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1 results

  • 4 - Trefftz Trial Functions for Computational Grains for Planar and 3D Problems

  • Leiting Dong, Beihang University, China, Satya N. Atluri, University of California, Irvine
  • Book:
    Computational Grains
    Published online:
    05 October 2023
    Print publication:
    19 October 2023, pp 45-71

  • Summary

    In this chapter, Trefftz trial functions which satisfy identically all the governing equations of linear elasticity in 2D and 3D problems are summarized. These Trefftz functions are later used in conjunction with boundary variational principles (since all the field equations are satisfied identically inside the Voronoi cell elements), to construct planar and 3D Computational Grains to directly model statistically equivalent representative volume elements (SERVEs) of heterogeneous materials at the microscale. In as much as the Trefftz functions are used as trial solutions, this modeling captures the correct and accurate stress solutions in the matrix, inclusions, and at their interfaces. The presented Trefftz solutions include: (1) Muskhelishvili’s complex functions for 2D problems,(2) Papkovich-Neubar solutions for 3 D problems,and (3) Harmonic functions in spherical coordinates, cylindrical coordinates, and ellipsoidal coordinates.