Optimized Design of Composite Material Samples: From Standardized Testing to Simulation-Assisted Identification

This article is based on the M2 internship work of Antoine Vintache in collaboration with François Hild from the Paris- Saclay Mechanics Laboratory (Laboratoire de Mécanique de Paris-Saclay) and was presented at the ECCM21 conference [1]. This work was supported by ArianeGroup.

In experimental mechanics, composite material testing is usually based on standardized specimen geometries. These geometries—uniaxial tension, three-point bending, Iosipescu shear, etc.—were designed in the context where measurement resources were limited to point sensors (mainly strain gauges). Therefore, these geometries were typically developed to maintain uniform deformation states. They are still widely used today, especially in standards. Digital image correlation (DIC) and other full-field strain measurement techniques have challenged the applicability of these geometries. These methods allow detailed observation of displacement and deformation fields over the entire surface. They open possibilities for deeper exploitation of each test. At the same time, using model updating tools based on test results, multiple material parameters can be identified from a single test. In this case, the specimen geometry becomes an optimization variable that can be adjusted to maximize the useful information extracted from the test.

Parameter identifiability issues

Not all parameters can be identified from a given test. Parameter sensitivity is defined as the relative change in the measured quantity (e.g., displacement field) caused by a change in that parameter. If this sensitivity is low relative to measurement uncertainty, parameter identification becomes difficult.
Therefore, tests conforming to existing standards may fail to identify certain constitutive parameters. Without explicit evaluation, this lack of identifiability may remain unnoticed, leading to identification errors, especially those related to coupling between parameters or over-reliance on numerical regularization.
In such cases, it is necessary to rely on quantitative methods to identify parameters, integrating measurement uncertainty and coupling between parameters. This not only allows assessment of the relevance of a given test but also enables optimization of its design to maximize the amount of exploitable information.

Model fitting method based on EikoTwin Digital Twin

Material parameter identification can be formulated as an inverse optimization problem: adjusting the parameters of a finite element model (FEM) to reduce the difference between simulation results and experimental measurements. The method used here is based on weighted finite element model updating (FEMU) integrated into the EikoTwin digital twin environment.
The objective is to minimize a cost function based on the Mahalanobis distance between measurement data and simulation data, explicitly considering experimental uncertainty:

where F represents force, U represents displacement field, indices m and FE refer to measurement data and simulation data respectively, and covariance matrices model the uncertainty of each data source.
Identification is solved using a Gauss-Newton algorithm based on an estimate of the Hessian matrix of the cost function. This Hessian matrix is composed of sensitivity vectors of simulation data with respect to material parameters.
When some parameters have low sensitivity, the problem becomes ill-posed. Tikhonov-type regularization is then introduced to stabilize the solution. It includes adding a penalty term to penalize deviation of parameters from initial values and iteratively adjusting weights.
This method fully exploits the measurement fields provided by image correlation tools while providing a rigorous basis for assessing parameter influence and their identifiability in a given test.

Sensitivity analysis as a design tool

Sensitivity analysis is a core element of the specimen geometry optimization process. It allows us to evaluate which material parameters significantly affect measurement data for a given test, considering experimental uncertainty. This can be performed using the EikoTwin digital twin or dedicated scripts.

Sensitivity analysis: Hessian matrix, eigenvalues, parameters, and identifiability classes.

The main tool is the Hessian matrix of the FEMU cost function, here expressed in weighted form according to the signal-to-noise ratio (SNR). Each diagonal term in this matrix corresponds to the sensitivity of a parameter, while off-diagonal terms reflect correlations between parameters.
To interpret this Hessian matrix, diagonalization is performed. This provides:

• an orthogonal basis of "eigenparameters" (linear combinations of initial parameters);
• a series of eigenvalues expressed on a logarithmic SNR scale, quantifying the identifiability of each "eigenparameter."

Based on these quantities, we propose dividing them into "identifiability classes" [1]:

• Eigenparameters with negative class (integer part of the decimal logarithm of eigenvalues) are considered unidentifiable (sensitivity < uncertainty);
• those with positive or zero class are identifiable.

These classes are then reprojected onto the initial material parameters, assigning an overall class to each parameter. This determines the number of identifiable parameters for each test specimen geometry. This approach is especially useful during the design phase. By exploring the geometric space (e.g., length of the loading unit, lamination angles), the number of identifiable parameters can be mapped and geometric configurations maximizing information extraction can be found.

Case study: Optimization of Iosipescu test for laminated composites

This case study involves an Iosipescu-type test on laminated composites composed of 20 symmetric plies arranged at two angles α and β. Its mechanical behavior is modeled as orthotropic and elastic, requiring identification of 9 constitutive parameters.

Schematic of the test and geometric parameters to be optimized (L, α, β)

Three geometric parameters of the specimen are used as optimization variables:

• L, the distance between the grips of the Iosipescu test;
• α, the direction of the first set of folds;
• β, the direction of the second set of folds.

The goal is to determine values of L, α, and β that maximize the identifiability of constitutive parameters based on the sensitivity analysis presented above. Rossi
Pierron [2] demonstrated a method for optimizing test geometries that improves identifiability by accurately replicating simulated test data. In their study, a complete identification procedure was performed for each possible geometry, whereas this research introduces a method to optimize test geometries before specimen manufacturing, with lower computational cost.

Exploration configurations

Three exploration plans have been identified:

1. (α,L) for unidirectional composites (β=0°);
2. (α,L) for 0°-90° “cross” composites (β=90°);
3. (α,β) where L=5 mm, a value considered favorable for identifiability.

For each configuration, a series of finite element simulations are conducted. The sensitivity of each constituent parameter to unit changes is calculated (10 simulations per configuration). This amounts to thousands of simulations and is analyzed according to sensitivity levels. The criterion used is the average of all parameter levels, which must be maximized to find the overall most favorable configuration for the 9 parameters.

Average category plots related to each exploration plan. (a) β = 0°; (b) β = 90°; (c) L = 5 mm. Red circles indicate optimal values (global maxima).

Analysis shows that lower L values are generally more favorable for identification. The (α,β) plane at L=5 mm can identify up to five parameters with categories greater than or equal to 0: E₁, E₂, ν₁₂, ν₂₃, and G₁₂.
By interpolating results on a fine grid, the optimal configuration is determined as L=5 mm, α=69°, β=46° (red dot in the figure above). This geometry is selected to generate virtual tests aimed at assessing the identification capability of the synthetic shell.

Virtual testing and identification

A virtual dataset is generated by deforming specimens with optimized geometry and randomly perturbed material parameters (standard deviation about 8%). Virtual images generated by EikoTwin Virtual are analyzed using EikoTwin DIC for digital image correlation to obtain displacement fields. Noise is also simulated and added.

Algorithm diagram for method validation

All material parameters are identified with regularization relaxed at each convergence. L-curve analysis is used to determine the regularization level to apply (here 10⁻³) to minimize two functionals (FEMU and regularization). With this regularization, FEMU identification recovers initial parameters with mean squared error reduced to about 5.7%. The most sensitive parameters (E₁, E₂, G₁₂) have lower identification errors (<1%), while other parameters remain close to their initial values under regularization.

Algorithm progress and L-curve

Prospects for real testing

The complete chain is based on three components:

• EikoTwin Virtual Generates realistic images through finite element simulation, enabling test preparation and measurement quality prediction.
• EikoTwin DIC Processes real images via digital image correlation to produce displacement fields consistent with simulated geometry.
• EikoTwin Digital Twin Performs sensitivity analysis, then resets material parameters by minimizing the FEMU cost function while considering uncertainties.

In expensive tests and complex material environments, this method enables:

• A priori evaluation of whether geometric configurations can be used to identify desired parameters;
• Adjustment of specimen geometry before testing;
• Reduction in the number of tests needed to achieve given objectives (recalibration, certification, model validation).

This method is especially suitable for anisotropic, heterogeneous, or complex performance materials, as standard tests often cannot sufficiently constrain all parameters of these materials. Based on existing research [3], we can also envision designing a scheme to identify several complementary geometries by simultaneously optimizing all these specimens, as they follow the same material laws. This would allow identification of some less sensitive parameters while still limiting the number of tests required for full identification of a material.

Conclusion

By integrating existing full-field measurement and numerical simulation capabilities from the outset, specimen design for mechanical parameter identification can be optimized. Traditional methods based on standard geometries cannot guarantee identifiability of target parameters, especially for orthotropic composites, and lead to tests involving many specimens.
Using the EikoTwin digital twin model, this identifiability can be quantitatively assessed through sensitivity analysis based on the Hessian matrix of the identification cost function. Classification of sensitivity levels directly indicates the information potential of each test and guides the selection of geometric parameters.

Examples show that by optimizing the geometry of the Iosipescu test, at least five parameters can be identified with accuracy consistent with experimental uncertainty, reducing the need for multiple tests.
Combined with EikoTwin tools, this method provides a systematic approach to designing efficient tests aimed at precise identification within a rigorous framework integrating simulation, uncertainty, and experimental measurement.

References

[1] A. Vintache et al., Test optimization for elastic and orthotropic parameter identification, ECCM21 Conference, 2024.
[2] M. Rossi, F. Pierron, "On the use of simulated experiments to design full-field measurement material characterization tests," International Journal of Solids and Structures, 49[3]:420-35, 2012.
[3] J. Neggers, F. Mathieu, F. Hild, S. Roux, "Simultaneous full-field multi-experiment identification," Mechanics of Materials, 133:71-84, 2019.