scholarly journals CUBATURE FORMULA FOR AN OCTAHEDRON OF THE SEVENTH ALGEBRAIC ORDER OF ACCURACY

2020 ◽  
Vol 3 (2-2) ◽  
pp. 184-193
Author(s):  
А.P. MOTAILO
Keyword(s):  
Cubature Formula ◽  
Order Of Accuracy ◽  
Algebraic Order

scholarly journals Constructing Steklov-type cubature formulas for a finite element in the shape of a bipyramid

2021 ◽  
Vol 4 (4(112)) ◽  
pp. 40-46
Author(s):  
Anzhelika Motailo ◽  
Galina Tuluchenko
Keyword(s):  
Finite Element ◽  
The Finite Element Method ◽  
Order Of Accuracy ◽  
Cubature Formulas ◽  
Compression Parameter ◽  
Algebraic Order ◽  
The Stability ◽  
Local Stiffness ◽  
Approximate Integration ◽  
Interpolation Nodes

This paper reports the construction of cubature formulas for a finite element in the form of a bipyramid, which have a second algebraic order of accuracy. The proposed formulas explicitly take into consideration the parameter of bipyramid deformation, which is important when using irregular grids. The cubature formulas were constructed by applying two schemes for the location of interpolation nodes along the polyhedron axes: symmetrical and asymmetrical. The intervals of change in the elongation (compression) parameter of a bipyramid semi-axis have been determined, within which interpolation nodes of the constructed formulas belong to the integration region, while the weight coefficients are positive, which warrants the stability of calculations based on these cubature formulas. If the deformation parameter of the bipyramid is equal to unity, then both cubature formulas hold for the octahedron and have a third algebraic order of accuracy. The resulting formulas make it possible to find elements of the local stiffness matrix on a finite element in the form of a bipyramid. When calculating with a finite number of digits, a rounding error occurs, which has the same order for each of the two cubature formulas. The intervals of change in the elongation (compression) parameter of the bipyramid semi-axis have been determined, which meet the requirements, which are employed in the ANSYS software package, for deviations in the volume of the bipyramid from the volume of the octahedron. Among the constructed cubature formulas for a bipyramid, the optimal formula in terms of the accuracy of calculations has been chosen, derived from applying a symmetrical scheme of the arrangement of nodes relative to the center of the bipyramid. This formula is invariant in relation to any affinity transformations of the local bipyramid coordinate system. The constructed cubature formulas could be included in libraries of methods for approximate integration used by those software suites that implement the finite element method.

Eighth-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs

2018 ◽  
Vol 29 (01) ◽  
pp. 1850002 ◽  
Author(s):  
J. M. Franco ◽  
L. Rández
Keyword(s):  
Scientific Literature ◽  
Hybrid Methods ◽  
Computational Cost ◽  
High Dispersion ◽  
Minimal Cost ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Order Of Accuracy ◽  
Algebraic Order

The construction of new two-step hybrid (TSH) methods of explicit type with symmetric nodes and weights for the numerical integration of orbital and oscillatory second-order initial value problems (IVPs) is analyzed. These methods attain algebraic order eight with a computational cost of six or eight function evaluations per step (it is one of the lowest costs that we know in the literature) and they are optimal among the TSH methods in the sense that they reach a certain order of accuracy with minimal cost per step. The new TSH schemes also have high dispersion and dissipation orders (greater than 8) in order to be adapted to the solution of IVPs with oscillatory solutions. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order explicit TSH methods are more efficient than other standard TSH or Numerov-type methods proposed in the scientific literature.

Nonlinear dispersion hyperbolic models of continuous media

2007 ◽  
Vol 5 ◽  
pp. 273-278
Author(s):  
V.Yu Liapidevskii
Keyword(s):  
Boussinesq Equations ◽  
Nonlinear Dispersion ◽  
Order Of Accuracy ◽  
Flow Models ◽  
Wave Approximation ◽  
Long Wave ◽  
Primary Model ◽  
Nonequilibrium Flows ◽  
Internal Inertia ◽  
Long Wave Approximation

Nonequilibrium flows of an inhomogeneous liquid in channels and pipes are considered in the long-wave approximation. Nonlinear dispersion hyperbolic flow models are derived allowing taking into account the influence of internal inertia during the relative motion of phases upon the structure of nonlinear wave fronts. The asymptotic derivation of dispersion hyperbolic models is shown on the example of classical Boussinesq equations. It is shown that the hyperbolic approximation of the equations has the same order of accuracy as the primary model.

scholarly journals 3D Unsteady Diffusion and Reaction-Diffusion with Singularities by GFEM with 27-Node Hexahedrons

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Estaner Claro Romão
Keyword(s):  
Finite Element ◽  
Variable Temperature ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Third Order ◽  
Order Of Accuracy ◽  
Finite Element Approximations ◽  
Temperature Solution ◽  
The One ◽  
Primary Variable

The Galerkin Finite Element Method (GFEM) with 8- and 27-node hexahedrons elements is used for solving diffusion and transient three-dimensional reaction-diffusion with singularities. Besides analyzing the results from the primary variable (temperature), the finite element approximations were used to find the derivative of the temperature in all three directions. This technique does not provide an order of accuracy compatible with the one found in the temperature solution; thereto, a calculation from the third order finite differences is proposed here, which provide the best results, as demonstrated by the first two applications proposed in this paper. Lastly, the presentation and the discussion of a real application with two cases of boundary conditions with singularities are proposed.

The Measurement of the Thickness of Metal Walls from One Surface Only, by an Electrical Method

1938 ◽  
Vol 140 (1) ◽  
pp. 349-398 ◽  
Author(s):  
B. M. Thornton ◽  
W. M. Thornton
Keyword(s):  
Potential Drop ◽  
Engineering Industry ◽  
Steel Plates ◽  
Electrical Method ◽  
Order Of Accuracy ◽  
Iron Castings ◽  
Engine Cylinder ◽  
External Power ◽  
Metal Wall ◽  
Core Shift

There is a definite need in the engineering industry for an instrument which will determine accurately the thickness of a metal wall from one side only, and which is portable, robust, independent of external power supply, and easy to operate. The instrument described in the paper fulfils all these conditions. The method is essentially that of comparing the resistance of a metal wall under test with that of a similarly shaped wall of the same material, of which the thickness is known. Current, usually under 10 amp., is supplied by a 6- or 12-volt battery to two current contacts held against the wall and usually spaced 2 or 3 inches apart. The current in the circuit is adjusted by varying the rheostat or the number of cells of the small car-battery employed, until the very small potential drop indicated by the deflexion of a galvanometer connected to two potential contacts reaches a predetermined fixed value. From a calibration curve the thickness of the wall corresponding to the current registered by the ammeter is determined. The contacts are made on the plates whose thickness is to be measured by means of Evershed and Vignoles spring-controlled rotating points that were originally designed for a four-point method of working. The flexible leads from these, sometimes 100 but usually 30 feet long, are rubber-covered, and are connected to the circuit in the box by plugs and sockets. These are removed for transport and carried in a separate box. The instrument was originally designed for measuring the thickness of boiler tubes, and examples are given showing that its accuracy in this respect is within a few thousandths of an inch. It was found in practice that the instrument could be used to measure the thickness of mild steel plates up to 1·25 inches and of iron castings, up to 3 inches, with the same order of accuracy. It is also used for the measurement of engine cylinder walls to detect core shift. Examples of these are given in the paper. One notable case was that of the examination of a bank of superheater tubes in a boiler that had given much trouble due to the use of an unsuitable chain-grate stoker. Thinned tubes were at once detected and on being cut out the thicknesses indicated by the instrument were confirmed. Examples are given of the regular use of the instrument in determining the thickness of boiler and superheater tubes, ships' hull plates, tank wagons, and various iron castings that were accessible from one side only.

Hybrid high algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives

2016 ◽  
Vol 27 (05) ◽  
pp. 1650049 ◽  
Author(s):  
Junyan Ma ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Truncation Error ◽  
Resonance Problem ◽  
Phase Lag ◽  
Step Method ◽  
Algebraic Order ◽  
Test Equation ◽  
The Stability ◽  
Scalar Test Equation

A hybrid tenth algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives are obtained in this paper. We will investigate • the construction of the method • the local truncation error (LTE) of the newly obtained method. We will also compare the lte of the newly developed method with other methods in the literature (this is called the comparative LTE analysis) • the stability (interval of periodicity) of the produced method using frequency for the scalar test equation different from the frequency used in the scalar test equation for phase-lag analysis (this is called stability analysis) • the application of the newly obtained method to the resonance problem of the Schrödinger equation. We will compare its effectiveness with the efficiency of other known methods in the literature. It will be proved that the developed method is effective for the approximate solution of the Schrödinger equation and related periodical or oscillatory initial value or boundary value problems.

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