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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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Consider a real-valued function defined on a closed interval. One of the fundamental objectives in calculus is to determine the area under the graph of such a function. When an exact computation is not readily available, this area can be estimated by dividing the interval into a finite number of equal subintervals. Each subinterval corresponds to a rectangle whose width is the length of the subinterval and whose height is determined by the value of the function at a selected point within that...
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The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing...
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Smart speakers process voice commands by modeling audio inputs as piecewise functions and analyzing them through integration against trigonometric functions, such as cosine. This mathematical approach is fundamental in signal processing, where complex sound waves are decomposed into simpler frequency components.Consider a definite integral involving a piecewise function multiplied by a cosine function. Because the function is defined differently over separate intervals, the integral is split...
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In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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Phase Contrast and Differential Interference Contrast DIC Microscopy
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A consistent full-field integrated DIC framework for HR-EBSD.

T Vermeij1, J P M Hoefnagels1

  • 1Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands.

Ultramicroscopy
|May 18, 2018
PubMed
Summary

A new Integrated Digital Image Correlation (IDIC) framework for high angular resolution Electron BackScatter Diffraction (HR-EBSD) offers accurate, robust strain and rotation measurements. This direct, one-step method rivals existing algorithms for materials analysis.

Keywords:
Electron backscatter diffractionFinite-strain formulationHR-EBSDHigh angular resolution EBSDHigh strain accuracyIntegrated DICVirtual experiments

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Area of Science:

  • Materials Science
  • Crystallography
  • Computational Mechanics

Background:

  • High angular resolution Electron BackScatter Diffraction (HR-EBSD) is crucial for analyzing material microstructures.
  • Accurate strain and rotation measurements are essential for understanding material deformation.
  • Existing HR-EBSD methods often involve complex, multi-step processes with potential for error accumulation.

Purpose of the Study:

  • To develop a general, transparent, finite-strain Integrated Digital Image Correlation (IDIC) framework for HR-EBSD.
  • To enable direct, one-step correlation of full-field-of-view Electron BackScatter Patterns (EBSPs).
  • To achieve high accuracy and robustness in strain and rotation measurements.

Main Methods:

  • Derivation of an optimization scheme from the brightness conservation equation and an EBSP formation model.
  • Implementation of a direct, one-step correlation of full-field-of-view EBSPs.
  • Validation using dynamically simulated EBSP patterns with varying strain, rotation, and image noise.

Main Results:

  • Errors in strain and rotation components are below 10⁻⁵ for small/medium strain and 3×10⁻⁵ for large strain.
  • The framework demonstrates high robustness against initial guess errors (up to 1° misorientation) and image noise (up to 20%).
  • Consistent convergence is achieved within 5 iterations for typical noise levels (5%).

Conclusions:

  • The proposed IDIC/HR-EBSD framework achieves accuracy and robustness comparable to state-of-the-art methods.
  • This general framework provides a foundation for future advancements in EBSP analysis and absolute HR-EBSD.
  • The direct, one-step approach simplifies and enhances the reliability of HR-EBSD measurements.