IsoGeometric Analysis: B´ ezier techniques in Numerical Simulations Ahmed Ratnani IPP, Garching, Germany July 30, 2015 1 / 1 A. Ratnani IGA – CEMRACS 2015 1/1
Outline � Motivations � Computer Aided Design (CAD) Zoology � B´ ezier curves � Tensor Product surfaces � B-Spline curves � B´ ezier triangular surfaces � Splines/NURBS Finite Elements � The IsoGeometric Approach � Impact of the k-refinement strategy � The discrete DeRham diagram � Maxwell equations � MultiGrid Methods � Adaptive meshes � B´ ezier techniques in Computational Plasma Physics 2 / 1 A. Ratnani IGA – CEMRACS 2015 2/1
Motivations Engineering Analysis Process � Finite Elements Analysis (FEA) models are created from CAD representations � Fixing CAD geometry and creating FEA models accounts more than 80% of overall analysis time and is a major engineering bottelneck � The geometry is approximated in the FEA mesh 3 / 1 A. Ratnani IGA – CEMRACS 2015 3/1
Motivations Engineering Analysis Process � Finite Elements Analysis (FEA) models are created from CAD representations � Fixing CAD geometry and creating FEA models accounts more than 80% of overall analysis time and is a major engineering bottelneck � The geometry is approximated in the FEA mesh 3 / 1 A. Ratnani IGA – CEMRACS 2015 3/1
Motivations 4 / 1 A. Ratnani IGA – CEMRACS 2015 4/1
Motivations Even if the p-form: C ( x ( t )) = ∑ n i = 0 t i P i , is a natural description for curves, it presents some disadvantages : � the curve is not necessary regular everywhere. ➠ non-efficient approximation, � the points ( P i ) 0 ≤ i ≤ n do not have any geometric interpretation, � unstable numerical evaluation. 5 / 1 A. Ratnani IGA – CEMRACS 2015 5/1
Motivations Even if the p-form: C ( x ( t )) = ∑ n i = 0 t i P i , is a natural description for curves, it presents some disadvantages : � the curve is not necessary regular everywhere. ➠ non-efficient approximation, � the points ( P i ) 0 ≤ i ≤ n do not have any geometric interpretation, � unstable numerical evaluation. Properties ➠ Bezier techniques provide a geometric-based method for describing and manipulating polynomial curves and surfaces. � brings sophisticated mathematical concepts into a highly geometric and intuitive form. � this form facilitates the creative design process. � Bezier techniques are an excellent choice in the context of numerical stability of floating point operations[Farouki & Rajan]. ➠ Bezier techniques are at the core of 3D Modeling or Computer Aided Geometric Design (CAGD). 5 / 1 A. Ratnani IGA – CEMRACS 2015 5/1
Computer Aided Design B´ ezier curves Rather than use { 1, t , · · · , t n } as a basis of Π < n + 1 , we can take Bernstein polynomials; this leads to the B´ ezier-form. Therefore, it is equivalent to the p-form and writes : n B n ∑ C ( x ( t )) = i ( t ) P i , 0 ≤ t ≤ 1 (1) i = 0 where B n i denote Bernstein polynomials : � n � n ! t i ( 1 − t ) n − i = B n i ! ( n − i ) ! t i ( 1 − t ) n − i , i ( t ) = 0 ≤ t ≤ 1 i (2) The sequence ( P i ) 0 ≤ i ≤ n is called control points. Conics can be descrived exactly using Non-Rational B´ ezier arcs. 6 / 1 A. Ratnani IGA – CEMRACS 2015 6/1
Computer Aided Design B´ ezier curves 7 / 1 A. Ratnani IGA – CEMRACS 2015 7/1
Computer Aided Design B´ ezier curves 7 / 1 A. Ratnani IGA – CEMRACS 2015 7/1
Computer Aided Design B´ ezier curves 7 / 1 A. Ratnani IGA – CEMRACS 2015 7/1
Computer Aided Design B´ ezier curves 7 / 1 A. Ratnani IGA – CEMRACS 2015 7/1
Computer Aided Design B´ ezier curves 7 / 1 A. Ratnani IGA – CEMRACS 2015 7/1
Computer Aided Design B´ ezier curves � Invariance under some transformations : rotation, translation, scaling; it is sufficient to transform the control points, � B n i ( t ) ≥ 0, ∀ 0 ≤ t ≤ 1 � partition of unity : ∑ n i = 0 B n i ( t ) = 1, ∀ 0 ≤ t ≤ 1 � B n 0 ( 0 ) = B n n ( 1 ) = 1 � each B n i has exactly one maximum in [ 0, 1 ] , at i n i are symmetric with respect to 1 � B n 2 � recursive property : B n i ( t ) = ( 1 − t ) B n − 1 ( t ) + tB n − 1 i − 1 ( t ) , and i B n i ( t ) = 0, if i < 0 or, i > n � deriving a curve : C ′ ( t ) = n { ∑ n − 1 i = 0 B n − 1 ( t ) ( P i + 1 − P i ) } , then : i C ′ ( 0 ) = n ( P 1 − P 0 ) C ′ ( 1 ) = n ( P n − P n − 1 ) (3) C ′′ ( 0 ) = n ( n − 1 ) ( P 0 − 2 P 1 + P 2 ) C ′′ ( 1 ) = n ( P n − 2 P n − 1 + P n − 2 ) (4) � DeCasteljau algorithm: C n ( t ; P 0 , · · · , P n ) = ( 1 − t ) C n − 1 ( t ; P 0 , · · · , P n − 1 ) + t C n − 1 ( t ; P 1 , · · · , P n ) (5) 8 / 1 A. Ratnani IGA – CEMRACS 2015 8/1
Computer Aided Design Tensor Product surfaces Bezier patchs of arbitrary degrees A B´ ezier patch of degrees ( p , q ) is defined as p , q x ( s , t ) = ∑ x ij B i ( s ) B j ( t ) , s , t ∈ [ 0, 1 ] (6) i , j = 0 where ( x ij ) 0 ≤ i ≤ p ,0 ≤ j ≤ q are called control points Properties � Endpoint interpolation: The patch passes through the four corner control points { ( s , t ) = ( 0, 0 ) , ( 0, 1 ) , ( 1, 0 ) , ( 1, 1 ) } , � Each boundary corredponds to a Bezier curve � Symmetry in the parametric domain � Affine invariance (applied to the control points) � Convex Hull, � C 1 patchs can be easily created by solving (local/global) linear systems, using the endpoints derivatives and moving some specific control points, 9 / 1 A. Ratnani IGA – CEMRACS 2015 9/1
Computer Aided Design Tensor Product surfaces Geometric Operations � (Exact) Subdivision, � (Exact) Degree Elevation, � (Inexact) Patchs merge, (Exact if a Spline description is used) � (Inexact) Degree Reduction Figure : A mapping as a cubic Bezier patch (left) parametric domain with its domain points, (right) the resulting physical domain 10 / 1 A. Ratnani IGA – CEMRACS 2015 10/1
Computer Aided Design B-Splines curves � For a fixed number of control points, we have a fixed number of ezier curve ( = p + 1). degrees of freedom to control the B´ � For a better control of the curve, one can subdivise it into a given number of B´ ezier curves (using a refinement algorithm). � How to insure that the local regularity of the curve is preserved, when controling these curves? Need the notion of Macro-Elements or Macro-Patchs, where given regularities are imposed between elements. 11 / 1 A. Ratnani IGA – CEMRACS 2015 11/1
Computer Aided Design B-Splines curves (a) (b) (a) Original curve given as B´ ezier curves. (a) The quadratic B-spline curve and its control points. The knot vector is T = { 000, 1 4 , 1 2 , 3 4 , 111 } . 12 / 1 A. Ratnani IGA – CEMRACS 2015 12/1
Computer Aided Design B-Splines curves Figure : (left) A quadratic B-Spline curve and its control points using the knot vector T = { 000 1 3 3 4 111 } , (right) the corresponding B-Splines. 2 4 13 / 1 A. Ratnani IGA – CEMRACS 2015 13/1
Computer Aided Design B-Splines To create a family of B-splines , we need a non-decreasing sequence of knots T = ( t i ) 1 � i � N + k , also called knot vector , with k = p + 1. Each set of knots T j = { t j , · · · , t j + p } will generate a B-spline N j . Definition (B-Spline serie) The j-th B-Spline of order k is defined by the recurrence relation: j N k − 1 j + 1 ) N k − 1 N k j = w k + ( 1 − w k j j + 1 where, x − t j N 1 w k j ( x ) = j ( x ) = χ [ t j , t j + 1 [ ( x ) t j + k − 1 − t j for k ≥ 1 and 1 ≤ j ≤ N . 14 / 1 A. Ratnani IGA – CEMRACS 2015 14/1
Computer Aided Design B-Splines 2 1 1 N 1 N 1 N 1 N 2 N 2 N 2 N 3 N 3 N 3 N 4 N 4 N 4 N 5 N 5 N 5 N 6 N 6 N 6 N 7 0.8 N 7 0.8 N 7 1.5 N 8 N 8 N 8 0.6 0.6 1 0.4 0.4 0.5 0.2 0.2 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure : B-splines functions associated to the knot vector T = { 000 1 2 3 44 555 } , of order k = 1, 2, 3 Figure : Quadratic B-Splines for T = { 000, 111 } , T = { 000, 1 2 , 111 } and 15 / 1 T = { 000, 1 2 , 3 3 4 , 111 } . 4 A. Ratnani IGA – CEMRACS 2015 15/1
Computer Aided Design B´ ezier triangular surfaces Barycentric coordinates Let T = { v 1 , v 2 , v 3 } be a non-degenerate triangle in the plance. Then for all point P in the plane, there exists τ = { τ 1 , τ 2 , τ 3 } such that P = ∑ 3 i = 1 τ i v i . � τ is unique if one add the normalization constraint ∑ 3 i = 1 τ i = 1 (will be assumed during this talk) � P ∈ T if and only if τ ≥ 0 � Affine invariance: if the triangle T together with the point P are transformed by an affine transformation, the transformed point has unchanged barycentric coordinates Bernstein polynomials on triangles Let λ be a multi-index such that | λ | = n , T a triangle, and x a point in the plane, with τ as barycentric coordinates with respect to T . Bernstein polynomials are defined using the barycentric coordinates. n ! λ 1 ! λ 2 ! λ 3 ! τ 1 λ 1 τ 2 λ 2 τ 3 λ 3 B n λ ( τ ) = (7) Let ξ ijk = i v 1 + j v 2 + k v 3 . d The set D d , T = { ξ ijk , i + j + k = d } is the set of domain-points. 16 / 1 A. Ratnani IGA – CEMRACS 2015 16/1
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