Rational Barycentrics Polygons and Polycons 1
Properties • Regular within each element • Degree-k polynomial basis within each element • Global continuity 2
Boundary • Convex polygon: linear sides • Well-set polycon: linear and conic sides • Well-set polypol: rational algebraic sides 3
Polygon construction • Mean-value approach with distances and areas • My 1975 approach with basic properties: Global continuity: zero on opposite sides and denominator from EIP. Linear on adjacent sides. Nodes for degree-k approximation consistent with adjacent factor construction. 4
Distances and linear forms • Mean-value: z to side by square root and perpendicular to side with proper sign • Alternative is to normalize side on which ax+by+c = 0 to (a 2 + b 2 ) 1/2 = 1 and positive within the polygon. Then the signed distance to z = (x,y) is just the value of the linear form at (x,y). 5
Barycentrics • Degree-one rational bases are barycentric coordinates with rigorous theoretical foundation. • The numerators are normalized products of opposite sides • The denominator is the sum of the numerators 6
Linear Basis on adjacent sides • W j = k j N j /Q where Q is the sum of the numerators and is known to be unique. • Linear on side (j,j+1) yields k recursion k j N j / (k j N j + k j+1 N j+1 ) on side (j, j+1) 7
GADJ Recursion on a Line • k j+1 from k j with k 1 = 1 • Cancel common factor of sides from j + 2 (perhaps through n) to j – 1: • k j (j+1;j+2)/[k j (j+1;j+2) + k j+1 (j; j-1)] • Numerator linear so denominator constant 8
Gradient at Vertex • Gradient increases as angle approaches 180 o 9
Isoparametrics • A convex n-gon may be mapped into another convex n-gon • Rational basis not maintained • Poor rationals as interior angle at a vertex approaches 180 o . • Map element with reasonable angles to element with angles closer to 180 o . 10
Failure of mapping • An attempt to map isoparametrically from a convex to a non-convex n-gon fails • A region interior to the convex n-gon maps into points exterior to the mapped element! • The mapping succeeds at 180 o vertices • A brickwork pattern may be modeled 11
Convex to 180 o • Isoparametric success limit 12
Isoparametric Basis • The isoparametric transformation from the unit diamond to the triangle yields: • W 1 = 1 + x – (4x+y 2 ) 1/2 • W 2 = .5[(4x+y 2 ) 1/2 – y – 2x], W 3 = x • W 4 = .5[(4x+y 2 ) 1/2 + y – 2x] 13
Brickwork • Convex rational, rectangular isoparametric • Concave mean-value 14
Polycon • Distance-area approach not easily generalized to polycons • Continuity approach succeeds. No distance from point to curve. Evaluate un-normalized forms at adjacent vertices for GADJ numerator normalization 15
Degree-one Polycon • Well-set replaces convex • Well-set if side curve has no point in element other than side itself and vertices all simple transverse intersections • Degree-one rather than barycentric since not positive over element. • Need one side node on each conic side 16
Side nodes • Linear 3 d.o.f. on conic • Line-conic vertex has adjacent factor: line through exterior l-c intersection and conic side node • Conic-conic adjacent factor from conic determined by 2 side nodes and 3 EIP of conics 17
Adjacent Factors • L- C and C-C adjacent factors at j 18
GADJ • The GADJ algorithm simplifies construction by giving constant k i for numerator N i = k i F i where F i is the opposite factor at vertex i. • GADJ algorithm based on linearity on sides adjacent to vertex i • Denominator is just sum of numerators. • F and k from continuity and linearity 19
GADJ on Conic Side • k j+1 = b j k j (j+1; j+2) j / b j+1 (j-1; j) j+1 • k j+1/2 = c j k j / c j+1/2 (j-1; j) j+1/2 • D = k j [(j+1/2, j+1) j +(j, j+1) j+1/2 +(j, j+1/2) j+1 ] = k j b and c from A.G. congruences 20
Why curved sides? • Replace sides at concave vertex with parabola: regular rational replaces irrational with no derivative at the concave vertex • A side node replaces the vertex • Valid for star-shaped elements • Linear on parabola replaces piecewise linear with concave vertex at joint 21
Star Polycon • Rational barycentrics 22
Curved sides-2 • Curved boundaries formerly modeled with piecewise linear now approximated by algebraic sides • Finite element quadrature must be addressed • Graphics application promising: no integration 23
Parabolas Replacing Adjacent Lines • A vertex of an element may have linear adjacent sides with an interior angle close to 180 o . • Rational bases are poorly conditioned and mean-value coordinates have been recommended. • A vertex with a large interior angle may be chosen as the midpoint of a parabolic side. 24
A 3-con with a parabolic side 25
Degree-one Basis • For general a with x(1.5) = -1/a: • Q = 2 + a – x • W 1 (x,y) = .5(1 – x + y)[1 + y + a(x + y)]/Q • W 2 (x,y) = .5(1 – x - y)[1 - y + a(x - y)]/Q • W 3 (x,y) = (1 + ax –y 2 )/Q • W 1.5 (x,y) = a[(1-x) 2 – y 2 ]/Q 26
Parabola to line x = 0 • When a increases to infinity: • W 1 = .5(x+y)(1 – x + y) • W 2 = .5(x – y)(1 – x – y) • W 3 = x • W 1.5 = (1 – x) 2 – y 2 27
Isoparametric Basis • The isoparametric transformation from the unit diamond to the triangle yields: • W 1 = 1 + x – (4x+y 2 ) 1/2 • W 2 = .5[(4x+y 2 ) 1/2 – y – 2x], W 3 = x • W 4 = .5[(4x+y 2 ) 1/2 + y – 2x] 28
Degree-one hybrid • The basis is degree-one. • Quadratic approximation is achieved on the linear side (1,1.5,2). • (1,1.5,2) on the 3-con is the parabola on which 1 + ax – y 2 = 0. As a increases. this side approaches the line (1,2). The hybrid triangle basis is identical to the limit of the degree-one basis for the 3-con at a = infinity. 29
Hybrid Degree-one Triangle • Degree two on side (1,2) 30
Degree-one bases on parabola • Now bases are quadratic instead of piecewise linear. • quadratic piecewise linear 31
Parabolic side • The right element is concave but well-set • The left element is convex and well-set • The linear intersection is degree-two 32
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