1. No category

The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
The stable GFEM. Convergence, accuracy and
Diffpack implementation
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
Daniel Alves Paladim
Sundararajan Natarajan
St´ephane Bordas
Pierre Kerfriden
Institute of Mechanics and Advanced Materials
Cardiff University
May 12, 2015
1 / 24
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
1
Context
2
Blending
3
Ill-conditioning
4
Numerical integration
5
Numerical examples
6
Conclusion
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
2 / 24
Context
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Diffpack is a commercial software library used for the
development numerical software, with main emphasis on
numerical solutions of partial differential equations. It was
developed in C++ following the object oriented paradigm.
The library is mostly oriented to the implementation of the
finite element method, however it has tools for other methods
such as finite volume and finite differences.
Conclusion
3 / 24
Context
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
The extended/generalized finite element method is usually
connected to the following issues:
Blending
Illconditioning
Numerical
integration
• Blending
• Ill-conditioning of the stiffness matrix
• Numerical integration
Numerical
examples
Conclusion
4 / 24
Blending
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Blending
Illconditioning
In the extended finite element method, there are 3 types of
elements.
• Elements with all its nodes enriched
• Elements that none of its nodes enriched.
• Elements that have both type of nodes (blending
elements).
In those elements, there is no partition of unity and the
convergence rate is degraded.
Numerical
integration
Numerical
examples
Conclusion
5 / 24
Blending solution
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
Babuska and Banerjee (2011) proposed the stable generalized
finite element method. In the SGFEM, the approximation has
the following form
X
X
uh (x) =
Ni (x)ui +
Ni (x)[ψi (x) − τ ψi (x)]ai
(1)
i∈I
i∈I ∗
where τ ψi is the finite element interpolation of ψi
X
τ ψi (x) =
Ni (x)ψi (xi )
(2)
i∈I
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Enriched basis function
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
3
0.5
2
0
1
−0.5
0
−1
−1
Context
−2
−1.5
−3
1
0.5
Blending
0
−0.5
−1
−0.5
−1
1
0.5
0
−2
1
0.5
0.5
0
−0.5
−1
Illconditioning
1
0
−1
1
0.5
0
0.8
1
−0.5
−1
0.6
0
−1
−0.5
0
0.5
1
0.4
0.2
−2
1
0.5
0
−0.5
Conclusion
0.5
1
2
−1
Numerical
examples
0
−0.5
3
Numerical
integration
−1
−0.5
−1
−1
−0.5
0
0.5
1
0
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
1
7 / 24
Remarks
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
• SGFEM has no blending problems.
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
• Easy to implement, especially when compared to the
corrected XFEM.
• The Kronecker delta property is still valid u(xi ) = ui .
• The SGFEM enriched basis function of the absolute value,
coincides with the modified absolute value enrichment
proposed by M¨
oes.
Conclusion
8 / 24
Ill-conditioning
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
Consider the following system of equations
Ax = b
If we consider a small change on right hand side, b0 , we are
interested in determining how this will affect the solution.
Defining
e = b0 − b
ex = x0 − x
Then, the relative change of the solution x is
|A−1 e| |b|
|A−1 e| |Ax|
|ex |/|x|
=
·
=
·
≤ kA−1 kkAk
|e|/|b|
|e|
|x|
|e|
|x|
Therefore, we define
Cond(A) = kA−1 kkAk
9 / 24
Ill-conditioning solution
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Blending
Illconditioning
Numerical
integration
a
• Scaled condition number Let hij = √a ija . Then the
ii jj
scaled condition number is defined as
κ(A) = kH −1 kkHk
• The scaled condition number of the FEM stiffness matrix
is O(h−2 ).
• The standard GFEM condition number is usually higher
than O(h−2 ).
Numerical
examples
• SGFEM condition number in 1D is O(h−2 ).
Conclusion
• For higher dimensions, if 2 assumptions hold, the condition
number of SGFEM also grows at the same rate as FEM.
10 / 24
The two assumptions
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Blending
Illconditioning
Assumption 1 There exist L1 and U1 ∈ R and indepedent of h
(element size) such that
• 0 < L1 ≤ U1
• L1 [kαk2 + kβk2 ] ≤ |a(α + β, α + β)| ≤ U1 [kαk2 + kβk2 ]
where α =
P
i ui Ni
and β =
P
j
vj Nj ψ
∀ui , vj ∈ R.
Numerical
integration
Numerical
examples
Conclusion
The space spanned by the standard FEM shape functions is
almost orthogonal to the space spanned by the enriched
shape functions.
11 / 24
The two assumptions
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Let A(i) be the scaled stiffness matrix of element i.
Assumption 2 There exist L2 and U2 ∈ R and indepedent of h
(element size) such that
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
• 0 < L2 ≤ U2
• Lkyk2 ≤ yT A(i) y ≤ U kyk2
∀y ∈ Rk
Provided that those 2 assumptions are fulfilled, the scaled
condition no. of the stiffness matrix is also O(h−2 ).
12 / 24
Numerical integration
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
• Numerical integration of discontinuous and singular
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
functions must be perfomed.
• The integration of the branch functions (singular
functions) is performed using a parabolic mapping (B´echet
et al. 2005).
• Integration of discontinuous and weakly discontinuous
functions is performed with subdivision of the elements.
Conclusion
13 / 24
Integration algorithm
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
1
2
The cut points between the interface and the element
edges are found.
A least squares plane is adjusted to the cut points.
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
(1)
(2)
14 / 24
Integration algorithm
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
3
4
Points are placed over the plane.
A polynomial interpolation in the normal direction is built
and solved.
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
(3)
(4)
15 / 24
Integration algorithm
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
5
Two sets of points are created. φi ≥ 0 and φi ≤ 0
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
(5)
16 / 24
Integration algorithm
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
6
Delaunay tetrahedralization is computed for both sets.
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
(6)
17 / 24
Integration algorithm
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
7
Gauss points are mapped into the tetrahedrons
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
(7)
18 / 24
Numerical Examples. Problem definition
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Context
Blending
Illconditioning
∇ · σ + b = 0 on Ω
σ · n = t on Γt
u = u on Γu
= 12 (∇u + (∇u)T ) on Ω
σ = C on Ω
t
Numerical
integration
Numerical
examples
Conclusion
b
u
19 / 24
Circular Inclusion
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
The absolute value of the level set is used as enrichment
function.
Circular inclusion
−1
10
Context
Illconditioning
Numerical
integration
Energy norm error
Blending
−2
10
Numerical
examples
Conclusion
−3
10
−3
10
−2
10
h
−1
10
20 / 24
Crack
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
Enriched with the “linear Heaviside” (H(x), H(x)x, H(x)y)
and the branch enrichemet functions.
0
10
Context
GFEM
SGFEM
Illconditioning
Numerical
integration
Error in energy−norm
Blending
−1
10
−2
10
Numerical
examples
Conclusion
−3
10
−3
10
−2
10
−1
h
10
0
10
21 / 24
Crack
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
12
10
SGFEM
GFEM
10
10
Blending
Illconditioning
Numerical
integration
Numerical
examples
Scaled condition number
Context
8
10
6
10
4
10
2
10
−3
10
−2
−1
10
10
0
10
h
Conclusion
22 / 24
Summary
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
• The stable generalized finite element is an easy to
Context
Blending
Illconditioning
Numerical
integration
Numerical
examples
implement solution to blending problems.
• At the moment, SGFEM is not a complete solution against
the ill-conditioning.
• Numerical integration perfomed through element
subdivision and parabolic mapping.
Conclusion
23 / 24
Literature
The stable
GFEM.
Convergence,
accuracy and
Diffpack implementation
• Babuska, I., Banerjee, U. (2012). Stable Generalized
Context
• Gupta, V., Duarte, C. a., Babuska, I., Banerjee, U. (2013).
Blending
Illconditioning
Numerical
integration
Numerical
examples
Conclusion
Finite Element Method (SGFEM). Computer Methods in
Applied Mechanics and Engineering
A Stable and Optimally Convergent Generalized FEM
(SGFEM) for Linear Elastic Fracture Mechanics.
Computer Methods in Applied Mechanics and Engineering
• Gupta, V., Duarte, C. a., Babuska, I., Banerjee, U. (2015).
Stable GFEM (SGFEM): Improved conditioning and
accuracy of GFEM/XFEM for three-dimensional fracture
mechanics. Computer Methods in Applied Mechanics and
Engineering
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