Belytschko T. - Introduction (779635), страница 98
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Thelinear problems were studied to examine the convergence rate of various forms of these andcompeting elements. Table 5 gives a complete listing of the names associated with theelements tested in this section. All use 1-point integration in the nonlinear problems exceptfor QUAD4, ASQBI(2pt), ASQBI(2x2).Table 5. Names and descriptions of elements tested in this sectionNameSectionDescriptionQUAD41.2Standard isoparametric element with full 2x2 integration.FB (0.1)1.3Perturbation hourglass stabilization with the hourglass controlfactor, α s = 0.1.
(Flanagan and Belytschko (1981))FB (0.3)1.3Same as FB (0.1), except with α s = 0.3OI1.4Mixed method Optimal Incompressible stabilization (Belytschko andBachrach (1986)).QBI1.4Mixed method Quintessential Bending and Incompressiblestabilization (Belytschko and Bachrach).ASOI1.5Assumed strain stabilization using the OI strain fieldASQBI1.5Assumed strain stabilization using the QBI strain filedADS1.5Assumed deviatoric strain stabilizationASMD1.5Assumed strain stabilization using the strain field associated with themean dilatation element (Nagtegaal et al.(1974)).ASSRI1.5Assumed strain stabilization using the strain field associated withselective reduced integration (Hughes(1980))ASQBI(2 pt) 1.5.6 The QBI strain field is used with two stress evaluations per elementASQBI(2x2) 1.5.6 The QBI strain filed is used with four stress evaluations per elementPian-SumiharaThe Pian-Sumihara (1984) hybrid element (the formulation does notappear in this paper)631.6.1 Static Beam.
A linear, elastic cantilever with a load at its end is shown in Fig. 13.M and P at the left end of the cantilever are reactions at the support.yxMPDPLFigure 13. Static cantilever beamThis problem is identical to that used by Belytschko and Bachrach (1986). The analyticalsolution from Timoshenko and Goodier (1970) is−Py[(6L− 3x)x + (2+ν)(y2 − 14D2 )]6EIPuy (x,y)=[3νy2 (L−x) + 14(4+5ν)D2 x + (3L−x)x2 ]6EI1 3I = 12 D(1.5.44b)Efor plane stressE = E/(1−ν 2 ) for plane strain(1.5.45a)ux (x,y)=whereν=ν{ ν/(1−ν)for plane stressfor plane strain(1.5.44a)(1.5.45b)The displacements at the support end, x=0, − 12 D ≤ y ≤ 12 D are nonzero except at the top,bottom, and midline (as shown in Fig.
14). Reaction forces are applied at the supportbased on the stresses corresponding to (1.5.46) at x=0, which arePyσ x = − I (L−x)σ y = 0.(1.5.46a)(1.5.46b)Pτ xy = 2I(14 D2 − y2 )(1.5.46c)The distribution of applied load to the nodes at x=L is also obtained from the closed-formstress fields. The coarsest mesh used is shown in Fig. 14. This problem is symmetric, soonly half the cantilever is modeled.yD/2 = 6L=48.Figure 14. Coarse mesh of rectangular elementsP/2=20000.64xPt. AAll meshes use elements with an aspect ratio of 2.
Only the top half of the cantileveris modeled since the problem is antisymmetric. The following isotropic elastic materialswere used:1. Plane stress, ν= 0.252. Plane strain, ν= 0.4999The displacement and energy error norms are plotted in Figs. 15 and 16. for ν=0.25,the rate of convergence of the displacement error norm is around 1.8 for all of the elementsexcept for QBI, ASQBI and Pian-Sumihara which converge at a rate of 2.
All have a rateof convergence of the energy error norm of 1. For ν=0.4999, the rate of convergence ofthe displacement error norms is around 1.7 to 1.8 and the and rate of convergence of theenergy error norms is 1.0 for all elements except QUAD4 which locks as expected andexhibits very slow convergence. For incompressible material, QBI and ASQBI are almostidentical to OI and ASOI, whereas ASMD has less absolute accuracy. For rectangularelements and any linear material, OI and ASOI are identical. Likewise, the Pian andSumihara (1984) element is identical to QBI and ASQBI.6512Log (energy norm)Log (displacement norm)0-1-210-3-4-1-0.50.00.51.01.5-0.50.0Log (H)QUAD40.51.0Log (H)ASMDQBI, ASQBI,andPian-SumiharaASOIand OIADSFigure 15.
Convergence of displacement and energy error norms; ν=0.25, plane stress1.512.001.5Log (energy norm)Log (displacement norm)66-1-2-31.00.50.0-4-0.50.00.51.01.5-0.5-0.50.0Log (H)QUAD40.51.01.5Log (H)ASMDQBI, ASQBI,andPian-SumiharaASOIand OIADSFigure 16. Convergence of displacement and energy error norms; ν=0.4999, plane strainTo assess the coarse mesh accuracy of the elements, the normalized enddisplacements (point A if Fig. 14) for the 1x4 element mesh are shown in Table 6. Acoarse 1x4 element mesh of skewed elements was also run and the normalized enddisplacements (point A in Fig. 17) are shown in Table 7.
Pian-Sumihara is slightly betterthan ASQBI for the skewed elements, but the difference is minor.yθxL/4 = 12 (typ.)Pt. A67Figure 17. Skewed coarse mesh with θ = 9.462°Table 6. dyFEM / dyAnalytical at point A of mesh in Fig. 14 (rectangular elements)MaterialQUAD4ASMD120.7080.0610.7970.935QBI, ASQBI, andPian-Sumihara0.9860.982OIASOI0.8620.982and ADS1.1551.205Table 7. dyFEM / dyAnalytical at point A of mesh in Fig. 17 (skewed elements)MaterialQUAD4ASMD120.6890.0610.7760.915ASQBI PianASOISumihara0.9480.9550.8340.9570.9600.957ADS1.1121.1701.6.2 Circular Hole in Plate. This problem was considered to evaluate the performance ofthese elements in a different setting. A plate with a hole, solved by R.
C. J. Howland(1930) is shown in Fig. 18. The solution is in the form of an infinite series and gives thestress field around the circular hole in the center of an axially loaded plane stress plate offinite width and of infinite length. The series converges only within a circular regionaround the hole. The diameter of this circular area is equal to the plate width. Thedisplacement field is not given so convergence of the displacement norm could not bechecked.68The shaded area indicatesthe region of convergenceof the series solution.yσxxσxWL=∞Figure 18.
Plate of finite width with a circular holeFor the finite element meshes, the plate length was taken to be twice the plate width.The nodes at which the load is applied are outside the region in which the analyticalsolution converges, so the analytical solution could not be used to determine the loaddistribution on the end of the plate. The nodal forces were therefore calculated byassuming the analytical stress field at infinity, which is uniaxial. The error due to the finitelength was checked by running meshes with lengths of 2 and 5 times the plate width.
Thedifference between these solutions was found to be negligible. Four different meshes wereused which are summarized in Table 8. Fig. 19 shows the dimensions and boundaryconditions of the finite element model, and Fig. 20 shows the discretization for mesh 3with 320 elements.
The problem is symmetric, so only one fourth of the plate wasmodeled.Table 8. Meshes used for Howland plate with hole problemMesh numberNumber of elementsTotal in mesh123420803201280In portion of mesh used tocalculate the energy norm124819276869yThe shaded part indicatesthe area used to calculatethe energy normσxW/2xPt. AW/2 = 1L/2 = 2R = 0.1σx = 1E = 3.0 x 107ν = 0.25RL2Figure 19. Finite element model of plate with a circular holeFigure 20.
Mesh 3 discretizationThe circular hole is approximated by elements with straight edges, so the hole isactually a polygon. As the number of elements is increased, the shape and area of the holechanges slightly.Because the analytical solution only converges in a region around the hole, a subsetof the total number of elements in the mesh was used to calculate the energy norm.
Thisarea, shaded in Fig. 19, was held constant as the mesh was refined, except for the changein the area of the hole.Table 9 shows the calculated stress concentration factor at point A on Fig. 19normalized by the analytical solution. At point A, σ x = 3.0361 according to the analyticalsolution. The stress concentration factor depends on both the constant and non-constantpart of the stress field. None of the elements can represent exactly the nonlinear stressfield in the area near the hole; however, some are better than others.
The ASQBI elementwas shown earlier to represent the pure bending mode of deformation better than the ASOIelements. This ability seems to help also in the calculation of the stress concentration factorat point A. For the ASMD and ADS elements (e1 = 1/2), the non-constant part of the strainis only half the magnitude that of the ASOI element (e1 = 1), so the stress concentrationfactor is lower.70Table 9.
σ xFEM /σ xAnalytical at point A in Fig. 19Mesh1234QUAD40.8880.9730.9941.000ASMD0.7210.8380.9000.946ASQBI0.8850.9610.9880.997Pian-S0.7780.9140.9710.993ASOI0.7720.8740.9260.963ADS0.7330.8310.9020.947Table 10 shows the normalized x-component of stress at the center of the element thatis nearest to the point of maximum stress (point A on Fig. 19). This value is independentof the nonconstant part of the stress field, so there is much less variation between theelements. The coordinates of the element center change as the mesh is refined, so theanalytical stress used to normalize the solutions is included in Table 10.Table 10. σ xFEM /σ xAnalytical at the center of the element nearest point A in Fig.
19Mesh1234Analyticalstress1.6712.0892.4622.717QUAD4 ASMDASQBIPian-SASOI1.0001.0101.0051.0021.0091.0131.0061.0031.0561.0381.0161.007.982.995.997.9991.0311.0291.0151.007ADS1.0401.0311.0121.008Fig. 21 shows the convergence of the error in the energy norm. All elements werefound to have convergence rates ranging from 0.92 to 0.98. Theoretically, theconvergence rate of the energy norm should go to 1 as the element size H→0. Note thatthe differences in the errors for the various elements are much smaller than in the beamproblem.
This is expected, since the nonconstant mode of deformation in this problem ismuch less significant than it is in bending.71-5.0-5.2Log(energy norm)-5.4QUAD4ASMDASQBIASOIADSPian-Sumihara-5.6-5.8-6.0-6.2-2.0-1.8-1.6 -1.4-1.2-1.0Log (H)Figure 21. Convergence of the error in the energy norm1.6.3 Dynamic Cantilever The rate form of stabilization was implemented in the twodimensional version of WHAMS (Belytschko and Mullen (1978)). An end loadedcantilever was modeled with both elastic and elastic-plastic materials as shown in Fig. 22.A similar problem is reported in Liu et al.
(1988). Two plane-strain isotropic materialswere used with ν=0.25, E=1x104, and the material density, ρ=1.(1) elastic(2) elastic-plastic with 1 plastic segment (σ y = 300; Et=0.01E)where σ y is the yield stress, Et is the plastic hardening modulus; a Mises yield surface andisotropic hardening were used.yxhyLDL = 25D=4hy = 15 1-y2 4 applied as astep function at time T=0.d = 1 (out of plane thickness)Figure 22. Dynamic cantilever beamTen meshes were considered. Six of them are composed of rectangular elements,while the other four are skewed. A coarse mesh called the 1x6 mesh has one elementthrough the beam depth and 6 along the length. The aspect ratio of these elements is nearly1.
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