-
Notifications
You must be signed in to change notification settings - Fork 1
/
eqpol.f90
393 lines (356 loc) · 14.7 KB
/
eqpol.f90
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
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
! EQUIVALENT POLYNOMIAL LIBRARY V 1.1
! BY GIULIO VENTURA
! giulio.ventura@polito.it
! www.equivalent-polynomials.net
!
! WHEN USING THIS LIBRARY PLEASE ALWAYS CITE
!
! Ventura G., On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method,
! International Journal for Numerical Methods in Engineering 66 (2006) 761–795.
!
! Ventura G., Benvenuti E., Equivalent polynomials for quadrature in heaviside function enriched elements,
! International Journal for Numerical Methods in Engineering 102 (2015) 688–710.
!
! USAGE
! Call first Heqpol_coefficients for a given element and discontinuity plane
! Then, at each Gauss point of coords x,y,z, evaluate HEqPol, passing the previously computed eqcv
! See the Library example file main.f90 for usage hints.
double precision function HEqPol(x,y,z,eqcv,etype)
! On input
!
! x,y,z coordinate of the evaluation point in the PARENT element reference system
! x,y,z in [0,1] for triangular and tetrahedral elements
! x,y,z in [-1,1] for quad and hexas
! the z coordinate is unused for 2D elements
!
! eqcv vector of polynomial coefficients as computed by eqpol_hex_coefficients
!
! etype the element type. It is:
! 20 linear triangle, required length for eqcv is 1
! 21 linear quadrilateral, required length for eqcv is 6
! 30 linear tetrahedron
! 31 linear hexahedron, required length for eqcv is 23
!
! On output
! the value of the equivalent polynomial at point x,y,z
implicit none
integer etype,l
double precision x,y,z
double precision eqcv(:)
double precision v(size(eqcv))
l = size(eqcv)
select case (etype)
case (20)
if (l.ne.1) stop ' Invalid lenght for vector eqcv'
v=(/ 1.D0 /)
case (21)
if (l.ne.6) stop ' Invalid lenght for vector eqcv'
v=(/ 1.D0, x, x**2, y, x*y, y**2 /)
case (30)
if (l.ne.1) stop ' Invalid lenght for vector eqcv'
v=(/ 1.D0 /)
case (31)
if (l.ne.23) stop ' Invalid lenght for vector eqcv'
v=(/ 1.D0, x, x**2, y, x*y, x**2*y, y**2, x*y**2, x**2*y**2, z, x*z, x**2*z, y*z, x*y*z, &
x**2*y*z, y**2*z, x*y**2*z, z**2, x*z**2, x**2*z**2, y*z**2, x*y*z**2, y**2*z**2 /)
case default
stop 'Invalid etype specified'
end select
HEqPol=dot_product(v,eqcv)
end function HEqPol
subroutine Heqpol_coefficients(a,b,c,d,eqcv,etype)
! On input: a,b,c,d, etype
! On output: eqcv
!
! This sub computed the vector of equivalent polynomial coefficients for given
! H discontinuity plane coefficients a,b,c,d in the parent element domain
! for 3D elements the discontinuity plane has equation a xi + b eta + c zeta + d = 0
! for 2D elements the discontinuity plane has equation a xi + b eta + c = 0
!
! On output the vector eqcv of polinomial coefficients w.r.t. the base
! defined in the function Hex_HeqPol
!
! etype the element type. It is:
! 20 linear triangle, required length for eqcv is 1
! 21 linear quadrilateral, required length for eqcv is 6
! 30 linear tetrahedron, required length for eqcv is 1
! 31 linear hexahedron, required length for eqcv is 23
implicit none
integer etype
double precision a,b,c,d,eqcv(:)
double precision BV(size(eqcv))
real, parameter :: tol = 1E-4 ! tolerance for vanishing plane coefficients
double precision, parameter :: InvA20(1,1) = reshape( (/ 2.d0 /), shape(InvA20))
double precision, parameter :: InvA21(6,6) = reshape( &
(/ 7.d0/8.d0, 0.d0, -15.d0/16.d0, 0.d0, 0.d0, -15.d0/16.d0, &
0.d0, 3.d0/4.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
-15.d0/16.d0, 0.d0, 45.d0/16.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 3.d0/4.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 9.d0/4.d0, 0.d0, &
-15.d0/16.d0, 0.d0, 0.d0, 0.d0, 0.d0, 45.d0/16.d0 &
/),shape(InvA21))
double precision, parameter :: InvA30(1,1) = reshape( (/ 6.d0 /), shape(InvA30))
double precision, parameter :: InvA31(23,23) = reshape( &
(/ 151.d0/128.d0, 0.d0, -(105.d0/64.d0), 0.d0, 0.d0, 0.d0, -(105.d0/64.d0), 0.d0, 225.d0/128.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, -(105.d0/64.d0), 0.d0, 225.d0/128.d0, 0.d0, 0.d0, 225.d0/128.d0, 0.d0, 21.d0/16.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, &
0.d0, -(105.d0/64.d0), 0.d0, 315.d0/64.d0, 0.d0, 0.d0, 0.d0, 225.d0/128.d0, 0.d0, -(675.d0/128.d0), 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 225.d0/128.d0, 0.d0, -(675.d0/128.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 21.d0/16.d0, &
0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(45.d0/32.d0), 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 81.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, -(135.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, -(45.d0/32.d0), 0.d0, 135.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(105.d0/64.d0), 0.d0, 225.d0/128.d0, 0.d0, 0.d0, 0.d0, &
315.d0/64.d0, 0.d0, -(675.d0/128.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 225.d0/128.d0, 0.d0, 0.d0, 0.d0, &
0.d0, -(675.d0/128.d0), 0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 135.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 225.d0/128.d0, 0.d0, -(675.d0/128.d0), 0.d0, 0.d0, &
0.d0, -(675.d0/128.d0), 0.d0, 2025.d0/128.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 21.d0/16.d0, 0.d0, -(45.d0/32.d0), 0.d0, 0.d0, &
0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
81.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(135.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, -(45.d0/32.d0), 0.d0, 135.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 81.d0/32.d0, 0.d0, -(135.d0/32.d0), 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 27.d0/8.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, -(135.d0/32.d0), 0.d0, 405.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 135.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(135.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 405.d0/32.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, -(105.d0/64.d0), 0.d0, 225.d0/128.d0, 0.d0, 0.d0, 0.d0, 225.d0/128.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 315.d0/64.d0, 0.d0, -(675.d0/128.d0), 0.d0, &
0.d0, -(675.d0/128.d0), 0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 135.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 225.d0/128.d0, 0.d0, -(675.d0/128.d0), 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(675.d0/128.d0), 0.d0, 2025.d0/128.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, -(45.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
135.d0/32.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(135.d0/32.d0), 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 405.d0/32.d0, 0.d0, 225.d0/128.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(675.d0/128.d0), &
0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, 0.d0, -(675.d0/128.d0), 0.d0, 0.d0, 0.d0, 0.d0, 2025.d0/128.d0 &
/),shape(InvA31))
integer i
double precision a2,a3,a4,a5,a6,b2,b3,b4,b5,b6,c2,c3,c4,c5,c6,d2,d3,d4,d5,d6
double precision t,am,bm,cm
double precision abs01,abs02,abs03,abs04,abs05,abs06,abs07,abs08,abs09,abs10,abs11,abs12
double precision abs13,abs14,abs15,abs16
! plane coefficient normalization
select case (etype)
case (20,21)
t=sqrt(a**2+b**2)
a=a/t; b=b/t; c=c/t
am=abs(a)
bm=abs(b)
case (30,31)
t=sqrt(a**2+b**2+c**2)
a=a/t; b=b/t; c=c/t; d=d/t
am=abs(a)
bm=abs(b)
cm=abs(c)
case default
stop 'Invalid etype'
end select
select case (etype)
case (20)
a2=a**2; b2=b**2
if ( am<bm*tol ) then
! a=0, b<>0
! b
BV(1) = (b2-(2*b+c)*abs(c)+(b+c)*abs(b+c))/(4.d0*b2)
else if ( bm<am*tol ) then
! a<>0, b=0
! a
BV(1) = (a2-(2*a+c)*abs(c)+(a+c)*abs(a+c))/(4.d0*a2)
else if (abs(a-b)<tol) then
! a<>0, b<>0, a=b
! a=b
BV(1) = (a2+c*abs(c)+(a-c)*abs(a+c))/(4.d0*a2)
else
! a<>0, b<>0
! a b
BV(1) = ((a-b)*c*abs(c)+b*(a+c)*abs(a+c)+a*((a-b)*b-(b+c)*abs(b+c)))/(4.d0*a*(a-b)*b)
end if
eqcv=matmul(InvA20,BV)
case (21)
a2=a**2; b2=b**2; c2=c**2
a3=a**3; b3=b**3; c3=c**3
if ( am<bm*tol ) then
! a=0, b<>0
! b
abs01=abs(b-c)
abs02=abs(b+c)
include 'quad_ph_6_b.f90'
else if ( bm<am*tol ) then
! a<>0, b=0
! a
abs01=abs(a-c)
abs02=abs(a+c)
include 'quad_ph_6_a.f90'
else
! a<>0, b<>0
! a b
abs01=abs(a-b-c)
abs02=abs(a+b-c)
abs03=abs(a-b+c)
abs04=abs(a+b+c)
include 'quad_ph_6_ab.f90'
end if
eqcv=matmul(InvA21,BV)
case (30)
a2=a**2; b2=b**2; c2=c**2; d2=d**2
a3=a**3; b3=b**3; c3=c**3; d3=d**3
if ( am<bm*tol .and. am<cm*tol .and. min(bm,cm)/max(bm,cm)>tol ) then
! a=0, b<>0, c<>0
! b c
if (abs(b-c)>tol) then
! b<>c
BV(1)=((b-c)*d*(c*d+b*(3*c+d))*abs(d)+c2*(b + d)**2*abs(b+d)+ &
b2*((b-c)*c2-(c+d)**2*abs(c+d)))/(12.d0*b2*(b-c)*c2)
else
! b=c
BV(1) = (b3 + d*(3*b + 2*d)*abs(d) + (b2 - b*d - 2*d2)*Abs(b + d))/(12.*b3)
end if
else if ( bm<am*tol .and. bm<cm*tol .and. min(am,cm)/max(am,cm)>tol ) then
! a<>0, b=0, c<>0
! a c
if (abs(a-c)>tol) then
! a<>c
BV(1)=((a-c)*d*(c*d+a*(3*c+d))*abs(d)+c2*(a+d)**2*abs(a+d)+ &
a2*((a-c)*c2-(c+d)**2*abs(c+d)))/(12.d0*a2*(a - c)*c2)
else
! a=c
BV(1) = (a3 + d*(3*a + 2*d)*abs(d) + (a2 - a*d - 2*d2)*abs(a + d))/(12.*a3)
end if
else if ( cm<am*tol .and. cm<bm*tol .and. min(am,bm)/max(am,bm)>tol ) then
! a<>0, b<>0, c=0
! a b
if (abs(a-b)>tol) then
! a<>b
BV(1)=((a-b)*d*(b*d+a*(3*b+d))*abs(d)+b2*(a+d)**2*abs(a+d)+ &
a2*((a-b)*b2-(b+d)**2*abs(b+d)))/(12.d0*a2*(a-b)*b2)
else
! a=b
BV(1) = (a3 + d*(3*a + 2*d)*abs(d) + (a2 - a*d - 2*d2)*abs(a + d))/(12.*a3)
end if
else if ( am<cm*tol .and. bm<cm*tol ) then
! a=0, b=0, c<>0
! c
BV(1)=(c3-(3*c2+3*c*d+d2)*abs(d)+(c+d)**2*abs(c+d))/(12.d0*c3)
else if ( bm<am*tol .and. cm<am*tol ) then
! a<>0, b=0, c=0
! a
BV(1)=(a3-(3*a2+3*a*d+d2)*abs(d)+(a+d)**2*abs(a+d))/(12.d0*a3)
else if ( am<bm*tol .and. cm<bm*tol ) then
! a=0, b<>0, c=0
! b
BV(1)=(b3-(3*b2+3*b*d+d2)*abs(d)+(b+d)**2*abs(b+d))/(12.d0*b3)
else
! a<>0, b<>0, c<>0
! a b c
if (abs(a-b)>tol .and. abs(a-c)>tol .and. abs(b-c)>tol) then
! a<>b, a<>c, b<>c
BV(1)=((-a+b)*(a-c)*(b-c)*d2*abs(d)+b*(b-c)*c*(a+d)**2*abs(a+d)+a*(c*(-a+c)*(b+d)**2*abs(b+d)+ &
(a-b)*b*((a-c)*(b-c)*c+(c+d)**2*abs(c+d))))/(12.d0*a*(a-b)*b*(a-c)*(b-c)*c)
else if (abs(a-b)<tol .and. abs(a-c)>tol .and. abs(b-c)>tol) then
! a=b, a<>c, b<>c
BV(1)=(-((a - c)**2*d2*abs(d)) + c*(a + d)*(a2 + c*d - 2*a*(c + d))*abs(a + d) + &
a2*((a - c)**2*c + (c + d)**2*abs(c + d)))/(12.*a2*(a - c)**2*c)
else if (abs(a-b)>tol .and. abs(a-c)<tol .and. abs(b-c)>tol) then
! a<>b, a=c, b<>c
BV(1)=(-((a - b)**2*d2*abs(d)) + b*(a + d)*(a2 + b*d - 2*a*(b + d))*abs(a + d) + &
a2*((a - b)**2*b + (b + d)**2*abs(b + d)))/(12.*a2*(a - b)**2*b)
else if (abs(a-b)>tol .and. abs(a-c)>tol .and. abs(b-c)<tol) then
! a<>b, a<>c, b=c
BV(1)=(-((a - b)**2*d2*abs(d)) + b2*(a + d)**2*abs(a + d) + &
a*((a - b)**2*b2 + (b + d)*(b*(b - 2*d) + a*(-2*b + d))*abs(b + d)))/ &
(12.*a*(a - b)**2*b2)
end if
end if
eqcv=matmul(InvA30,BV)
case (31)
a2=a**2; b2=b**2; c2=c**2; d2=d**2
a3=a**3; b3=b**3; c3=c**3; d3=d**3
a4=a**4; b4=b**4; c4=c**4; d4=d**4
a5=a**5; b5=b**5; c5=c**5; d5=d**5
a6=a**6; b6=b**6; c6=c**6; d6=d**6
if ( am<bm*tol .and. am<cm*tol .and. min(bm,cm)/max(bm,cm)>tol ) then
! a=0, b<>0, c<>0
! b c
abs01=abs(b+c+d)
abs02=abs(b+c-d)
abs03=abs(b-c+d)
abs04=abs(b-c-d)
abs05=abs(-b+c+d)
abs06=abs(-b+c-d)
abs07=abs(-b-c+d)
abs08=abs(-b-c-d)
include 'hex_ph_23_bc.f90'
else if ( bm<am*tol .and. bm<cm*tol .and. min(am,cm)/max(am,cm)>tol ) then
! a<>0, b=0, c<>0
! a c
abs01=abs(a+c+d)
abs02=abs(a+c-d)
abs03=abs(a-c+d)
abs04=abs(a-c-d)
abs09=abs(-a+c+d)
abs10=abs(-a+c-d)
abs11=abs(-a-c+d)
abs12=abs(-a-c-d)
include 'hex_ph_23_ac.f90'
else if ( cm<am*tol .and. cm<bm*tol .and. min(am,bm)/max(am,bm)>tol ) then
! a<>0, b<>0, c=0
! a b
abs01=abs(a+b+d)
abs02=abs(a+b-d)
abs05=abs(a-b+d)
abs06=abs(a-b-d)
abs09=abs(-a+b+d)
abs10=abs(-a+b-d)
abs13=abs(-a-b+d)
abs14=abs(-a-b-d)
include 'hex_ph_23_ab.f90'
else if ( am<cm*tol .and. bm<cm*tol ) then
! a=0, b=0, c<>0
! c
abs01=abs(c+d)
abs02=abs(c-d)
abs03=abs(-c+d)
abs04=abs(-c-d)
include 'hex_ph_23_c.f90'
else if ( bm<am*tol .and. cm<am*tol ) then
! a<>0, b=0, c=0
! a
abs01=abs(a+d)
abs02=abs(a-d)
abs09=abs(-a+d)
abs10=abs(-a-d)
include 'hex_ph_23_a.f90'
else if ( am<bm*tol .and. cm<bm*tol ) then
! a=0, b<>0, c=0
! b
abs01=abs(b+d)
abs02=abs(b-d)
abs05=abs(-b+d)
abs06=abs(-b-d)
include 'hex_ph_23_b.f90'
else
! a<>0, b<>0, c<>0
! a b c
abs01=abs(a+b+c+d)
abs02=abs(a+b+c-d)
abs03=abs(a+b-c+d)
abs04=abs(a+b-c-d)
abs05=abs(a-b+c+d)
abs06=abs(a-b+c-d)
abs07=abs(a-b-c+d)
abs08=abs(a-b-c-d)
abs09=abs(-a+b+c+d)
abs10=abs(-a+b+c-d)
abs11=abs(-a+b-c+d)
abs12=abs(-a+b-c-d)
abs13=abs(-a-b+c+d)
abs14=abs(-a-b+c-d)
abs15=abs(-a-b-c+d)
abs16=abs(-a-b-c-d)
include 'hex_ph_23_abc.f90'
end if
eqcv=matmul(InvA31,BV)
case default
stop 'Invalid etype specified'
end select
end subroutine Heqpol_coefficients