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Simultaneous identification of the parameters in the plasticity function for power hardening materials : A Bayesian approach

Salih Tatar, Mohamed BenSalah

TL;DR

This work addresses the simultaneous identification of the strain hardening exponent $\kappa$, yield stress $\xi_0^2$, and shear modulus $G$ in a nonlinear elasto-plastic torsion model by formulating a Bayesian inverse problem based on torque data $\mathcal{T}(\varphi)$. A Lipschitz-continuous input-output map for the forward problem is established, ensuring well-posedness of the Bayesian formulation. The authors extend the iterative regularizing ensemble Kalman method (IREKM) to efficiently sample and update the posterior over $\theta=(\kappa,\xi_0^2,G)$ without adjoint computations, and validate the approach with numerical experiments showing accurate recovery under noisy data and limited measurements. The results demonstrate the method’s robustness and potential for reliable material parameter identification in nonlinear PDE-based torsion models, with practical implications for material characterization under strain hardening.

Abstract

In this paper, we study simultaneous determination of the strain hardening exponent, the shear modulus and the yield stress in an inverse problem. First, we analyze the direct and the inverse problems. Then we formulate the inverse problem in the Bayesian framework. After solving the direct problem by an iterative approach, we propose a numerical method based on a Bayesian approach for the numerical solution of the inverse problem. Numerical examples with noisy data illustrate applicability and accuracy of the proposed method to some extent.\

Simultaneous identification of the parameters in the plasticity function for power hardening materials : A Bayesian approach

TL;DR

This work addresses the simultaneous identification of the strain hardening exponent , yield stress , and shear modulus in a nonlinear elasto-plastic torsion model by formulating a Bayesian inverse problem based on torque data . A Lipschitz-continuous input-output map for the forward problem is established, ensuring well-posedness of the Bayesian formulation. The authors extend the iterative regularizing ensemble Kalman method (IREKM) to efficiently sample and update the posterior over without adjoint computations, and validate the approach with numerical experiments showing accurate recovery under noisy data and limited measurements. The results demonstrate the method’s robustness and potential for reliable material parameter identification in nonlinear PDE-based torsion models, with practical implications for material characterization under strain hardening.

Abstract

In this paper, we study simultaneous determination of the strain hardening exponent, the shear modulus and the yield stress in an inverse problem. First, we analyze the direct and the inverse problems. Then we formulate the inverse problem in the Bayesian framework. After solving the direct problem by an iterative approach, we propose a numerical method based on a Bayesian approach for the numerical solution of the inverse problem. Numerical examples with noisy data illustrate applicability and accuracy of the proposed method to some extent.\

Paper Structure

This paper contains 6 sections, 3 theorems, 30 equations, 13 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Analysis of the direct and the inverse problems
  3. Iterative regularizing ensemble Kalman method
  4. Numerical experiments
  5. Numerical solution of the direct problem
  6. Reconstruction Results for the Inverse Problem

Key Result

Theorem 2.1

Let the conditions in (S1-3) hold. Then the approximate solution $u^{ (n)} \in \mathring{H}^{1}(\Omega)$, defined by the iteration scheme linearization, of the nonlinear problem (govequationfunctional) converges to unique exact solution $u\in \mathring{H}^{1}(\Omega)$ of the problem (govequationfunc

Figures (13)

  • Figure 1: Variation of the exact solution $u_{ex}$
  • Figure 2: Regions $\Omega_1$ and $\Omega_2$ within the domain, along with the plasticity function $g(|\nabla u_{ex}|^2)$ and the forcing term $F$ associated to the exact solution $u_{ex}$ (Test 1).
  • Figure 3: Convergence analysis of the iterative process: Norms of successive iterations and error with respect to the exact solution (Test 1).
  • Figure 4: The numerically approximated solution alongside the absolute errors (Test 1).
  • Figure 5: Variations of the plasticity function $g(|\nabla u_{ex}|^2)$ and the forcing term $F$ associated to the exact solution $u_{ex}$ (Test 2).
  • ...and 8 more figures

Theorems & Definitions (5)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3