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Novel anisotropic continuum-discrete damage model capable of representing localized failure of massive structures. Part II: identification from tests under heterogeneous stress field
A. Kucerova, D. Brancherie, A. Ibrahimbegovic, J. Zeman, Z. Bittnar
TL;DR
Identifying parameters for a localized-failure model from structural tests is challenging, particularly when three-point bending produces heterogeneous fields and simple tension is difficult to perform reproducibly. The paper uses sequential optimization and reports a worst-case reliability of 76% across 100 optimization processes, while noting that experimental validation remains necessary.
Problem
Method
The paper uses a sequential identification strategy in which three stages optimize parameter groups using only part of the test simulation at each stage.
Source: §7 · p. 15Results
76% was the worst-case algorithm reliability across 100 independent optimization processes for each objective function.
Source: §7 · p. 15Takeaways & Limitations
The sequential stages use only part of each test simulation, leading to substantial computational time savings.
Source: §7 · p. 15The method was verified against a non-noisy reference load-deflection diagram, and experimental validation is still necessary for its full acceptance.
Source: §7 · p. 16
Abstract
from arXiv · showhide
In Part I of this paper we have presented a simple model capable of describing the localized failure of a massive structure. In this part, we discuss the identification of the model parameters from two kinds of experiments: a uniaxial tensile test and a three-point bending test. The former is used only for illustration of material parameter response dependence, and we focus mostly upon the latter, discussing the inverse optimization problem for which the specimen is subjected to a heterogeneous stress field.
1 Introduction
The paper identifies model parameters from structural experiments, focusing on three-point bending because it is easier to perform than simple tension but produces heterogeneous stress and strain fields. It uses the model’s physically interpretable parameters to structure a sequential calibration strategy and adopts neural-network-based smoothing with a genetic algorithm for optimization.
1 Introduction
- The model represents diffuse microcracking and macrocrack propagation, with physically interpretable parameters whose effects dominate recognizable material-behavior stages.This parameter structure is used to simplify calibration.Source: §1 · p. 1
- Three-point bending is easier to perform than simple tension, but its heterogeneous stress and strain fields make the load-deflection curve insufficient for direct parameter determination.The curve integrates responses from specimen regions in different elastic and inelastic regimes.Source: §1 · p. 2
- The proposed calibration divides identification into three sequential optimization stages, improving efficiency and allowing early stages to use only subsets of simulations.The stages exploit the model’s structure.Source: §1 · p. 2
- History dependence and parameter interactions make sensitivities difficult to compute, while large simulation time steps can make response-based objectives nonsmooth and gradient methods less suitable.Evolutionary algorithms can handle complex objectives but are often time-consuming; the paper characterizes this problem primarily by nonsmoothness rather than multiple optima.Source: §1 · p. 2
- The optimization approach adaptively smooths the objective with a Radial Basis Function Network whose minima are dynamically evolved by a real-encoded genetic algorithm.The paper also assesses the reliability of its proposed sequential stochastic strategy.Source: §1 · p. 2
2 A brief description of the identified model
This section briefly describes the model underlying parameter identification, whose continuum and discrete components represent distinct forms of damage and dissipation. Its parameter bounds and model ingredients frame the calibration task.
2 A brief description of the identified model
- The model represents bulk dissipation from distributed microcracks with continuum damage and surface dissipation from collapse-related macrocracks with a strong discontinuity model.The discrete component uses a traction-displacement jump relation.Source: §2 · p. 3
- The continuum and displacement-discontinuity models share a thermodynamic framework, and their main ingredients are summarized in Tables 1 and 2.The section provides a brief model description, with further detail in Part I.Source: §2 · p. 3, Table 1 · p. 3, Table 2 · p. 4
- Six independent material parameters are identified after fixing the limit tangential traction at 0.1 times the limit normal traction.They comprise elastic parameters E and ν, continuum damage parameters σ̄f and K̄, and discrete damage parameters σ̄̄f and β̄̄.Source: §2 · p. 4
- The parameter bounds are kept wide because the identification method does not assume an expert can provide initial parameter estimates.Table 3 presents the limits for the model parameters.Source: §2 · p. 4, Table 3 · p. 4
3 Tensile test
The tensile test offers a largely homogeneous response that permits staged identification from recognizable portions of its load-displacement curve. However, performing the test reproducibly is difficult, motivating the paper’s focus on three-point bending.
3 Tensile test
- The specimen’s stress and strain fields remain homogeneous until localized failure, making its structural response close to that of a material point.Figure 1 shows the load-deflection diagram and lateral contraction evolution.Source: §3 · pp. 4–5, Figure 1 · p. 5
- The load-displacement response has recognizable elastic, hardening, and softening portions that support sequential identification of the corresponding parameter groups.The elastic portion identifies E and ν, the hardening portion identifies σ̄f and K̄, and the softening branch identifies σ̄̄f and β̄̄.Source: §3 · p. 5
- Identifying Poisson’s ratio requires measuring lateral contraction in addition to the structural load-displacement curve.The lateral-contraction measurement is the local measurement shown in Figure 1b.Source: §3 · p. 5
- Despite robust and accurate calibration, a simple tension test is difficult to perform reproducibly, so the paper turns to three-point bending.The bending test is described as simpler to perform in a laboratory.Source: §3 · p. 5
4 Three-point bending test
The three-point bending procedure identifies model parameters sequentially despite heterogeneous stress and strain fields, using separate elastic, hardening, and softening stages. Because experimental data are unavailable here, reference-simulation values provide the target data.
4 Three-point bending test
- Reference-simulation values from Part I provide the target data because experimental data are unavailable.The calibration retains the three-stage sequence despite the bending test’s global response not mapping simply to the material’s elastic, hardening, and softening stages.Source: §4 · p. 5
4.1 Identification of elastic parameters
- Young’s modulus and Poisson’s ratio are identified from the linear load-deflection response and specimen expansion ∆l = v2 − v1.The elastic objective evaluates load and expansion at u = 0.01 mm, when only these parameters are active, allowing simulations to stop at that deflection.Source: §4.1 · pp. 6–7
4.2 Identification of hardening parameters
- The hardening stage fits limit stress ¯σf and parameter ¯K using the limit displacement and a secant slope from two load-deflection points.The points must be far enough from the limit displacement to limit pseudo-time discretization error, but close enough to avoid entering softening; solutions that fail to exceed ¯σf are penalized, and the objective is noisier than in the elastic stage.Source: §4.2 · pp. 7–8, Figure 5 · p. 8
4.3 Identification of softening parameters
- The softening stage identifies limit normal traction ¯¯σf and softening parameter ¯¯β using local notch-corner displacements and load after softening begins.The local estimate of the displacement marking this event corresponds rather well to the global quantity, and cases that do not reach softening are penalized.Source: §4.3 · pp. 8–9
5 Optimization method
The optimization method combines an adaptive neural-network approximation with an evolutionary algorithm to search the objective functions. GRADE and CERAF guide the search, while the RBFN surrogate is updated as new points are evaluated.
5 Optimization method
- The objective function is approximated by an RBFN and optimized with GRADE; new points, including prior approximate optima, adaptively improve the surrogate.Penalized points are added only when they are optima found by the evolutionary algorithm, to avoid discontinuities the smooth RBFN cannot capture.Source: §5 · pp. 9–10, §5.2 · p. 13
5.1 Evolutionary algorithm GRADE extended by strategy CERAF
- GRADE uses mutation and cross-over to generate candidate solutions, evaluates them, and applies tournament selection to remove the worse of two sampled solutions.The population starts with ten times as many chromosomes as optimized variables; cross-over creates 90% of new individuals.Source: §5.1 · p. 11
- CERAF adds a multi-start strategy: it stores a stagnant best point and rejects new solutions inside a surrounding ellipsoidal zone.The zone diameters are set to 50% of the search-domain size and shrink after cross-over solutions fall inside it.Source: §5.1 · p. 11
5.2 Radial Basis Function Network
- The RBFN uses a weighted sum of radial basis functions as an approximation of the objective function, with its synaptic weights determined by a linear system.The approximation is repeatedly evaluated during stochastic search; the computationally expensive objective is evaluated only when adding neurons.Source: §5 · p. 10, §5.2 · pp. 12–13
6 Identification procedure verification
The verification study evaluates stochastic optimization reliability and parameter accuracy under different stopping precisions. The identified parameters reproduce the reference response more closely with tighter tolerances, and mesh refinement reduces estimation-related errors.
6 Identification procedure verification
- The maximum errors for elastic parameters E and ν are below 3%, while raising hardening-stage stopping precision to 10^-3 reduces parameter errors by roughly 50% and yields an almost perfect reference-curve fit.Figure 11a shows that the coarser precision is insufficient for the hardening response.Source: §6 · p. 14, Figure 11 · p. 14
- For softening parameters σ̄̄_f and β̄̄, stopping precision 10^-2 is too coarse; reducing it to 3 × 10^-3 improves parameter accuracy, as shown in the load-deflection comparison.The study assesses the influence of stopping precision on identified-parameter accuracy relative to the parameter intervals.Source: §6 · pp. 13–14, Figure 11 · p. 14, Table 6 · p. 14
- Mesh refinement reduces errors produced by parameter estimation in the force-displacement response.Figure 12 compares coarse- and fine-mesh computations using both exact and estimated parameter values.Source: §6 · pp. 14–15, Figure 12 · p. 15
7 Conclusion
The proposed three-stage identification procedure extends from uniaxial tension to three-point bending, using model-informed objective functions and soft-computing optimization. Tests across 100 independent optimization processes per objective function found worst-case reliability of 76%, while sequential error accumulation and the need for experimental validation remain concerns.
7 Conclusion
- The sequential identification approach extends to three-point bending, and using only part of each test simulation at each stage yields substantial computational time savings.The three stages are linked to the structure of the constitutive model.Source: §7 · p. 15
- Physical insight into the model supports simple objective functions F1, F2, and F3 that are highly sensitive to relevant parameters; robust soft-computing methods optimize their non-smooth, non-convex forms.The functions' sensitivity to relevant parameters is the stated basis for their construction.Source: §7 · p. 15
- Worst-case reliability was 76% across 100 independent optimization processes per objective function, with the low call limit set by termination conditions contributing to this result.The authors describe this reliability as satisfactory in light of their experience with evolutionary algorithms.Source: §7 · p. 15
- Because errors accumulate across sequential stages, identified parameter values need greater accuracy than the 5% usually required in applications.The authors also compare this accuracy requirement with that achievable by neural network-based inverse analysis.Source: §7 · p. 15
- The major challenge is identifying the three stages of structural behavior; verification succeeded with a noiseless reference load-deflection diagram, but experimental validation remains necessary.The authors present experimental validation as necessary for full acceptance of the procedure.Source: §7 · p. 16