gemlab 2.0.0

Geometry and meshes laboratory for finite element analyses
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283

use russell_lab::{Matrix, Vector};

/// Defines a hexahedron with 20 nodes (quadratic faces)
///
/// ![hex20](https://raw.githubusercontent.com/cpmech/gemlab/main/data/figures/test_draw_shape_hex20.svg)
///
/// # Local IDs of nodes
///
/// ```text
///            4_______15_______7            r     s     t             r     s     t
///          ,'|              ,'|    p:0 [-1.0, -1.0, -1.0]   p:8  [ 0.0, -1.0, -1.0]
///       12'  |            ,'  |    p:1 [ 1.0, -1.0, -1.0]   p:9  [ 1.0,  0.0, -1.0]
///      ,'    16         ,14   |    p:2 [ 1.0,  1.0, -1.0]   p:10 [ 0.0,  1.0, -1.0]
///    ,'      |        ,'      19   p:3 [-1.0,  1.0, -1.0]   p:11 [-1.0,  0.0, -1.0]
///  5'=====13========6'        |                             
///  |         |      |         |    p:4 [-1.0, -1.0,  1.0]   p:12 [ 0.0, -1.0,  1.0]
///  |         |      |         |    p:5 [ 1.0, -1.0,  1.0]   p:13 [ 1.0,  0.0,  1.0]
///  |         0_____ | _11_____3    p:6 [ 1.0,  1.0,  1.0]   p:14 [ 0.0,  1.0,  1.0]
/// 17       ,'       |       ,'     p:7 [-1.0,  1.0,  1.0]   p:15 [-1.0,  0.0,  1.0]
///  |     8'        18     ,'       
///  |   ,'           |   ,10       p:16 [-1.0, -1.0,  0.0]
///  | ,'             | ,'          p:17 [ 1.0, -1.0,  0.0]
///  1_______9________2'            p:18 [ 1.0,  1.0,  0.0]
///                                 p:19 [-1.0,  1.0,  0.0]
/// ```
///
/// # Local IDs of edges
///
/// ```text
///                     7
///             +----------------+          p0  p1 p2
///           ,'|              ,'|     e:0  [0, 1,  8],
///       4 ,'  |8          6,'  |     e:1  [1, 2,  9],
///       ,'    |          ,'    |     e:2  [2, 3, 10],
///     ,'      |   5    ,'      |11   e:3  [3, 0, 11],
///   +'===============+'        |     e:4  [4, 5, 12],
///   |         |      |         |     e:5  [5, 6, 13],
///   |         |      |  3      |     e:6  [6, 7, 14],
///   |         .- - - | -  - - -+     e:7  [7, 4, 15],
///  9|       ,'       |       ,'      e:8  [0, 4, 16],
///   |    0,'         |10   ,'        e:9  [1, 5, 17],
///   |   ,'           |   ,' 2        e:10 [2, 6, 18],
///   | ,'             | ,'            e:11 [3, 7, 19],
///   +----------------+'
///           1
/// ```
///
/// * The order of edge nodes corresponds to **Lin3** nodes.
///
/// # Local IDs of faces
///
/// ```text
///           4-------15-------7
///         ,'|              ,'|
///      12' 16  ___       14  |
///     ,'    |,'5,'  [0],'    |
///   ,'      |~~~     ,'     19   f:0 [0, 4, 7, 3, 16, 15, 19, 11]
/// 5'======13=======6'  ,'|   |   f:1 [1, 2, 6, 5,  9, 18, 13, 17]
/// |   ,'|   |      |   |3|   |   f:2 [0, 1, 5, 4,  8, 17, 12, 16]
/// |   |2|   |      |   |,'   |   f:3 [2, 3, 7, 6, 10, 19, 14, 18]
/// |   |,'   0- - - | 11 - - -3   f:4 [0, 3, 2, 1, 11, 10,  9,  8]
/// 17      ,'      18       ,'    f:5 [4, 5, 6, 7, 12, 13, 14, 15]
/// |     8' [1]  ___|     10
/// |   ,'      ,'4,'|   ,'
/// | ,'        ~~~  | ,'
/// 1-------9--------2'
/// ```
///
/// # Notes
///
/// * The reference coordinates range from -1 to +1 with the geometry centred @ 0
/// * The order of face nodes is such that the normals are outward
/// * The order of face nodes corresponds to [super::Qua8] nodes
/// * This shape is a higher-order version of [super::Hex8]
pub struct Hex20 {}

impl Hex20 {
    pub const GEO_NDIM: usize = 3;
    pub const NNODE: usize = 20;
    pub const NEDGE: usize = 12;
    pub const NFACE: usize = 6;
    pub const EDGE_NNODE: usize = 3;
    pub const FACE_NNODE: usize = 8;
    pub const FACE_NEDGE: usize = 4;

    #[rustfmt::skip]
    pub const EDGE_NODE_IDS: [[usize; Hex20::EDGE_NNODE]; Hex20::NEDGE] = [
        [0, 1,  8],
        [1, 2,  9],
        [2, 3, 10],
        [3, 0, 11],
        [4, 5, 12],
        [5, 6, 13],
        [6, 7, 14],
        [7, 4, 15],
        [0, 4, 16],
        [1, 5, 17],
        [2, 6, 18],
        [3, 7, 19],
    ];

    #[rustfmt::skip]
    pub const FACE_NODE_IDS: [[usize; Hex20::FACE_NNODE]; Hex20::NFACE] = [
        [0, 4, 7, 3, 16, 15, 19, 11],
        [1, 2, 6, 5,  9, 18, 13, 17],
        [0, 1, 5, 4,  8, 17, 12, 16],
        [2, 3, 7, 6, 10, 19, 14, 18],
        [0, 3, 2, 1, 11, 10,  9,  8],
        [4, 5, 6, 7, 12, 13, 14, 15],
    ];

    #[rustfmt::skip]
    pub const FACE_EDGE_NODE_IDS: [[[usize; Hex20::EDGE_NNODE]; Hex20::FACE_NEDGE]; Hex20::NFACE] = [
        [[0, 4, 16], [4, 7, 15], [7, 3, 19], [3, 0, 11]],
        [[1, 2,  9], [2, 6, 18], [6, 5, 13], [5, 1, 17]],
        [[0, 1,  8], [1, 5, 17], [5, 4, 12], [4, 0, 16]],
        [[2, 3, 10], [3, 7, 19], [7, 6, 14], [6, 2, 18]],
        [[0, 3, 11], [3, 2, 10], [2, 1,  9], [1, 0,  8]],
        [[4, 5, 12], [5, 6, 13], [6, 7, 14], [7, 4, 15]],
    ];

    #[rustfmt::skip]
    pub const NODE_REFERENCE_COORDS: [[f64; Hex20::GEO_NDIM]; Hex20::NNODE] = [
        [-1.0, -1.0, -1.0],
        [ 1.0, -1.0, -1.0],
        [ 1.0,  1.0, -1.0],
        [-1.0,  1.0, -1.0],

        [-1.0, -1.0,  1.0],
        [ 1.0, -1.0,  1.0],
        [ 1.0,  1.0,  1.0],
        [-1.0,  1.0,  1.0],

        [ 0.0, -1.0, -1.0],
        [ 1.0,  0.0, -1.0],
        [ 0.0,  1.0, -1.0],
        [-1.0,  0.0, -1.0],

        [ 0.0, -1.0,  1.0],
        [ 1.0,  0.0,  1.0],
        [ 0.0,  1.0,  1.0],
        [-1.0,  0.0,  1.0],

        [-1.0, -1.0,  0.0],
        [ 1.0, -1.0,  0.0],
        [ 1.0,  1.0,  0.0],
        [-1.0,  1.0,  0.0],
    ];

    /// Computes the interpolation functions
    ///
    /// # Output
    ///
    /// * `interp` -- interpolation function evaluated at ksi (nnode)
    ///
    /// # Input
    ///
    /// * `ksi` -- reference coordinates with length ≥ geo_ndim
    pub fn calc_interp(interp: &mut Vector, ksi: &[f64]) {
        let (r, s, t) = (ksi[0], ksi[1], ksi[2]);

        let rm = 1.0 - r;
        let sm = 1.0 - s;
        let tm = 1.0 - t;
        let rp = 1.0 + r;
        let sp = 1.0 + s;
        let tp = 1.0 + t;
        let rr = 1.0 - r * r;
        let ss = 1.0 - s * s;
        let tt = 1.0 - t * t;

        interp[0] = rm * sm * tm * (-2.0 - r - s - t) / 8.0;
        interp[1] = rp * sm * tm * (-2.0 + r - s - t) / 8.0;
        interp[2] = rp * sp * tm * (-2.0 + r + s - t) / 8.0;
        interp[3] = rm * sp * tm * (-2.0 - r + s - t) / 8.0;
        interp[4] = rm * sm * tp * (-2.0 - r - s + t) / 8.0;
        interp[5] = rp * sm * tp * (-2.0 + r - s + t) / 8.0;
        interp[6] = rp * sp * tp * (-2.0 + r + s + t) / 8.0;
        interp[7] = rm * sp * tp * (-2.0 - r + s + t) / 8.0;
        interp[8] = rr * sm * tm / 4.0;
        interp[9] = rp * ss * tm / 4.0;
        interp[10] = rr * sp * tm / 4.0;
        interp[11] = rm * ss * tm / 4.0;
        interp[12] = rr * sm * tp / 4.0;
        interp[13] = rp * ss * tp / 4.0;
        interp[14] = rr * sp * tp / 4.0;
        interp[15] = rm * ss * tp / 4.0;
        interp[16] = rm * sm * tt / 4.0;
        interp[17] = rp * sm * tt / 4.0;
        interp[18] = rp * sp * tt / 4.0;
        interp[19] = rm * sp * tt / 4.0;
    }

    /// Computes the derivatives of interpolation functions with respect to the reference coordinates
    ///
    /// # Output
    ///
    /// * `deriv` -- derivatives of the interpolation function with respect to
    ///   the reference coordinates ksi, evaluated at ksi (nnode,geo_ndim)
    ///
    /// # Input
    ///
    /// * `ksi` -- reference coordinates with length ≥ geo_ndim
    pub fn calc_deriv(deriv: &mut Matrix, ksi: &[f64]) {
        let (r, s, t) = (ksi[0], ksi[1], ksi[2]);

        let rm = 1.0 - r;
        let sm = 1.0 - s;
        let tm = 1.0 - t;
        let rp = 1.0 + r;
        let sp = 1.0 + s;
        let tp = 1.0 + t;
        let rr = 1.0 - r * r;
        let ss = 1.0 - s * s;
        let tt = 1.0 - t * t;

        // with respect to r
        deriv.set(0, 0, -rm * sm * tm / 8.0 - sm * (-2.0 - r - s - t) * tm / 8.0);
        deriv.set(1, 0, rp * sm * tm / 8.0 + sm * (-2.0 + r - s - t) * tm / 8.0);
        deriv.set(2, 0, rp * sp * tm / 8.0 + sp * (-2.0 + r + s - t) * tm / 8.0);
        deriv.set(3, 0, -rm * sp * tm / 8.0 - sp * (-2.0 - r + s - t) * tm / 8.0);
        deriv.set(4, 0, -rm * sm * tp / 8.0 - sm * (-2.0 - r - s + t) * tp / 8.0);
        deriv.set(5, 0, rp * sm * tp / 8.0 + sm * (-2.0 + r - s + t) * tp / 8.0);
        deriv.set(6, 0, rp * sp * tp / 8.0 + sp * (-2.0 + r + s + t) * tp / 8.0);
        deriv.set(7, 0, -rm * sp * tp / 8.0 - sp * (-2.0 - r + s + t) * tp / 8.0);
        deriv.set(8, 0, -r * sm * tm / 2.0);
        deriv.set(9, 0, ss * tm / 4.0);
        deriv.set(10, 0, -r * sp * tm / 2.0);
        deriv.set(11, 0, -ss * tm / 4.0);
        deriv.set(12, 0, -r * sm * tp / 2.0);
        deriv.set(13, 0, ss * tp / 4.0);
        deriv.set(14, 0, -r * sp * tp / 2.0);
        deriv.set(15, 0, -ss * tp / 4.0);
        deriv.set(16, 0, -sm * tt / 4.0);
        deriv.set(17, 0, sm * tt / 4.0);
        deriv.set(18, 0, sp * tt / 4.0);
        deriv.set(19, 0, -sp * tt / 4.0);

        // with respect to s
        deriv.set(0, 1, -rm * sm * tm / 8.0 - rm * (-2.0 - r - s - t) * tm / 8.0);
        deriv.set(1, 1, -rp * sm * tm / 8.0 - rp * (-2.0 + r - s - t) * tm / 8.0);
        deriv.set(2, 1, rp * sp * tm / 8.0 + rp * (-2.0 + r + s - t) * tm / 8.0);
        deriv.set(3, 1, rm * sp * tm / 8.0 + rm * (-2.0 - r + s - t) * tm / 8.0);
        deriv.set(4, 1, -rm * sm * tp / 8.0 - rm * (-2.0 - r - s + t) * tp / 8.0);
        deriv.set(5, 1, -rp * sm * tp / 8.0 - rp * (-2.0 + r - s + t) * tp / 8.0);
        deriv.set(6, 1, rp * sp * tp / 8.0 + rp * (-2.0 + r + s + t) * tp / 8.0);
        deriv.set(7, 1, rm * sp * tp / 8.0 + rm * (-2.0 - r + s + t) * tp / 8.0);
        deriv.set(8, 1, -rr * tm / 4.0);
        deriv.set(9, 1, -rp * s * tm / 2.0);
        deriv.set(10, 1, rr * tm / 4.0);
        deriv.set(11, 1, -rm * s * tm / 2.0);
        deriv.set(12, 1, -rr * tp / 4.0);
        deriv.set(13, 1, -rp * s * tp / 2.0);
        deriv.set(14, 1, rr * tp / 4.0);
        deriv.set(15, 1, -rm * s * tp / 2.0);
        deriv.set(16, 1, -rm * tt / 4.0);
        deriv.set(17, 1, -rp * tt / 4.0);
        deriv.set(18, 1, rp * tt / 4.0);
        deriv.set(19, 1, rm * tt / 4.0);

        // with respect to s
        deriv.set(0, 2, -rm * sm * (-2.0 - r - s - t) / 8.0 - rm * sm * tm / 8.0);
        deriv.set(1, 2, -rp * sm * (-2.0 + r - s - t) / 8.0 - rp * sm * tm / 8.0);
        deriv.set(2, 2, -rp * sp * (-2.0 + r + s - t) / 8.0 - rp * sp * tm / 8.0);
        deriv.set(3, 2, -rm * sp * (-2.0 - r + s - t) / 8.0 - rm * sp * tm / 8.0);
        deriv.set(4, 2, rm * sm * (-2.0 - r - s + t) / 8.0 + rm * sm * tp / 8.0);
        deriv.set(5, 2, rp * sm * (-2.0 + r - s + t) / 8.0 + rp * sm * tp / 8.0);
        deriv.set(6, 2, rp * sp * (-2.0 + r + s + t) / 8.0 + rp * sp * tp / 8.0);
        deriv.set(7, 2, rm * sp * (-2.0 - r + s + t) / 8.0 + rm * sp * tp / 8.0);
        deriv.set(8, 2, -rr * sm / 4.0);
        deriv.set(9, 2, -rp * ss / 4.0);
        deriv.set(10, 2, -rr * sp / 4.0);
        deriv.set(11, 2, -rm * ss / 4.0);
        deriv.set(12, 2, rr * sm / 4.0);
        deriv.set(13, 2, rp * ss / 4.0);
        deriv.set(14, 2, rr * sp / 4.0);
        deriv.set(15, 2, rm * ss / 4.0);
        deriv.set(16, 2, -rm * sm * t / 2.0);
        deriv.set(17, 2, -rp * sm * t / 2.0);
        deriv.set(18, 2, -rp * sp * t / 2.0);
        deriv.set(19, 2, -rm * sp * t / 2.0);
    }
}