Identifying the Cause of and Fixing Ill-Conditioned Matrices in Nuclear Analysis Codes

2020 ◽  
Author(s):  
Lance C. Larsen
Keyword(s):  
Nuclear Industry ◽  
Numerical Error ◽  
System Of Equations ◽  
Geometric Problem ◽  
Nuclear Analysis ◽  
Geometric Concepts ◽  
The Matrix ◽  
Linearized System ◽  
Ill Conditioning ◽  
Ill Conditioned Matrices

Abstract Many of the analytical codes used in the nuclear industry, such as TRACE, RELAP5, and PARCS, approximate the equations that model the physics via a linearized system of equations. One common difficulty when solving linearized systems is that an accurately formulated system of equations may be ill-conditioned. Ill-conditioned matrices can result in significant amplification of error leading to poor, or even invalid, results. Ill-conditioned matrices lead to some challenging issues for the analytical code developers: • An ill-conditioned matrix is often solvable, and there may be no obvious indication numerically that something has gone wrong even though numerical error is large. Thus, how can ill-conditioning be effectively detected for a matrix? • When ill-conditioning is detected, how can the source of the ill-conditioning be determined so that it can be analyzed and corrected? Ill-conditioning is fundamentally a geometric problem that can be understood with geometric concepts associated with matrices and vectors. Geometric concepts and tools, useful for understanding the cause of ill-conditioning of a matrix, are presented. A geometric understanding of ill-conditioning can point to the rows or columns of the matrix that most contribute to ill-conditioning so that the source of ill-conditioning can be analyzed and understood, and leads to techniques for building matrix preconditioners to improve the solvability of the matrix.

Research of stamping of unequal channel bars. Part 3. Force parameters and shape change of the billet when extruding channels. 1. Kinematic and stress states of the billet

2021 ◽  
pp. 65-71
Author(s):  
A.L. Vorontsov
Keyword(s):  
Plastic Flow ◽  
Complete System ◽  
Shape Change ◽  
Plane Deformation ◽  
System Of Equations ◽  
Flow Rates ◽  
Die Forging ◽  
The Matrix ◽  
Stress States ◽  
Theory Of Plastic Flow

On the basis of the complete system of equations of the theory of plastic flow, the kinematic and stress states of the billet are determined when the channels are extruded under conditions of plane deformation of the misaligned position of the punch and the matrix. Keywords: die forging, extrusion, misaligned position, punch, matrix, plane deformation, plastic flow rates, stresses. [email protected]

Research of stamping of unequal channel bars. Part 3. Force parameters and shape change of the billet when extruding channels. 2. Determination of the extrusion force, maximum pressure on the matrix wall and the heights of the walls, taking into account the elastic deformation jf the matrix

2021 ◽  
pp. 63-69
Author(s):  
A.L. Vorontsov
Keyword(s):  
Plastic Flow ◽  
Elastic Deformation ◽  
Shape Change ◽  
Plane Deformation ◽  
Maximum Pressure ◽  
Extrusion Force ◽  
System Of Equations ◽  
The Matrix ◽  
Theory Of Plastic Flow

On the basis of the system of equations of the theory of plastic flow, the forces, the maximum pressure on the wall of the matrix and the heights of the obtained walls when extruding channels are determined, taking into account the elastic deformation of the matrix. Keywords: die forging, extrusion, misalignment, punch, matrix, plane deformation, stresses. [email protected]

Singular analysis of stereoscopic serif

2021 ◽  
Vol 966 (12) ◽  
pp. 21-30
Author(s):  
E.G. Voronin
Keyword(s):  
System Of Equations ◽  
Stereo Images ◽  
Geometric Conditions ◽  
Basis Plane ◽  
The Matrix ◽  
Spatial Coordinates ◽  
Singular Numbers ◽  
Singular Analysis

The article is devoted to the development of a methodological apparatus for evaluating the influence of geometric survey conditions on the ratio of errors in the planned and altitude components of determining the terrain points’ spatial coordinates. For this purpose, an approach based on the singular analyzing the matrix of the system of equations for intersecting lines in the basis plane is used. It is shown that singular numbers characterize the ellipse of coordinate definition errors in plane and height. They are calculated for the most typical cases of stereo surveying. It is noted that the geometric conditions of stereo photography are mainly determined by the serif angle. By interpolating the results of calculating singular numbers, the formulas characterizing the direct and inverse dependence of the ratio of the stereo-cut in plan and height on the serif angle errors’ are obtained. The author considers the practical issues of using the developed methodological apparatus to justify the solutions related to the assessment of the ratio between the errors in determining the plane coordinates and the heights of terrain points from stereo images.

Formation of Change Support Teams As a New Tool for Personnel Management in the Organization

2019 ◽  
Vol 8 (5) ◽  
pp. 5-12
Author(s):  
S. Brykalov ◽  
A. Balyberdin ◽  
E. Krylova
Keyword(s):  
Change Management ◽  
Personnel Management ◽  
Nuclear Industry ◽  
Organization Of Work ◽  
Support Teams ◽  
Industrial Enterprises ◽  
The Matrix ◽  
Principles Of Change

The article describes the main principles of Change management in the organization, the main barriers rising out of the Change management implementation in terms of employees’ attitude to innovations. Approaches to change management in the organization according to John Kotter are analyzed as well. This article is devoted to describing the original experience of forming change support teams at one of the industrial enterprises of the nuclear industry (JSC Afrikantov OKBM), reviewing the original organization of work and the matrix of distribution of functions between the main participants, as well as presenting practical results and economic eff ects from the work of the TOC.

Synthesis of Crystalline Ceramics for Actinide Immobilisation

2007 ◽  
Author(s):  
B. Burakov ◽  
V. Gribova ◽  
A. Kitsay ◽  
M. Ojovan ◽  
N. C. Hyatt ◽  
...  
Keyword(s):  
Hot Isostatic Pressing ◽  
Industrial Applications ◽  
Nuclear Industry ◽  
Synthesis Methods ◽  
Khlopin Radium Institute ◽  
Radium Institute ◽  
Joint Project ◽  
Ceramic Synthesis ◽  
The Matrix ◽  
Processing Plants

Methods for the synthesis of ceramic wasteforms for the immobilization of actinides are common to those for non-radioactive ceramics: hot uniaxial pressing (HUP); hot isostatic pressing (HIP); cold pressing followed by sintering; melting (for some specific ceramics, such as garnet/perovskite composites). Synthesis of ceramics doped with radionuclides is characterized with some important considerations: all the radionuclides should be incorporated into crystalline structure of durable host-phases in the form of solid solutions and no separate phases of radionuclides should be present in the matrix of final ceramic wasteform; all procedures of starting precursor preparation and ceramic synthesis should follow safety requirements of nuclear industry. Synthesis methods that avoid the use of very high temperatures and pressures and are easily accomplished within the environment of a glove-box or hot cell are preferable. Knowledge transfer between the V. G. Khlopin Radium Institute (KRI, Russia) and Immobilisation Science Laboratory (ISL, UK) was facilitated in the framework of a joint project supported by UK Royal Society. In order to introduce methods of precursor preparation and ceramic synthesis we selected well-known procedures readily deployable in radiochemical processing plants. We accounted that training should include main types of ceramic wasteforms which are currently discussed for industrial applications.

scholarly journals II. A memoir on the theory of matrices

1858 ◽  
Vol 148 ◽  
pp. 17-37 ◽  
Keyword(s):  
Rational Function ◽  
General Expression ◽  
Linear Equations ◽  
Integral Function ◽  
The Other ◽  
System Of Equations ◽  
General Sense ◽  
Condensed Form ◽  
Restricted Sense ◽  
The Matrix

The term matrix might be used in a more general sense, but in the present memoir I consider only square and rectangular matrices, and the term matrix used without qualification is to be understood as meaning a square matrix; in this restricted sense, a set of quantities arranged in the form of a square, e. g . ( a, b, c ) | a', b', c' | | a", b", c" | is said to be a matrix. The notion of such a matrix arises naturally from an abbreviated notation for a set of linear equations, viz. the equations X = ax + by + cz , Y = a'x + b'y + c'z , Z = a"x + b"y + c"z , may be more simply represented by ( X, Y, Z)=( a, b, c )( x, y, z ), | a', b', c' | | a", b", c" | and the consideration of such a system of equations leads to most of the fundamental notions in the theory of matrices. It will be seen that matrices (attending only to those of the same order) comport themselves as single quantities; they may be added, multiplied or compounded together, &c.: the law of the addition, of matrices is precisely similar to that for the addition of ordinary algebraical quantities; as regards their multiplication (or composition), there is the peculiarity that matrices are not in general convertible; it is nevertheless possible to form the powers (positive or negative, integral or fractional) of a matrix, and thence to arrive at the notion of a rational and integral function, or generally of any algebraical function, of a matrix. I obtain the remarkable theorem that any matrix whatever satisfies an algebraical equation of its own order, the coefficient of the highest power being unity, and those of the other powers functions of the terms of the matrix, the last coefficient being in fact the determinant; the rule for the formation of this equation may be stated in the following condensed form, which will be intelligible after a perusal of the memoir, viz. the determinant, formed out of the matrix diminished by the matrix considered as a single quantity involving the matrix unity, will be equal to zero. The theorem shows that every rational and integral function (or indeed every rational function) of a matrix may be considered as a rational and integral function, the degree of which is at most equal to that of the matrix, less unity; it even shows that in a sense, the same is true with respect to any algebraical function whatever of a matrix. One of the applications of the theorem is the finding of the general expression of the matrices which are convertible with a given matrix. The theory of rectangular matrices appears much less important than that of square matrices, and I have not entered into it further than by showing how some of the notions applicable to these may be extended to rectangular matrices.

A new analytic procedure to determine hypocentral parameters of local seismic events

1984 ◽  
Vol 74 (2) ◽  
pp. 655-667
Author(s):  
D. Caccamo ◽  
G. Neri
Keyword(s):  
Least Squares ◽  
Iterative Process ◽  
Classical Method ◽  
Earth Model ◽  
Seismic Events ◽  
System Of Equations ◽  
Analytic Procedure ◽  
The Earth ◽  
The Matrix ◽  
Hypocentral Parameters

Abstract A new procedure for locating local earthquakes is proposed. Essentially, this procedure consists in solving—by means of least-squares technique—a system of equations which is formally analogous to that of Geiger (Ax = b) but different from his in the values of the matrix A and vector b elements. This difference makes our procedure more reliable than Geiger's because it significantly reduces the cases of both iterative process divergence and low precision. Among the factors contributing to this progress the lesser possibility that the matrix A contains columns proportional (or nearly proportional) one to another is considered of particular influence. More than 20,000 hypocentral calculations have been performed on simulated shocks: significant differences in the number of good locations were revealed between our procedure and the classical method of Geiger (1910). Ours is more precise, particularly when few stations were used or networks with an unsatisfactory geometry. The earth model also influences the observed differences as it contributes to generating those analytical conditions which make the calculation convergence more problematic when applying Geiger's method. Further applications are currently carried out in order to verify the procedure features for velocity laws and station configuration different from those used in this study.

scholarly journals The Zeeman effect and spherical harmonics

1927 ◽  
Vol 115 (770) ◽  
pp. 1-19 ◽  
Keyword(s):  
Recent Work ◽  
Spherical Harmonics ◽  
Strong Field ◽  
Zeeman Effect ◽  
Simple System ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Wave Mechanics ◽  
Matrix Methods ◽  
The Matrix

1. The chief object of the present paper is to present a simple system of equations which are competent to determine the frequencies and intensities of the lines in the standard Zeeman effect. By the standard Zeeman effect is meant the type where the terms are given by Landé’s " g "and “γ” formulæ, and where the multiplicity of the two sets of terms is the same. It will probably be held that the theory of such multiplets has been fairly completely understood for the last two years owing to the works of Sommerfeld, Heisenberg, Landé and Pauli, and the main new contribution of the present work is that, whereas these writers gave formulæ valid only either in weak or strong field (except for doublets, where Heisenberg and Jordan give the value for any strength), here we have complete formulæ from which the intensity of any component in any strength of field can be obtained merely from the solution of rather simple algebraic equations. The proper attack on this problem would undoubtedly be by way of the recent work of Heisenberg on the helium spectrum, and his still more recent work on complex spectra; but this theory is still in the making, so that it has not been practicable to apply it here, and to this extent our results are unverified. Their validity rests firstly on a complete verification of all known facts connected with both weak and strong fields (and intermediate fields for doublets), and also on a conformity with the general features of wave mechanics. The work of Heisenberg and Jordan could readily have been adapted to give all the results of the present paper; but it would have been harder to follow because the matrix methods are not so easy for most readers as are spherical harmonics.

Development of a GSP Integrated Formative Assessment System on Geometric Creativity

2010 ◽  
Vol 108-111 ◽  
pp. 979-984
Author(s):  
Yuan Chen ◽  
Pi Hsia Hung
Keyword(s):  
Problem Solving ◽  
Formative Assessment ◽  
Assessment System ◽  
Intervention Effect ◽  
Geometric Problem ◽  
Interactive Exploration ◽  
Geometric Concepts ◽  
Scoring Rubrics ◽  
On Line ◽  
Exploration Environment

A GSP integrated learning and assessment system (LAS-GSP) is developed to provide students an interactive exploration environment and on-line feedbacks on geometric problem solving. Three tasks of maximal segmentation are developed to investigate the applicability and intervention effect of the system. There are four scoring elements for students’ on line assignments: (1) systematic approaches, (2) correct solutions, (3) originality of representation, and (4) function derived. The characteristics of students’ learning progress are discussed by the scoring rubrics applied. The results suggest abstract geometric concepts can be visualized, internalized, and enhanced at an earlier age, if mind-tool can be effectively implemented.

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