I want to solve a small deformation solid structure problem applying periodic boundary conditions in FEM. The geometry is a square and the equations are: $$ \text{div} \, \sigma = 0 \\ \sigma = f(\epsilon)\\ \epsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i}) \\ u^+ - u^- = c \\ \sigma^+ \cdot \hat{n} + \sigma^- \cdot \hat{n} = 0 $$

The variables with the $.^+$ or $.^-$ correspond to opposite sides in the square.

A weak form the equilibrium equation can be written as: $$ \int_{V} \text{div} \, \sigma: \epsilon(\delta v) \,dV = 0 \\ $$ where $\delta v$ is a test function. Discretizing by FE and applying Newton-Raphson iterative scheme I wrote a residual of the form:

$$ r = \left[ \begin{array}{c} \int_{V} \text{div} \, \sigma: \epsilon(\delta v_j) \,dV \quad \text{ for } j = 1...N_\text{int} - N_\text{ext}/2\\ u_j^+ - u_j^- - c \quad \text{ for } j = 1...N_\text{ext}/2\\ \end{array} \right] $$

Question: The weak formulation above does not have the surface contribution so the correct form is: $$ \int_{V} \text{div} \, \sigma: \epsilon(\delta v) \,dV + \int_{\partial V} \sigma \cdot \hat{n} \, \,dS = 0 \\ $$ If you take the condition $\sigma^+ \cdot \hat{n} + \sigma^- \cdot \hat{n} = 0$, how does the condition $\sigma^+ \cdot \hat{n} + \sigma^- \cdot \hat{n} = 0$ affects that term in order to by enforce ? (It seems that that term should vanish but that did not work for me)

If this is not the correct way how do I enforce the conditions $\sigma^+ \cdot \hat{n} + \sigma^- \cdot \hat{n} = 0$ ?

Note: Here I have the octave code run_periodic.m for you to see. The ass_periodic.m function do the job of assembly the residual and jacobian and tries to enforce the boundary conditions.

  • $\begingroup$ That boundary condition should be satisfied implícitly $\endgroup$ – nicoguaro Jan 27 '18 at 17:58
  • $\begingroup$ Indeed. Your weak form is wrong: it needs to contain boundary terms that will allow you to include the periodic boundary conditions. $\endgroup$ – Wolfgang Bangerth Jan 27 '18 at 20:43
  • $\begingroup$ I edited the question again adding the surface force contribution. I tried to enforce that "anti-periodic" boundary condition vanishing that term but that did not work. $\endgroup$ – gagiuntoli Jan 28 '18 at 9:41
  • $\begingroup$ What concretely does "did not work" mean? $\endgroup$ – Wolfgang Bangerth Jan 29 '18 at 22:54
  • $\begingroup$ That I do not get periodic boundary conditions by doing that. I solved the problem applying Lagrange multipliers. Then I will edited the solution to the question. $\endgroup$ – gagiuntoli Jan 30 '18 at 15:45

I could arrive to the solution at the end applying the *Lagrange Multiplier Method. The surface term at the weak form: $$ \int_{V} \text{div} \, \sigma: \epsilon(\delta v) \,dV + \int_{\partial V} \sigma \cdot \hat{n} \, \,dS = 0 \\ $$ should vanish, i.e.: $$ \int_{V} \text{div} \, \sigma: \epsilon(\delta v) \,dV = 0 \\ $$ because in these problem we are not applying Neumann or Robin Boundary conditions, we are a applying a Dirichlet type: $$ u^+ - u^- = c $$ that we don't know at first the value of $u^+$ and $u^-$. The other thing we know is that the external force applied should be the same : $$\sigma^+ \cdot \hat{n} + \sigma^- \cdot \hat{n} = 0$$ This mean that is logical to add a Lagrange multiplier in the force term (in the correct degrees of freedom and with the right sign) as:

$$ \int_{V} \text{div} \, \sigma: \epsilon(\delta v_j) \,dV - \lambda_j = 0 \text{ for } j = 1...P $$ By this way we have added $P$ more unknown $\lambda_j$ to the system. To solve the problem we add the displacement periodic boundary conditions: $$u_j^+ - u_j^- = c_j \text{ for } j = 1...P.$$ There are other techniques that can be used such as the Master Slave Method, Penalty Method, Augmented Lagrangian, between others. I didn't go deeper with them yet.

Here I give a picture of how a body is deformed under a certain value of $c$ using the Lagrange Multiplier Method :

Body deformed using periodic boundary conditions

The code to obtained that solution can be seen Here and can be executed as:

octave run_periodic_lm.m

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