NSF Workshop on Barycentric Coordinates | Columbia University, NY | July 26, 2012

S. E. Mousavia and N. Sukumarb

a University of Texas, Austinb University of California, Davis

Efficient Numerical Integration in

Polygonal Finite Element Method

Mousavi and Sukumar 2

Me =

Z

­0

N(»; ´)T N(»; ´)det(J)d»d´

Need for Integration

Ke =

Z

­0

(J¡1B(»; ´))T (J¡1B(»; ´))det(J)d»d´

Fe =

Z

­0

N(»; ´)T f(»; ´)det(J)d»d´

Mass matrix:

Stiffness matrix:

Force vector:

Mousavi and Sukumar 3

Me =

Z

­0

N(»; ´)T N(»; ´)det(J)d»d´

Need for Integration

Fe =

Z

­0

N(»; ´)T f(»; ´)det(J)d»d´

Mass matrix:

Stiffness matrix:

Force vector:

Ke =

Z

­0

(J¡1B(»; ´))T (J¡1B(»; ´))det(J)d»d´

Mousavi and Sukumar 4

Goal

n Constructing Gaussian-like quadratures for n-gons, n>3

(polynomial precision)

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Goal

n Constructing Gaussian-like quadratures for n-gons, n>3

(polynomial precision)

n Weighted quadratures: quadratures for polygonal basis

functions (rational polynomials)

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Outline

q Moment equations

q Node elimination algorithm

q Quadratures on the fly

q Weighted quadratures

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For a set of basis functions over the domain , find

the quadrature such that:

Moment Equations

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Moment Equations

For a set of basis functions over the domain , find

the quadrature such that:

q Newton iterations

q Least squares solution

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Moment Equations

For a set of basis functions over the domain , find

the quadrature such that:

q Newton iterations

o time-consuming, fewer points: n ≈ m/(d+1)

q Least squares solution

o faster, more points: n ≈ m

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Node Elimination Algorithm

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Node Elimination Algorithm

[Xiao and Gimbutas, Comp. Math. App., 2010]

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Node Elimination Algorithm

sj = !j

mX

i=1

Á2i (xj)

q Expected number of integration points:

(in two dimensions)

q Start from a quadrature over the partitions

q Significance factor:

m=3

[Xiao and Gimbutas, Comp. Math. App., 2010]

Mousavi and Sukumar 13

Polygonal Quadratures: AccuracyNumber of integration points for different n-gons

err =

qP(kij ¡ kref

ij )2

jkijjmax

[PolyMesher: Talischi, Paulino, et al., 2012]

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Displacement Patch Test

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Quadratures on the Fly

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Quadratures on the Fly

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q For strong discontinuity: replace the weight function with the

generalized Heaviside function

q For weak discontinuity: construct two quadratures on the two sides

of the interface

Quadratures on the Fly

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Homogeneous Quadratures

For a real, q-homogeneous function,

, we have:

[Lasserre, Proc. AMS, 1998]

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Homogeneous Quadratures

For a real, q-homogeneous function,

, we have:

(convex domains)

[Lasserre, Proc. AMS, 1998]

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Discontinuous Quadratures

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Discontinuous Quadratures

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Discontinuous Quadratures

[Mousavi and Sukumar, Comp. Mech., 2011]

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Discontinuous Quadratures

[Mousavi and Sukumar, Comp. Mech., 2011]

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Weighted Quadratures: mass matrix

Me =

Z

­0

N(»; ´)T N(»; ´)det(J)d»d´

J =

"@x@»

@y@»

@x@´

@y@´

#

x =nX

i=1

Ni(»; ´)xi; y =nX

i=1

Ni(»; ´)yi

Ni =p(»; ´)

Q(e.g., for hexagon: Q = 3 ¡ »2 ¡ ´2)

Typical element mass matrix

=) det(J) » 1

Q3

Jacobian of the transformation

Polygonal basis functions

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Ke =

Z

­

B(x; y)T B(x; y)dxdy

=

Z

­0

(J¡1B(»; ´))T (J¡1B(»; ´))det(J)d»d´

det(J) =P(»; ´)

Q3

) J¡1 =Q3

P(»; ´)[p(»; ´)

Q2] =

Q

P(»; ´)[p(»; ´)] (2 £ 2)

Weighted Quadratures: stiffness matrix

=) KeIJ =

Z

­0

p(»; ´)

P(»; ´)Q5d»d´

Element stiffness matrix

Inverse of the Jacobian

Bi =@Ni

@»=

@p@» Q ¡ @Q

@» p

Q2Basis function derivative

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Weighted Quadratures: AccuracyNumber of integration points for different n-gons

err =

qP(kij ¡ kref

ij )2

jkijjmax

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Displacement Patch Test

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Displacement Patch Test

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Conclusions

q Polygonal quadratures are more efficient than triangulation

q Polygonal quadratures need one level of mapping, whereas

partitioning requires two levels of mapping

q For higher accuracies, weighted quadratures are more

efficient than polynomial precision polygonal quadratures