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

 

Welcome to
Rapid Reliable Solution of the µPDEs of Continuum Mechanics and Transport

This site has both a research and a related educational component. The research component is focused on reduced basis (RB) approximation and a posteriori error estimation for rapid and reliable solution of parametrized Partial Differential Equations (µPDEs) in the many-query and real-time contexts. The educational component is focused on application of the RB methodology to illustrative "worked problems" in continuum mechanics and transport — steady and time-dependent heat transfer, solid mechanics, acoustics, and fluid mechanics.  

The research component provides a general methodology for the rapid certified solution of parametrized partial differential equations in the many-query and real-time contexts. The key ingredients are reduced basis approximations over a "parametric manifold;" rigorous a posteriori error bounds for the outputs of interest; and Offline-Online computational decomposition strategies. The integration of these three components (in conjunction with effective sampling procedures) yields a technique that guarantees the high accuracy of a refined Finite Element (FE) model but at the very low Online cost of a reduced order model. The methodology has many applications in computational engineering: parameter estimation, inverse problems, and detection; embedded diagnostics and health monitoring; adaptive design, mission planning, and optimization; and real-time optimal control. 

The educational component is realized in the application of the general methodology to particular engineering "worked problems" in continuum mechanics and transport. Despite years of promise, numerical simulations often remain either too unreliable or too slow to serve educational needs: our goal is to ensure both finite element fidelity and real-time (typically, Online millisecond) response. The pedagogical possibilities are interactive in-class visualizations and parametric exploration, assessment of classical engineering approximations (from thermal fins to structural beams to acoustic waveguides to fluidic elements), and incorporation of simulation and more realistic models in homework assignments and design projects.

The Website is intended for "Developers," "Users," and "End-Users."
 "Developers" — numerical analysts and computational tool-builders — interested in further investigation, extension, and improvement of reduced basis methods and associated a posteriori error estimators are directed to the methodology page: primary resource is the Technical Papers and Book (in progress); secondary resource is the rbMIT © MIT Software package (in progress) which implements in Matlab® all the general RB algorithms. For "Developers," the rbMIT package can serve as a platform for numerical tests and methodological extensions; we provide the full Matlab® source code for all our RB software.
"Users" — computational engineers and educators — interested in the application of the RB methodology to new problems are also directed to the methodology page: primary resource is the rbMIT © MIT Software package (in progress) which implements in Matlab® all the general RB algorithms; secondary resource is the Book (in progress) which also serves as documentation for the rbMIT software. For any new problem of interest, the rbMIT software permits quick and "automatic" (i) application of RB Offline tools, for (ii) generation of RB Offline/Online datasets, and finally (iii) real-time exercise of the RB Online RB Evaluator. Some but not extensive knowledge of both FE methods and RB methods is required of the "User."
"End Users" — students/teachers of engineering, in particular mechanical engineering — interested in visualizing the fields of continuum mechanics and transport, assessing the validity of various engineering approximations, and integrating more advanced models into characterization, design, and optimization projects are directed to the Worked Problems page. For each worked problem — defined by governing equation, parameter domain, and outputs of interest — we provide the RB Online datasets to which the real-time RB Online Evaluator (of the rbMIT © MIT package) can then be directly applied (either through our Webserver or downloaded to the End-User's site). No knowledge of either FE methods or RB methods is required of the "End-User."

We acknowledge our many former and current collaborators and sponsors as cited in the Technical Papers and Book acknowledgements.

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