variational form as : U = (1/2) ∫Ω [a]T [K] [a]

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Comparing both the above equations , we get :

[K]nxn = ∫c [α [∂F/ ∂u]T [∂F/ ∂u] + β[∂2F/ ∂u2]T [∂2F/ ∂u2] ]

So, for the B-spline curve, the new control points can be obtained by :

[K]nxn [P]nx3 = [f]nx3

where [f] = force vector defined by user

This equation can be simplified into three independent equations given by:

[K] [Px] = [fx] , [K] [Py] = [fy] , [K] [Pz] = [fz]

Solving these equations, the new control point positions can be obtained .

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Calculation of [K] matrix

To calculate [K] matrix, Gaussian quadrature is used Each curve segment is regarded as an element

From the Gauss quadrature method,

we can find all entries of [K] matrix.

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Results1. Order of curve, p=3

knot vector,t=[0 0 0 0.25 0.5 0.75 1 1 1]

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2. Order of curve, p=4 knot vector,t=[0 0 0 0 0.3 0.7 1 1 1 1] Initial control points= [(0.2, 0.3) , (0.3, 0.51), (0.49, 0.57), (0.72, 0.73),

(0.85, 0.46)]

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B-Spline SurfacesA B-Spline surface patch can be represented as :

m

i

n

jjiij

m

i

n

jjiij vNuNPvuNPvuS

0 00 0, )()(),(),(

Possible ways of modifying the surface : By changing knot vector By moving the control points Changing the weights

Shape modification of B-spline surface with point constraint

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Surface modification with geometric constraints

• For each element r(u,v), we have :-

where, N = [N0,4(u)N0,4(v), N1,4(u)N1,4(v),….., N3,4(u)N3,4(v)] and, P = [P0,0 , P1,0 , P2,0 ,……….., P1,3 , P2,3 , P3,3 ]T

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Element Stiffness matrix (Ks)

Element force vector(Fs)

Assembling above B-spline surface element matrices and vector gives :-[K][P]=[F]

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Results 1. 4th-order B-Spline surface

t=[0,0,0,0,0.3,0.7,1,1,1,1]

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2. 3rd-order B-Spline surfacet=[0,0,0,0.25,0.5.0.75,1,1,1]

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References1. Direct manipulations of B-spline and NURBS curves, M.

Pourazady*, X. Xu2. Modifying the shape of NURBS surfaces with geometric

constraints , CHENG Si-yuan, ZHAO Bin, ZHANG Xiang-wei.3. Constraint-Based NURBS Surfaces Manipulation, Xiaoyan LIU,

Feng Feng.