# 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. ![Cantilevered-beam](https://raw.githubusercontent.com/UM-Bridge/benchmarks/main/benchmarks/muq-beam/BeamDrawing.png "Drawing of cantilevered beam") ## Authors - [Matthew Parno](mailto:matthew.d.parno@dartmouth.edu) - [Linus Seelinger](mailto:linus.seelinger@iwr.uni-heidelberg.de) ## 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 Gradient | True ApplyJacobian | True ApplyHessian | True Config | Type | Default | Description ---|---|---|--- None | | | ## Mount directories Mount directory | Purpose ---|--- None | ## Source code [Model sources here.](https://github.com/UM-Bridge/benchmarks/tree/main/benchmarks/muq-beam-propagation) ## 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. ![model-distribution](https://raw.githubusercontent.com/UM-Bridge/benchmarks/main/benchmarks/muq-beam-propagation/samples.png "Model output distribution") The quantity of interest's components have the following distributions, as computed from $10^5$ simple Monte Carlo samples. ![qoi-distribution](https://raw.githubusercontent.com/UM-Bridge/benchmarks/main/benchmarks/muq-beam-propagation/qoi_dist.png "Quantity of interest distribution")