At the highest level, the process of engineering design is: given a desired functionality, synthesize, refine, and describe a device which robustly exhibits the desired function. To build computational environments which support the designer and engineer in this task --- as the complexity of engineering continues to grow --- requires new, integrated modeling, simulation, visualization, and design environments. To be truly predictive they must be mathematically well-founded, scale gracefully to large problem sizes, and reliably model physical behavior across a wide range of scales. The research team represents computer science, mathematics, mechanical engineering, control and dynamical systems, and engineering design, to develop wholly new representations and algorithms, capable of addressing these issues in concert. The core integrating principle they employ is the mathematics of "multiresolution," i.e., descriptions at different levels of resolution, for geometry, numerical solvers, mechanics, and conceptual design. The technical approach is based on the use of subdivision geometry, coupled with thin shell equation dynamics. Integrating automatic model reduction techniques --- encompassing geometry, topology, and mechanics --- based on rigorous mathematics and dynamics principles allows them to accurately and efficiently model the behavior of complex assemblies across a wide range of physical scales from microscopic to gross behavior. Fluidly and accurately moving across many physical scales makes these representations and algorithms highly suitable for engineering design. Coupling them with set-based design methods allows designers and engineers to rapidly explore large design spaces.
The objective of the proposed research is to significantly advance modeling, simulation, and design environments as needed in, for example, the aircraft and automobile industries (among many others). Current industry practice is based on separately developed, often incompatible computational modules for geometric modeling, physical simulation, and analysis. With the increasing complexity of modern engineering, this approach incurs ever larger overhead and ultimately yields results whose physical accuracy is questionable. Reconsidering the mathematical, physical, and computer science foundations of such systems from scratch, employing an integrated team of researchers and students, can radically advance the state of the art in engineering design. Cross fertilization among the team members is already yielding a new approach to simulating the behavior of thin sheets of metal, removing a long standing problem in mechanics simulation, with the potential to revolutionize the use of computer simulations in this area. To ensure the relevance of their work to real industrial practice the researchers are collaborating closely with aircraft industry and vendors of computer modeling applications. Postdoctoral scholars, graduate students and undergraduates are integrated into this crossdisciplinary program. A new generation of researchers and practitioners is being educated who are equally at ease with questions of mathematics, computer science, and mechanical engineering.
Funding for this activity will be provided by the Division of Mathematical Sciences, the MPS Office of Multidisciplinary Activities, and by DARPA.
