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Bayesian identification of material parameters in viscoelastic structures as an inverse problem in a semigroup setting

Rebecca Rothermel, Thomas Schuster

TL;DR

The paper tackles the inverse problem of identifying viscoelastic material parameters when the number of Maxwell elements is unknown, and thus the parameter count is itself part of the solution. It develops a Bayesian inversion framework with a binomial prior on the Maxwell-element count $n$ and couples it with a Tikhonov-regularized parameter recovery for each fixed $n$, yielding an iterative Bayes algorithm that alternates between updating $n$ and the parameter vector $x$ in the semigroup setting $\mathbb{N}\times \ell^2(\mathbb{N})$. The authors prove existence, stability, and convergence of the regularized solutions and demonstrate robust performance on synthetic relaxation data, including comparisons to clustering and residual minimization under noise; the prior parameter $q$ controls sparsity and model complexity. The work extends regularization theory to semigroup-structured forward operators and shows that Bayesian priors can effectively regularize complex, nonlinear inverse problems in viscoelastic identification, with practical implications for determining model order and material parameters from stress-relaxation data.

Abstract

The article considers the nonlinear inverse problem of identifying the material parameters in viscoelastic structures based on a generalized Maxwell model. The aim is to reconstruct the model parameters from stress data acquired from a relaxation experiment, where the number of Maxwell elements, and thus the number of material parameters themselves, are assumed to be unknown. This implies that the forward operator acts on a Cartesian product of a semigroup (of integers) and a Hilbert space and demands for an extension of existing regularization theory. We develop a stable reconstruction procedure by applying Bayesian inversion to this setting. We use an appropriate binomial prior which takes the integer setting for the number of Maxwell elements into account and at the same time computes the underlying material parameters. We extend the regularization theory for inverse problems to this special setup and prove existence, stability and convergence of the computed solution. The theoretical results are evaluated by extensive numerical tests.

Bayesian identification of material parameters in viscoelastic structures as an inverse problem in a semigroup setting

TL;DR

The paper tackles the inverse problem of identifying viscoelastic material parameters when the number of Maxwell elements is unknown, and thus the parameter count is itself part of the solution. It develops a Bayesian inversion framework with a binomial prior on the Maxwell-element count and couples it with a Tikhonov-regularized parameter recovery for each fixed , yielding an iterative Bayes algorithm that alternates between updating and the parameter vector in the semigroup setting . The authors prove existence, stability, and convergence of the regularized solutions and demonstrate robust performance on synthetic relaxation data, including comparisons to clustering and residual minimization under noise; the prior parameter controls sparsity and model complexity. The work extends regularization theory to semigroup-structured forward operators and shows that Bayesian priors can effectively regularize complex, nonlinear inverse problems in viscoelastic identification, with practical implications for determining model order and material parameters from stress-relaxation data.

Abstract

The article considers the nonlinear inverse problem of identifying the material parameters in viscoelastic structures based on a generalized Maxwell model. The aim is to reconstruct the model parameters from stress data acquired from a relaxation experiment, where the number of Maxwell elements, and thus the number of material parameters themselves, are assumed to be unknown. This implies that the forward operator acts on a Cartesian product of a semigroup (of integers) and a Hilbert space and demands for an extension of existing regularization theory. We develop a stable reconstruction procedure by applying Bayesian inversion to this setting. We use an appropriate binomial prior which takes the integer setting for the number of Maxwell elements into account and at the same time computes the underlying material parameters. We extend the regularization theory for inverse problems to this special setup and prove existence, stability and convergence of the computed solution. The theoretical results are evaluated by extensive numerical tests.

Paper Structure

This paper contains 25 sections, 7 theorems, 98 equations, 7 figures, 17 tables.

Table of Contents

  1. Introduction
  2. A rheological model for viscoelastic material behavior
  3. Viscoelastic materials
  4. The forward operator
  5. The inverse problem
  6. A Bayes inversion algorithm
  7. Introduction to statistical inversion theory
  8. Application to the inverse problem
  9. Impact of $q$ to the minimization
  10. The Bayes algorithm
  11. Convergence, stability and regularization of Algorithm \ref{['alg:Bayes']}
  12. Ill-posedness and regularization in Banach spaces
  13. Local ill-posedness
  14. Tikhonov regularization
  15. Convergence, stability and regularization in $\mathbb{N}\times \ell^2 (\mathbb{N})$
  16. ...and 10 more sections

Key Result

Theorem 3.1

Let $X$ be a random variable with values in $\mathbb{R}^n$ and prior $\rho_0(x)$. In addition, let $Y$ be a random variable with values in $\mathbb{R}^k$ and realization $Y=y_{mess}$ with $\rho(y_{mess})>0$. Then, the a posteriori probability distribution of $X$ conditional on the data $y_{mess}$ is

Figures (7)

  • Figure 1: Strain curve $\varepsilon(t)$ with strain rate $\dot{\varepsilon}$ and maximum strain value $\bar{\varepsilon}$
  • Figure 2: A Hooke and a Newtonian element
  • Figure 3: Standard specimen and generalized Maxwell model with unknown number of Maxwell elements $n$ and $2n+1$ material parameters
  • Figure 4: The binomial distribution $\rho_0(n)$ and $\phi_0(n)$ for $q=0.1$ (left picture) and $q=0.5$ (right picture)
  • Figure 5: Strain- and stress-data for different displacement rates $\dot{\varepsilon}_u$
  • ...and 2 more figures

Theorems & Definitions (13)

  • Theorem 3.1: Theorem of Bayes
  • Definition 4.1: Local ill-posedness
  • Definition 4.2: Regularization method
  • Definition 4.3: $\Omega$-minimizing solution
  • Theorem 4.1: Existence
  • Theorem 4.2: Stability
  • Theorem 4.3: Convergence
  • Definition 4.4
  • Theorem 4.4
  • Definition 4.5
  • ...and 3 more