Modeling Guide

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Modeling Guide

Introduction


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Modeling GuideOverview of Model Generation

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Overview of Model Generation

This document gives a general overview of the steps used to create a finite element analysis model. Fordetailed information on commands, refer to the SimXpert Quick Reference. You may also refer to theExample Problems or Workspace Guides for sample problems with step by step instructions.


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Model Generation


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Modeling GuideGeometry Overview

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Geometry Overview

Geometry for FEA is created for the purpose of describing the physical shape of the model to be analyzed.It forms the base upon which the finite element mesh will be created.

Geometry can be created in SimXpert, imported from CAD, or directly accessed from CAD files with alive link to the CAD code.

Units

SimXpert interprets all dimensions and input data with respect to a system of units. It is important to setthe appropriate units prior to importing any unitless analysis files (such as a Nastran Bulk Data file) orcreating materials, properties, or loads. You can control the system of units by selecting Units Managerfrom the Options Editor in the Tools menu. If you import a file that contains units, SimXpert willconvert them into those specified in the Units Manager.

Create

Point

In addition to Geometric points, vertices on curves, surfaces, and solids are also considered points. Youcan also create Mesh Control Points which mark locations where nodes will be placed when meshing isperformed.

Points can be suppressed so that a node will not be forced to be placed at that location when meshing.

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5Model GenerationElements and Meshing

Elements and Meshing

This section describes the different element types supported in SimXpert, as well as the procedures usedto create these elements.

Check your Workspace Guide to determine which elements are supported for your analysis type.

Element Types

There are four basic element groups in SimXpert:

Scalar (0-D)

Line (1-D)

Surface (2-D)

Solid (3-D) elements.

There are also a number of specialty elements. A list of the elements currently supported by SimXpert,including a brief description of each, follows.

Scalar (0D) Elements

Concentrated Mass - Three dimensional mass and/or inertia element located at a node. Used torepresent parts of a structure which contain mass but do not add stiffness.

Spring - Used to provide a specified stiffness between a nodal degree of freedom and ground.

Damper - Used to provide viscous damping between a nodal degree of freedom and ground.

Line (1D) Elements

Bar - Uniaxial element with tension, compression, torsion and bending. It is used to modelgeneral beam/frame structures for linear analysis.

Beam - Uniaxial element with tension, compression, torsion and bending. It can be tapered andhave different properties along its axis. The shear center can be offset from the neutral axis.Used to model both linear and nonlinear beam/frame structures and beams with open sections.

Curved Beam or Pipe - Curved beam, pipe, or elbow element with an arc for the neutral axis.Useful for modeling piping systems.

Bushing - Vibration control device that has impedance values (stiffness and damping) that arefrequency dependent and they recommended as a replacement for spring elements.

Damper - Used to provide viscous damping between two selected degrees of freedom orbetween one degree of freedom and ground.

Spring - Used to provide a specified stiffness between two selected degrees of freedom. Otherproperties can include a damping coefficient and a stress coefficient to be used in stressrecovery. It is recommended to create this element between two coincident nodes.

Gap - Nonlinear element with different tension, compression and shear stiffness. It is used torepresent surfaces or points which can separate, close or slide relative to each other.


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Modeling GuideElements and Meshing

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Scalar Mass - Useful for the selective representation of inertia properties, such as occurs when aconcentrated mass is effectively isolated for motion in one direction only.

Rod - Uniaxial element with tension, compression and torsional stiffness. No bending or shear.Generally, used to model trusses.

Tube - Rod element with tubular cross section which is often used to model pipes.

Plot Only - Non-structural element used to represent structural features that are not beinganalyzed but aid in the visualization of the model.

Surface (2D) Elements

Plate/Shell - Two dimensional plate or shell element used to represent any three dimensionalstructure. Resists membrane, shear and bending forces. Used to model structures comprised ofthin plate shells.

Tria - Three or six noded element (linear or quadratic element, respectively).

Quad - Four or eight noded element (linear or quadratic element, respectively).

Cohesive - Four or eight noded element used to simulate the process of delamination (linearor quadratic element, respectively).

Axisymmetric - (Three or six) or (four or eight) noded element. The element can be linear orfully nonlinear. Also, the element can be cohesive, for which it is either four or eight noded(linear or quadratic element, respectively).

Shear Panel - Resists only shear forces. Used to model structures which contain very thin elasticsheets, typically supported by stiffeners. This is a four noded element.

Acoustic Absorber - Defines a frequency dependent acoustic absorber element. Used in coupledfluid structural analysis.

Solid (3D) Elements

Solid - Three dimensional solid element used to represent any three dimensional structure.

Specialty Elements

Rigid - Include both rigid and interpolation elements. General use rigid elements consist ofRBAR, RROD, RBE2, RBE1, and RSSCON. Of these elements, the RBAR and RBE2 are the

most commonly used. Interpolation elements consist of the RBE3 and RSPLINE. The RBE3 is alinear interpolation element often used to distribute either loading or mass. It is can also be usedto interpolate the motion of a set of independent nodes and assign the motion from theinterpolation to a reference node (dependent node). The RSPLINE is often used to model meshtransitions.

Weld - General purpose connector element which can connect non congruent meshes. Used inlinear analysis.

Contact - Used to identify bodies which are potential candidates for contact during analysis.Bodies can be rigid or deformable. Self contact is also permitted.


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elements without the drawback of the potential internal constraint. Furthermore, since a spring elementneeds to be defined for each connected degree of freedom, several spring elements can be reduced to a

single bushing element.

Mesh Size Control

Mesh sizes can be set interactively using Seed in the Automesh group under the Meshing tab. It ispossible to specify a default mesh size for surfaces, or the size of an element at a vertex. For curves thereare several options for creating mesh seeds. The options are as follows:

Uniform: Causes elements to be distributed evenly along a curve. One Way Bias: Specifies the mesh sizing for a given curve with an increasing or decreasing

element edge length.

Two Way Bias: Specifies the mesh sizing for a given curve with symmetric non-uniformelement edge lengths.

Curvature Based: Specifies the mesh sizing for a given curve with the element edge lengthcontrolled by curvature of the curve.

Tabular: For specifying either parametric locations (place seeds at parametric locations of thecurve), arc length ratio locations (place seeds at arc length ratio locations of the curve arc), ornode/point locations (place seeds at the location of selected nodes or points) for mesh seeds.

A mesh seed consists of two adjacent control points. To create linear finite elements while meshing , amesh seed defines a single element edge. The two control points cause the creation of two finite elementnodes at opposite ends of the element edge.

When meshing to create parabolic finite elements (elements with mid-side nodes), two mesh seeds areused to define one element edge. An example of this is a geometric solid that is seeded with Number ofElements = 10 and then is meshed with parabolic tetrahedral elements, five elements are created alongthe seeded edge of the solid.


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In addition, it is possible to define mesh control points on curves or surfaces to ensure that a node is

placed at that location. This is done using the FEM Tool Control > Points/Curves in the Misc groupunder the Meshing tab.

Mesh should have high density in areas of large stress gradients.

Nodes along common edges of adjoining geometry entities may need to match. If these nodes are notcoincident, equivalencing nodes will not cause all of the adjacent elements to become connected. If theadjacent elements are not connected, the model will have free edges or free faces at these locations. Fora linear analysis, always merge coincident nodes before analyzing the model using Equivalence in the

Modify group under the Nodes/Elements tab.

Element Creation

In SimXpert you can use a number of methods to create elements:

Automesh

Mesh Geometry: Used to create a set of elements on curves, surfaces, or solids. Curves: Ageometric curve is defined parametrically (one parameter). The meshing is done in

parametric space using 1D elements.

Mesh seed oncurve, Numberof elements =10


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Surfaces: A geometric surface is defined either parametrically or non-parametrically. Themesher either follows the parametric directions (surface defined parametrically), or initiallyfollows the perimeter of the surface, subsequently meshing inward following the inner mostperimeter defined by the created element edges (surface defined non-parametrically).

Solids: Controls automatic meshing of solid geometry or volumes (in space). The meshing canbe tetrahedral, or by sweeping, it can be hexahedral. Tetrahedral meshing is for a geometricsolid or a set of contiguous tria elements that completely enclose a volume. The meshing isdone, for a solid, by surface meshing its faces with tria elements, then creating one layer oftetrahedral elements that are congruent to the tria elements. Then, another layer of tetrahedralelements is created that is congruent to the first layer of tetrahedral elements. This is continued

until the solid is filled with tetrahedral elements. The meshing for a set of contiguous triaelements is similar to that for a geometric solid, but the mesher begins with the set of triaelements.

Hexahedral meshing is for a geometric solid, where the meshing is done by sweeping a set of2D quadrilateral elements from one solid face (starting face) to the opposing face (endingface). This means that the solid must be sweepable, or 2 1/2/ D. The starting and ending face(opposing faces), are connected with a set of four edged faces (linking faces). Each of the fouredged faces must have two of its opposing edges congruent with an edge of the starting orending face. The mesher creates on one of the opposing faces a 2D quadrilateral elementmesh, then sweeps it to the other opposing face, creating a hexahedral element mesh betweenthe opposing faces.

Non-geometry meshing commands.

Skin - Creates shell elements on the free faces of 3D solid elements.

2-3-4 line mesh - Produces mesh between curves.

3-4 Point mesh - Produces mesh between virtual curves created by selecting end points. On Mesh - Used to modify an existing mesh. Useful for fixing or cleaning-up a distorted

area of a mesh .

Create elements one at a time. Useful for simple models, line elements, and to fix areas of distortedelements.

Copy existing elements using the Transform option of the Tools menu.

Project elements onto plane, mesh, curve, surface. Reflect (mirror) elements through a plane.

Reorient - Move elements with respect to coordinate systems.

Scale - Create a scaled copy of the elements around a given location.

Rotate - Move elements a specified angle.

Translate - Move elements a specified distance.

Sweep Shells into Solids or Node pairs into Quads using the FEM based group under the Meshing tab.

Revolve - Revolve around a vector.


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Extrude - Drag along a vector.

Glide - Drag along a curve.

Loft - Create linear mesh between destination and source elements.

Normal - Drag elements along normals.

Flange on Mesh - Drag node pairs along a specified angle.

Surface Meshing Guidelines

Examine your model before you begin the meshing process to determine which method of meshgeneration is most applicable. Following are the guidelines for meshing surfaces.

For Surface from Automesh under the Meshing tab, there are several methods of meshing.

Auto Decide: Meshes the surface(s) and uses the meshing algorithm which gives the bestquality elements.

Paver: The mesh will be a paved mesh. The paving algorithm creates elements around anyedges and boundaries of the surface first, then fills with elements in the interior of the

surface, and finally smooths the mesh. Mapped: The mesh will be a mapped (iso) mesh. Elements are created in rows and

columns using intersecting iso-lines.

Minimal: Using this will cause the creation of the smallest surface mesh (fewest number of2D elements) possible, considering adjacent meshes and mesh seeds. Only one surface canbe meshed at a time for this option. Currently (2010), minimal meshing only works withCATIA geometry (surfaces and curves).

Most meshes involve creating geometry first. If you have small features in your geometry thatare not critical to the analysis, you can suppress them in SimXpert.

You can use the FEM Based meshing Revolve / Extrude / Glide commands to generate Surfaceelements from node pairs.

Utilize symmetry whenever possible to reduce model size. Model size (and therefore run time)is significantly reduced if the loading/constraints are also symmetrical. If loads/constraints arenot symmetrical, you can use the Tools > Transform > Reflect command to mirror the mesh

through a plane. Remember, you may also want to use Translate from the Element group of the Nodes/Elements

tab or one of the commands under Tools > Transform to replicate elements instead ofperforming more surface or boundary meshing.

Use the Equivalence command in the Modify group of the Nodes/Elements tab to mergecoincident nodes and connect the meshes.

Use the View, Highlight FE Boundary command to verify that you do not have any unwanted

free edges in the model.


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13Model Generation


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For hard curves, they need to divide the surface being meshed into a number of patch shapes. Each patchshape should be a three or four sided region. If seeds or existing nodes are defined for the hard curves,

the numbers of seeds or nodes on the opposite sides of each patch should be equal. A node will be createdat the intersection of two hard curves. There can be problems if the two ends of a hard curve do not touchthe perimeter of the surface -- the ends of a hard curve are outside or inside of the surface perimeter.

Currently (2010), minimal meshing only works with CATIA geometry -- all the surfaces and curvesbeing minimally meshed must be of CATIA type.

A typical scenario is for the aerospace industry, that involves meshing a fuselage or wing model. Here,a single (large) surface represents the entire top skin of the wing. Similarly, a single surface represents

the entire bottom skin of the wing. Between the two skins are individual ribs and spars (represented bycurves) that act as stiffeners.

When meshing the top or bottom surface (skin) of the wing, usually it is desired to have only a singleelement span the surface (skin) between the stiffener locations (at each skin and rib, or skin and sparsections). In such cases, it is important that only one element be generated on the surface (skin) betweenthe stiffeners. The minimal mesh option satisfies this requirement.

Users want to create a minimal mesh on a model with a single tool and the least possible number of

picking and selecting.

Element Shape Quality

Once you have created a surface mesh or solid mesh, check the quality of the elements using the toolsunder the Quality tab. This can be done using the Plots, Checks, orEdit/Fix Element groups (e.g. selectQuality Index from the Plots group under the Quality tab).

Using the methods from the Plots group causes the elements in the model canvas to be colored dependingon the method used and the quality of the elements. The methods that can be used are described asfollows:

Quality Index: Color-codes the model elements based on a weighted composite (weighted sum)of all the selected quality checks results. The quality checks tests that are to be used are specifiedin the form which is accessed by selecting Check Quality from the Checks group under theQuality tab. The name of the form is ShowMeshQuality.

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To use this form specify 1) Application Region (Model, Scene, Select Elements), 2)entity/element types (i.e. Geometry, Rigid Elem, 2D Elem, 3D Elem), and 3) the Values (Forexample, Geomerty Verification Values, Quality Values) that are to be used (that is select All,Min. Edge Length, Min. Surface Area, Aspect Ratio, Jacobian Ratio). The Values that areselected will cause the corresponding test to be performed when performing the Quality Indextest (select Quality Index from the Plots group under the Quality tab). The Quality Index iscalculated as the normalized sum of the product ofthe Quality Results (quality test results forQuality Values) and the Weight factor; Quality Index = normalized ((activated QualityResults) * (Weight factor)).

Aspect Ratio: The aspect ratio is generally the ratio of the length to the width of an element.The ratio is calculated differently depending on the type of element.

Tria:

This is the same as the Patran evaluator. Let V1, V2 and V3 be three vertices of a triangle, andM1, M2 and M3 be the bisectors of three edges on the triangle. Let Edge 1 be the edge fromV1 to V2, and Segm 1 be the line segment from V3 to M1. First, we calculate two intermediateaspect ratios on Edge1.

The first intermediate aspect ratio is the ratio of length of Edge 1, l1, to the height, h1. Theratio is then multiplied by (Sin 600) such that a perfect element in the shape of anequilateral triangle will equal one. The ratio is inverted if it is less than one.

The second intermediate aspect ratio is the ratio of the distance from M3 to the Segm 1, h2, tothe length of Segm 1, l2. The ratio is then multiplied by 4 * . It is inverted if it is lessthan one.

The aspect ratio on Edge 1 is the maximal value of these two intermediate ratios.

3 2

3 2

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This procedure is repeated for the remaining two edges of the triangle, and the largest valueis retained as the aspect ratio for the triangle.

If Normalize is selected on the verification form, then the aspect ratio is inverted such that itbecomes less than or equal to one. This inverted aspect ratio is subtracted from one to yieldthe aspect factor. An equilateral triangle will have an aspect factor of 0.

Quad:

This is the same as the Patran evaluator. The aspect ratio for a quad is derived from one testproposed by Robinson and Haggenmacher. This test is based on projection plane created byfirst bisecting the four element edges, creating a point on the plane at the vector average ofthe corners. The x-axis extends from the point to the bisector on Edge 2. The ratio isdetermined as the ratio of the length from the origin to the bisector of Edge 2 and the lengthfrom the origin to the bisector of Edge 3. If the ratio is less than 1.0, it is inverted.

If Normalize is selected on the verification form, then the aspect ratio is inverted such that itbecomes less than or equal to one. This inverted aspect ratio is subtracted from one to yieldthe normalized aspect ratio. A square element will have a normalized aspect ratio of 0.

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gElements and Meshing

Tet:

This is the same as the Nastran evaluator. The aspect ratio for a tet element is computed by

taking the ratio of the height of a vertex to the square root of the area of the opposing face.This value is then manipulated in one of two ways, depending on whether the Normalizeparameter is selected on the verification form.

If Normalize is NOT selected, the maximum height to area value is multiplied by a factor, which is the ratio of height to edge length for an equilateral tetrahedron. This

result is reported as the Aspect Ratio. An equilateral tet will report a value of 1.Aspect Ratio = .

If Normalize IS selected, the maximum height to area value is inverted and subtracted from 1.Aspect Factor =

Hex:

This is the same as the Nastran evaluator. The aspect ratio is calculated as the ratio of the

distance between opposing faces. This distance is determined by treating each HEX face as ifit were a warped quadrilateral. Each face is processed to produce a projected plane. Thedistances between the centerpoints of all three pairs of opposing faces are compared.Theaspect ratio is determined by taking the maximum distance between any two faces anddividing it by the minimum distance between any two faces.

If Normalize is selected on the verification form, then the aspect ratio is inverted such that itbecomes less than or equal to one. This inverted aspect ratio is subtracted from one to yieldthe aspect factor. A cubic element will have an aspect factor of 0.

0.805927=

ax Cf hi Ai , i = 1, 2, 3,

1 1 MaxCf hi Ai , i = 1, 2, 3,

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Wedge:

This is the same as the Nastran evaluator. Natran averages the two triangular faces of thewedge element to obtain a mid-surface. The aspect ratio of this triangular mid-surface iscomputed . Next the height (h1) of the wedge is compared to the maximum edgelength of the mid-surface (h4).

If the height of the wedge is greater than the maximum edge length then the aspect ratio forthe wedge element equals the mid-surface aspect ratio multiplied by the maximum edgelength divided by the distance between the triangular faces (h3).

If the height of the wedge is less than the maximum edge length then the aspect ratio for thewedge element equals either the mid-surface aspect ratio or the maximum edge length dividedby the distance between the triangular faces, whichever is greater.

If Normalize is selected on the verification form, then the aspect ratio is inverted such that it

becomes less than or equal to one. This inverted aspect ratio is subtracted from one to yieldthe aspect factor. An equilateral wedge element will have an aspect factor of 0.

3h2 2h1


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Jacobian Ratio: This is approximately the ratio of the maximum value of | J | to the minimumvalue of | J | , where | J | is the determinant of the Jacobian matrix, [ J(r,s,t) ] , of an element.

Tria, Quad, Tet, Hex, Wedge -

The Jacobian Ratio for all element topologies is computed as follows:

Jmax and Jmin are respectively the maximum and minimum values of the determinant of theJacobian matrix, [ J ] , computed at the integration points. The Jacobian ratio is morecomplicated than indicated by the above equation for the Jacobian Ratio. Hence, theapproximately = symbol is used. The Jacobian matrix, [ J ] , for a 2D element is the 2x2 matrixin the following equation relating the partial derivatives of the shape function, N, with respect

to the element parametric coordinates andto the derivatives of the shape function withrespect to the element cartesian coordinates x and y.

A similar relationship exists for a 3D element.

For Jacobian Minimum:Tria:

See note for Tet element.

JacobianRatio Jmax Jmin

N N

x y x y

N xN y

=



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Quad:


Tet:

Relies upon same general Jacobian evaluator used in Patran and defined in the followingbook:

O.C. Zienkiewicz, "The Finite Element Method" McGraw-Hill, 1985. 3rd edition. Chapter8 : Curved, Isoparametric Elements and Numerical Integration. Section 8.5 : "Evaluation ofElement Matrices (Transformation in Xi, Eta, Zeta Coordinates). Pages 189-192.

Minimum value of the Jacobian is for both the Gauss-Legendre quadrature integration pointsand nodal positions.

Wedge:


Hex:


Skew - this is a measure of how much an element or element face is skewed. A skewed element

or element face looks like it has been sheared.

Tria -

This is the same as the Sofy evaluator. It is calculated as the minimum of the angles formedby each of the vertices with the mid-points of its opposite edge with that edge.

Quad -

The skew for a quadrilateral element is computed as follows:

Skew = 90 - If < 30o a skew message is issued.

is an acute angle formed by two intersecting lines, with each line connecting the mid-pointof the opposite edges. is the same as obtained by the Nastran evaluator.

Tet -

There is no corresponding evaluator in SimXpert.

Hex -



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Wedge:


Taper: This is a measure of how much an element or element face is tapered.Tria:


Quad: The taper ratio for a quadralateral element is computed as follows:

This is the same as the Nastran evaluator. The taper ratio test compares the area of the trianglesformed at the four corner nodes (a triangle is formed from a corner node and the two nodesattached to it via the two element edges connected to the corner node) to the total area of theelement. The maximum of the four values is used as the taper ratio.

The first two triangles, and corresponding areas, are formed from Node 1 and 3. For example,Triangle 1: Node 1, 2, 4; with area A1.

The second two triangles, and corresponding areas, are formed from Node 2 and 4.

Taper (i) = | A i / (Aquad element / 2) - 1| , i = 1,2,3,4.

A i is the i-th area (A1, A2, A3, or A4); and Aquad element is the total area of the quadralateral

element.

The maximum of the four values is used as the taper ratio.

1 2

34

A1

A3

1 2

34

A2

A4



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Tet -


Hex -


Wedge -


Warpage - this is a measure of how much an element or element face is out of plane.

Tria -


Quad -

There is a distinction between: Warp Angle and Warp Factor.

Warp Factor:

This is the same as the Nastran evaluator. This test, for a quadralateral element, evaluates howfar out of plane the four corner node points are by determining the distance of each node point

from a "mean" plane passing near the locations of the four node points.The corner node points are assumed to be, alternately, H units above, then H units below themean plane.


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If the lengths of the diagonals of the element are denoted by D1 and D2, the warpingcoefficient is obtained from the equation WF = H / ((D1+D2) / 2). If this value exceeds thetolerance, an informational message is produced.

Warp Angle:The highest angle between the normals of triangular areas of the Quad as divided by a diagonalof opposing corner nodes. This is the same as the sofy evaluator.

Tet:


Hex:

There is no corresponding evaluator in SimXpert.Wedge:


Length (W) - this color-codes the element edges based on edge length.

Min Length = minimum edge length of any side (neglect mid side node for higher-orderelements).

Max Length = maximum edge length of any side (neglect mid side node for higher-orderelements).

Length (Max) - this color-codes the element edges based on maximum edge length.

Max Length = maximum edge length of any side (neglect mid side node for higher-orderelements).

Length (Min) - this color-codes the element edges based on minimum edge length.

Min Length = minimum edge length of any side (neglect mid side node for higher-order

elements). Quad Min Angle - color-codes based on the minimum internal angle for quadralateral elements.

H

HH

H



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Angles are computed for each corner (neglecting mid side nodes of higher order elements)from the edge vectors meeting at each corner (vertex of face).

Quad Max Angle - color-codes based on the maximum internal angle for quadralateral elements.Angles are computed for each corner (neglecting mid side nodes of higher order elements)from the edge vectors meeting at each corner (vertex of face).

Tri MinAngle - color-codes based on the minimum internal angle for triangular elements.


Tri Max Angle - color-codes based on the maximum internal angle for triangular elements.


Weld Norm - color codes based on the weld connection normality. Weld refers to a 2 nodedelement (1D element) that is connected at each of its ends (nodes) to a shell element. The weldconnection normality is defined as the angle between the axis of the 1D element and the surfaceof the shell elements. If the weld connection normality is less than its ideal value of 90, thevalue of the Plate RZ Stiffness Factor may have to be adjusted. This is generally considered to

be a bad modeling practice.

Chordal Deviation - color-codes based on the maximum distance between the center of theelement edge and a true arc.

The deviation is measured at the center of the element edge

Dependency Loops - displays the number of rigid loops in the model in the message region.

The message region displays the quality check results.

Using the methods from the Checks group causes information about the quality of the model elementsto be displayed in forms and/or the message window. The methods that can be used are described asfollows:

Check Quality - this form (ShowMeshQuality) is used to specify the parameter values thatdefine elements of various quality (i.e. Good, Failing), and display the quality of the elements.The form is displayed using Check Quality from the Check group under the Quality tab. Anapplication region (domain containing elements whos quailty is to be checked/verified) must be

specified. There are three choices for this; they are Model (a model consisting of all theelements), Scene (all the elements in a particular scene), and Select Elements.

After this is done, select the tabs that correspond to the quality tests that needs to be performed(i.e. Geometry, 2D Elem, Elem Length). Shown below is the form for 2D Elem.


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In the first column are the names of the types of tests that can be selected (i.e. Warp Angle Quad,

Warp Factor Quad, Aspect Ratio, Jacobian Ratio). The tests that are to be performed must beselected using the checkbox. The second column (Ideal) has the assigned ideal value for each test(i.e. Warp Angle Quad =0.0, Warp Factor Quad=0.0, Aspect Ratio=1.0, Jacobian Ratio=1.0).


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For example, for the Taper Quad test out of 95 elements 80 were good, and 15 were poor. Also,quality results are displayed in the history window.



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More information can be obtained by right-clicking in a cell for a quality test (e.g. Taper Quad)for Good elements, then selecting, among several options, Highlight.


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The elements in the canvas that are of good taper quad quality are highlighted. It is easy toidentify the 15 elements that are of poor taper quad quality.

Aspect Ratio - calculate the aspect ratio for 2D or 3D elements. This is done using Aspect Ratiofrom the Checks group under the Quality tab. The results are displayed in the history window.



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Tri int angle - calculate the minimum internal angle for triangular elements. This is done usingTri int angle from the Checks group under the Quality tab. The results are displayed in thehistory window.

Quad int angle - calculate the minimum internal angle for quadralateral elements. This is done

using Quad int angle from the Checks group under the Quality tab. The results are displayed inthe history window.

Jacobian - calculate the Jacobian for any type of element. This is done using Jacobian from theChecks group under the Quality tab. The results are displayed in the history window.

Length - calculate the minimum and maximum length for the set of all the elements. This is doneusing Length from the Checks group under the Quality tab. The results are displayed in thehistory window.

It may be necessary to set the values for ideal, good, and failing in the ShowMeshQuality / ElemLength form.

Skew - calculate the skew for the elements. This is done using Skew from the Checks groupunder the Quality tab. The results are displayed in the history window.

Taper - calculate the taper for the elements. This is done using Taper from the Checks groupunder the Quality tab. The results are displayed in the history window.

Warpage - calculate the warp for the elements. This is done using Warpage from the Checks

group under the Quality tab. The results are displayed in the history window.


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Angle Degen - calculates all the angular measures (e.g. Warp Angle Quad) for the elements, thencompare the results with the Angle Degeneracy value set in the ShowMeshQuality form. This isdone using Angle Degen from the Checks group under the Quality tab. The results are displayedin the history window.

Length Degen - calculates all the length measures (e.g. Min. Length) for the elements, thencompare the results with the Length Degeneracy value set in the ShowMeshQuality form. This isdone using Length Degen from the Checks group under the Quality tab. The results are displayedin the history window.

Connectivity - determines if there are badly connected elements. The results are displayed in thehistory window.

Dependency - determines if there are any elements with bad dependency. The results aredisplayed in the history window.

Duplicates - determines if there are duplicate elements. An element is a duplicate element if it

uses all the nodes that another element is using. A message is shown in the history window. Also,the duplicate elements are hightlighted in the model canvas.



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Free - determines if there are free elements. An element is free if it is not connected to otherelements, even if is coincident with another element. A message is shown in the history window.Also, the free elements are hightlighted in the model canvas.

Nastran GeomCheck - using this causes the quality of the elements to be determined using theNastran geometry checker code. The results are displayed in the history window.

Tet10 Linearization - linearization of tet10 elements causes the elements midside nodes to bemoved so the corresponding edges become straight. The elements are still tet10s, but theiredges are straight as for tet4 elements. The tet10 linearization occurs only for the tet10 elementsthat fail the Jacobian tests (Jacobian Minimum and Jacobian Ratio).

Miscellaneous - using this causes the results of various checks to be displayed in a window, andthe SimXpert history window.


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% Tria - using this causes the percent of triangular elements of the total number of elements tobe calculated, and the quailty of the triangular elements to be determined. For this example thereare 16 triangular elements of a total of 103 shell (quad and tria) elements.

Show - display, or hide, elements that have quality within a specified range.

Specify the minimum and maximum quality for the range. Specify if the elements are to bedisplayed or hidden. Click Apply. Show the element labels.


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As can be seen, for this example, the elements with quality from 51 to 100 are displayed, and theelements with quality less than 51 are not displayed.

Geom Assoc - using this causes a check to determine if some elements are not associated togeometry, or some elements nodes are not asscociated to geometry descendants.

Specify how the elements are to be selected under Element Selection Method; 1) Direct - selectby elements, or 2) By Geometry Association - select by geometry, i.e. curve, surface. Select

elements. Click Apply. Observe any message in the history window.

Find Faces - highlights the faces of solid elements that are on the faces of geometric solids.



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Select the faces of geometric solids. Click Apply. Observe the highlighted solid element faces.

Using the methods from the Edit/Fix Elements group causes the quality of the elements to be improved.The methods that can be used are described as follows:

Fix Elements

Normals - this form allows the user to perform various operations involving element normals.

The options are as follows:

Action Type: Display

Show the element normals


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Hide the element normals

Action Type: Edit

Fix normals using reference element

Reference element normal is up, and the other normals are down.

Click Apply.



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Reverse normals of set of elements

Enhance Quality by Smoothing - this form allows the user to increase the quality of the elementsby performing a smoothing operation.


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Fix Angle Degeneracy - this form allows the user to reduce the element angle degeneracy.

Fix Length Degeneracy - this form allows the user to reduce the element length degeneracy.

Manual - this form allows the user to move the position of a node manually.



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1. Under Move Node Along select how you would like to move the node:

Element: move the node within a plane of connected elements.

Node Normal: move the node along a vector that is the average of connected element normals.If desired, enter the maximum allowable distance to move the node in the Max. normal lengthtext box.

Element Surface: move the node within a virtual surface that is fit thought pertinent elements.

2. Click in the Node text box and select the node you would like to drag.

3. Drag to reposition the node.

4. Click the middle mouse button (MMB) to finalize the node position.

Notes and Tips

1. You may only move a node within the outside boundary surrounding all of its adjacent elements.

2. By default, nodes cannot be dragged across model features. Feature lines can be displayed byselecting View > Highlight Boundaries. If you would like to drag nodes across model featuresyou can toggle from Fix Geometry to Change Geometry by using the pull-down menu for theElement Fringes icon on the Element Render toolbar. Feature Angle can be set using UserOptions (Tools > Options > Workspaces > Structures > Entity Options > Element.Structures).

3. The selected node will follow the cursor location until you click the middle mouse button.

4. The Location entry will only change as the position of the node is changed by moving the mousecuursor. It is not possible to manually change the Location entry.

5. This command works well when elements are color-coded based on a selected quality parameter.As you drag the node, the element color will update in real time to reflect the current elementquality. You can select a specific quality parameter or a weighted composite of quality checks tocolor-code your model using a selection (i.e. Quality Index (Plots-Quality Index) , Aspect Ratio,Jacobian) from the Plots group under the Quality tab.

Before moving node.


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Moving the node.


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Laminate Verification - verifies laminate composites.

Weld Length - shows messages about weld length.

Weld Normal - shows messages about weld normal.

All By Element - shows the quality of a single element.


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44

Improve Quality by Parts or Sets of Elements


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Improve Quality by Parts or Sets of Elements

Improves the quality of elements. The element quality is improved by remeshing the domain of the

elements which have poor quality. UseRepair

in theEdit

group under theMeshing

tab. The parametersthat are used are

Type: Part (elements associated to the Part) or Elements (set of elements)

Parts: Select parts that have associated meshes that need to be replaced by remeshing the domainthey are in. The meshes may have some elements of poor quality that may need to be replaced.

Elements: Select the elements that are to be replaced by remeshing the domain they are in.

Remesh Triangles: If checked, the current triangular elements are to be replaced either by quad

elements if possible, or by better quality tria elements.

Remesh Hole Rings: If checked, the elements around holes are replaced by better qualityelements.

Split

Splits selected elements into multiple elements, thus possibly creating a better quality mesh. Use Split in

the Edit group under the Meshing tab. The parameters that are used are:

Split: Elements (splits the selected elements along either direction or both, depending on thePattern selected) or Along Line (splits elements along a line you define).

Option: On Element(splits the elements on the element surface) or On Surface (forces splitelements to adhere to the original geometric surface).

Elements: Select the elements that are to be split.

Mesh Transition - If checked, the adjoining elements are split to keep the mesh congruent. Pattern Type: Select the pattern the elements that are created by splitting are to have. This is only

used for Split: Elements.

Pattern (button) - Click this button to see what the elements will look like after the splitting.

Find and Fix Cracks

Determines if a shell mesh is not continuous for all elements in the current scene and highlights problemelements. Optionally, merges nodes or splits elements to fix cracks. Use Cracks in the Edit group underthe Meshing tab. The parameters that are used are:

Maximum edge distance: Maximum boundary/edge-to-boundary/edge distance. If the (actual)distance is less than this parameter value, the boundary/edge-to-boundary/edge is not consideredto be a crack, otherwise it is a crack.

Maximum edge angle: Two boundary edges must be reasonably parallel to be considered to be a

crack. If the angle between two adjacent edges is greater than the Maximum edge angle value,the space between the edges is not considered to be a crack.


Fix Cracks: If checked merge nodes or split elements to fix cracks; or if unchecked cracks in


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Fix Cracks: If checked, merge nodes or split elements to fix cracks; or if unchecked, cracks inthe model are highlighted, but not fixed.

Auto Geometry Cleanup

The Auto cleanup command in the Misc group under the Meshing tab is used to suppress curves at theinterface between two adjacent surfaces. The curves are suppressed so the surface mesher will notautomatically create nodes along them. The surfaces must be three or four sided. The parameters that areused are:

Surfaces: Select surfaces, some of whose curves are to be suppressed.

Element Size: Specify the size of the finite elements that are to be created during meshing.

Surface cross section length tolerance: For two congruent, adjacent surfaces, each with fouredges, SimXpert will determine the smallest edge length for the four edges emanating from theinterface, L. If L is less than the specified tolerance, the curve along the interface of the twosurfaces will be suppressed.

Small surface area tolerance: For two congruent, adjacent surfaces, each with four edges,SimXpert will determine the area of both surfaces. The smallest area, S, will be determined. If Sis less than the specified tolerance, the curve along the interface of the two surfaces will besuppressed.

Short curve length tolerance: If the length of curves associated to the selected surfaces is lessthan the specified tolerance, they will be suppressed.

Small angle tolerance: For each surface in the list, SimXpert will determine the smallest interiorangle, A. If A is less than the specified tolerance, the shortest edge at the vertex of each anglewill be suppressed.

Break Angle: For two congruent, adjacent surfaces, each with four edges, SimXpert willdetermine the intersection angle, A. If A is less than the specified Break Angle, the curve alongthe interface of the two surfaces will be suppressed.

Lock / Unlock

The Control > Lock / Unlock command in the Misc group under the Meshing tab is used to prevent

geometric edges/curves from being modified. For example, if a curve is locked then selected forsuppression, it cannot be suppressed. Subsequently, if the same curve is unlocked, it can be suppressed.

Collapse / Expand Edges

The Edit Edges command in the Edit group under the Meshing tab is used to

Merge two nodes that define an edge or

Expand (split) along the edge defined by the two nodes. The method to use is Method:Collapse or Expand. The parameters that are used are:


46

Nodes/Edge: Select two nodes to define an edge or select an edge. The nodes can be coincident.


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The nodes do not have to belong to the same finite element edge.

Method:

Collapse: Collapses the edge formed by the two nodes.

Expand: Ssplits the edge formed by the two nodes.

Propagate: If checked, extends the collapse/expansion along all connected elements until the endof the mesh path or a feature boundary is reached.

Force Editing: If checked, if a node that is associated with a hard point (vertex) can be moved,otherwise the node cannot be moved.

Creating a Surface Mesh From an Existing Surface Mesh

The On Mesh command in the Automesh group under the Meshing tab is used to create a surface meshfrom an existing surface mesh. The parameters that are used are:

Elements: select the set of surface elements that are to be re-meshed.

Element Size: edge length of the new elements.

Mesh Type: Mixed (Quad and Tria), Tria only.

Mesh Method: Paving, Mapped.

Seed Type:

Uniform: The mesher will create new boundary nodes based on input global edge length.

Existing Boundary: All boundary edges on input mesh will be preserved.

Defined Boundary: The mesher will use all the nodes selected in the Feature Selection sub-

form to define the boundary of the output mesh. No other boundary nodes will be created. Delete Input Mesh: Option to delete original elements.

Element property: Option to specify the property to be associated to the new elements.

Add to part: Option to associate the new mesh to a desired part.

Curvature check: Option to adjust the mesh density to control the deviation between the inputmesh and the straight element edges on the output mesh.

Feature recognition: Option to define features on the input mesh automatically based on featureedge angles and vertex angles, and preserve them on the output mesh.

Feature selection: Allows for manual selection of entities to be preserved in the new mesh.

Washers around holes: Used to create layers of elements that are well shaped around holes.

Remesh Elements Retaining Features

The Remesh command in the Edit group under the Meshing tab is used to create a new mesh on anexisting shell mesh without removing features, hard points, or connected rigid elements. The parametersthat are used are:


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The form for modifying the element properties of the selected elements will appear.

Selecting a multiple number of elements to modify their properties will display only common properties,and the modified properties will be applied to all the selected elements.

Instead of selecting Edit > Properties from the top menu, you can right-click the selected elements, thenselect Properties form the drop-down menu.

Modeling GuideConnections

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Connections


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Connections can be created in the Connection group under the LBCs tab.

A connection in SimXpert defines the location of the connection between parts. Before creating aconnection, a Connection Type must first be selected. Creating a Connection alone does not create anyelements. Once a Connection has been created, it must be Rendered to produce a specific connectiontype such as a RBE2 or CWELD.

Connection Groups

Connections are stored in Connection Groups. A connection group references up to four parts that willbe connected when the Connections in it are rendered. A Maximum Length tolerance defines themaximum distance between parts for which to generate connections.

The diagram above is an analogy between Parts and Connection Groups. Parts are similar to ConnectionGroups. Parts can contain geometry entities. Connection Groups contain Connections. Geometry issimilar to Connections in that neither are exported to a solver file. Geometry and Connections can be re-used across different disciplines (e.g. Structures and Crash). Also, as compared in the boxed regions ofthe diagram, meshing is similar to rendering. Meshing has several algorithms - some for surfaces andsome for solids, while rendering has several methods for each connection type. Meshing and Renderingare both procedures that create actual elements that can be exported to a solver file.

Along with the various Connection Types, there are also two Connection Group Options to choose fromwhen creating connections.

Elements Elements

RenderMeshing

Parts ConnectionGroup

GeometryConnections


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53Model GenerationConnections

Poissons Ratio - Material property for the adhesive region


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The Mesh Width and Length are assumed from the element selection. The Mesh Width Offset is zero.

Trim Mass

Trim Mass represents a lumped mass connected to various nodes.

To create a trim mass connection select a location for the mass element followed by the locations toconnect the mass to the structure.

For Trim Mass, there are various rendering options such as:

Mass - The lumped mass Search Tolerance - Distance to search for nodes from the selected locations

Stiffness Type - Select from RBE2 (rigid) or RBE3 (interpolation) element to connect the massto the structure

Modeling GuideCoordinate Systems

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Coordinate Systems


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Before describing the geometry of a structure, a coordinate system must exist. SimXpert has a built-inrectangular Cartesian system called the basic coordinate system. It is the default coordinate system andhas an ID of zero. The orientation of the basic coordinate system is indicated by the triad in the lower leftcorner of the graphics window. The triad shows orientation only, it is not located at the origin of the basiccoordinate system.

Sometimes it is convenient to use local coordinate systems for specifying loads, and or boundaryconditions. For example, a certain node may have a roller support placed in an inclined plane. A localcoordinate system with one of its axes normal to the inclined plane needs to be created and used to specifythe fixity (SPC) of the displacement component along the direction normal to the inclined plane.

Basiccoordinatesystemorientation

CONSTRAINTCONSTRAINT


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Modeling GuideCoordinate Systems

56

In a cartesian coordinate system, the principal axes 1, 2 and 3 correspond to the X, Y and Z axes,respectively. Points in space are entered in the order: x-coordinate, y-coordinate and z-coordinate. The

i i l f t i di t t d i t P( ) h i Fi 0 2


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principal axes of a cartesian coordinate system and a point, P(x, y,z) are shown in Figure0-2.

Figure 0-2 Cartesian Coordinate System

The graphical representation for creating a coordinate system using 3 Points method is shown inFigure0-3.

Figure 0-3 Coordinate system using 3 Points method

In Figure 0-4, the normal method to create a cartesian coordinate system is graphically represented. Theorigin of the coordinate system is at a node on a surface. The z-axis direction is normal to the surface byusing right-hand rule and crossing the surfaces U direction with the V direction.The x-axis is parallel tothe U direction.

x

y

z

Axis 3

Axis 2

Axis 1

Y

Z

X

P = (x, y, z)

Second, a point location on the specifiedaxis

First, a point location at the OriginThird, a point location in the speci-fied plane


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Modeling GuideLoads and Boundary Conditions

58

Loads and Boundary Conditions

Loads and boundary conditions describe the physical environment of the model to be analyzed LBCs


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Loads and boundary conditions describe the physical environment of the model to be analyzed. LBCscan be created under the LBCs tab.


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61Model GenerationMaterial

Table 0-1 Common Material Characteristics (continued)

Material Characteristics Examples Models


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Constitutive Models

A single material may contain multiple constitutive models. Each constitutive model characterizesdistinct ranges of the materials response. The constitutive models in MSC Nastran Implicit

Nonlinear contain a range of linear and nonlinear material models that can address or approximate thematerial response of most commonly encountered materials. The constitutive models in MSC NastranImplicit Nonlinear can be accessed by any of the solid or structural elements. The models are assessedindependently at each constitutive calculation point (the numerical integration points in the elements).

Thus, the constitutive models are concerned only with a single calculation point. The element thenprovides an estimate of the kinematic solution to the problem at the point under consideration.

Hyperelastic

(MATHE)

Stress function of instantaneousstrain. Nonlinear load-displacement relation. Unloadingpath same as loading.

Rubber Mooney

Ogden

Arruda-Boyce

Gent

Hypoelastic Rate form of stress-strain law Concrete Buyukozturk

Viscoelastic

(MATVE)

Time dependence of stresses inelastic material under loads. Fullrecovery after unloading.

Rubber,

Glass, industrial

plastics

Simo Model

Narayanaswamy

Viscoplastic

(MATVP)

Combined plasticity and creepphenomenon

Metals

Powder

Power law

Shima Model

Modeling GuideMaterial

62

MSC Nastran Implicit Nonlinear Material Entries

The following material bulk data entries are available in SOL 600. Each of these options are summarizedin the sections of this chapter and detailed in the MSC NASTRAN QRG. All standard MSC Nastran
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in the sections of this chapter and detailed in the MSC NASTRAN QRG. All standard MSC Nastran

materials are also available in SOL 600.

The following sections describe how to model material behavior in MSC Nastran Implicit Nonlinear.Modeling material behavior consists of both specifying the constitutive models used to describe thematerial behavior and defining the actual material data necessary to represent the material. Directionaldependency can be included for materials other than isotropic materials. Data for the materials can beentered into MSC Nastran Implicit Nonlinear either directly through the input file or by user subroutines,or material models may be defined in the SimXpert Materials Application. Each section of this chapterdiscusses various options for organizing material data for input. Each section also discusses the

constitutive (stress-strain) relation and graphic representation of the models and includesrecommendations and cautions concerning the use of the models.

Bulk Data Entry Description

MATEP Specifies elasto-plastic material properties.

--MATTEP Specifies temperature-dependent elasto-plastic material properties.

MATF Specifies failure model properties for linear elastic materials.

MATG Specifies gasket material properties to be used in MSC Nastran Implicit

Nonlinear (SOL 600) only.--MATTG Specifies gasket material property temperature variation to be used in MSCNastran Implicit Nonlinear (SOL 600) only.

MATHE Specifies hyperelastic (rubber-like) material properties for nonlinear (large strainand large rotation) analysis in MSC Nastran Implicit Nonlinear (SOL 600) only.

--MATTHE Specifies temperature-dependent properties of hyperelastic (rubber-like)materials (elastomers) in MSC Nastran Implicit Nonlinear (SOL 600) only.

MATED Specifies damage model properties for hyperelastic materials in MSC Nastran

Implicit Nonlinear (SOL 600) only.

MATORT Specifies elastic orthotropic material properties for 3-dimensional and planestrain behavior for linear and nonlinear analyses in MSC Nastran ImplicitNonlinear (SOL 600) only.

--MATTORT Specifies temperature-dependent properties of elastic orthotropic materials forlinear and nonlinear analyses used in MSC Nastran Implicit Nonlinear(SOL 600) only.

MATVE Specifies isotropic visco-elastic material properties in MSC Nastran ImplicitNonlinear (SOL 600) only.

--MATTVE Specifies temperature-dependent visco-elastic material properties in terms ofThermo-Rheologically Simple behavior in MSC Nastran Implicit Nonlinear(SOL 600) only.

MATVP Specifies viscoplastic or creep material properties to be used for quasi-staticanalysis in MSC Nastran Implicit Nonlinear (SOL 600) only.


Linear Elastic

MSC Nastran Implicit Nonlinear is capable of handling problems with any combination of isotropic,
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orthotropic, or anisotropic linear elastic material behavior.The linear elastic model is the model most commonly used to represent engineering materials. Thismodel, which has a linear relationship between stresses and strains, is represented by Hookes Law.Figure0-6 shows that stress is proportional to strain in a uniaxial tension test. The ratio of stress to strainis the familiar definition of modulus of elasticity (Youngs modulus) of the material.

E(modulus of elasticity) = (axial stress)/(axial strain) (0-1)

Figure 0-6 Uniaxial Stress-Strain Relation of Linear Elastic Material

Experiments show that axial elongation is always accompanied by lateral contraction of the bar. The ratio

for a linear elastic material is:

= (lateral contraction)/(axial elongation) (0-2)

This is known as Poissons ratio. Similarly, the shear modulus (modulus of rigidity) is defined as:

(shear modulus) = (shear stress)/(shear strain) (0-3)

A Poissons ratio of 0.5, which would be appropriate for an incompressible material, can be used for the

following elements: Herrmann, plane stress, shell, truss, or beam. A Poissons ratio which is close (butnot equal) to 0.5 can be used for constant dilation elements and reduced integration elements in situationswhich do not include other severe kinematic constraints. Using a Poissons ratio close to 0.5 for all otherelements usually leads to behavior that is too stiff. A Poissons ratio of 0.5 can also be used with theupdated Lagrangian formulation in the multiplicative decomposition framework using the standarddisplacement elements. In these elements, the treatment for incompressibility is transparent.

Stress

Strain

E

1

v

G


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68

(0-7)

where is a function of the mechanical strain and is a function of the temperature.

ij Lijkl

klgij+=

L g


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The stress and strains are true stresses and logarithmic strains, respectively, when used in conjunctionwith the updated Lagrange and large displacement options.

When used in conjunction with the large displacement option only, Equation (0-7)is expressed as

(0-8)

where are the Green-Lagrangian strain and second Piola-Kirchhoff stress, respectively.

This model can be used with any stress element, including Herrmann formulation elements.

The tensors and may be defined by user subroutine HYPELA. In order to provide an accurate

solution, should be a tangent stiffness evaluated at the beginning of the iteration. In addition, the total

stress should be defined as its exact value at the end of the increment. This allows the residual loadcorrection to work effectively.

In user subroutine HYPELA2, besides the functionality of HYPELA, additional information is available

regarding the kinematics of deformation. In particular, the deformation gradient ( ), rotation tensor

( ), and the eigenvalues ( ) and eigenvectors ( ) to form the stretch tensor ( ) are also provided.

This information is available only for the continuum elements namely: plane strain, generalized planestrain, plane stress, axisymmetric, axisymmetric with twist, and three-dimensional cases.

Hyperelastic - Isotrop icHyperelastic models are specified using either the MATHP or MATHE bulk data entries and are used todescribe the behavior of materials that exhibit elastic response up to large strains, such as rubber, solidpropellant, and other elastomeric materials. These materials are described in terms of a strain energypotential, U, which defines the strain energy stored in the material per unit of volume in the initialconfiguration as a function of the strain at that point in the material.

Elastomeric materials are elastic in the classical sense. Upon unloading, the stress-strain curve is retraced

and there is no permanent deformation. Elastomeric materials are initially isotropic. Figure0-7 shows atypical stress-strain curve for an elastomeric material.

S

ij LijklE

klgij+=

E S

L g

L

F

R N U

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Figure 0-7 A Typical Stress-Strain Curve for an Elastomeric Material

Calculations of stresses in an elastomeric material requires an existence of a strain energy function whichis usually defined in terms of invariants or stretch ratios. Significance and calculation of these kinematicquantities is discussed next.

Characteristics of Elastomeric Materials

Most solid rubberlike materials are nearly incompressible. Their bulk modulus is several orders ofmagnitude larger than their shear modulus. For applications where the material is not highly confined,the assumption that the material is fully incompressible is usually a good approximation. In cases wherethe material is highly confined (such as in an O-ring), modeling the compressibility can be important forobtaining accurate results. In either case, the use of hybrid (mixed formulation) elements is

recommended for this type of material in all but plane stress cases.Elastomeric foams on the other hand are elastic but very compressible.

Elastomeric materials are considered to be isotropic in nature with random orientation of the longchain molecules.

Strain Energy Potential and Representative Models

Calculations of stresses in an elastomeric material requires an existence of a strain energy function which

is usually defined in terms of invariants or stretch ratios.

In the rectangular block in Figure0-8, , , and are the principal stretch ratios along the edges

of the block defined by

(0-9)

,

Stress

, Strain

100%

1 2 3

i Li ui+ Li=


70


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Figure 0-8 Rectangular Rubber Block

In practice, the material behavior is (approximately) incompressible, leading to the constraint equation

the strain invariants are defined as

(0-10)

Depending on the choice of configurations, for example, reference (at ) or current ( ),

you obtain total or updated Lagrange formulations for elasticity. The kinematic measures for the twoformulations are discussed next.

Total Lagrangian Formulation

The strain measure is the Green-Lagrange strain defined as:

(0-11)

where is the right Cauchy-Green deformation tensor defined as:

(0-12)

in which is the deformation gradient (a two-point tensor) written as:

L3

2L2

L2

L1

1L1Undeformed

Deformed

3L3

123 1=

I1 12

22 3

2+ +=

I2 12

2

2 2

23

2 3

21

2+ +=

I3 12

2

23

2=

t 0= t n 1+=

Eij 12--- Cij ij =

Cij

Cij FkiFkj=

Fkj


(0-13)

h bi i d fi d

FkjxkXj--------=

J


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The Jacobian is defined as:

(0-14)

Thus, the invariants can be written as:

(0-15)

in which is the permutation tensor. Also, using spectral decomposition theorem,

(0-16)

in which the stretches are the eigenvalues of the right Cauchy-Green deformation tensor, and

the eigenvectors are .

Updated Lagrange Formulation

The strain measure is the true or logarithmic measure defined as:

(0-17)

where the left Cauchy-Green or finger tensor is defined as:

(0-18)

Thus, using the spectral decomposition theorem, the true strains are written as:

(0-19)

J

J 123 det Cij

12---

= =

I1 Cii=

I2CijCij Cii

2

2---------------------------------------=

I316---eij kepqrCipCjqCkr det Cij = =

(implied sum on i)

eijk

Cij A2Ni

ANj

A=

A2

Cij

NiA

ij12---lnbij=

bij

bij FikFjk=

ij1

2--- Aln niA

nj

A

=


72

where is the eigenvectors in the current configuration. It is noted that the true strains can also be

approximated using first Pad approximation, which is a rational expansion of the tensor, as:

(0 20)

niA

2 V V + 1


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(0-20)

where a polar decomposition of the deformation gradient is done into the left stretch tensor and

rotation tensor as:

The Jacobian is defined as:

(0-21)

and the invariants are now defined as:

(0-22)

It is noted that eitherEquation (0-15) orEquation (0-22) gives the same strain energy since it is scalarand invariant. Also, to account for the incompressibility condition, in both formulations, the strain energyis split into deviatoric and volumetric parts as:

(0-23)

Mooney-Rivlin Model

The generalized Mooney-Rivlin model for nearly-incompressible elastomeric materials is written as:

(0-24)

where and are the first and second deviatoric invariants.

ij 2 Vij ij Vij ij+ 1=

Fij Vij

Rij

Fij VikRkj=

J

J 123 det bij

12---

= =

I1 bii=

I212--- bijbij bii

2 =

I316---eijkepqrbipbjqbkr det bij = =and

W Wdeviatoric Wvolumetric+=

Wdeviatoricgmr

Cmn I1 3 m

I2 3 n

n 1=

m 1=

=

I1 I2


Jamus-Green-Simpson Model

A particular form of the generalized Mooney-Rivlin model, namely the third order deformation (tod)model, is implemented in MSC Nastran Implicit Nonlinear (SOL 600). This is one of the few places

where the formulation for SOLs 106 and 129 may be more appropriate because they can use up to fifth


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where the formulation for SOLs 106 and 129 may be more appropriate because they can use up to fifthorder terms. However, the Ogden formulation (below) is usually better for large strain behavior than eventhe fifth order Mooney-Rivlin.

(0-25)

where is the deviatoric third order deformation form strain energy function,

are material constants obtained from experimental data.

Simpler and popular forms of the above strain energy function are obtained as:

(0-26)

Ogden Model

The form of strain energy for the Ogden model in MSC Nastran Implicit Nonlinear is,

(0-27)

where are the deviatoric stretch ratios while , , and are the

material constants obtained from the curve fitting of experimental data.

The Ogden model is usually applied to slightly compressible materials. If no bulk modulus is given, it istaken to be virtually incompressible. This model is different from the Mooney model in several respects.The Mooney material model is with respect to the invariants of the right or left Cauchy-Green straintensor and implicitly assumes that the material is incompressible. The Ogden formulation is with respectto the eigenvalues of the right or left Cauchy-Green strain, and the presence of the bulk modulus impliessome compressibility. Using a two-term series results in identical behavior as the Mooney mode if:

, , , and

Wdevratorictod

C10 I1 3 C01 I2 3 C11 I1 3 I2 3 C20 I1 3 2

C30 I1 3 3

+ + +

+

=

Wdeviatorictod

C10 C01 C11 C20 C30

Wdeviatoricmr

C10 I1 3 C01 I2 3 += Mooney-RivlinWdeviatoric

nh

C10 I1 3

= Neo-Hookean

Wdeviatoricogden

kk------ 1

k 2k 3

k3+ +

k 1=

=

ik

Jk3-----

ik

= Cmn k k

1 2

C10

= 1 2

= 2 2

C01

= 2 2

=


74

Ar ruda-Boyce Model

In the Arruda-Boyce strain energy model, the underlying molecular structure of elastomer is representedby an eight chain model to simulate the non-Gaussian behavior of individual chains in the network. The

two parameters and ( is the chain density is the Boltzmann constant is thenk N n k


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two parameters, and ( is the chain density, is the Boltzmann constant, is thetemperature, and is the number of statistical links of length l in the chain between chemical crosslinks)

representing initial modules and limiting chain extensibility and are related to the molecular chainorientation thus representing the physics of network deformation.

As evident in most models describing rubber deformation, the strain energy function constructed byfitting experiment data obtained from one state of deformation to another fails to accurately describe thatdeformation mode. The Arruda-Boyce model ameliorates this defect and is unique since the standard

tensile test data provides sufficient accuracy for multiple modes of deformation.

Figure 0-9 Eight Chain Network in Stretched Configuration

The model is constructed using the eight chain network as follows:

Consider a cube of dimension with an unstretched network including eight chains of length

, where the fully extended chain has an approximate length ofNl. A chain vector from the

center of the cube to a corner can be expressed as:

(0-28)

Using geometrical considerations, the chain vector length can be written as:

(0-29)

and

nk N n k N

20

30

10

C1

k

j

i

0

r0 Nl=

C102

------ 1 i02

------ 2 j02

------ 3 k+ +=

rchain1

3

------- Nl 12 2

2 32+ +

1 2=


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76

The constant is independent of molecular length and, hence, of degree of crosslinking. The model

is attractive due to its simplicity, but yet captures the main behavior of a network of extensible moleculesover the entire range of possible strains.

The volumetric part of the strain energy is for all the rubber models in MSC Nastran Implicit

EIm


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p gy pNonlinear is:

(0-37)

when is the bulk modulus. It can be noted that the particular form of volumetric strain energy is chosensuch that:

1. The constraint condition is satisfied for incompressible deformations only; for example:

(0-38)

2. The constraint condition does not contribute to the dilatational stiffness.

This yields the constraint function as:

(0-39)

upon substitution ofEquation (0-39) in Equation (0-35) and taking the first variation of the variationalprinciple, you obtain the pressure variable as:

(0-40)

The equation has a physical significance in that for small deformations, the pressure is linearly related to

the volumetric strains by the bulk modulus .

The discontinuous or continuous damage models discussed in the models section on damage can beincluded with the generalized Mooney-Rivlin, Ogden, Arruda-Boyce, and Gent models to simulateMullins effect or fatigue of elastomers when using the updated Lagrangian approach. In the totalLagrangian framework however, this is available for the Ogden model only.

Wvolumetric9K2

------- J

13---

1

2

=

K

I3

> 0 if I3 0

0= if I3 1=

< 0 if I3 0

I3 3 I3

1

6---

1 =

p 3K J

13---

1

=

K


Foam Model

Sometimes elastomeric materials show large volumetric deformations. For this type of behavior, themodels discussed above are not appropriate. Instead, the foam model expressed by:


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(0-41)

should be used. In contrast to the Ogden model, the first part of the foam strain energy function is not

purely deviatoric. The material constants provide additional flexibility to describe the material

behavior also for a large amount of compressibility.

Updated Lagrange Formulation for Nonlinear Elasticity

The Mooney-Rivlin, Ogden, Arruda-Boyce, Gent and Foam models may be used either in the totalLagrange or updated Lagrange framework. This is selected using the PARAM,MARUPDAT. For planestress analysis the total Lagrange procedure will always be used.

The updated Lagrangian rubber elasticity capability can be used in conjunction with both continuous aswell as discontinuous damage models. Thermal, as well as viscoelastic, effects can be modeled with thecurrent formulation. While the Mooney model can account for the temperature dependent material

properties, the Ogden model does not support the temperature dependence at this time. The singularityratio of the system is inversely proportional to the order of bulk modulus of the material due to thecondensation procedure.

A consistent linearization has been carried out to obtain the tangent modulus. The singularity for the caseof two- or three-equal stretch ratios is analytically removed by application of LHospitals rule. Thecurrent framework with an exact implementation of the finite strain kinematics along with the split of

strain energy to handle compressible and nearly incompressible response is eminently suitable forimplementation of any nonlinear elastic as well as inelastic material models. In fact, the finite

deformation plasticity model based on the multiplicative decomposition, is

implemented in the same framework.

To simulate elastomeric materials, incompressible element(s) are used for plane strain, axisymmetric,and three-dimensional problems for elasticity in total Lagrangian framework. These elements can be

used with each other or in combination with other elements. For plane stress, beam, plate or shellanalysis, conventional elements can be used. For updated Lagrangian elasticity, both conventionalelements (as well as Herrmann elements) can be used for plane strain, axisymmetric, and three-dimensional problems.

Experimental Determination of Hyperelastic Material Parameters

In order to determine the material parameters to be used, like Mooney coefficients, Ogden moduli,relaxation times, etc., experiments must be carried out. In this section, the laboratory tests of which data

can be used to fit the material parameters will be described. Once the test data is available the

Wnn------ 1

n 2n 3

n3+ +

nn----- 1 J

n

n 1=

+n 1=

=

n

F FeF

F

p=


78

Experimental Data Fitting module in MSC SimXpert can be used to calculate appropriate coefficientvalues.

For a homogeneous material, homogeneous deformation modes suffice to characterize the material

constants. MSC Nastran Implicit Nonlinear accepts test data from the following deformation modes:


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Uniaxial tension and compression. Biaxial tension and compression. Planar tension and compression (also known as pure shear). Simple Shear Volumetric tension and compression


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80

Probably the most popular test is the uniaxial test (see Figure0-11). This test can be used in tension aswell as in compression, both for incompressible and (slightly) compressible elastomeric materials. Theshape of the specimen used in compression will usually be less slender than the shape used in tension.Within the region indicated by the dashed line, the state of deformation will be homogeneous, where the

deformation can be described by:


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, (0-42)

while the corresponding engineering stresses are given by:

, (0-43)

in which is the applied force and is the cross sectional area of the undeformed specimen in the

- -plane, within the region indicated by the dashed line.

Figure 0-11 Uniaxial (Tensile) Test

Necessary input for the curve fitting program in MSC SimXpert consists of at least engineering strain

( ) versus engineering stress ( ) data points. In case of (slightly) compressible materials,

information about the volume changes is also needed. This data can be given either in terms of the area

ratio or the volume ratio. The area ratio is defined by the current cross sectional area over the original

cross sectional area . Similarly, the volume ratio is defined by the current volume over the

undeformed volume . Notice that the volume ratio and the area ratio are related by:

1 1 e11+= = 2 3 J = =

11 F

A0

------= = 22 33 0= =

F A0

E2 E3

E1E3

E2

F F

e11 11

A

A0 V

V0


If, for a particular elastomeric material, both a tensile and a compression test have been performed, allthe data points should be collected into one data file. The layout of a data file containing uniaxial test

V

V0------ J

A

A0------ 1 e11+ = =


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p y gdata is given in the figure below. The columns may be separated by either spaces or commas. For (nearly)incompressible material behavior, the third column can be omitted.

Figure 0-12 Layout of Data File for a Uniaxial Test

Equi-Biaxial Test

The equi-biaxial tensile test outlined in Figure 0-13 can be used to obtain, within the region indicated bythe dashed line, a homogeneous state of deformation defined by:

Figure 0-13 Equi-biaxial (Tensile) Test

, (0-44)

with corresponding engineering stresses:

, (0-45)

with being the original cross sectional area of the elastomeric sheet in the direction perpendicular

to the applied forces, which is assumed to be the same in the - -plane and the - -plane.

For compressible elastomers, volumetric information is needed. For the equi-biaxial test, this can begiven in terms of a thickness ratio or, similar to the uniaxial test, a volume ratio. The thickness ratio is

defined as the current sheet thickness over the original sheet thickness . The relation between the

thickness ratio and the volume ratio is:

e11 11 A A0 e11 11 V V0

or

1 2 1 e11+ 1 e22+= = = = 3 J 2=

11 22 F

A0------= = = 33 0=

A0

E1 E3 E2 E3

t t0


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, (0-48)

in which is the cross sectional area of the undeformed specimen in the - -plane. Notice that

the engineering strain is zero, but that the corresponding engineering stress depends on the

material behavior.

(0-49)

(0-50)

Simple Shear Test

A test which, compared to the above mentioned tests, leads to a more complex kinematic description, is

the simple shear test (see Figure0-15).Upon introducing the shear strain , the coordinates in the

deformed configuration are given by:

, , (0-51)

which yields for the deformation gradient:

FF

E1E3

E2

11 F

A0------= = 33 0=

A0 E1 E3

e22 22

U TSS=

TS S

U 2 S S3

I1U

I2U

+

= =

1 X1 X2+= x2 X2= x3 X3=

Modeling GuideMaterial84

(0-52)F

1 00 1 0

0 0 1

=


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Figure 0-15 Simple Shear Test

Notice that , irrespective of the value of , from which it can be concluded that a simple

shear test is a constant volume test.

Based on Equation (0-51), Equation (0-52) andFigure 0-15, the engineering strain tensor and the rightCauchy-Green strain tensor can be evaluated as:

(0-53)

(0-54)

According to Equation (0-54), the principal stretch ratios follow from the princip

Modeling Guide

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