Element Selection CriteriaAppendix 1

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

Elements in ABAQUS

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

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.

special-purpose elements like springs,

dashpots, and masses

continuum (solid elements) shell elementsbeam elements

rigid elements membrane elements

infinite elementstruss elements

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

First-order interpolation

Second-order interpolation

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

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

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.

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.

First-order interpolation

Full integration

Second-orderinterpolation

Reduced integration

Elements in ABAQUS•Element naming conventions: examples 单元命名约定

B21: Beam, 2-D, 1st-order interpolation

CAX8R: Continuum, AXisymmetric, 8-node, Reduced integration

DC3D4: Diffusion (heat transfer), Continuum, 3-D, 4-node

S8RT: Shell, 8-node, Reduced integration, Temperature

DC1D2E: Diffusion (heat transfer), Continuum, 1-D, 2-node, Electrical

CPE8PH: Continuum, Plane strain, 8-node, Pore pressure, Hybrid

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 typmany analysis typeses 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.

Structural Elements (Shells and Beams) vs. Continuum Elements

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

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

3-D continuum surface model

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

line model

framed structure modeled using beam elements

3-D continuum

Modeling Bending Using Continuum Elements

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

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.

Lines that are initially vertical do not change length (implies yy=0).

Because the element edges can assume a curved shape, the angle between the deformed isoparametric lines remains equal to 90o (implies xy=0).

isoparametric lines

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”).

Because the element edges must remain straight, the angle between the deformed isoparametric lines is not equal to 90o

(implies ). 0xy

Integration point

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.

Modeling Bending Using Continuum Elements

Four elements through the thickness

No hourglassing

• Hourglassing is not a problem if you use multiple elements—at least four through the thickness.– Each element captures either co

mpressive or tensile axial strains, but not both.

– The axial strains are measured correctly.

– The thickness and shear strains are zero.

– Cheap and effective elements.

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.

Same load and displacement magnification (1000×)

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.

Internal energy

Artificial energy Artificial energy

Internal 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•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.

Modeling Bending Using Continuum Elements•Example: Cantilever beam with distorted elements

Parallel distortion Trapezoidal distortion

Modeling Bending Using Continuum Elements

Element type xx yy xy Notes

Physical behavior 0 0 0

Second-order 0 0 0 OK

First-order, full integration

0 0 0 Shear locking

First-order, reduced integration

0 0 0 Hourglassing if too few elements through thickness

0 0 0 OK if enough elements through the thickness

Incompatible mode 0 0 0 OK if not overly distorted

Summary

Stress Concentrations

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.

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

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.

ideal okay bad

distortedundistorted

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

P

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.

Coarse mesh Fine meshCPS3 55.06 76.87CPS4 71.98 91.2CPS4I 63.45 84.37CPS4R 43.67 60.6CPS6 96.12 101.4CPS8 91.2 100.12CPS8R 92.56 97.16

yyat D (Target=100.0)Element type

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.

Contact

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.