The stable GFEM. Convergence, The stable GFEM. Convergence, accuracy and accuracy and Diffpack im- plementation Diffpack implementation Context Blending Daniel Alves Paladim Ill- Sundararajan Natarajan conditioning St´ ephane Bordas Numerical integration Pierre Kerfriden Numerical examples Institute of Mechanics and Advanced Materials Conclusion Cardiff University May 12, 2015 1 / 24
The stable GFEM. Convergence, 1 Context accuracy and Diffpack im- plementation 2 Blending Context Blending 3 Ill-conditioning Ill- conditioning 4 Numerical integration Numerical integration Numerical examples 5 Numerical examples Conclusion 6 Conclusion 2 / 24
Context The stable GFEM. Convergence, accuracy and Diffpack im- plementation Diffpack is a commercial software library used for the development numerical software, with main emphasis on Context numerical solutions of partial differential equations. It was Blending developed in C++ following the object oriented paradigm. Ill- conditioning Numerical The library is mostly oriented to the implementation of the integration finite element method, however it has tools for other methods Numerical examples such as finite volume and finite differences. Conclusion 3 / 24
Context The stable GFEM. Convergence, accuracy and Diffpack im- plementation The extended/generalized finite element method is usually Context connected to the following issues: Blending • Blending Ill- conditioning • Ill-conditioning of the stiffness matrix Numerical • Numerical integration integration Numerical examples Conclusion 4 / 24
Blending In the extended finite element method, there are 3 types of The stable GFEM. elements. Convergence, accuracy and • Elements with all its nodes enriched Diffpack im- plementation • Elements that none of its nodes enriched. • Elements that have both type of nodes (blending Context elements). Blending In those elements, there is no partition of unity and the Ill- conditioning convergence rate is degraded. Numerical integration Numerical examples Conclusion 5 / 24
Blending solution The stable GFEM. Convergence, accuracy and Babuska and Banerjee (2011) proposed the stable generalized Diffpack im- plementation finite element method. In the SGFEM, the approximation has the following form Context u h ( x ) = � � Blending N i ( x ) u i + N i ( x )[ ψ i ( x ) − τψ i ( x )] a i (1) Ill- i ∈ I i ∈ I ∗ conditioning Numerical where τψ i is the finite element interpolation of ψ i integration Numerical examples � τψ i ( x ) = N i ( x ) ψ i ( x i ) (2) Conclusion i ∈ I 6 / 24
Enriched basis function The stable GFEM. Convergence, accuracy and Diffpack im- plementation 3 0.5 2 0 1 − 0.5 0 − 1 − 1 Context − 2 − 1.5 − 3 − 2 0.5 1 1 0.5 1 Blending 0.5 0.5 1 0 0.5 0 0 − 0.5 0 − 0.5 0 − 0.5 − 0.5 − 1 − 1 − 1 − 1 Ill- − 0.5 conditioning − 1 1 3 1 0.5 1 0.5 Numerical 0 2 0.8 0 − 0.5 − 0.5 1 integration 0.6 − 1 − 1 0 0.4 − 1 Numerical 0.2 − 2 0 1 examples 1 0.5 1 0.5 1 0.5 0 0.5 0 0 − 0.5 0 − 0.5 − 0.5 − 0.5 − 1 − 1 Conclusion − 1 − 1 7 / 24
Remarks The stable GFEM. Convergence, accuracy and Diffpack im- plementation • SGFEM has no blending problems. • Easy to implement, especially when compared to the Context corrected XFEM. Blending Ill- • The Kronecker delta property is still valid u ( x i ) = u i . conditioning • The SGFEM enriched basis function of the absolute value, Numerical integration coincides with the modified absolute value enrichment Numerical examples proposed by M¨ oes. Conclusion 8 / 24
Ill-conditioning Consider the following system of equations The stable GFEM. Convergence, accuracy and Ax = b Diffpack im- plementation If we consider a small change on right hand side, b ′ , we are interested in determining how this will affect the solution. Context Defining Blending e = b ′ − b e x = x ′ − x Ill- conditioning Then, the relative change of the solution x is Numerical integration | e | / | b | = | A − 1 e | | x | = | A − 1 e | Numerical | e x | / | x | · | b | · | Ax | ≤ � A − 1 �� A � examples | e | | e | | x | Conclusion Therefore, we define Cond ( A ) = � A − 1 �� A � 9 / 24
Ill-conditioning solution The stable GFEM. Convergence, accuracy and a ij • Scaled condition number Let h ij = √ a ii a jj . Then the Diffpack im- plementation scaled condition number is defined as κ ( A ) = � H − 1 �� H � Context • The scaled condition number of the FEM stiffness matrix Blending is O ( h − 2 ) . Ill- conditioning • The standard GFEM condition number is usually higher Numerical than O ( h − 2 ) . integration Numerical • SGFEM condition number in 1D is O ( h − 2 ) . examples 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 im- Assumption 1 There exist L 1 and U 1 ∈ R and indepedent of h plementation (element size) such that • 0 < L 1 ≤ U 1 Context • L 1 [ � α � 2 + � β � 2 ] ≤ | a ( α + β, α + β ) | ≤ U 1 [ � α � 2 + � β � 2 ] Blending Ill- where α = � i u i N i and β = � j v j N j ψ ∀ u i , v j ∈ R . conditioning Numerical integration The space spanned by the standard FEM shape functions is Numerical examples almost orthogonal to the space spanned by the enriched Conclusion shape functions. 11 / 24
The two assumptions The stable GFEM. Convergence, accuracy and Diffpack im- Let A ( i ) be the scaled stiffness matrix of element i . plementation Assumption 2 There exist L 2 and U 2 ∈ R and indepedent of h (element size) such that Context Blending Ill- • 0 < L 2 ≤ U 2 conditioning • L � y � 2 ≤ y T A ( i ) y ≤ U � y � 2 Numerical ∀ y ∈ R k integration Numerical Provided that those 2 assumptions are fulfilled, the scaled examples condition no. of the stiffness matrix is also O ( h − 2 ) . Conclusion 12 / 24
Numerical integration The stable GFEM. Convergence, accuracy and Diffpack im- plementation • Numerical integration of discontinuous and singular functions must be perfomed. Context • The integration of the branch functions (singular Blending Ill- functions) is performed using a parabolic mapping (B´ echet conditioning et al. 2005). Numerical integration • Integration of discontinuous and weakly discontinuous Numerical functions is performed with subdivision of the elements. examples Conclusion 13 / 24
Integration algorithm The stable 1 The cut points between the interface and the element GFEM. Convergence, edges are found. accuracy and Diffpack im- 2 A least squares plane is adjusted to the cut points. plementation Context Blending Ill- conditioning Numerical integration Numerical examples Conclusion (1) (2) 14 / 24
Integration algorithm The stable 3 Points are placed over the plane. GFEM. Convergence, 4 A polynomial interpolation in the normal direction is built accuracy and Diffpack im- and solved. plementation Context Blending Ill- conditioning Numerical integration Numerical examples Conclusion (3) (4) 15 / 24
Integration algorithm The stable GFEM. Convergence, accuracy and 5 Two sets of points are created. φ i ≥ 0 and φ i ≤ 0 Diffpack im- plementation Context Blending Ill- conditioning Numerical integration Numerical examples (5) Conclusion 16 / 24
Integration algorithm The stable GFEM. 6 Delaunay tetrahedralization is computed for both sets. Convergence, accuracy and Diffpack im- plementation Context Blending Ill- conditioning Numerical integration Numerical examples Conclusion (6) 17 / 24
Integration algorithm The stable GFEM. 7 Gauss points are mapped into the tetrahedrons Convergence, accuracy and Diffpack im- plementation Context Blending Ill- conditioning Numerical integration Numerical examples Conclusion (7) 18 / 24
Numerical Examples. Problem definition ∇ · σ + b = 0 on Ω The stable GFEM. σ · n = t on Γ t Convergence, accuracy and u = u on Γ u Diffpack im- plementation 2 ( ∇ u + ( ∇ u ) T ) on Ω ǫ = 1 σ = Cǫ on Ω Context Blending t Ill- conditioning Numerical integration u Numerical b examples Conclusion 19 / 24
Circular Inclusion The stable GFEM. Convergence, The absolute value of the level set is used as enrichment accuracy and Diffpack im- function. plementation Circular inclusion −1 10 Context Blending Ill- Energy norm error conditioning −2 10 Numerical integration Numerical examples Conclusion −3 10 −3 −2 −1 10 10 10 h 20 / 24
Crack The stable GFEM. Convergence, Enriched with the “linear Heaviside” ( H ( x ) , H ( x ) x , H ( x ) y ) accuracy and Diffpack im- and the branch enrichemet functions. plementation 0 10 Context GFEM SGFEM Blending Ill- −1 10 Error in energy−norm conditioning Numerical integration −2 10 Numerical examples Conclusion −3 10 −3 −2 −1 0 10 10 10 10 h 21 / 24
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