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In fluid mechanics, shear stresses arise from viscosity, which represents a fluid's internal resistance to deformation. For low-viscosity fluids, like water, these stresses are minimal, simplifying flow analysis by allowing the fluid to be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent, leaving only normal stresses, which act perpendicularly to fluid elements. Notably, pressure — defined as the negative of the normal stress — remains uniform across...
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An optimized finite element extrapolating method for 2D viscoelastic wave equation.

Hong Xia1, Zhendong Luo2

  • 1School of Control and Computer Engineering, North China Electric Power University, No. 2, Bei Nong Road, Changping District, Beijing, 102206 China.

Journal of Inequalities and Applications
|October 6, 2017
PubMed
Summary
This summary is machine-generated.

A new optimized finite element extrapolating (OFEE) method was developed for the 2D viscoelastic wave equation. This method demonstrates feasibility and efficiency, validated by numerical examples matching theoretical results.

Keywords:
classical finite element methoderror estimateoptimized finite element extrapolating methodproper orthogonal decomposition method

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Area of Science:

  • Computational Mathematics
  • Wave Phenomena
  • Viscoelasticity

Background:

  • The 2D viscoelastic wave equation models complex material behaviors.
  • Classical finite element (FE) methods require robust analysis for stability and convergence.
  • Efficient numerical techniques are crucial for simulating wave propagation in viscoelastic media.

Purpose of the Study:

  • To introduce and analyze a classical finite element (FE) method for the 2D viscoelastic wave equation.
  • To develop and validate an optimized FE extrapolating (OFEE) method using proper orthogonal decomposition (POD).
  • To confirm the feasibility and efficiency of the OFEE method through numerical verification.

Main Methods:

  • Classical finite element (FE) analysis for existence, stability, and convergence.
  • Optimized FE extrapolating (OFEE) method incorporating proper orthogonal decomposition (POD).
  • Implementation procedure detailing the OFEE method's steps.
  • Numerical example for validating theoretical results.

Main Results:

  • The classical FE method's existence, stability, and convergence were analyzed.
  • The OFEE method's existence, stability, and convergence were established.
  • Numerical results confirmed the OFEE method's accuracy and efficiency.
  • The OFEE method was shown to be a feasible approach for the 2D viscoelastic wave equation.

Conclusions:

  • The classical FE method provides a foundation for analyzing the 2D viscoelastic wave equation.
  • The OFEE method, enhanced by POD, offers a significant improvement in solving this equation.
  • Numerical verification confirms the OFEE method's practical applicability and efficiency.