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References from Monolithic, partially staggered, and fully staggered schemes for thermomechanically coupled phase-field crack simulations: Solver comparison and performance evaluation. Local targets link to admitted publications; unresolved targets remain external evidence.
Revisiting brittle fracture as an energy minimization problem
10.1016/s0022-5096(98)00034-9 · 1998 · External reference
Numerical experiments in revisited brittle fracture
10.1016/s0022-5096(99)00028-9 · 2000 · External reference
The variational approach to fracture
10.1007/s10659-007-9107-3 · 2008 · External reference
Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations
10.1002/nme.2861 · 2010 · External reference
A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits
10.1016/j.cma.2010.04.011 · 2010 · External reference
A review on phase-field models of brittle fracture and a new fast hybrid formulation
10.1007/s00466-014-1109-y · 2015 · External reference
A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach
10.1016/j.cma.2015.03.009 · 2015 · External reference
A unified phase-field theory for the mechanics of damage and quasi-brittle failure
10.1016/j.jmps.2017.03.015 · 2017 · External reference
Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation
10.1016/j.cma.2017.07.026 · 2017 · External reference
On the BFGS monolithic algorithm for the unified phase field damage theory
10.1016/j.cma.2019.112704 · 2020 · External reference
A multi-resolution approach to hydraulic fracture simulation
10.1007/s10704-022-00662-y · 2022 · External reference
Gradient projection method for enforcing crack irreversibility as box constraints in a robust monolithic phase-field scheme
10.1016/j.cma.2024.117622 · 2025 · External reference
Study the dynamic crack path in brittle material under thermal shock loading by phase field modeling
10.1007/s10704-017-0220-4 · 2017 · External reference
An h-adaptive thermo-mechanical phase field model for fracture
10.1016/j.finel.2017.09.003 · 2018 · External reference
Fracture of thermo-elastic solids: Phase-field modeling and new results with an efficient monolithic solver
10.1016/j.cma.2020.113648 · 2021 · External reference
Functionally graded ultra-high temperature ceramics for hypersonic applications: A numerical study of fracture under high-temperature extremes
10.1016/j.engfracmech.2025.111794 · 2026 · External reference
Thermal-conductivity degradation across cracks in coupled thermo-mechanical systems modeled by the phase-field fracture method
10.1016/j.jmps.2019.103861 · 2020 · External reference
A phase-field fracture formulation for generalized standard materials: The interplay between thermomechanics and damage
10.1016/j.jmps.2025.106154 · 2025 · External reference
Fracture behavior of anisotropic 3D-printed parts: experiments and numerical simulations
10.1016/j.jmrt.2022.05.068 · 2022 · External reference
A thermo-mechanical phase-field fracture model: Application to hot cracking simulations in additive manufacturing
10.1016/j.jmps.2022.105169 · 2023 · External reference
A variational framework to model diffusion induced large plastic deformation and phase field fracture during initial two-phase lithiation of silicon electrodes
10.1016/j.cma.2016.05.007 · 2016 · External reference
Phase field modeling of electrochemically induced fracture in li-ion battery with large deformation and phase segregation
10.1002/gamm.201610006 · 2016 · External reference
Phase field modeling of chemomechanical fracture of intercalation electrodes: Role of charging rate and dimensionality
10.1016/j.jmps.2019.103696 · 2019 · External reference
A phase field-based framework for electro-chemo-mechanical fracture: Crack-contained electrolytes, chemical reactions and stabilisation
10.1016/j.cma.2023.116235 · 2023 · External reference
A length-scale insensitive cohesive phase-field interface model: Application to concurrent bulk and interface fracture simulation in lithium-ion battery materials
10.1016/j.jmps.2024.106013 · 2025 · External reference
A phase field formulation for hydrogen assisted cracking
10.1016/j.cma.2018.07.021 · 2018 · External reference
A phase-field regularized cohesive zone model for hydrogen assisted cracking
10.1016/j.cma.2019.112614 · 2020 · External reference
A generalised, multi-phase-field theory for dissolution-driven stress corrosion cracking and hydrogen embrittlement
10.1016/j.jmps.2022.104951 · 2022 · External reference
Computational predictions of hydrogen-assisted fatigue crack growth
10.1016/j.ijhydene.2024.05.264 · 2024 · External reference
A phase field model of electromechanical fracture
10.1016/j.jmps.2010.06.013 · 2010 · External reference
Phase-field modeling of electromechanical fracture in piezoelectric solids: Analytical results and numerical simulations
10.1016/j.cma.2021.114125 · 2021 · External reference
Crack nucleation and propagation of electromagneto-thermo-mechanical fracture in bulk superconductors during magnetization
10.1016/j.jmps.2022.105168 · 2023 · External reference
A generalised phase field model for fatigue crack growth in elastic–plastic solids with an efficient monolithic solver
10.1016/j.cma.2021.114286 · 2022 · External reference
A coupled phase field formulation for modelling fatigue cracking in lithium-ion battery electrode particles
10.1016/j.jpowsour.2022.231805 · 2022 · External reference
Phase field modelling of fracture and fatigue in shape memory alloys
10.1016/j.cma.2020.113504 · 2021 · External reference
Linear and nonlinear solvers for variational phase-field models of brittle fracture
10.1002/nme.5300 · 2017 · External reference
An accelerated staggered scheme for variational phase-field models of brittle fracture
10.1016/j.cma.2021.113822 · 2021 · External reference
A phase-field model of thermo-elastic coupled brittle fracture with explicit time integration
10.1007/s00466-020-01820-6 · 2020 · External reference
Interior-point methods for the phase-field approach to brittle and ductile fracture
10.1016/j.cma.2020.113612 · 2021 · External reference
An efficient and robust monolithic approach to phase-field quasi-static brittle fracture using a modified Newton method
10.1016/j.cma.2021.114091 · 2021 · External reference
A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation
10.1016/j.cma.2022.114927 · 2022 · External reference
A recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture
10.1016/j.cma.2019.112720 · 2020 · External reference
A line search assisted monolithic approach for phase-field computing of brittle fracture
10.1016/j.cma.2015.12.017 · 2016 · External reference
Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme
10.1016/j.tafmec.2019.102446 · 2020 · External reference
A novel phase-field monolithic scheme for brittle crack propagation based on the limited-memory BFGS method with adaptive mesh refinement
10.1002/nme.7572 · 2024 · External reference
Updating Quasi-Newton matrices with limited storage
10.1090/s0025-5718-1980-0572855-7 · 1980 · External reference
On the limited memory BFGS method for large scale optimization
10.1007/bf01589116 · 1989 · External reference
Representations of quasi-Newton matrices and their use in limited memory methods
10.1007/bf01582063 · 1994 · External reference
Gradient damage models and their use to approximate brittle fracture
10.1177/1056789510386852 · 2011 · External reference
Morphogenesis and propagation of complex cracks induced by thermal shocks
10.1103/physrevlett.112.014301 · 2014 · External reference
Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling
10.1016/j.jmps.2013.09.003 · 2014 · External reference
Crack surface energy: Temperature and force dependence
10.2320/matertrans.mer2007238 · 2008 · External reference
A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems
10.1002/nme.1620350408 · 1992 · External reference
Unresolved reference
2008 · External reference
Unresolved reference
2006 · External reference
Phase-field fracture analysis: A gradient-based line search strategy for the L-BFGS algorithm
10.1016/j.cma.2025.118170 · 2025 · External reference
Adaptive phase field method using novel physics based refinement criteria
10.1016/j.cma.2021.113874 · 2021 · External reference
A novel adaptive mesh refinement scheme for the simulation of phase-field fracture using trimmed hexahedral meshes
10.1002/nme.6587 · 2021 · External reference
An adaptive mesh refinement algorithm for phase-field fracture models: Application to brittle, cohesive, and dynamic fracture
10.1016/j.cma.2022.115347 · 2022 · External reference
The deal.II library, version 9.7
10.1515/jnma-2025-0115 · 2025 · External reference
A study of the mechanism of formation and numerical simulations of crack patterns in ceramics subjected to thermal shock
10.1016/j.actamat.2012.05.020 · 2012 · External reference
Phase field modeling of electrochemically induced fracture in li-ion battery with large deformation and phase segregation
10.1002/gamm.201610006 · ExternalCitation · doi-reference
A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems
10.1002/nme.1620350408 · ExternalCitation · doi-reference
Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations
10.1002/nme.2861 · ExternalCitation · doi-reference
Linear and nonlinear solvers for variational phase-field models of brittle fracture
10.1002/nme.5300 · ExternalCitation · doi-reference
A novel adaptive mesh refinement scheme for the simulation of phase-field fracture using trimmed hexahedral meshes
10.1002/nme.6587 · ExternalCitation · doi-reference
A novel phase-field monolithic scheme for brittle crack propagation based on the limited-memory BFGS method with adaptive mesh refinement
10.1002/nme.7572 · ExternalCitation · doi-reference
Representations of quasi-Newton matrices and their use in limited memory methods
10.1007/bf01582063 · ExternalCitation · doi-reference
On the limited memory BFGS method for large scale optimization
10.1007/bf01589116 · ExternalCitation · doi-reference
A review on phase-field models of brittle fracture and a new fast hybrid formulation
10.1007/s00466-014-1109-y · ExternalCitation · doi-reference
A phase-field model of thermo-elastic coupled brittle fracture with explicit time integration
10.1007/s00466-020-01820-6 · ExternalCitation · doi-reference
The variational approach to fracture
10.1007/s10659-007-9107-3 · ExternalCitation · doi-reference
Study the dynamic crack path in brittle material under thermal shock loading by phase field modeling
10.1007/s10704-017-0220-4 · ExternalCitation · doi-reference
A multi-resolution approach to hydraulic fracture simulation
10.1007/s10704-022-00662-y · ExternalCitation · doi-reference
A study of the mechanism of formation and numerical simulations of crack patterns in ceramics subjected to thermal shock
10.1016/j.actamat.2012.05.020 · ExternalCitation · doi-reference
A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits
10.1016/j.cma.2010.04.011 · ExternalCitation · doi-reference
A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach
10.1016/j.cma.2015.03.009 · ExternalCitation · doi-reference
A line search assisted monolithic approach for phase-field computing of brittle fracture
10.1016/j.cma.2015.12.017 · ExternalCitation · doi-reference
A variational framework to model diffusion induced large plastic deformation and phase field fracture during initial two-phase lithiation of silicon electrodes
10.1016/j.cma.2016.05.007 · ExternalCitation · doi-reference
Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation
10.1016/j.cma.2017.07.026 · ExternalCitation · doi-reference
A phase field formulation for hydrogen assisted cracking
10.1016/j.cma.2018.07.021 · ExternalCitation · doi-reference
A phase-field regularized cohesive zone model for hydrogen assisted cracking
10.1016/j.cma.2019.112614 · ExternalCitation · doi-reference
On the BFGS monolithic algorithm for the unified phase field damage theory
10.1016/j.cma.2019.112704 · ExternalCitation · doi-reference
A recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture
10.1016/j.cma.2019.112720 · ExternalCitation · doi-reference
Phase field modelling of fracture and fatigue in shape memory alloys
10.1016/j.cma.2020.113504 · ExternalCitation · doi-reference
Interior-point methods for the phase-field approach to brittle and ductile fracture
10.1016/j.cma.2020.113612 · ExternalCitation · doi-reference
Fracture of thermo-elastic solids: Phase-field modeling and new results with an efficient monolithic solver
10.1016/j.cma.2020.113648 · ExternalCitation · doi-reference
An accelerated staggered scheme for variational phase-field models of brittle fracture
10.1016/j.cma.2021.113822 · ExternalCitation · doi-reference
Adaptive phase field method using novel physics based refinement criteria
10.1016/j.cma.2021.113874 · ExternalCitation · doi-reference
An efficient and robust monolithic approach to phase-field quasi-static brittle fracture using a modified Newton method
10.1016/j.cma.2021.114091 · ExternalCitation · doi-reference
Phase-field modeling of electromechanical fracture in piezoelectric solids: Analytical results and numerical simulations
10.1016/j.cma.2021.114125 · ExternalCitation · doi-reference
A generalised phase field model for fatigue crack growth in elastic–plastic solids with an efficient monolithic solver
10.1016/j.cma.2021.114286 · ExternalCitation · doi-reference
A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation
10.1016/j.cma.2022.114927 · ExternalCitation · doi-reference
An adaptive mesh refinement algorithm for phase-field fracture models: Application to brittle, cohesive, and dynamic fracture
10.1016/j.cma.2022.115347 · ExternalCitation · doi-reference
A phase field-based framework for electro-chemo-mechanical fracture: Crack-contained electrolytes, chemical reactions and stabilisation
10.1016/j.cma.2023.116235 · ExternalCitation · doi-reference
Gradient projection method for enforcing crack irreversibility as box constraints in a robust monolithic phase-field scheme
10.1016/j.cma.2024.117622 · ExternalCitation · doi-reference
Phase-field fracture analysis: A gradient-based line search strategy for the L-BFGS algorithm
10.1016/j.cma.2025.118170 · ExternalCitation · doi-reference
Functionally graded ultra-high temperature ceramics for hypersonic applications: A numerical study of fracture under high-temperature extremes
10.1016/j.engfracmech.2025.111794 · ExternalCitation · doi-reference
An h-adaptive thermo-mechanical phase field model for fracture
10.1016/j.finel.2017.09.003 · ExternalCitation · doi-reference
Computational predictions of hydrogen-assisted fatigue crack growth
10.1016/j.ijhydene.2024.05.264 · ExternalCitation · doi-reference
A phase field model of electromechanical fracture
10.1016/j.jmps.2010.06.013 · ExternalCitation · doi-reference
Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling
10.1016/j.jmps.2013.09.003 · ExternalCitation · doi-reference
A unified phase-field theory for the mechanics of damage and quasi-brittle failure
10.1016/j.jmps.2017.03.015 · ExternalCitation · doi-reference
Phase field modeling of chemomechanical fracture of intercalation electrodes: Role of charging rate and dimensionality
10.1016/j.jmps.2019.103696 · ExternalCitation · doi-reference
Thermal-conductivity degradation across cracks in coupled thermo-mechanical systems modeled by the phase-field fracture method
10.1016/j.jmps.2019.103861 · ExternalCitation · doi-reference
A generalised, multi-phase-field theory for dissolution-driven stress corrosion cracking and hydrogen embrittlement
10.1016/j.jmps.2022.104951 · ExternalCitation · doi-reference
Crack nucleation and propagation of electromagneto-thermo-mechanical fracture in bulk superconductors during magnetization
10.1016/j.jmps.2022.105168 · ExternalCitation · doi-reference
A thermo-mechanical phase-field fracture model: Application to hot cracking simulations in additive manufacturing
10.1016/j.jmps.2022.105169 · ExternalCitation · doi-reference
A length-scale insensitive cohesive phase-field interface model: Application to concurrent bulk and interface fracture simulation in lithium-ion battery materials
10.1016/j.jmps.2024.106013 · ExternalCitation · doi-reference
A phase-field fracture formulation for generalized standard materials: The interplay between thermomechanics and damage
10.1016/j.jmps.2025.106154 · ExternalCitation · doi-reference
Fracture behavior of anisotropic 3D-printed parts: experiments and numerical simulations
10.1016/j.jmrt.2022.05.068 · ExternalCitation · doi-reference
A coupled phase field formulation for modelling fatigue cracking in lithium-ion battery electrode particles
10.1016/j.jpowsour.2022.231805 · ExternalCitation · doi-reference
Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme
10.1016/j.tafmec.2019.102446 · ExternalCitation · doi-reference
Revisiting brittle fracture as an energy minimization problem
10.1016/s0022-5096(98)00034-9 · ExternalCitation · doi-reference
Numerical experiments in revisited brittle fracture
10.1016/s0022-5096(99)00028-9 · ExternalCitation · doi-reference
Updating Quasi-Newton matrices with limited storage
10.1090/s0025-5718-1980-0572855-7 · ExternalCitation · doi-reference
Morphogenesis and propagation of complex cracks induced by thermal shocks
10.1103/physrevlett.112.014301 · ExternalCitation · doi-reference
Gradient damage models and their use to approximate brittle fracture
10.1177/1056789510386852 · ExternalCitation · doi-reference
The deal.II library, version 9.7
10.1515/jnma-2025-0115 · ExternalCitation · doi-reference
Crack surface energy: Temperature and force dependence
10.2320/matertrans.mer2007238 · ExternalCitation · doi-reference
