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## Non-linear Iteration (non-linear + iteration)
## Selected Abstracts## Spectral estimation on a sphere in geophysics and cosmology GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2008F. A. DahlenSUMMARY We address the problem of estimating the spherical-harmonic power spectrum of a statistically isotropic scalar signal from noise-contaminated data on a region of the unit sphere. Three different methods of spectral estimation are considered: (i) the spherical analogue of the one-dimensional (1-D) periodogram, (ii) the maximum-likelihood method and (iii) a spherical analogue of the 1-D multitaper method. The periodogram exhibits strong spectral leakage, especially for small regions of area A, 4,, and is generally unsuitable for spherical spectral analysis applications, just as it is in 1-D. The maximum-likelihood method is particularly useful in the case of nearly-whole-sphere coverage, A, 4,, and has been widely used in cosmology to estimate the spectrum of the cosmic microwave background radiation from spacecraft observations. The spherical multitaper method affords easy control over the fundamental trade-off between spectral resolution and variance, and is easily implemented regardless of the region size, requiring neither non-linear iteration nor large-scale matrix inversion. As a result, the method is ideally suited for most applications in geophysics, geodesy or planetary science, where the objective is to obtain a spatially localized estimate of the spectrum of a signal from noisy data within a pre-selected and typically small region. [source] ## Regional teleseismic tomography of the western Lachlan Orogen and the Newer Volcanic Province, southeast Australia GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2002Frank M. GraeberSummary From 1998 May to September a portable array of 40 short-period digital seismograph stations was operated in western Victoria, southeast Australia, across the western end of the mid-Paleozoic Lachlan Foldbelt and the Newer Volcanic Province. Consisting of four parallel, almost W,E-oriented receiver lines, the array covered an area of about 270 × 150 km2. The major aim of the LF98 (Lachlan Foldbelt survey 1998) project is to map lateral variations in P -wave speeds (Vp) in the crust and upper mantle using teleseismic arrival time tomography, primarily in order to investigate whether the major surface structural zones are associated with seismic velocity signatures at depth. Little a priori information from seismic profiling is available. We invert 4067 relative arrival time residuals for a minimum structure Vp model in the upper few hundred km using non-linear iteration and 3-D ray tracing. The most prominent negative anomaly (,3.8 per cent) in Vp is found at a depth of about 45 km underneath the eastern part of the Newer Volcanic Province. It correlates spatially with the highest density of Pliocene and Pleistocene eruption centres northwest of Melbourne, and is therefore interpreted as a hotspot-related high-temperature anomaly causing reduced mantle velocities. The related coherent volume of significantly lower than average velocities extends down to depths greater than 100 km in the east, and extends west underneath the Newer Volcanic Province. A strong velocity contrast, with average velocities ,2 per cent greater in the west, is found down to about 100 km across the Moyston Fault Zone, which forms the major structural boundary between the early-Paleozoic Delamerian Orogen in the west and the Lachlan Orogen in the east. This result suggests that the Moyston Fault Zone should be seen as a major lithospheric boundary. In the south this boundary is also expressed by a distinct discontinuity in Sr-isotopic ratios of xenoliths (the so-called Mortlake discontinuity) and a change in the geochemistry of plutons of similar age. However, if the east to west velocity contrast originally existed in this southern zone, it is now overprinted by the thermally reduced mantle velocities beneath the Newer Volcanic Province. [source] ## Compressible flow SUPG parameters computed from element matrices INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2005L. CatabrigaAbstract We present, for the SUPG formulation of inviscid compressible flows with shocks, stabilization parameters defined based on the element-level matrices. These definitions are expressed in terms of the ratios of the norms of the matrices and take into account the flow field, the local length scales, and the time step size. Calculations of these stabilization parameters are straightforward and do not require explicit expressions for length or velocity scales. We compare the performance of these stabilization parameters, accompanied by a shock-capturing parameter introduced earlier, with the performance of a stabilization parameter introduced earlier, accompanied by the same shock-capturing parameter. We investigate the performance difference between updating the stabilization and shock-capturing parameters at the end of every time step and at the end of every non-linear iteration within a time step. We also investigate the influence of activating an algorithmic option that was introduced earlier, which is based on freezing the shock-capturing parameter at its current value when a convergence stagnation is detected. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Time,accurate Modular CFD-CSD Coupling for Aeroelastic Rotor Simulations PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003A. Altmikus Dipl.-Ing.This paper addresses the timewise accuracy of different coupling approaches applied to instationary aeroelastic simulations of rotors in forward flight. Two different approaches which are widely discussed in literature are examined: the tight or strong coupling, and the fully integrated or monolithic coupling. Strong coupling means an exchange of fluid loads and structural deformations at each time step which is effectuated in a fully modular manner. We will address aspects of conservativity and time-accuracy, and will present results for a helicopter forward flight scenario. However, objections concerning the correct solution of the global non-linear three field problem , structure, grid deformation, aerodynamics , remain. These objections are normally rejected by the monolithic approach. Here, a common set of partial differential equations is derived and solved in a single code. However, a truly monolithic system of equations is only needed for stability analysis, and it can be decomposed in a three field problem respecting appropriate boundary conditions for each domain. Thus, modularity can be maintained, conceiving a quasi-monolithic procedure, when both domains are simultaneously solved in a common non-linear iteration loop on a per time-step basis. First results will be shown for a 2D flutter testcase. [source] ## Design spaces, measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design,illustration to structural dynamics INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2005X. ZhouAbstract For the first time, for time discretized operators, we describe and articulate the importance and notion of design spaces and algorithmic measures that not only can provide new avenues for improved algorithms by design, but also can distinguish in general, the quality of computational algorithms for time-dependent problems; the particular emphasis is on structural dynamics applications for the purpose of illustration and demonstration of the basic concepts (the underlying concepts can be extended to other disciplines as well). For further developments in time discretized operators and/or for evaluating existing methods, from the established measures for computational algorithms, the conclusion that the most effective (in the sense of convergence, namely, the stability and accuracy, and complexity, namely, the algorithmic formulation and algorithmic structure) computational algorithm should appear in a certain algorithmic structure of the design space amongst comparable algorithms is drawn. With this conclusion, and also with the notion of providing new avenues leading to improved algorithms by design, as an illustration, a novel computational algorithm which departs from the traditional paradigm (in the sense of LMS methods with which we are mostly familiar with and widely used in commercial software) is particularly designed into the perspective design space representation of comparable algorithms, and is termed here as the forward displacement non-linearly explicit L-stable (FDEL) algorithm which is unconditionally consistent and does not require non-linear iterations within each time step. From the established measures for comparable algorithms, simply for illustration purposes, the resulting design of the FDEL formulation is then compared with the commonly advocated explicit central difference method and the implicit Newmark average acceleration method (alternately, the same conclusion holds true against controllable numerically dissipative algorithms) which pertain to the class of linear multi-step (LMS) methods for assessing both linear and non-linear dynamic cases. The conclusions that the proposed new design of the FDEL algorithm which is a direct consequence of the present notion of design spaces and measures, is the most effective algorithm to-date to our knowledge in comparison to the class of second-order accurate algorithms pertaining to LMS methods for routine and general non-linear dynamic situations is finally drawn through rigorous numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## A-scalability and an integrated computational technology and framework for non-linear structural dynamics. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003Part 1: Theoretical developments, parallel formulationsAbstract For large-scale problems and large processor counts, the accuracy and efficiency with reduced solution times and attaining optimal parallel scalability of the entire transient duration of the simulation for general non-linear structural dynamics problems poses many computational challenges. For transient analysis, explicit time operators readily inherit algorithmic scalability and consequently enable parallel scalability. However, the key issues concerning parallel simulations via implicit time operators within the framework and encompassing the class of linear multistep methods include the totality of the following considerations to foster the proposed notion of A-scalability: (a) selection of robust scalable optimal time discretized operators that foster stabilized non-linear dynamic implicit computations both in terms of convergence and the number of non-linear iterations for completion of large-scale analysis of the highly non-linear dynamic responses, (b) selecting an appropriate scalable spatial domain decomposition method for solving the resulting linearized system of equations during the implicit phase of the non-linear computations, (c) scalable implementation models and solver technology for the interface and coarse problems for attaining parallel scalability of the computations, and (d) scalable parallel graph partitioning techniques. These latter issues related to parallel implicit formulations are of interest and focus in this paper. The former involving parallel explicit formulations are also a natural subset of the present framework and have been addressed previously in Reference 1 (Advances in Engineering Software 2000; 31: 639,647). In the present context, of the key issues, although a particular aspect or a solver as related to the spatial domain decomposition may be designed to be numerically scalable, the totality of the aforementioned issues simultaneously play an important and integral role to attain A-scalability of the parallel formulations for the entire transient duration of the simulation and is desirable for transient problems. As such, the theoretical developments of the parallel formulations are first detailed in Part 1 of this paper, and the subsequent practical applications and performance results of general non-linear structural dynamics problems are described in Part 2 of this paper to foster the proposed notion of A-scalability. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction,convection,diffusion processes INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2005A. M. P. ValliAbstract We propose two timestep selection algorithms, based on feedback control theory, for finite element simulation of steady state and transient 2D viscous flow and coupled reaction,convection,diffusion processes. To illustrate performance of the schemes in practice, we solve Rayleigh,Benard,Marangoni flows, flow across a backward-facing step, unsteady flow around a circular cylinder and chemical reaction systems. Numerical experiments confirm that the feedback controllers produce in some cases a very smooth stepsize variation, suggesting that robust control algorithms are possible. These experiments also show that parameter selection can improve timesteps when co-ordinated with the convergence control of non-linear iterations. Further, computational cost of the selection procedures is negligible, since they involve only storing a few extra vectors, computation of norms and evaluation of kinetic energy. Copyright © 2004 John Wiley & Sons, Ltd. [source] |