Cantilevered tapered membrane#

問題の説明#

テーパのある片持ち梁を膜要素を用いて固有値解析を実施した際の6次モードまでの固有振動数を比較するベンチマークです。The Standard NAFEMS BenchmarksにFV32として掲載されています。

条件#

ベンチマークに使用するメッシュは次の通りです。1次要素と2次要素でメッシュのパターンを分けています。

../../../_images/cantilevered_tapered_membrane_mesh_LOE.png
../../../_images/cantilevered_tapered_membrane_mesh_HOE.png

条件図を次に示します。図の左端を並進拘束（XY拘束）し、全節点は奥行方向の並進拘束（Z拘束）しています。

../../../_images/cantilevered_tapered_membrane_bc.png

入力する物性値は次の通りです。

入力した材料物性・その他特性#

材料物性

入力値

ヤング率

200E+9 N/m^2

ポアソン比

0.3

密度

8000 kg/m^3

結果と考察#

比較する結果は各モード次数での固有振動数の値です。比較結果を示します。

Results (Mode1)#

Solver

Type

Order

Shape

Result

Error

Reference value

1

44.905

‐

Reference value

2

44.636

‐

Commercial code

M3D4

1

quad

44.783

99.73%

Commercial code

M3D8

2

quad

44.636

100.00%

CalculiX 2.18

M3D4

1

quad

44.946

100.09%
../../../_images/cantilevered_tapered_membrane_commercial_order1_mode1.png

Commercial code 1st order elements (mode1)#

../../../_images/cantilevered_tapered_membrane_commercial_order2_mode1.png

Commercial code 2nd order elements (mode1)#

../../../_images/cantilevered_tapered_membrane_calculix_order1_mode1.png

CalculiX 1st order elements (mode1)#

Results (Mode2)#

Solver

Type

Order

Shape

Result

Error

Reference value

1

132.12

‐

Reference value

2

130.14

‐

Commercial code

M3D4

1

quad

130.63

98.87%

Commercial code

M3D8

2

quad

130.14

100.00%

CalculiX 2.18

M3D4

1

quad

132.23

100.08%
../../../_images/cantilevered_tapered_membrane_commercial_order1_mode2.png

Commercial code 1st order elements (mode12)#

../../../_images/cantilevered_tapered_membrane_commercial_order2_mode2.png

Commercial code 2nd order elements (mode2)#

../../../_images/cantilevered_tapered_membrane_calculix_order1_mode2.png

CalculiX 1st order elements (mode2)#

Results (Mode3)#

Solver

Type

Order

Shape

Result

Error

Reference value

1

162.83

‐

Reference value

2

162.72

‐

Commercial code

M3D4

1

quad

162.59

99.85%

Commercial code

M3D8

2

quad

162.72

100.00%

CalculiX 2.18

M3D4

1

quad

162.81

99.99%
../../../_images/cantilevered_tapered_membrane_commercial_order1_mode3.png

Commercial code 1st order elements (mode3)#

../../../_images/cantilevered_tapered_membrane_commercial_order2_mode3.png

Commercial code 2nd order elements (mode3)#

../../../_images/cantilevered_tapered_membrane_calculix_order1_mode3.png

CalculiX 1st order elements (mode3)#

Results (Mode4)#

Solver

Type

Order

Shape

Result

Error

Reference value

1

252.99

‐

Reference value

2

246.63

‐

Commercial code

M3D4

1

quad

246.79

97.55%

Commercial code

M3D8

2

quad

246.63

100.00%

CalculiX 2.18

M3D4

1

quad

253.20

100.08%
../../../_images/cantilevered_tapered_membrane_commercial_order1_mode4.png

Commercial code 1st order elements (mode4)#

../../../_images/cantilevered_tapered_membrane_commercial_order2_mode4.png

Commercial code 2nd order elements (mode4)#

../../../_images/cantilevered_tapered_membrane_calculix_order1_mode4.png

CalculiX 1st order elements (mode4)#

Results (Mode5)#

Solver

Type

Order

Shape

Result

Error

Reference value

1

393.31

‐

Reference value

2

382.02

‐

Commercial code

M3D4

1

quad

379.14

96.40%

Commercial code

M3D8

2

quad

382.02

100.00%

CalculiX 2.18

M3D4

1

quad

393.00

99.92%
../../../_images/cantilevered_tapered_membrane_commercial_order1_mode5.png

Commercial code 1st order elements (mode5)#

../../../_images/cantilevered_tapered_membrane_commercial_order2_mode5.png

Commercial code 2nd order elements (mode5)#

../../../_images/cantilevered_tapered_membrane_calculix_order1_mode5.png

CalculiX 1st order elements (mode5)#

Results (Mode6)#

Solver

Type

Order

Shape

Result

Error

Reference value

1

396.26

‐

Reference value

2

391.55

‐

Commercial code

M3D4

1

quad

389.83

98.38%

Commercial code

M3D8

2

quad

391.55

100.00%

CalculiX 2.18

M3D4

1

quad

396.53

100.07%
../../../_images/cantilevered_tapered_membrane_commercial_order1_mode6.png

Commercial code 1st order elements (mode6)#

../../../_images/cantilevered_tapered_membrane_commercial_order2_mode6.png

Commercial code 2nd order elements (mode6)#

../../../_images/cantilevered_tapered_membrane_calculix_order1_mode6.png

CalculiX 1st order elements (mode6)#

CalculiX1次要素のモード5と6は画像の掲載間違いではなく、出力されているモードは図の通りである。