Rapid Reliable Solution
of the Parametrized Partial Differential Equations
of Continuum Mechanics and Transport


Worked Problems

 

The site comprises a number of illustrative Worked Problems — defined by governing equations, parameter domain, and outputs of interest — in solid mechanics, heat transfer, acoustics, and fluid dynamics. Each problem is characterized by a vector of P input parameters and an associated input domain $\mathcal{D}$; and a scalar output representing the engineering quantity of interest.
For each Worked Problem, the End-User can rapidly and reliable predict the parametric dependence of the output: Given an input parameter value, the RB Online Evaluator returns — typically in milliseconds — both (a) an accurate RB prediction for the output, and (b) a certificate of fidelity that rigorously bounds the error in the RB prediction relative to a highly accurate finite element solution. We thus achieve educationally (and design) relevant "interactive timescales" with little or no compromise in accuracy or certainty.
The software also includes a RB Visualizer which, given an input parameter value, (a) renders the relevant field variable, and (b) provides an associated rigorous bound for the error in the RB field relative to the finite element field. This feature is somewhat slower and more memory-intensive, but also more "physical" and colorful.
The End-User can access the RB Online Evaluator and RB Visualizer for a particular Worked Problem in two fashions: with limited functionality and some degradation in performance — but very simply — through our Matlab® webserver; or with full functionality and optimal performance — but with a bit more effort — downloaded to the End-User's own computer.
Note for the download option, the user must first download the necessary rbMIT Software and documentation. Then, for each Worked Problem available (see below or sidebar), we provide for download the necessary user input file. The End-User must execute several commands indicated in Sections I., II., and III. of the rbMIT documentation to generate the RB Online Evaluator and RB Visualizer functions (for any particular Worked Problem); the End-User is not required to understand the software, make any choices, or provide any arguments — except ultimately the desired parameter values for output evaluation or field visualization.

The Worked Problems available are:

Solid Mechanics

CenterCrack

Heat Transfer

Heterogeneous Thermal Block
Thermal Contraction Resistance
Thermal Fin
The "Graetz" Problem
Nonhomogeneous Semi-Infinite Body
A Transient Heat Treatment Problem
The Transient Convection-Diffusion Problem Around a Cylinder
Transient Forced Convection: Heating Duct

Acoustics

Fluid Mechanics

The "addedmass" Problem in Hydrodynamics
Venturi: Potential Flow
Circular Bend: Potential Flow


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