Nonlinear elastodynamic material identification of heterogeneous isogeometric Bernoulli-Euler beams

TL;DR

This work develops a nonlinear elastodynamic FEMU framework to identify heterogeneous material distributions in planar Bernoulli--Euler beams using a rotation-free isogeometric formulation. The approach sequentially identifies elastic properties from quasi-static data and then density from modal data, leveraging three independent discretizations: an isogeometric FE mesh, a high-resolution experimental grid, and a low-order Lagrange material mesh. Analytical derivatives for both nonlinear statics and modal dynamics enable efficient gradient-based optimization with regularization to combat ill-posedness and overfitting. Numerical experiments on straight and curved beams demonstrate robustness to noise and indicate practical guidelines for mesh choice and data requirements, while highlighting the benefits of hybrid discretizations (B2M1) to mitigate locking. The framework is modular and extensible to shells and 3D continua, providing a scalable pathway for nondestructive identification of complex material distributions in slender structures.

Abstract

This paper presents a Finite Element Model Updating framework for identifying heterogeneous material distributions in planar Bernoulli-Euler beams based on a rotation-free isogeometric formulation. The procedure follows two steps: First, the elastic properties are identified from quasi-static displacements; then, the density is determined from modal data (low frequencies and mode shapes), given the previously obtained elastic properties. The identification relies on three independent discretizations: the isogeometric finite element mesh, a high-resolution grid of experimental measurements, and a material mesh composed of low-order Lagrange elements. The material mesh approximates the unknown material distributions, with its nodal values serving as design variables. The error between experiments and numerical model is expressed in a least squares manner. The objective is minimized using local optimization with the trust-region method, providing analytical derivatives to accelerate computations. Several numerical examples exhibiting large displacements are provided to test the proposed approach. To alleviate membrane locking, the B2M1 discretization is employed when necessary. Quasi-experimental data is generated using refined finite element models with random noise applied up to 4%. The method yields satisfactory results as long as a sufficient amount of experimental data is available, even for high measurement noise. Regularization is used to ensure a stable solution for dense material meshes. The density can be accurately reconstructed based on the previously identified elastic properties. The proposed framework can be straightforwardly extended to shells and 3D continua.

Figures (22)

• Figure 1: Example of mapping $\xi\mapsto\bar{\xi}$. Here, $n_\mathrm{el} = 4$, $\bar{n}_\mathrm{el} = 2$, $m = 2$, and $e = -1$.
• Figure 2: Flow chart of the inverse identification algorithm: Given the experimental data, constitutive law, and the initial guess $\mathbf{q}_0$, the algorithm calculates the optimal solution of the material parameters $\mathbf{q}_\mathrm{opt}$ for chosen FE and material meshes. Source: Borzeszkowski2022.
• Figure 3: The inverse analysis is based on three separately discretized fields. The resolution of the FE analysis mesh and the experimental grid influences the reconstruction of the unknown material parameters of the material mesh.
• Figure 4: Uniaxial stretching of a bar: a. undeformed configuration with boundary conditions; b. deformed configuration, colored by stretch $\lambda$; c. material mesh with the reference distribution for $E\!A$; d. FE convergence of the discrete $L^2$ error w.r.t. the FE solution for 1024 elements.
• Figure 5: Uniaxial stretching of a bar, Case 1.3: a. columns of the Jacobian, $\mathbf{J}_{I}=\partial \bar{\mathbf{U}}_\mathrm{R} / \partial E\!A_{I}$, at the optimal $\mathbf{E}\!\mathbf{A}$, plotted on the experimental grid. The values are normalized by $E\!A_\mathrm{ref}$. Colors ranging from blue, through green, to gray correspond to $E\!A_1$--$E\!A_{16}$; b. correlation matrix for the optimal $\mathbf{E}\!\mathbf{A}$, derived from the covariance approximation $(\mathbf{J}^\mathrm{T}\mathbf{J})^{-1}$, see Hansen2013. For the sake of the correlation matrix, measurement errors are assumed to be uncorrelated, uniform, and Gaussian.