In FEM with bubble functions, the field ($\boldsymbol{u}$) is approximated as a linear combination of the standard one ($\tilde{\boldsymbol{u}}$) plus the bubble field ($\boldsymbol{u}^b$). That is, \begin{equation} \boldsymbol{u} = \tilde{\boldsymbol{u}} + \boldsymbol{u}^b. \end{equation}

Considering the addition of new basis functions and DOFs, I wonder about the partition of unity property.

Will it be violated? If yes, what are the potential consequences? If not, are the basis functions scaled accordingly? Appreciate any insights and concerned references.

  • 2
    $\begingroup$ What is your definition of the partition of unity property? That the constant function $u(x) = 1$ is in the span of the basis functions? If so, then enriching a FE basis that already had the partition of 1 property with bubble functions doesn't violate it -- you can just take whatever basis coefficients give you back the constant function 1 and set all the bubble DoFs to 0. $\endgroup$ Jan 6, 2023 at 20:25
  • $\begingroup$ Thank you @DanielShapero! I was referring to $\sum_i N_i=1$, $N_i$ as the basis functions. My confusion, which led to this question, was that the basis functions are used without scaling in many papers, leading to incorrect mass matrices. For example, doi.org/10.1016/j.cma.2007.10.007 $\endgroup$
    – Chenna K
    Jan 6, 2023 at 23:48
  • 1
    $\begingroup$ Gotcha, I understand now. I think the most important thing, for example to ensure global conservation, is to have that $1 \in \text{span}\{N_i\}$. (I'm mostly sure about this but I could be wrong.) The condition that $\sum_iN_i = 1$ is sufficient but not necessary to guarantee that $1 \in \text{span}\{N_i\}$ and indeed adding bubble functions, scaled or not, violates the one condition but not the other. $\endgroup$ Jan 7, 2023 at 1:04
  • $\begingroup$ I think if the partition of unity is satisfied, then correct rigid body motion and mass conservation are ensured automatically. If the basis functions are not scaled/modified appropriately, then the PU property is violated. While it might still capture the constant function as you mentioned, mass conservation is lost. I am surprised to see that many papers don't scale the basis functions correctly. $\endgroup$
    – Chenna K
    Jan 7, 2023 at 10:53
  • $\begingroup$ Can you explain how mass can be not conserved in Lagrangian (material) formulation? Paper referenced above is for such a case? $\endgroup$
    – likask
    Jan 8, 2023 at 8:55


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