Data-Driven FEM: Adaptive D-Refinement

• Data-Driven Finite Element Method (DDFEM) is a computational approach that solves boundary value problems using direct experimental or simulation data without traditional regression models.
• It integrates standard FEM with adaptive d-refinement, activating data-driven computations only in regions where nonlinear behavior exceeds a defined threshold.
• The method employs efficient algorithms such as k-d trees for nearest-neighbor searches and alternating-projection solvers, achieving high accuracy with significant speedup over full DDCM.

A data-driven finite element method (DDFEM) is a class of computational algorithms that numerically solve boundary value problems in solid mechanics directly from experimental or simulation data, without requiring explicit regression-based constitutive models. The mesh d-refinement framework, as presented by Wattel et al., combines model-free data-driven computational mechanics (DDCM) with adaptive refinement to enable efficient and accurate capture of localized nonlinear material response in otherwise linear domains (Wattel et al., 2022).

1. Mathematical Foundations of Model-Free Data-Driven FEM

The DDCM paradigm is formulated in the product phase space: $$\mathbf{z}_e = (\boldsymbol{\epsilon}_e,\, \boldsymbol{\sigma}_e) \in \mathbb{R}^{2N_c}$$ for each element $e$, with $N_c$ being the number of independent strain (and stress) components. The global phase space is

$Z = Z_1 \times Z_2 \times \dots \times Z_{N_e}$

and the admissible set $\mathrm{E} \subset Z$ consists of all points simultaneously satisfying FE compatibility, equilibrium, and boundary conditions.

The material is described by a discrete database

$$\mathrm{D} = \mathrm{D}_1 \times \cdots \times \mathrm{D}_{N_e}$$

where each $\mathrm{D}_e$ is a finite collection of phase-space points from experiments or lower-scale simulation (e.g., $(\epsilon_e^*, \sigma_e^*)$).

A local phase-space distance metric is introduced via any SPD matrix $\mathbf{C}_e$: $$|\mathbf{z}_e|^2 = \tfrac{1}{2} \epsilon_e^\top \mathbf{C}_e \epsilon_e + \tfrac{1}{2} \sigma_e^\top \mathbf{C}_e^{-1} \sigma_e$$ yielding the global distance

$$d(\mathbf{z}, \mathbf{y}) = \Bigl( \sum_{e=1}^{N_e} w_e\, d^2(\mathbf{z}_e, \mathbf{y}_e) \Bigr)^{1/2}$$

with $w_e$ denoting the element's weight.

The DDCM solution is defined as the minimizer of the global functional

$$\Pi(\mathbf{z}, \mathbf{z}^*, \boldsymbol{\eta}) = d^2(\mathbf{z}, \mathbf{z}^*) + \boldsymbol{\eta}^\top \left[ \sum_{e\in S_2} w_e B_e^\top \sigma_e + \sum_{e\in S_1} w_e B_e^\top D_e B_e \mathbf{u} - \mathbf{f} \right]$$

where $S_2$ are data-driven (DD) elements, $S_1$ are standard FEM elements, $B_e$ is the compatibility matrix, $D_e$ is the elastic stiffness, and $\boldsymbol{\eta}$ are Lagrange multipliers enforcing equilibrium.

2. D-Refinement: Adaptive Elementwise Data-Driven Substitution

The mesh d-refinement strategy exploits the empirical fact that many structural materials remain linear (or nearly so) up to a known strain or stress threshold $\sigma_\text{lim}$. Only elements predicted to enter the nonlinear regime are adaptively marked for conversion to DDCM.

An element $e$ is flagged for data-driven refinement if

$$f(\mathbf{z}_e) > \varepsilon_{\rm th}~=~0.9\,\sigma_\text{lim}$$

with $f$ a scalar measure such as von Mises stress.

The refinement algorithm:

• Initialize: $S_2 = \emptyset$ (no DD elements); $S_1 = \{1,\dots,N_e\}$.
• For each load increment or a posteriori, solve the FEM problem on $S_1$, or, if $S_2\ne\emptyset$, the DDCM/FEM coupled problem for current $S_2$.
• For each $e \in S_1$, compute $f(\mathbf{z}_e)$: if above threshold, move $e$ to $S_2$ and assign its initial $\mathbf{z}_e^*$ as either $P_D(\mathbf{z}_e)$ (nearest neighbor) or $0$.
• Iterate until $S_2$ converges (no new elements flagged).

Pseudocode summary:

j = 1 S2(j) = ∅ ; S1(j) = all elements for load steps l=1..Nsteps: if S2(j)==∅: solve linear FEM for u else: DDCM-FEM alternating-projection until data assignments stabilize for e in S1(j): compute z_e = (B_e u, D_e B_e u) if f(z_e)>ε_th: S2(j+1).add(e) S1(j+1).remove(e) initialize z_e^* (as above) if |S2(j+1)|==|S2(j)|: stop else: j←j+1, loop

This process localizes the expensive DDCM machinery to the smallest set of elements necessary for accurate nonlinear prediction.

3. Computational Matching and Solver Architecture

The dominant cost in DDCM is the per-element nearest-neighbor search within the data cloud $\mathrm{D}_e$. This is handled by k-d trees, yielding $O(\log|\mathrm{D}_e|)$ scaling per projection. When $|S_2|>12$, spatial parallelization is used.

Once an element enters $S_2$, its contribution to the global system switches from the standard $w_e B_e^\top D_e B_e \mathbf{u}$ term to the DDCM Lagrange-multiplier coupling $-w_e B_e^\top C_e B_e \boldsymbol{\eta}$, realized within a global fixed-point (alternating-projection) iterative process. The remainder of the mesh remains within the standard linear-FEM bulk system.

This hybrid assembly ensures that data-driven searching and projection are only leveraged in the critical subset of the mesh, while all other elements use precomputed linear solves.

4. Performance, Resource Footprint, and Scaling

Table: Representative wall times for the "hole-in-plate" example (database size $|D|=886$, $p=100$ MPa, $10$ load increments):

Method	Wall Time (s)
d-refinement	20.3
NR (tol=1e-3)	38.5
NR (tol=1e-5)	72.8
Pure DDCM (	D

Accuracy, measured by the normalized global phase-space distance

$$\frac{\|\mathbf{z}^{d} - \mathbf{z}^{NR}\|}{\|\mathbf{z}^d\|} \approx 0.034$$

demonstrates that, with ~10% of elements data-driven, less than 4% error is observed in phase space. There is no visible degradation in resolved displacement or stress fields relative to Newton-Raphson reference.

The d-refinement approach thus achieves $2\times$–$4\times$ speedup over highly accurate NR, and $>40\times$ over full DDCM, with negligible fidelity loss.

5. Illustrative Application: Multiscale Metamaterial Bridging

The framework directly supports bridging of microstructural effects in architected materials to macroscopic mechanical response.

At the microscale (RVE), a dense stress–strain database for a constituent (e.g., TMPTA with $$\sigma(\epsilon) = \sigma_f \tanh(\frac{E_0}{\sigma_f} \epsilon)$$) is generated by sampling $\epsilon\in[-0.2,0.2]$. Pure DDCM is then run on a unit cell to extract:

• Effective compliance $S$ for small strains (supplying the FE stiffness $D_e$).
• Nonlinear homogenized $(\epsilon,\sigma)$ pairs for use as local datasets $\mathrm{D}_e$ in the macroscale mesh.

Macroscopic analysis of a cracked block employs a linear pre-analysis to flag elements, after which d-refinement assigns DD status only to a small band near the crack tip (the process zone). Linear FEM alone yields a singular $1/r$ field; d-refinement regularizes the tip zone, producing a realistic opening stress profile with the singularity eliminated. The remaining bulk continues to use linear-FEM. This demonstrates physically accurate, mesh-agnostic resolution of nonlinearity with minimal computational overhead.

6. Significance, Limitations, and Integration

The mesh d-refinement methodology delivers several notable outcomes:

• Localized adaptivity: Data-driven modeling is only activated in regions where the linear hypothesis is demonstrably invalid, avoiding the need for global data coverage or dense sampling where it is unnecessary.
• Computational efficiency: The adaptive projection and k-d tree structure confine the high cost of nearest-neighbor queries to a minimal set ($S_2$), while linear segments remain in fast, standard assembly and solve.
• Accuracy/fidelity: With less than 4% global phase-space error and no observable loss in physical fields, the method provides robust predictive accuracy, even in critical multiscale or fracture-dominated scenarios.
• Scaling and legacy compatibility: The d-refinement scheme is compatible with any legacy FEM code base and integrates seamlessly with existing global load-step/incremental solution workflows.

A remaining consideration is that sharp detection thresholds (e.g., $\varepsilon_\text{th}=0.9\sigma_\text{lim}$) depend on prior knowledge of material limits and the structure of the data cloud. For materials lacking a well-defined linear regime or exhibiting complex, path-dependent response beyond the initial nonlinearity, further extension of the refinement criteria and phase-space representation (possibly with internal variables) would be needed.

7. Relation to Broader Data-Driven Mechanics Paradigms

The d-refinement approach sits within the larger context of DDCM (Kirchdoerfer et al., 2015), DDFEM for generalized (multi-field) states (2002.04446), and hybrid data–model coupling at scale (Korzeniowski et al., 2021). It leverages the core alternating-projection solver structure, but addresses performance bottlenecks and data scarcity by minimizing the number of elements requiring expensive projections. Comparisons with pure DDCM and standard Newton solvers in the literature validate its practical acceleration and accuracy characteristics.

The success of d-refinement underlines the importance of adaptivity and local modeling in data-driven FE, creating a framework capable of integrating heterogeneous data sources, multiscale effects, and critical nonlinearities without regression bias or global modeling assumptions.