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Revisiting brittle fracture as an energy minimization problem. +Journal of the Mechanics and Physics of Solids, 46(8), 1319–1342. +https://doi.org/10.1016/S0022-5096(98)00034-9 + + + Parallel distributed computing using +Python + Dalcin + Advances in Water Resources + 9 + 34 + 10.1016/j.advwatres.2011.04.013 + 0309-1708 + 2011 + Dalcin, L. D., Paz, R. R., Kler, P. +A., & Cosimo, A. (2011). Parallel distributed computing using +Python. Advances in Water Resources, 34(9), 1124–1139. +https://doi.org/10.1016/j.advwatres.2011.04.013 + + + The issues of the uniqueness and the +stability of the homogeneous response in uniaxial tests with gradient +damage models + Pham + Journal of the Mechanics and Physics of +Solids + 6 + 59 + 10.1016/j.jmps.2011.03.010 + 0022-5096 + 2011 + Pham, K., Marigo, J.-J., & +Maurini, C. (2011). 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Software, 31(3), +351–362. https://doi.org/10.1145/1089014.1089019 + + + La mécanique de l’endommagement au secours de +la mécanique de la rupture : L’évolution de cette idée en un +demi-siècle + Marigo + Comptes Rendus Mécanique + 10.5802/crmeca.156 + 2023 + Marigo, J.-J. (2023). La mécanique de +l’endommagement au secours de la mécanique de la rupture : L’évolution +de cette idée en un demi-siècle. Comptes Rendus Mécanique. +https://doi.org/10.5802/crmeca.156 + + + Dolfiny: Python wrappers for +DOLFINx + Habera + 2024 + Habera, M., & Zilian, A. (2024). +Dolfiny: Python wrappers for DOLFINx. +https://github.com/michalhabera/dolfiny + + + DOLFINx: The next generation FEniCS problem +solving environment + Baratta + 10.5281/zenodo.10447666 + 2023 + Baratta, I. A., Dean, J. P., Dokken, +J. S., Habera, M., Hale, J. S., Richardson, C. N., Rognes, M. E., +Scroggs, M. W., Sime, N., & Wells, G. N. (2023). DOLFINx: The next +generation FEniCS problem solving environment. Zenodo. +https://doi.org/10.5281/zenodo.10447666 + + + PETSc/TAO users manual + Balay + 10.2172/2205494 + 2023 + Balay, S., Abhyankar, S., Adams, M. +F., Benson, S., Brown, J., Brune, P., Buschelman, K., Constantinescu, +E., Dalcin, L., Dener, A., Eijkhout, V., Faibussowitsch, J., Gropp, W. +D., Hapla, V., Isaac, T., Jolivet, P., Karpeev, D., Kaushik, D., +Knepley, M. G., … Zhang, J. (2023). PETSc/TAO users manual (ANL-21/39 - +Revision 3.20). Argonne National Laboratory. +https://doi.org/10.2172/2205494 + + + Numerical bifurcation and stability analysis +of variational gradient-damage models for phase-field +fracture + León Baldelli + Journal of the Mechanics and Physics of +Solids + 152 + 10.1016/j.jmps.2021.104424 + 2021 + León Baldelli, A. A., & Maurini, +C. (2021). Numerical bifurcation and stability analysis of variational +gradient-damage models for phase-field fracture. Journal of the +Mechanics and Physics of Solids, 152, 104424. +https://doi.org/10.1016/j.jmps.2021.104424 + + + Rate-independent systems + Mielke + 10.1007/978-1-4939-2706-7 + 0066-5452 + 2015 + Mielke, A., & Roubíček, A. +(2015). Rate-independent systems. Springer New York, NY. +https://doi.org/10.1007/978-1-4939-2706-7 + + + Cone-constrained eigenvalue problems: theory +and algorithms + Pinto da Costa + Computational Optimization and +Applications + 1 + 45 + 10.1007/s10589-008-9167-8 + 2010 + Pinto da Costa, A., & Seeger, A. +(2010). Cone-constrained eigenvalue problems: theory and algorithms. +Computational Optimization and Applications, 45(1), 25–57. +https://doi.org/10.1007/s10589-008-9167-8 + + + Numerical optimization + Jorge Nocedal + 10.1007/978-0-387-40065-5 + 0387303030 + 1999 + Jorge Nocedal, S. W. (1999). +Numerical optimization. Springer. +https://doi.org/10.1007/978-0-387-40065-5 + + + Décomposition orthogonale d’un espace +hilbertien selon deux cônes mutuellement polaires + Moreau + Comptes rendus hebdomadaires des séances de +l’Académie des sciences + 255 + 1962 + Moreau, J. J. (1962). Décomposition +orthogonale d’un espace hilbertien selon deux cônes mutuellement +polaires. Comptes Rendus Hebdomadaires Des séances de l’Académie Des +Sciences, 255, 238–240. + + + + + + From bd2dc4b786acd4949927f7ba3301f8ea58f174f4 Mon Sep 17 00:00:00 2001 From: The Open Journals editorial robot <89919391+editorialbot@users.noreply.github.com> Date: Thu, 5 Dec 2024 15:22:15 +0000 Subject: [PATCH 3/4] Creating 10.21105.joss.06897.jats --- .../paper.jats/10.21105.joss.06897.jats | 777 ++++++++++++++++++ 1 file changed, 777 insertions(+) create mode 100644 joss.06897/paper.jats/10.21105.joss.06897.jats diff --git a/joss.06897/paper.jats/10.21105.joss.06897.jats b/joss.06897/paper.jats/10.21105.joss.06897.jats new file mode 100644 index 0000000000..577e859718 --- /dev/null +++ b/joss.06897/paper.jats/10.21105.joss.06897.jats @@ -0,0 +1,777 @@ + + +
    + + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +6897 +10.21105/joss.06897 + +Variational Solvers for Irreversible Evolutionary +Systems + + + +https://orcid.org/0000-0002-3019-602X + +León Baldelli +Andrés A + + +* + + +https://orcid.org/0000-0002-0304-9382 + +Cesana +Pierluigi + + +* + + + +Sorbonne Université, CNRS, Institut Jean Le Rond +d’Alembert, F-75005 Paris, France + + + + +Institute of Mathematics for Industry, Kyushu University, +744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan. + + + + +* E-mail: +* E-mail: + + +1 +8 +2024 + +9 +104 +6897 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2024 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +Python +evolutions +stability +bifurcation +irreversibility +nonlinear +nonconvex +singular perturbations + + + + + + Summary +

    We study irreversible evolutionary processes with a general + energetic notion of stability. With this contribution, we release + three nonlinear variational solvers as modular components (based on + FEniCSx/dolfinx) that address three mathematical optimisation + problems. They are general enough to apply, in principle, to + evolutionary systems with instabilities, jumps, and emergence of + patterns. Systems with these qualities are commonplace in diverse + arenas spanning from quantum to continuum mechanics, economy, social + sciences, and ecology. Our motivation proceeds from fracture + mechanics, with the ultimate goal of deploying a transparent numerical + platform for scientific validation and prediction of large scale + natural fracture phenomena. Our solvers are used to compute + one solution to a problem encoded in a system of two + inequalities: one (pointwise almost-everywhere) constraint of + irreversibility and one global energy statement.

    +
    + + Statement of need +

    Quasi-static evolution problems arising in fracture are strongly + nonlinear + (Marigo, + 2023), + (Bourdin + et al., 2008). They can admit multiple solutions, or none + (León + Baldelli & Maurini, 2021). This demands both a functional + theoretical framework and practical computational tools for real case + scenarios. Due to the lack of uniqueness of solutions, it is + fundamental to leverage the full variational structure of the problem + and investigate solutions up to second order, to detect nucleation of + stable modes and transitions of unstable states. The stability of a + multiscale system along its nontrivial evolutionary paths in phase + space is a key property that is difficult to check: numerically, for + real case scenarios with several length scales involved, and + analytically, in the infinite-dimensional setting. Despite the concept + of unilateral stability is classical in the variational theory of + irreversible systems + (Mielke + & Roubíček, 2015) and the mechanics of fracture + (Francfort + & Marigo, 1998) (see also Nguyen + (2000)), + few studies have explored second-order criteria for crack nucleation + and evolution. Although sporadic, these studies are significant, + including + (Pham + et al., 2011), + (Pham + & Marigo, 2013), + (Sicsic + et al., 2014), + (León + Baldelli & Maurini, 2021), and + (Zolesi + & Maurini, 2024). The current literature in computational + fracture mechanics predominantly focuses on unilateral first-order + criteria, systematically neglecting the exploration of higher-order + information for critical points. To the best of our knowledge, no + general numerical tools are available to address second-order criteria + in evolutionary nonlinear irreversible systems and fracture + mechanics.

    +

    To fill this gap, our nonlinear solvers offer a flexible toolkit + for advanced stability analysis of systems which evolve with + constraints.

    +
    + + <bold>Functionality</bold> +

    We attack the following abstract problem which encodes a selection + principle:

    +

    + 0, \text{ find an } \text{ irreversible-constrained evolution } y_t + ]]> + P(0): Given T>0, find an irreversible-constrained evolution yt

    +

    + + yt:t[0,T]Xt such that

    +

    + + [Unilateral Stability]E(yt)E(yt+z),zV0×K0+[1]

    +

    Above, + + T + defines a horizon of events. The system is represented by its total + energy + + E + and + + Xt + is the time-dependent space of admissible states. A generic element of + + + Xt + contains a macroscopic field that can be externally driven (or + controlled, e.g. via boundary conditions) and an internal field (akin + to an internal degree of order). In the applications of fracture, the + kinematic variable is a vector-valued displacement + + + u(x) + and the degree of order + + α(x) + controls the softening of the material. Irreversibility applies to the + internal variable, hence an irreversible-constrained evolution is a + mapping parametrised by + + t + such that + + αt(x) + is non-decreasing with respect to + + t. + The kinematic variable is subject to bilateral variations belonging to + a linear subset of a Sobolev vector space + + + V0, + whereas the test space for the internal order parameter + + + K0+ + only contains positive fields owing to the irreversibility constraint. + The main difficulties are to correctly enforce unilateral constraints + and to account for the changing nature of the space of variations.

    +

    HybridSolver (1) + BifurcationSolver, (2) and + StabilitySolver (3) address the solution of [1] + in three stages:

    + + +

    A constrained variational inequality; that is first order + necessary conditions for unilateral equilibrium.

    +
    + +

    A singular variational eigen-problem in a vector space; that is + a bifurcation problem indicating uniqueness (or lack thereof) of + the evolution path.

    +
    + +

    A constrained eigen-inequality in a convex cone; originating + from a second order eigenvalue problem indicating stabilty of the + system (or lack thereof).

    +
    +
    +

    These numerical tools can be used to study general evolutionary + problems formulated in terms of fully nonlinear functional operators + in spaces of high or infinite dimension. In this context, systems can + have surprising and complicated behaviours such as symmetry breaking + bifurcations, endogenous pattern formation, localisations, and + separation of scales. Our solvers can be extended or adapted to a + variety of systems described by an energetic principle formulated as + in [1].

    + + Software +

    Our solvers are written in Python and are + built on DOLFINx, an expressive and + performant parallel distributed computing environment for solving + partial differential equations using the finite element method + (Baratta + et al., 2023). It enables us wrapping high-level functional + mathematical constructs with full flexibility and control of the + underlying linear algebra backend. We use PETSc + (Balay + et al., 2023), petsc4py + (Dalcin + et al., 2011), SLEPc.EPS + (Hernandez + et al., 2005), and dolfiny + (Habera + & Zilian, 2024) for parallel scalability.

    +

    Our solver’s API receives an abstract energy functional, a + user-friendly description of the state of the system as a dictionary + {"u": u, "alpha": alpha}, + where the first element is associated to the reversible field and + the second to the irreversible component, the associated constraints + on the latter, and the solver’s parameters (see an example in the + Addendum). + Solvers can be instantiated calling

    + solver = {Hybrid,Bifurcation,Stability}Solver( + E, # An energy (dolfinx.fem.form) + state, # A dictionary of fields describing the system + bcs, # A list of boundary conditions + [bounds], # A list of bounds (lower and upper) for the order parameter + parameters # A dictionary of numerical parameters + ) +

    where [bounds]=[lower, upper] are required + for the HybridSolver. Calling + solver.solve(<args>) triggers the + solution of the corresponding variational problem. Here, + <args> depend on the solver (see the + documentation for details).

    +

    HybridSolver solves a (first order) + constrained nonlinear variational inequality, implementing a + two-phase hybrid strategy which is ad hoc for + energy models typical of applications in damage and fracture + mechanics. The first phase (iterative alternate minimisation) is + based on a de-facto industry standard, conceived to exploit the + (partial, directional) convexity of the underlying mechanical models + (Bourdin + et al., 2000). Once an approximate-solution enters the + attraction set around a critical point, the solver switches to + perform a fully nonlinear step solving a block-matrix problem via + Newton’s method. This guarantees a precise estimation of the + convergence of the first-order nonlinear problem based on the norm + of the (constrained) residual.

    +

    BifurcationSolver is a variational + eigenvalue solver which uses SLEPc.EPS to explore the lower part of + the spectrum of the Hessian of the energy, automatically computed + performing two directional derivatives. Constraints are accounted + for by projecting the full Hessian onto the subspace of inactive + constraints + (Jorge + Nocedal, 1999). The relevance of this approach is typical of + systems with threshold laws. Thus, the solve + method returns a boolean value indicating whether the restricted + Hessian is positive definite. Internally, the solver stores the + lower part of the operators’ spectrum as an array.

    +

    StabilitySolver solves a constrained + variational eigenvalue inequality in a convex cone, to check whether + the (restricted) nonlinear Hessian operator is positive therein. + Starting from an initial guess + + z0*, + it iteratively computes (eigenvalue, eigenvector) pairs + + + (λk,zk) + converging to a limit + + (λ*,z*) + (as + + k), + by implementing a simple projection and scaling algorithm + (Moreau, + 1962), + (Pinto + da Costa & Seeger, 2010). The positivity of + + + λ* + (the smallest eigenvalue) allows to conclude on the stability of the + current state (or lack thereof), hence effectively solving P(0). + Notice that, if the current state is unstable + ( + + λ*<0), + the minimal eigenmode indicates the direction of energy + decrease.

    +

    We dedicate a separate contribution to illustrate how the three + solvers are algorithmically combined to solve problem P(0) in the + case of fracture. + [fig:convergence] + illustrates the numerical convergence properties of the + StabilitySolver in a 1d verification + test.

    +

    In a + supplementary + document, we perform a thorough verification of the code + through parametric benchmark for investigating the stability of a 1D + mechanical system, providing analytical expressions used for + comparison with numerical solutions, as well as all parameters + (numerical and physical) employed in the calculations. Accuracy and + reliability of the solvers is shown by the close agreement between + numerical and analytic solutions in a benchmark minimisation of (a + constrained) Rayleigh ratio, a key problem for applications in + structural mechanics and stability analysis.

    + +

    Rate of convergence for + StabilitySolver in 1d (cf. benchmark + problem in the + Addendum). + Targets are the eigenvalue + + limkλk=:λ* + (pink) and the associated eigen-vector + + + x* + (error curve in blue). Note that the residual vector (green) for + the cone problem need not be zero at a + solution.

    + +
    +
    + + Acknowledgements +

    ALB acknowledges the students of MEC647 (Complex Crack + Propagation in Brittle Materials) of the + Modélisation Multiphysique Multiéchelle des Matériaux et des Structures + master program at ENSTA Paris Tech/École Polytechnique for their + contributions, motivation, and feedback; Yves Capdeboscq, + Jean-Jacques Marigo, Sebastien Neukirch, and Luc Nguyen, for + constructive discussions and one key insight that was crucial for + this project. The work of PC was supported by the JSPS Innovative + Area grant JP21H00102 and JSPS Grant-in-Aid for Scientific Research + (C) JP24K06797. PC holds an honorary appointment at La Trobe + University and is a member of GNAMPA. +

    +
    +
    + + + + + + + + SicsicPaul + MarigoJean-Jacques + MauriniCorrado + + Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling + Journal of the Mechanics and Physics of Solids + 2014 + 63 + 10.1016/j.jmps.2013.09.003 + 256 + 284 + + + + + + BažantZdeněk P. + + Stable States and Paths of Stmuctures with Plasticity or Damage + Journal of Engineering Mechanics + 1988 + 114 + 10.1061/(ASCE)0733-9399(1988)114:12(2013) + + + + + + BourdinBlaise + FrancfortGilles A + MarigoJean-Jacques + + The variational approach to fracture + Springer + 2008 + 91 + 10.1007/978-1-4020-6395-4 + 5 + 148 + + + + + + PetrykH. + ThermannK. + + Second-order bifurcation in elastic-plastic solids + Journal of the Mechanics and Physics of Solids + 1985 + 33 + 6 + 10.1016/0022-5096(85)90004-3 + 577 + 593 + + + + + + NguyenQuoc Son + + Standard dissipative systems and stability analysis + Continuum thermodynamics: The art and science of modeling matter’s behavior + + MauginGérard + DrouotRaymonde + SidoroffFrançois + + Springer + 2000 + 10.1007/0-306-46946-4 + 343 + 354 + + + + + + NguyenQuoc Son + + Bifurcation and stability in dissipative media (plasticity, friction, fracture) + Applied Mechanics Reviews + 199401 + 47 + 1 + 0003-6900 + 10.1115/1.3111068 + 1 + 31 + + + + + + ZolesiCamilla + MauriniCorrado + + Stability and crack nucleation in variational phase-field models of fracture: Effects of length-scales and stress multi-axiality + Journal of the Mechanics and Physics of Solids + 2024 + 192 + 105802 + + + + + + + PhamKim + MarigoJean-Jacques + + Stability of Homogeneous States with Gradient Damage Models: Size Effects and Shape Effects in the Three-Dimensional Setting + Journal of Elasticity + 2013 + 110 + 1 + 10.1007/s10659-012-9382-5 + 63 + 93 + + + + + + FrancfortG. A. + MarigoJ.-J. + + Revisiting brittle fracture as an energy minimization problem + Journal of the Mechanics and Physics of Solids + 1998 + 46 + 8 + 0022-5096 + 10.1016/S0022-5096(98)00034-9 + 1319 + 1342 + + + + + + DalcinLisandro D. + PazRodrigo R. + KlerPablo A. + CosimoAlejandro + + Parallel distributed computing using Python + Advances in Water Resources + 2011 + 34 + 9 + 0309-1708 + 10.1016/j.advwatres.2011.04.013 + 1124 + 1139 + + + + + + PhamKim + MarigoJean-Jacques + MauriniCorrado + + The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models + Journal of the Mechanics and Physics of Solids + 2011 + 59 + 6 + 0022-5096 + 10.1016/j.jmps.2011.03.010 + 1163 + 1190 + + + + + + BourdinBlaise + FrancfortGilles A + MarigoJean-Jacques + + Numerical experiments in revisited brittle fracture + Journal of the Mechanics and Physics of Solids + 2000 + 48 + 4 + 10.1016/S0022-5096(99)00028-9 + 797 + 826 + + + + + + HernandezVicente + RomanJose E. + VidalVicente + + SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems + ACM Trans. Math. Software + 2005 + 31 + 3 + 10.1145/1089014.1089019 + 351 + 362 + + + + + + MarigoJean-Jacques + + La mécanique de l’endommagement au secours de la mécanique de la rupture : L’évolution de cette idée en un demi-siècle + Comptes Rendus Mécanique + 2023 + 10.5802/crmeca.156 + + + + + + HaberaMichal + ZilianAndreas + + Dolfiny: Python wrappers for DOLFINx + 2024 + https://github.com/michalhabera/dolfiny + + + + + + BarattaIgor A. + DeanJoseph P. + DokkenJørgen S. + HaberaMichal + HaleJack S. + RichardsonChris N. + RognesMarie E. + ScroggsMatthew W. + SimeNathan + WellsGarth N. + + DOLFINx: The next generation FEniCS problem solving environment + Zenodo + 202312 + 10.5281/zenodo.10447666 + + + + + + BalaySatish + AbhyankarShrirang + AdamsMark F. + BensonSteven + BrownJed + BrunePeter + BuschelmanKris + ConstantinescuEmil + DalcinLisandro + DenerAlp + EijkhoutVictor + FaibussowitschJacob + GroppWilliam D. + HaplaVáclav + IsaacTobin + JolivetPierre + KarpeevDmitry + KaushikDinesh + KnepleyMatthew G. + KongFande + KrugerScott + MayDave A. + McInnesLois Curfman + MillsRichard Tran + MitchellLawrence + MunsonTodd + RomanJose E. + RuppKarl + SananPatrick + SarichJason + SmithBarry F. + ZampiniStefano + ZhangHong + ZhangHong + ZhangJunchao + + PETSc/TAO users manual + Argonne National Laboratory + 2023 + 10.2172/2205494 + + + + + + León BaldelliAndrés A. + MauriniCorrado + + Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture + Journal of the Mechanics and Physics of Solids + 2021 + 152 + 10.1016/j.jmps.2021.104424 + 104424 + + + + + + + MielkeA. + RoubíčekA. + + Rate-independent systems + Springer New York, NY + 2015 + 0066-5452 + 10.1007/978-1-4939-2706-7 + + + + + + Pinto da CostaA. + SeegerA. + + Cone-constrained eigenvalue problems: theory and algorithms + Computational Optimization and Applications + 2010 + 45 + 1 + 10.1007/s10589-008-9167-8 + 25 + 57 + + + + + + Jorge NocedalStephen Wright + + Numerical optimization + Springer + 1999 + 0387303030 + 10.1007/978-0-387-40065-5 + + + + + + MoreauJean Jacques + + Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires + Comptes rendus hebdomadaires des séances de l’Académie des sciences + 1962 + 255 + 238 + 240 + + + + +
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