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Divergence and Stokes' Theorems01:06

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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
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The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as the volume tends to zero. More practically, divergence measures how much a vector field spreads out or diverges from a given point. For an outgoing flux, conventionally, the divergence is positive. The diverging point is often called the "source" of the field. Meanwhile, the negative divergence of a vector field at a point means that the...
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An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...
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The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows:
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The divergence of a vector is a measure of how much the vector spreads out (diverges) from a point. For example, an electric field vector diverges from the positive charge and converges at the negative charge. The divergence of an electric field is derived using Gauss's law and is equal to the charge density divided by the permittivity of space. Mathematically, it is expressed as
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Singularity Functions for Bending Moment01:18

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Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented...
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LOCAL DIVERGENCE-FREE IMMERSED FINITE ELEMENT-DIFFERENCE METHOD COMPOSITE B-SPLINES を使った. このメソッドは,複合B-SPLINES を使った.

Lianxia Li1, Cole Gruninger1, Jae H Lee1

  • 1Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA.

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まとめ
この要約は機械生成です。

複合Bスライン (CBS) カーネルは,体積保存を維持し,精度を高め,計算コストを削減することにより,流体構造の相互作用シミュレーションを改善します. これらのカーネルは,追加の安定化なしで弾性固体を変形させるための安定的かつ効率的なシミュレーションを提供します.

キーワード:
浸水した境界プライマリ: 58F15, 58F17二次: 53C35複合Bスライン・カーネル (CBS)流体構造の相互作用浸された有限要素-有限差法イソトロピック・カーネルボリューム保存容積安定化

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科学分野:

  • 計算式流体力学 (CFD)
  • 流体構造相互作用 (FSI)
  • 数学的方法

背景:

  • 浸水境界 (IB) 方法は,流体と固体領域間の情報転送のために正規化されたデルタ関数に依存しています.
  • IB方法における従来の同位体カーネルは,往々にして,離散のない状態を維持できず,固体力学の非圧縮性エラーにつながります.
  • 既存の体積安定化技術は,圧縮不可能な弾性構造における大きな変形のシミュレーションに複雑さを加える.

研究 の 目的:

  • 複合Bスプライン (CBS) カーネルの性能を浸水境界シミュレーションで評価する.
  • 従来の同位体 (IBとB-spline) カーネルの体積保存と精度を比較する.
  • 大きな変形のためにCBSカーネルを使用する際に,追加の体積安定化の必要性を評価する.

主な方法:

  • 埋め込まれた境界枠内で複合Bスプライン (CBS) カーネルを実装し,テストしました.
  • 弾性帯,圧縮膜,圧縮ブロック,クック膜,傾いたチャネルフロー,チューレック・ホロン問題など,様々なフローシナリオを用いたベンチマーク試験を実施した.
  • パルス複製器で生体義肢心弁の流体構造相互作用モデルで 方法論を検証した.

主要な成果:

  • CBSカーネルは,従来の同位体カーネルと比較して,より優れた体積保存を示し,明示的な体積安定化の必要性を否定した.
  • 粗いグリッドのCBSカーネルの精度は,より細いグリッドのIBとB-splineカーネルの精度と比べられる.
  • CBSカーネルは,より小さなマッシュ比因数で性能を改善し,同位体カーネルよりも相対的なグリッド間隔の変動に敏感ではありませんでした.

結論:

  • 複合Bスラインカーネルは,流体構造相互作用の浸水境界シミュレーションのための安定した,正確な,効率的な代替案を提供します.
  • CBSカーネルは本質的に離散性のない性質を維持し,弾性固体の大きな変形を含むシミュレーションを簡素化します.
  • この研究は,複雑なFSI問題における特殊な体積安定化処理を避けるために,CBSのカーネルの使用を提唱している.