Sparse Matrix Techniques Research Articles

Real-time analysis of mobile machines using sparse matrix technique

The use of modern multibody simulation techniques enables the description of complex products, such as mobile machinery, with a high level of detail while still solving the equations of motion in real time. Using the appropriate modelling and implementation techniques, the accuracy of real-time simulation can be improved considerably. Conventionally, in multibody system dynamics, equations of motion are implemented using the full matrices approach that does not consider the sparsity feature of matrices. With this implementation approach, numerical efficiency decreases when sparsity increases. In this study, a numerical procedure based on semi-recursive and augmented Lagrangian methods for real-time dynamic simulation is introduced. To enhance computing efficiency, an equation of motion is implemented by employing the sparse matrix technique.

A sparse stiff chemistry solver based on dynamic adaptive integration for efficient combustion simulations

A sparse stiff chemistry solver based on dynamic adaptive hybrid integration (AHI-S) is developed and demonstrated for efficient combustion simulations. In a previous study, a dynamic adaptive method for hybrid integration (AHI) was developed to speed up the time integration of chemically reacting flows with detailed chemistry. The AHI method solves the fast subcomponent of chemistry implicitly and the slow subcomponent of chemistry and transport explicitly, and it was shown that AHI is more accurate and efficient than the operator-splitting schemes when there are significant radical sources from the transport term. In the present study, the AHI method is first improved to minimize the number of nontrivial entries in the Jacobian. Sparse matrix techniques are further integrated into AHI to achieve high computational efficiency. The performance of the new AHI-S solver is investigated in constant-pressure auto-ignition systems using different mechanisms that consist of 9–2878 species. It is shown that the computational cost of the AHI-S solver is overall linearly proportional to the mechanism size and is comparable to that of evaluating reaction rates using CHEMKIN-II subroutines. The AHI-S solver achieves speed-up factors ranging from approximately 10, for the 9-species hydrogen mechanism, to approximately 3000, for the 2878-species biodiesel mechanism, compared with the fully implicit VODE solver with Jacobian evaluated through numerical perturbations and factorized with dense matrix operations. It is further found that for mechanisms with less than approximately 100 species, the time saving of AHI-S is primarily attributed to the reduced size of the implicit core of the governing equations, while for mechanisms with more than 100 species, the computational cost of VODE is dominated by the dense LU factorization, such that the time saving of AHI-S is mostly attributed to the sparse LU factorization. The AHI-S solver is then applied to unsteady perfectly stirred reactors involving extinction and re-ignition. Speed-up factors from 50 to 30,000 are achieved compared with the Strang splitting scheme with the chemistry substeps implicitly integrated with VODE, while speed-up factors of 10–100 are achieved compared with the Strang splitting scheme implemented with the sparse stiff LSODES solver. In the end, the performance of AHI-S is investigated in one-dimensional (1-D) unsteady freely propagating laminar premixed flames for a methane/air mixture, for which the time step size in AHI-S is limited by the fastest transport process. A speed-up factor of approximately 200 is achieved compared with the Strang splitting scheme for fixed time step sizes between 10−8s and 10−6s.

Three‐phase power flow calculations using direct Z BUS method for large‐scale unbalanced distribution networks

In this study, a three-phase power flow solution method using graph theory, injection current, and sparse matrix techniques for large-scale unbalanced distribution networks is proposed. In the bus frame of reference, a direct iterative method is adopted. To integrate the electric characteristics of transformers and step voltage regulators into the proposed method, the existing component models are modified by equivalent injected currents. To validate the performance and effectiveness of the proposed method, four three-phase IEEE test systems and random test systems are used for comparison purposes. As the size of the network increases, the proposed direct Z BUS method drastically shows its superiority over the other methods. The results reveal that the proposed method has good potential for improving the computational efficiency of optimal planning and design as well as real-time power dispatch applications in large-scale distribution systems.

Solving Scattering From Multiple Bodies of Revolution by Modal Characteristic Basis Function Method With Sparse Matrix Technique

Modal Characteristic Basis Function method for bodies of revolution (BoR-CBFM) is a fast solver for scattering from multiple BoRs. By using of BoR-CBFM, the modal characteristic basis functions (MCBFs) are constructed, and the size of impedance matrix is greatly reduced. However, the matrix filling, especially the filling of mutual-interaction matrix, is still a huge burden for numerical computation. To further enhance the efficiency of BoR-CBFM and reduce the cost of matrix filling, we present a sparse technique for BoR-CBFM. Through this technique, those weak elements in mutual-interaction matrix that contribute little to the linear system are neglected, and a sparse impedance matrix is achieved. As a result, this technique leads to a reduction on the cost of matrix filling. Numerical results are presented to prove the efficiency of this technique.

Numerical methods for spin-dependent transport calculations and spin bound states analysis in Rashba waveguides

Numerical methods are developed in the quantum transport calculations for electron in the waveguides with spin–orbital (Rashba) interaction. The methods are based on a hybrid mode-matching scheme in which the wavefunctions are expressed as the superposition of eigenmodes in the lead regions and in the device region the wavefunction is expressed on the discrete basis. Two versions are presented for the lead without and with the Rashba interaction. In the latter case the eigenmodes are obtained from a quadratic eigenproblem calculation. These methods are suitable for the systems with variable geometries or arbitrary potential profiles. The computation can be effectively accelerated by the sparse matrix technique. We also investigate the Fano–Rashba bound states in the Rashba waveguides by some nonlinear eigenstate calculation. This calculation is based on a mode-matching method and self-consistent results are obtained in our calculations.

Real-Time Dynamic Simulations of Large Road Vehicles Using Dense, Sparse, and Parallelization Techniques

This article presents three multibody formulations with improved efficiency in order to achieve real-time simulations for the forward dynamic of two real-life road vehicles. The bigger is a semitrailer truck with 40 degrees of freedom (DOF). Two topological and semirecursive formulations are used as well as a global formulation based on the use of Euler parameters and flexible joints. The first semirecursive formulation carries out a double velocity transformation and the integration is done by means of the explicit fourth order Runge–Kutta method. The second semirecursive formulation and the global one use a penalty scheme at position level and orthogonal projections at velocity and acceleration levels. In both cases the integrator was the implicit Hilbert–Huges–Taylor (HHT) method. The double velocity transformation method involves the coordinate partitioning of the constraint Jacobian matrix which leads to the costly solution of a redundant but consistent with the constraints linear system of equations. The choice of a unique set of independent coordinates may not be valid for a complete simulation and additional repartitioning would be required. Based on previous experience and as the examples show in this article, a careful initial choice of the independent coordinates can remain valid for complete simulations involving common maneuvers. This represents a numerical advantage for dense matrix methods and can be further exploited if sparse matrix techniques are employed. This has been the case for both of the vehicles used, reaching real-time simulations even with the semitrailer truck. The implicit semirecursive formulation involves the numerical evaluation of the stiffness and damping matrices, which hamper obtaining real-time simulations. For the semitrailer truck, this computation represents the 76% of the total simulation time. The numerical computation of these matrices is carried out by columns and its algorithm is straightforwardly parallelizable. Using a quad-core processor and with a simple and efficient OpenMP implementation, it has been possible to achieve a speedup of 3.25 reducing the simulation times under the real-time limit. The sparse matrices of Euler parameters formulation show very different sparsity degrees, difference that grows with the size of the multibody model. This poses a challenge to sparse matrix implementations in order to be able to efficiently perform matrix operations without increasing fillings or handling zero entries. This has been successfully accomplished using a new sparse matrix representation. This one is not a feature of general purpose sparse software, requiring at some stages the implementation of our own algorithms. Reductions in time of three orders of magnitude have led to real-time simulations even with the semitrailer truck.

Modeling optical properties of silicon clusters by first principles: From a few atoms to large nanocrystals

Time dependent density functional tight binding (TDDFTB) method is implemented with sparse matrix techniques and improved parallelization algorithms. The method is employed to calculate the optical properties of various Si nanocrystals (NCs). The calculated light absorption spectra of small Si NCs from TDDFTB were found to be comparable with many body perturbation methods utilizing planewave basis sets. For large Si NCs (more than a thousand atoms) that are beyond the reach of conventional approaches, the TDDFTB method is able to produce reasonable results that are consistent with prior experiments. We also employed the method to study the effects of surface chemistry on the optical properties of large Si NCs. We learned that the optical properties of Si NCs can be manipulated with small molecule passivations such as methyl, hydroxyl, amino, and fluorine. In general, the shifts and profiles in the absorption spectra can be tuned with suitably chosen passivants.

Compressible modes in a square lid-driven cavity

This paper studies the effect of compressibility on the linear stability of a two-dimensional lid-driven cavity flow in the subsonic regime. The base flow is generated by high fidelity direct numerical simulation and a biglobal mode instability analysis is carried out by a matrix forming approach. The eigenvalue problem is discretized by high-order finite differences and Arnoldi algorithm is used to reduce the size of the problem. The solution procedure uses sparse matrix techniques. Influence of Mach number on the modes known from incompressible calculations is presented, showing that compressibility has a stabilizing effect. New modes that appear only for compressible flows are presented and their relationship with duct acoustics is investigated.

Sparse Matrix for ECG Identification with Two-Lead Features.

Electrocardiograph (ECG) human identification has the potential to improve biometric security. However, improvements in ECG identification and feature extraction are required. Previous work has focused on single lead ECG signals. Our work proposes a new algorithm for human identification by mapping two-lead ECG signals onto a two-dimensional matrix then employing a sparse matrix method to process the matrix. And that is the first application of sparse matrix techniques for ECG identification. Moreover, the results of our experiments demonstrate the benefits of our approach over existing methods.

Gain Matrix Distributed Computing Technique for Power System State Estimation

The Electric Power System State Estimation problem involves large sparse matrices. The Jacobian matrix is highly sparse in nature and the computational efforts can be enhanced by avoiding arithmetic operations resulting in ‘zero’. The researchers have introduced sparse matrix techniques so as to store only non-zero elements of the matrix and thereby reducing the huge dynamic memory requirements, which intern reduce the computational time. A few such techniques [2],[3], [4] are listed in the reference. The primary focuses of these sparse techniques are on the memory/storage space reduction. This paper elaborates a different technique to obtain the “effective operation” with the focus on the computational time and the storage space reduction. The “effective operation” can be achieved without applying conventional compact storage techniques to find the Jacobian product. A different style for multiplication of two large sparse Jacobian matrices is adopted to obtain this novel approach. As a result, computational time is reduced and also Jacobian array size is reduced form two dimensional array to single dimensional array. The solution gives scope for distributed/parallel computing without disturbing the network structure [6].

The Influence of Wind Power Grid to Power System Small Disturbance Stability Study

Setting up a practical power system containing wind farm small disturbance stability mathematical model, and the linearized equation of wind turbines and original equations of linear system, power system containing wind turbines augmented state formation matrix, turn the inverse iteration with PSASP software Rayleigh (Rayleigh quotient iterative algorithm and sparse matrix technique for large state matrix eigenvalues. In this paper, the Inner Mongolia power grid, wind power access to power system oscillation mode and the influence of the oscillation characteristics and meet the requirements of small disturbance stability of wind scale and operation control scheme. The influence of the wind farm is proposed for power system damping characteristics will not greater than the same capacity of synchronous generator's point of view.

Homotopy Method for Finding the Steady States of Oscillators

Shooting, finite difference, or harmonic balance techniques in conjunction with a damped Newton method are widely employed for the numerical calculation of limit cycles of (free-running, autonomous) oscillators. In some cases, however, nonconvergence occurs when the initial estimate of the solution is not close enough to the exact one. Generally, the higher the quality factor of the oscillator the tighter are the constraints for the initial estimate. A 2-D homotopy method is presented in this paper that overcomes this problem. The resulting linear set of equations is underdetermined, leading to a nullspace of rank two. This underdetermined system is solved in a least squares sense for which a rigorous mathematical basis can be derived. An efficient algorithm for solving the least squares problem is derived where sparse matrix techniques can be used. As continuation methods are only employed for obtaining a sufficient initial guess of the limit cycle, a coarse grid discretization is sufficient to make the method runtime efficient.

Node Ordering Scheme of Large Scale Power Systems Using Sparse Matrix Techniques

Power flow is the basic tool for power system analysis which reveals the system operation in a steadystate mode for evaluation of the power system planning and operations. The accuracy, simulation time, computer storage size and convergence of any model used depend largely on the size of the bus admittance matrix of the system under study. This paper, therefore, presents the study of the bus admittance matrix of the different systems with sparse techniques. And also analyze the number of zeros and non-zeros element in the different systems with minimum ordering schemes. The proposed method is validated using a 5-bus, 30-bus, 118-bus and 300-bus systems. The results are presented in graphical form and discussed. The sparse Matrix techniques show that as the system is increasing in size, the percentage of stored bus admittance elements decreases and changing the order of the nodes gives more impact on the size of the system. Thus, an appreciable reduction in the computer memory required to store the bus admittance matrix and in turn reduces the overall simulation time.

%HPGLIMMIX: A High-PerformanceSASMacro for GLMM Estimation

Generalized linear mixed models (GLMMs) comprise a class of widely used statistical tools for data analysis with fixed and random effects when the response variable has a conditional distribution in the exponential family. GLMM analysis also has a close relationship with actuarial credibility theory. While readily available programs such as the GLIMMIX procedure in SAS and the lme4 package in R are powerful tools for using this class of models, these progarms are not able to handle models with thousands of levels of fixed and random effects. By using sparse-matrix and other high performance techniques, procedures such as HPMIXED in SAS can easily fit models with thousands of factor levels, but only for normally distributed response variables. In this paper, we present the %HPGLIMMIX SAS macro that fits GLMMs with large number of sparsely populated design matrices using the doubly-iterative linearization (pseudo-likelihood) method, in which the sparse-matrix-based HPMIXED is used for the inner iterations with the pseudo-variable constructed from the inverse-link function and the chosen model. Although the macro does not have the full functionality of the GLIMMIX procedure, time and memory savings can be large with the new macro. In applications in which design matrices contain many zeros and there are hundreds or thousands of factor levels, models can be fitted without exhausting computer memory, and 90% or better reduction in running time can be observed. Examples with a Poisson, binomial, and gamma conditional distribution are presented to demonstrate the usage and efficiency of this macro.

Exploring the potential of hierarchical generalized linear models in animal breeding and genetics

Exploring the potential of hierarchical generalized linear models in animal breeding and genetics

Global Optimization of Non-Convex Hydro-Thermal Coordination Based on Semidefinite Programming

In order to reduce the fuel cost of thermal units, a semidefinite programming (SDP) method is used to solve a hydrothermal coordination (HTC) optimization problem. By manipulating the structure of decision variable matrix, the original nonconvex problem is reformulated into a convex SDP relaxation model without sacrificing the nonlinear relation among hydropower generation, reservoir storage volume, and discharge water. A global minimum is therefore guaranteed and well-developed convex optimization theories can thus be employed to solve the problem. Both sparse matrix techniques and a simplified SDP model are discussed to reduce computational cost. One mostly used HTC case is employed to test the performance of the proposed method. Detailed comparisons between the proposed and other methods show that the final result of SDP model is by far the best result ever. In addition, a large-sized HTC case shows the potential of SDP in practical use. Finally, we prove that as a relaxation technique, the SDP solution, which satisfies all the constraints, is indeed the optimal solution of the original nonconvex problem.

A parametric generalized coordinate formulation for mechanical joint elements in dynamic systems

The efficient and easy computational implementation of multibody dynamic formulations is an important issue. In this paper, a parametric generalized coordinate formulation is proposed as a new approach to formulate joint constraint equations. By introducing the parametric generalized coordinates in the constraint equations, the complexity of the equations is significantly reduced. The number of arithmetic operations required to compute the derivatives of the constraints is drastically decreased depending on the type of joint involved (i.e. especially when the proposed method is applied to joints having lower degrees of freedom). Furthermore, the second derivatives of the constraint equations tend to have a large portion of zero sub-matrices, which simplifies the system matrix of the governing equations and allows for application of efficient sparse matrix techniques. Although the proposed approach has the drawback of a larger set of coordinates than a conventional Cartesian coordinate approach resulting in slightly less efficient computation, a systematic and easy formulation is achieved, which may relieve the implementation complexity for program developers. The dynamic analysis of a multibody slider-crank system is carried out to demonstrate the validity of the proposed formulation.

Minimal Split Checkerboard Method for Exponentiating Sparse Matrices and Its Applications in Quantum Statistical Mechanics

Multiplication of matrix exponentials is one of the computational kernels in simulations of quantum statistical mechanics, in which matrices are symmetric and sparse. Although the matrix is not particularly large, its multiplication needs to be performed millions of times and thus becomes a performance bottleneck. For a sparse symmetric matrix $A$, the checkerboard method splits $A=A_1+A_2+\cdots + A_k$ and approximates $e^A$ by $e^{A_1}e^{A_2}\cdots e^{A_k}$, in which each $e^{A_i}$ is sparse. When combined with sparse matrix techniques, the checkerboard method can significantly reduce the expense of storing and multiplying $e^A$. However, the accuracy of the checkerboard method degrades as the number of split matrices increases. In this paper, the Minimal Split Checkerboard method (MSCKB) is proposed to reduce the number of split matrices of the checkerboard method. In addition, the Block Checkerboard method (BlkCKB) and the Low-Rank Checkerboard method (LRCKB) are presented to further enhance the performance. All the ideas can be extended to exponentiate skew-symmetric matrices and general matrices. Experiments based on the simulations of quantum statistical mechanics demonstrate the effectiveness of the proposed methods.

Performance Analysis of a Kitting Process as a Paired Queue

Nowadays, customers request more variation in a company’s product assortment leading to an increased amount of parts moving around on the shop floor. To cope with this tendency, a kitting process can be implemented. Kitting is the operation of collecting the necessary parts for a given end product in a specific container, called a kit, prior to arriving at an assembly unit. As kitting performance is critical to the overall cost and performance of the manufacturing system, this paper analyses a two-part kitting process as a Markovian model. In particular, kitting is studied as a paired queue, thereby accounting for stochastic part arrivals, and kit assembly times. Using sparse matrix techniques, we assess the impact of kitting interruptions, bursty part arrivals and phase-type distributed kit assembly times on the behaviour of the part buffers. Finally, a cost-profit analysis of kitting processes is conducted and an approximation for a two-part kitting process is established.

A simple and efficient finite difference frequency domain method for the analysis and design of leaky-wave antennas

The finite difference frequency domain method for the analysis of periodic structures will lead to a large-scale nonsymmetrical generalized eigenvalue problem, especially when the open periodic structures are analyzed.In this article, the method is greatly simplified, because the large-scale sparse matrix technique and the shift-and-invert Arnoldi technique, which must be applied to significantly reduce the memory need and increase simulation speed, are implemented by means of some existed codes, for example, matlab codes. Besides, the difference equations of this method are much simpler than those of the previous method, because the longitudinal field components do not have to be eliminated to approximately transform the generalized eigenvalue problem to a standard eigenvalue problem. As a result, the method proposed here is quite simple and efficient for the analysis of leaky-wave antenna, because a leaky-wave antenna can be regarded as direct geometrical evolution from a guided-wave structure that permits to leak energy along its longitudinal direction. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2814–2817, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.27216