Modeling A ============== Characteristics of modeling ----------------------------------- 3D elements (PENTA15 + HEXA20) .. image:: images/Shape1.gif .. _RefSchema_Shape1.gif: Modeling: 1/4 of the cylinder according to the circumference .. csv-table:: "2 zones:", "zone 1 = lower part", ":math:`(0\le z\le L/2)`" "", "zone 2 = upper part", ":math:`(L/2\le Z\le L)`" Cutting: 20 elements depending on the length 16 elements depending on the circumference 2 elements in the thickness Coordinates of points :math:`(r,\theta ,z)` .. csv-table:: "", "A", "G", "B", "B", "E", "G1", "F", "A2 A'2", "H H'", "B2 B'2", "E2 E'2", "H1 H'1", "F2 F'2", "A3", "I", "B3", "B3", "E3", "I1", "F3" ":math:`r` ", "Ri", "R", "R", "Re", "Ri", "Ri", "R", "R", "R", "R", "R", "Re", "Ri", "Ri", "Ri", "Ri", "Ri", "R", "R", "Ri", "R", "R", "Ri", "Ri", "Ri", "Ri", "R", "Ri", "Ri", "Ri", "Ri", "Ri" ":math:`\theta` ", "0. ", "0. ", "0. ", "90. ", "90. ", "90. ", "0. ", "0", ".0. ", "90. ", "90. ", "90. ", "0. ", "0. ", "0. ", "90. ", "90. ", "90." ":math:`z` ", "0. ", "0. ", "0. ", "0. ", "0. ", "0. ", "L/2", "L/2", "L/2", "L/2", "L/2", "L/2", "L/2", "L", "L", "L", "L", "L", "L" :math:`\mathrm{Ri}` = inner radius :math:`\text{Re}` = outer radius The :math:`\mathrm{A2},H,\mathrm{B2},\mathrm{E2},\mathrm{H2},\mathrm{F2}` points are in section :math:`z=L/2` of zone 1 Points :math:`A’\mathrm{2,}H’,B’\mathrm{2,}E’\mathrm{2,}H’\mathrm{2,}F’2` are the opposite sides in zone 2 Boundary conditions: * Support conditions :math:`w=0` at the base (section :math:`z=0.`) introduced by the keyword LIAISON_OBLIQUE * Symmetry conditions :math:`v=0.` on the :math:`\mathrm{AB}` side introduced by the LIAISON_OBLIQUE keyword * Symmetry conditions :math:`u=0.` on the :math:`\mathrm{EF}` side introduced by the LIAISON_OBLIQUE keyword * Identification of the nodes common to the 2 zones (section :math:`z=L/2`) by the LIAISON_GROUP keyword. Charging: Surface load :math:`p=q/h=500000N/\mathrm{m2}`, along the axis, or in global coordinate system: :math:`\mathrm{Fx}=0.` :math:`\mathrm{Fy}=p/2` :math:`\mathrm{Fz}=p\frac{\sqrt{3}}{2}` Node name: .. csv-table:: "plan :math:`z=0.` "," :math:`A=N1` "," :math:`B=N321` "," "," :math:`E=N1740` "," "," :math:`F=N1541` ","" :math:`G=N1540` "", "", "", "", "", "", "" "plan :math:`z=2` (zone 1)", ":math:`\mathrm{A2}=N961` "," :math:`\mathrm{B2}=N993` "," "," :math:`\mathrm{E2}=N2141` "," :math:`\mathrm{F2}=N2122` "," :math:`H=N962` "," :math:`\mathrm{H1}=N2121`" "", "", "", "", "", "", "" "plan :math:`z=2` (zone 2)", ":math:`A’2=N3361` "," :math:`B’2=N3364` "," "," :math:`E’2=N2159` "," :math:`F’2=N2155` "," :math:`H’=N3360` "," :math:`H’1=N2156`" "", "", "", "", "", "", "" "plan :math:`z=4` "," :math:`\mathrm{A3}=N3359` "," :math:`\mathrm{B3}=N3355` "," "," :math:`I=N3356` "," "," :math:`\mathrm{E3}=N2151` "," :math:`\mathrm{F3}=N2154` "," :math:`\mathrm{I1}=N2150`" Characteristics of the mesh ---------------------------- Number of knots: 4298 Number of meshes and types: 160 HEXA20, 320 PENTA15 Tested values --------------- :math:`U,V,W` displacement values read from file .. csv-table:: "**Location**", "**Value type**", "**Reference**" "Point :math:`G` "," :math:`U(m)` ", "—7.143 x 10—7" "", ":math:`V(m)` ", "0." "", ":math:`W(m)` ", "0." "Point :math:`H,H’` "," :math:`U(m)` ", "—7.143 x 10—7" "Point :math:`I` "," :math:`U(m)` ", "—7.143 x 10—7" "Point :math:`\mathrm{G1}` "," :math:`U(m)` ", "0." "Points :math:`\mathrm{H1},H’1` "," :math:`U(m)` ", "0." Values of :math:`u,v,{u}_{r}` movements in local coordinate system calculated from :math:`U,V,W` .. csv-table:: "**Location**", "**Value type**", "**Reference**" "Point :math:`G` "," :math:`{u}_{r}(m)` ", "—7.143 x 10—7" "", ":math:`v(m)` ", "0." "Point :math:`H,H’` "," :math:`{u}_{r}(m)` ", "—7.143 x 10—7" "", ":math:`v(m)` ", "0." "Point :math:`I` "," :math:`{u}_{r}(m)` ", "—7.143 x 10—7" "", ":math:`v(m)` ", "0." "Point :math:`\mathrm{A2},A’2` Points :math:`\mathrm{B2},B’2` "," :math:`v(m)` ", "0." "Point :math:`\mathrm{G1}` "," :math:`u(m)` ", "0." "", ":math:`{u}_{r}(m)` ", "—7.143 x 10—7" "Points :math:`\mathrm{H1},H’1` "," :math:`u(m)` ", "0." "", ":math:`{u}_{r}(m)` ", "—7.143 x 10—7" "Point :math:`\mathrm{I1}` "," :math:`u(m)` ", "0." "", ":math:`{u}_{r}(m)` ", "—7.143 x 10—7" "Points :math:`\mathrm{E2},E’2` "," :math:`u(m)` ", "0." "Points :math:`\mathrm{F2},F’2` "," :math:`u(m)` ", "0." "Points :math:`A,B,G` :math:`\mathrm{A2},\mathrm{B2},H` :math:`A’\mathrm{2,}B’\mathrm{2,}H’` :math:`\mathrm{A3},\mathrm{B3},I` "," :math:`{\sigma }_{\mathrm{YY}}(\mathrm{Pa})` ", "1.25 x 105" "Points :math:`A,B,G` :math:`\mathrm{A2},\mathrm{B2},H` :math:`A’\mathrm{2,}B’\mathrm{2,}H’` :math:`\mathrm{A3},\mathrm{B3},I` "," :math:`{\sigma }_{\mathrm{ZZ}}(\mathrm{Pa})` ", "3.75 x 105" notes --------- * Radial displacement :math:`\mathrm{ur}` is obtained with good precision. * The symmetry conditions on face :math:`\mathrm{AB}` (:math:`v=0` locally, i.e. :math:`\frac{\sqrt{3}}{2}V–05W=0`) are verified at the points :math:`\mathrm{A2},A’\mathrm{2,}G,\mathrm{B2},B’\mathrm{2,}H,H’,I` considered. Likewise, the symmetry conditions on face :math:`\mathrm{EF}` (:math:`u=U=0`) are verified at the envisaged points :math:`\mathrm{E2},E’\mathrm{2,}\mathrm{F2},F’\mathrm{2,}\mathrm{G1},\mathrm{H1},H’\mathrm{1,}\mathrm{I1}`. The LIAISON_OBLIQUE keyword is thus validated. * The identification of the nodes common to the 2 zones by the keyword LIAISON_GROUP is also validated: the movements :math:`U,V,W` are identical to the points :math:`A’\mathrm{2,}B’\mathrm{2,}H’,E’\mathrm{2,}F’\mathrm{2,}H’1` in comparison with the movements to the respective opposite sides :math:`\mathrm{A2},\mathrm{B2},H,\mathrm{E2},\mathrm{F2},\mathrm{H1}`.