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Linear and nonlinear solvers for variational phase-field models of brittle fracture
Patrick E. Farrell, Corrado Maurini
TL;DR
Variational phase-field fracture models capture complex crack patterns but require expensive solves of large, ill-conditioned nonlinear variational inequalities. The paper accelerates alternate minimization with over-relaxation and Newton composition, and develops preconditioners for the associated linear systems. These improvements reduce time to solution by 5–6× on the surfing and thermal shock test cases.
Problem
Complex crack simulations require solving large, non-convex, constrained problems with small damage length scales, slow alternate minimization, and ill-conditioned linear subproblems.
Method
The paper applies nonlinear Gauss-Seidel over-relaxation, composes alternate minimization with active-set Newton methods, and designs scalable preconditioners for both solver types.
Results
5–6× lower time to solution is reported for the surfing and thermal shock test cases after combining the proposed alternate-minimization improvements.
Takeaways & Limitations
Over-relaxation is inexpensive to implement, while combining it with Newton-type methods further reduces runtime at greater development cost.
Takeaways & Limitations
The relaxation parameter is selected through numerical experimentation on coarser problems because its optimal value is difficult to identify a priori.
Abstract
from arXiv · showhide
The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution; and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is 5--6$\times$ faster on a majority of the test cases
1. INTRODUCTION
Variational phase-field fracture models make complex crack evolution tractable through regularization, but their non-convex, constrained optimization problems remain expensive because small damage bands require fine meshes and ill-conditioned linear solves. This paper develops faster nonlinear and linear solvers for the discretized problem.
- 1. INTRODUCTION: Variational fracture formulations recast crack nucleation and propagation as energy minimization, with phase-field regularization replacing discontinuous cracks by smooth damage fields.The damage field localizes in bands of thickness on the order of the internal length ℓ, providing a smeared crack representation.
- 1. INTRODUCTION: The regularized model enforces damage irreversibility through bound constraints, so each loading step requires solving a variational inequality.The constraint prevents self-healing and makes the damage update unilateral.
- 1. INTRODUCTION: The numerical focus is the simplest small-deformation, isotropic, quasi-static model, excluding geometrical nonlinearities and asymmetric traction-compression behavior.Extensions involving more complex physical effects are outside the scope of the work.
- 1. INTRODUCTION: The discretized problem is difficult because it is non-convex, resolves damage bands much smaller than the domain, and produces badly conditioned linear systems.Rapid stiffness variation from the undamaged value toward zero contributes to the conditioning problem.
- 1. INTRODUCTION: Alternate minimization separates displacement and damage updates, but its subproblems involve spatially varying linear elasticity and generalized Helmholtz-type variational inequalities.Its slow convergence motivates active-set and semismooth Newton alternatives, although Newton methods can be unreliable from poor initial guesses.
- 1. INTRODUCTION: The paper accelerates alternate minimization with nonlinear Gauss-Seidel over-relaxation, combines it with active-set Newton iterations, and designs scalable solvers for the resulting linear systems.The stated contributions target both the alternate-minimization subproblems and coupled Jacobians in active-set Newton methods.
2. NONLINEAR SOLVERS
The paper accelerates nonlinear solution by combining over-relaxed alternate minimization with an active-set Newton method. Alternate minimization supplies robust progress toward Newton’s convergence basin, while Newton iterations provide rapid final convergence.
- Composite nonlinear solver: The composite ORAM-N solver combines robust alternate minimization with an active-set Newton method and backtracking line search.Alternate minimization reduces the residual until Newton is effective; Newton then solves the nonlinear problem in a handful of iterations.
- Over-relaxed alternate minimization: Alternate minimization is reinterpreted as nonlinear block Gauss–Seidel, enabling over-relaxation of displacement and damage updates.The relaxation parameter satisfies ω ∈ (0, 2), while damage updates are additionally restricted to remain feasible.
- Choosing the relaxation parameter ω: Over-relaxed alternate minimization can significantly reduce iterations, but the best relaxation parameter is difficult to determine a priori.The proposed strategy selects ω through numerical experimentation on coarser problems rather than an automated rule.
- Composite nonlinear solver: Even with over-relaxation, difficult problems may require hundreds or thousands of iterations for tight convergence.The method is therefore used to enter Newton’s basin rather than to fully converge the optimization problem by itself.
- Reduced-space active set method: The active-set method zeros active search components and approximately solves a Newton system on inactive components.A line search minimizes the NCP residual norm, with steepest descent used if the Newton direction fails.
3. LINEAR SOLVERS AND PRECONDITIONERS
The linear subproblems are solved with iterative Krylov methods and algebraic multigrid preconditioners. For coupled Newton systems, the paper approximates the dense Schur complement by the damage block, yielding a practical block preconditioner.
- Linear subproblems: The solver addresses three linear subproblems: elasticity, a Helmholtz-like damage operator, and submatrices of the coupled Jacobian.All require scalable iterative solvers and preconditioners for finely discretized problems.
- Linear elastic subproblem: The elasticity problem uses conjugate gradients with GAMG smoothed aggregation, but damage localization makes it poorly conditioned.The stiffness variation induced by damage motivates algebraic multigrid preconditioning.
- Linear elastic subproblem: Additional damage-dependent near-nullspace vectors improve multigrid convergence but are difficult to compute and are omitted.With a Krylov accelerator, the iteration-count ratio between full and partial near-nullspaces decreases from approximately 10 to approximately 2.
- Damage subproblem: The inactive damage submatrix is solved with conjugate gradients and ML smoothed aggregation multigrid using the constant vector as near-nullspace.
- The Newton step: For the coupled Jacobian, the dense Schur complement S = C − B^T A^-1 B is approximated by C.The resulting preconditioner requires one C^-1 action and two A^-1 actions per application, with both approximated by two algebraic-multigrid V-cycles.
4. TEST CASES
Three benchmark problems assess the proposed solvers on smooth crack propagation, crack nucleation, and complex multiple-crack evolution. The tests vary loading, geometry, thermal shock intensity, and discretization to probe solver behavior under increasingly difficult fracture patterns.
- Test cases: The benchmarks cover smooth propagation of a single crack, uniaxial crack nucleation, and thermal-shock-driven nucleation and propagation of multiple cracks.The thermal-shock case is designed to test ill-conditioned elastic subproblems, nonlinear convergence under bifurcations and snap-backs, and irreversibility constraints.
- Surfing: smooth crack propagation: The smooth-propagation test uses a rectangular slab with controlled Dirichlet loading that drives a preexisting crack at constant speed when KI = 1.0.The setup uses L = 2, H = 1, Lc = 0.05, and v = 1, with the crack expected to propagate along x2 = 0.
- Surfing: smooth crack propagation: The smooth-propagation test checks that dissipated energy is independent of ℓ and equals crack length times Gc, requiring mesh refinement with h = ℓ/5 in the propagation band.A non-uniform mesh is used in the benchmark to satisfy this resolution condition efficiently.
- Traction of a bar: The uniaxial bar test evaluates crack nucleation under imposed displacement and compares the computed response with an analytically characterized critical load.For sufficiently large L relative to ℓ, the purely elastic state persists before tc, while a one-crack solution is expected after the threshold.
- Thermal shock: The thermal-shock test varies ∆T and mesh size for a slab with L = 20 and ℓ = 1, using thermally induced stresses from an approximate diffusion solution.The model neglects the influence of cracks on thermal diffusivity and uses E = Gc = β = 1.
- Thermal shock: For ∆T > ∆Tc, thermal shock produces a homogeneous damage band, periodic crack bands, and period-doubling propagation; stronger shocks increase crack spacing and wavelength.The reported cases use ∆T/∆Tc equal to 2, 4, and 8.
- Thermal shock: Energy evolution is smooth for intense shocks but exhibits dissipation jumps and snap-backs for mild shocks such as ∆T = 2∆Tc.These snap-backs force the minimization algorithm to search for a new solution that may be far from the previous time step.
5. RESULTS OF NUMERICAL EXPERIMENTS
The experiments compare over-relaxation, Newton composition, and preconditioning across surfing, traction, and thermal-shock problems. Over-relaxation and Newton composition improve convergence or runtime mainly in the harder cases, while the full-Jacobian preconditioner scales favorably as problem difficulty increases.
- All test problems use an absolute l2 residual tolerance of 10^-7 while varying internal length or thermal-shock loading intensity.
- Thermal shock: For thermal shock, dissipated energy has two time jumps at ∆T = 2∆Tc but evolves smoothly at ∆T = 8∆Tc.The jumps correspond to snapbacks and sudden crack growths.
- Over-relaxation: Over-relaxation reduces iterations by a factor between 1/2 and 3/4 in surfing and thermal-shock cases, with larger benefits as difficulty increases.Traction is an exception: standard alternate minimization is already efficient, and other relaxation parameters slow convergence.
- Newton composition: Newton composition yields worthwhile and consistent runtime reductions for surfing and thermal-shock tests, but only marginal gains for traction.The experiments apply Newton after alternate minimization reduces the l2 residual by 10^-1 and use optimal over-relaxation for the relevant cases.
- Preconditioning: The full-Jacobian preconditioner remains practical as meshes are refined and physical difficulty increases, with iteration counts growing slowly or staying approximately constant.Practical algebraic-multigrid inner solvers add measurable cost in thermal shock, but the resulting performance still degrades slowly.
6. CONCLUSION
The paper combines over-relaxed alternate minimization, Newton-type methods, and scalable preconditioners to improve solver efficiency for variational fracture models. These improvements reduce runtime in tested cases, while the analysis remains limited to simple fracture settings and leaves convergence questions open.
- 5–6× lower time to solution was achieved for the surfing and thermal shock test cases.
- Over-relaxation reduces alternate-minimization iterations cheaply, while composition with Newton-type methods provides further runtime reduction at higher development cost.
- Preconditioners were proposed for alternate-minimization linear subproblems and coupled Jacobians in monolithic active-set Newton iterations.
- A convergence analysis of block over-relaxed nonlinear Gauss–Seidel remains a direction for future research, including the choice of the over-relaxation parameter.
- For surfing, the coupled-Jacobian preconditioner depends weakly on mesh refinement and internal length with ideal inner solvers, and practical AMG solvers hardly affect convergence.
- For thermal shock, the coupled-Jacobian preconditioner depends weakly on mesh refinement and Griffith length with ideal inner solvers, while practical AMG solvers somewhat degrade convergence.
- The study assumes small deformations and symmetric traction–compression behavior, and broader models require suitable preconditioners for the elastic subproblem and coupled Jacobian.