KerMor  0.9
Model order reduction for nonlinear dynamical systems and nonlinear approximation
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HexahedronSerendipity.m
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1 namespace fem{
2 
3 
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17 
18 class HexahedronSerendipity
19  :public fem.BaseFEM {
27  public:
28 
29  HexahedronSerendipity(geo) {
30  if nargin < 1
31  geo = fem.geometry.Cube20Node;
32  end
33  this = this@fem.BaseFEM(geo);
34  }
35 
36 
37  function Nx = N(colvec<double> x) {
38  Nx = [(1-x(1,:)).*(1-x(2,:)).*(1-x(3,:)).*(-x(1,:)-x(2,:)-x(3,:)-2)/8;... /* C1 */
39 
40  (1-x(1,:).^2).*(1-x(2,:)).*(1-x(3,:))/4;... /* E2 */
41 
42  (1+x(1,:)).*(1-x(2,:)).*(1-x(3,:)).*(x(1,:)-x(2,:)-x(3,:)-2)/8;... /* C3 */
43 
44  (1-x(2,:).^2).*(1-x(1,:)).*(1-x(3,:))/4;... /* E4 */
45 
46  (1-x(2,:).^2).*(1+x(1,:)).*(1-x(3,:))/4;... /* E5 */
47 
48  (1-x(1,:)).*(1+x(2,:)).*(1-x(3,:)).*(-x(1,:)+x(2,:)-x(3,:)-2)/8;... /* C6 */
49 
50  (1-x(1,:).^2).*(1+x(2,:)).*(1-x(3,:))/4;... /* E7 */
51 
52  (1+x(1,:)).*(1+x(2,:)).*(1-x(3,:)).*(x(1,:)+x(2,:)-x(3,:)-2)/8;... /* C8 */
53 
54  (1-x(3,:).^2).*(1-x(1,:)).*(1-x(2,:))/4;... /* E9 */
55 
56  (1-x(3,:).^2).*(1+x(1,:)).*(1-x(2,:))/4;... /* E10 */
57 
58  (1-x(3,:).^2).*(1-x(1,:)).*(1+x(2,:))/4;... /* E11 */
59 
60  (1-x(3,:).^2).*(1+x(1,:)).*(1+x(2,:))/4;... /* E12 */
61 
62  (1-x(1,:)).*(1-x(2,:)).*(1+x(3,:)).*(-x(1,:)-x(2,:)+x(3,:)-2)/8;... /* C13 */
63 
64  (1-x(1,:).^2).*(1-x(2,:)).*(1+x(3,:))/4;... /* E14 */
65 
66  (1+x(1,:)).*(1-x(2,:)).*(1+x(3,:)).*(x(1,:)-x(2,:)+x(3,:)-2)/8;... /* C15 */
67 
68  (1-x(2,:).^2).*(1-x(1,:)).*(1+x(3,:))/4;... /* E16 */
69 
70  (1-x(2,:).^2).*(1+x(1,:)).*(1+x(3,:))/4;... /* E17 */
71 
72  (1-x(1,:)).*(1+x(2,:)).*(1+x(3,:)).*(-x(1,:)+x(2,:)+x(3,:)-2)/8;... /* C18 */
73 
74  (1-x(1,:).^2).*(1+x(2,:)).*(1+x(3,:))/4;... /* E19 */
75 
76  (1+x(1,:)).*(1+x(2,:)).*(1+x(3,:)).*(x(1,:)+x(2,:)+x(3,:)-2)/8]; /* C20 */
77 
78  }
95  function dNx = gradN(colvec<double> x) {
96  dNx = [[-(1-x(2,:)).*(1-x(3,:)).*(-x(1,:)-x(2,:)-x(3,:)-2)-(1-x(1,:)).*(1-x(2,:)).*(1-x(3,:)) -(1-x(1,:)).*(1-x(3,:)).*(-x(1,:)-x(2,:)-x(3,:)-2)-(1-x(1,:)).*(1-x(2,:)).*(1-x(3,:)) -(1-x(1,:)).*(1-x(2,:)).*(-x(1,:)-x(2,:)-x(3,:)-2)-(1-x(1,:)).*(1-x(2,:)).*(1-x(3,:))]/8;... /* 1 */
97 
98  [-2*x(1,:).*(1-x(2,:)).*(1-x(3,:)) -(1-x(1,:).^2).*(1-x(3,:)) -(1-x(1,:).^2).*(1-x(2,:))]/4;...
99  [(1-x(2,:)).*(1-x(3,:)).*(x(1,:)-x(2,:)-x(3,:)-2)+(1+x(1,:)).*(1-x(2,:)).*(1-x(3,:)) -(1+x(1,:)).*(1-x(3,:)).*(x(1,:)-x(2,:)-x(3,:)-2)-(1+x(1,:)).*(1-x(2,:)).*(1-x(3,:)) -(1+x(1,:)).*(1-x(2,:)).*(x(1,:)-x(2,:)-x(3,:)-2)-(1+x(1,:)).*(1-x(2,:)).*(1-x(3,:))]/8;...
100  [-(1-x(2,:).^2).*(1-x(3,:)) -2*x(2,:).*(1-x(1,:)).*(1-x(3,:)) -(1-x(2,:).^2).*(1-x(1,:))]/4;...
101  [(1-x(2,:).^2).*(1-x(3,:)) -2*x(2,:).*(1+x(1,:)).*(1-x(3,:)) -(1-x(2,:).^2).*(1+x(1,:))]/4;...
102  [-(1+x(2,:)).*(1-x(3,:)).*(-x(1,:)+x(2,:)-x(3,:)-2)-(1-x(1,:)).*(1+x(2,:)).*(1-x(3,:)) (1-x(1,:)).*(1-x(3,:)).*(-x(1,:)+x(2,:)-x(3,:)-2)+(1-x(1,:)).*(1+x(2,:)).*(1-x(3,:)) -(1-x(1,:)).*(1+x(2,:)).*(-x(1,:)+x(2,:)-x(3,:)-2)-(1-x(1,:)).*(1+x(2,:)).*(1-x(3,:))]/8;...
103  [-2*x(1,:).*(1+x(2,:)).*(1-x(3,:)) (1-x(1,:).^2).*(1-x(3,:)) -(1-x(1,:).^2).*(1+x(2,:))]/4;...
104  [(1+x(2,:)).*(1-x(3,:)).*(x(1,:)+x(2,:)-x(3,:)-2)+(1+x(1,:)).*(1+x(2,:)).*(1-x(3,:)) (1+x(1,:)).*(1-x(3,:)).*(x(1,:)+x(2,:)-x(3,:)-2)+(1+x(1,:)).*(1+x(2,:)).*(1-x(3,:)) -(1+x(1,:)).*(1+x(2,:)).*(x(1,:)+x(2,:)-x(3,:)-2)-(1+x(1,:)).*(1+x(2,:)).*(1-x(3,:))]/8;...
105  [-(1-x(3,:).^2).*(1-x(2,:)) -(1-x(3,:).^2).*(1-x(1,:)) -2*x(3,:).*(1-x(1,:)).*(1-x(2,:))]/4;...
106  [(1-x(3,:).^2).*(1-x(2,:)) -(1-x(3,:).^2).*(1+x(1,:)) -2*x(3,:).*(1+x(1,:)).*(1-x(2,:))]/4;... /* E10 */
107 
108 
109  [-(1-x(3,:).^2).*(1+x(2,:)) (1-x(3,:).^2).*(1-x(1,:)) -2*x(3,:).*(1-x(1,:)).*(1+x(2,:))]/4;...
110  [(1-x(3,:).^2).*(1+x(2,:)) (1-x(3,:).^2).*(1+x(1,:)) -2*x(3,:).*(1+x(1,:)).*(1+x(2,:))]/4;...
111  [-(1-x(2,:)).*(1+x(3,:)).*(-x(1,:)-x(2,:)+x(3,:)-2)-(1-x(1,:)).*(1-x(2,:)).*(1+x(3,:)) -(1-x(1,:)).*(1+x(3,:)).*(-x(1,:)-x(2,:)+x(3,:)-2)-(1-x(1,:)).*(1-x(2,:)).*(1+x(3,:)) (1-x(1,:)).*(1-x(2,:)).*(-x(1,:)-x(2,:)+x(3,:)-2)+(1-x(1,:)).*(1-x(2,:)).*(1+x(3,:))]/8;...
112  [-2*x(1,:).*(1-x(2,:)).*(1+x(3,:)) -(1-x(1,:).^2).*(1+x(3,:)) (1-x(1,:).^2).*(1-x(2,:))]/4;...
113  [(1-x(2,:)).*(1+x(3,:)).*(x(1,:)-x(2,:)+x(3,:)-2)+(1+x(1,:)).*(1-x(2,:)).*(1+x(3,:)) -(1+x(1,:)).*(1+x(3,:)).*(x(1,:)-x(2,:)+x(3,:)-2)-(1+x(1,:)).*(1-x(2,:)).*(1+x(3,:)) (1+x(1,:)).*(1-x(2,:)).*(x(1,:)-x(2,:)+x(3,:)-2)+(1+x(1,:)).*(1-x(2,:)).*(1+x(3,:))]/8;...
114  [-(1-x(2,:).^2).*(1+x(3,:)) -2*x(2,:).*(1-x(1,:)).*(1+x(3,:)) (1-x(2,:).^2).*(1-x(1,:))]/4;...
115  [(1-x(2,:).^2).*(1+x(3,:)) -2*x(2,:).*(1+x(1,:)).*(1+x(3,:)) (1-x(2,:).^2).*(1+x(1,:))]/4;...
116  [-(1+x(2,:)).*(1+x(3,:)).*(-x(1,:)+x(2,:)+x(3,:)-2)-(1-x(1,:)).*(1+x(2,:)).*(1+x(3,:)) (1-x(1,:)).*(1+x(3,:)).*(-x(1,:)+x(2,:)+x(3,:)-2)+(1-x(1,:)).*(1+x(2,:)).*(1+x(3,:)) (1-x(1,:)).*(1+x(2,:)).*(-x(1,:)+x(2,:)+x(3,:)-2)+(1-x(1,:)).*(1+x(2,:)).*(1+x(3,:))]/8;...
117  [-2*x(1,:).*(1+x(2,:)).*(1+x(3,:)) (1-x(1,:).^2).*(1+x(3,:)) (1-x(1,:).^2).*(1+x(2,:))]/4;...
118  [(1+x(2,:)).*(1+x(3,:)).*(x(1,:)+x(2,:)+x(3,:)-2)+(1+x(1,:)).*(1+x(2,:)).*(1+x(3,:)) (1+x(1,:)).*(1+x(3,:)).*(x(1,:)+x(2,:)+x(3,:)-2)+(1+x(1,:)).*(1+x(2,:)).*(1+x(3,:)) (1+x(1,:)).*(1+x(2,:)).*(x(1,:)+x(2,:)+x(3,:)-2)+(1+x(1,:)).*(1+x(2,:)).*(1+x(3,:))]/8];
119  }
120 
121 
122  public: /* ( Static ) */
123 
124  static function res = test_QuadraticBasisFun() {
125  q = fem.HexahedronSerendipity;
126  res = fem.BaseFEM.test_BasisFun(q);
127 
128  /* test for correct basis function values on nodes */
129  [X,Y,Z] = ndgrid(-1:1:1,-1:1:1,-1:1:1);
130  p = [X(:) Y(:) Z(:)]^t;
131  /* Remove 7 (inner points not used here) */
132  p(:,[5,11,13,14,15,17,23]) = [];
133  res = res && isequal(q.N(p),eye(20));
134  }
135 
136 
137 
138 };
139 }
140 
141 
142 
fem.BaseFEM
FEMBASE Summary of this class goes here Detailed explanation goes here.
Definition: BaseFEM.m:18
fem.HexahedronSerendipity.test_QuadraticBasisFun
static function res = test_QuadraticBasisFun()
Definition: HexahedronSerendipity.m:124
X
#define X(i, j)
Definition: dontuse_evaluate.c:5
fem.HexahedronSerendipity
Triquatratic: Quadratic ansatz functions on cube with 20 nodes per cube.
Definition: HexahedronSerendipity.m:18
fem.HexahedronSerendipity.N
function Nx = N(colvec< double > x)
Triquadratic basis functions.
Definition: HexahedronSerendipity.m:37
fem.HexahedronSerendipity.gradN
function dNx = gradN(colvec< double > x)
Definition: HexahedronSerendipity.m:95
Y
#define Y(i, j)
Definition: dontuse_evaluate.c:6
t
* t
Definition: Gordon66SarcoForceLength.m:116