Large-scale structural analysis using domain decomposition method on distributed parallel computing environment

Distributed parallel computing, which uses general-purpose workstations connected by a network as a large parallel computing resource, is one of the most promising trends in parallel computing. Over the last two decades, the growth in the use of such parallel computing environment has ensured the interest on the parallel finite element method. In this work, parallel finite element method using domain decomposition technique has been adapted to the distributed parallel environment of networked workstations and the supercomputer. Using the developed code, several model problems are solved and the parallel performances are analyzed on the network of eight Pentium IV PC cluster and GP7000F workstations. Finally, very large size of problem over 11 million DOFs has been solved using 1024 SR8000 supercomputer.

Recently, acoustic analysis techniques are used in the design of acoustic environments both inside and outside of rooms containing a sound source. However, when the analysis area is expanded or the frequency is increased, unknowns of a solving liner equation increase, and large-scale analyses becomes necessary. Therefore, in this study, a large-scale acoustic analysis method has been developed based on a parallel finite element method. In the proposed large-scale analysis method, an iterative method is applied to the interface problem as the iterative domain decomposition method, which is known to be an effective parallelization method. To improve the convergence of the iterative method, balancing domain decomposition has been applied as a preconditioner step, and its effectiveness in large-scale acoustic analysis is demonstrated. The acoustic analysis code is developed in this study that is shown to be capable of the large-scale analysis of finite element models on the order of tens of millions of elements with high accuracy.

Most of researches for large-scale parallel structural analysis using finite element method have focused on iterative solution methods since direct solution methods generally have many difficulties and disadvantages for large-scale problems. However, due to the numerical robustness of direct methods that guarantees the solution to be obtained within estimated time, direct methods are much more desirable for general application of large-scale structural analysis, as well as due to ease of use which makes most of structural engineers prefer direct methods. In this research, we propose a parallel direct solver for large-scale finite element analysis that can overcome most of difficulties and disadvantages of direct methods by using domain-wise approach for direct methods. By using the proposed solver with our own structural analysis code, we can show good scalability as a direct method and can solve the largest problem ever solved by direct solvers.

Parallel computing of the transient radiative transfer process in the three-dimensional homogeneous and nonhomogeneous participating media is studied with an integral equation model. The model can be used for analyzing the ultra-short light pulse propagation within the highly scattering media. Two numerical quadratures are used: the discrete rectangular volume (DRV) method and YIX method. The parallel versions of both methods are developed for one-dimensional and three-dimensional geometries, respectively. Both quadratures achieve good speedup in parallel performance. Because the integral equation model uses very small amount of memory, the parallel computing can take advantage of having each compute node or processor store the full spatial domain information without using the typical domain decomposition parallelism, which will be necessary in other solution methods, e.g., discrete ordinates and finite volume methods, for large scale simulations. The parallel computation is conducted by assigning different portion of the quadrature to different compute node. In DRV method, a variation of the spatial domain decomposition is used. In the case of YIX scheme, the angular quadrature is divided up according to the number of compute nodes, instead of the spatial domain being divided. This parallel scheme minimizes the communications overhead. The only communication needed is at the end of each time step when each node shares the partial integrated result of the current time step with all other compute nodes. The angular quadrature decomposition approach leads to very good parallel efficiency. Two new discrete ordinate sets are used in the YIX angular quadrature and their parallel performances are discussed. One of the discrete ordinates sets, called spherical ring set, is also suitable for use in the conventional discrete ordinates method.

Computer Solution of Large Linear Systems

A domain decomposition method is used to construct a new type of block matrix incomplete factorization method. The properties of this method are such that it can be used as an efficient (i.e. with low computational complexity) and robust, corrector on a coarse mesh. Since the cost of it is of optimal order of computational complexity there is no need to use any further levels of grids as it is in a classical multigrid method. Combined with a smoother on the fine mesh the method turns out to perform as well on difficult problems as on model type problems and with a complexity about as low as that for a classical multigrid method on the model problems. The method is well suited for vector- and parallel Computers. The smoothing-correction forms a V-cycle step which can be used as a preconditioner for a conjugate gradient method, thus guaranteeing convergence. However, the method is so efficient that there is rarely any need for convergence acceleration.

Summary form only given. Time-domain electric field integral equations (TD-EFIE) are widely used for analyzing transient scattering from perfect electrically conducting surfaces. These equations often are solved via marching-on-in-time (MOT) methods. Unfortunately, the computational cost of classically formulated MOT-based TD-EFIE solvers scales as O(N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) where N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> and N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> are the number of time steps and spatial surface unknowns, respectively. This cost prevents these solvers from being used to study high-frequency phenomena involving large-scale structures.In the past, a variety of methods for accelerating MOT-based TD-EFIE solvers have been developed; they include, but are not limited to, hybrid TD-EFIE and time domain physical optics (TDPO) methods, and fast summation schemes such as the time-domain adaptive integral method (TD-AIM). In the hybrid TD-EFIE-PO method, structures are partitioned into EFIE and PO regions. In the EFIE region, surface currents are updated by MOT; incident fields comprise those produced by external sources and currents in the PO region. In the PO region, currents are updated using incident fields produced by external sources and currents in the EFIE region (W. Luo, W. Y. Yin, M. D. Zhu, J. Y. Zhao, and J. F. Mao, IEEE Trans. Electromagn. Compat., 54, 1006-1016, 2012). The asymptotic computational complexity of the TD-EFIE-PO scheme scales like that of the TD-EFIE scheme albeit with a far smaller leading constant, enabling the solution of much larger problems. The TD-AIM constitutes the extension of the frequency domain adaptive integral method to the time domain, and reduces the computational complexity of a TD-EFIE solver from O(NtNs ) to 2 O ( N t N s log N s ) without any (high-frequency) 3/2 2 assumptions on the current (A. E. Yılmaz, J. M. Jin, and E. Michielssen, IEEE Trans. Antennas Propag., 52, 2692-2708, 2004). This paper presents a solver that hybridizes the TD-EFIE-PO and TD-AIM methods. In the hybrid scheme, the TD-AIM is used to propagate fields within the EFIE region as well as from the EFIE to the PO region, and back. The hybrid scheme is computationally far more efficient than its standalone constituents. Depending on the nature of the structure considered, the TD-AIM method uses a single grid that covers both the EFIE and PO regions, or separate and nonoverlapping grids for both regions. We apply the proposed scheme to the analysis of interactions of high-power electromagnetic pulses (HP-EMP) with large-scale platforms. Numerical results demonstrate the efficiency, accuracy, and stability of the proposed method.

With the development of computer hardware, the size of finite element analysis is getting larger and larger. Still, it is practically impossible to perform structural analysis of large scale structure with only one analysis. Structural analysis of large scale structures like ship is usually carried out on the models of several different scale level, varying from whole ship, cargo hold parts and to the detail structures. Currently, the finite element models for different analysis level is generated from different geometric model, mainly because the unit geometry of mesh generation is different in each level of the analysis. In this study, a prototype system is developed that automatically generates finite element models for different analysis level from single ship geometry data. (See Fig. 1)

Analysis of the time-Schwarz DDM on the heat PDE

A hierarchical partition model for adaptive finite element computation

Parallel processing and FEM: Fulfilling the promise

Parallel finite element method for the analysis of quasi-three dimensional shallow water flow is presented. The mode splitting technique and the sigma coordinate (generalized coordinate) are employed to use parallel computers effectively. Parallel implementation of the unstructured grid-based formulation is carried out on the Hitachi parallel-super computer SR2201. The tidal flow of Tokyo Bay is simulated for a numerical example. The speed-up ratio and the efficiency of the parallelization are investigated. The present method is shown to be a useful and powerful tool for the large scale computation of shallow water flows.

The purpose of this thesis is to define efficient parallel preconditioners based on the domain decomposition paradigm and to apply them to the solution of the steady compressible Euler equations. In the first part we propose and analyse various domain decomposition preconditioners of both overlapping (Schwarz) and non-overlapping (Schur complement-based) type. For the former, we deal with two-level methods, with an algebraic formulation of the coarse space. This approach enjoys several interesting properties not always shared by more standard twolevel methods. For the latter, we introduce a class of preconditioners based on a peculiar decomposition of the computational domain. The domain is decomposed in such a way that one subdomain is connected to all the others, which are in fact disconnected components. A class of approximate Schur complement preconditioners is also considered. Theoretical and numerical results are given for a model problem. In the second part we consider the application of the previous domain decomposition preconditioners to the compressible Euler equations. The discretisation procedure, based on multidimensional upwind residual distribution schemes, is outlined. We introduce a framework that combines non-linear system solvers, Krylov accelerators, domain decomposition preconditioners, as well as mesh adaptivity procedures. Several numerical tests of aeronautical interest are carried out in order to assess both the discretisation schemes and the mesh adaptivity procedures. In the third part we consider the parallel aspects inherent in the numerical solution of the compressible Euler equations on parallel computers with distributed memory. All the main kernels of the solution algorithm are analysed. Many numerical tests are carried out, with the aim of investigating the performance of the domain decomposition preconditioners proposed in the first part of the thesis, in the applications addressed in the second part.

There has been a recent emergence of many interesting and highly efficient hierarchical (multilevel) algorithms (e.g. multigrid, domain decomposition, wavelets, multilevel preconditioning, the fast multipole algorithms, etc.) for solving numerical problems in scientific computing. These algorithms derive their efficiency from using a hierarchical approach to capture the sharing of global information which is inherent in the physical processes being modelled. In addition to being computationally efficient, these algorithms also possess relatively high degrees of parallelism. I therefore argue that the architectures of parallel computers (especially massively parallel ones) should be designed to support hierarchical communication and synchronization needs of these algorithms. Hierarchical architectures are also more universal because they go beyond supporting a particular class of algorithms to supporting the underlying physical processes being modelled. Finally, I propose that algorithm designers take a critical look at traditional kernel algorithms and re-examine their cost-effectiveness in a massively parallel computing environment.

Parallel FEM based on level set method for free-surface flow using PC cluster