# Euler-Bernoulli Beam¶

## Overview¶

This benchmark models the deformation of an Euler-Bernoulli beam with a spatially variable stiffness parameter. The PDE describing the beam deformation is solved with a finite difference method and the beam stiffness is defined at each of the (typically 31) nodes in the discretization. The displacement of the beam is also returned at these 31 points.

This docker container uses MUQ and UM-Bridge to evaluate the model. A realization of a Gaussian process is used to define a the load $$f(x)$$. More details are provided below.

## Run¶

docker run -it -p 4242:4242 linusseelinger/model-muq-beam:latest


## Properties¶

Model

Description

forward

Forward evaluation of the beam model

### forward¶

Mapping

Dimensions

Description

inputSizes



The stiffness $$E(x)$$ at each finite difference node in the discretization.

outputSizes



The vertical displacement $$u(x)$$ at each finite difference node.

Feature

Supported

Evaluate

True

True (via finite difference)

ApplyJacobian

True (via finite difference)

ApplyHessian

True (via finite difference)

Config

Type

Default

Description

None

Mount directory

Purpose

None

## Source code¶

Model sources here.

## Description¶

Let $$u(x)$$ denote the vertical deflection of the beam and let $$f(x)$$ denote the vertical 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

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

where $$E(x)$$ is an effective stiffness and $$r$$ is the beam radius. This model takes in $$E(x)$$ at $$N$$ finite difference nodes and returns the value of $$u(x)$$ at those nodes. The beam radius is set to $$r=0.1$$ and the value of $$f(x)$$ is fixed to a precomputed realization of a Gaussian process (the value of $$f$$ will not change between model evaluations).

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

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

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