NUMERICAL INTEGRATION of HOMOGENEOUS FUNCTIONS and POLYNOMIALS on - - PowerPoint PPT Presentation
UC DAVIS Eric B. Chin Jean B. Lasserre LAAS-CNRS N. Sukumar UC DAVIS Atlanta, GA // October 28, 2015 Research support of the NSF is gratefully acknowledged NUMERICAL INTEGRATION of HOMOGENEOUS FUNCTIONS and POLYNOMIALS on POLYTOPES POEMs
and POLYNOMIALS on
POEMs 2015
UC DAVIS
LAAS-CNRS
UC DAVIS
Atlanta, GA // October 28, 2015
Research support of the NSF is gratefully acknowledged
2/33
Determine
▶ f (x) is a homogeneous function ▶ P is a convex or nonconvex polytope
3/33
Three methods to integrate functions on polytopes Triangulation Divergence theorem Moment fjtting ∫
V ∇ · F dV =
∫
S F · n dS
x x x x x x x
4/33
Euler’s homgeneous function theorem and Generalized Stokes’s theorem
Integration over convex and nonconvex polytopes
4/33
Euler’s homgeneous function theorem and Generalized Stokes’s theorem
Integration over convex and nonconvex polytopes
4/33
Euler’s homgeneous function theorem and Generalized Stokes’s theorem
Integration over convex and nonconvex polytopes
4/33
Euler’s homgeneous function theorem and Generalized Stokes’s theorem
Integration over convex and nonconvex polytopes
4/33
Euler’s homgeneous function theorem and Generalized Stokes’s theorem
Integration over convex and nonconvex polytopes
Background 5/33
Background 6/33
A continuously difgerentiable function f (x) is said to be positive homogeneous of degree q if:
and then it also satisfjes:
⟨·, ·⟩: inner product d: dimension
Background 7/33
By defjnition, a homogeneous function of degree q has the property λqf(x) = f(λx) Defjne x′ := λx and calculate
∂ ∂λ:
qλq−1f(x) = ∂f ∂x′ · ∂x′ ∂λ = ∂f ∂x′ · x Let λ = 1: qf(x) = ∂f ∂x · x = ⟨∇f(x), x⟩
▶ Converse is also readily established
Background 8/33
Background 8/33
2:
1 √r,
x
2:
Background 9/33
(See: Taylor, PDEs: Basic Theory, 2011) ∫
M
dω = ∫
∂M
ω
∫
M
(∇ · X) f (x) dx + ∫
M
X · ∇f (x) dx = ∫
∂M
(X · n) f (x) dσ
▶ X: vector fjeld ▶ M: region of integration ▶ dσ: Lebesgue measure on ∂M
Extension of Lasserre’s method 10/33
Extension of Lasserre’s method 11/33
▶ Method developed by Lasserre (Proc. Am. Math.
Soc., 1998, 1999) for convex regions
▶ First applied to X-FEM by Mousavi and S (Comp.
Mech., 2011)
▶ Extended to nonconvex regions by Chin et al. (Comp.
Mech., 2015, doi 10.1007/s00466-015-1213-7) [PDF]
▶ Method uses properties of homogeneous functions and
generalized Stokes’s theorem
Main results 12/33
Main results 13/33
Apply Stokes’s theorem with X := x and f (x) is q-homogeneous: d ∫
P
f (x) dx + ∫
P
⟨∇f (x) , x⟩dx =
m
∑
i=1
∫
Fi
(x · ni) f (x) dσ P: polygon Fi: boundary facets Apply Euler’s homogeneous fn. theorem, qf (x) = ⟨∇f (x) , x⟩: d ∫
P
f (x) dx + q ∫
P
f (x) dx =
m
∑
i=1
∫
Fi
(x · ni) f (x) dσ ∫
P
f (x) dx = 1 d + q
m
∑
i=1
∫
Fi
(x · ni) f (x) dσ
Main results 14/33
∫
P
f (x) dx = 1 d + q
m
∑
i=1
∫
Fi
(x · ni)f (x) dσ
▶ Fi ⊂ ai · x = bi: equation of a hyperplane ▶ ni = ai ∥ai∥: unit normal to hyperplane ▶ x · ni = x · ai ∥ai∥ = x·ai ∥ai∥ = bi ∥ai∥
∴ ∫
P
f (x) dx = 1 d + q
m
∑
i=1
∫
Fi
bi ∥ai∥f (x) dσ (*)
▶ Using (*), one can reduce integration to the boundary of
the polytope
Main results 15/33
Question: x = 1 and −x = −1 produce the same line, yet only
Answer: Given the vertices of the polygon, travel around the polygon in a clockwise direction.
P F1 start 1 2
Main results 15/33
Question: x = 1 and −x = −1 produce the same line, yet only
Answer: Given the vertices of the polygon, travel around the polygon in a clockwise direction.
P F1 F2 1 2 3
Main results 15/33
Question: x = 1 and −x = −1 produce the same line, yet only
Answer: Given the vertices of the polygon, travel around the polygon in a clockwise direction.
P F1 F2 F3 1 2 3 4
Main results 15/33
Question: x = 1 and −x = −1 produce the same line, yet only
Answer: Given the vertices of the polygon, travel around the polygon in a clockwise direction.
P F1 F2 F3 F4 1 2 3 4 5
Main results 15/33
Question: x = 1 and −x = −1 produce the same line, yet only
Answer: Given the vertices of the polygon, travel around the polygon in a clockwise direction.
P F1 F2 F3 F4 F5 1 2 3 4 5 6
Main results 15/33
Question: x = 1 and −x = −1 produce the same line, yet only
Answer: Given the vertices of the polygon, travel around the polygon in a clockwise direction.
P F1 F2 F3 F4 F5 F6 1 2 3 4 5 6 7
Main results 15/33
Question: x = 1 and −x = −1 produce the same line, yet only
Answer: Given the vertices of the polygon, travel around the polygon in a clockwise direction.
P F1 F2 F3 F4 F5 F6 F7 finish 1 2 3 4 5 6 7
Main results 16/33
det
y 1 x1 y1 1 x2 y2 1
ai = {y1 − y2, x2 − x1}T bi = −(x1y2 − y1x2)
Main results 17/33
∫
M
(∇ · X) f (x) dx + ∫
M
X · ∇f (x) dx = ∫
∂M
(X · n) f (x) dσ Select:
▶ M as Fi ▶ ∂M as vij (vertices in I
R2 – intersection of Fi and Fj)
▶ X := x − x0 (x0: any point on hyperplane containing Fi) ▶ f (x) is a homogeneous function of degree q
Main results 18/33
∫
M
(∇ · X) f (x) dx + ∫
M
X · ∇f (x) dx = ∫
∂M
(X · n) f (x) dσ
∫
Fi
f (x) dσ = 1 d + q − 1 [ 2 ∑
j=1
dijf (vij) + ∫
Fi
⟨∇f (x) , x0⟩ dσ ] (**)
▶ dij := ⟨x − x0, nij⟩ – algebraic distance from vij to x0 ▶ ∫ Fi ⟨∇f (x) , x0⟩ dσ can be applied recursively
Main results 19/33
∫
P
f (x) dx = 1 d + q
m
∑
i=1
∫
Fi
bi ∥ai∥f (x) dσ (*) ∫
Fi
f (x) dσ = 1 d + q − 1 [ 2 ∑
j=1
dijf (vij) + ∫
Fi
⟨∇f (x) , x0⟩ dσ ] (**)
▶ These formulas can be used to reduce integration to the
vertices of the polytope
▶ Further, a closed-form cubature rule can be developed ▶ If the partial derivatives of f(x) eventually vanish, this
cubature rule is exact
Main results 20/33
Combine (*) with (**): ∫
P
f(x) dx = ∑m
i=1 bi ∥ai∥
∑
j̸=i dijI(vij)
(q + 2)(q + 1) where I(vij) :=
q
∑
k=0
Qk(vij) (q
k
) and Qk(vij) := ∑
|α|=q−k
D|α|f(vij) α!
2
∏
ℓ=1
(x0ℓ)αℓ
▶ α is an n-tuple of nonnegative integers ▶ D is the difgerential operator in multi-index notation
Main results 21/33
Consider a region V bounded by homogeneous functions hi (x) = bi (i = 1, . . . , m) of degree qi Apply Stokes’s theorem: ∫
V
f (x) dx = 1 d + q
m
∑
i=1
qibi ∫
Ai
∥∇hi∥−1f (x) dσ Polar transformation: ∫
V
f (x) dx = 1 2 + q
m
∑
i=1
∫ βi
αi
H2
i (θ)f (x(θ)) dθ ▶ Region bounded by equations of the form r = Hi (θ)
Numerical examples 22/33
Numerical examples 23/33
Integrate f(x) = x2 + xy + y2 over convex and nonconvex polygons
−5 5 −4 −3 −2 −1 1 2 3
∫
P f(x) dx ≈ 323.1821
−5 5 −2 −1 1 2 3 4
∫
P f(x) dx ≈ 80.95348 ▶ Results verifjed in LattE (De Loera et al., Comput
Geom, 2013)
Numerical examples 24/33
Integrate f(x) = x2 + y2 + z2 over nonconvex polyhedra Octahedron 5-compound ∫
P f(x) dx ≈
0.353553 Echidnahedron ∫
P f(x) dx ≈
253.5696 Cube 5-compound ∫
P f(x) dx ≈
1.250000
▶ Shape data from PolyhedronData[] in Mathematica
Numerical examples 25/33
A := {r ∈ [0, 1], θ ∈ [π/4, π/2], r ≥ cos θ, r ≤ sin θ, θ ≤ π/2} ∫
A
1 √ x2 + y2 = √ 2 − 1
1 2 3 4 5 6 7 8 10
−16
10
−12
10
−8
10
−4
10 Total number of integration points Relative error of integration
▶ Weakly singular integrand at the origin ▶ Using equation derived for a curved region, domain
integral is transformed to 1D line integrals
▶ With 6 quadrature points, integration error is close to
machine precision
Application: X-FEM (Chin & S, work in progress, 2015) 26/33
Application: X-FEM (Chin & S, work in progress, 2015) 27/33
Numerical integration in elements with discontinuous and weakly singular integrands Current approach
x1 x2 y2 y1 x y
Ωe
+
2 1 4 3
Ωe
partitioning!
Application: X-FEM (Chin & S, work in progress, 2015) 28/33
▶ Use (*) to reduce integration, then apply quadrature ▶ f (x) = sin(θ/2)/√r: discontinuous and weakly singular ▶ Biunit square centered at (0.5, 0.5)
10 20 30 40 50 60 70 10
−10
10
−8
10
−6
10
−4
10
−2
10 Total number of integration points Relative error of integration Polar Cartesian Tensor
Application: X-FEM (Chin & S, work in progress, 2015) 29/33
Strong form ∇ · σ = 0 in Ω u = ¯ u on Γu σ · n = ¯ t on Γt σ · n = 0 on Γc Weak form a(u, δu) = ℓ(δu) ∀δu ∈ U0, a(u, δu) := ∫
Ω
σ : δϵ dx, ℓ(δu) := ∫
Γt
¯ t · δu dS
Application: X-FEM (Chin & S, work in progress, 2015) 30/33
Standard FEM u(x) = ∑
i∈I
Ni(x)ui Extended FEM [X-FEM] (Moës et al., IJNME, 1999) u(x) = ∑
i∈I
Ni(x)ui+ ∑
j∈J⊆I
Nj(x)φ(x)aj+ ∑
k∈Kt⊆I
Nk(x)
2
∑
t=1 4
∑
α=1
Fαt(x)bkαt ui, aj, bkαt - degrees of freedom (DOFs) φ(x) - discont. enrichment
( usually ϕ(x) := H(x) = { 1 x ∈ Ω+
e
−1 x ∈ Ω−
e
)
Fαt(x) - crack-tip enrichment
Application: X-FEM (Chin & S, work in progress, 2015) 31/33
σ1 σ1
2
σ
2
σ β a a x 2w 2w x2
1
β K1 (exact) K1 (GD) K1 (CLS) 0◦ 1.2533 1.2547 1.2549 15◦ 1.3373 1.3379 1.3378 30◦ 1.5666 1.5666 1.5662 45◦ 1.8800 1.8864 1.8861 60◦ 2.1933 2.1936 2.1935 75◦ 2.4227 2.4255 2.4251 90◦ 2.5066 2.5095 2.5088
▶ Integrand in the interaction integral are homogeneous
functions ⇒ can use Lassere’s approach to compute SIFs
▶ For the results shown above, Triangulation is used to
compute the interaction integral in cracked elements
Conclusions and outlook 32/33
Conclusions and outlook 33/33
▶ Presented a method to integrate homogeneous functions
Euler’s homogeneous function theorem and generalized Stokes’s theorem Cubature rules for polynomials on polytopes in Accurate and effjcient integration of weakly singular functions was realized Method was shown to be applicable to curved regions Showcased an application in elastic fracture using the X-FEM New integration scheme is well-suited for adoption in novel discretization techniques such as MFD, VEM, WG, DG, PUFEM, embedded interface methods,
Conclusions and outlook 33/33
▶ Presented a method to integrate homogeneous functions
Euler’s homogeneous function theorem and generalized Stokes’s theorem
▶ Cubature rules for polynomials on polytopes in I
Rd Accurate and effjcient integration of weakly singular functions was realized Method was shown to be applicable to curved regions Showcased an application in elastic fracture using the X-FEM New integration scheme is well-suited for adoption in novel discretization techniques such as MFD, VEM, WG, DG, PUFEM, embedded interface methods,
Conclusions and outlook 33/33
▶ Presented a method to integrate homogeneous functions
Euler’s homogeneous function theorem and generalized Stokes’s theorem
▶ Cubature rules for polynomials on polytopes in I
Rd
▶ Accurate and effjcient integration of weakly singular
functions was realized Method was shown to be applicable to curved regions Showcased an application in elastic fracture using the X-FEM New integration scheme is well-suited for adoption in novel discretization techniques such as MFD, VEM, WG, DG, PUFEM, embedded interface methods,
Conclusions and outlook 33/33
▶ Presented a method to integrate homogeneous functions
Euler’s homogeneous function theorem and generalized Stokes’s theorem
▶ Cubature rules for polynomials on polytopes in I
Rd
▶ Accurate and effjcient integration of weakly singular
functions was realized
▶ Method was shown to be applicable to curved regions
Showcased an application in elastic fracture using the X-FEM New integration scheme is well-suited for adoption in novel discretization techniques such as MFD, VEM, WG, DG, PUFEM, embedded interface methods,
Conclusions and outlook 33/33
▶ Presented a method to integrate homogeneous functions
Euler’s homogeneous function theorem and generalized Stokes’s theorem
▶ Cubature rules for polynomials on polytopes in I
Rd
▶ Accurate and effjcient integration of weakly singular
functions was realized
▶ Method was shown to be applicable to curved regions ▶ Showcased an application in elastic fracture using the
X-FEM New integration scheme is well-suited for adoption in novel discretization techniques such as MFD, VEM, WG, DG, PUFEM, embedded interface methods,
Conclusions and outlook 33/33
▶ Presented a method to integrate homogeneous functions
Euler’s homogeneous function theorem and generalized Stokes’s theorem
▶ Cubature rules for polynomials on polytopes in I
Rd
▶ Accurate and effjcient integration of weakly singular
functions was realized
▶ Method was shown to be applicable to curved regions ▶ Showcased an application in elastic fracture using the
X-FEM
▶ New integration scheme is well-suited for adoption in
novel discretization techniques such as MFD, VEM, WG, DG, PUFEM, embedded interface methods, . . .