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Related Concept Videos

Finding Volume Using Cross-Sectional Area01:24

Finding Volume Using Cross-Sectional Area

For solids whose cross-sectional areas vary in a predictable way, volume can be determined by integrating these areas along an axis perpendicular to the slices. This approach is particularly useful for polyhedral solids, where classical geometric formulas may not be immediately applicable. A tetrahedron provides a clear example of how cross-sectional integration can be applied to a three-dimensional object with continuously changing geometry.Consider a tetrahedron with height h and a base that...
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The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
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Volume calculation often begins with simple geometric solids. For example, the volume of a rectangular box is obtained by multiplying the area of its base by its height. This straightforward approach relies on the fact that the cross-sectional area of the box remains constant throughout its length. Many real-world objects, however, do not have uniform cross-sections, and their volumes cannot be determined using elementary geometric formulas.To address this limitation, the Slicing Method...
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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
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The volume of a fuel tank mounted on the wing of a jet aircraft can be modeled using the concept of solids of revolution. In this case, the tank is formed by rotating a two-dimensional region, defined by a mathematical function, about the x-axis. The region extends along the axis from zero to two meters, and the resulting three-dimensional shape is symmetric about the axis of rotation. Because the boundary curve lies directly against the axis, the disk method is an appropriate technique for...
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The midpoint rule for a double integral provides a practical method for estimating volume over a rectangular region when the surface height varies continuously. In civil engineering, this method is useful for approximating the amount of soil to be moved when planning a road across uneven terrain. The road footprint is represented as a rectangle in the xy-plane. At the same time, the terrain elevation above a flat reference level is described by a continuous height function f(x,y). The objective...

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An efficient algorithm for computing hypervolume contributions.

Karl Bringmann1, Tobias Friedrich

  • 1Universität des Saarlandes, Saarbrücken, Germany. s9kabrin@stud.uni-saarland.de

Evolutionary Computation
|June 22, 2010
PubMed
Summary
This summary is machine-generated.

This study introduces faster algorithms for multi-objective evolutionary algorithms (MOEAs) using the hypervolume indicator. The new methods efficiently identify solutions with minimal hypervolume contribution, improving computational speed for complex optimization problems.

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Area of Science:

  • Computational intelligence
  • Optimization algorithms
  • Multi-objective evolutionary algorithms (MOEAs)

Background:

  • The hypervolume indicator is crucial for sorting solutions in MOEAs.
  • Current MOEAs typically remove solutions with the smallest hypervolume loss.
  • Efficiently calculating hypervolume contributions is computationally challenging.

Purpose of the Study:

  • To develop novel algorithms for determining minimal hypervolume contributions in MOEAs.
  • To improve the computational efficiency of hypervolume-based selection processes.
  • To address the computational expense of selecting multiple solutions (lambda > 1) for removal.

Main Methods:

  • Developed a new algorithm to find a solution with minimal hypervolume contribution in O(n(d/2) log n) time for d > 2.
  • Introduced the first hypervolume algorithm to directly calculate contributions for sets of lambda solutions.
  • Analyzed algorithms that remove lambda > 1 solutions, showing inefficiencies in iterative selection.

Main Results:

  • The new algorithm offers significant speed improvements over existing methods: a factor of n for d > 3 and √n for d = 3.
  • The algorithm for calculating contributions of sets of lambda solutions adds an O(n(lambda)) term to the runtime, instead of a multiplicative factor.
  • Demonstrated that iterative selection of lambda solutions can be suboptimal compared to optimal set selection.

Conclusions:

  • The proposed algorithms substantially enhance the efficiency of hypervolume-based MOEAs.
  • These advancements enable faster and more effective multi-objective optimization.
  • The findings provide a more computationally tractable approach to hypervolume calculation in complex optimization scenarios.