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.

Authors¶

Matthew Parno

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	[31]	The stiffness \(E(x)\) at each finite difference node in the discretization.
outputSizes	[31]	The vertical displacement \(u(x)\) at each finite difference node.

Feature	Supported
Evaluate	True
Gradient	True (via finite difference)
ApplyJacobian	True (via finite difference)
ApplyHessian	True (via finite difference)

Config	Type	Default	Description
None

Mount directories¶

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.