Rational Barycentrics Polygons and Polycons 1 Properties - - PowerPoint PPT Presentation

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Rational Barycentrics

Polygons and Polycons

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Properties

• Regular within each element
• Degree-k polynomial basis within each element
• Global continuity

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Boundary

• Convex polygon: linear sides
• Well-set polycon: linear and conic sides
• Well-set polypol: rational algebraic sides

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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.

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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 (a2 + b2)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).

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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

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Linear Basis on adjacent sides

• Wj = kj Nj /Q where Q is the sum of the numerators and is known to

be unique.

• Linear on side (j,j+1) yields k recursion

kj Nj / (kj Nj + kj+1 Nj+1) on side (j, j+1)

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GADJ Recursion on a Line

• k j+1 from kj with k1 = 1
• Cancel common factor of sides from j + 2 (perhaps through n) to j –

1:

• k j(j+1;j+2)/[kj (j+1;j+2) + kj+1 (j; j-1)]
• Numerator linear so denominator constant

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Gradient at Vertex

• Gradient increases as angle approaches 180o

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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 180o.
• Map element with reasonable angles to element with angles closer

to 180o.

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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 180o vertices
• A brickwork pattern may be modeled

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Convex to 180o

• Isoparametric success limit

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Isoparametric Basis

• The isoparametric transformation from the unit diamond to the

triangle yields:

• W1 = 1 + x – (4x+y2)1/2
• W2 = .5[(4x+y2)1/2 – y – 2x], W3 = x
• W4 = .5[(4x+y2)1/2 + y – 2x]

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Brickwork

• Convex rational, rectangular isoparametric
• Concave mean-value

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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

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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

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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

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Adjacent Factors

• L- C and C-C adjacent factors at j

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GADJ

• The GADJ algorithm simplifies construction by giving constant ki for

numerator Ni = ki Fi where Fi 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

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GADJ on Conic Side

• kj+1 = bjkj (j+1; j+2)j / bj+1 (j-1; j)j+1
• k j+1/2 = cj kj / cj+1/2 (j-1; j)j+1/2
• D = kj [(j+1/2, j+1)j+(j, j+1)j+1/2+(j, j+1/2)j+1] = kj b and c from A.G.

congruences

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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

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Star Polycon

• Rational barycentrics

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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

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Parabolas Replacing Adjacent Lines

• A vertex of an element may have linear adjacent sides with an

interior angle close to 180o.

• 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
• f a parabolic side.

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A 3-con with a parabolic side

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Degree-one Basis

• For general a with x(1.5) = -1/a:
• Q = 2 + a – x
• W1(x,y) = .5(1 – x + y)[1 + y + a(x + y)]/Q
• W2(x,y) = .5(1 – x - y)[1 - y + a(x - y)]/Q
• W3(x,y) = (1 + ax –y2)/Q
• W1.5(x,y) = a[(1-x)2 – y2]/Q

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Parabola to line x = 0

• When a increases to infinity:
• W1 = .5(x+y)(1 – x + y)
• W2 = .5(x – y)(1 – x – y)
• W3 = x
• W1.5 = (1 – x)2 – y2

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Isoparametric Basis

• The isoparametric transformation from the unit diamond to the

triangle yields:

• W1 = 1 + x – (4x+y2)1/2
• W2 = .5[(4x+y2)1/2 – y – 2x], W3 = x
• W4 = .5[(4x+y2)1/2 + y – 2x]

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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 – y2 = 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.

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Hybrid Degree-one Triangle

• Degree two on side (1,2)

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Degree-one bases on parabola

• Now bases are quadratic instead of piecewise linear.
• quadratic piecewise linear

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Parabolic side

• The right element is concave but well-set
• The left element is convex and well-set
• The linear intersection is degree-two