# Uncertainty propagation of material properties of a cantilevered beam¶

## Overview¶

This is an uncertainty propagation problem modeling the effect of uncertain material parameters on the displacement of an Euler-Bernoulli beam with a prescribed load.

The Young’s modulus is piecewise constant over three regions as shown below, and the three constants are assumed to each be uncertain with a uniform distribution.

## Run¶

docker run -it -p 4243:4243 linusseelinger/benchmark-muq-beam-propagation:latest


## Properties¶

Model

Description

forward

Forward model

### forward¶

Mapping

Dimensions

Description

input

[3]

The value of the beam stiffness in each lumped region.

output

[31]

The resulting beam displacement at 31 equidistant grid nodes.

Feature

Supported

Evaluate

True

True

ApplyJacobian

True

ApplyHessian

True

Config

Type

Default

Description

None

Mount directory

Purpose

None

## Source code¶

Model sources here.

## Description¶

### Forward model¶

Let $$u(x)$$ denote the vertical deflection of the beam and let $$m(x)$$ denote the vertial force acting on the beam at point $$x$$ (positive for upwards, negative for downwards). We assume that the displacement can be well approximated using Euler-Bernoulli beam theory and thus satisfies the PDE

$\frac{\partial^2}{\partial x^2}\left[ \exp(m(x)) \frac{\partial^2 u}{\partial x^2}\right] = f(x),$

where $$m(x) = \log E(x)$$ is the log of an effective stiffness $$E(x)$$ that depends both on the beam geometry and material properties.

For a beam of length $$L$$, the cantilever boundary conditions take the form

$u(x=0) = 0,\quad \left.\frac{\partial u}{\partial x}\right|_{x=0} = 0$

and

$\left.\frac{\partial^2 u}{\partial x^2}\right|_{x=L} = 0, \quad \left.\frac{\partial^3 u}{\partial x^3}\right|_{x=L} = 0.$

Discretizing this PDE with finite differences (or finite elements, etc…), we obtain a linear system of the form

$K(m)\hat{u} = \hat{f},$

where $$\hat{u}\in\mathbb{R}^N$$ and $$\hat{f}\in\mathbb{R}^N$$ are vectors containing approximations of $$u(x)$$ and $$m(x)$$ at finite difference nodes.

We assume that $$m(x)$$ is piecwise constant over $$P$$ nonoverlapping intervals on $$[0,L]$$. More precisely,

$m(x) = \sum_{i=1}^P m_i \,I\left(x\in [a_i, a_{i+1})\right),$

where $$I(\cdot)$$ is an indicator function.

As our forward model, we define $$F : \mathbb{R}^3 \rightarrow \mathbb{R}^N$$ mapping a parameter vector $$m = [m_1, m_2, m_3]$$ onto the corresponding solution vector $$\hat{u}$$.

Finally, the quantity of interest $$Q : \mathbb{R}^N \rightarrow \mathbb{R}^2$$ simply picks the solution at the 10th and 25th node, i.e. $$Q(\hat{u}) := \left(\begin{matrix} \hat{u}_{10} \\ \hat{u}_{25} \end{matrix} \right)$$.

The container implements the mapping $$F$$; implementing $$Q$$, i.e. picking out the respective solution entries, is up to the user.

### Uncertainty propagation¶

For the prior, we assume the material parameter to be a uniformly distributed random variable

$M \sim U_{[1, 1.05]^3}.$

The goal is to identify the distribution of the resulting quantity of interest $$Q(F(M))$$. Samples from the model output $$F(M)$$ are illustrated in the following figure.

The quantity of interest’s components have the following distributions, as computed from $$10^5$$ simple Monte Carlo samples.