Asynchronous optimized Schwarz methods with and without overlap

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

In this work we consider the optimized Schwarz method designed for computational domains that feature spherical or almost spherical interfaces. In the first part, we consider the diffusion-reaction problem. We provide a convergence analysis of the generalized Schwarz method and, following [G. Gigante and C. Vergara, Numer. Math., 131 (2015), pp. 369--404], we discuss an optimization procedure for constant interface parameters leading to a Robin--Robin scheme. Finally, we present some numerical results both in spherical and in ellipsoidal domains. In the second part of the work, we address the fluid-structure interaction problem. Again, we provide a convergence analysis and discuss optimized choices of constant interface parameters. Finally, we present three-dimensional numerical results inspired by hemodynamic applications, to validate the proposed choices in the presence of large added mass effect. In particular, we consider numerical experiments both in an ideal spherical domain and in a realistic abdominal aortic aneurysm.

Harmonic functions attain their pointwise maximum on the boundary of the domain. In this article, we analyze the relationship between various norms of nearly harmonic functions and we show that the trace norm is maximized on the boundary of the domain. One application is that the Optimized Schwarz Method with two subdomains converges for all Robin parameters α > 0.

The propagation of the action potential in the heart chambers is accurately described by the Bidomain model, which is commonly accepted and used in the specialistic literature. However, its mathematical structure of a degenerate parabolic system entails high computational costs in the numerical solution of the associated linear system. Domain decomposition methods are a natural way to reduce computational costs, and Optimized Schwarz Methods have proven in the recent years their effectiveness in accelerating the convergence of such algorithms. The latter are based on interface matching conditions more efficient than the classical Dirichlet or Neumann ones. In this paper we analyze an Optimized Schwarz approach for the numerical solution of the Bidomain problem. We assess the convergence of the iterative method by means of Fourier analysis, and we investigate the parameter optimization in the interface conditions. Numerical results in 2D and 3D are given to show the effectiveness of the method.

Dynamic responses considering fluid structure interaction (FSI) is important in many engineering fields and some of the FSI phenomena are treated as an acoustic fluid and structure interaction (AFSI) problem. The dynamic interactions between the fluid and structure can change dynamic characteristics of structures and their responses to external excitation such as seismic loading. The authors have developed a coupled simulation system for the large scale AFSI problems using an iterative partitioned coupling technique. In the system, the authors employed ADVENTURE system which adopted an efficient preconditioned iterative linear algebraic solver, and ADVENTURE Coupler is used to handle interface variable efficiently on various parallel computational environments. The authors employ Broyden method for updating interface accelerations to obtain the robust and fast convergence property of fixed point iterations. This paper presents the overview of the coupled analysis system and the results of its application to several AFSI problems are shown. The system runs efficiently in a parallel environment and it is capable for analyses of complex shaped three dimensional structures with more than 20 million degrees of freedom model.

This article presents a review of various hardware and software parallel computing environments so far developed for the implementation of parallel algorithms to solve complex and computationally dominated optimization problems. These parallel environments are composed of parallel programming tools, performance evaluation tools, debuggers and optimization libraries. The environments can be classified based on the ease of implementation, hardware independence, ease to understand, and guarantee of performance.

The goal of this dissertation is two-fold. The first part concerns the development of a numerical method for solving Maxwell's equations on unstructured hexahedral grids that employs both high order spatial and high order temporal discretizations. The second part involves the use of this method as a computational tool to perform high fidelity simulations of various electromagnetic devices such as optical transmission lines and photonic crystal structures to yield a level of accuracy that has previously been computationally cost prohibitive. This work is based on the initial research of Daniel White who developed a provably stable, charge and energy conserving method for solving Maxwell's equations in the time domain that is second order accurate in both space and time. The research presented here has involved the generalization of this procedure to higher order methods. High order methods are capable of yielding far more accurate numerical results for certain problems when compared to corresponding h-refined first order methods , and often times at a significant reduction in total computational cost. The first half of this dissertation presents the method as well as the necessary mathematics required for its derivation. The second half addresses the implementation of the method in a parallel computational environment, its validation using benchmark problems, and finally its use in large scale numerical simulations of electromagnetic transmission devices.

Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers

Towards modular CFD using the cerfacs parallel utilities

We describe total congestion 1 embeddings of complete binary trees into three dimensional grids with expansion ratios 1.172 and 1.25. That is, we give a one-to-one embedding of any complete binary tree into a hexahedron shaped grid such that: the number of nodes in the grid is at most 1.172 (1.25) times the number of nodes in the tree; and no tree nodes or edges occupy the same grid positions. The first strategy embeds trees into cube shaped 3D grids. That is, 3D grids in which all dimensions are roughly equal in size. The second strategy embeds trees into flat 3D grid shapes. That is, it maps complete binary trees into 3D grids with a fixed, small number of layers. An application suggests which embedding to use. For simulations in parallel computer environments, or possibly graph drawing, a cube shaped 3D grid is appropriate. For the sake of VLSI, or other graph drawing applications, embeddings with a small number of layers are better.

During the development and parametrization of 2D image-processing algorithms for surface inspection uses, you need to test a huge amount of image-data for each modification of the algorithms or parameters. For algorithm runtimes up to several seconds, this will take a long time. To speed up this process it is recommended to distribute the computation in a parallel computation environment. Compute Grids, which use the unused resources of existing hardware are the most cost efficient way to solve this problem. The most existing Grid-Concepts are based on flat connection structures with a scheduler on the top; for high job-rates the scheduler becomes the bottleneck of the whole system. Concepts to solve this problem organize the nodes in tree-structures to discharge the central scheduler. In heterogeneous Desktop-Grids where the different nodes are widely distributed the usually used random arrangement of the nodes in the tree-structure can be counterproductive, because the bandwidthes and latencies in a Grid can be varying. In this paper we will show a solution to arrange the nodes of the grid optimized by bandwidth and latency, using modified spanning-tree algorithms, so that the average response time is reduced and in result of this the job-throughput of the Compute-Grid is increased.

The loss of orthogonality in the Gram-Schmidt orthogonalization process

The component-mode method in a parallel computer environment

Nowadays, research and development for optimal setting control (OSC) of complex industrial processes is becoming a hot topic. However, there is no configurable software for OSC design. In this paper, a novel software platform is proposed. The major superiority is that it not only integrates intelligent and data-driven mainstream algorithms within a unified framework, but also provides a powerful controller design tool and employs a Petri net engine to schedule control units in a parallel computation environment. The overall framework and key technologies are introduced in detail. The platform is verified and validated through a case of application to the design and development of OSC system of grinding process in a hardware-in-the-loop simulation environment.

Application of the parallel diagonal dominant algorithm for the incompressible Navier-Stokes equations