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

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

Direct manipulations of B-spline curves A B-spline curve is defined by the following equation: where are B-spline basis functions P i (u) are control point vectors p = order of curve Another way of representing B-spline curve is : C(u) = { F 1,p (u) F 2,p (u) ………..F n,p (u) } {P 1 P 2 ……….. P n } T = [F][P] where [F] = Blending matrix, [P] = matrix of control points of order (n x 3)

Deformable models The 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 Initial B-spline curve. ___ Modified curve.

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. {t i, t i+1 } is an element. For a curve, the energy functional is given by – U c = (1/2) ∫ c [α (∂C(u)/ ∂u) 2 + β(∂ 2 C(u)/ ∂u 2 ) 2 ] du ………….. (i) where, U c 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 U c, the shape of a deformable model can be obtained. So, putting C(u) in eq. (i), we get U c = (1/2) ∫ c [α [P] T [∂F/ ∂u] T [∂F/ ∂u] [P] + β [P] T [∂ 2 F/ ∂u 2 ] T [∂ 2 F/ ∂u 2 ] [P] ] du = (1/2) [P] T [ ∫ c [α [∂F/ ∂u] T [∂F/ ∂u] + β[∂ 2 F/ ∂u 2 ] T [∂ 2 F/ ∂u 2 ] ] du ]. [P] This equation resembles with the variational form as : U = (1/2) ∫ Ω [a] T [K] [a]

Comparing both the above equations, we get : [K] nxn = ∫ c [α [∂F/ ∂u] T [∂F/ ∂u] + β[∂ 2 F/ ∂u 2 ] T [∂ 2 F/ ∂u 2 ] ] 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] [P x ] = [f x ], [K] [P y ] = [f y ], [K] [P z ] = [f z ] 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.

Results 1.Order of curve, p=3 knot vector,t=[ ]

2. Order of curve, p=4 knot vector,t=[ ] 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 constraint

Surface modification with geometric constraints For each element r(u,v), we have :- where, N = [N 0,4 (u)N 0,4 (v), N 1,4 (u)N 1,4 (v),….., N 3,4 (u)N 3,4 (v)] and,P = [P 0,0, P 1,0, P 2,0,……….., P 1,3, P 2,3, P 3,3 ] T

Element Stiffness matrix (K s ) Element force vector(F s ) Assembling above B-spline surface element matrices and vector gives :- [K][P]=[F]

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

2. 3rd-order B-Spline surface t=[0,0,0,0.25, ,1,1,1]

References 1.Direct manipulations of B-spline and NURBS curves, M. Pourazady*, X. Xu 2.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.