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

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B-Spline Surface Deformations

B-Spline Constrained Deformations Submitted by:- Course Instructor:-Avinash Kumar (10105017) Prof. Bhaskar DasguptaPiyush Rai (10105070)(ME 751)ObjectiveTo 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 surfacesDirect manipulations of B-spline curvesA 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 .. Pn}T = [F][P] where [F] = Blending matrix , [P] = matrix of control points of order (n x 3)

Deformable modelsThe extent of a curves deformation depends on two factors:The external forces and constraints. Point constraint Boundary constraint2. The physical properties of the curve, e.g. and terms, where represents resistance to stretching, and represents resistance to bending.

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

Deformation Energy functional Finite Element approachFirst, 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 [ (C(u)/ u)2 + (2C(u)/ u2)2 ] du .. (i)where, Uc is the deformation energy of the curveC(u) is the arbitrary point on the curve = Stretching stiffness , = bending stiffnessBy minimizing the energy functional Uc , the shape of a deformable model can be obtained.So, putting C(u) in eq. (i) , we getUc = (1/2) c [ [P]T [F/ u]T [F/ u] [P] + [P]T [2F/ u2]T [2F/ u2] [P] ] du = (1/2) [P]T [c [ [F/ u]T [F/ u] + [2F/ u2]T [2F/ u2] ] du].[P]

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

Comparing both the above equations , we get :[K]nxn = c [ [F/ u]T [F/ u] + [2F/ u2]T [2F/ u2] ] So, for the B-spline curve, the new control points can be obtained by :[K]nxn [P]nx3 = [f]nx3where [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 .

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.

ResultsOrder of curve, p=3knot vector,t=[0 0 0 0.25 0.5 0.75 1 1 1]

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)]

B-Spline Surfaces

A B-Spline surface patch can be represented as :

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 constraintSurface 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

Element Stiffness matrix (Ks)Element force vector(Fs)

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

Results 1. 4th-order B-Spline surfacet=[0,0,0,0,0.3,0.7,1,1,1,1]

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

ReferencesDirect manipulations of B-spline and NURBS curves, M. Pourazady*, X. XuModifying the shape of NURBS surfaces with geometric constraints , CHENG Si-yuan, ZHAO Bin, ZHANG Xiang-wei.Constraint-Based NURBS Surfaces Manipulation, Xiaoyan LIU, Feng Feng.

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