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Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme
Philip K. Kristensen, Emilio Martínez-Pañeda
TL;DR
The paper addresses convergence difficulties in monolithic phase field fracture solution schemes, including indefinite Newton Jacobians and the need for very small load steps. It presents a quasi-Newton monolithic scheme with adaptive time stepping, reporting robust convergence and substantial computational acceleration.
Problem
Monolithic minimisation remains extremely challenging, while indefinite Jacobians in Newton’s method hinder convergence and may require very small load steps.
Method
The paper presents a quasi-Newton monolithic solution scheme using the BFGS algorithm, enhanced with a new adaptive time-stepping scheme.
Results
Monolithic quasi-Newton schemes are robust, and Newton computations are 10 to 100 times faster in all the problems considered.
Takeaways & Limitations
Retaining unconditional stability while using quasi-Newton monolithic schemes yields drastically reduced computation times.
Takeaways & Limitations
Monolithic minimisation remains extremely challenging and may require very small load steps.
Abstract
from arXiv · showhide
We investigate the potential of quasi-Newton methods in facilitating convergence of monolithic solution schemes for phase field fracture modelling. Several paradigmatic boundary value problems are addressed, spanning the fields of quasi-static fracture, fatigue damage and dynamic cracking. The finite element results obtained reveal the robustness of quasi-Newton monolithic schemes, with convergence readily attained under both stable and unstable cracking conditions. Moreover, since the solution method is unconditionally stable, very significant computational gains are observed relative to the widely used staggered solution schemes. In addition, a new adaptive time increment scheme is presented to further reduces the computational cost while allowing to accurately resolve sudden changes in material behavior, such as unstable crack growth. Computation times can be reduced by several orders of magnitude, with the number of load increments required by the corresponding staggered solution being up to 3000 times higher. Quasi-Newton monolithic solution schemes can be a key enabler for large scale phase field fracture simulations. Implications are particularly relevant for the emerging field of phase field fatigue, as results show that staggered cycle-by-cycle calculations are prohibitive in mid or high cycle fatigue. The finite element codes are available to download from www.empaneda.com/codes.
1. Introduction
Phase field fracture couples displacement and phase-field variables in a non-convex problem that challenges robust monolithic solution. The paper proposes quasi-Newton monolithic schemes with adaptive time stepping and evaluates them across quasi-static, fatigue, and dynamic fracture.
- Motivation: The coupled displacement–phase-field problem is non-convex, making the Newton Jacobian indefinite and hindering monolithic convergence.The displacement field u and phase field ϕ are solved simultaneously in monolithic schemes.
- Motivation: Staggered schemes are robust but computationally demanding, requiring many iterations and small load steps to track equilibrium.Their alternating minimization makes the energy convex in the variable being solved, but removes unconditional stability.
- Contributions: The paper combines quasi-Newton methods with a monolithic scheme for standard phase field fracture and introduces adaptive time stepping.The approach is tested on quasi-static fracture, phase field fatigue, and dynamic fracture.
- Contributions: The adaptive criterion is designed to resolve sudden material changes, including unstable crack growth, while reducing computational cost.The study applies the scheme to paradigmatic problems spanning quasi-static, fatigue, and dynamic cracking.
- Results: Computation times are up to 100 times smaller than with the widely used staggered solution for the same result.The reported gains support quasi-Newton monolithic implementations for phase field fracture and fatigue modelling.
2. The phase field fracture method
The phase field fracture method regularizes unknown crack surfaces using an auxiliary damage-like phase field, converting fracture-surface work into a computationally tractable volume formulation. Coupled displacement and phase-field equations are derived from variational energy balances with prescribed boundary conditions.
- The fracture surface is unknown in the discrete formulation, but phase-field regularization makes the fracture-energy problem computationally tractable through volume integrals.
- The phase field ϕ tracks crack interfaces as a damage-like variable, ranging from 0 in intact material points to 1 in fully cracked points.
- The degradation function g(ϕ) = (1 − ϕ)2 reduces material stiffness as damage evolves.
- The reported performance applies to PF-CZM and AT2 regularizations, while the AT1 model remains unaddressed.
- The model combines displacement and phase fields subject to Dirichlet- and Neumann-type boundary conditions, including prescribed crack conditions.
- The governing coupled balances follow from kinetic and potential energies and yield residual equations for displacement and phase-field evolution.
3. Finite element implementation
The finite element implementation discretizes the coupled displacement–phase-field problem and solves its residual equations using monolithic or staggered strategies. Quasi-Newton BFGS updates, standard convergence controls, and adaptive increment reduction address computational efficiency and sudden material changes.
- Finite element discretization produces residuals and stiffness matrices for the displacement and crack phase fields.
- The implementation uses an Abaqus user element subroutine, with Abaqus2Matlab employed to preprocess input files.
- The findings concern the incrementally linear hybrid approach of Ambati et al.; performance in other models remains unaddressed.
- Monolithic schemes solve displacement and phase fields simultaneously, whereas staggered schemes solve them sequentially.
- Staggered schemes are not unconditionally stable and require sufficiently small increments, while monolithic schemes retain unconditional stability and permit larger increments.
- Quasi-Newton methods retain an approximate stiffness matrix across iterations and update it after a set number of nonconverged iterations.
- The BFGS update couples displacement and phase fields while retaining symmetry and positive definiteness when present initially, with significant computational savings.
- The adaptive scheme reduces increments when material response changes significantly, such as during unstable fracture, while very large increments remain possible otherwise.
4. Results
Numerical experiments evaluate the approach across quasi-static fracture, phase field fatigue, and dynamic crack branching. The results emphasize convergence and computational cost, including fatigue calculations that are prohibitive with staggered schemes.
- The numerical experiments cover stable and unstable quasi-static cracking, cycle-by-cycle phase field fatigue, and dynamic crack branching.
- Cycle-by-cycle phase field fatigue calculations are computationally prohibitive for staggered schemes, whereas the quasi-Newton method can enable them.
4.1. Quasi-static fracture
Quasi-static benchmarks show that quasi-Newton monolithic schemes robustly capture stable and unstable fracture, while substantially reducing increments, iterations, and computation time relative to staggered schemes.
- Uniaxial tension: Unstable crack growth is captured within a single increment without convergence problems using the quasi-Newton monolithic scheme.The complete crack growth extent is resolved in the critical increment.
- Uniaxial tension: 105 staggered increments are required to recover the accurate monolithic result, compared with 30 monolithic increments.Coarser staggered stepping produces substantial sensitivity and approximately 20% error in the critical displacement.
- Uniaxial tension: The monolithic implementation is roughly 100 times faster than the widely used staggered scheme across mesh densities.The time difference is mainly attributed to the reduced number of increments in the monolithic calculation.
- Adaptive time stepping: The adaptive scheme uses large increments outside cracking and drastically reduces increment size during unstable crack growth.It recovers large loading steps when few iterations are needed and captures the response with only 30 load increments.
- Shear: 105 or more staggered increments are needed for accurate shear results, and the staggered iteration count is roughly two orders of magnitude larger.Using 103 increments substantially changes both the force-displacement response and predicted crack path.
- Robustness: Convergence is attained in all cases without viscous dissipation parameters, whereas conventional Newton methods fail even with 105 increments.Adding a line-search algorithm does not prevent the reported conventional Newton failures.
4.2. Phase field fatigue
The phase field fatigue study applies quasi-Newton monolithic methods to a fatigue framework and finds substantially lower computational cost than staggered cycle-by-cycle calculations.
- Framework and validation: The study evaluates quasi-Newton monolithic performance within an emerging phase field fatigue modelling framework.The implementation is validated against results from Carrara and co-workers before comparison with staggered approaches.
- Theoretical framework: The fatigue model introduces a degradation function f(α(t)) depending on a cumulative history variable α.The selected simple history variable is independent of mean load, and degradation vanishes asymptotically.
- Fatigue response: Quasi-Newton predictions show relatively good agreement with isotropic and spectral decomposition results.The spectral split requires roughly twice as many cycles to trigger complete fracture because compressive cycles do not contribute to damage.
- Fatigue response: The volumetric-deviatoric quasi-Newton monolithic implementation converges without problems through final cyclic loading at R = −1.The loading ratio corresponds to equal compression and tension loads.
- Computational performance: 5 times larger computation times are required by the most precise staggered calculation, which remains far from the equilibrium result.More than 1000 increments per cycle may be needed to approximate the monolithic result.
- Computational performance: Mid-, high-, and very-high-cycle fatigue cannot be addressed with staggered schemes under the reported cycle-by-cycle requirements.The authors identify quasi-Newton methods as a potential route for phase field fatigue calculations.
4.3. Dynamic results
Dynamic fracture simulations show that quasi-Newton monolithic schemes reproduce crack branching while converging faster than staggered schemes and generating complex crack patterns.
- Dynamic crack branching: The dynamic benchmark uses a rectangular specimen with an initially sharp crack and an instantaneously applied vertical tensile traction.The simulation uses specified density, elastic properties, phase-field length scale, fracture energy, and linear quadrilateral elements.
- Dynamic solution: Backward Euler solves the dynamic system without requiring HHT or Newmark’s β-method to achieve convergence.The formulation includes inertia terms and off-diagonal matrices with larger relative weight.
- Dynamic crack branching: Both staggered and monolithic approaches predict crack growth followed by branching qualitatively.The shared qualitative result is reported for the dynamic crack branching benchmark.
- Computational performance: 4–6 times more iterations per increment are required by the staggered approach after crack widening begins, making total computation time almost 3 times longer.The comparison indicates faster convergence for the monolithic quasi-Newton approach in the dynamic problem.
- Method versatility: Complex crack patterns are obtained with the monolithic quasi-Newton implementation in an additional dynamic configuration.The reported example uses Gc = 0.5 J/m2, quadratic elements, and an initial crack prescribed through ϕ.
5. Conclusions
The paper presents a BFGS-based quasi-Newton monolithic scheme with adaptive time stepping for phase field fracture problems. It reports robust convergence, substantial computational savings over staggered schemes, and potential for cycle-by-cycle fatigue and large-scale simulations.
- 5. Conclusions: The framework combines a Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton monolithic scheme with a new adaptive time stepping algorithm.It addresses stable and unstable quasi-static fracture, phase field fatigue, and dynamic crack branching.
- 5. Conclusions: Monolithic quasi-Newton schemes solve benchmark problems of varying complexity robustly without the convergence problems reported for standard Newton monolithic frameworks.The conclusion identifies robustness across the considered problem set as a main finding.
- 5. Conclusions: Monolithic quasi-Newton computations are 10 to 100 times faster than staggered solution schemes in all problems considered.The reported computational gain is attributed alongside retention of unconditional stability.
- 5. Conclusions: Accurate phase field fatigue predictions require only 4 increments per cycle, several orders of magnitude below the requirements of staggered schemes.This reduction supports cycle-by-cycle fatigue calculations that would otherwise be computationally prohibitive.
- 5. Conclusions: The authors expect monolithic quasi-Newton schemes to enable physically based cycle-by-cycle fatigue calculations and large-scale phase field fracture problems.They identify performance in other challenging applications, including large strains and nearly incompressible materials, as future work.