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Efficient isogeometric Boundary Element simulation of elastic domains containing thin inclusions

Vincenzo Mallardo, Christian Dunser, Gernot Beer

TL;DR

The paper addresses efficient and accurate elastic simulations of domains containing thin, highly-anisotropic inclusions using an isogeometric boundary element framework. It introduces an initial-stress approach to model material contrasts, NURBS-based geometry and field discretisation, and robust volume and surface integration strategies that handle singular and nearly-singular kernels, especially for extreme aspect ratios. The methodology is validated through three test cases, including curved geometries and curved inclusions, demonstrating accurate displacement and stress results and favorable comparison with analytical solutions or FEM. The work advances practical, boundary-only modeling of complex multi-material elastic problems and highlights the method’s suitability for reinforced civil/mechanical systems and underground constructions, with potential extension to elasto-plastic behavior via the initial-stress formulation.

Abstract

This paper is concerned with the Boundary Element simulation of elastic domains that contain thin inclusions that have elastic material properties, which are different to the domain. With thin inclusions we mean inclusions with extreme aspect ratios, i.e. where one dimension is much smaller than the other ones. Examples of this are reinforcements in civil/mechanical engineering or concrete linings in underground construction. The fact that an inclusion has an extreme aspect ratio poses a challenge to the numerical integration of the arising singular integrals and novel approaches are presented to deal with it. Several examples demonstrate the efficiency and accuracy of the proposed methods and show that the results are in good agreement with analytical and other numerical solutions.

Efficient isogeometric Boundary Element simulation of elastic domains containing thin inclusions

TL;DR

The paper addresses efficient and accurate elastic simulations of domains containing thin, highly-anisotropic inclusions using an isogeometric boundary element framework. It introduces an initial-stress approach to model material contrasts, NURBS-based geometry and field discretisation, and robust volume and surface integration strategies that handle singular and nearly-singular kernels, especially for extreme aspect ratios. The methodology is validated through three test cases, including curved geometries and curved inclusions, demonstrating accurate displacement and stress results and favorable comparison with analytical solutions or FEM. The work advances practical, boundary-only modeling of complex multi-material elastic problems and highlights the method’s suitability for reinforced civil/mechanical systems and underground constructions, with potential extension to elasto-plastic behavior via the initial-stress formulation.

Abstract

This paper is concerned with the Boundary Element simulation of elastic domains that contain thin inclusions that have elastic material properties, which are different to the domain. With thin inclusions we mean inclusions with extreme aspect ratios, i.e. where one dimension is much smaller than the other ones. Examples of this are reinforcements in civil/mechanical engineering or concrete linings in underground construction. The fact that an inclusion has an extreme aspect ratio poses a challenge to the numerical integration of the arising singular integrals and novel approaches are presented to deal with it. Several examples demonstrate the efficiency and accuracy of the proposed methods and show that the results are in good agreement with analytical and other numerical solutions.

Paper Structure

This paper contains 42 sections, 107 equations, 22 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Governing integral equations
  4. Discretisation of the boundary integrals
  5. Geometry definition of finite patches
  6. Geometry definition of infinite patches
  7. Geometrical discretisation of Volume integrals
  8. Mapping
  9. Local coordinate system for defining stresses/strains
  10. Approximation of the unknown boundary values
  11. Approximation of values inside an inclusion
  12. Discretised integral equations
  13. Computation of values at grid points inside the inclusion
  14. Solution procedure
  15. Iterative solution for elastic effects
  16. ...and 27 more sections

Figures (22)

  • Figure 1: A finite patch with control points (numbered squares). Left: in the global, right: in the local coordinate system. Also shown is the "outward normal".
  • Figure 2: Example of an infinite patch. Left in the global and right in the local coordinate system
  • Figure 3: An inclusion showing coloured bounding surfaces. Right: in local $\boldsymbol{s}$ coordinate system. Left: in global $\boldsymbol{x}$ coordinate system with control points and local axes.
  • Figure 4: Example of generation of basis functions for approximation of the unknown. Left: Linear basis functions for the geometry, Right: Refined functions by 2 knot insertions at 0.4 and 0.6
  • Figure 5: Example of generation of basis functions for approximation of the unknown. Left: Linear basis functions for the geometry, Right: Refined functions by one order elevation
  • ...and 17 more figures