diff --git a/paper/paper.bib b/paper/paper.bib index ec80665f..c9a8f3d2 100644 --- a/paper/paper.bib +++ b/paper/paper.bib @@ -1,226 +1,13 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for pierluigi at 2024-08-02 17:21:35 +0900 +%% Created for pierluigi at 2024-07-30 08:23:53 +0900 %% Saved with string encoding Unicode (UTF-8) -@article{SICSIC, - abstract = {This paper studies the initiation of cracks in the thermal shock problem through the variational analysis of the quasi-static evolution of a gradient damage model. We consider a two-dimensional semi-infinite slab with an imposed temperature drop on its free surface. The damage model is formulated in the framework of the variational theory of rate-independent processes based on the principles of irreversibility, stability and energy balance. In the case of a sufficiently severe shock, we show that damage immediately occurs and that its evolution follows first a fundamental branch without localization. Then it bifurcates into another branch in which damage localization will take place finally to generate cracks. The determination of the time and mode of that bifurcation allows us to explain the periodic distribution of the so-initiated cracks and to calculate the crack spacing in terms of the material and loading parameters. Numerical investigations complete and quantify the analytical results.}, - author = {Paul Sicsic and Jean-Jacques Marigo and Corrado Maurini}, - date-added = {2024-08-02 17:20:53 +0900}, - date-modified = {2024-08-02 17:21:17 +0900}, - doi = {10.1016/j.jmps.2013.09.003}, - journal = {Journal of the Mechanics and Physics of Solids}, - keywords = {Damage mechanics, Gradient damage model, Thermal shock, Variational methods, Energy balance, Stability and bifurcation, Rayleigh ratio}, - pages = {256-284}, - title = {Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling}, - volume = {63}, - year = {2014}, - bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0022509613001786}, - bdsk-url-2 = {https://doi.org/10.1016/j.jmps.2013.09.003}} - -@article{bazant, - abstract = {We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.}, - author = {Zden\v{e}k P. Ba\v{z}ant}, - date-added = {2024-08-02 17:17:28 +0900}, - date-modified = {2024-08-02 17:18:29 +0900}, - doi = {10.1061/(ASCE)0733-9399(1988)114:12(2013)}, - journal = {Journal of Engineering Mechanics}, - title = {Stable States and Paths of Stmuctures with Plasticity or Damage}, - volume = {114}, - year = {1988}, - bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S002250961100055X}, - bdsk-url-2 = {https://doi.org/10.1016/j.jmps.2011.03.010}} - -@book{bourdin:2008-the-variational, - abstract = {We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.}, - author = {Bourdin, Blaise and Francfort, Gilles A and Marigo, Jean-Jacques}, - date-added = {2024-08-02 17:16:54 +0900}, - date-modified = {2024-08-02 17:16:54 +0900}, - doi = {10.1007/978-1-4020-6395-4}, - journal = {Journal of Elasticity}, - number = {6}, - pages = {5--148}, - publisher = {Springer}, - title = {The variational approach to fracture}, - volume = {91}, - year = {2008}, - bdsk-url-1 = {https://doi.org/10.1007/978-1-4020-6395-4}} - -@article{PETRYK, - abstract = {Second-order rate constitutive equations are formulated for a time-independent elastic-plastic material, obeying the normality flow rule with a smooth yield surface. Under specified regularity restrictions imposed on the involved fields, the regular second-order rate boundary value problem with quasistatic accelerations as unknowns is posed. It is shown that every solution of this generally non-linear rate problem is governed by a variational principle and that the corresponding functional reaches a strict absolute minimum, provided the solution satisfies a sufficient uniqueness condition. With the same incrementally linear comparison solid, Hill's exclusion condition rules out not only a first- but also a second-order bifurcation. The criticality of the exclusion condition is discussed and conditions are indicated under which a second-order bifurcation becomes possible, while the first-order rate problem is still uniquely solvable.}, - author = {H. Petryk and K. Thermann}, - date-added = {2024-08-02 17:10:29 +0900}, - date-modified = {2024-08-02 17:11:16 +0900}, - doi = {10.1016/0022-5096(85)90004-3}, - journal = {Journal of the Mechanics and Physics of Solids}, - number = {6}, - pages = {577-593}, - title = {Second-order bifurcation in elastic-plastic solids}, - volume = {33}, - year = {1985}, - bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0022509685900043}, - bdsk-url-2 = {https://doi.org/10.1016/0022-5096(85)90004-3}} - -@incollection{Quoc2002, - author = {Nguyen, Quoc Son}, - booktitle = {Continuum thermodynamics: the art and science of modeling matter's behavior}, - date-added = {2024-08-02 17:04:28 +0900}, - date-modified = {2024-08-02 17:06:32 +0900}, - doi = {10.1007/0-306-46946-4\_26}, - editor = {G\'erard Maugin and Raymonde Drouot and Fran\c{c}ois Sidoroff}, - hal_id = {hal-00112276}, - hal_version = {v1}, - keywords = {generalized standard model ; standard dissipative system ; static or dynamic stability ; bifurcation ; stability criterion}, - pages = {343-354}, - publisher = {Springer}, - title = {Standard dissipative systems and stability analysis}, - year = {2000}, - bdsk-url-1 = {https://hal.science/hal-00112276}, - bdsk-url-2 = {https://doi.org/10.1007/0-306-46946-4%5C_26}} - -@article{Quoc, - abstract = {{This paper addresses stability and bifurcation analysis for common systems of solids in the framework of plasticity, of friction and of fracture mechanics. Although physically different, usual time-independent laws adopted in these domains lead to a certain mathematical similarity concerning the quasi-static behaviour of materials and structures. These mechanical systems can be described practically in the same mathematical manner concerning their quasi-static evolution and in particular concerning the stability of their response. Our objective is to present within this framework a review of principal results of the recent literature on these subjects in relation with some energy-related considerations and with an unified description based upon energy and dissipation analysis.}}, - author = {Nguyen, Quoc Son}, - date-added = {2024-08-02 17:02:15 +0900}, - date-modified = {2024-08-02 17:02:41 +0900}, - doi = {10.1115/1.3111068}, - issn = {0003-6900}, - journal = {Applied Mechanics Reviews}, - month = {01}, - number = {1}, - pages = {1-31}, - title = {Bifurcation and Stability in Dissipative Media (Plasticity, Friction, Fracture)}, - volume = {47}, - year = {1994}, - bdsk-url-1 = {https://doi.org/10.1115/1.3111068}} - -@unpublished{camilla, - author = {Camilla Zolesi, Corrado Maurini}, - date-added = {2024-08-02 16:57:06 +0900}, - date-modified = {2024-08-02 16:58:32 +0900}, - journal = {preprint, \url{https://hal.sorbonne-universite.fr/hal-04552309}}, - title = {Stability and crack nucleation in variational phase-field models of fracture: effects of length-scales and stress multi-axiality}} - -@article{Pham2013aa, - abstract = {Considering a family of gradient-enhanced damage models and taking advantage of its variational formulation, we study the stability of homogeneous states in a full three-dimensional context. We show that gradient terms have a stabilizing effect, but also how those terms induce structural effects. We emphasize the great importance of the type of boundary conditions, the size and the shape of the body on the stability properties of such states.}, - author = {Pham, K. and Marigo, J. -J.}, - date = {2013/01/01}, - date-added = {2024-08-02 14:24:09 +0900}, - date-modified = {2024-08-02 14:24:22 +0900}, - doi = {10.1007/s10659-012-9382-5}, - id = {Pham2013}, - isbn = {1573-2681}, - journal = {Journal of Elasticity}, - number = {1}, - pages = {63--93}, - title = {Stability of Homogeneous States with Gradient Damage Models: Size Effects and Shape Effects in the Three-Dimensional Setting}, - volume = {110}, - year = {2013}, - bdsk-url-1 = {https://doi.org/10.1007/s10659-012-9382-5}} - -@article{MIEHE20102765, - abstract = {The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. Following our recent work [C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically-consistent phase field models of fracture: Variational principles and multi-field fe implementations, International Journal for Numerical Methods in Engineering DOI:10.1002/nme.2861] on phase-field-type fracture, we propose in this paper a new variational framework for rate-independent diffusive fracture that bases on the introduction of a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a new algorithmic treatment of diffusive fracture. Here, we propose an extremely robust operator split scheme that successively updates in a typical time step the history field, the crack phase field and finally the displacement field. A regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added. The proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples.}, - author = {Christian Miehe and Martina Hofacker and Fabian Welschinger}, - date-added = {2024-08-02 13:05:42 +0900}, - date-modified = {2024-08-02 13:05:52 +0900}, - doi = {10.1016/j.cma.2010.04.011}, - issn = {0045-7825}, - journal = {Computer Methods in Applied Mechanics and Engineering}, - keywords = {Fracture, Crack propagation, Phase fields, Gradient-type damage, Incremental variational principles, Finite elements, Coupled multi-field problem}, - number = {45}, - pages = {2765-2778}, - title = {A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits}, - volume = {199}, - year = {2010}, - bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0045782510001283}, - bdsk-url-2 = {https://doi.org/10.1016/j.cma.2010.04.011}} - -@article{Miehe10, - abstract = {Abstract The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright {\copyright} 2010 John Wiley \& Sons, Ltd.}, - author = {Miehe, C. and Welschinger, F. and Hofacker, M.}, - date-added = {2024-08-02 13:05:32 +0900}, - date-modified = {2024-08-02 13:06:20 +0900}, - doi = {10.1002/nme.2861}, - journal = {International Journal for Numerical Methods in Engineering}, - keywords = {fracture, crack propagation, phase-fields, gradient-type damage, incremental variational principles, finite elements, coupled multi-field problems}, - number = {10}, - pages = {1273-1311}, - title = {Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations}, - volume = {83}, - year = {2010}, - bdsk-url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.2861}, - bdsk-url-2 = {https://doi.org/10.1002/nme.2861}} - -@article{FRANCFORT, - abstract = {A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith's theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.}, - author = {G.A. Francfort and J.-J. Marigo}, - date-added = {2024-08-02 11:48:12 +0900}, - date-modified = {2024-08-02 11:48:30 +0900}, - doi = {10.1016/S0022-5096(98)00034-9}, - issn = {0022-5096}, - journal = {Journal of the Mechanics and Physics of Solids}, - number = {8}, - pages = {1319-1342}, - title = {Revisiting brittle fracture as an energy minimization problem}, - volume = {46}, - year = {1998}, - bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0022509698000349}, - bdsk-url-2 = {https://doi.org/10.1016/S0022-5096(98)00034-9}} - -@article{dalcinpazklercosimo2011, - author = {Lisandro D. Dalcin and Rodrigo R. Paz and Pablo A. Kler and Alejandro Cosimo}, - date-added = {2024-08-02 11:42:21 +0900}, - date-modified = {2024-08-02 11:42:21 +0900}, - doi = {10.1016/j.advwatres.2011.04.013}, - issn = {0309-1708}, - journal = {Advances in Water Resources}, - note = {New Computational Methods and Software Tools}, - number = {9}, - pages = {1124 - 1139}, - title = {Parallel distributed computing using Python}, - volume = {34}, - year = {2011}, - bdsk-url-1 = {https://doi.org/10.1016/j.advwatres.2011.04.013}} - -@article{moreau:hal-01867187, - author = {Moreau, Jean Jacques}, - date-added = {2024-07-31 15:03:36 +0900}, - date-modified = {2024-07-31 15:03:36 +0900}, - hal_id = {hal-01867187}, - hal_version = {v1}, - journal = {{Comptes rendus hebdomadaires des s{\'e}ances de l'Acad{\'e}mie des sciences}}, - pages = {238-240}, - pdf = {https://hal.science/hal-01867187/file/D%C3%A9composition_orthogonale_espace_hilbertien_Moreau_CRAS_1962.pdf}, - publisher = {{Gauthier-Villars}}, - title = {{D{\'e}composition orthogonale d'un espace hilbertien selon deux c{\^o}nes mutuellement polaires}}, - url = {https://hal.science/hal-01867187}, - volume = {255}, - year = {1962}, - bdsk-url-1 = {https://hal.science/hal-01867187}} - -@article{pham:2011-the-issues, - abstract = {We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.}, - author = {Kim Pham and Jean-Jacques Marigo and Corrado Maurini}, - date-added = {2024-07-31 14:57:29 +0900}, - date-modified = {2024-07-31 14:57:29 +0900}, - doi = {10.1016/j.jmps.2011.03.010}, - issn = {0022-5096}, - journal = {Journal of the Mechanics and Physics of Solids}, - keywords = {Damage mechanics, Gradient damage model, Variational methods, Stability, Bifurcation}, - number = {6}, - pages = {1163-1190}, - title = {The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models}, - url = {https://www.sciencedirect.com/science/article/pii/S002250961100055X}, - volume = {59}, - year = {2011}, - bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S002250961100055X}, - bdsk-url-2 = {https://doi.org/10.1016/j.jmps.2011.03.010}} - @article{bourdin:2000-numerical, abstract = {The numerical implementation of the model of brittle fracture developed in Francfort and Marigo (1998. J. Mech. Phys. Solids 46 (8), 1319--1342) is presented. Various computational methods based on variational approximations of the original functional are proposed. They are tested on several antiplanar and planar examples that are beyond the reach of the classical computational tools of fracture mechanics.}, author = {Bourdin, Blaise and Francfort, Gilles A and Marigo, Jean-Jacques}, @@ -264,13 +51,30 @@ @article{marigo:2023-la-mecanique bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0022509621001010}, bdsk-url-2 = {https://doi.org/10.1016/j.jmps.2021.104424}} +@article{pham:2011-the-issues, + abstract = {We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.}, + author = {Kim Pham and Jean-Jacques Marigo and Corrado Maurini}, + date-added = {2024-07-29 23:12:53 +0000}, + date-modified = {2024-07-29 23:13:24 +0000}, + doi = {10.1016/j.jmps.2011.03.010}, + issn = {0022-5096}, + journal = {Journal of the Mechanics and Physics of Solids}, + keywords = {Damage mechanics, Gradient damage model, Variational methods, Stability, Bifurcation}, + number = {6}, + pages = {1163-1190}, + title = {The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models}, + url = {https://www.sciencedirect.com/science/article/pii/S002250961100055X}, + volume = {59}, + year = {2011}, + bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S002250961100055X}, + bdsk-url-2 = {https://doi.org/10.1016/j.jmps.2011.03.010}} + @url{Habera:aa, author = {Michal Habera and Andreas Zilian}, date-added = {2024-03-04 16:19:39 +0100}, date-modified = {2024-03-04 16:21:00 +0100}, title = {dolfiny: Python wrappers for DOLFINx}, url = {https://github.com/michalhabera/dolfiny}, - year = {2024}, bdsk-url-1 = {https://github.com/michalhabera/dolfiny}} @misc{dolfinx2023preprint, @@ -335,15 +139,19 @@ @techreport{balay:2017-petsc year = {2017}, bdsk-file-1 = {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}} -@book{mielke, - author = {Mielke, A. and Roub\'{i}\v{c}ek, A.}, +@article{dalcinpazklercosimo2011, + author = {Lisandro D. Dalcin and Rodrigo R. Paz and Pablo A. Kler and Alejandro Cosimo}, date-added = {2024-03-01 16:00:17 +0100}, - date-modified = {2024-08-02 11:46:39 +0900}, - doi = {10.1007/978-1-4939-2706-7}, - issn = {0066-5452}, - publisher = {Springer New York, NY}, - title = {Rate-Independent Systems}, - year = {2015}, + date-modified = {2024-03-01 16:00:17 +0100}, + doi = {10.1016/j.advwatres.2011.04.013}, + issn = {0309-1708}, + journal = {Advances in Water Resources}, + note = {New Computational Methods and Software Tools}, + number = {9}, + pages = {1124 - 1139}, + title = {Parallel distributed computing using Python}, + volume = {34}, + year = {2011}, bdsk-url-1 = {https://doi.org/10.1016/j.advwatres.2011.04.013}} @article{pinto-da-costa:2010-cone-constrained, @@ -365,6 +173,19 @@ @article{pinto-da-costa:2010-cone-constrained bdsk-file-1 = {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}, bdsk-url-1 = {https://doi.org/10.1007/s10589-008-9167-8}} +@article{bourdin:2008-the-variational, + author = {Bourdin, Blaise and Francfort, Gilles A and Marigo, Jean-Jacques}, + date-added = {2015-07-07 15:15:16 +0000}, + date-modified = {2022-08-10 11:03:49 +0200}, + journal = {Journal of elasticity}, + number = {1-3}, + pages = {5--148}, + publisher = {Springer}, + title = {The variational approach to fracture}, + volume = {91}, + year = {2008}, + bdsk-file-1 = {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}} + @book{jorge-nocedal:1999-numerical, author = {Jorge Nocedal, Stephen Wright}, date-added = {2019-11-27 14:27:43 +0100}, @@ -404,7 +225,7 @@ @comment{BibDesk group name _PAPER_JOSS_2023 keys - jorge-nocedal:1999-numerical,balay:2017-petsc,petsc-web-page,petsc-user-ref,Habera:aa,mielke,moreau:1962-decomposition,dolfinx2023preprint,pinto-da-costa:2010-cone-constrained,leon-baldelli:2021-numerical + jorge-nocedal:1999-numerical,balay:2017-petsc,petsc-web-page,petsc-user-ref,bourdin:2008-the-variational,Habera:aa,dalcinpazklercosimo2011,moreau:1962-decomposition,dolfinx2023preprint,pinto-da-costa:2010-cone-constrained,leon-baldelli:2021-numerical group name @@ -416,13 +237,13 @@ @comment{BibDesk group name APPL_F2M_ICEICE keys - leon-baldelli:2021-numerical + bourdin:2008-the-variational,leon-baldelli:2021-numerical group name APPL_F2M_PICASSO keys - leon-baldelli:2021-numerical + leon-baldelli:2021-numerical,bourdin:2008-the-variational group name @@ -446,7 +267,7 @@ @comment{BibDesk group name CONFERENCE ICTAM 2024 keys - leon-baldelli:2021-numerical + bourdin:2008-the-variational,leon-baldelli:2021-numerical group name @@ -458,7 +279,7 @@ @comment{BibDesk group name Eval CNRS keys - + bourdin:2008-the-variational group name @@ -476,7 +297,7 @@ @comment{BibDesk group name HNF keys - + bourdin:2008-the-variational group name @@ -572,7 +393,7 @@ @comment{BibDesk group name TALK_FENICS keys - leon-baldelli:2021-numerical + bourdin:2008-the-variational,leon-baldelli:2021-numerical group name