Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.

A survey of direct methods for sparse linear systems

Carbon fiber reinforced polymer in drilling: From damage mechanisms to suppression

In this paper we provide a robust reordering scheme for sparse matrices. The scheme relies on the notion of multisection, a generalization of bisection. The reordering strategy is demonstrated to have consistently good performance in terms of fill reduction when compared with multiple minimum degree and generalized nested dissection. Experimental results show that by using multisection, we obtain an ordering which is consistently as good as or better than both for a wide spectrum of sparse problems.

Robust Ordering of Sparse Matrices using Multisection

Most state-of-the-art ordering schemes for sparse matrices are a hybrid of a bottom-up method such as minimum degree and a top-down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottom-up and top-down methods. In our methodology vertex separators are interpreted as the boundaries of the remaining elements in an unfinished bottom-up ordering. As a consequence, we are using bottom-up techniques such as quotient graphs and special node selection strategies for the construction of vertex separators. Once all separators have been found, we are using them as a skeleton for the computation of several bottom-up orderings. Experimental results show that the orderings obtained by our scheme are in general better than those obtained by other ordering codes.

Towards a Tighter Coupling of Bottom-Up and Top-Down Sparse Matrix Ordering Methods

An arbitrary Lagrangian-Eulerian finite element method for solving three-dimensional free surface flows

Moisture strongly affects the quality and mechanical specificity of carbon fiber reinforced plastic (CFRP) when using lubrication fluids during machining, and the significant impact of the cutting tool geometry and cryogenic gas cooling on CFRP machining capabilities are observed. The main body of this paper aims at making decisions about the optimum parameter of the drilling process while machining on CFRP base on the grey relational coefficient embed to the technique for order of preference by similarity to an ideal solution (Grey-TOPSIS). The entropy method was used to determine the weight of decision-making for handling a multiple measure decision-making response. The twist angle of the tool drill, lubrication, and feed rate were used as the input variables, and were analyzed while taking into account several multi-response outputs, such as the surface roughness, uncut fiber, and delamination. The result showed that a feed rate of 228 mm/min, the high-helix twist angle, and cryogenic CO2 lubrication leads the calculated value to close the relative value, which minimizes the value of the surface roughness, the uncut fiber, and the delamination. Finally, verification of the valid effect of each parameter process was conducted using analysis of variance. The results indicated that the lubrication was the highest remarkable criterion on the uncut fiber, the delamination, and the surface roughness. By integrating the advantage of grey systems theory, and the technique for order preference by similarity to an ideal solution, to evaluate and optimize the machining parameter, the results indicate that the proposed model is useful to facilitate the multi-criteria decision-making problem under the environment of uncertainty and vagueness. This relatively advanced approach is very effectual in rejecting process variation and a great assistive strategy than other multi-criteria decision-making approaches.

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Drilling Process on CFRP: Multi-Criteria Decision-Making with Entropy Weight Using Grey-TOPSIS Method

The tungsten inert gas (TIG) welding method most commonly used to weld ferrous metals, nonferrous metals, and other metals since it is simple, easily implemented, and achieves consistent high-quality welds. In this study, butt joints produced between aluminum alloy A6061-T6 and stainless steel SUS304L have been achieved by using TIG welding with ER4047 filler metal. The macrostructure and microstructure of the resulting specimens were analyzed by means of an optical microscope (OM), a scanning electron microscope (SEM), and an energy dispersive X-ray spectrometer (EDS). A uniform intermetallic layer was found at the interface between the stainless steel and the weld seam having a thickness of 2 µm, and the intermetallic compound (IMC) includes Fe4Al13, Fe2Al5, and FeAl3 phases. The micro-hardness and tensile strength of the weld joints were also investigated. Due to Si content in the compensating metal, there was a prevention of iron diffusion into the aluminum, thus hindering the development of the IMC layer and reducing its thickness in such a way that the weld joint strength increases. The analyzed results show that the average micro-hardness of the stainless steel, weld seam, aluminum alloys, and IMC layer were 218 HV, 88.3 HV, 63.3 HV, and 411 HV, respectively. The fracture occurred at the brazed interface, and the ultimate tensile strength value reached 225 MPa.

Microstructure and Mechanical Properties of Butt Joints between Stainless Steel SUS304L and Aluminum Alloy A6061-T6 by TIG Welding.

Solution methods are presented for the large systems of linear equations resulting from the implicit, coupled solution of the Navier-Stokes equations in three dimensions. Two classes of methods for such solution have been studied: direct and iterative methods.

For direct methods, sparse matrix algorithms have been investigated and a Gauss elimination, optimized for vector-parallel processing, has been developed. Sparse matrix results indicate that reordering algorithms deteriorate for rectangular, i.e. M × M × N, grids in three dimensions as N gets larger than M. A new local nested dissection reordering scheme that does not suffer from these difficulties, at least in two dimensions, is presented. The vector-parallel Gauss elimination is very efficient for processing on today's supercomputers, achieving execution rates exceeding 2.3 Gflops the Cray YMP-8 and 9.2 Gflops on the NEC on SX3.

For iterative methods, two approaches are developed. First, conjugate-gradient-like methods are studied and good results are achieved with a preconditioned conjugate gradient squared algorithm. Convergence of such a method being sensitive to the preconditioning, a hybrid viscosity method is adopted whereby the preconditioner has an artificial viscosity that is gradually lowered, but frozen at a level higher than the dissipation introduced in the physical equations. The second approach is a domain decomposition one in which overlapping domain and side-by-side methods are tested. For the latter, a Lagrange multiplier technique achieves reasonable rates of convergence.

Large‐scale computational fluid dynamics by the finite element method

A new ordering scheme is presented for sparse matrices associated with rectangular grid problems. The new scheme combines features of both the nested dissection and natural orderings to obtain orderings superior to either of these methods for such problems. Asymptotic bounds and experimental results on sparse factorization operations are provided to demonstrate the advantages of the new ordering.

Van Nhat Nguyen

Papers

Drilling Process on CFRP: Multi-Criteria Decision-Making with Entropy Weight Using Grey-TOPSIS Method

Microstructure and Mechanical Properties of Butt Joints between Stainless Steel SUS304L and Aluminum Alloy A6061-T6 by TIG Welding.

Large‐scale computational fluid dynamics by the finite element method

A note on nested dissection for rectangular grids

Large-scale computational fluid dynamics by the finite element method