Localized tensor-based solvers for interactive finite element applications using C++ and Java

This paper presents techniques for solving systems of equations arising in finite element applications using a localized, tensor-based approach. The approach is localized in that a major part of the solution responsibility is delegated to vector degree-of-freedom objects, rather than residing solely in a global solver working on a monolithic data structure. The approach is tensor-based in that the fundamental quantities used for computation are considered to be second-order tensors. The localized data structure strategy provides the benefits of an efficient sparse and symmetric storage scheme without requiring complex implementation code. The tensor-based aspect of the approach can bring substantial performance benefits by increasing the granularity at which solution algorithms deal with their data. Java and C++ implementations of interactive finite element programs are used to demonstrate performance that is competitive with other available solvers, especially in the case of problems for which interactive analysis is feasible on commonly available hardware.