Convergence Of Iterative Solution Research Articles

A VSC-based Model for Power Flow Assessment of Multi-terminal VSC-HVDC Transmission Systems

This paper puts forward a new practical voltage source converter (VSC) based AC-DC converter model suitable for conducting power flow assessment of multi-terminal VSC-based high-voltage direct current (VSC-MTDC) systems. The model uses an advanced method to handle the operational limits and control modes of VSCs into the power flow formulation. The new model is incorporated into a unified framework encompassing AC and DC power grids and is solved by using the Newton-Raphson method to enable quadratically convergent iterative solutions. The use of complementarity constraints, together with the Fischer-Burmeister function, is proposed to enable the seamless incorporation of operational control modes of VSC and automatic enforcement of any converter's operational limits that become violated during the iterative solution process. Thus, a dedicated process for checking limits is no longer required. Furthermore, all existing relationships between the VSC control laws and their operational limitsare considered directly during the solution of the power flow problem. The applicability of the new model is demonstrated with numerical examples using various multi-terminal AC-DC transmission networks, one of which is a utility-sized power system.

Optimal grid size for precipitators using finite difference method based on full multi-grid method

The paper presents a novel approach for tracking the optimal grid size of the computational domain for modeling the corona problem within the wire-duct precipitators, which in turn helps in decreasing the experimental efforts. The Finite Difference Method (FDM) is used to model the corona problem using the full multi-grid method (FMG) as a powerful convergent iterative solution for Poisson equation particularly on finer computational domains. The full multi-grid method is examined against successive over relaxation (SOR) strategy and the latter is effectively transcendent in terms of timing performance. Indeed, using finer grids is a double ended weapon; on one hand it reduces the truncation error of the Finite Difference Method which reflects in getting more accurate view for the corona problem in precipitators. While on the other hand, the round off error will be increased which might give un-accurate results. Accordingly, the issue of choosing the optimal grid size arises. The full multi-grid method tracked the optimal grid size that gives the appropriate results for the potential and current density that well matched the previous published experimental measurements.

Analysis of heat generation/absorption in thermally stratified Sutterby fluid flow with Cattaneo–Christov theory

In this study, heat generation (or absorption) in flow of stagnant Sutterby fluid past over a linearly shrinking sheet is analyzed. Heat transfer features are explored by implementing thermal stratification phenomenon. Cattaneo–Chrostov theory is also implemented to analyze the heat flow behavior instead of conventional law of Fourier. The constitutive flow laws are transmuted into non-linear dimensionless ordinary differential equations by the suitable variables. Homotopic procedure is adopted to solve the flow equations for the convergent iterative solutions. The behavior of fluid temperature and velocity profile are analyzed and discussed graphically for the eminent parameters involved in the governing laws. In this attempt the major found is that decreasing trend is observed in temperture distribution for thermal stratification parameter, heat generation (or absorption) paramere and thermal relaxation parameter. Morover, graphical results illustrate that behavior of velocity and temperature distributions are more dominant for $$\hbox {n}=2$$ as compared to $$\hbox {n}=1$$ .

Fast corona discharge solver for precipitators using multi‐grid methods on fine grids

A novel approach for modelling the corona problem in the wire-duct precipitators using the finite difference technique integrated with multi-grid methods is adopted in this study. The multi-grid method is applied as a fast convergent iterative solution for the finite difference technique to solve Poisson equation especially on finer grids. Two schemes of the multi-grid method, mainly the V-cycle and the full multi-grid method, are adopted in the present study. Compared with Gauss–Seidel method, the above-mentioned schemes are successfully transcendent owing to timing performance. By using finer grids, the proposed algorithm allows to get a more accurate picture about the performance of the precipitators in the design stage without suffering from the excessive computational time. Accurate results for the potential and current density computations, closed to the previous published experimental measurements, are obtained in comparison with earlier numerical techniques for several design parameters of the precipitators. Finally, the effect of changing the effective ion mobility and the surface roughness factor on the voltage–current density is considered.

VSIE-Based Domain Decomposition Method With Simplified Prism Vector Basis Functions for Planar Thin Dielectric-Conductor Composite Objects

A nonoverlapping and nonconforming volume-sur-face integral equation (VSIE) domain decomposition method (DDM) using simplified prism vector (SPV) basis functions is presented for complex electrically large composite planar thin dielectric-perfect electric conductor objects. Compared with the recently developed SPV-based VSIE, this scheme considerably accelerates the convergence of iterative solutions by an effective preconditioner resulted from the DDM. Moreover, this DDM does not require imposing continuity conditions at the interfaces between adjacent subdomains, and the use of discontinuous Galerkin approach allows the nonconforming volume discretizations. Due to the nonconforming nature, subdomains can be built up separately, resulting in a flexible and convenient geometrical modeling. For composite subdomains involving thin dielectric layers, the full/half SPV basis is applied so that, in comparison with the generally used Schaubert–Wilton–Glisson basis, the number of unknowns is remarkably reduced. Several numerical examples are reported to illustrate the efficiency and fast convergence of this scheme.

VoxHenry: FFT-Accelerated Inductance Extraction for Voxelized Geometries

VoxHenry, a fast Fourier transform (FFT)-accelerated integral-equation-based simulator for extracting frequency-dependent inductances and resistances of structures discretized by voxels, is presented. VoxHenry shares many features with the popular inductance extractor, FastHenry. Just like FastHenry, VoxHenry solves a combination of the electric volume integral equation and the current continuity equation, but with three distinctions that make VoxHenry suitable and extremely efficient for analyzing voxelized geometries: 1) it leverages a carefully selected set of piecewise-constant and piecewise-linear basis functions; 2) it exploits FFTs to accelerate the matrix–vector multiplications during the iterative solution of system of equations; and 3) it employs a sparse preconditioner to ensure the rapid convergence of iterative solution. VoxHenry is capable of accurately computing frequency-dependent inductances and resistances of arbitrarily shaped and large-scale structures on a desktop computer. The accuracy, efficiency, and applicability of VoxHenry are demonstrated through inductance analysis of various structures, including square and circular coils as well as arrays of RF inductors (situated over ground planes).

Adaptive approximation of the Monge–Kantorovich problemviaprimal-dual gap estimates

The Monge–Kantorovich problem arises as a special case for linear cost functionals in optimal transportation problems. It leads to a convex minimization problem with limited regularity properties. The convergent finite element discretization and iterative solution of the problem and its dual are addressed. Based on these approximations a computable upper bound for the primal-dual gap is derived which is suitable for efficient local mesh refinement. Numerical experiments reveal a significant improvement of related adaptive methods.

Current-Based Volume Integral Equation Formulation for Bianisotropic Materials

The electric and magnetic current volume integral equation formulation is developed for the electromagnetic scattering analysis of arbitrarily shaped 3-D bianisotropic objects. The eigenvalue spectrum of the operator is derived and analyzed in the case of bi-isotropic media. The effect of inhomogeneity and bianisotropy on the spectrum and convergence of iterative solutions is investigated numerically.

Analytical and computational methods for a class of nonlinear singular integral equations

We consider a class of nonlinear singular Hammerstein Volterra integral equations. In general, these equations will have kernels containing both an end point and an Abel-type singularity, with exact solutions being typically nonsmooth. Under certain conditions, a uniformly convergent iterative solution is obtained on a small interval near the origin. In this work, two product integration methods are proposed and analyzed where the integral over a small initial interval is calculated analytically, allowing the optimal convergence rates to be achieved. This is illustrated by some numerical examples.

Parameter investigation with line-implicit lower–upper symmetric Gauss–Seidel on 3D stretched grids

An implicit lower–upper symmetric Gauss–Seidel (LU-SGS) solver has been implemented as a multigrid smoother combined with a line-implicit method as an acceleration technique for Reynolds-averaged Navier–Stokes (RANS) simulation on stretched meshes. The computational fluid dynamics code concerned is Edge, an edge-based finite volume Navier–Stokes flow solver for structured and unstructured grids. The paper focuses on the investigation of the parameters related to our novel line-implicit LU-SGSsolver for convergence acceleration on 3D RANS meshes. The LU-SGS parameters are defined as the Courant–Friedrichs–Lewy number, the left-hand side dissipation, and the convergence of iterative solution of the linear problem arising from the linearisation of the implicit scheme. The influence of these parameters on the overall convergence is presented and default values are defined for maximum convergence acceleration. The optimised settings are applied to 3D RANScomputations for comparison with explicit and line-implicit Runge–Kutta smoothing. For most of the cases, a computing time acceleration of the order of 2 is found depending on the mesh type, namely the boundary layer and the magnitude of residual reduction.

Investigation of Multigrid Preconditioner for Integral Equation Fast Analysis of Electromagnetic Scattering Problems

This paper proposes a multigrid preconditioner for integral equation fast analysis of electromagnetic scattering problems. The multigrid model is constructed by recursively dividing every triangular element on the coarse level into 4 small ones. The Rao-Wilton-Glisson (RWG) basis functions defined on the coarse level are expressed with linear combinations of those on the fine level, and the expansion coefficients of which are used to construct the restriction and interpolation operator. To further accelerate the convergence of iterative solution, a set of solenoidal basis functions on the coarse level are constructed additionally, which are also expressed with linear combinations of RWG basis functions on the fine level and adopted to construct the second restriction and interpolation operator. The two sets of coarsening techniques are then combined together to remove the low-frequency components of the iterative error. Numerical results show that the proposed method can reduce both the steps of iteration and CPU time compared with the iterative solver without preconditioning.

Leveraging Anderson Acceleration for improved convergence of iterative solutions to transport systems

In this note we demonstrate that using Anderson Acceleration (AA) in place of a standard Picard iteration can not only increase the convergence rate but also make the iteration more robust for two transport applications. We also compare the convergence acceleration provided by AA to that provided by moment-based acceleration methods. Additionally, we demonstrate that those two acceleration methods can be used together in a nested fashion. We begin by describing the AA algorithm. At this point, we will describe two application problems, one from neutronics and one from plasma physics, on which we will apply AA. We provide computational results which highlight the benefits of using AA, namely that we can compute solutions using fewer function evaluations, larger time-steps, and achieve a more robust iteration.

An iterative finite element time-domain method for simulating three-dimensional electromagnetic diffusion in earth

Geophysical Journal International An iterative finite element time-domain method for simulating three-dimensional electromagnetic diffusion in earth Evan Schankee Um, 1 Jerry M. Harris 2 and David L. Alumbaugh 3 1 Earth Sciences Division, Geophysics Department, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. E-mail: evanum@gmail.com USA 3 Chevron Energy Technology, 6001 Bollinger Canyon Road, San Ramon, CA 94583, USA 2 Geophysics Department, Stanford University, Stanford, CA, SUMMARY An iterative finite element time-domain (FETD) method has been developed for simulating transient electromagnetic fields in 3-D diffusive earth media and has been verified through comparisons with analytic and finite-difference time-domain solutions. The adaptive time step doubling (ATSD) method plays an important role in reducing solution run-time by allowing large time steps in late time when high-frequency electric fields are increasingly attenuated in the Earth. We demonstrate that for the ATSD method to work effectively, the conductivity of the air and a drop tolerance of a preconditioner should be carefully selected. The conductivity of the air should not be too large for accurate simulations but also not too small for avoid- ing ill conditioning that results in error amplification in the ATSD method. A proper drop tolerance keeps the eigenvalues of the preconditioned FETD matrices clustered when a time step size is successively doubled during the ATSD processes, resulting in the convergence of iterative solutions with the reasonable number of iterations. A rule of thumb for determining the conductivity of the air and the drop tolerance has been presented. We also present the simultaneous multiple-sources modelling (SM 2 ) approach. The SM 2 approach simultaneously advances the electric fields excited by multiple individual sources in a single time stepping loop. This approach allows multiple sources to share the same preconditioner in the time stepping loop and improves the simulation efficiency per a survey line. Key words: Numerical solutions; Electromagnetic theory; Marine electromagnetics. I N T R O D U C T I O N Transient electromagnetic (TEM) methods have been widely used in a wide range of geophysical research such as deep crustal stud- ies (e.g. Ho¨ rdt et al. 1992; Ho¨ rdt et al. 2002), energy explorations (e.g. Strack 1992), environmental investigations (e.g. Auken et al. ´ rnason et al. 2010). In- 2006) and geothermal explorations (e.g. A terpretations of TEM data in complex geological environments in- creasingly resort to multidimensional inverse modelling. Because forward modelling is a major numerical bottleneck for the inverse modelling, it is important to develop efficient and accurate forward modelling algorithms. Introduced by Yee (1966) in electrical engineering, finite- difference time-domain (FDTD) methods have become one of the standard tools used to simulate TEM fields in geophysics. Their popularity is mainly based on the fact that FDTD methods are relatively easy to implement, easy to use and can provide rea- sonably accurate solutions over a wide range of geophysics prob- lems. For example, Goldman & Stoyer (1983) present 2-D finite- difference modelling of TEM fields in simplified axially symmetric earth media. Wang & Hohmann (1993) develop a 3-D FDTD al- gorithm that advances TEM fields in time using an explicit time stepping approach. Commer & Newman (2004) introduce the par- allel version of Wang & Hohmann (1993) along with new algo- rithmic features such as parallel upward continuation procedures and parallel time stepping. The parallel version has been em- bedded to a 3-D TEM inversion algorithm (Newman & Commer Despite their popularity, the 3-D FDTD methods also have well known disadvantages. For example, their explicit time stepping ap- proach requires small time step sizes to satisfy stability conditions especially when there are large conductivity contrasts. To overcome this issue, Haber et al. (2004) use an unconditionally stable implicit time discretization scheme in their finite volume approach (we also employ an implicit time discretization scheme in this paper). Be- sides, in FDTD modelling, complex geological structures such as seafloor bathymetry, reservoirs and salt domes need to be approx- imated by small stair steps when the structures do not confirm to rectangular cells. Therefore, this approximation approach can quickly increase FDTD problem sizes; FDTD problems resulting from realistic 3-D earth models are usually solved in massively parallel computing environments (Commer & Newman 2004; Newman & Commer 2005; Commer et al. 2008). Furthermore, such stair steps can introduce discretization errors into numerical modelling results especially when sources and receivers are placed on or very close to the complex surface described by the stair steps.

AN EFFICIENT HIGH ORDER MULTILEVEL FAST MULTIPOLE ALGORITHM FOR ELECTROMAGNETIC SCATTERING ANALYSIS

An e-cient higher order MLFMA is developed by using an \extended-tree. With this extended-tree, the size of the box at the flnest level is reduced, and the cost associated with the aggregation and disaggregation operations is signiflcantly decreased. The sparse approximate inverse (SAI) preconditioner is utilized to accelerate the convergence of iterative solutions. The Cholesky factorization, instead of the often used QR factorization, is employed to construct the SAI preconditioner for cavity scattering analysis, by taking advantage of the symmetry of the matrix arising from electric fleld integral equation. Numerical experiments show that the higher order MLFMA is more e-cient than its low-order counterpart.

Accelerating convergence in iterative solution for large‐scale complete active space self‐consistent‐field calculations

AbstractAn algorithm that accelerates the convergence of the iterative optimization of the complete active space self‐consistent field (CASSCF) wavefunction so as to find a optimum solution in fewer macroiterations is described. The algorithm is oriented to large‐scale CASSCF problems that are to be solved with a combination of density matrix renormalization group (DMRG) method for the configuration interaction (CI) process. The algorithm is based on the alternating (or two‐step) CASSCF optimization in which the CI and molecular orbital (MO) parameters are optimized separately. Convergence ratio is improved by finding further optimized MOs from a linear extrapolation of the MO sets of the iteration history. The acceleration results in fewer diagonalizations in a total CASSCF calculation to save a considerable computational cost. The convergence performance is examined in a couple of realistic applications on SiC3 and poly(phenyl)carbenes. For poly(phenyl)carbenes, the large‐size CASSCF calculations with CAS(30e,30o) that entails full π valence space as well as sp2 orbital space of carbenes are performed by using the practical implementation of DMRG‐CASSCF in conjunction with the acceleration technique. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Divide-and-conquer density functional theory on hierarchical real-space grids: Parallel implementation and applications

A linear-scaling algorithm based on a divide-and-conquer (DC) scheme has been designed to perform large-scale molecular-dynamics (MD) simulations, in which interatomic forces are computed quantum mechanically in the framework of the density functional theory (DFT). Electronic wave functions are represented on a real-space grid, which is augmented with a coarse multigrid to accelerate the convergence of iterative solutions and with adaptive fine grids around atoms to accurately calculate ionic pseudopotentials. Spatial decomposition is employed to implement the hierarchical-grid DC-DFT algorithm on massively parallel computers. The largest benchmark tests include $11.8\ifmmode\times\else\texttimes\fi{}{10}^{6}$-atom ($1.04\ifmmode\times\else\texttimes\fi{}{10}^{12}$ electronic degrees of freedom) calculation on 131 072 IBM BlueGene/L processors. The DC-DFT algorithm has well-defined parameters to control the data locality, with which the solutions converge rapidly. Also, the total energy is well conserved during the MD simulation. We perform first-principles MD simulations based on the DC-DFT algorithm, in which large system sizes bring in excellent agreement with x-ray scattering measurements for the pair-distribution function of liquid Rb and allow the description of low-frequency vibrational modes of graphene. The band gap of a CdSe nanorod calculated by the DC-DFT algorithm agrees well with the available conventional DFT results. With the DC-DFT algorithm, the band gap is calculated for larger system sizes until the result reaches the asymptotic value.

Convergent iterative solutions of Schroedinger equation for a generalized double well potential

We present an explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with a generalized double well potential V = g 2 2 ( x 2 - 1 ) 2 ( x 2 + a ) . The condition for the convergence of the iteration procedure and the dependence of the shape of the groundstate wave function on the parameter a are discussed.

Improving Conditioning of Electromagnetic Surface Integral Equations Using Normalized Field Quantities

When the surface integral equation method is applied to study electromagnetic scattering by dielectric or composite metallic and dielectric objects, the unknowns, i.e., the electric and magnetic surface current densities, and the elements of the system matrix, are often of the very different scales. As a consequence, the system matrix may have a high (singular value) condition number. An efficient method is presented to balance the unknowns and the integral equations, and the elements of the system matrix, too. The method is based on the use of normalized field quantities and unknowns, and carefully chosen scaling factors. In the case of dielectric and composite objects the condition numbers of the SIE matrices can be reduced with several orders of magnitudes by the developed method. In the case of high contrast objects, or if the frequency is very low, the developed method leads also to a clear improvement on the convergence of iterative solutions

An Improved Finite Element Computational Scheme for Transient Field-Circuit Coupled Systems

Time stepping simulation of transient field-circuit coupled problems is a computation-intensive area. In general, finite element (FE) modeling and simulation of a large class of engineering applications require heavy computations that are often repetitive in nature. Any deeper insight into the dependence of the FE solution or its characterization may allow significant reduction of computational efforts. This paper illustrates the possibility of characterization of the time-varying FE solutions for transient field-circuit coupled problems. The characterization leads to a significantly smaller FE model and an improved modeling and simulation scheme for such problems. Construction of the reduced FE model for the field is illustrated, along with its coupling to a general external electric circuit. Relation of the proposed approach to the well-known substructural approach based on Schur Complements is discussed. Possibilities of further improvement of the numerical properties of the FE equations, possibly leading to improved convergence of iterative solutions, are indicated. Finally, extension of the approach to nonlinear field-circuit coupled problems is illustrated. Numerical simulations using the proposed approach for a typical electromagnetic system with solid bars connected to external circuit are presented. The results are compared to those from a commercial software, and a relative assessment of the computational effort involved in the proposed and conventional methods is also presented.

Convergent iterative solutions for a Sombrero-shaped potential in any space dimension and arbitrary angular momentum

We present an explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with an N-dimensional radial potential V = g 2 2 ( r 2 - 1 ) 2 and an angular momentum l. For g large, the rate of convergence is similar to a power series in g −1.