A Gauss-Kronrod-Trapezoidal integration scheme for modeling biological tissues with continuous fiber distributions

Chieh Hou, Gerard A. Ateshian

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Fibrous biological tissues may be modeled using a continuous fiber distribution (CFD) to capture tension–compression nonlinearity, anisotropic fiber distributions, and load-induced anisotropy. The CFD framework requires spherical integration of weighted individual fiber responses, with fibers contributing to the stress response only when they are in tension. The common method for performing this integration employs the discretization of the unit sphere into a polyhedron with nearly uniform triangular faces (finite element integration or FEI scheme). Although FEI has proven to be more accurate and efficient than integration using spherical coordinates, it presents three major drawbacks: First, the number of elements on the unit sphere needed to achieve satisfactory accuracy becomes a significant computational cost in a finite element (FE) analysis. Second, fibers may not be in tension in some regions on the unit sphere, where the integration becomes a waste. Third, if tensed fiber bundles span a small region compared to the area of the elements on the sphere, a significant discretization error arises. This study presents an integration scheme specialized to the CFD framework, which significantly mitigates the first drawback of the FEI scheme, while eliminating the second and third completely. Here, integration is performed only over the regions of the unit sphere where fibers are in tension. Gauss–Kronrod quadrature is used across latitudes and the trapezoidal scheme across longitudes. Over a wide range of strain states, fiber material properties, and fiber angular distributions, results demonstrate that this new scheme always outperforms FEI, sometimes by orders of magnitude in the number of computational steps and relative accuracy of the stress calculation.

Original languageEnglish (US)
Pages (from-to)883-893
Number of pages11
JournalComputer methods in biomechanics and biomedical engineering
Issue number8
StatePublished - Jun 10 2016
Externally publishedYes

Bibliographical note

Funding Information:
Research reported in this publication was supported by the National Institute of General Medical Sciences of the National Institutes of Health under award numbers R01GM083925. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institutes of Health.

Publisher Copyright:
© 2015 Taylor & Francis.


  • Soft tissue mechanics
  • fiber density distribution
  • fibrous tissues
  • finite element analysis
  • integration on the sphere


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