Element Selection Criteria Appendix 1

1. Element Selection Criteria Appendix 1

2. 内容提要 • Elements in ABAQUS • Structural Elements (Shells and Beams) vs. Continuum Elements • Modeling Bending Using Continuum Elements 用实体单元模拟弯曲 • Stress Concentrations 应力集中 • Contact 接触 • Incompressible Materials 不可压缩材料 • Mesh Generation 网格生成 • Solid Element Selection Summary

3. Elements in ABAQUS

4. Elements in ABAQUS • ABAQUS单元库中提供广泛的单元类型，适应不同的结构和几何特征The wide range of elements in the ABAQUS element library provides flexibility in modeling different geometries and structures. • Each element can be characterized by considering the following:单元特性： • Family 单元类型 • Number of nodes 节点数 • Degrees of freedom 自由度数 • Formulation 公式 • Integration 积分

5. continuum (solid elements) shell elements beam elements rigid elements membrane elements infinite elements special-purpose elements like springs, dashpots, and masses truss elements • Elements in ABAQUS • 单元类型（Family） • A family of finite elements is the broadest category used to classify elements. • 同类型单元有很多相同的基本特。Elements in the same family share many basic features. • 同种类单元又有很多变化：There are many variations within a family.

6. First-order interpolation Second-order interpolation • Elements in ABAQUS • Number of nodes节点数(interpolation) • An element’s number of nodes determines how the nodal degrees of freedom will be interpolated over the domain of the element. • ABAQUS includes elements with both first- and second-order interpolation. 插值函数阶数可以为一次或者两次

7. Elements in ABAQUS • 自由度数目Degrees of freedom • The primary variables that exist at the nodes of an element are the degrees of freedom in the finite element analysis. • Examples of degrees of freedom are: • Displacements 位移 • Rotations 转角 • Temperature 温度 • Electrical potential 电势

8. Elements in ABAQUS • 公式Formulation • The mathematical formulation used to describe the behavior of an element is another broad category that is used to classify elements. • Examples of different element formulations: • Plane strain 平面应变 • Plane stress 平面应力 • Hybrid elements 杂交单元 • Incompatible-mode elements 非协调元 • Small-strain shells 小应变壳元 • Finite-strain shells 有限应变壳元 • Thick shells 后壳 • Thin shells 薄壳

9. Elements in ABAQUS • 积分Integration • 单元的刚度和质量在单元内的采样点进行数值计算，这些采样点叫做“积分点” The stiffness and mass of an element are calculated numerically at sampling points called “integration points” within the element. • 数值积分的算法影响单元的行为The numerical algorithm used to integrate these variables influences how an element behaves. • ABAQUS包括完全积分和减缩积分。ABAQUS includes elements with both “full” and “reduced” integration.

10. Full integration Reduced integration First-order interpolation Second-orderinterpolation • Elements in ABAQUS • Full integration:完全积分 • The minimum integration order required for exact integration of the strain energy for an undistorted element with linear material properties. • Reduced integration:简缩积分 • The integration rule that is one order less than the full integration rule.

11. Elements in ABAQUS • Element naming conventions: examples 单元命名约定 B21: Beam, 2-D, 1st-order interpolation S8RT: Shell, 8-node, Reduced integration, Temperature CAX8R: Continuum, AXisymmetric, 8-node, Reduced integration CPE8PH: Continuum, Plane strain, 8-node, Pore pressure, Hybrid DC3D4: Diffusion (heat transfer), Continuum, 3-D, 4-node DC1D2E: Diffusion (heat transfer), Continuum, 1-D, 2-node, Electrical

12. Elements in ABAQUS • ABAQUS/Standard 和 ABAQUS/Explicit单元库的对比 • Both programs have essentially the same element families: continuum, shell, beam, etc. • ABAQUS/Standard includes elements for many analysis types in addition to stress analysis: 热传导, 固化soils consolidation, 声场acoustics, etc. • Acoustic elements are also available in ABAQUS/Explicit. • ABAQUS/Standard includes many more variations within each element family. • ABAQUS/Explicit 包括的单元绝大多数都为一次单元。 • 例外: 二次▲单元和四面体单元 and 二次 beam elements • Many of the same general element selection guidelines apply to both programs.

13. Structural Elements (Shells and Beams) vs. Continuum Elements

14. Structural Elements (Shells and Beams) vs. Continuum Elements • 实体单元建立有限元模型通常规模较大，尤其对于三维实体单元 • 如果选用适当的结构单元 (shells and beams) 会得到一个更经济的解决方案 • 模拟相同的问题，用结构体单元通常需要的单元数量比实体单元少很多 • 要由结构体单元得到合理的结果需要满足一定要求： the shell thickness or the beam cross-section dimensions should be less than 1/10 of a typical global structural dimension, such as: • The distance between supports or point loads • The distance between gross changes in cross section • The wavelength of the highest vibration mode

15. 3-D continuum surface model • Structural Elements (Shells and Beams) vs. Continuum Elements • Shell elements • Shell elements approximate a three-dimensional continuum with a surface model. • 高效率的模拟面内弯曲Model bending and in-plane deformations efficiently. • If a detailed analysis of a region is needed, a local three-dimensional continuum model can be included using multi-point constraints or submodeling. • 如果需要三维实体单元模拟细节可以使用子模型 Shell model of a hemispherical dome subjected to a projectile impact

16. framed structure modeled using beam elements • Structural Elements (Shells and Beams) vs. Continuum Elements • Beam elements • 用线简化三维实体。Beam elements approximate a three-dimensional continuum with a line model. • 高效率模拟弯曲，扭转，轴向力。 • 提供很多不同的截面形状 • 截面形状可以通过工程常数定义 3-D continuum line model

17. Modeling Bending Using Continuum Elements

18. Modeling Bending Using Continuum Elements • Physical characteristics of pure bending • The assumed behavior of the material that finite elements attempt to model is:纯弯状态： • Plane cross-sections remain plane throughout the deformation. 保持平面 • The axial strain xx varies linearly through the thickness. • The strain in the thickness direction yy is zero if =0. • No membrane shear strain. • Implies that lines parallel to the beam axis lie on a circular arc. xx

19. Lines that are initially vertical do not change length (implies yy=0). isoparametric lines Because the element edges can assume a curved shape, the angle between the deformed isoparametric lines remains equal to 90o (implies xy=0). • Modeling Bending Using Continuum Elements • Modeling bending using second-order solid elements (CPE8, C3D20R, …) 二次单元模拟 • Second-order full- and reduced-integration solid elements model bending accurately: • The axial strain equals the change in length of the initially horizontal lines. • The thickness strain is zero. • The shear strain is zero.

20. Integration point Because the element edges must remain straight, the angle between the deformed isoparametric lines is not equal to 90o (implies ). • Modeling Bending Using Continuum Elements • Modeling bending using first-order fully integrated solid elements (CPS4, CPE4, C3D8) • These elements detect shear strains at the integration points. • Nonphysical; present solely because of the element formulation used. • Overly stiff behavior results from energy going into shearing the element rather than bending it (called “shear locking”).

21. Modeling Bending Using Continuum Elements • Modeling bending using first-order reduced-integration elements (CPE4R, …) • These elements eliminate shear locking. • However, hourglassing is a concern when using these elements. • Only one integration point at the centroid. • A single element through the thickness does not detect strain in bending. • Deformation is a zero-energy mode (有应变形但是没有应变能的现象 called “hourglassing”). Change in length is zero (implies no strain is detected at the integration point). Bending behavior for a single first-order reduced-integration element.

22. Modeling Bending Using Continuum Elements • Hourglassing is not a problem if you use multiple elements—at least four through the thickness. • Each element captures either compressive or tensile axial strains, but not both. • The axial strains are measured correctly. • The thickness and shear strains are zero. • Cheap and effective elements. Four elements through the thickness No hourglassing

23. Same load and displacement magnification (1000×) • Modeling Bending Using Continuum Elements • Detecting and controlling hourglassing • Hourglassing can usually be seen in deformed shape plots. • Example: Coarse and medium meshes of a simply supported beam with a center point load. • ABAQUS has built-in hourglass controls that limit the problems caused by hourglassing. • Verify that the artificial energy used to control hourglassing is small (<1%) relative to the internal energy.

24. Internal energy Internal energy Artificial energy Artificial energy Two elements through the thickness: Ratio of artificial to internal energy is 2%. Four elements through the thickness: Ratio of artificial to internal energy is 0.1%. • Modeling Bending Using Continuum Elements • Use the X–Y plotting capability in ABAQUS/Viewer to compare the energies graphically. • Use the X–Y plotting capability in ABAQUS/Viewer to compare the energies graphically.

25. Modeling Bending Using Continuum Elements • Modeling bending using incompatible mode elements (CPS4I, …) • Perhaps the most cost-effective solid continuum elements for bending-dominated problems. • Compromise in cost between the first- and second-order reduced-integration elements, with many of the advantages of both. • Model shear behavior correctly—no shear strains in pure bending. • Model bending with only one element through the thickness. • No hourglass modes and work well in plasticity and contact problems. • The advantages over reduced-integration first-order elements are reduced if the elements are severely distorted; however, all elements perform less accurately if severely distorted.

26. Modeling Bending Using Continuum Elements • Example: Cantilever beam with distorted elements Parallel distortion Trapezoidal distortion

27. Modeling Bending Using Continuum Elements Summary

28. Stress Concentrations

29. Stress Concentrations • 二次单元处理应力集中问题，明显优于一次单元Second-order elements clearly outperform first-order elements in problems with stress concentrations and are ideally suited for the analysis of (stationary) cracks. • W无论是完全积分还是减缩积分都可以很好的反映应力集中Both fully integrated and reduced-integration elements work well. • 减缩积分效率更高，而且计算结果往往优于完全积分。Reduced-integration elements tend to be somewhat more efficient—results are often as good or better than full integration at lower computational cost.

30. Stress Concentrations • 二次单元可以以更少的单元更好的反应结构的几何特征Second-order elements capture geometric features, such as curved edges, with fewer elements than first-order elements. Physical model Model with first-order elements—element faces are straight line segments Model with second-order elements—element faces are quadratic curves

31. ideal okay bad undistorted distorted • Stress Concentrations • Both first- and second-order quads and bricks become less accurate when their initial shape is distorted. • First-order elements are known to be less sensitive to distortion than second-order elements and, thus, are a better choice in problems where significant mesh distortion is expected. • Second-order triangles and tetrahedra are less sensitive to initial element shape than most other elements; however, well-shaped elements provide better results.

32. Stress Concentrations • A typical stress concentration problem, a NAFEMS benchmark problem, is shown at right. The analysis results obtained with different element types follow. P elliptical shape

33. Element s at D (Target=100.0) yy type Coarse mesh Fine mesh 55.06 76.87 CPS3 71.98 91.2 CPS4 63.45 84.37 CPS4I 43.67 60.6 CPS4R 96.12 101.4 CPS6 91.2 100.12 CPS8 92.56 97.16 CPS8R • Stress Concentrations • First-order elements (including incompatible mode elements) are relatively poor in the study of stress concentration problems. • Second-order elements such as CPS6, CPS8, and CPS8R give much better results.

34. Stress Concentrations • Well-shaped, second-order, reduced-integration quadrilaterals and hexahedra can provide high accuracy in stress concentration regions. • Distorted elements reduce the accuracy in these regions.

35. Contact

36. Contact • Almost all element types are formulated to work well in contact problems, with the following exceptions: • Second-order quad/hex elements • “Regular” second-order tri/tet elements (as opposedto “modified” tri/tet elementswhose names end with the letter “M”) • The directions of the consistent nodal forces resulting from a pressure load are not uniform.