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Hooman December 3, 2012 13:21

FVM for Solids
Can anyone suggest a good paper or book on the use of finite volume methods for solids?

Rami December 4, 2012 04:25


Originally Posted by Hooman (Post 395511)
Can anyone suggest a good paper or book on the use of finite volume methods for solids?

  1. Which equations are you wishing to solve? Heat transfer? Stress analysis? Other?
  2. Usually FEM is more common than FVM for solids, although FVM may certainly be used too.

Engr.RZA December 4, 2012 12:10


I think FVM is preferably right choice to analyze solid structures.

Most of the FVM literature is available in the perspective of computational fluid dynamics. So you should consult CFD books.

1. Computational Fluid Dynamics: Principles & Applications By J. Blazek

2. Computational Fluid Dynamics By Hoffman

Hooman December 5, 2012 05:04

I am interested in the governing equations and their discretizations for stress analysis problems. I'm familiar with fvm for fluids, solid equation are somewhat different. In particular I am interested in the stress term ...

Rami December 5, 2012 06:09

Hi Hooman,

The equations used in stress analysis for solids are nearly identical to those of the momentum equations for viscous fluid flow, except that for solids u_i stands for the displacement vector, whereas for fluids - it is the velocity (the stress is related to strain for solid and to strain-rate in fluids).

You should also choose whether you use an Eulerian or Lagrangian formulation. The Lagrangian formulation in Cartesian coordinates is simply \rho Du_i / Dt = \sigma_{ik,k} + \rho G_i (using tensor notation where D/Dt is the material derivative, comma is the covariant derivative - which degenerates to simple spatial derivative in Cartesian coordinates, u_i is the displacement, \sigma_{ik} is the stress, G_i is the body force and \rho is the density).

If you treat small displacement and small strain, the various stress and strain measures (Cauchy, Piola-Kirchhoff, Green, Lagrange, etc.) are identical and their relation to each other are practically the same as those for fluids, e.g. for elasticity \sigma_{ik} =  \lambda \epsilon_{mm}  \delta_{ik} + 2 \mu  \epsilon_{ik}, where \lambda and \mu are the Lame modulii and \epsilon_{ik} = 1/2 ( u_{i,k} + u_{k,i} ).

Nevertheless, I still suggest to look into FEM, which has much more literature on solids. It is in many ways similar (and more consistent and general) to FVM. Actually, FVM can be viewed as a special case of FEM, using piecewise-constant shape-functions and some additional minor approximations.

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