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Corrigendum to “A coupled electro-thermal Discontinuous Galerkin method” [J. Comput. Phys. 348 (2017) 231–258]
JID:YJCPH AID:7594 /ERR
[m3G; v1.222; Prn:27/09/2017; 13:34] P.1 (1-3)
Journal of Computational Physics ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Computational Physics www.elsevier.com/locate/jcp
Corrigendum
Corrigendum to “A coupled electro-thermal Discontinuous Galerkin method” [J. Comput. Phys. 348 (2017) 231–258] L. Homsi a , C. Geuzaine b , L. Noels a,∗ a
Computational & Multiscale Mechanics of Materials (CM3), Department of Aerospace and Mechanical Engineering, University of Liège, Quartier Polytech 1, Allée de la Découverte 9, B-4000 Liège, Belgium b Department of Electrical Engineering and Computer Science, University of Liège, Quartier Polytech 1, Allée de la Découverte 10, B-4000 Liège, Belgium
a r t i c l e
i n f o
Article history: Received 5 September 2017 Accepted 21 September 2017 Available online xxxx
The authors regret that an error was introduced when unifying the expressions of the symmetrization and stability terms on domain interfaces and on the Dirichlet boundary in the weak form (45). Indeed on the Dirichlet boundary, the tensor ¯ , leading to Z (M ¯ ). Equation (45) of the published version of the paper Z (M ) should be evaluated at the constrained value M 1 should thus be finding M ∈ X+ such that : 1
δM Tj¯dS −
¯ )∇δM dS + ¯ Z (M M n T
∂N h
∂D h
=
∇δM Tj (M , ∇M )d +
h
+
r
+
δM Ti d +
MnT Z (M )∇δM dS +
∂I h
r
δMnT
z B hs
δMnT
∂D h
h
z
r
MnT
B hs
r
¯) M ¯ n dS Z (M
δMnT
z
∂I h ∪∂D h
z
¯ )∇δM dS Z (M
∂D h
r z B ¯ ) JMn K dS ∀δM ∈ X1 . Z (M ) JMn K dS + Z (M δMnT
∂I h
hs
∂D h
As a result, some modifications arise in intermediate demonstration steps:
• The weak form (46)–(48) becomes finding M ∈ X+ 1 such that:
* 1
j (M , ∇M ) dS
DOI of original article: http://dx.doi.org/10.1016/j.jcp.2017.07.028. Corresponding author. E-mail address: [email protected] (L. Noels). In the online version, the modified terms appear in red.
http://dx.doi.org/10.1016/j.jcp.2017.09.022 0021-9991/© 2017 Elsevier Inc. All rights reserved.
(1)
JID:YJCPH AID:7594 /ERR
[m3G; v1.222; Prn:27/09/2017; 13:34] P.2 (1-3)
L. Homsi et al. / Journal of Computational Physics ••• (••••) •••–•••
2
¯ , δM ) − a(M , δM ) = b(M
δM Ti d ∀δM ∈ X1 ,
(2)
h
with
r
∇δM Tj (M , ∇M )d +
a(M , δM ) = h
r
MnT
+
z
∂I h
r
+
δMnT
δMnT
∂I h ∪∂D h
r
Z (M )∇δM dS +
z B hs
MnT
∂D h
Z (M ) JMn K dS +
∂I h
z
j (M , ∇M ) dS
z
¯ )∇δM dS Z (M
r
δMnT
z B hs
(3)
¯ ) JMn K dS, Z (M
∂D h
and
¯ , δM ) = b(M
δM Tj¯dS −
∂N h
¯ )∇δM dS + ¯ Z (M M n T
∂D h
δMnT
B hs
¯) M ¯ n dS, Z (M
(4)
∂D h
in which the Dirichlet boundary terms of a(M , δM ) differ from the published version of the paper. • When deriving the interface nodal forces (60)–(62) of the published version of the paper in Section 3.2, the contributions on ∂D h can be directly deduced by removing the factor (1/2) accordingly to the definition of the average flux on ¯ ). the Dirichlet boundary and by substituting Z (M h ) by Z (M • When demonstrating the consistency in Section 4.2, Eqs. (67)–(69) of the published version of the paper should read:
δM Tj¯dS −
∂N h
T
∂D h
∇δM Tj (M e , ∇M e )d +
= h
r
B hs
¯) M ¯ n dS Z (M
∂D h
z δMnT j (M e , ∇M e )dS
∂D h
(5)
¯ )∇δM dS Mne Z (M T
δMnT j (M e , ∇M e )dS− ∂D h
δMnT
+
δMnT
∂I h
−
¯ )∇δM dS + ¯ Z (M M n
B hs
¯ )M e dS ∀δM ∈ X . Z (M n
∂D h
Integrating the first term of the right hand side by parts leads to
e
∇δM Tj (M e , ∇M e )d = −
e
e
e
δM T ∇j (M e , ∇M e )d+
e
(6)
δMnT j (M e , ∇M e )dS,
∂e
and Eq. (5) becomes
δM Tj¯dS −
∂N h
¯ )∇δM dS + ¯ Z (M M n T
∂D h
−
e
−
∂D h
δMnT
∂D h
¯ )∇δM dS + Mne Z (M T
hs
¯) M ¯ n dS = Z (M
δM T ∇j (M e , ∇M e )d +
e
B
δMnT j (M e , ∇M e )dS
∂N h
δMnT
B hs
¯ )Mne dS ∀δM ∈ X, Z (M
∂D h
which still allows recovering the set of conservation laws and the boundary conditions of the strong form.
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3
• When deriving the non-self-adjoint linear elliptic problem in Section 4.3.1, Eq. (80) of the published version e of the is zobtained by subtracting a(Mrh , δM h ) from r paper z a(M , δM h), and adding and subtracting successively B T T eT e eT j ∇M (M )∇δM h dS and ∂ Mn − M h j (M e ) JδM hn K dS, which yield ∂I h Mn − M hn hs ∇ M I h n 0 = a(M e , δM h ) − a(M h , δM h )
= h
∇δM Th j (M e , ∇M e ) − j (M h , ∇M h ) d
r
+ ∂I h ∪∂D h
r
δM Thn T
∂I h ∪∂D h
r
∂I h
(8)
T
z B
Mne − M Thn
+
r
j ∇ M (M e )∇δM h dS
z
∂I h
r
z
T
Mne − M Thn
−
j (M e , ∇M e ) − j (M h , ∇M h ) dS
Mne − M Thn
+ −
z
j ∇M (M e ) − j ∇M (M h ) ∇δM h dS
hs
T
Mne − M Thn
j ∇M (M e ) − j ∇M (M h )
z B hs
∂I h ∪∂D h
JδM hn K dS
j ∇M (M e ) JδM hn K dS ∀δM h ∈ Xk .
This relation could not be derived in the published version of the paper.
• Finally, using this last result, the non-linear term N (M e , M h ; δM h ), Eq. (84) of the published version of the paper, now reads
N (M e , M h ; δM h ) = h
∇δM Th (R¯ j (M e − M h , ∇M e − ∇M h ))d
r
+
δM Thn
∂I h ∪∂D h
+
r
z
T
z B
∂I h
+
R¯ j (M e − M h , ∇M e − ∇M h ) dS
T
Mne − M Thn
r
z
Mne − MThn
j ∇M (M e ) − j ∇M (M h ) ∇δM h dS
hs
j∇M (Me ) − j∇M (Mh )
JδMhn K dS,
∂I h
in which only domain interfaces are involved in the last two terms. The remaining derivations, demonstrations, and numerical results are unchanged. The authors would like to apologize for any inconvenience caused.
(9)
[m3G; v1.222; Prn:27/09/2017; 13:34] P.1 (1-3)
Journal of Computational Physics ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Computational Physics www.elsevier.com/locate/jcp
Corrigendum
Corrigendum to “A coupled electro-thermal Discontinuous Galerkin method” [J. Comput. Phys. 348 (2017) 231–258] L. Homsi a , C. Geuzaine b , L. Noels a,∗ a
Computational & Multiscale Mechanics of Materials (CM3), Department of Aerospace and Mechanical Engineering, University of Liège, Quartier Polytech 1, Allée de la Découverte 9, B-4000 Liège, Belgium b Department of Electrical Engineering and Computer Science, University of Liège, Quartier Polytech 1, Allée de la Découverte 10, B-4000 Liège, Belgium
a r t i c l e
i n f o
Article history: Received 5 September 2017 Accepted 21 September 2017 Available online xxxx
The authors regret that an error was introduced when unifying the expressions of the symmetrization and stability terms on domain interfaces and on the Dirichlet boundary in the weak form (45). Indeed on the Dirichlet boundary, the tensor ¯ , leading to Z (M ¯ ). Equation (45) of the published version of the paper Z (M ) should be evaluated at the constrained value M 1 should thus be finding M ∈ X+ such that : 1
δM Tj¯dS −
¯ )∇δM dS + ¯ Z (M M n T
∂N h
∂D h
=
∇δM Tj (M , ∇M )d +
h
+
r
+
δM Ti d +
MnT Z (M )∇δM dS +
∂I h
r
δMnT
z B hs
δMnT
∂D h
h
z
r
MnT
B hs
r
¯) M ¯ n dS Z (M
δMnT
z
∂I h ∪∂D h
z
¯ )∇δM dS Z (M
∂D h
r z B ¯ ) JMn K dS ∀δM ∈ X1 . Z (M ) JMn K dS + Z (M δMnT
∂I h
hs
∂D h
As a result, some modifications arise in intermediate demonstration steps:
• The weak form (46)–(48) becomes finding M ∈ X+ 1 such that:
* 1
j (M , ∇M ) dS
DOI of original article: http://dx.doi.org/10.1016/j.jcp.2017.07.028. Corresponding author. E-mail address: [email protected] (L. Noels). In the online version, the modified terms appear in red.
http://dx.doi.org/10.1016/j.jcp.2017.09.022 0021-9991/© 2017 Elsevier Inc. All rights reserved.
(1)
JID:YJCPH AID:7594 /ERR
[m3G; v1.222; Prn:27/09/2017; 13:34] P.2 (1-3)
L. Homsi et al. / Journal of Computational Physics ••• (••••) •••–•••
2
¯ , δM ) − a(M , δM ) = b(M
δM Ti d ∀δM ∈ X1 ,
(2)
h
with
r
∇δM Tj (M , ∇M )d +
a(M , δM ) = h
r
MnT
+
z
∂I h
r
+
δMnT
δMnT
∂I h ∪∂D h
r
Z (M )∇δM dS +
z B hs
MnT
∂D h
Z (M ) JMn K dS +
∂I h
z
j (M , ∇M ) dS
z
¯ )∇δM dS Z (M
r
δMnT
z B hs
(3)
¯ ) JMn K dS, Z (M
∂D h
and
¯ , δM ) = b(M
δM Tj¯dS −
∂N h
¯ )∇δM dS + ¯ Z (M M n T
∂D h
δMnT
B hs
¯) M ¯ n dS, Z (M
(4)
∂D h
in which the Dirichlet boundary terms of a(M , δM ) differ from the published version of the paper. • When deriving the interface nodal forces (60)–(62) of the published version of the paper in Section 3.2, the contributions on ∂D h can be directly deduced by removing the factor (1/2) accordingly to the definition of the average flux on ¯ ). the Dirichlet boundary and by substituting Z (M h ) by Z (M • When demonstrating the consistency in Section 4.2, Eqs. (67)–(69) of the published version of the paper should read:
δM Tj¯dS −
∂N h
T
∂D h
∇δM Tj (M e , ∇M e )d +
= h
r
B hs
¯) M ¯ n dS Z (M
∂D h
z δMnT j (M e , ∇M e )dS
∂D h
(5)
¯ )∇δM dS Mne Z (M T
δMnT j (M e , ∇M e )dS− ∂D h
δMnT
+
δMnT
∂I h
−
¯ )∇δM dS + ¯ Z (M M n
B hs
¯ )M e dS ∀δM ∈ X . Z (M n
∂D h
Integrating the first term of the right hand side by parts leads to
e
∇δM Tj (M e , ∇M e )d = −
e
e
e
δM T ∇j (M e , ∇M e )d+
e
(6)
δMnT j (M e , ∇M e )dS,
∂e
and Eq. (5) becomes
δM Tj¯dS −
∂N h
¯ )∇δM dS + ¯ Z (M M n T
∂D h
−
e
−
∂D h
δMnT
∂D h
¯ )∇δM dS + Mne Z (M T
hs
¯) M ¯ n dS = Z (M
δM T ∇j (M e , ∇M e )d +
e
B
δMnT j (M e , ∇M e )dS
∂N h
δMnT
B hs
¯ )Mne dS ∀δM ∈ X, Z (M
∂D h
which still allows recovering the set of conservation laws and the boundary conditions of the strong form.
(7)
JID:YJCPH AID:7594 /ERR
[m3G; v1.222; Prn:27/09/2017; 13:34] P.3 (1-3)
L. Homsi et al. / Journal of Computational Physics ••• (••••) •••–•••
3
• When deriving the non-self-adjoint linear elliptic problem in Section 4.3.1, Eq. (80) of the published version e of the is zobtained by subtracting a(Mrh , δM h ) from r paper z a(M , δM h), and adding and subtracting successively B T T eT e eT j ∇M (M )∇δM h dS and ∂ Mn − M h j (M e ) JδM hn K dS, which yield ∂I h Mn − M hn hs ∇ M I h n 0 = a(M e , δM h ) − a(M h , δM h )
= h
∇δM Th j (M e , ∇M e ) − j (M h , ∇M h ) d
r
+ ∂I h ∪∂D h
r
δM Thn T
∂I h ∪∂D h
r
∂I h
(8)
T
z B
Mne − M Thn
+
r
j ∇ M (M e )∇δM h dS
z
∂I h
r
z
T
Mne − M Thn
−
j (M e , ∇M e ) − j (M h , ∇M h ) dS
Mne − M Thn
+ −
z
j ∇M (M e ) − j ∇M (M h ) ∇δM h dS
hs
T
Mne − M Thn
j ∇M (M e ) − j ∇M (M h )
z B hs
∂I h ∪∂D h
JδM hn K dS
j ∇M (M e ) JδM hn K dS ∀δM h ∈ Xk .
This relation could not be derived in the published version of the paper.
• Finally, using this last result, the non-linear term N (M e , M h ; δM h ), Eq. (84) of the published version of the paper, now reads
N (M e , M h ; δM h ) = h
∇δM Th (R¯ j (M e − M h , ∇M e − ∇M h ))d
r
+
δM Thn
∂I h ∪∂D h
+
r
z
T
z B
∂I h
+
R¯ j (M e − M h , ∇M e − ∇M h ) dS
T
Mne − M Thn
r
z
Mne − MThn
j ∇M (M e ) − j ∇M (M h ) ∇δM h dS
hs
j∇M (Me ) − j∇M (Mh )
JδMhn K dS,
∂I h
in which only domain interfaces are involved in the last two terms. The remaining derivations, demonstrations, and numerical results are unchanged. The authors would like to apologize for any inconvenience caused.
(9)