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Mesh analysis becomes simpler when analyzing circuits with current sources, whether independent or dependent. The presence of current sources reduces the number of equations required for analysis. Two cases illustrate this:
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In the analysis of structural systems, it is common to encounter members subjected to various forces and couple moments. Simplifying these systems can make the analysis more manageable and easier to understand. One approach to achieve this simplification is by moving a force to a point O that does not lie on its line of action and adding a couple with a moment equal to the moment of the force about point O.
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The Meshless Analysis of Scale-Dependent Problems for Coupled Fields.

Jan Sladek1, Vladimir Sladek1, Pihua H Wen2

  • 1Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia.

Materials (Basel, Switzerland)
|June 6, 2020
PubMed
Summary
This summary is machine-generated.

The meshless local Petrov-Galerkin method analyzes 2D flexoelectricity and thermoelectricity, incorporating scale-dependent size effects. This advanced numerical technique solves complex multiphysical problems using higher-grade continuum theories.

Keywords:
MLS approximationgradients of electric intensity vectorgradients of strainshigher-grade heat flux

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Area of Science:

  • Computational Mechanics
  • Materials Science
  • Solid Mechanics

Background:

  • Flexoelectricity and higher-grade thermoelectricity are multiphysical phenomena exhibiting scale-dependent behavior.
  • Classical continuum theories often fail to capture size effects crucial in these advanced material behaviors.

Purpose of the Study:

  • To develop and apply the meshless local Petrov-Galerkin (MLPG) method for analyzing 2D flexoelectric and higher-grade thermoelectric problems.
  • To incorporate scale-dependent size effects through strain/electric field gradients and higher-grade heat flux.

Main Methods:

  • Derivation of governing equations using a variational principle within higher-grade continuum theory.
  • Numerical implementation using the MLPG method with local weak-forms on fictitious subdomains.
  • Approximation of physical fields via the moving least-squares (MLS) scheme.

Main Results:

  • Successfully applied the MLPG method to solve coupled partial differential equations (PDEs) for the analyzed phenomena.
  • Obtained a system of algebraic equations for nodal unknowns by applying spatial approximations to local integral equations and boundary conditions.

Conclusions:

  • The MLPG method provides an effective numerical framework for analyzing multiphysical, scale-dependent problems in flexoelectricity and higher-grade thermoelectricity.
  • The inclusion of higher-order gradients in the governing equations and numerical treatment is essential for capturing size effects.