Solution of Plane Problems of Elasticity Utilizing Partitioning Concepts

1973 ◽  
Vol 40 (3) ◽  
pp. 767-772 ◽  
Author(s):  
O. L. Bowie ◽  
C. E. Freese ◽  
D. M. Neal
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Collocation Method ◽  
Stress Function ◽  
Function Theory ◽  
Series Expansions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Power Series Expansions

A partitioning plan combined with the modified mapping-collocation method is presented for the solution of awkward configurations in two-dimensional problems of elasticity. It is shown that continuation arguments taken from analytic function theory can be applied in the discrete to “stitch” several power series expansions of the stress function in appropriate subregions of the geometry. The effectiveness of such a plan is illustrated by several numerical examples.

A Method of Nonlinear Modal Superposition for Weakly Nonlinear Autonomous Systems

2014 ◽  
Vol 670-671 ◽  
pp. 1321-1325
Author(s):  
Jian Guo Wei ◽  
Lan Min Zhao
Keyword(s):  
General Solution ◽  
Power Series ◽  
Autonomous Systems ◽  
Series Expansions ◽  
Initial Condition ◽  
Mathematical Transformation ◽  
Weakly Nonlinear ◽  
Numerical Examples ◽  
Modal Superposition ◽  
Power Series Expansions

For a weakly nonlinear autonomous systems, the nonlinear coordinate transformation is provided in the form of power series expansions and the mathematical transformation from the physics system coordinate to the modal coordinate is presented. And then, according to the relations between physics system coordinate and modal coordinate, a method of nonlinear modal superposition is obtained. Using this method, the general solution of the vibration under any initial condition can be answered. In order to explain the applications of the method, numerical examples are presented in this paper.

scholarly journals Parallel Software to Offset the Cost of Higher Precision

2021 ◽  
Vol 40 (2) ◽  
pp. 59-64
Author(s):  
Jan Verschelde
Keyword(s):  
Power Series ◽  
Parallel Algorithms ◽  
Series Expansions ◽  
Use Case ◽  
Double Precision ◽  
Algebraic Space ◽  
Space Curves ◽  
Computational Overhead ◽  
Power Series Expansions ◽  
The Cost

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

scholarly journals Free Function Theory Through Matrix Invariants

2017 ◽  
Vol 69 (02) ◽  
pp. 408-433 ◽  
Author(s):  
Igor Klep ◽  
Špela Špenko
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Function Theory ◽  
Rational Functions ◽  
Generalized Polynomials ◽  
Direct Sums ◽  
Real Analytic Functions ◽  
Power Series Expansions ◽  
Noncommutative Polynomials ◽  
Free Function

Abstract This paper concerns free function theory. Freemaps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of g noncommuting variables is a function on g-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invarianttheoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involutionfree counterparts.

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