On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

Advanced Modeling and Simulation in Engineering Sciences

Table 4 Expressions of F(x) for weakly discontinuous enrichments with linear and quadratic approximations resulting from Table 3 for the one dimensional bi-material bar problem

	Sukumar [14]	Moes [27]	Belytschko [28]
Linear F(x)	\(\left\| {x-x_0 } \right\| \)	\(x+x_0 -2\frac{xx_0 }{L}-\left\| {x-x_0 } \right\| \)	\(\left\| {x-x_0 } \right\| -\left\| {x_J -x_0 } \right\| \)
Quadratic F(x)	\(\left\| {x-x_0 } \right\| \)	\(\bullet \,\,\hbox {If }\hbox {L}/2\ge x_{0}\) \(x_0 +\frac{x}{L}\left( {L-6x_0 } \right) +4x_0 \left( {\frac{x}{L}} \right) ^{2}-\left\| {x-x_0 } \right\| \)	\(\left\| {x-x_0 } \right\| -\left\| {x_J -x_0 } \right\| \)
Quadratic F(x)	\(\left\| {x-x_0 } \right\| \)	\( \bullet \,\,\hbox {If }\hbox {L}/2\le x_{0}\) \(x_0 +\frac{x}{L}\left( {2x_0 -3L} \right) +4\left( {L-x_0 } \right) \left( {\frac{x}{L}} \right) ^{2}-\left\| {x-x_0 } \right\| \)	\(\left\| {x-x_0 } \right\| -\left\| {x_J -x_0 } \right\| \)

	Sukumar [14]	Moes [27]	Belytschko [28]
Linear F(x)	\(\left\| {x-x_0 } \right\| \)	\(x+x_0 -2\frac{xx_0 }{L}-\left\| {x-x_0 } \right\| \)	\(\left\| {x-x_0 } \right\| -\left\| {x_J -x_0 } \right\| \)
Quadratic F(x)	\(\left\| {x-x_0 } \right\| \)	\(\bullet \,\,\hbox {If }\hbox {L}/2\ge x_{0}\) \(x_0 +\frac{x}{L}\left( {L-6x_0 } \right) +4x_0 \left( {\frac{x}{L}} \right) ^{2}-\left\| {x-x_0 } \right\| \)	\(\left\| {x-x_0 } \right\| -\left\| {x_J -x_0 } \right\| \)
Quadratic F(x)	\(\left\| {x-x_0 } \right\| \)	\( \bullet \,\,\hbox {If }\hbox {L}/2\le x_{0}\) \(x_0 +\frac{x}{L}\left( {2x_0 -3L} \right) +4\left( {L-x_0 } \right) \left( {\frac{x}{L}} \right) ^{2}-\left\| {x-x_0 } \right\| \)	\(\left\| {x-x_0 } \right\| -\left\| {x_J -x_0 } \right\| \)