Introduction to the DLM: The First-Order Polynomial Model

1989 ◽  
pp. 37-74
Author(s):  
Mike West ◽  
Jeff Harrison
Keyword(s):  
Polynomial Model ◽  
First Order ◽  
Order Polynomial

scholarly journals Accuracy evaluation of 2D geometric correction models: Iraq as a case study

2018 ◽  
Vol 162 ◽  
pp. 03015
Author(s):  
Abdul Razzak Ziboon ◽  
Israa Mohammed
Keyword(s):  
Mathematical Models ◽  
Two Dimensions ◽  
Second Order ◽  
Polynomial Model ◽  
Accuracy Evaluation ◽  
Geometric Correction ◽  
Flat Area ◽  
First Order ◽  
High Resolution Satellite Images ◽  
Order Polynomial

One of the most important preprocessing methods for remote sensing data and geometrical distrortion is the geometric correction. In this research, several mathematical models were used in two dimensions case and were compared to each other to determine the accuracy of each mathematical model when used in three different regions with different terrains. Three high-resolution satellite images of the QuickBird satellite (a flat area, a hilly area and a mountain area) of Iraq have been used in this work. The flat area is chosen in Baghdad, while the hilly and mountain areas are chosen in Irbil in the north of Iraq. In this research, the mathematical models used are the first and second order polynomials, as well as the projective transform. All of these models were applied to all different topographic areas and their accuracy was assessed based on the Matlab program. The results of the models in the three areas studied indicate that the best precision is achieved with the second order polynomial model, while the worst precision is obtained with the first order polynomial model. On the other hand, the precision of the projective transformation is almost similar to the precision of the first order polynomial.

Statistical method of steepest improvement of response

2018 ◽  
Vol 84 (11) ◽  
pp. 74-87
Author(s):  
V. B. Bokov
Keyword(s):  
Statistical Method ◽  
Linear Model ◽  
Response Prediction ◽  
Initial Experiment ◽  
First Order ◽  
Prediction Response ◽  
Conditional Extremum ◽  
Order Polynomial ◽  
Limiting Values

A new statistical method for response steepest improvement is proposed. This method is based on an initial experiment performed on two-level factorial design and first-order statistical linear model with coded numerical factors and response variables. The factors for the runs of response steepest improvement are estimated from the data of initial experiment and determination of the conditional extremum. Confidence intervals are determined for those factors. The first-order polynomial response function fitted to the data of the initial experiment makes it possible to predict the response of the runs for response steepest improvement. The linear model of the response prediction, as well as the results of the estimation of the parameters of the linear model for the initial experiment and factors for the experiments of the steepest improvement of the response, are used when finding prediction response intervals in these experiments. Kknowledge of the prediction response intervals in the runs of steepest improvement of the response makes it possible to detect the results beyond their limits and to find the limiting values of the factors for which further runs of response steepest improvement become ineffective and a new initial experiment must be carried out.

scholarly journals Thematic Maps for the Variation of Bearing Capacity of Soil Using SPTs and MATLAB

Geosciences ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 329
Author(s):  
Mahdi O. Karkush ◽  
Mahmood D. Ahmed ◽  
Ammar Abdul-Hassan Sheikha ◽  
Ayad Al-Rumaithi
Keyword(s):  
Bearing Capacity ◽  
Mean Squared Error ◽  
Ground Level ◽  
The Other ◽  
Thematic Maps ◽  
First Order ◽  
Squared Error ◽  
Order Polynomial ◽  
Interpolation Polynomials ◽  
Penetration Tests

The current study involves placing 135 boreholes drilled to a depth of 10 m below the existing ground level. Three standard penetration tests (SPT) are performed at depths of 1.5, 6, and 9.5 m for each borehole. To produce thematic maps with coordinates and depths for the bearing capacity variation of the soil, a numerical analysis was conducted using MATLAB software. Despite several-order interpolation polynomials being used to estimate the bearing capacity of soil, the first-order polynomial was the best among the other trials due to its simplicity and fast calculations. Additionally, the root mean squared error (RMSE) was almost the same for the all of the tried models. The results of the study can be summarized by the production of thematic maps showing the variation of the bearing capacity of the soil over the whole area of Al-Basrah city correlated with several depths. The bearing capacity of soil obtained from the suggested first-order polynomial matches well with those calculated from the results of SPTs with a deviation of ±30% at a 95% confidence interval.

Study on Fillet Stresses Distribution of Crankshaft of High-Duty Diesel Engine

2011 ◽  
Vol 52-54 ◽  
pp. 1364-1368
Author(s):  
Wen Li ◽  
Ri Dong Liao ◽  
Zheng Xing Zuo
Keyword(s):  
Diesel Engine ◽  
Second Order ◽  
First Order ◽  
Fourier Expansions ◽  
Order Polynomial ◽  
Polynomial Expansions ◽  
Regional Stresses

Stresses distribution in crankshaft fillet region are mainly discussed in this paper. FEM is used for calculating the stresses. With the analysis of the regional stresses distribution, we concludes that the stresses can be fitted by first order of Fourier expansions, while all the parameters in the expansions can be expressed as second order polynomial expansions.

Deformation in Multilayer Stacked Assemblies

1990 ◽  
Vol 112 (1) ◽  
pp. 30-34 ◽  
Author(s):  
Tsung-Yu Pan ◽  
Yi-Hsin Pao
Keyword(s):  
Thermal Loading ◽  
Bending Moment ◽  
Radius Of Curvature ◽  
Element Analysis ◽  
First Order ◽  
Strain Behavior ◽  
Linear Elastic ◽  
Nonlinear Stress ◽  
Order Polynomial ◽  
Vertical Displacements

A linear-elastic analytical model has been developed to describe the deformed geometry of a multi-layered stack assembly subject to thermal loading. The model is based on Timoshenko’s bimetal thermostat analysis [1] and consists of a series of first-order polynomial equations. The radius of curvature, bending moment, force, horizontal and vertical displacements can be determined numerically. These quantities match well with finite element analysis. Calculations for silicon power transistor stacks are presented in order to demonstrate the model capability. The results from this analyitcal model have been found to correlate well with experimental measurements when an appropriate secant modulus is used to represent the nonlinear stress-strain behavior of solder.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Beam Theory ◽  
Second Order ◽  
Governing Equations ◽  
Deflection Curve ◽  
Order Analysis ◽  
First Order ◽  
Finite Difference Approximations ◽  
Order Polynomial ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally, the finite difference approximations are derived based on fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection curve; the FOPH is made for the fourth and third derivative of the deflection curve while the SOPH is made for its second and first derivative. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, the FOPH was made for all of the derivatives of the deflection curve, and additional points were introduced at the beam’s ends and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid. Further research will be needed to investigate polynomial refinements (higher-order polynomials such as fifth-order, sixth-order…) of the deflection curve; the polynomial refinements aimed to increase the accuracy, whereby non-centered finite difference approximations at beam’s ends and positions of discontinuity would be used.

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