Shape functions

This chapter describes the shape functions for the finite elements.

Understanding shape function notations

The notations used in shape functions are listed below:

• u: displacement in x (or s) direction.
• v: displacement in y (or t) direction.
• w: displacement in z (or s) direction.
• ${\theta }_x$: Rotation about x direction.
• ${\theta }_y$: Rotation about y direction.
• ${\theta }_z$: Rotation about z direction.
• $A_x$: x-component of vector magnetic potential.
• $A_y$: y-component of vector magnetic potential.
• $A_z$: z-component of vector magnetic potential.
• C: Concentration.
• P: Pressure.
• T: Temperature.
• V: Electric potential or source current.
• For the shell element, the u and v represent in-plane motions, and w denotes the out-of-plane motion.

3D shell elements

This section describes the shape functions for 3D shell elements that are applied in the WELSIM application.

3-Node triangle

The shape functions for the 3-node triangular shell elements are:

$$u=u_0{L}_0+u_1{L}_1+u_2{L}_2$$

$$v=v_0{L}_0+v_1{L}_1+v_2{L}_2$$

$$w=w_0{L}_0+w_1{L}_1+w_2{L}_2$$

$$A_x=A_{x0}{L}_0+A_{x1}{L}_1+A_{x2}{L}_2$$

$$A_y=A_{y0}{L}_0+A_{y1}{L}_1+A_{y2}{L}_2$$

$$A_z=A_{z0}{L}_0+A_{z1}{L}_1+A_{z2}{L}_2$$

$$T=T_0{L}_0+T_1{L}_1+T_2{L}_2$$

$$V=V_0{L}_0+V_1{L}_1+V_2{L}_2$$

6-Node triangle

The shape functions for the 6-node triangular shell elements are:

$$u=u_0(2L_0-1)L_0+u_1(2L_1-1)L_1+u_2(2L_2-1)L_2+u_3(4L_0{L}_1)+u_4(4L_1{L}_2)+u_5(4L_2{L}_0)$$

$$v=v_0(2L_0-1)L_0+v_1(2L_1-1)L_1+v_2(2L_2-1)L_2+v_3(4L_0{L}_1)+v_4(4L_1{L}_2)+v_5(4L_2{L}_0)$$

$$w=w_0(2L_0-1)L_0+w_1(2L_1-1)L_1+w_2(2L_2-1)L_2+w_3(4L_0{L}_1)+w_4(4L_1{L}_2)+w_5(4L_2{L}_0)$$

3D solid elements

This section describes the shape functions for the 3D solid elements that are applied in the WELSIM application.

4-Node tetrahedra

The 4-node tetrahedra is also called liner tetrahedra element. The shape functions are:

$$u=u_0{L}_0+u_1{L}_1+u_2{L}_2+u_3{L}_3$$

$$v=v_0{L}_0+v_1{L}_1+v_2{L}_2+v_3{L}_3$$

$$w=w_0{L}_0+w_1{L}_1+w_2{L}_2+w_3{L}_3$$

10-Node tetrahedra

The 10-node tetrahedra is also called bilinear tetrahedra element. The shape functions are:

$$u=u_0(2L_0-1)L_0+u_1(2L_1-1)L_1+u_2(2L_2-1)L_2+u_3(2L_3-1)L_3+4u_4{L}_0{L}_1+u_5{L}_1{L}_2+u_6{L}_0{L}_2+u_7{L}_0{L}_3+u_8{L}_1{L}_3+u_9{L}_2{L}_3$$

$$v=...\text{(analogous to u)}$$

$$w=...\text{(analogous to u)}$$