KerMor  0.9
Model order reduction for nonlinear dynamical systems and nonlinear approximation
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HexahedronTriquadratic.m
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1 namespace fem{
2 
3 
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17 
18 class HexahedronTriquadratic
19  :public fem.BaseFEM {
27  public:
28 
29  HexahedronTriquadratic(geo) {
30  if nargin < 1
31  geo = fem.geometry.Cube27Node;
32  end
33  this = this@fem.BaseFEM(geo);
34  }
35 
36 
37  function Nx = N(colvec<double> x) {
38  Nx = [((x(1,:)-1).*x(1,:) .* (x(2,:)-1).*x(2,:) .* (x(3,:)-1).*x(3,:))/8; ... /* 1 */
39 
40  ((1-x(1,:).^2) .* (x(2,:)-1).*x(2,:) .* (x(3,:)-1).*x(3,:))/4; ... /* 2 */
41 
42  ((1+x(1,:)).*x(1,:) .* (x(2,:)-1).*x(2,:) .* (x(3,:)-1).*x(3,:))/8; ... /* 3 */
43 
44  ((x(1,:)-1).*x(1,:) .* (1-x(2,:).^2) .* (x(3,:)-1).*x(3,:))/4; ... /* 4 */
45 
46  ((1-x(1,:).^2) .* (1-x(2,:).^2) .* (x(3,:)-1).*x(3,:))/2; ... /* 5 */
47 
48  ((1+x(1,:)).*x(1,:) .* (1-x(2,:).^2) .* (x(3,:)-1).*x(3,:))/4; ... /* 6 */
49 
50  ((x(1,:)-1).*x(1,:) .* (1+x(2,:)).*x(2,:) .* (x(3,:)-1).*x(3,:))/8; ... /* 7 */
51 
52  ((1-x(1,:).^2) .* (1+x(2,:)).*x(2,:) .* (x(3,:)-1).*x(3,:))/4; ... /* 8 */
53 
54  ((1+x(1,:)).*x(1,:) .* (1+x(2,:)).*x(2,:) .* (x(3,:)-1).*x(3,:))/8; ... /* 9 */
55 
56  ((x(1,:)-1).*x(1,:) .* (x(2,:)-1).*x(2,:) .* (1-x(3,:).^2))/4; ... /* 10 */
57 
58  ((1-x(1,:).^2) .* (x(2,:)-1).*x(2,:) .* (1-x(3,:).^2))/2; ... /* 11 */
59 
60  ((1+x(1,:)).*x(1,:) .* (x(2,:)-1).*x(2,:) .* (1-x(3,:).^2))/4; ... /* 12 */
61 
62  ((x(1,:)-1).*x(1,:) .* (1-x(2,:).^2) .* (1-x(3,:).^2))/2; ... /* 13 */
63 
64  ((1-x(1,:).^2) .* (1-x(2,:).^2) .* (1-x(3,:).^2))/1; ... /* 14 */
65 
66  ((1+x(1,:)).*x(1,:) .* (1-x(2,:).^2) .* (1-x(3,:).^2))/2; ... /* 15 */
67 
68  ((x(1,:)-1).*x(1,:) .* (1+x(2,:)).*x(2,:) .* (1-x(3,:).^2))/4; ... /* 16 */
69 
70  ((1-x(1,:).^2) .* (1+x(2,:)).*x(2,:) .* (1-x(3,:).^2))/2; ... /* 17 */
71 
72  ((1+x(1,:)).*x(1,:) .* (1+x(2,:)).*x(2,:) .* (1-x(3,:).^2))/4; ... /* 18 */
73 
74  ((x(1,:)-1).*x(1,:) .* (x(2,:)-1).*x(2,:) .* (1+x(3,:)).*x(3,:))/8; ... /* 19 */
75 
76  ((1-x(1,:).^2) .* (x(2,:)-1).*x(2,:) .* (1+x(3,:)).*x(3,:))/4; ... /* 20 */
77 
78  ((1+x(1,:)).*x(1,:) .* (x(2,:)-1).*x(2,:) .* (1+x(3,:)).*x(3,:))/8; ... /* 21 */
79 
80  ((x(1,:)-1).*x(1,:) .* (1-x(2,:).^2) .* (1+x(3,:)).*x(3,:))/4; ... /* 22 */
81 
82  ((1-x(1,:).^2) .* (1-x(2,:).^2) .* (1+x(3,:)).*x(3,:))/2; ... /* 23 */
83 
84  ((1+x(1,:)).*x(1,:) .* (1-x(2,:).^2) .* (1+x(3,:)).*x(3,:))/4; ... /* 24 */
85 
86  ((x(1,:)-1).*x(1,:) .* (1+x(2,:)).*x(2,:) .* (1+x(3,:)).*x(3,:))/8; ... /* 25 */
87 
88  ((1-x(1,:).^2) .* (1+x(2,:)).*x(2,:) .* (1+x(3,:)).*x(3,:))/4; ... /* 26 */
89 
90  ((1+x(1,:)).*x(1,:) .* (1+x(2,:)).*x(2,:) .* (1+x(3,:)).*x(3,:))/8]; /* 27; */
91 
92  }
102  function dNx = gradN(colvec<double> x) {
103  dNx = [[((2*x(1,:)-1).*(x(2,:)-1).*x(2,:).*(x(3,:)-1).*x(3,:)) ((x(1,:)-1).*x(1,:).*(2*x(2,:)-1).*(x(3,:)-1).*x(3,:)) ((x(1,:)-1).*x(1,:).*(x(2,:)-1).*x(2,:).*(2*x(3,:)-1))]/8; ...
104  [((-2*x(1,:)).*(x(2,:)-1).*x(2,:).*(x(3,:)-1).*x(3,:)) ((1-x(1,:).^2).*(2*x(2,:)-1).*(x(3,:)-1).*x(3,:)) ((1-x(1,:).^2).*(x(2,:)-1).*x(2,:).*(2*x(3,:)-1))]/4; ...
105  [((1+2*x(1,:)).*(x(2,:)-1).*x(2,:).*(x(3,:)-1).*x(3,:)) ((1+x(1,:)).*x(1,:).*(2*x(2,:)-1).*(x(3,:)-1).*x(3,:)) ((1+x(1,:)).*x(1,:).*(x(2,:)-1).*x(2,:).*(2*x(3,:)-1))]/8; ...
106  [((2*x(1,:)-1).*(1-x(2,:).^2).*(x(3,:)-1).*x(3,:)) ((x(1,:)-1).*x(1,:).*(-2*x(2,:)).*(x(3,:)-1).*x(3,:)) ((x(1,:)-1).*x(1,:).*(1-x(2,:).^2).*(2*x(3,:)-1))]/4; ...
107  [((-2*x(1,:)).*(1-x(2,:).^2).*(x(3,:)-1).*x(3,:)) ((1-x(1,:).^2).*(-2*x(2,:)).*(x(3,:)-1).*x(3,:)) ((1-x(1,:).^2).*(1-x(2,:).^2).*(2*x(3,:)-1))]/2; ...
108  [((1+2*x(1,:)).*(1-x(2,:).^2).*(x(3,:)-1).*x(3,:)) ((1+x(1,:)).*x(1,:).*(-2*x(2,:)).*(x(3,:)-1).*x(3,:)) ((1+x(1,:)).*x(1,:).*(1-x(2,:).^2).*(2*x(3,:)-1))]/4; ...
109  [((2*x(1,:)-1).*(1+x(2,:)).*x(2,:).*(x(3,:)-1).*x(3,:)) ((x(1,:)-1).*x(1,:).*(1+2*x(2,:)).*(x(3,:)-1).*x(3,:)) ((x(1,:)-1).*x(1,:).*(1+x(2,:)).*x(2,:).*(2*x(3,:)-1))]/8; ...
110  [((-2*x(1,:)).*(1+x(2,:)).*x(2,:).*(x(3,:)-1).*x(3,:)) ((1-x(1,:).^2).*(1+2*x(2,:)).*(x(3,:)-1).*x(3,:)) ((1-x(1,:).^2).*(1+x(2,:)).*x(2,:).*(2*x(3,:)-1))]/4; ...
111  [((1+2*x(1,:)).*(1+x(2,:)).*x(2,:).*(x(3,:)-1).*x(3,:)) ((1+x(1,:)).*x(1,:).*(1+2*x(2,:)).*(x(3,:)-1).*x(3,:)) ((1+x(1,:)).*x(1,:).*(1+x(2,:)).*x(2,:).*(2*x(3,:)-1))]/8; ...
112  [((2*x(1,:)-1).*(x(2,:)-1).*x(2,:).*(1-x(3,:).^2)) ((x(1,:)-1).*x(1,:).*(2*x(2,:)-1).*(1-x(3,:).^2)) ((x(1,:)-1).*x(1,:).*(x(2,:)-1).*x(2,:).*(-2*x(3,:)))]/4; ...
113  [((-2*x(1,:)).*(x(2,:)-1).*x(2,:).*(1-x(3,:).^2)) ((1-x(1,:).^2).*(2*x(2,:)-1).*(1-x(3,:).^2)) ((1-x(1,:).^2).*(x(2,:)-1).*x(2,:).*(-2*x(3,:)))]/2; ...
114  [((1+2*x(1,:)).*(x(2,:)-1).*x(2,:).*(1-x(3,:).^2)) ((1+x(1,:)).*x(1,:).*(2*x(2,:)-1).*(1-x(3,:).^2)) ((1+x(1,:)).*x(1,:).*(x(2,:)-1).*x(2,:).*(-2*x(3,:)))]/4; ...
115  [((2*x(1,:)-1).*(1-x(2,:).^2).*(1-x(3,:).^2)) ((x(1,:)-1).*x(1,:).*(-2*x(2,:)).*(1-x(3,:).^2)) ((x(1,:)-1).*x(1,:).*(1-x(2,:).^2).*(-2*x(3,:)))]/2; ...
116  [((-2*x(1,:)).*(1-x(2,:).^2).*(1-x(3,:).^2)) ((1-x(1,:).^2).*(-2*x(2,:)).*(1-x(3,:).^2)) ((1-x(1,:).^2).*(1-x(2,:).^2).*(-2*x(3,:)))]/1; ...
117  [((1+2*x(1,:)).*(1-x(2,:).^2).*(1-x(3,:).^2)) ((1+x(1,:)).*x(1,:).*(-2*x(2,:)).*(1-x(3,:).^2)) ((1+x(1,:)).*x(1,:).*(1-x(2,:).^2).*(-2*x(3,:)))]/2; ...
118  [((2*x(1,:)-1).*(1+x(2,:)).*x(2,:).*(1-x(3,:).^2)) ((x(1,:)-1).*x(1,:).*(1+2*x(2,:)).*(1-x(3,:).^2)) ((x(1,:)-1).*x(1,:).*(1+x(2,:)).*x(2,:).*(-2*x(3,:)))]/4; ...
119  [((-2*x(1,:)).*(1+x(2,:)).*x(2,:).*(1-x(3,:).^2)) ((1-x(1,:).^2).*(1+2*x(2,:)).*(1-x(3,:).^2)) ((1-x(1,:).^2).*(1+x(2,:)).*x(2,:).*(-2*x(3,:)))]/2; ...
120  [((1+2*x(1,:)).*(1+x(2,:)).*x(2,:).*(1-x(3,:).^2)) ((1+x(1,:)).*x(1,:).*(1+2*x(2,:)).*(1-x(3,:).^2)) ((1+x(1,:)).*x(1,:).*(1+x(2,:)).*x(2,:).*(-2*x(3,:)))]/4; ...
121  [((2*x(1,:)-1).*(x(2,:)-1).*x(2,:).*(1+x(3,:)).*x(3,:)) ((x(1,:)-1).*x(1,:).*(2*x(2,:)-1).*(1+x(3,:)).*x(3,:)) ((x(1,:)-1).*x(1,:).*(x(2,:)-1).*x(2,:).*(1+2*x(3,:)))]/8; ...
122  [((-2*x(1,:)).*(x(2,:)-1).*x(2,:).*(1+x(3,:)).*x(3,:)) ((1-x(1,:).^2).*(2*x(2,:)-1).*(1+x(3,:)).*x(3,:)) ((1-x(1,:).^2).*(x(2,:)-1).*x(2,:).*(1+2*x(3,:)))]/4; ...
123  [((1+2*x(1,:)).*(x(2,:)-1).*x(2,:).*(1+x(3,:)).*x(3,:)) ((1+x(1,:)).*x(1,:).*(2*x(2,:)-1).*(1+x(3,:)).*x(3,:)) ((1+x(1,:)).*x(1,:).*(x(2,:)-1).*x(2,:).*(1+2*x(3,:)))]/8; ...
124  [((2*x(1,:)-1).*(1-x(2,:).^2).*(1+x(3,:)).*x(3,:)) ((x(1,:)-1).*x(1,:).*(-2*x(2,:)).*(1+x(3,:)).*x(3,:)) ((x(1,:)-1).*x(1,:).*(1-x(2,:).^2).*(1+2*x(3,:)))]/4; ...
125  [((-2*x(1,:)).*(1-x(2,:).^2).*(1+x(3,:)).*x(3,:)) ((1-x(1,:).^2).*(-2*x(2,:)).*(1+x(3,:)).*x(3,:)) ((1-x(1,:).^2).*(1-x(2,:).^2).*(1+2*x(3,:)))]/2; ...
126  [((1+2*x(1,:)).*(1-x(2,:).^2).*(1+x(3,:)).*x(3,:)) ((1+x(1,:)).*x(1,:).*(-2*x(2,:)).*(1+x(3,:)).*x(3,:)) ((1+x(1,:)).*x(1,:).*(1-x(2,:).^2).*(1+2*x(3,:)))]/4; ...
127  [((2*x(1,:)-1).*(1+x(2,:)).*x(2,:).*(1+x(3,:)).*x(3,:)) ((x(1,:)-1).*x(1,:).*(1+2*x(2,:)).*(1+x(3,:)).*x(3,:)) ((x(1,:)-1).*x(1,:).*(1+x(2,:)).*x(2,:).*(1+2*x(3,:)))]/8; ...
128  [((-2*x(1,:)).*(1+x(2,:)).*x(2,:).*(1+x(3,:)).*x(3,:)) ((1-x(1,:).^2).*(1+2*x(2,:)).*(1+x(3,:)).*x(3,:)) ((1-x(1,:).^2).*(1+x(2,:)).*x(2,:).*(1+2*x(3,:)))]/4; ...
129  [((1+2*x(1,:)).*(1+x(2,:)).*x(2,:).*(1+x(3,:)).*x(3,:)) ((1+x(1,:)).*x(1,:).*(1+2*x(2,:)).*(1+x(3,:)).*x(3,:)) ((1+x(1,:)).*x(1,:).*(1+x(2,:)).*x(2,:).*(1+2*x(3,:)))]/8];
130  }
131 
132 
133  public: /* ( Static ) */
134 
135  static function res = test_QuadraticBasisFun() {
136  q = fem.HexahedronTriquadratic;
137  res = fem.BaseFEM.test_BasisFun(q);
138 
139  /* test for correct basis function values on nodes */
140  [X,Y,Z] = ndgrid(-1:1:1,-1:1:1,-1:1:1);
141  p = [X(:) Y(:) Z(:)]^t;
142  res = res && isequal(q.N(p),eye(27));
143  }
144 
145 
146  static function generateN_dN() {
147  N = [" (x(1,:)-1).*x(1,:) ", " (1-x(1,:).^2) ", " (1+x(1,:)).*x(1,:) "
148  " (x(2,:)-1).*x(2,:) ", " (1-x(2,:).^2) ", " (1+x(2,:)).*x(2,:) "
149  " (x(3,:)-1).*x(3,:) ", " (1-x(3,:).^2) ", " (1+x(3,:)).*x(3,:) "];
150 
151  dN = [" (2*x(1,:)-1) ", " (-2*x(1,:)) ", " (1+2*x(1,:)) "
152  " (2*x(2,:)-1) ", " (-2*x(2,:)) ", " (1+2*x(2,:)) "
153  " (2*x(3,:)-1) ", " (-2*x(3,:)) ", " (1+2*x(3,:)) "];
154 
155  [x,y,z] = ndgrid(1:3);
156  idx = [x(:) y(:) z(:)]^t;
157  fac = 2.^sum(idx ~= 2);
158  s = ;
159  ds = ;
160  for k = 1:27
161  s = [s sprintf(" (%s .* %s .* %s)/%d; ... %%%d\n ",N[1,idx(1,k)],N[2,idx(2,k)],N[3,idx(3,k)],fac(k),k)];/* #ok */
162 
163  dn1 = sprintf(" (%s.*%s.*%s) ",dN[1,idx(1,k)],N[2,idx(2,k)],N[3,idx(3,k)]);
164  dn2 = sprintf(" (%s.*%s.*%s) ",N[1,idx(1,k)],dN[2,idx(2,k)],N[3,idx(3,k)]);
165  dn3 = sprintf(" (%s.*%s.*%s) ",N[1,idx(1,k)],N[2,idx(2,k)],dN[3,idx(3,k)]);
166  ds = [ds sprintf(" [%s %s %s]/%d; ...\n ",dn1,dn2,dn3,fac(k))];/* #ok */
167 
168  end
169  disp(s);
170  disp(ds);
171  N = eval(sprintf(" @(x)[%s]; ",s(1:end-9)));
172  gradN = eval(sprintf(" @(x)[%s]; ",ds(1:end-6)));
173  [x,y,z] = ndgrid(-1:1);
174  pos = [x(:) y(:) z(:)]^t;
175  sum(N(pos))
176  sum(gradN(pos))
177  }
186 };
187 }
188 
189 
190 
fem.BaseFEM
FEMBASE Summary of this class goes here Detailed explanation goes here.
Definition: BaseFEM.m:18
fem.HexahedronTriquadratic
Triquatratic: Quadratic ansatz functions on cube with 20 nodes per cube.
Definition: HexahedronTriquadratic.m:18
fem.HexahedronTriquadratic.generateN_dN
static function generateN_dN()
Little hack to generate the appropriate basis function expressions.
Definition: HexahedronTriquadratic.m:146
fem.HexahedronTriquadratic.test_QuadraticBasisFun
static function res = test_QuadraticBasisFun()
Definition: HexahedronTriquadratic.m:135
handle.disp
disp
Handle object disp method which is called by the display method. See the MATLAB disp function...
Definition: class_substitutes.c:122
X
#define X(i, j)
Definition: dontuse_evaluate.c:5
fem.HexahedronTriquadratic.gradN
function dNx = gradN(colvec< double > x)
Definition: HexahedronTriquadratic.m:102
Y
#define Y(i, j)
Definition: dontuse_evaluate.c:6
k
*for k
Definition: Gordon66SarcoForceLength.m:114
fem.HexahedronTriquadratic.N
function Nx = N(colvec< double > x)
Triquadratic basis functions.
Definition: HexahedronTriquadratic.m:37
t
* t
Definition: Gordon66SarcoForceLength.m:116