BRIDGE: Bridging Routine Inputs and Digital modelling for Geotechnical Engineering

1.1. Motivation

• Mechanical properties needed for design are measured in relatively few samples, whereas basic index properties are routinely available across larger datasets, creating the characterization gap.The stated requirements for addressing the gap are prediction from basic properties, physical consistency within and beyond the source data, and limited experimental or training data.Source: §1.1 · p. 2

1.2.1. Data-Driven Approaches

• Data-driven correlations and machine-learning models use basic properties, but mineralogical variability and out-of-domain behavior undermine reliability, while unconstrained predictions and large training needs limit physical consistency and data efficiency.Undrained strength at the plastic limit varies by more than an order of magnitude across fine-grained soils due to clay mineralogy.Source: §1.2.1 · pp. 2–3

1.2.2. Direct Physics-Based Modeling

• Direct constitutive calibration enforces physical consistency within the selected model's assumptions, but does not use routine basic properties to predict mechanical properties.Proposed correlations between constitutive parameters and plasticity index are based on reconstituted clays and therefore characterize intrinsic rather than natural behavior.Source: §1.2.2 · p. 3

1.2.3. Hybrid and Physics-Informed Approaches

• Hybrid inverse-analysis and physics-informed approaches can enforce mechanical constraints, but neither provides a direct route from routine basic properties to mechanical properties while ensuring limited data needs.Physics-informed methods may train on experimental measurements, full-field observations, or synthetic constitutive responses rather than routine basic properties.Source: §1.2.3 · pp. 3–4

1.3. Research Gap and Proposed Framework

• BRIDGE screens outliers and non-physical responses, bins samples into Cells using basic properties and undrained strength, and selects calibration sets of no more than 20% per Cell.The remaining samples are reserved for validation, and this work uses a finite-element Extended Cam-Clay model calibrated through a multi-stage optimization process.Source: §1.3 · p. 4
• Converged Cell calibrations define ECC parameter ranges and a predictive envelope; a Cell is validated when most validation samples fall within that envelope, enabling range predictions for new in-Cell samples.The mechanical surrogate applies to new samples whose basic properties lie within the Cell's basic-property limits.Source: §1.3 · p. 4
• BRIDGE predicts mechanical properties from routine basic properties using physically constrained Cell surrogates calibrated from a limited number of samples.The framework is modular, allowing the source database, mechanical model, calibration parameters, binning criterion, and sampling method to vary.Source: §1.3 · p. 4
• BRIDGE’s components may each be changed independently without altering its underlying architecture.Source: §1.3 · p. 4

2. Overview of the BRIDGE Framework

• Basic inputs are LL, PL, w, e, Sr, and γt; validated Cells predict ranges of E50, po, and Cc, with Su also used to locate a Cell.Su can be estimated rapidly and inexpensively in the field, so it is considered for binning.Source: §2 · pp. 6, 9
• Prediction-stage consistency is assessed by checking whether ranges enclosing a fraction of calibration values also enclose approximately the same fraction of validation samples.The overview gives 50% and 80% prediction ranges as examples.Source: §2 · p. 7
• The source database, binning, sampling, mechanical model, and parameter sets can be adapted without changing BRIDGE's overall architecture.The framework is intended to preserve the sequence of binning, calibration, validation, and prediction across applications.Source: §2 · pp. 7–8
• BRIDGE screens the marine-clay database, bins retained samples into Cells, then partitions each Cell into calibration and validation sets.The framework uses one- or two-level binning to form Cells.Source: §2 · pp. 6, 9
• Repeated finite-element triaxial and oedometer simulations calibrate ECC parameters κ, λ, M, and a0 against reference responses, retaining converged calibrations to form Cell-specific ranges and envelopes.The fixed set e, ρ, k, and ν is held constant during calibration.Source: §2 · p. 9
• Validation maps measured samples onto each Cell's envelope; a majority classified pass or marginal validates the Cell, enabling prediction for new samples assigned by basic properties and Su.Samples outside the convex hull but within the bounding box are marginal; those outside the bounding box fail.Source: §2 · p. 9

3.1. Dataset overview

• The working dataset contains 2,766 samples, with six basic properties and four laboratory-measured mechanical properties.The mechanical properties are Su, E50, po, and Cc; PI is derived from LL and PL.Source: §3.1 · p. 10
• PI is strongly associated with LL (ρ = +0.97), water content and void ratio (ρ = +0.81 each), and total unit weight (ρ = −0.81), but its LL association is partly by construction.PI is derived from the Atterberg limits; samples with similar PI span nearly the full observed ranges of Su, E50, and po.Source: §3.1 · pp. 10–11
• The ten-parameter correlation matrix separates basic properties from Su, E50, and po, which correlate strongly with one another but weakly with the basic-property block.Void ratio and natural water content have ρ = +0.99; each has ρ = −0.99 with total unit weight.Source: §3.1 · p. 12

3.2. Dataset binning options

• The squared correlation ratio η² ranks binning variables by how well bins reduce target scatter and separate their means.It is dimensionless and bounded from zero to one, allowing targets measured in different units to be compared.Source: §3.2 · p. 13
• Equal-count partitioning is adopted because equally spaced bins can leave outer bins with only one or two samples, unsuitable for calibration.The comparison evaluates five candidate variables against four mechanical properties using K = 8 bins.Source: §3.2 · pp. 13–14
• Su is the strongest binning criterion for three of four mechanical targets, while void ratio best captures Cc (η² = 0.70).Su attains η² = 0.67 for E50 and 0.53 for po; the resulting two-level scheme uses Su first and e within each Su bin.Source: §3.2 · p. 14

4. Preparation of Calibration and Validation Datasets

• Preparation comprises sequential screening, binning into Cells with comparable property ranges, and sampling calibration and validation datasets from each Cell.These datasets feed the subsequent calibration and validation stages, respectively.Source: §4 · p. 14

4.1. Data screening

• Five physical-admissibility filters leave 2,609 of 2,766 samples; the largest removals are 123 for e and 33 for E50 ≥ Su violations.The filters require 0.5 ≤ e ≤ 3.0, LL > PL, Sr ≤ 105%, Su > 0 and E50 > 0, and E50 ≥ Su; violations are treated as measurement or recording errors.Source: §4.1 · pp. 14–15

4.2. Binning and sub-binning

• Retained samples are partitioned first by undrained shear strength Su and then by void ratio e, with equal-count edges defining the Cells.The two-level scheme is intended to reduce within-Cell spread beyond primary binning alone.Source: §4.2 · p. 15

4.3. Sampling

• Each Cell contributes no more than 20% of its samples to calibration, with the remaining samples reserved for validation.Five candidate sampling methods select using basic properties alone; sampling affects computational cost and how well the calibration set represents its Cell.Source: §4.3 · pp. 15–16

5. Multi-Stage Calibration

• Successful calibrations define Cell-specific calibration-set parameter ranges and predictive envelopes for validating surrogates and predicting properties of new samples.Four parameters—λ, κ, M, and a0—are updated during the procedure, while each sample’s initial void ratio, mass density, permeability, and Poisson’s ratio remain fixed.Source: §5 · p. 16, §5.1 · p. 16

5.1. Forward Model

• The forward model uses two single-element ABAQUS/Standard simulations, one for each laboratory test—the undrained triaxial and oedometer—and derives Su from half peak deviatoric stress and E50 from the corresponding curve slope.Source: §5.1 · pp. 16–17

5.2. Staged Calibration

• Five sequential stages calibrate ECC parameters against triaxial and oedometer responses, ordering parameter updates by their influence on mechanical properties.M is calibrated first, followed by λ and κ, then a refinement of M and finally a0; each stage uses a 20% relative-error threshold.Source: §5.2 · pp. 19–20
• Each of five stages uses SciPy’s COBYLA from a fixed initial guess and is capped at 25 forward evaluations; failures prompt Nelder–Mead then Brent retries, while in-stage stagnation restarts from a perturbed point.The initial guess is κ = 0.075, λ = 0.5, M = 1.0, and a0 = 2×10^5 Pa; the restart uses an enlarged trust region.Source: §5.2 · p. 20
• A calibration passes only when all four mechanical properties are within 20% of their reference values, checked after each stage and at each cycle’s end.Samples that still fail are flagged and excluded from envelope construction.Source: §5.2 · p. 21
• Appendix C safeguards are enforced to maintain physical meaningfulness and numerical stability during simulations.The five stages are repeated for up to three cycles, each defined as one pass through the sequence in Table 3.Source: §5.2 · pp. 20–21

• Non-converged calibration samples are flagged and excluded from the predictive envelope.Source: §17 · p. 21

6. Predictive Envelope

• Non-converged calibration samples are excluded before the envelope is constructed from converged samples plotted in the mechanical-property space for each test.The spaces are Su and E50 for triaxial tests and po and Cc for oedometer tests.Source: §6 · pp. 21–22
• The convex hull encloses the calibration points and lies within their bounding box, making the hull the stricter validation criterion.The convex hull is computed with the Quickhull algorithm.Source: §6 · p. 22
• Validation samples pass inside the convex hull, are marginal inside the bounding box only, and fail outside both regions.Source: §6 · p. 22
• Envelope coverage measures how well converged calibration samples represent the Cell; higher accuracy for a property indicates greater coverage of its Cell range.Accuracy is defined for each mechanical property and averaged across the properties considered.Source: §6 · p. 22

7. Demonstration of the BRIDGE workflow

• Cell B1S4 contains 163 samples, of which 30 (18.4%) were selected for calibration and 133 for validation.The demonstration partitions the screened dataset into 16 Cells using four Su bins, each subdivided into four e sub-bins.Source: §7 · p. 23

7.1. Automated multi-stage calibration

• The five calibration stages tune M, λ, κ, refine M, and tune a0 to match Su, Cc, E50, Su, and po, respectively.The κ adjustment affects the Su match, so the subsequent stage refines M to restore it.Source: §7.1 · pp. 23–24
• 27 of 30 selected samples converged without manual intervention; the other three terminated with non-convergence flags.Their residual mismatches were E50 (41.7%), po (50.7%), and Cc (21.2%), respectively.Source: §7.1 · p. 24

7.2. From calibrated samples to the predictive envelope

• The 27 converged calibration samples alone define the Cell’s bounding box and convex hull; the three non-converged samples are flagged and excluded.The resulting predictive envelope is constructed in the triaxial and oedometer assessment planes.Source: §7.2 · p. 25, Figure 15 · p. 27

7.3. Validation mapping and the calibrated output

• 85.0% of validation samples passed and 8.3% were marginal in the triaxial plane; in the oedometer plane, 85.7% passed and 7.5% were marginal, validating Cell B1S4.The corresponding fail fractions were 6.8% in both planes; a majority classified as pass or marginal meets the validation criterion.Source: §7.3 · p. 27, Figure 16 · p. 28

7.4. Prediction

• For new samples, BRIDGE assigns a Cell using measured e and Su, then predicts mechanical-property ranges spanned by that Cell’s converged calibration samples.This part introduces construction of 50% and 80% ranges, including a method based on percentiles about the median.Source: §7.4 · pp. 28–29
• Across Cell and property combinations, median-based 50% and 80% ranges contain 40% and 74% of prediction samples on average, versus 54% and 83% for mean-based ranges.Table 6 summarizes predictions for the three Cells and three mechanical properties.Source: §7.4 · p. 29, Table 6 · p. 32
• For median-based ranges, 14 of 18 cases capture 38–61% of prediction samples within the 50% range, and all 18 capture at least 65% within the 80% range.For the 50% ranges, only two of the four remaining cases fall below 30%.Source: §7.4 · p. 29
• Predicted ranges remain physically wide because samples vary within Cells and E50 and po correlate only weakly with basic properties.In B1S4, the median-based 80% ranges span 0.30–2.41 MPa for E50 and 17–91 kPa for po; Su varies from 0.76 to 26.9 kPa, while the absolute correlations for E50 and po do not exceed 0.38. Reducing range width remains future work.Source: §7.4 · pp. 29–30
• In Cell B1S4, all five checked prediction samples fall within the 80% ranges for E50, po, and Cc, while S2, S3, and S4 also fall within the 50% ranges.The five samples span the Cell’s e range, supporting the use of predicted ranges to estimate mechanical properties from Su and basic properties alone.Source: §7.4 · p. 32, Figure 20 · p. 33

9. Conclusions

• BRIDGE calibrates Extended Cam-Clay parameters, builds envelopes from converged samples, classifies validation as pass, marginal, or fail, and outputs Cell-specific parameter ranges for simulation bounds.Source: §9 · pp. 41–42
• Binning by Su matches the coverage of binning by e at lower computational cost.Source: §9 · p. 42
• Sub-binning helps maintain accuracy at lower mean computational cost in Study B.Source: §9 · p. 42
• Sampling method and sample budget affect mean accuracy; Puchwein sampling with ns = 30 was adopted, but the choice remains user-defined rather than universally optimal.Source: §9 · p. 42
• Calibration converged for 95.8% of 1,410 instances, with median errors of 9 to 14%.The authors report these reliability and accuracy results in Section 8.4.Source: §9 · p. 42
• BRIDGE uses only 10 to 20% of the total samples to produce a validated mechanical surrogate for a Cell.The framework selects a small calibration dataset from each Cell, with the remaining samples used for validation.Source: §9 · p. 42
• Future work includes evaluating sampling and budgets across Cells and investigating non-convergence in low-Su, low-e Cells, especially for E50 and Cc.Source: §9 · p. 42
• More than 90% of validation samples are represented by predictive envelopes built from physically consistent finite element simulations.Once validated, a Cell’s mechanical property ranges can predict the ranges for new samples assigned to that Cell.Source: §9 · p. 42

Appendix A. Constitutive Model: Extended Cam-Clay

• ECC is the BRIDGE constitutive model, and its calibration set comprises κ, λ, M, and a0.It generalizes Modified Cam-Clay and is used as the Clay Plasticity model in ABAQUS, fitting the marine clays in the dataset.Source: Appendix A · p. 43
• Isotropic compression is bilinear in void ratio e against log mean effective pressure, with κ for swelling lines and λ for the virgin compression line.The slopes are de = −κ d(ln p) and de = −λ d(ln p), respectively.Source: Appendix A · p. 43
• At critical state, shear deformation continues at constant volume and effective stress, and q = Mp relates deviatoric stress q to stress ratio M.The critical state is characterized by that stress relation.Source: Appendix A · p. 43
• Yielding begins on an elliptical surface bounded by a0, the initial yield-surface size, while associated plastic flow expands or contracts the surface with plastic volumetric strain.The model is implemented through ABAQUS finite element analysis.Source: Appendix A · p. 43

Appendix B. Pearson and Spearman correlations

• Pearson correlation quantifies the strength of the linear association between two parameters.Its calculation uses paired sample values and their means.Source: Appendix B · p. 43
• Spearman correlation is Pearson correlation evaluated on parameter ranks, so it quantifies monotonic rather than linear association.Ranks run from 1 for the smallest value to n for the largest, with ties assigned their average rank.Source: Appendix B · pp. 43–44
• The Spearman coefficient uses the mean rank, denoted R̄ = (n + 1) /2.The appendix defines both Pearson and Spearman correlations.Source: Appendix B · pp. 43–44

Appendix C. Physical and Numerical Safeguards

• The elastic slope is constrained to κ < 0.925 λ, maintaining a 7.5% gap below the plastic slope.This safeguard is intended to prevent non-physical parameter values.Source: Appendix C · p. 44
• The overconsolidation ratio is fixed at 1.5 because a ratio of unity leaves no elastic zone and makes κ irrelevant.This is one of the calibration safeguards.Source: Appendix C · p. 44
• Poisson’s ratio is clamped to ν ≥ (3−M)/(6+M)+0.02 so the oedometer’s one-dimensional elastic path does not cross the critical state line and trigger instability.The condition is described as a wet-side condition.Source: Appendix C · p. 44
• The safeguards turn frequent numerical instability causes into admissible corrections or explicit non-convergence flags, keeping parameter sets used in FEM simulations physically meaningful.The procedure is designed to ensure physical meaningfulness and numerical stability.Source: Appendix C · p. 44