Sparse direct solvers, which have been for a long time the main workhorses of commercial finite element software, continue to play an important role in these simulation codes. However, with the pressing need for higher-fidelity three-dimensional finite element structural models with millions of degrees of freedom, and the demands placed by direct solvers on computer resources for such large problems, a large segment of the computational mechanics community is increasingly leaning towards iterative solvers. These include some software development houses. Presently ANSYS and NISA, among others, feature powerful iterative solvers. Other vendors will likely follow this trend.
The significant progress achieved during the last decade in the development of fast and robust iterative algorithms for the solution of solid mechanics, plate and shell, and acoustic scattering problems, and the advent of commercial parallel hardware, are other key factors for this change in directions. These factors are interrelated because iterative methods are more amenable to parallel processing than direct algorithms. Their development has benefited and continue to flourish under research programs such as the National Science Foundation's HPCC (High Performance Computing & Communications) and the Department of Energy's ASCI (Accelerated Strategic Computing Initiative), which have emphasized challenging applications of parallel computation.
Krylov-based iterative substructuring methods, also known as domain decomposition (DD) methods, have emerged as powerful iterative solvers on both sequential and parallel computing platforms. When equipped with an appropriate subdomain level preconditioner, a DD method can achieve numerical scalability with respect to problem size; that is, delivering a convergence rate that is asymptotically independent of the total number of degrees of freedom. Furthermore, when equipped with a "coarse space" preconditioner, it can also achieve numerical scalability with respect to the number of subdomains, delivering a convergence rate that is independent of that number. Usually, a DD method is implemented on a parallel platform by mapping one or several subdomains onto each CPU. For this reason, numerical scalability with respect to the number of subdomains is paramount to parallel scalability: the ability to deliver a linear, or almost linear, speedup in the number of processors.
In my lecture, I will overview a specific class of DD methods that employ discrete and user-transparent Lagrange multipliers for subdomain interconnection. I will discuss their numerical and parallel scalability properties, as well as their implementation on sequential, parallel, and massively parallel computing platforms. I will also highlight their application to the solution of large-scale linear and nonlinear, static and dynamic, structural and acoustic scattering problems. Finally I will address some issues pertaining to multipoint constraints and unilateral contact.