Spectral methods have been shown to be effective in approximating the high gradient bands which occur during failure processes. These methods are effective in solid mechanics when applied in small patches or domains, which correspond to the high gradient regions such as shear bands, coupled with a finite element discretization of the rest of the problem. Here, adaptive procedures will be developed to determine the location and extent of these spectral domains from properties of the solution, for example the strain field. These numerical methods will be employed in large scale computations to study the structure of shear bands and factors influencing their growth. The use of spectral patches or subdomains provides a high degree of resolution of high gradient bands. Mappings within the subdomains and patches will be employed to adaptively enhance the resolution. These methods, because of their employment of subdomains, lend themselves naturally to MIMD parallel computers. Implementations designed to exploit this parallelism, will be developed. These methods will be implemented and applied in large scale computational studies relating to materials forming processes. In particular the role of imperfections and thermal coupling in the development of shear bands will be studied. Additionally, these methods will be applied in computational studies of the dynamics of gaseous combustion with multiple stage reaction mechanisms.
