Band Sparse Matrix Research Articles

Structure Dynamic Analysis of C-9 Parachute Using Absolute Nodal Coordinate Formulation Thin Shell Element

In parachute fluid structure interaction (FSI) simulation, traditional parachute structure discrete methods including classic finite element method and mass-spring-damp method suffers from its large computational cost and deficiency in describing material property. This paper develops a generic shell element based on absolute nodal coordinate formulation (ANCF) to model parachute canopy to reduce computation resources and retain membrane strain/stress analyzing capability. We conduct multiple validation cases of ANCF plate and shell element and compare with current literature, in which large deformation configuration and stress distribution have been verified. A 1/4 C-9 parachute canopy inflation process by uniform pressure is simulated by discretizing into 7 ANCF shell and 84 shell element, for which the number of element discretization reduces to 1/10 than classic FEM and computational cost is considerably reduced using band sparse matrix solving technique.

A three-dimensional Petrov-Galerkin finite element interface method for solving inhomogeneous anisotropic Maxwell's equations in irregular regions

In this paper, a three-dimensional Petrov-Galerkin finite element interface (3D PGFEI) method with non-body-fitted grids is proposed for solving the inhomogeneous anisotropic time-harmonic Maxwell's equations in irregular regions. We derive the weak variational form of the anisotropic Maxwell interface problem. The irregular region is discretized by uniform tetrahedral grids while the media interface is not aligned with the grids. The level-set functions are applied to capture the discontinuity on the anisotropic media interface. Meanwhile, we construct the special basis functions according to the jump conditions across the inhomogeneous interface, where the permittivity and permeability are discontinuous three-dimensional tensors. The resulting linear system is a banded sparse matrix and demands fewer memory resources than dense matrix. Numerical experiments illustrate the efficiency and accuracy of the proposed 3D PGFEI method and the solutions can achieve second-order accuracy in terms of the relative L∞(Ω) and relative L2(Ω) error norms.

Substructure exploitation of a nonsmooth Newton method for large-scale optimal control problems with full discretization

We investigate the application of full discretization and a nonsmooth Newton method to large-scale optimal control problems. Based on a first discretize, then optimize approach, we discretize the state and control variables in time following a collocation method. Then, a nonsmooth Newton method combined with a line search globalization strategy is used to find a solution to the resulting finite-dimensional nonlinear optimization problem. In order to reduce the computational effort of solving the linear systems that arise from the application of the nonsmooth Newton method, we propose a structure exploitation strategy that results in a sparse banded matrix. We propose as well a substructure exploitation strategy based on a block LU decomposition. The different exploitation strategies combined with the use of appropriate linear solvers are demonstrated and compared for a quadratic 2D heat equation control problem discretized with the method of lines, and the approach that proved to be the most efficient is applied to a nonlinear version of the problem.

A novel Green element method by mixing the idea of the finite difference method

This paper proposes a novel Green element method (GEM) by mixing the idea of finite difference method (FDM). In the novel method, We come to the original formula when boundary integral equation is applied to an element, and use difference quotient of the central nodal value on two sides of the shared edge of adjoining elements to approximate the boundary integration ∫ΓG∇p · nds. This treatment is similar to FDM, and the integral operator relevant to element size controls the estimated error. The novel GEM makes the numerical solution correspond to the actual physical meaning, and the coefficient matrix of the global matrix is a banded sparse matrix with larger bandwidth than previous GEMs. Meanwhile, the instability of the original GEM is illuminated. We have proven it by theoretical error analysis and five numerical examples that, the accuracy of the novel GEM is three-order higher than the original GEM, and the novel GEM has a good convergence and stability, which is the property that the original GEM does not have.Indeed, the novel GEM proposed in this paper is essentially a new numerical method mixed with the idea of boundary element method (BEM), finite element method (FEM), and FDM. In contrast with BEM, FEM, FDM and previous GEM, the characteristics of our novel GEM include:(i) Compared with FEM and FDM, the novel GEM has the accuracy of BEM and can better accord with material balance.(ii) Compared with BEM, the novel GEM can solve nonlinear problems with heterogeneous media, which are hard to be handled by BEM.(iii) Compared with previous GEMs, the novel GEM has a three-order accuracy, and has a better convergence that the calculation error can be well controlled by the element size.

Large-Size Local-Domain Basis Functions with Phase Detour and Fresnel Zone Threshold for Sparse Reaction Matrix in the Method of Moments

Locality in high frequency diffraction is embodied in the Method of Moments (MoM) in view of the method of stationary phase. Local-domain basis functions accompanied with the phase detour, which are not entire domain but are much larger than the segment length in the usual MoM, are newly introduced to enhance the cancellation of mutual coupling over the local-domain; the off-diagonal elements in resultant reaction matrix evanesce rapidly. The Fresnel zone threshold is proposed for simple and effective truncation of the matrix into the sparse band matrix. Numerical examples for the 2-D strip and the 2-D corner reflector demonstrate the feasibility as well as difficulties of the concept; the way mitigating computational load of the MoM in high frequency problems is suggested.

A Compressed Diagonals Remapping Technique for Dynamic Data Redistribution on Banded Sparse Matrix

In many scientific applications, dynamic data redistribution is used to enhance algorithm performance and achieve data locality in parallel programs on distributed memory multi-computers. In this paper, we present a new method, Compressed Diagonals Remapping (CDR) technique aims to the efficiency of runtime data redistribution on banded sparse matrices. The main idea of the proposed technique is first to compress the source matrix into a Compressed Diagonal Matrix (CDM) form. Based on the compressed diagonal matrix, a one-dimensional local and global index transformation method can be applied to carry out data redistribution on the compressed diagonal matrix. This process is identical to redistribute data in the two-dimensional banded sparse matrix. The CDR technique uses an efficient one-dimensional indexing scheme to perform data redistribution on banded sparse matrix. A significant improvement of this approach is that a processor does not need to determine the complicated sending or receiving data sets for dynamic data redistribution. The indexing cost is reduced significantly. The second advantage of the present techniques is the achievement of optimal packing/unpacking stages consequent upon the consecutive attribute of column elements in a compressed diagonal matrix. Another contribution of our methods is the ability to handle sparse matrix redistribution under two disjoint processor grids in the source and destination phases. A theoretical model to analyze the performance of the proposed technique is also presented in this paper. To evaluate the performance of our methods, we have implemented the present techniques on an IBM SP2 parallel machine along with the v2m algorithm and a dense redistribution strategy. The experimental results show that our technique provides significant improvement for runtime data redistribution of banded sparse matrices in most test samples.

Stieltjes continued fraction and QD algorithm: scalar, vector, and matrix cases

The definition, in previous studies, of vector Stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series. The subject was more undefined in the matrix case. A new version of the QD algorithm for matrix problem, allows to extend to the vector and matrix cases the result of Stieltjes, expansion of a (scalar) function in terms of a Stieltjes continued fraction. Beside this connection, it solves the inverse Miura transform and gives interesting identities between general band matrix and sparse band matrix. Finally, as a consequence, we extend to some dynamical systems a method known for Toda lattices.

Stieltjes continued fraction and QD algorithm: scalar, vector, and matrix cases

The definition, in previous studies, of vector Stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series. The subject was more undefined in the matrix case. A new version of the QD algorithm for matrix problem, allows to extend to the vector and matrix cases the result of Stieltjes, expansion of a (scalar) function in terms of a Stieltjes continued fraction. Beside this connection, it solves the inverse Miura transform and gives interesting identities between general band matrix and sparse band matrix. Finally, as a consequence, we extend to some dynamical systems a method known for Toda lattices.

NEW IMPEDANCE BOUNDARY CONDITIONS IN TWO-DIMENSIONAL LTL-FD TREATMENT OF SCATTERING PROBLEMS

In this paper, two approximate and efficient boundary conditions in the two dimensional Long Transmission Line-Frequency Domain (LTL-FD) circuits [1] are introduced. The first is the circuit representation of the symmetry when passing through the centers of nodes. The second is an accurate absorbing impedance boundary condition to represent open boundaries which is proved to be an efficient absorbing boundary. The validity of these conditions is first tested in the problem of radiation by a line source. The method is then applied to the problem of plane wave scattering by conducting scatterers using the null field method. A sparse banded matrix is obtained and is solved for the required scattered field. Results for square and circular conducting scatterers are given and are compared with available analytical and numerical solutions. It is found that accurate results are obtained when the distance d between the source and the outer boundary is as small as kd ≈ 1.0, where k is the wavenumber. This considerably reduces the matrix dimensions and hence the computation time.

Vector Stieltjes Continued Fraction and Vector QD Algorithm

The definition, in previous studies, of vector Stieltjes continued fractions in connexion with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series. A new version of the vector QD algorithm allows to extend to the vector case the result which was known for one scalar function. Beside this connexion, it solves the inverse Miura transform and gives interesting identities between general band matrix and sparse band matrix. It gives also some effective computations of coefficients linked to vector orthogonality.

An efficient mixed algorithm of L-MEI and DDM for the wave scattering by a concave cylinder

In this paper, a localized MEI method (L-MEI) is developed and combined with the domain decomposition method (DDM) for the simulation of scattering by a concave cylinder. In the L-MEI, the whole domain is decomposed into many subdomains. Different from the conventional MEI method, the MEI coefficients of the L-MEI method in each subdomain are only dependent on the localized metrons that are defined in the subdomain. The localization of metrons has the following advantages: (1) speeding up the calculation of MEI coefficients and saving memory, (2) making the MEI method available for concave structures, and (3) obtaining a band sparse matrix directly without any modification.

Numerical study on the vortex motion patterns around a rotating circular cylinder and their critical characters

A hybrid finite difference and vortex method (HFDV), based on the domain decomposition method (DDM), is used for calculating the flow around a rotating circular cylinder at Reynolds number Re=1000, 200 and the angular-to-rectilinear speed ratio α∈(0.5, 3.25) respectively. A fully implicit third-order eccentric finite difference scheme is adopted in the finite difference method, and the deduced large broad band sparse matrix equations are solved by a highly efficient modified incomplete LU decomposition conjugate gradient method (MILU-CG). The long-time, fully developed features about the variations of the vortex patterns in the wake, as well as the drag and lift forces on the cylinder, are given. The calculated streamline contours are in good agreement with the experimentally visualized flow pictures. The existence of the critical state is confirmed again, and the single side shed vortex pattern at the critical state is shown for the first time. Also, the optimized lift-to-drag force ratio is obtained near the critical state. Copyright © 1999 John Wiley & Sons, Ltd.

Highly accurate calculation of the energy levels of the molecular ion

In this paper we present a new numerical method for the calculation of energy levels of the non-relativistic molecular ion , which is `exact', i.e. beyond the Born-Oppenheimer approximation. It relies on the choice of a suitable basis using the dynamical symmetries of the system, in which the Hamiltonian is a sparse banded matrix. The numerical diagonalization of the Hamiltonian produces well converged energy levels, with a typical accuracy of , and also highly accurate wavefunctions.

Numerical Solution of the Schrödinger Equation in a Wavelet Basis for Hydrogen-like Atoms

An iterative method is proposed to solve the Schrödinger eigenvalue problem in a wavelet framework. Orthonormal wavelets are used to represent the corresponding operator as a sparse band matrix. This representation, called the nonstandard (NS) form, is obtained by means of the Beylkin--Coifman--Rokhlin (BCR) algorithm and simplifies the numerical calculations. Problems due to the one-dimensional mathematical model and to the discretization process receive special attention.

A recursive doubling algorithm for inverting tridiagonal matrices

Evans [2, 3] introduced the method of recursive point partitioning algorithm for the solution of sparse banded matrix systems and investigated the “one-line at a time” strategy for the solution of tridiagonal linear systems. Recursive block partitioning schemes resulting from variation in the size of the block structure using “two-lines at a time” have been investigated for both the tridiagonal and the quindiagonal matrix systems in Okolie [6]. The case of partitioning strategy for an nth order system has been considered by Evans and Okolie [4] resulting in a recursive decoupling algorithm for tridiagonal linear systems. Following the recursive point partitioning algorithm of Evans [2, 3], Chawla et al [1] developed a recursive partitioning algorithm for inverting tridiagonal matrices. In the present paper we present a method for inverting tridiagonal matrices by adopting the strategy resulting in a recursive doubling algorithm; the present algorithm has a highly parallel structure.

Program F. MATINV for the solution of the sparse linear system [formula omitted] including normalisation

A program F. MATINV for the solution of the sparse linear system Ax = b has been designed by Zisserman and Caldwell. This paper describes an amendment to the program which incorporates an extra section in the Gaussian elimination subroutine MATINV. The aim is to make further savings on computer storage requirements by introducing the process of row normalisation. A comparison of CPU and storage requirements is included for a sparse band matrix with a range of dimensions.