Detection of Model Uncertainty in the Dynamic Linear-Elastic Model of Vibrations in a Truss

Alexander Matei,Stefan Ulbrich

doi:10.1007/978-3-030-77256-7_22

Abstract

AbstractDynamic processes have always been of profound interest for scientists and engineers alike. Often, the mathematical models used to describe and predict time-variant phenomena are uncertain in the sense that governing relations between model parameters, state variables and the time domain are incomplete. In this paper we adopt a recently proposed algorithm for the detection of model uncertainty and apply it to dynamic models. This algorithm combines parameter estimation, optimum experimental design and classical hypothesis testing within a probabilistic frequentist framework. The best setup of an experiment is defined by optimal sensor positions and optimal input configurations which both are the solution of a PDE-constrained optimization problem. The data collected by this optimized experiment then leads to variance-minimal parameter estimates. We develop efficient adjoint-based methods to solve this optimization problem with SQP-type solvers. The crucial test which a model has to pass is conducted over the claimed true values of the model parameters which are estimated from pairwise distinct data sets. For this hypothesis test, we divide the data into k equally-sized parts and follow a k-fold cross-validation procedure. We demonstrate the usefulness of our approach in simulated experiments with a vibrating linear-elastic truss.

Highlights

In science and technology, dynamic processes are often described by time-variant mathematical models
The mathematical models used to describe and predict time-variant phenomena are uncertain in the sense that governing relations between model parameters, state variables and the time domain are incomplete
The best setup of an experiment is defined by optimal sensor positions and optimal input configurations which both are the solution of a partial differential equations (PDE)-constrained optimization problem

Summary

Introduction

Dynamic processes are often described by time-variant mathematical models. In [12] we developed an algorithm to identify this model uncertainty using parameter estimation, optimal experimental design and classical hypothesis testing. It is the aim of this paper to extend this approach to dynamic models. Optimal sensor placement has been addressed within the PDE-context in [1,2,23] and optimal input configuration has been extensively analyzed for both linear and nonlinear ordinary differential equations in various engineering applications [6,17,21,22,28] In these cases the problem dimension is small compared to a (discretized) time-variant PDE and gradient-based optimization with a sensitivity approach, as suggested by [4] and [19], works fine.

Model Equations of Transient Linear Elasticity and Their Discretization

Lame-Parameter Estimation and the Optimal Experimental Design Problem

Derivative and Adjoint Computation

Computational Remarks

Detection of Uncertainty in Dynamic Models

Numerical Results for Simulated Vibrations of a Truss

Conclusion