Shape Functions Solid 186

12.8. 3-D Solids This section contains shape functions for 3-D solid elements. These elements are available in a number of configurations, including certain combinations of the following features:

element shapes may be tetrahedra, pyramids, wedges, or bricks (hexahedra).

- if wedges or bricks, with or without extra shape functions (ESF)

with or without rotational degrees of freedom (RDOF)

with or without midside nodes

The wedge elements with midside nodes (15 node wedges) are either a condensation of the 20 node brick element or are based on wedge shape functions.

12.8.1. 4 Node Tetrahedra

This element is a condensation of an 8 node brick element such as SOLID5, FLUID30, SOLID45, or FLUID142

Figure 12.11 3-D Solid Elements

The resulting effective shape functions are:

(12–152)

(12–153)

(12–154)

(12–155)

(12–156)

(12–157)

(12–158)

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12.8.2. 10 Node Tetrahedra

These shape functions are for 10 node tetrahedron elements such as SOLID98 and SOLID92:

Figure 12.12 10 Node Tetrahedra Element

12.8.3. 5 Node Pyramids

This element is a condensation of an 8 node brick element.

Figure 12.13 8 Node Brick Element

(12–159)

(12–160)

(12–161)

(12–162)

(12–163)

(12–164)

(12–165)

(12–166)

(12–167)

(12–168)

(12–169)



The resulting effective shape functions are:

12.8.4. 13 Node Pyramids

These shape functions are for 13 node pyramid elements which are based on a condensation of a 20 node brick element such as SOLID95:

Figure 12.14 13 Node Pyramid Element

(12–170)

(12–171)



12.8.5. 6 Node Wedges without ESF

Figure 12.15 6 Node Wedge Element

The 6 node wedge elements are a condensation of an 8 node brick such as SOLID5, FLUID30, or SOLID45. These shape functions are for 6 node wedge elements without extra shape functions:

(12–172)

(12–173)

(12–174)

(12–175)

(12–176)

(12–177)

(12–178)

(12–179)

(12–180)



12.8.6. 6 Node Wedges with ESF

The 6 node wedge elements are a condensation of an 8 node brick such as SOLID5, FLUID30, or SOLID45. (Please see Figure 12.15: "6 Node Wedge Element".) These shape functions are for 6 node wedge elements with extra shape functions:

12.8.7. 15 Node Wedges as a Condensation of 20 Node Brick

Figure 12.16 15 Node Wedge Element (SOLID90)

These shape functions are for 15 node wedge elements such as SOLID90 that are based on a condensation of a 20 node brick element Equation 12–209.

12.8.8. 15 Node Wedges Based on Wedge Shape Functions

Figure 12.17 15 Node Wedge Element (SOLID95)

(12–181)

(12–182)

(12–183)

(12–184)

(12–185)



Elements such as SOLID95 in a wedge configuration use shape functions based on triangular coordinates and the r coordinate going from -1.0 to +1.0.

12.8.9. 8 Node Bricks without ESF


(12–186)

(12–187)

(12–188)

(12–189)

(12–190)



These shape functions are for 8 node brick elements without extra shape functions such as SOLID5 with KEYOPT(3) = 1, FLUID30, SOLID45 with KEYOPT(1) = 1, or FLUID142:

(12–191)

(12–192)

(12–193)

(12–194)

(12–195)

(12–196)

(12–197)

(12–198)

(12–199)

(12–200)

(12–201)

(12–202)

(12–203)



12.8.10. 8 Node Bricks with ESF

(Please see Figure 12.18: "8 Node Brick Element") These shape functions are for 8 node brick elements with extra shape functions such as SOLID5 with KEYOPT(3) = 0 or SOLID45 with KEYOPT(1) = 0:

12.8.11. 20 Node Bricks


These shape functions are used for 20 node solid elements such as SOLID90 or SOLID95.

(12–204)

(12–205)

(12–206)

(12–207)

(12–208)

(12–209)



12.8.12. 8 Node Infinite Bricks

Figure 12.20 3-D 8 Node Brick Element

These Lagrangian isoparametric shape functions and “mapping” functions are for the 3-D 8 node solid brick infinite elements such as INFIN111:

12.8.12.1. Lagrangian Isoparametric Shape Functions

(12–210)

(12–211)

(12–212)

(12–213)

(12–214)



12.8.12.2. Mapping Functions

12.8.13. 3-D 20 Node Infinite Bricks

Figure 12.21 20 Node Solid Brick Infinite Element

(12–215)

(12–216)

(12–217)

(12–218)

(12–219)

(12–220)

(12–221)

(12–222)

(12–223)



These Lagrangian isoparametric shape functions and “mapping” functions are for the 3-D 20 node solid brick infinite elements such as INFIN111:

12.8.13.1. Lagrangian Isoparametric Shape Functions

(12–224)

(12–225)

(12–226)

(12–227)



12.8.13.2. Mapping Functions

The shape and mapping functions for the nodes U, V, W, X, Y, Z, A, and B are deliberately set to zero.

Prev 12.7. Axisymmetric Harmonic Solids

(12–228)

(12–229)

(12–230)

(12–231)

(12–232)



Shape Functions Solid 186

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