Belytschko T. - Introduction (779635), страница 99
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Meshes of 2x12, 4x24, and 8x48 elements are generated from the 1x6 mesh by72subsequent divisions of each element into 4 smaller elements. Two meshes of elongatedelements, 2x6(E) and 4x12(E) were made of elements with aspect ratios of slightly morethan 2. Finally four meshes are made up of skewed elements. Two of them, 2x12(S) and4x24(S), are formed by skewing 2x12 and 4x24; the other two, 2x6(ES) and 4x12(ES),are formed by skewing 2x6(E) and 4x12(E). Figures (23a-g) show 7 of the meshes.yPoint AxFigure 23a. 1x6 meshyxFigure 23b. 4x24 meshyxFigure 23c. 4x12(E) meshFigure 23d.
2x12(S) mesh73Figure 23e. 4x24(S) meshFigure 23f. 2x6(ES) meshFigure 23g. 4x12(ES) meshThe problem involves very large displacement (of order one third the length of thebeam). No analytical solutions is available, so the results are not normalized; however, amore refined meshes of 32x192 elements were run using a 1-point element with ADSstabilization in an attempt to find a converged solution. The end displacements at point Ain Fig. 23(a) are listed in Tables 11a through 11d. Fig.
24 is a typical deformed meshwhich shows the large strain and rotation that occurs. Figs. (25a-e) are time plots of the ycomponent of the displacement at the end of the cantilever. The first three demonstrate theconvergence of the elastic-plastic solution with mesh refinement for ASQBI and ADSstabilization, and for the ASQBI(2pt) element.
These plots also include the elastic solutionand the 32x192 element elastic-plastic solution using ADS stabilization for comparison.The last two time plots each show a solution of a single mesh by ADS and ASQBIstabilization, and the ASQBI (2pt) and ASQBI (2x2) elements. These plots also includethe elastic and 32x192 element solution for comparison.Table 12 lists the percentage of the strain energy that is associated with the hourglassmode of deformation at the time of maximum end displacement for some of the runs withelastic-plastic material.
As expected, nearly all the strain energy is in the hourglass modefor the coarse (1x6) mesh. As the mesh is refined, the percentage of strain energy in thehourglass mode decreases rapidly, so the importance of accurately calculating the hourglassstrains also decreases.74Figure 24. Deformed 4x24 mesh showing maximum end displacement (elastic-plasticmaterial)With all of the elements, the onset of plastic deformation is significantly retardedwhen the mesh is too coarse.
This is most evident in the QBI elements which are flexuralsuperconvergent for elastic material. The ADS or FB (0.1) elastic solutions are tooflexible, which tends to mask the error caused by too few integration points. The only sureway to reduce the error in solutions that involve elastic-plastic bending is to increase thenumber of integration points. This can be accomplished by mesh refinement or by usingmultiple integration points in each element, as with the 2 point and 2x2 integration. If themesh is refined, not only are the number of integration points increased, but the amount ofstrain energy that is in the hourglass mode of deformation decreases (Table 12), so theaccuracy of the coarse mesh solution becomes less relevant.
When multiple integrationpoints are used, the energy in the nonconstant modes of deformation remains significant,so an accurate strain field such as ASQBI is more important.With two and four stress evaluations per element respectively, ASQBI(2 pt) andASQBI(2x2) give similar results to ADS stabilization when the mesh is refined to 8x48elements. These elements are also have flexural-superconvergence with elastic material.The improvement over a 1-point element with ASQBI stabilization is similar to theimprovement obtained by one level of mesh refinement, and it is significantly lesscomputationally expensive.
Each level of mesh refinement slows the run by a factor of 8,while additional integration points slow it by less than 2 for ASQBI (2 pt) and 4 for ASQBI(2x2). For this problem with a fairly simple constitutive relationship, the additional c.p.utime needed for an a second stress evaluation is largely offset by the elimination of the needfor stabilization, so ASQBI(2 pt) solutions are less than 10% slower than the stabilized 1point element.7510Displacement832x192 mesh (ADS)8x48 mesh4x24 mesh2x12 meshElastic solution6420024681012TimeFigure 25a. End displacement of elastic-plastic cantilever; ASQBI stabilization10Displacement832x192 mesh (ADS)8x48 mesh4x24 mesh2x12 meshElastic solution6420024681012TimeFigure 25b.
End displacement of elastic-plastic cantilever; ASQBI (2 pt) element7610Displacement832x192 mesh8x48 mesh4x24 mesh2x12 meshElastic solution6420024681012TimeFigure 25c. End displacement for elastic-plastic cantilever; ADS stabilization7710Displacement832x192 mesh (ADS)ADSASQBIASQBI (2x2)ASQBI (2 pt)Elastic solution6420024681012TimeFigure 25d. End displacement for 4x12(E) mesh (elastic-plastic)10Displacement832x192 mesh (ADS)ADSASQBIASQBI (2x2)ASQBI (2 pt)Elastic solution64200Figure25e.24End6Timedisplacement810for4x2412elementmesh(elastic-plastic)78Table 11a. Maximum end displacement of elastic cantileverElementQUAD4 (2x2)FB (0.1)FB (0.3)OIASOIQBIASQBIASQBI (2x2)ASQBI (2 pt)ADSASMDASSRI1x64.6915.97.684.784.786.896.896.896.8914.28.496.052x126.148.127.046.176.176.866.866.866.857.957.206.634x246.687.176.936.706.706.886.886.886.887.136.976.828x486.846.976.916.856.856.906.906.906.906.966.926.882x6(E)4.927.225.356.116.116.796.796.796.787.875.595.234x12(E)6.256.976.4326.666.666.866.866.866.857.116.516.38Table 11b.
Maximum end displacement of elastic cantilever for the meshes of skewedelements; solutions are normalized by the solutions from Table 11a for the correspondingmeshes of rectangular elementsElementQUAD4 (2X2)FB (0.1)FB (0.3)OIASOIQBIASQBIASQBI (2x2)ASQBI (2 pt)ADSASMDASSRI2x12(S)0.991.010.990.991.000.990.990.990.991.001.000.994x24(S)0.991.001.000.991.000.990.990.990.991.001.001.002x6(ES)0.970.990.990.970.980.970.970.970.960.990.990.994x12(ES)0.980.990.990.980.990.980.980.980.980.990.990.9979Table 11c.
Maximum end displacement and residual displacement (in parentheses) ofelastic-plastic cantilever; a solution by ADS stabilization with a 32x192 element mesh givesa maximum displacement of 8.17, and a residual displacement of 5.24.ElementQUAD4 (2x2)FB (0.1)FB (0.3)OIASOIQBIASQBIASQBI (2x2)ASQBI (2 pt)ADSASMDASSRI1x64.69(0.11)15.9(0.00)7.68(0.12)4.78(0.05)4.78(0.05)6.89(0.11)6.89(0.11)6.98(1.79)7.00(1.75)14.2(0.00)8.49(0.13)6.05(0.09)2x126.30(1.79)8.39(3.40)7.05(1.15)6.17(0.20)6.17(0.20)6.86(0.89)6.86(0.87)7.52(3.62)7.53(3.54)8.15(3.03)7.21(1.38)6.63(0.60)4x247.31(3.69)8.18(4.88)7.59(3.74)7.17(3.13)7.17(3.16)7.53(3.69)7.54(3.72)7.86(4.53)7.87(4.57)8.12(4.77)7.73(4.05)7.42(3.54)8x487.85(4.65)8.14(5.04)7.92(4.67)7.76(4.41)7.76(4.40)7.90(4.64)7.90(4.64)8.05(4.99)8.06(5.01)8.12(5.01)7.97(4.77)7.86(4.57)2x6(E)4.94(0.78)7.22(1.05)5.35(0.13)6.11(0.16)6.11(0.16)6.79(0.34)6.79(0.34)7.27(3.10)7.28(3.14)7.94(1.89)5.59(0.14)5.23(0.12)4x12(E)6.61(2.76)7.67(3.82)6.69(2.41)7.00(2.63)7.00(2.63)7.34(3.16)7.34(3.16)7.68(4.17)7.69(4.21)7.94(4.19)6.83(2.58)6.60(2.21)80Table 11d.
Maximum end displacement and residual end displacement (in parentheses) ofelastic-plastic cantilever for the meshes of skewed elements; solutions are normalized by thesolutions from Table 11c for the corresponding meshes of rectangular elementsElementQUAD4 (2x2)FB (0.1)FB (0.3)OIASOIQBIASQBIASQBI (2x2)ASQBI (2 pt)ADSASMDASSRI2x12(S)1.08(0.62)1.04(1.18)1.00(1.23)0.99(2.40)1.00(2.45)0.99(1.21)0.99(1.28)0.98(0.96)0.98(1.03)1.03(1.17)1.00(1.30)0.99(1.62)4x24(S)0.98(0.96)0.99(0.99)0.99(0.99)0.98(0.98)0.99(0.98)0.98(0.99)0.98(0.97)0.98(0.97)0.98(0.97)0.99(0.99)0.98(0.98)0.98(0.97)2x6(ES)0.98(1.21)1.02(1.78)0.99(2.28)0.97(3.61)0.98(3.66)0.98(3.07)0.98(3.12)0.98(1.03)0.96(0.96)1.03(1.48)0.99(3.15)0.99(1.53)4x12(ES)0.98(1.02)0.99(1.05)0.99(1.04)0.98(0.97)0.98(0.96)0.98(0.97)0.98(0.98)0.98(0.98)0.98(0.97)0.99(1.02)0.99(1.04)0.99(1.05)Table 12.
Hourglass energy in the mesh when the end displacements maximum(normalized by total strain energy)Mesh1x62x124x248x48FB (0.1)0.9820.1080.0330.011ASOI0.9750.3270.1100.035ASQBI0.9810.2470.0790.026ADS0.9880.1240.0360.012ASMD0.9840.2070.0650.021REMARK 6.1 The QUAD4 element performs no better that the stabilized one-pointelementsREMARK 6.2 The value of α s has a significant effect on the solution of bending problemsusing perturbation stabilization (FB) when the mesh is coarseREMARK 6.3 Those elements that do not project out the nonconstant part of the strainfield, (QUAD4, ASMD, and ASSRI) stiffen significantly more than the others when theelements are elongated as with 2x6(E) and 4x12(E) solutions. Perturbation stabilization(FB) is also sensitive since it is not responsive to the element aspect ratio.81REMARK 6.4 Skewing the elements seems to have little effect on any of the elements.This may be a little deceptive since this is a large deformation problem.
The elements of allthe meshes skew noticeably when deformed (Fig. 24) so the initially skewed meshes onlyintroduce additional skewing. The elastic-plastic 2x6(ES) results are of dubioussignificance, since the elastic-plastic 2x6(E) solutions are quite inaccurate.REMARK 6.5 Another set of runs was made using an elastic-plastic material with a largerplastic modulus (Et=0.1E). The results were similar to those for (Et=0.01E) and are notshown.1.6.4 Cylindrical Stress Wave.
A two dimensional domain with a circular hole at its centerwas modeled with 4876 quadrilateral elements as shown in Figs 26 and 27. A compressiveload with the time history shown in Fig. 28 was applied to the hole and the dynamicevolution was obtained until t=0.09. The domain is large enough to prevent the wave fromreflecting from the outer boundary. Elastic and elastic plastic materials were used.To provide an estimate of the error in the 2D results, solutions were obtained for thesame domain and load history using 3600 axisymmetric, 1D elements. The radial strain ε rrfor the elastic and elastic-plastic solutions at t=0.09 is shown in Fig.
29. The normalizedL2 norms of the error in displacements at time t=0.09 along the radial lines at θ=0 andθ=π/4 are given in Tables 13a and 13b. All of the elements have the same magnitude oferror.Elastic material:Young's modulus, E=1x106Density, ρ=1.0rθr°=10100Figure 26. 4 node quad. mesh dimensions100Elastic-plastic material:Young's modulus, E=1x106Density, ρ=1.0Yield stress, σy =1x104Plastic modulus, E t = E 1682Figure 27. Discretization of infinite domain with a hole15Pressure load atr=10 in.
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