A Virtual Fields Method-Genetic Algorithm (VFM-GA) calibration framework for isotropic hyperelastic constitutive models with application to an elastomeric foam material

TL;DR

This work presents VFM-GA, a framework that combines the Virtual Fields Method with a Boltzmann/Genetic Algorithm to identify material parameters in isotropic hyperelastic models using either full-field DIC data or conventional stress–strain inputs. The method formulates an implicit objective from the quasi-static balance of momentum and incorporates stability checks (ellipticity and monotonicity) to ensure physically plausible calibrations. Applied to a compressible elastomeric foam model, the framework demonstrates improved predictive capability over manual fitting across homogeneous and inhomogeneous deformations, with two functional variants enabling broad applicability. The approach is implemented in Fortran 90 with MATLAB interfaces and is shown to be robust to measurement noise, making it practical for automated material characterization in complex hyperelastic systems.

Abstract

This work introduces a calibration framework for material parameter identification in isotropic hyperelastic constitutive models. The framework synergizes the Virtual Fields Method (VFM) to define an objective function with a Genetic Algorithm (GA) as the optimization method to facilitate automated calibration. The formulation of the objective function uses experimental displacement fields measured from Digital Image Correlation (DIC) synchronized with load cell data and can accommodate data from experiments involving homogeneous or inhomogeneous deformation fields. The framework places no restrictions on the target isotropic hyperelastic constitutive model, accommodating models with coupled dependencies on deformation invariants and specialized functional forms with a number of material parameters, and assesses material stability, eliminating sets of material parameters that potentially lead to non-physical behavior for the target hyperelastic constitutive model. To minimize the objective function, a GA is deployed as the optimization tool due to its ability to navigate the intricate landscape of material parameter space. The VFM-GA framework is evaluated by applying it to a hyperelastic constitutive model for compressible elastomeric foams. The evaluation process entails a number of tests that employ both homogeneous and inhomogeneous displacement fields collected from DIC experiments on open-cell foam specimens. The results outperform manual fitting, demonstrating the framework's robust and efficient capability to handle material parameter identification for complex hyperelastic constitutive models.

Figures (11)

• Figure 1: Schematic of the mechanics framework based on the Virtual Fields Method (VFM). The inputs for the mechanics framework are a material parameter set $\boldsymbol{\theta}$ and experimental DIC and load cell data, and the outputs are an objective function value, which may be used in optimization. Starting from the experimental input side (left), (1) shows a typical experimental setup. In (2), a camera images the specimen over the experimental load path, and, in (3), a load cell collects reaction force data synchronized with the camera. The camera outputs images at each load step with index $m$ in (4). With a DIC algorithm, displacement fields may be calculated in (5) and sampled using an undeformed mesh in (6). The sampled displacement fields are interpolated using shape functions in (7), and deformation gradient fields are computed in (8) to obtain logarithmic strain fields. Then, starting from the right side, we have (9) where a given material parameter set $\boldsymbol{\theta}$ is plugged into the designated hyperelastic constitutive model $\psi$. Combining (8) and (9), stress fields at each load step $m$ are then computed in (10). Combining load cell data (3) and the computed stress fields (10), the balances of (11) external forces and (12) internal forces may be computed at each load step $m$. In (13), the two balance residuals are combined, normalized, and summed over the load step index $m$ to compute the objective function ${\rm obj}(\boldsymbol{\theta})$. Before passing the objective function value into the optimization method (here, a genetic algorithm), it passes through a stability filter (14) where material stability associated with the given material parameter set is assessed.
• Figure 2: Schematic of the GA. The minimum and maximum considered values for each material parameter are first specified in $\boldsymbol{\theta}_{\rm min}$ and $\boldsymbol{\theta}_{\rm max}$, respectively. In (1), the binary representation of an initial population $\boldsymbol{\Theta}^{(0)}$ is generated, which consists of $n_{\rm t}$ random material parameter sets $\boldsymbol{\theta}^{(\beta)}$ with $\beta\in\{1,2,\ldots,n_{\rm t}\}$, each within the prescribed range. In (2), the evolution cycle begins, in which $gen\in\{1,2,\ldots,n_{\rm gen}\}$ denotes the generation index. Each material parameter set in the population is passed to the VFM framework to compute the corresponding objective function ${\rm obj}(\boldsymbol{\theta}^{(\beta)})$. In (3), Boltzmann selection is used to select two parent sets, $\boldsymbol{\theta}_{\rm p1}$ and $\boldsymbol{\theta}_{\rm p2}$, from the population $\boldsymbol{\Theta}^{(gen)}$. The probability of being selected depends on the objective function value computed for each material parameter set ${\rm obj}(\boldsymbol{\theta}^{(\beta)})$. The selected parents are then passed into (4): single point crossover, in which a random bit-wise operation is performed on the parent sets to produce two child material parameter sets, $\boldsymbol{\theta}_{\rm c1}$ and $\boldsymbol{\theta}_{\rm c2}$. The two children are passed to (5): bit flip mutation. In this step, a random bit-wise operation is applied to alter the parameters, resulting in the mutated children, $\boldsymbol{\theta}_{\rm cm1}$ and $\boldsymbol{\theta}_{\rm cm2}$. The two mutated child sets are assigned to the next generation's population $\boldsymbol{\Theta}^{(gen+1)}$ in (6). Steps (3)-(6) are repeated until $n_{\rm t}$ mutated child sets are produced, completing $\boldsymbol{\Theta}^{(gen+1)}$. The generation then ends, overwriting $\boldsymbol{\Theta}^{(gen)}$ with $\boldsymbol{\Theta}^{(gen+1)}$, and the GA repeats the evolution cycle in steps (2)-(6). Between two consecutive generations, an elitism strategy is applied to prevent divergence. When $gen$ reaches the target number of generations $n_{\rm gen}$, the material parameter set corresponding to the minimum objective function value in the population is saved in (7). Steps (1)-(7) complete the process for one independent population. Our implementation involves $n_{\rm pop}$ independent initial populations that perform steps (1)-(7) in parallel. In the end, the GA outputs the best performing material parameter set found across all independent populations.
• Figure 3: Example of the convergence behavior of the GA for seven independent initial populations selected from $n_{\rm pop} = 500$ populations from the application of VFM-GA to calibrating high-density Poron XRD foam in Section \ref{['sec:1st_1']}. Convergence is characterized using the population-wise minimum objective function $\min({\mathbb O}^{({gen})})$ versus the generation index $gen$ in the evolution cycles of the GA.
• Figure 4: Data conversion preprocessor for the 1st-func. (a) Engineering stress-strain curve with experimental data for the high density Poron XRD foam shown by the solid black curve and the sampling load steps indicated by the red dots. The increment between two sampling load steps along the paths of compression and tension are defined by an equal arclength along the respective engineering stress-strain curves. The sampling points are mapped onto (b) the engineering lateral-axial strain curves at the same sampled axial strain values. The axial and lateral kinematics and the stress at each load step are projected onto (c) a four-element undeformed mesh consisting of quadrilateral elements, where (d) the reaction forces at the boundaries and the displacement field are computed assuming homogeneous deformation.
• Figure 5: Results for the 1st-func in simple compression and tension. Axial engineering stress versus axial engineering strain (left column) and lateral engineering strain versus axial engineering strain (right column) for Poron XRD foams of (a) low density, (b) medium density, and (c) high density. Solid black lines with shaded error regions (one standard deviation, nine experiments for (a) and (b); one standard deviation, four experiments for (c)) indicate experimental data. Dashed orange lines indicate constitutive model fits using VFM-GA material parameters, and dashed blue lines indicate constitutive model fits using the manually fit material parameters from landauer2019experimental.