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A partitioned model order reduction approach to rationalise computational expenses in multiscale fracture mechanics

Pierre Kerfriden, Olivier Goury, Timon Rabczuk, Stephane Pierre-Alain Bordas

arXiv:1208.3940v2physics.class-ph

TL;DR

Parametric multiscale fracture simulations are expensive because damage zones resist low-dimensional approximation and repeated fine-scale solutions are costly. The paper combines domain decomposition with local POD reduction, selectively retaining richer or unreduced models near damage. The resulting partitioned strategy provides significant speed-up, especially when highly damaged regions are small relative to the structure.

  • Problem

    Multiscale fracture problems are difficult to reduce because damage initiation and propagation occur in regions that are not accurately represented by low-dimensional spaces, while global POD offline computations can be expensive.

  • Method

    The paper couples primal Schur-complement domain decomposition with local snapshot-POD projection, assigning independent reduced spaces to subdomains and switching unsuitable subproblems to full-scale solves.

  • Results

    The partitioned strategy produces significant speed-up for parametric fracture mechanics problems, with greater gains when highly damaged zones are small relative to the structure.

  • Takeaways & Limitations

    Local reduced dimensions focus computational effort near highly damaged zones while preserving a flexible treatment of subproblems with different reducibility.

  • Takeaways & Limitations

    The Galerkin version produces insignificant speed-ups because evaluating reduced nonlinear forces still requires costly integrations over the full spatial domain.

Abstract

from arXiv · show

We propose in this paper an adaptive reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projection-based model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates. No \textit{a priori} knowledge of the damage pattern is required, the extraction of the corresponding spatial regions being based solely on algebra. The efficiency of the proposed approach is demonstrated numerically with an example relevant to engineering fracture.

1 Introduction

Fracture simulations are costly because localized damage must be resolved within multiscale structures and across many parameter values. The paper addresses this by partitioning the domain and applying local POD reduction selectively, especially where the solution remains reducible.

  • 1 Introduction: Localized fracture phenomena create large numerical models, while design studies require many solutions across parameter values.Fine-scale gradients and explicit mesostructure representation drive the computational burden.
  • 1 Introduction: Projection-based model order reduction replaces many fine-scale unknowns with a few reduced variables, but constructing global bases remains expensive.The approach is attractive when costly offline computations can support repeated online interaction.
  • 1 Introduction: Damage zones are poorly represented by low-dimensional spaces, so reduction must be restricted to subdomains where accuracy is not sacrificed.The proposed strategy partitions structural components and treats highly nonlinear regions differently from more reducible regions.
  • 1 Introduction: The method couples Schur-based domain decomposition with POD and is presented as the first systematic partitioned reduction approach for multiscale fracture.Local reduced bases can be constructed independently, with dimensions adapted to subproblem nonlinearity.
  • 1 Introduction: Local POD bases support parallel offline construction and allow unreducible subproblems to switch to full-scale online solves.The partitioned framework reuses substructured offline information during the online stage.

2 General problem statement

The paper considers a parametrised, quasi-static nonlinear continuum-mechanics problem with rate-independent damage or plasticity. A fine space-time discretisation produces a history-dependent system solved sequentially with Newton iterations.

  • 2 General problem statement: The model describes quasi-static structural evolution with nonlinear constitutive laws for progressive failure, including plasticity or damage.Damage processes are assumed to be rate-independent and depend on parameters in a real design space.
  • 2 General problem statement: Finite-element spatial discretisation yields coupled nonlinear equations for the parametric evolution of the state variables.Internal forces depend on the current state, its history, and potentially design-dependent material and loading quantities.
  • 2 General problem statement: A time-discrete sequence over T_h is solved sequentially from an initially undamaged structure at rest.The fine discretisation is treated as the truth solution that reduced models approximate.
  • 2 General problem statement: At each time step, the nonlinear system is solved with Newton–Raphson iterations until the relative residual norm falls below ε_new.Each iteration requires solving a tangent linear system involving the displacement increment, tangent operator, and residual.

3 Model Order Reduction and Proper Orthogonal Decomposition

Projection-based MOR assumes that solutions over the parameter domain lie near a low-dimensional subspace. POD constructs such a space from fine-scale truth solutions, after which the evolution problem is solved approximately in that reduced space.

  • 3 Model Order Reduction and Proper Orthogonal Decomposition: The method targets parametric problems whose solution variations can be represented accurately in an attractive low-dimensional subspace.A reduced basis is obtained offline from truth computations and used for other parameter values.

3.1 Projection-based model order reduction

Projection-based MOR approximates the full state in a basis of reduced vectors and determines reduced coordinates online through residual projection or minimisation. Its accuracy depends on the reduced space, while global nonlinear integrations can erase computational gains.

  • 3.1 Projection-based model order reduction: The full solution is approximated as a linear combination of a few basis vectors, with reduced coordinates determined at each time and parameter point.The basis matrix has dimensions R^nu × R^nc, and the reduced space may depend on time and parameters.
  • 3.1 Projection-based model order reduction: Galerkin projection and least-squares minimisation provide online choices for determining the reduced variables.The nonlinear reduced problem is solved with a Newton algorithm, whose linearisation produces a reduced tangent system.
  • 3.1 Projection-based model order reduction: The well-posedness and accuracy of the tangent problems strongly depend on the selected reduced space.A useful reduction therefore requires a basis that supports both solution accuracy and stability.
  • 3.1 Projection-based model order reduction: Global evaluation of nonlinear residuals and tangents can make Galerkin reduction computationally ineffective despite fewer degrees of freedom.The integrations still depend on the size of the original spatial problem, leading to insignificant speed-up.

3.2 Proper Orthogonal Decomposition in projection-based model order reduction

POD constructs a low-dimensional, time- and parameter-independent spatial basis from fine-scale solution snapshots. Truncating this basis gives an optimal discrete representation, but its quality depends on snapshot coverage and can degrade for fracture-induced topological changes.

  • POD formulation: POD separates the solution into spatial modes and time-parameter weighting functions, with the spatial basis independent of time and parameters.The modes are ordered by decreasing importance and can be truncated to obtain a reduced space.
  • POD formulation: The truncated POD basis minimises the summed representation distance over the time-parameter domain among bases of the same order.The resulting distance is expected to decrease rapidly as the decomposition order increases.
  • Snapshot POD: Snapshot POD approximates the full-domain transform using truth solutions at selected parameter values and computes modes through the snapshot covariance operator or its SVD.The snapshot set contains n_s = n_t × n_µ solutions, so its size limits the available rank.
  • Snapshot POD: The reduced model discards snapshot weighting functions and computes modal amplitudes online through projection for parameter values not included in the snapshot set.Explicit interpolation of the weighting functions is described as suboptimal between precomputed parameter values.
  • Computational cost: Snapshot construction requires cost-intensive fine-scale simulations, although domain decomposition can make those truth solutions feasible on parallel architectures.This offline expense is assumed acceptable because the target evolution problem has many spatial and temporal degrees of freedom.

3.3 System approximation

System approximation reduces the cost of evaluating nonlinear reduced models by replacing full-domain nonlinear evaluations with approximate operators or selected spatial evaluations. The paper focuses on the gappy technique while using partitioned POD to address fracture-specific reduced-space construction.

  • Approximation strategies: Nonlinear reduced models use either approximate operators or evaluations at selected spatial points to avoid full truth-discretisation costs.These strategies target the nonlinear function evaluation required after selecting a reduced space.
  • Approximation strategies: Gappy methods reconstruct the nonlinear function from evaluations at selected degrees of freedom, while collocation methods enforce selected equations without reconstructing the function over the domain.The selection operator must balance optimality, stability, and tractability.
  • Chosen strategy: The paper selects the gappy technique because its main focus is partitioned POD rather than the system approximation strategy.The method is presented as widely used and studied, with nonlinear terms represented through a truncated POD basis and selected evaluations.
  • Gappy approximation: The gappy approach reduces online complexity by evaluating only a restriction of the nonlinear function at selected spatial degrees of freedom.The stated complexity no longer depends on the truth discretisation size.
  • Chosen strategy: The selected system approximation is combined with the partitioned POD approach for computing and using relevant reduced spaces in fracture mechanics.The paper defers the broader application of the nonlinear approximation technique to its final section.

3.4 Example of application of the POD in fracture mechanics

The fracture example uses a parametrised, damageable lattice whose crack pattern changes with loading angle. Global POD struggles with these changing patterns, but the projection error remains localised, motivating reduction only in selected domain regions.

  • Damage model: The damage model uses a variable d ranging from safe material at 0 to completely damaged material at 1, with an irreversible evolution driven by Y.The lattice regularises localisation through beam length, and shorter beams or higher order reduce dissipated crack energy.
  • Parametrised fracture problem: The numerical problem is a 1071-node, 4070-bar lattice with an initial notch, ten homogeneous load steps, and load angle θ ∈ [15°, 45°] as its parameter.The right-side displacement is gradually applied in tension, while the left side is fixed and body forces are null.
  • Snapshot-POD prediction: The goal is to predict the damage state for arbitrary load angles using a snapshot-POD space built from homogeneously distributed parameter samples including both angle endpoints.The online prediction is intended to avoid solving the truth model for every queried angle.
  • Global POD limitation: Each load angle produces a distinct crack pattern, so a limited global snapshot basis cannot generally represent the evolving topological changes efficiently.The crack or damage zone propagates approximately orthogonally to the load direction.
  • Localisation of reduction error: Projection error is concentrated near the crack, while regions farther away remain well approximated by low-dimensional bases, enabling reduction only over selected domain regions.This localisation motivates the subsequent domain-decomposition strategy with independently selected reduced subdomains.
  • Model scope: The initial crack is an idealised stress-concentration source, whereas realistic fracture may initiate at joints, supports, free edges, corners, or highly stressed interior regions.The example therefore represents one idealised initiation setting rather than all engineering initiation sites.

4 Partitioned model order reduction approach

The partitioned approach combines non-overlapping primal Schur domain decomposition with selective local reduction. It solves interface problems globally while retaining local subdomain operators and parallelisable computations.

  • Schur formulation: Primal Schur decomposition eliminates interior degrees of freedom and condenses the linearised balance equations onto interface degrees of freedom.The resulting interface problem is solved iteratively, while local subdomain problems provide the condensed operators.
  • Domain partitioning: The domain is split into non-overlapping subdomains, with each lattice bar assigned to one subdomain and shared nodes treated as interface nodes.The example shown partitions the domain into ten subdomains.
  • Interface coupling: Interface displacement continuity is enforced through a shared global interface vector, while local reaction forces satisfy global interface equilibrium.Assembly and trace operators connect local subdomain variables to the interface conditions.
  • Newton solution: The method linearises local subdomain problems at Newton iterations and assembles their condensed contributions to obtain the global interface problem.Local tangent stiffnesses, residuals, and displacement increments enter the linearised systems.
  • Iterative solution: The assembled interface system can be solved with Krylov methods without assembling the global Schur complement.Matrix-vector products and dot products are performed from local subdomain computations in parallel.

4.2 Formulation of reduced order modelling in the domain decomposition framework

The method applies local POD reduction to subdomain interiors, projects and condenses the resulting systems on shared interfaces, and permits selected subdomains to remain unreduced when local nonlinearities make reduction inefficient.

  • 4.2.1 Local snapshot POD reduced spaces: Local POD bases are computed independently from snapshots of each subdomain’s interior degrees of freedom.The local bases and truncation errors are defined separately for every subdomain.
  • 4.2.1 Local snapshot POD reduced spaces: Local bases, projection, and condensation preserve subdomain independence while coupling subproblems through the interface.The method therefore separates local reduced computations from the globally assembled interface solve.
  • 4.2.2 Local projection: The online approximation represents each subdomain’s interior state in its local reduced space and obtains reduced equilibria through Galerkin projection.The projected local system is linear, square, and symmetric after enforcing residual orthogonality to the local reduced space.
  • 4.2.2 Local projection: The local formulation is introduced for the linearised problem but can equivalently be obtained by reducing and then linearising the nonlinear partitioned problem.This provides the stated route from the nonlinear partitioned formulation to the reduced online system.
  • 4.2.3 Condensed interface problem: Reduced local systems are condensed by eliminating reduced interior variables, then assembled into an interface problem using unchanged interface kinematic and equilibrium conditions.The condensed interface problem can be solved in parallel with a Krylov algorithm when snapshot data are distributed.
  • 4.2.3 Condensed interface problem: The approach partitions subdomains into reduced and unreduced sets, solving damage-affected local problems directly when POD reduction would require too many basis vectors.The resulting hybrid condensed problem combines reduced local models with direct local solves.

4.3 Local error estimation by Cross-Validation

The paper uses local cross-validation to select reduced dimensions and identify non-reducible subdomains without prespecifying the damage pattern. Applied to fracture, the estimate converges rapidly away from damage but more slowly in damaged regions.

  • 4.3.1 Principle: The local scheme independently selects each reduced-space dimension and tests whether the corresponding subproblem is POD-reducible.It targets predefined accuracy while distinguishing reducible from non-reducible subproblems.
  • 4.3.1 Principle: The method assumes an initially relevant snapshot that sufficiently explores the parameter domain; generating additional informative snapshots is left for future work.The paper explicitly addresses local dimension and reducibility, not adaptive snapshot regeneration.
  • 4.3.1 Principle: Leave-One-Out cross-validation estimates projection error using the same snapshots while omitting each snapshot in turn from the local POD basis.This emulates separate training and validation sets without additional fine-scale solutions.
  • 4.3.2 Application: The fracture test uses five parameter values on a regular grid and applies the same procedure to all subdomains without using prior knowledge of damage locations.The four illustrated subdomains include damaged, crack-tip, and remote regions.
  • 4.3.2 Application: A LOOCV threshold of 10^-3 is reached with 4 to 5 local POD vectors in subdomains far from the crack.Their error curves converge quickly, although they do not show a clear elbow.
  • 4.3.2 Application: Subdomain 4, containing the crack tip over part of the parameter range, requires 7 local POD basis vectors to reach the target accuracy.The damaged-region convergence curves are flatter than those of remote subdomains.
  • 4.3.2 Application: The resulting local spaces are used in the subsequent numerical demonstration of the partitioned model order reduction approach.The cross-validation procedure thereby supplies both local dimensions and identified non-reducible subproblems.

5 System approximation in the partitioned model order reduction approach

The system-approximation stage uses partitioned gappy reconstruction to reduce online assembly costs by evaluating nonlinear contributions only on selected control-node regions, while retaining local reduced models and interface coupling.

  • 5.1 Local gappy approximations: Partitioned gappy reconstruction approximates the Galerkin projection framework for local nonlinear subproblems in reduced subdomains.The approximation is applied after localisation and linearisation of the nonlinear tangent systems.
  • 5.1 Local gappy approximations: Static reduced bases for internal-force vectors are constructed from snapshot information and used to interpolate local forces along reduced kinematic trajectories.Interpolation coefficients are obtained by minimising a sampled distance between approximate and exact local force vectors.
  • 5.1 Local gappy approximations: Gappy reconstruction removes symmetry from the condensed interface problem, so the paper proposes GMRes to alleviate this issue.The trade-off accompanies the reduction of online tangent and residual evaluations.
  • 5.1 Local gappy approximations: Only elements connected to selected control nodes contribute explicitly to online internal-force assembly, defining a reduced integration domain independent of the truth-system unknown count.The remaining contributions are reconstructed through interpolation in static reduced spaces.
  • 5.1 Local gappy approximations: The reduced integration domain includes all elements of unreduced subdomain 6 and all elements connected to unreduced interfaces, with remaining controlled nodes selected by partitioned DEIM.This is the configuration illustrated in Figure 7.
  • 5.1 Local gappy approximations: Control nodes are selected by a partitioned DEIM-inspired greedy procedure that uses an SVD-ordered basis and repeatedly targets the largest gappy interpolation error.The selected nodes define the local boolean sampling operator.
  • 5.1 Local gappy approximations: The local SVD truncation rank is chosen by relating gappy reconstruction error to displacement error so it can be compared with kinematic POD truncation error.A predefined tolerance or cross-validation estimate is also identified as a simpler alternative.

6 Results

The partitioned reduced-order model is evaluated on fracture problems with different lattice resolutions and load angles, comparing runtime–accuracy trade-offs against reference and alternative reduced models. System approximation provides substantial speed-up in its reachable-accuracy range, while unresolved interface costs limit further gains.

  • 6.1 Online numerical costs (“speed-up”): The experiments use four lattice resolutions across 10 subdomains and evaluate unseen load angles θ = 40° and θ = 27°.The reduced model omits reduction in the damage-affected subdomain 6 and uses Petrov–Galerkin system approximation elsewhere.
  • 6.1 Online numerical costs (“speed-up”): Significant speed-up is obtained with partitioned POD plus system approximation, although its error cannot fall below 2 × 10^-3 in one tested case.The reachable accuracy is limited by snapshot approximation and low-order truncation; larger local reduced spaces can improve it.
  • 6.1 Online numerical costs (“speed-up”): Galerkin partitioned POD produces insignificant speed-ups because reduced generalised-force integrations still span the spatial domain.Although the number of degrees of freedom decreases, the integration cost prevents computational gains over the reference model.
  • 6.1 Online numerical costs (“speed-up”): Speed-up increases with the reference problem’s degrees of freedom in the accuracy range reachable by POD-based reduced models.The reduced-space dimensions depend on parametric statistics rather than mesh refinement, but implementation overhead prevents ideal scaling.
  • 6.2 Remarks about the numerical efficiency of the system approximation: Speed-up is directly related to the reduced integration domain size, identifying unreduced interface integrations as the main barrier to further computational gains.Figure 12 plots speed-up against the ratio of total lattice elements to elements connected to control nodes.

7 Conclusion and perspectives

The paper concludes that partitioned model order reduction flexibly concentrates computational effort near damaged zones while retaining direct local solvers where reduction is inefficient. It achieves significant speed-up, but interface reduction, parallel load balancing, and error estimation remain limitations or open directions.

  • 7 Conclusion and perspectives: The partitioned strategy uses primal Schur-complement coupling with independently generated snapshot-POD local reduced spaces to reduce nonlinear fracture problems.Independent local spaces can have different dimensions, enabling spatially targeted reduction.
  • 7 Conclusion and perspectives: Local spaces become richer near highly damaged zones, automatically concentrating numerical effort where topological changes require greater model complexity.The local POD transforms generate relatively large reduced dimensions near highly damaged regions.
  • 7 Conclusion and perspectives: The domain decomposition framework transparently switches between reduced and direct local solvers, retaining direct solves in process zones when reduced modeling would be more expensive.This also supports simultaneous reduced computation, snapshot generation, and model refinement across different subdomains.
  • 7 Conclusion and perspectives: Significant speed-up is obtained for parametric fracture problems, increasing when the highly damaged zone is small relative to the structure.The result is attributed to flexible use of reduced and direct local computations across the partition.
  • 7 Conclusion and perspectives: The method remains suboptimal because the interface problem is not reduced, and parallel execution can suffer load imbalance across differently reduced subdomains.The authors propose jointly designing domain partitioning and model reduction to distribute computational load more evenly.
  • 7 Conclusion and perspectives: Current error estimation uses global Euclidean norms and cross-validation, which requires a sufficiently fine snapshot space and does not identify parameter-domain non-smoothness.The authors identify goal-oriented error criteria and improved snapshot selection as directions for further work.
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