Solid and shell elements

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1 Solid and shell elements Theodore Sussman, Ph.D. ADINA R&D, Inc,

2 Overview 2D and 3D solid elements Types of elements Effects of element distortions Incompatible modes elements u/p elements for incompressible analysis Membrane elements Shell elements Director vectors and rotational degrees of freedom Types of elements Large strain shell elements ADINA R&D, Inc,

3 2D and 3D solid elements 2D elements axisymmetric plane stress plane strain 3D plane stress (membrane, we will discuss these elements later) 3D elements ADINA R&D, Inc,

4 Types of 2D elements: 2D solid elements 2D elements can lie in one of several planes: ADINA R&D, Inc,

5 Some uses of 2D solid elements Axisymmetric seal Thin plate with holes, membrane action, plane stress assumed Long tunnel, plane strain assumed ADINA R&D, Inc,

6 Theory of 2D and 3D solid elements Based on the isoparametric concept. x(,) rs h(,) L L rsx L u(,) rs h(,) L L rsu L h1 1 at node 1 = 0 at other nodes x(,,) rst h(,,) L L rstx L u(,,) rst h(,,) L L rstu L ADINA R&D, Inc,

7 Theory of 2D and 3D solid elements Numerical integration, using Gaussian integration. The strains and stresses are calculated at the integration points. ADINA R&D, Inc,

8 Theory of 2D and 3D solid elements Elements must be compatible. Adjacent element faces must be topologically identical, with the same number of nodes ADINA R&D, Inc,

9 Theory of 2D and 3D solid elements Example solution with incompatible meshing. Beam in pure bending M M ADINA R&D, Inc,

.. Element results are computed at the integration points Element results need to be extrapolated to the element

10 Types of results Nodal results: displacements, velocities, accelerations, temperature,... Nodal results are continuous between adjacent elements Element results: strains, stresses,... Element results are computed at the integration points Element results need to be extrapolated to the element boundaries In general, element results are discontinuous between adjacent elements Nodal result: u node 351 Element result: τ int pt 22 in el 25 ADINA R&D, Inc,

Element results By default, the solver passes

If the command RESULTS-ELEMENT LOCATION=CORNER is used

integration results to the corner nodes, then passes

The post-processor can smooth the integration point or

11 Element results By default, the solver passes integration point results to the post-processor. If the command RESULTS-ELEMENT LOCATION=CORNER is used in the model definition, the solver extrapolates the integration results to the corner nodes, then passes the extrapolated results to the post-processor. The post-processor can smooth the integration point or corner node results, to obtain results at the nodes. ADINA R&D, Inc,

strain Inexpensive, need many elements for a given accuracy

12 Types of 2D elements First-order Second-order Linear displacement, "constant" strain Quadratic displacement, linear strain Inexpensive, need many elements for a given accuracy Relatively expensive, need fewer elements for a given accuracy ADINA R&D, Inc,

for a given accuracy Quadratic displacement, linear strain

13 Types of 3D elements First-order Second-order Linear displacement, "constant" strain Inexpensive, need many elements for a given accuracy Quadratic displacement, linear strain Relatively expensive, need fewer elements for a given accuracy ADINA R&D, Inc,

First and second-order elements First order elements Less computationally expensive for a given mesh size Less memory required for a given mesh

Second order elements More computationally expensive for a given mesh size More memory required for a given mesh size Accurate results in pure

14 First and second-order elements First order elements Less computationally expensive for a given mesh size Less memory required for a given mesh size Too stiff in bending Does not resolve stress gradients well Fine mesh needed for an accurate solution Well suited for contact analysis Second order elements More computationally expensive for a given mesh size More memory required for a given mesh size Accurate results in pure bending Stress gradients are resolved well Coarse mesh can give an accurate solution Not as well suited for contact analysis ADINA R&D, Inc,

15 Cost of elements The cost of an element consists of two parts: Cost of assembly integration points element stiffness matrix element degrees of freedom For the 27-node element, there are 81 element degrees of freedom For the 27-node element, there are 27 integration points element degrees of freedom The cost of assembly is not influenced by the arrangement of elements within the mesh. ADINA R&D, Inc,

degrees of freedom in model, can be very large (>10 ) (Inside the band,

16 Cost of elements Cost of solution of system of equations KU = R Schematic of global stiffness matrix K (simplified): 7 Number of degrees of freedom in model, can be very large (>10 ) (Inside the band, there are many zeros, and these zeros are not stored either.) ADINA R&D, Inc,

17 Cost of elements Assume that the sparse solver is used to solve the following models Memory Nm CPU time Nm N m 2 number of degrees of freedom average number of nonzeros per column N ADINA R&D, Inc,

18 Cost of elements, using the sparse solver 1 million 8-node bricks # equations N Memory CPU time Block 3 M 146 GB 4 hr Plate 6 M 26 GB 11 min Bar 12 M 12 GB 50 sec 1 million 20-node bricks # equations N Memory CPU time Block 12 M 1335 GB 7 days Plate 21 M 155 GB 2 hr Bar 36 M 56 GB 8 min The data in the tables are estimates obtained by scaling the results of smaller problems computed on an Intel Xeon CPU 3.10 GHz (4 cores SMP) ADINA R&D, Inc,

19 Cost of elements, using different solvers 1 million 20-node bricks, 3D-iterative solver # equations N Memory CPU time Block 12 M 127 GB 1 hr Plate 21 M 48 GB 9 min Bar 36 M 41 GB 6 min 1 million 20-node bricks, sparse solver # equations N Memory CPU time Block 12 M 1335 GB 7 days Plate 21 M 155 GB 2 hr Bar 36 M 56 GB 8 min ADINA R&D, Inc,

In 3D, the situation is even worse, since the contact forces at corner nodes

20 2nd order elements and contact The 2nd order elements don't work as well in contact as do the 1st order elements. In 3D, the situation is even worse, since the contact forces at corner nodes point in the "wrong" direction for the 20-node brick. We will see this later on in the contact lecture. ADINA R&D, Inc,

21 Effect of element distortions In practical analysis the elements are often distorted. All elements include the constant stress state, so if the mesh is fine enough, the solution is accurate. All elements contain the displacement shape ui A Bx Cy Dz where x, y, z are the global coordinates and A, B, C, D are constants. ADINA R&D, Inc,

element does not. Therefore the 9- node element is less sensitive to distortions than the 8-node element.

22 But what about the shape Effect of element distortions u A Bx Cy Dz Ex Fxy... Jz i 2 2 In 2D, the 9-node quad element contains this shape when the element has an angular distortion, but the 8-node quad element does not. Therefore the 9- node element is less sensitive to distortions than the 8-node element. In 3D the 27-node brick element contains this shape, but the 20-node brick element does not; so the 27-node brick element is less sensitive to distortions than the 20-node brick element. ADINA R&D, Inc,

23 Effect of element distortions What happens if the element is overdistorted? The Jacobian determinant (transformation from global volume to isoparametric coordinates volume) becomes negative within the element. ADINA Structures prints a warning about overdistorted elements, however ADINA Structures continues the analysis. The results cannot be trusted within the overdistorted elements. ADINA R&D, Inc,

24 Element locking Elements cannot exactly represent the (displacement) shapes in the exact solution. Example: shear locking in 4-node elements. Nonzero shear strain leads to nonzero shear stresses, element is too stiff in bending. ADINA R&D, Inc,

25 Incompatible modes elements Extra shapes are included in 4-node elements (and also in 8-node 3D elements). Incompatible modes work well when elements are square and in bending: ADINA R&D, Inc,

Very sensitive to element distortions When an assemblage of long slender

26 Incompatible modes elements However, incompatible modes elements are unreliable in general analysis. Very sensitive to element distortions When an assemblage of long slender elements is stressed under geometrically nonlinear conditions, the assemblage exhibits spurious modes: ADINA R&D, Inc,

27 Analysis of a power screw ADINA R&D, Inc,

28 Incompatible modes elements The incompatible modes feature does not apply to 3-node triangles and 4- node tet elements. Due to the unreliability of incompatible modes elements in general nonlinear analysis, the incompatible modes elements are not the default in ADINA Structures. If the structure undergoes significant bending, the structure should be modeled with higher-order elements, e.g. 9-node quad elements in 2D or 27- node brick elements in 3D. ADINA R&D, Inc,

29 Volumetric locking in incompressible materials Incompressible materials are those in which the ratio of bulk to shear modulus is high. Examples are Rubber-like materials, including the three-network model Plastic materials, when undergoing plastic deformations For a linear isotropic material, the bulk to shear modulus ratio becomes high as the Poisson's ratio approaches 0.5. Under plane stress conditions, the element can change thickness to model the incompressibility. Hence there are no difficulties when using plane stress elements in the analysis of incompressible materials. Plane strain, axisymmetric and 3D elements exhibit volumetric locking in the analysis of incompressible materials. ADINA R&D, Inc,

Considering only finite element shapes without volumetric strain, there are too few shapes remaining to give an accurate solution

30 Volumetric locking in incompressible materials In volumetric locking, the exact incompressible solution has no volumetric strain everywhere. Most finite element shapes have volumetric strain. Considering only finite element shapes without volumetric strain, there are too few shapes remaining to give an accurate solution (unless a very fine mesh is taken). The remedy is to separately interpolate the pressure (conjugate to the volumetric strain), using a lower order for the pressure. This is called the u/p formulation. ADINA R&D, Inc,

31 Example showing volumetric locking ADINA R&D, Inc,

32 Example showing volumetric locking ADINA R&D, Inc,

33 u/p formulation By default, the u/p formulation is used whenever any of the following material models are employed: Rubberlike materials Arruda-Boyce Eight-chain Mooney-Rivlin Ogden Sussman-Bathe Three-network model Inelastic materials Anand Creep Creep-irradiation Creep-variable Mroz-bilinear Multilinear-plastic-creep Multilinear-plastic-creep-variable Plastic-bilinear Plastic-creep Plastic-creep-variable Plastic-cyclic Plastic-multilinear Plastic-orthotropic Viscoelastic ADINA R&D, Inc,

34 u/p elements The following tables give the u/p elements employed when the u/p formulation is used. 2D elements 3/1 triangle 4/1 quad * 6/3 triangle 7/3 triangle * 8/3 quad 9/3 quad * * = recommended element 3D elements 4/1 tet 5/1 pyramid 6/1 wedge 8/1 brick * 10/4 tet 11/4 tet * 13/4 pyramid 14/4 pyramid 15/4 wedge 20/4 brick 21/4 brick 21/4 wedge 27/4 brick * ADINA R&D, Inc,

35 Element recommendations for 3D solids If second-order brick elements are to be used, the 27-node element is preferred over the 20-node element. The 27-node element is less sensitive to element distortions. The 27-node element is better suited to the u/p formulation (the 27/4 element does not lock, but the 20/4 element locks). The 27-node element does not have tensile contact forces. But the 27-node element is more expensive, for a given mesh size, than the 20-node element. If second-order tet elements are to be used, usually the 10-node element is preferred over the 11-node element. The 10-node element is much less expensive (5 int pts) than the 11- node element (17 int pts). But in incompressible analysis, the 11 node element is preferred, since the 11/4 element does not lock. ADINA R&D, Inc,

36 Membrane elements The 2D solid element has a membrane option: 3D plane stress. With this option, the element need not initially lie in a plane. The element need not be initially flat. The element can have from 3 to 9 nodes, just like the other 2D solid elements. ADINA R&D, Inc,

37 Assumptions in membrane elements Membrane action in the plane of the element, no out-of-plane bending Plane stress through the thickness of the element Each node has three displacement degrees of freedom. As a consequence, unless the element is carrying tensile stress, the element has no stiffness in the transverse direction. ADINA R&D, Inc,

x y z in transverse direction parallel to N1-N2 line to y, in plane of element The

38 Local coordinate system Stresses and strains in the membrane element are output in the element local coordinate system. x y z in transverse direction parallel to N1-N2 line to y, in plane of element The local system can change during the analysis, and can be different from point to point within the element. ADINA R&D, Inc,

stresses. Plane stress assumption in membrane element is consistent with zero normal traction on free surface.

39 Uses of membrane element Modeling of rubber sheets or balloons, modeling of fabrics. Modeling of coatings. If thin coating is applied to a free surface, then the coating stresses can be used to obtain the surface stresses. Plane stress assumption in membrane element is consistent with zero normal traction on free surface. Integration points in 3D element are in interior, integration points in membrane element are on surface. ADINA R&D, Inc,

membrane action. Can be initially flat or warped.

40 Introduction to shell elements A surface element that has stiffness in bending, transverse shear and membrane action. Can be initially flat or warped. Thickness can be constant or varying. ADINA R&D, Inc,

41 Shell element assumptions The fundamental shell element assumptions are Material particles that originally lie on a straight line parallel to the director vector remain on that straight line during deformations. The stress in the director vector direction is zero. The director vector direction need not be the same as the midsurface normal direction. The following geometric quantities are used: The nodal point locations, which usually (but not always) lie on the shell midsurface The director vectors The shell thicknesses, measured in the director vector directions ADINA R&D, Inc,

42 Shell-shell intersections The element director vectors might or might not be shared between neighboring elements: Shared between elements, shell is smooth. Not shared between elements, neighboring shell elements meet at an angle. Not shared between elements, shell-shell intersection. ADINA R&D, Inc,

43 Director vector options Director vector shared: flat shell, smoothly curved shell Director vector not shared: shell-shell intersection ADINA R&D, Inc,

44 5 and 6 DOF nodes The director vector options are controlled by the choice of number of degrees of freedom for each shell midsurface node: 5 DOF node, three translations, two rotations: director vectors are shared 6 DOF node, three translations, three rotations: director vectors not shared 5 DOF node 6 DOF node ADINA R&D, Inc,

45 5 and 6 DOF nodes On a flat or smoothly curved shell, there is no stiffness associated with rotations about the director vector, so we use only 5 DOF nodes with two rotations. The AUI automatically sets the number of degrees of freedom at a shell node to 5, unless: Shell elements intersect at an angle (> 5 o by default). The shell node is connected to other structural elements that provide rotational stiffness, e.g. iso-beams in the modeling of stiffened shells. The shell node is connected to rigid links. Rotational boundary conditions are assigned to the shell node. Prescribed moments are applied to the shell node. You can explicitly assign the number of degrees of freedom to the shell node, to override the AUI. ADINA R&D, Inc,

46 5 and 6 DOF nodes No zero stiffness rotations Nodes attached to rigid links must have 6 DOFs, zero stiffness rotations ADINA R&D, Inc,

An alternative is to connect the 6 DOF shell nodes to neighboring nodes using soft beam elements (weld elements),

47 Drilling stiffness options MASTER SINGULARITY-STIFFNESS=YES Places a small stiffness onto those rotational DOFs with zero stiffness that are associated with rigid links, beams, pipes, etc. This is the default. An alternative is to connect the 6 DOF shell nodes to neighboring nodes using soft beam elements (weld elements), and to not use the singularity-stiffness option. Singularity-stiffness option used Singularity-stiffness option not used ADINA R&D, Inc,

Need to alleviate shear and membrane locking in these elements.

48 Types of shell elements In addition to the cost vs accuracy tradeoffs of the solid elements, there are additional considerations in the shell elements. Need to alleviate shear and membrane locking in these elements. The MITC approach is used to reduce the shear and membrane locking. Although we list the MITC8 element here, this element is much worse than the MITC9, and we discourage the use of the MITC8. ADINA R&D, Inc,

49 Types of shell elements The MITC3 is also worse than the MITC4, and should be used as little as possible. There is ongoing research to improve the MITC3. Although ADINA includes the higher-order shell elements, most often the MITC4 element is used, and this element is the default. ADINA R&D, Inc,

50 Numerical integration in the shell elements Gauss integration is used in the plane of the shell. Either Gauss, Newton-Cotes or trapezoidal rule integration is used in the out-of-plane direction (default is 2 point Gauss). ADINA R&D, Inc,

point options is to be able to model the spread of

51 Numerical integration in the shell elements One reason for including all of the out-of-plane integration point options is to be able to model the spread of plasticity through the shell thickness. ADINA R&D, Inc,

In large strain analysis, the strain through the thickness affects the geometry.

52 3D-shell element for large strains It is not convenient to use the "zero stress through shell thickness assumption" in a large strain shell element. The material law affects the strain through the thickness. In large strain analysis, the strain through the thickness affects the geometry. It is better for all of the strains to be computed from the geometry (and not from the material law). In pure bending, the neutral surface and midsurface no longer coincide. Pure bending, as seen from the side ADINA R&D, Inc,

53 3D-shell element for large strains In the 3D-shell element, the director vectors are replaced by control vectors. The control vectors can change length independently. ADINA R&D, Inc,

linear thickness strain Constant thickness strain affects both control vectors equally.

54 3D-shell element for large strains At each node, the degrees of freedom are three translations, as usual either two or three rotations, as usual constant thickness strain linear thickness strain Constant thickness strain affects both control vectors equally. Linear thickness strain stretches one control vector and shrinks the other. ADINA R&D, Inc,

55 3D-shell element for large strains Shear locking is alleviated in a manner similar to that used in the MITC4 element, but is generalized for large strains. The fully 3D stress-strain material law is used. For incompressible materials, the u/p formulation is employed to reduce volumetric locking. The 3D-shell element is available only for the 3- and 4-node elements. The 3D-shell element can be used with the linear isotropic elastic, rubber-like materials and the plastic-cyclic material models. Applications of the element include crash/crush of tubes, metal forming, and analysis of thin rubber components in which bending effects are important. ADINA R&D, Inc,

Guidelines for proper use of Plate elements

Guidelines for proper use of Plate elements In structural analysis using finite element method, the analysis model is created by dividing the entire structure into finite elements. This procedure is known