Domain decomposition methods

Maday, Y., Mavripilis, C. and Petera, A. T., 1980. Nonconforming mortar element methods application to. spectral discretizations. In Chan, T. F, Glowinski, R., Perianx, J. and Widlund, O. B. (eds). Domain Decomposition Methods, SIAM, Philadelphia, pp. 392- 418. 

Vereurth, R., Non-overlapping domain decomposition methods, in Bristeau,... 

A. A. Zick and G. M. Homsy, Stokes flow through periodic arrays of spheres, J. Fluid Mech. 115, 13-26 (1982) A. S. Sangani and A. Acrivos, Slow flow through a periodic array of spheres, Int. J. Multiphase Flow 8, 343-60 (1982) G. Liu and K. E. Thompson, A domain decomposition method for modelling Stokes flow in porous materials, Int. J. Numer. Meth. Fluids 38, 1009-25 (2002). 

Vol. 1809 O. Steinbach, Stability Estimates for Hybrid Coupled Domain Decomposition Methods (2003)... 

Katafygiotis, L. S. and Cheung, S. H. Domain decomposition method for calculating the failure probabOity of linear dynamic systems subjected to Gaussian stochastic loads. Journal of Engineering Mechanics (ASCE) 132(5) (2006), 475-486. 

Shyy W, Francois M, Udaykumar HS (2001) Cartesian and curvilinear grid methods for multi-domain, moving boundary problems. In Debit M et al (eds) Thirteenth international conference on domain decomposition method. CIMNE, Barcelona... 

Spectral domain-decomposition methods are suggested as an efficient way of resolving the limitation of conventional spectral methods. The basic idea behind the methods is to partition the whole domain into several subdomains and then to simultaneously solve the differential equations in each subdomain and appropriate matching conditions on each interface. These methods have very attractive features, such as rapid convergence, geometric flexibility, and suitability to parallel implementation. Thus, they are appropriate when particular regions need to be resolved or when flows in complex geometries are to be simulated. 

In general, two classes of spectral domain-decomposition methods have been proposed in the literature patching methods and variational... 

Most of spectral methods have been developed paying attention to simple geometries, and, thus, one would suffer from many difficulties in applying them to more complex geometries. Recently, there have been a number of developments on the use of spectral schemes in more complex geometries, for example, the advent of spectral domain-decomposition methods. The basic idea behind these methods is to partition the whole domain into several subdomains and then to simultaneously solve differential equations in... 

Because ofthe material non-linearities, the mechanical contacts with friction, the large number of elements, many iteration steps, and the choice of 500 Monte Carlo simulations, four parallel computers (with 26 CPU) were used to handle the large computational requirements for this problem. The FETI Domain Decomposition Method (i.e. apphcation of parallel computers) was used, see Figure 11. 

Figure 11. FETI domain decomposition method used for application of 4 CPU.

Q.V. Dinh Glowinski and J. Periaux, Domain Decomposition Methods for Nonlinear Problems in Fluid Dynamics, Computer Methods in Applied Mechanics and Engineering, 40, 27-109 (1983). 

This article reviews some of the progress made in using parallel processor systems to study macromolecules. After an initial introduction to the key concepts required to understand parallelisation, the main part of the article focuses on molecular dynamics. It is shown that simple replicated data methods can be used to carry out molecular dynamics effectively, without the need for major changes from the approach used in scalar codes. Domain decomposition methods are then introduced as a path toward reducing inter-processor communication costs further to produce truly scalable simulation algorithms. Finally, some of the methods available for carrying out parallel Monte Carlo simulations are discussed. 

Farhat, C. 1991. Saddle-point principle domain decomposition method for the solution of solid mechanics problems. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Norfolk, VA, USA, May 6-8. 

Rixena, D. Magoules, F. 2007. Domain decomposition methods Recent advances and new challenges in engineering. Computer Methods in Applied Mechanics and Engineering, 196(8) 1345-1346. 

MD simulations of polymer systems, in particular, require computation of two kinds of interactions bonded forces (bond length stretching, bond angle bending, torsional) and nonbonded forces (van der Waals and Coulombic). Parallel techniques developed [31-33] include the atom-decomposition (or replicated-datd) method, the force-decomposition method, and the spatial (domain)-decomposition method. The three methods differ only in the way atom coordinates are distributed among the processors to perform the necessary computations. Although all methods scale optimally with respect to computation, their different data layouts incur different interprocessor communication costs which affect the overall performance of each method. 

In general, two classes of spectral domain-decomposition methods have been proposed in the literature patching methods and variational methods. The difference is in the way how the interface conditions are imposed. To solve second-order partial differential equations as an exanple, the interface condition is typically enforced by requiring that the solution and its first normal derivative be continuous on each interface. In patching methods, the continuity conditions on each interface are discretized by enforcing them at selected points, and thus are satisfied exactly by any approximation. In variational methods, on the other hand, the continuity conditions are enforced implicitly or variationaUy with differential... 

HI Couplings for an Overlapping Domain Decomposition Method using Lagrange Multiphers. 

Farhat C, Lesoinne M, Pierson K (2000) A scalable dual-primal domain decomposition method. Numer Linear Algebra Appl 7 687-714 Gao X, Castaneda N, Dyke SJ (2013) Real time hybrid simulation from dynamic system, motion control to experimental error. Earthq Eng Struct Dyn 42 815-832... 

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