Efficient strain-space hyperreduction in large-deformation solid mechanics

Technical Summary: Efficient Strain-Space Hyperreduction in Large-Deformation Solid Mechanics

Problem Statement
Model Order Reduction (MOR) techniques operating in strain space have recently demonstrated superior performance in computational homogenisation, particularly for Representative Volume Elements (RVEs). However, their application has been largely confined to this specific context. Extending these techniques to general large-deformation solid mechanics problems presents significant challenges. Unlike displacement-based formulations, strain-space methods cannot directly enforce Dirichlet boundary conditions (BCs) because integration point strains at the boundary are not explicitly known, and boundary values are typically prescribed as displacements, not strains. Furthermore, standard strain-space approaches often rely on periodic BCs, which are not applicable to macro-scale problems with arbitrary boundary values.

Methodology
The authors propose a generalized framework to apply strain-space MOR to problems with arbitrary, parameterized Dirichlet BCs. The core methodology involves three main components:

1. Boundary Handling via Lifting: To satisfy non-zero Dirichlet BCs, the authors employ a lifting strategy. A boundary-consistent reference field is computed offline for unit displacement values at each independent boundary. The actual solution is then decomposed into this reference field (scaled by the actual boundary values) and a "fluctuation field." The fluctuation field vanishes at all Dirichlet boundaries by construction, allowing the reduced basis (derived from Proper Orthogonal Decomposition, POD) to satisfy BCs a priori. This approach transforms inhomogeneous BCs into homogeneous ones for the reduced solver.

2. Strain-Space Formulations: The paper generalizes three specific hyperreduction techniques to this new setting:

   • Strain-Space ECM (Empirical Cubature Method): Reformulated to perform reduced integration directly at Gauss points, bypassing element-level operations.
   • Strain-Space E3C (Empirically Corrected Cluster Cubature): Generalized to arbitrary mechanical problems. E3C optimizes integration points and weights in strain space, allowing them to be distinct from the original mesh Gauss points.
   • Strain-Space EMSL (Empirical Material Sampling and Linearisation): Extended beyond RVEs. EMSL linearizes the material behavior around cluster-wise expected average strains. Unlike other methods that approximately integrate an exact material law, EMSL exactly integrates a cluster-wise linear approximation, resulting in an affine problem per load step that requires no iterative Newton-Raphson solver during the online phase.

3. Comparison Baseline: These strain-space methods are benchmarked against the standard displacement-space Energy Conserving Sampling and Weighting (ECSW) method, which serves as a robust, structure-preserving baseline.

Key Contributions

• Generalization of Strain-Space MOR: The work successfully extends strain-space MOR techniques (ECM, E3C, EMSL) from computational homogenisation to general large-deformation solid mechanics with arbitrary Dirichlet BCs.
• Novel Boundary Treatment: The introduction of a lifting-based fluctuation field formulation enables the imposition of arbitrary boundary conditions in strain space without requiring boundary strain data.
• Comprehensive Benchmarking: The authors provide a rigorous comparison of strain-space versus displacement-space hyperreduction on two hyperelastic examples involving inverse parameter estimation, varying material parameters, and complex loading paths (tension, shear, and mixed).

Results
The methods were tested on two hyperelastic plate problems with elliptical holes:

1. Example 1 (Moderate Deformation): A plate under tensile loading with up to 20% strain.
2. Example 2 (Large Deformation & Complex Loading): The same geometry subjected to tension (up to 50%), shear (up to 25%), and mixed loading, with a finer mesh and higher computational cost.

Performance Findings:

• Accuracy vs. Runtime: In both examples, strain-space methods consistently outperformed the displacement-space ECSW in the trade-off between accuracy and runtime.
• Speedups:
  • E3C: Achieved speedups of approximately 10,000-fold (Example 1) and 30,000-fold (Example 2) compared to the full-order model while maintaining high accuracy (e.g., <1% error). E3C generally provided the highest accuracy for a given computational budget among the strain-space methods when sufficient integration points were used.
  • EMSL: Demonstrated exceptional runtime efficiency, achieving speedups of up to 100,000-fold (Example 1) and 300,000-fold (Example 2). EMSL excelled when online and offline budgets were very limited, often outperforming E3C in speed but sometimes at the cost of slightly lower accuracy in highly nonlinear regimes compared to E3C with many integration points.
  • ECM: Performed better than ECSW but generally lagged behind E3C and EMSL in the specific metrics tested.
• Offline Costs: While E3C has a more expensive offline training phase due to iterative optimization, the authors note this is manageable and significantly less than the time required to generate training data. EMSL has the lowest offline cost.

Significance and Claims
The paper claims that the proposed methodology justifies further research into strain-space MOR techniques beyond computational homogenisation. The authors highlight that these techniques offer:

• Superior Efficiency: Dramatic reductions in online runtime (orders of magnitude) while retaining high accuracy.
• Non-Intrusiveness: The methods require only displacement, strain, and stress data (easily extracted from black-box solvers) and a material routine. They bypass element-level operations and do not require knowledge of the discretization details (e.g., element types) or residual vectors.
• Applicability: The successful handling of arbitrary Dirichlet BCs removes a major barrier to applying these efficient techniques to macro-scale structural problems.

The authors conclude that while challenges remain for highly nonlinear problems (such as elasto-plasticity) and the estimation of cluster-wise strains in EMSL, the demonstrated performance suggests these methods could enable new engineering applications in control, optimization, and material model fitting where computational cost is currently prohibitive.