Table 1 Strong discontinuity enrichments comparison
X-FEM [7] | Domain enrich. [33] | Shifted enrich. [26] | |
|---|---|---|---|
Linear shape functions on enriched nodes |
 |
 |
 |
 |
 |
 | |
Displacement approximation | \({\begin{array}{l} \left. {u^{h}} \right| _{\Omega +} =\sum \limits _{j\in I} {a_j \varPhi _j +\sum \limits _{j\in I_H } {b_j \varPhi _j } } \\ \left. {u^{h}} \right| _{\Omega -} =\sum \limits _{j\in I} {a_j \varPhi _j -\sum \limits _{j\in I_H } {b_j \varPhi _j } } \\ \end{array}}\) | \({\begin{array}{l} \left. {u^{h}} \right| _{\Omega +} =\sum \limits _{i\in I/I_H } {a_i } \varPhi _i +\sum \limits _{j\in I_H } {\alpha _{j,1} \varPhi _{j,1} } \\ \left. {u^{h}} \right| _{\Omega -} =\sum \limits _{i\in I/I_H } {a_i } \varPhi _i +\sum \limits _{j\in I_H } {\alpha _{j,2} \varPhi _{j,2} } \\ \end{array}}\) | \({\begin{array}{l} \left. {u^{h}} \right| _{\Omega +} =\sum \limits _{i\in I} {c_i } \varPhi _i -\sum \limits _{j\in I_H \cap \Omega -} {2d_j \varPhi _{j,2} } \\ \left. {u^{h}} \right| _{\Omega -} =\sum \limits _{i\in I} {c_i } \varPhi _i +\sum \limits _{j\in I_H \cap \Omega +} {2d_j \varPhi _{j,1} } \\ \end{array}}\) |
Jump approximation | \(\Delta u^{h}=\sum \limits _{j\in I_H } {2b_j \varPhi _j } \) | \(\Delta u^{h}=\sum \limits _{j\in I_H } {\alpha _{j,1} \varPhi _{j,1} -\alpha _{j,2} \varPhi _{j,2} } \) | \(\Delta u^{h}=-\sum \limits _{j\in I_H \cap \Omega -} {2d_j \varPhi _{j,2} } -\sum \limits _{j\in I_H \cap \Omega +} {2d_j \varPhi _{j,1}} \) |