Alejandro Marcos Aragon
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Apr
13
comment vector-valued function space definition except for measure zero
@PaulSinclair I don't think so, and thus I ended up asking this question as I don't understand what's going on. One example: "An extended finite element method with higher-order elements for curved cracks" by Stazi et al., for the variational formulation they just say $\mathbf{u} \in \mathcal{U}, \mathcal{U} = \left\{ \left. \mathbf{u} \right| \mathbf{u} \in C^0 \text{ except on }\Gamma_{\text {cr}}, \mathbf{u} = \bar{\mathbf{u}} \text{ on } \Gamma_{\text u} \right\}$. You see they just say the function needs to be continuous except on the crack. Also $\delta \mathbf{v} \in \mathcal{U}_0 $.
Apr
13
comment vector-valued function space definition except for measure zero
@PaulSinclair there's an entire research community (XFEM) that models cracks with finite element meshes that are independent of the geometry of the meshes. They don't enforce anything at the cracks, but just use functions that are discontinuous in the displacement (by adding to the finite element approximation a Heaviside function). You can even model cracks with the standard finite element method by having double nodes along the cracks, and again, you're not enforcing anything there (so I guess what's enforced is a zero traction).
Apr
12
comment vector-valued function space definition except for measure zero
@PaulSinclair I understand what you mean now. The problem is that I am not expecting to enforce BCs on $B$, but to use functions that have a jump in the displacement field. So I'm still confused by how to define $\mathcal{V} \left( \Omega \right)$ to do this. According to your message, this space can still be defined as $ \mathcal{V} = \left\{\mathbf{v}|v_i \in H^1\left(\Omega\right)\right\}$ because $B$ has zero measure, but I'm not sure this is strictly correct. Something's missing since I'm expecting to use functions that are discontinuous across the crack. Does this make any sense?
Apr
11
comment vector-valued function space definition except for measure zero
Hi Paul, you lost me. I don't see how $\Omega \setminus \cup_j \Gamma_j^c$ includes $\cup_j \Gamma_j^c$. For me $\Gamma = \overline \Omega \setminus \Omega$ is the external boundary (without the cracks), and thus $\Omega$ contains the cracks, doesn't it? Then $\Omega \setminus \cup_j \Gamma_j^c$ is the solid (with no cracks). Am I wrong?
Apr
11
asked vector-valued function space definition except for measure zero
Dec
16
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Dec
16
asked priority between set subtraction and set union
Feb
19
comment Less restrictive set intersection
Hi vonbrand, thanks for your answer. Got the book, but I can't seem to find such magical equation. Could you please point out which one you're talking about?
Feb
19
comment Less restrictive set intersection
Given sets $A_1, \dots, A_n$, I would like to find the set of elements that belong to any intersection of sets taken by two, i.e., $\bigcup_{i \neq j} A_i \cap A_j$. Is there a name for this?
Feb
19
revised Less restrictive set intersection
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Feb
19
asked Less restrictive set intersection