2.3. A basic FE method for the acoustic wave equation#
We follow the method of lines approach for Galerkin methods to discretize the second order wave equation (2.1) on a spacial domain \(\Omega:=(0,1)\times(0,1)\).
In NGSolve the domain \((0,1)\times(0,1)\) is readily available as unit_square
# import ngsolve and webgui
from ngsolve import *
from ngsolve.webgui import Draw
# draw the unit_square shape
Draw(unit_square.shape);
but can also be generated by hand as follows
# import occ geometry tools
from netgen.occ import *
unit_square_wp = Rectangle(1,1) # this actually returns a WorkPlane object
Draw(unit_square_wp.Face());
Specifically we pick a so-called finite-element method for the spacial discretization. In our context this means that the domain \(\Omega\) is split up into a family of disjoint open subdomains \(\mathcal T\) such that
\[\bar{\Omega} = \bigcup_{T\in\mathcal T}\bar T\]
The subdomains \(T\) are usually triangles or quadrilaterals (in 2d) or tetrahedra or hexahedra (in 3d) but in theory can take any shape. The decomposition into subdomains is called meshing and the set \(\mathcal T\) is called mesh. In NGSolve this can be done as follows where the parameter maxh defines the size of the triangles.
geo = OCCGeometry(unit_square.shape, dim = 2) # explicitely state dimension, otherwise dx integrals will vanish
mesh = Mesh(geo.GenerateMesh(maxh = 0.05))
Draw(mesh);
Next we need to define basis functions for the discrete space \(V\). In this example this is done using the command H1 as follows.
V = H1(mesh)
A function \(u\in V\) is called GridFunction and is represented by a coefficient vector.
gfu = GridFunction(V)
print(gfu.vec)
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By setting single entries of gfu.vec to one we obtain the respective basis functions.
scene = Draw (gfu)
from time import sleep
#sleep(5)
for i in range(V.ndof):
gfu.vec[:] = 0
gfu.vec[i] = 1
scene.Redraw()
#sleep(1)
We want to solve the semi-discrete weak problem to find \(u_h\in C^2([0,T];V)\)
(2.6)#\[\begin{split}\begin{aligned}
\int_\Omega \partial_t^2 u_h(t,x)u_h'(x)dx+\int_\Omega\nabla u_h(t,x)\cdot\nabla u'_h(x)dx&=0,\\
u_h(0,\cdot) &= u_{h,0}
\end{aligned}\end{split}\]
for all \(u_h'\in V\). Or equivalently if \(u_h,u_{h,0}\) are represented by the coefficient vectors \(\mathbf u(t)\), \(\mathbf u^0\)
(2.7)#\[\begin{split}\begin{aligned}
\frac{d^2}{dt}\mathbf M \mathbf u(t)+\mathbf S\mathbf u(t) &= 0\\
\mathbf u(0) &= \mathbf u^0
\end{aligned}\end{split}\]
where the matrices \(\mathbf M,\mathbf S\) are defined by
\[\begin{aligned}
(\mathbf M)_{i,j}&:=\int_\Omega b_j(x)b_i(x)dx,&
(\mathbf S)_{i,j}&:=\int_\Omega \nabla b_j(x)\cdot \nabla b_i(x)dx
\end{aligned}\]
for a basis \(b_0,\ldots,b_N\) of \(V\).
To this end we need to assemble the matrices \(\mathbf S,\mathbf M\). In NGSolve this can be done by
u,u_ = V.TnT() # get test and trial functions
M = BilinearForm(u*u_*dx).Assemble()
S = BilinearForm(grad(u)*grad(u_)*dx).Assemble()
print(M.mat)
Row 0: 0: 0.000208333 4: 0.000104167 79: 0.000104167
Row 1: 1: 0.000342367 22: 8.54832e-05 23: 8.57001e-05 98: 0.000171183
Row 2: 2: 0.000208333 41: 0.000104167 42: 0.000104167
Row 3: 3: 0.000341315 60: 8.54003e-05 61: 8.52571e-05 135: 0.000170657
Row 4: 0: 0.000104167 4: 0.000763462 5: 0.000126639 79: 0.000255092 80: 0.000277565
Row 5: 4: 0.000126639 5: 0.00073938 6: 0.000102417 80: 0.000267273 81: 0.000243051
Row 6: 5: 0.000102417 6: 0.000594258 7: 9.30758e-05 81: 0.000204053 82: 0.000194712
Row 7: 6: 9.30758e-05 7: 0.000547278 8: 8.94372e-05 82: 0.000184202 83: 0.000180563
Row 8: 7: 8.94372e-05 8: 0.000531263 9: 8.81101e-05 83: 0.000177521 84: 0.000176194
Row 9: 8: 8.81101e-05 9: 0.000525946 10: 8.75873e-05 84: 0.000175386 85: 0.000174863
Row 10: 9: 8.75873e-05 10: 0.000523909 11: 8.73887e-05 85: 0.000174566 86: 0.000174367
Row 11: 10: 8.73887e-05 11: 0.000523275 12: 8.73798e-05 86: 0.000174258 87: 0.000174249
Row 12: 11: 8.73798e-05 12: 0.000523056 13: 8.73503e-05 87: 0.000174178 88: 0.000174148
Row 13: 12: 8.73503e-05 13: 0.000522952 14: 8.73609e-05 88: 0.000174115 89: 0.000174126
Row 14: 13: 8.73609e-05 14: 0.000522603 15: 8.72745e-05 89: 0.000174027 90: 0.000173941
Row 15: 14: 8.72745e-05 15: 0.000521021 16: 8.69036e-05 90: 0.000173607 91: 0.000173236
Row 16: 15: 8.69036e-05 16: 0.000518299 17: 8.64754e-05 91: 0.000172674 92: 0.000172246
Row 17: 16: 8.64754e-05 17: 0.000515322 18: 8.60603e-05 92: 0.000171601 93: 0.000171185
Row 18: 17: 8.60603e-05 18: 0.000513015 19: 8.57385e-05 93: 0.000170769 94: 0.000170447
Row 19: 18: 8.57385e-05 19: 0.0005117 20: 8.53148e-05 94: 0.000170535 95: 0.000170111
Row 20: 19: 8.53148e-05 20: 0.000498635 21: 8.20445e-05 95: 0.000167273 96: 0.000164003
Row 21: 20: 8.20445e-05 21: 0.000464266 22: 7.7567e-05 96: 0.000154566 97: 0.000150088
Row 22: 1: 8.54832e-05 21: 7.7567e-05 22: 0.000469087 97: 0.000149061 98: 0.000156977
Row 23: 1: 8.57001e-05 23: 0.000471385 24: 7.82184e-05 98: 0.000157474 99: 0.000149993
Row 24: 23: 7.82184e-05 24: 0.000470957 25: 8.35656e-05 99: 0.000151913 100: 0.00015726
Row 25: 24: 8.35656e-05 25: 0.000513677 26: 8.82753e-05 100: 0.000168563 101: 0.000173273
Row 26: 25: 8.82753e-05 26: 0.000535098 27: 8.94693e-05 101: 0.00017808 102: 0.000179274
Row 27: 26: 8.94693e-05 27: 0.000536947 28: 8.91474e-05 102: 0.000179326 103: 0.000179004
Row 28: 27: 8.91474e-05 28: 0.000532249 29: 8.84193e-05 103: 0.000177705 104: 0.000176977
Row 29: 28: 8.84193e-05 29: 0.000527853 30: 8.78759e-05 104: 0.000176051 105: 0.000175507
Row 30: 29: 8.78759e-05 30: 0.000524819 31: 8.75913e-05 105: 0.000174818 106: 0.000174533
Row 31: 30: 8.75913e-05 31: 0.000523938 32: 8.7557e-05 106: 0.000174412 107: 0.000174378
Row 32: 31: 8.7557e-05 32: 0.000524859 33: 8.77186e-05 107: 0.000174711 108: 0.000174872
Row 33: 32: 8.77186e-05 33: 0.000526046 34: 8.79714e-05 108: 0.000175051 109: 0.000175304
Row 34: 33: 8.79714e-05 34: 0.000528212 35: 8.83514e-05 109: 0.000175755 110: 0.000176135
Row 35: 34: 8.83514e-05 35: 0.000531596 36: 8.88682e-05 110: 0.00017693 111: 0.000177447
Row 36: 35: 8.88682e-05 36: 0.000535362 37: 8.95445e-05 111: 0.000178137 112: 0.000178813
Row 37: 36: 8.95445e-05 37: 0.000541101 38: 9.04358e-05 112: 0.000180115 113: 0.000181006
Row 38: 37: 9.04358e-05 38: 0.000548879 39: 9.13642e-05 113: 0.000183076 114: 0.000184004
Row 39: 38: 9.13642e-05 39: 0.000553215 40: 9.11462e-05 114: 0.000185461 115: 0.000185243
Row 40: 39: 9.11462e-05 40: 0.000546593 41: 8.86457e-05 115: 0.000184651 116: 0.00018215
Row 41: 2: 0.000104167 40: 8.86457e-05 41: 0.000634166 42: 0.000228437 116: 0.000212916
Row 42: 2: 0.000104167 41: 0.000228437 42: 0.000816728 43: 8.28747e-05 116: 0.000221323 117: 0.000179927
Row 43: 42: 8.28747e-05 43: 0.000505887 44: 9.52776e-05 117: 0.000157666 118: 0.000170069
Row 44: 43: 9.52776e-05 44: 0.000585882 45: 9.90607e-05 118: 0.000193881 119: 0.000197664
Row 45: 44: 9.90607e-05 45: 0.00059059 46: 9.6029e-05 119: 0.000199266 120: 0.000196234
Row 46: 45: 9.6029e-05 46: 0.000571322 47: 9.38961e-05 120: 0.000191765 121: 0.000189632
Row 47: 46: 9.38961e-05 47: 0.000560722 48: 9.28762e-05 121: 0.000187485 122: 0.000186465
Row 48: 47: 9.28762e-05 48: 0.000556108 49: 9.23946e-05 122: 0.00018566 123: 0.000185178
Row 49: 48: 9.23946e-05 49: 0.00055317 50: 9.1926e-05 123: 0.000184659 124: 0.000184191
Row 50: 49: 9.1926e-05 50: 0.000550369 51: 9.15149e-05 124: 0.00018367 125: 0.000183258
Row 51: 50: 9.15149e-05 51: 0.000547693 52: 9.10881e-05 125: 0.000182758 126: 0.000182331
Row 52: 51: 9.10881e-05 52: 0.000544246 53: 9.04602e-05 126: 0.000181663 127: 0.000181035
Row 53: 52: 9.04602e-05 53: 0.000539253 54: 8.95555e-05 127: 0.000180071 128: 0.000179166
Row 54: 53: 8.95555e-05 54: 0.000532511 55: 8.84734e-05 128: 0.000177782 129: 0.0001767
Row 55: 54: 8.84734e-05 55: 0.000524765 56: 8.72511e-05 129: 0.000175131 130: 0.000173909
Row 56: 55: 8.72511e-05 56: 0.000517857 57: 8.6335e-05 130: 0.000172593 131: 0.000171677
Row 57: 56: 8.6335e-05 57: 0.00051289 58: 8.54452e-05 131: 0.000171 132: 0.00017011
Row 58: 57: 8.54452e-05 58: 0.000498016 59: 8.2056e-05 132: 0.000166952 133: 0.000163563
Row 59: 58: 8.2056e-05 59: 0.000463572 60: 7.75398e-05 133: 0.000154246 134: 0.00014973
Row 60: 3: 8.54003e-05 59: 7.75398e-05 60: 0.000468148 134: 0.000148674 135: 0.000156534
Row 61: 3: 8.52571e-05 61: 0.000467977 62: 7.75132e-05 135: 0.000156475 136: 0.000148732
Row 62: 61: 7.75132e-05 62: 0.000465576 63: 8.2443e-05 136: 0.000150345 137: 0.000155275
Row 63: 62: 8.2443e-05 63: 0.000505007 64: 8.65328e-05 137: 0.000165971 138: 0.00017006
Row 64: 63: 8.65328e-05 64: 0.000522515 65: 8.72903e-05 138: 0.000173967 139: 0.000174725
Row 65: 64: 8.72903e-05 65: 0.000522866 66: 8.69429e-05 139: 0.00017449 140: 0.000174143
Row 66: 65: 8.69429e-05 66: 0.000519598 67: 8.65832e-05 140: 0.000173216 141: 0.000172856
Row 67: 66: 8.65832e-05 67: 0.000517374 68: 8.65102e-05 141: 0.000172177 142: 0.000172104
Row 68: 67: 8.65102e-05 68: 0.000518479 69: 8.68042e-05 142: 0.000172435 143: 0.000172729
Row 69: 68: 8.68042e-05 69: 0.000520881 70: 8.72388e-05 143: 0.000173202 144: 0.000173636
Row 70: 69: 8.72388e-05 70: 0.000523074 71: 8.76698e-05 144: 0.000173867 145: 0.000174298
Row 71: 70: 8.76698e-05 71: 0.00052663 72: 8.81634e-05 145: 0.000175151 146: 0.000175645
Row 72: 71: 8.81634e-05 72: 0.000528806 73: 8.86626e-05 146: 0.000175741 147: 0.00017624
Row 73: 72: 8.86626e-05 73: 0.000532448 74: 8.92744e-05 147: 0.00017695 148: 0.000177561
Row 74: 73: 8.92744e-05 74: 0.000536116 75: 9.01279e-05 148: 0.00017793 149: 0.000178784
Row 75: 74: 9.01279e-05 75: 0.000543681 76: 9.17085e-05 149: 0.000180132 150: 0.000181713
Row 76: 75: 9.17085e-05 76: 0.000560497 77: 9.55241e-05 150: 0.000184724 151: 0.00018854
Row 77: 76: 9.55241e-05 77: 0.000607397 78: 0.000104691 151: 0.000199007 152: 0.000208174
Row 78: 77: 0.000104691 78: 0.00074982 79: 0.000128453 80: 0.000270219 152: 0.000246457
Row 79: 0: 0.000104167 4: 0.000255092 78: 0.000128453 79: 0.000767089 80: 0.000279378
Row 80: 4: 0.000277565 5: 0.000267273 78: 0.000270219 79: 0.000279378 80: 0.00188566 81: 0.000267144 152: 0.000269667 154: 0.000254411
Row 81: 5: 0.000243051 6: 0.000204053 80: 0.000267144 81: 0.00133548 82: 0.00019646 153: 0.000196546 154: 0.000228232
Row 82: 6: 0.000194712 7: 0.000184202 81: 0.00019646 82: 0.00111475 83: 0.000178929 153: 0.000183733 220: 0.000176713
Row 83: 7: 0.000180563 8: 0.000177521 82: 0.000178929 83: 0.00105908 84: 0.000174964 155: 0.000173087 220: 0.00017401
Row 84: 8: 0.000176194 9: 0.000175386 83: 0.000174964 84: 0.00104738 85: 0.000174169 155: 0.000173329 156: 0.000173342
Row 85: 9: 0.000174863 10: 0.000174566 84: 0.000174169 85: 0.00104404 86: 0.000173763 156: 0.000173392 157: 0.000173283
Row 86: 10: 0.000174367 11: 0.000174258 85: 0.000173763 86: 0.00104209 87: 0.000173586 157: 0.000173089 158: 0.000173022
Row 87: 11: 0.000174249 12: 0.000174178 86: 0.000173586 87: 0.00104123 88: 0.000173484 158: 0.00017288 159: 0.000172849
Row 88: 12: 0.000174148 13: 0.000174115 87: 0.000173484 88: 0.00104123 89: 0.000173564 159: 0.000172901 160: 0.000173014
Row 89: 13: 0.000174126 14: 0.000174027 88: 0.000173564 89: 0.00104033 90: 0.000173195 160: 0.000172844 161: 0.000172574
Row 90: 14: 0.000173941 15: 0.000173607 89: 0.000173195 90: 0.00103777 91: 0.000172678 161: 0.000172267 162: 0.000172084
Row 91: 15: 0.000173236 16: 0.000172674 90: 0.000172678 91: 0.0010333 92: 0.000171681 162: 0.000171735 163: 0.0001713
Row 92: 16: 0.000172246 17: 0.000171601 91: 0.000171681 92: 0.00102539 93: 0.000170071 163: 0.000170379 164: 0.000169414
Row 93: 17: 0.000171185 18: 0.000170769 92: 0.000170071 93: 0.0010159 94: 0.000168801 164: 0.000167965 165: 0.000167112
Row 94: 18: 0.000170447 19: 0.000170535 93: 0.000168801 94: 0.00101642 95: 0.000170315 165: 0.000167449 166: 0.000168875
Row 95: 19: 0.000170111 20: 0.000167273 94: 0.000170315 95: 0.00102144 96: 0.000168338 166: 0.000172271 167: 0.000173132
Row 96: 20: 0.000164003 21: 0.000154566 95: 0.000168338 96: 0.000955878 97: 0.000145874 167: 0.000168062 168: 0.000155035
Row 97: 21: 0.000150088 22: 0.000149061 96: 0.000145874 97: 0.000811087 98: 0.000126342 168: 0.000129113 169: 0.000110609
Row 98: 1: 0.000171183 22: 0.000156977 23: 0.000157474 97: 0.000126342 98: 0.000848508 99: 0.000126729 169: 0.000109803
Row 99: 23: 0.000149993 24: 0.000151913 98: 0.000126729 99: 0.000818322 100: 0.000148158 169: 0.000111011 171: 0.00013052
Row 100: 24: 0.00015726 25: 0.000168563 99: 0.000148158 100: 0.000983636 101: 0.000175371 170: 0.000175096 171: 0.000159186
Row 101: 25: 0.000173273 26: 0.00017808 100: 0.000175371 101: 0.00107959 102: 0.000181897 170: 0.000184622 172: 0.000186341
Row 102: 26: 0.000179274 27: 0.000179326 101: 0.000181897 102: 0.00108689 103: 0.000180279 172: 0.000183892 173: 0.000182221
Row 103: 27: 0.000179004 28: 0.000177705 102: 0.000180279 103: 0.00107117 104: 0.000176919 173: 0.000179663 174: 0.000177602
Row 104: 28: 0.000176977 29: 0.000176051 103: 0.000176919 104: 0.00105416 105: 0.000174803 174: 0.000175299 175: 0.00017411
Row 105: 29: 0.000175507 30: 0.000174818 104: 0.000174803 105: 0.00104402 106: 0.000173454 175: 0.000173048 176: 0.000172389
Row 106: 30: 0.000174533 31: 0.000174412 105: 0.000173454 106: 0.00104102 107: 0.000173435 176: 0.000172541 177: 0.000172643
Row 107: 31: 0.000174378 32: 0.000174711 106: 0.000173435 107: 0.0010425 108: 0.000174166 177: 0.000172705 178: 0.000173103
Row 108: 32: 0.000174872 33: 0.000175051 107: 0.000174166 108: 0.00104583 109: 0.000174536 178: 0.000173509 179: 0.0001737
Row 109: 33: 0.000175304 34: 0.000175755 108: 0.000174536 109: 0.0010507 110: 0.000175487 179: 0.000174556 180: 0.000175057
Row 110: 34: 0.000176135 35: 0.00017693 109: 0.000175487 110: 0.00105733 111: 0.000177092 180: 0.000175436 181: 0.000176246
Row 111: 35: 0.000177447 36: 0.000178137 110: 0.000177092 111: 0.00106619 112: 0.000178474 181: 0.000177174 182: 0.000177865
Row 112: 36: 0.000178813 37: 0.000180115 111: 0.000178474 112: 0.0010786 113: 0.000181034 182: 0.000179455 183: 0.000180713
Row 113: 37: 0.000181006 38: 0.000183076 112: 0.000181034 113: 0.00109899 114: 0.000185696 183: 0.000182791 184: 0.000185384
Row 114: 38: 0.000184004 39: 0.000185461 113: 0.000185696 114: 0.00113102 115: 0.000190822 184: 0.000190686 185: 0.000194354
Row 115: 39: 0.000185243 40: 0.000184651 114: 0.000190822 115: 0.0011394 116: 0.000188192 185: 0.000196265 186: 0.000194227
Row 116: 40: 0.00018215 41: 0.000212916 42: 0.000221323 115: 0.000188192 116: 0.00115559 117: 0.000176686 186: 0.000174321
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Row 254: 192: 0.000185186 193: 0.000184283 253: 0.000185692 254: 0.00110935 255: 0.000184275 305: 0.000185217 307: 0.000184702
Row 255: 193: 0.000183817 194: 0.000183457 254: 0.000184275 255: 0.00110436 256: 0.00018388 307: 0.000184485 308: 0.00018445
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Row 259: 197: 0.000174379 198: 0.000170939 258: 0.000176101 259: 0.00103697 260: 0.000168994 311: 0.000175113 312: 0.000171446
Row 260: 198: 0.000167593 199: 0.000162147 259: 0.000168994 260: 0.000982572 261: 0.000157061 312: 0.000166631 313: 0.000160145
Row 261: 199: 0.000158704 200: 0.000159951 260: 0.000157061 261: 0.00089574 262: 0.000146253 313: 0.000142913 314: 0.000130857
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Row 401: 337: 0.000161335 360: 0.00015419 393: 0.000166051 401: 0.000952681 406: 0.000150267 433: 0.000164738 446: 0.000156099
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Approximating the initial conditions (we choose \(u_0=\exp(-10(x^2+y^2))\)) can be done using GridFunction.Set
gfu.Set(exp(-10*(x**2+y**2)))
Draw(gfu);
It remains to solve (2.7) using an appropriate time stepping.
We choose a variant of the Newmark time-stepping which is given for problems of the form
\[A\frac{d^2}{dt} x + B\frac{d}{dt}x+Cx=r,\quad
x(0)=x_0,\quad \frac{d}{dt}x(0)=0\]
by the approximations \(x^j\approx x(\tau j)\), \(\dot x^j\approx \frac{d}{dt} x(\tau j)\) and a time step size \(\tau>0\)
\[\begin{split}\begin{aligned}
x^0&=x_0\\
\dot x^0&=0\\
x^{j+1}&=x^j+\frac{\tau}{2}(\dot x^{j+1}+\dot x^{j})\\
A\dot x^{j+1}&=A\dot x^j+\frac{\tau}{2}\left(r^{j+1}+r^j-C(x^{j+1}+x^{j})-B(\dot x^{j+1}+\dot x^{j})\right),
\end{aligned}\end{split}\]
where we also set \(r^j=r(\tau j)\). Eliminating \(x^{j+1}\) on the right hand side of the last equation and collecting the coefficients of \(\dot x^{j+1}\) leads to
\[\left(A+\frac{\tau}{2}B+\frac{\tau^2}{4}C\right)\dot x^{j+1}=A\dot x^j+\frac{\tau}{2}\left(r^{j+1}+r^j-B\dot x^j-C(2x^j+\frac{\tau}{2}\dot x^j)\right)
\]
Note that in our example \(A=\mathbf M\), \(C=\mathbf S\), \(r=0\), \(B=0\).
We precompute the inverse of the matrix \(\mathbf M^*=M+\frac{\tau^2}{4}\mathbf S\)
tau = 0.01
mstarinv = BilinearForm(u*u_*dx+tau**2/4* grad(u)*grad(u_)*dx).Assemble().mat.Inverse()
We define the approximation to the time derivative as a vector (and not a GridFunction) only and start the time loop
T = 1
udot = gfu.vec.CreateVector()
udot[:] = 0.
scene = Draw(gfu, deformation=True)
for j in range(int(T/tau)):
gfu.vec.data += tau/2 * udot
udot.data -= tau * mstarinv*(S.mat*gfu.vec)
gfu.vec.data += tau/2 * udot
scene.Redraw()
Boundary conditions#
Since we assumed \(\nabla u\cdot n=0\) we ommitted the boundary term from our weak formulation (2.2). If we impose boundary conditions \(\nabla u_h(t,\cdot)\cdot n=g_h(t,\cdot)\) on (a part of) the boundary \(\Gamma=\partial\Omega\) we obtain
\[\int_{\Gamma}\nabla u_h(t,x)\cdot n(x) u_h'(x) dS(x)=
\int_{\Gamma}g_h(t,x) u_h'(x) dS(x)\]
and thus the semi-discrete weak formulation
(2.8)#\[\begin{split}\begin{aligned}
\int_\Omega \partial_t^2 u_h(t,x)u_h'(x)dx+\int_\Omega\nabla u_h(t,x)\cdot\nabla u'_h(x)dx&=\int_{\Gamma}g_h(t,x) u_h'(x) dS(x),\\
u_h(0,\cdot) &= u_{h,0}
\end{aligned}\end{split}\]
for all \(u_h'\in V\). Note that the boundary term is now independent of the unknown function \(u_h\) and thus acts as a right-hand-side.
To implement this in NGSolve we first need to make sure that we can adress the part of the boundary \(\Gamma\) correctly. We pick the left edge of the unit_square, which is conveniently already named correctly.
print(mesh.GetBoundaries())
('bottom', 'right', 'top', 'left')
Boundaries can also be named manually, e.g.
# import occ geometry tools
from netgen.occ import *
square = Rectangle(1,1).Face() # this actually returns a WorkPlane object
square.edges.Min(X).name = "left"
Choosing \(g=cos(\omega t)\exp(-10(y-0.5)^2)\) for some \(\omega>0\) we assemble the spacial part of the boundary term as a LinearForm similar to the BilinearForm. The boundary integral can be evaluated using ds as follows:
f = LinearForm(exp(-10*(y-0.5)**2)*u_*ds("left")).Assemble()
It remains to include the right-hand-side in the time-stepping:
T = 1
udot = gfu.vec.CreateVector()
udot[:] = 0.
gfu.vec[:] = 0.
omega = 15
scene = Draw(gfu, deformation=True)
for j in range(int(T/tau)):
gfu.vec.data += tau/2 * udot
udot.data += tau * mstarinv*(1/2*(cos(j*tau*omega)+cos((j+1)*tau*omega))*f.vec-S.mat*gfu.vec)
gfu.vec.data += tau/2 * udot
scene.Redraw()