
[Sponsors] 
How can be calculate a surface integral in the finite element method? 

LinkBack  Thread Tools  Display Modes 
September 9, 2010, 16:46 
How can be calculate a surface integral in the finite element method?

#1 
Member
Hector Redal
Join Date: Aug 2010
Location: Madrid, Spain
Posts: 93
Rep Power: 8 
Hi,
I would like to post the following question about how to calculate the integral of a surface in the finite element method. As far as I know, when calculating, for example the stiff matrix, in the finite element formulation, the integral for such calculation is extended over the whole domain, for example a cube. I am developing a software for cfd that uses the finite element formulation, and there are some integrals that should be performed over the surface of the domain (These integrals are surface integrals). For integrals that are applied over the domain, the jacobi is used for differential transformation. My doubt arises when trying to calculate the surface integrals. As far as I know, the jacobi could not be used. Which kind of differential transformation should be used? I would appreciate any kind of help / suggestion. Thanks in advance, Hector. 

September 13, 2010, 05:25 

#2 
Senior Member
Rami BenZvi
Join Date: Mar 2009
Posts: 148
Rep Power: 9 
Hello Hector,
I assume (for simplicity) you use brick elements of any order. You integrate as usual (see below for details), except the following: Suppose you use the isoparametric coordinates (r,s,t) with the appropriate transformation to the (x,y,z) coordinates, and suppose your surface is defined by, say, r=1. The you simply use the shape functions (and/or their derivatives) as appearing in the integrand with the substitution of r=1, and instead of the Jacobian relating dV (the volume differential) to dr*ds*dt you will had a relation between dA (the area differential) and ds*dt. That's it. Now integrate either exactly (if possible) or numerically (e.g., Gauss quadrature on the square 1<s<1, 1<t<1). Good luck, Rami 

September 13, 2010, 15:58 

#3 
Member
Hector Redal
Join Date: Aug 2010
Location: Madrid, Spain
Posts: 93
Rep Power: 8 
Hello Rami,
Thank you for your answer. I will try to follow your instructions. Thanks, Hector. 

March 13, 2015, 09:25 

#4 
New Member
Samir
Join Date: Jun 2014
Posts: 4
Rep Power: 4 
hey !
I have been trying to formulate the FEM equation to calculate the temperature across the 2D cross section.. but the extended volume is not in a form of cube, but in the form of a quadrant. I can't use axisymmetric elements... Any hints ? 

Thread Tools  
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Comparison: Finite Volume Method vs. Analytic Method  mfry  Main CFD Forum  1  April 20, 2010 14:40 
Derivation of Momentum Equation in Integral Form  Demonwolf  Main CFD Forum  2  October 29, 2009 20:53 
Calculate velocity inflow on a 2D surface in 3D  quarkz  Main CFD Forum  4  May 10, 2009 05:54 
SolidLiquid Two Phase Flow Numerical Simulation with the Finite Element Method  Qing Hao  Main CFD Forum  2  January 9, 1999 00:31 
What is C.V. based finite element method  CH Kuo  Main CFD Forum  3  November 5, 1998 10:07 