김경종
263 lines
21 KiB
Markdown
263 lines
21 KiB
Markdown
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penalty stiffness, Stiffness scale factor: factor, Initial/Final stiffness ratio: ratio of initial penalty stiffness over final penalty stiffness, Upper quadratic limit scale factor: upper quadratic limit scale factor, Lower quadratic limit ratio: lower quadratic limit ratio, Clearance at which contact pressure is zero: clearance at zero pressure
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Scaling the penalty stiffness on a step-by-step basis
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You can also scale the penalty stiffness on a step-by-step basis, which will act as an additional multiplier on any scale factor specified as part of the surface behavior definition.
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Input File Usage: To scale the penalty stiffness on a step-by-step basis: \*CONTACT CONTROLS, STIFFNESS SCALE FACTOR=factor
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Abaqus/CAE Usage: To scale the penalty stiffness on a step-by-step basis: Interaction module: Abaqus/Standard contact controls editor: Augmented Lagrange: Stiffness scale factor: factor
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Adjusting the penalty stiffness across iterations of the first increment
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It is common to have convergence difficulties in the first increment of an analysis if the contact status changes over a large portion of the contact area upon initial loading. An approach that tends to improve convergence behavior without sacrificing accuracy is to use a reduced penalty stiffness in the early iterations of the first increment and return to the default penalty stiffness for the final iterations of the first increment and all iterations of subsequent increments. Use of a reduced penalty stiffness in early iterations helps to robustly find an approximate contact status distribution, and the goal of later iterations is to then find an accurate solution, which is reported as the converged solution for the first increment.
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Input File Usage: To scale the penalty stiffness within the first increment: \*CONTACT CONTROLS, STIFFNESS SCALE FACTOR=USER ADAPTIVE
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# Limitations of the penalty method
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The penalty method cannot be used for debonded surfaces.
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If the penalty method is specified, Lagrange multipliers are always used during analysis steps with the following procedures:
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• Design sensitivity analysis (see “Design sensitivity analysis,” Section 19.1.1)
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• Direct steady-state dynamic analysis (see “Direct-solution steady-state dynamic analysis,” Section 6.3.4)
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• Quasi-Newton method (see “Convergence criteria for nonlinear problems,” Section 7.2.3)
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If surface elements have been used to define a contact surface on the exterior of a substructure (see “Contact modeling if substructures are present,” Section 36.3.9), Abaqus/Standard interprets the underlying element stiffness to be zero. This can lead to difficulty in determining the default penalty stiffness and may cause numerical problems during the analysis.
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# Augmented Lagrange method
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The linear penalty method can be used within an augmentation iteration scheme that drives down the penetration distance. This so-called augmented Lagrange method applies only to hard pressure-overclosure relationships. The following describes the sequence that occurs in each increment with this approach:
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1. Abaqus/Standard finds a converged solution with the penalty method.
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2. If a slave node penetrates the master surface by more than a specified penetration tolerance, the contact pressure is “augmented” and another series of iterations is executed until convergence is once again achieved.
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3. Abaqus/Standard continues to augment the contact pressure and find the corresponding converged solution until the actual penetration is less than the penetration tolerance.
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The augmented Lagrange method may require additional iterations in some cases; however, this approach can make the resolution of contact conditions easier and avoid problems with overconstraints, while keeping penetrations small. The augmented Lagrange method is used by default for three-dimensional self-contact using node-to-surface discretization.
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The default penetration tolerance is one-tenth of a percent of the characteristic interface length except in the following cases:
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• if you specify a penalty stiffness scaling factor, $f _ { k }$ , of less than 1.0 (using the interface discussed below), Abaqus/Standard will automatically scale the default penetration tolerance by a factor of $\scriptstyle { \frac { 1 } { \sqrt { f _ { k } } } }$ (which will be greater than or equal to 1.0);
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• the default penetration tolerance for finite-sliding, surface-to-surface contact is five percent of the characteristic interface length, subject to the scaling discussed in the previous bullet point.
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The default penalty stiffness for the augmented Lagrange method is 1000 times the representative underlying element stiffness. Lagrange multipliers are used for the augmented Lagrange method if the penalty stiffness exceeds 1000 times the representative underlying element stiffness computed by Abaqus/Standard; otherwise, no Lagrange multipliers are used. Therefore, Lagrange multipliers are not used for the augmented Lagrange method with the default penalty stiffness.
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Input File Usage: Use both of the following options:
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\*SURFACE INTERACTION, NAME=interaction\_property\_name
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\*SURFACE BEHAVIOR, AUGMENTED LAGRANGE
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Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Normal Behavior: Constraint enforcement method: Augmented Lagrange (Standard)
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# Modifying the penetration tolerance for the augmented Lagrange method
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You can modify the penetration tolerance for the augmented Lagrange method on a step-by-step basis by specifying an absolute or relative penetration tolerance. The relative penetration tolerance is specified with respect to a characteristic length computed by Abaqus/Standard. The default penetration tolerance was discussed above. The default penetration tolerance is increased automatically if you set the penalty
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stiffness scale factor to a value less than 1.0 (also discussed above); however, Abaqus/Standard will not adjust any directly specified penetration tolerance. Choosing a very small penetration tolerance may result in an excessive number of augmentation iterations.
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Input File Usage: To specify an absolute penetration tolerance:
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\*CONTACT CONTROLS, ABSOLUTE PENETRATION TOLERANCE=tolerance
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To specify a relative penetration tolerance:
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\*CONTACT CONTROLS, RELATIVE PENETRATION TOLERANCE=tolerance
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Abaqus/CAE Usage: Interaction module: Abaqus/Standard contact controls editor:
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Augmented Lagrange: Penetration tolerance: Absolute: tolerance or Relative: tolerance
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# Modifying the penalty stiffness for the augmented Lagrange method
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As with the penalty method, you can specify the penalty stiffness, shift the pressure-overclosure relationship by specifying the clearance at which the contact pressure is zero, or scale the default or specified penalty stiffness by a factor as part of the surface behavior definition. You can also scale the penalty stiffness on a step-by-step basis, which will act as an additional multiplier on any scale factor specified as part of the surface behavior definition. Choosing a very low penalty stiffness may result in an excessive number of augmentation iterations.
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Input File Usage: To modify the penalty behavior in the surface behavior definition:
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\*SURFACE BEHAVIOR, AUGMENTED LAGRANGE
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penalty stiffness, clearance at zero pressure, factor
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To scale the penalty stiffness on a step-by-step basis:
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\*CONTACT CONTROLS, STIFFNESS SCALE FACTOR=factor
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Abaqus/CAE Usage: To modify the penalty behavior in the surface behavior definition:
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Interaction module: contact property editor: Mechanical→Normal Behavior:
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Constraint enforcement method: Augmented Lagrange (Standard),
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Stiffness value: Specify: penalty stiffness, Stiffness scale factor: factor, Clearance at which contact pressure is zero: clearance at zero pressure
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To scale the penalty stiffness on a step-by-step basis:
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Interaction module: Abaqus/Standard contact controls editor: Augmented
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Lagrange: Stiffness scale factor: factor
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# Modifying the number of allowed augmentations for the augmented Lagrange method
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You can define the number of allowed augmentations for the augmented Lagrange method.
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Input File Usage: \*CONTROLS, PARAMETERS=TIME INCREMENTATION
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<!-- source-page: 784 -->
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Abaqus/CAE Usage: Defining the number of allowed augmentations for the augmented Lagrange method is not supported in Abaqus/CAE.
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# Limitations of the augmented Lagrange method
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The augmented Lagrange method cannot be used for debonded surfaces.
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If the augmented Lagrange method is specified, Lagrange multipliers are always used during analysis steps with the following procedures:
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• Design sensitivity analysis (see “Design sensitivity analysis,” Section 19.1.1)
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• Direct steady-state dynamic analysis (see “Direct-solution steady-state dynamic analysis,” Section 6.3.4)
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• Quasi-Newton method (see “Convergence criteria for nonlinear problems,” Section 7.2.3)
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If surface elements have been used to define a contact surface on the exterior of a substructure (see “Contact modeling if substructures are present,” Section 36.3.9), Abaqus/Standard interprets the underlying element stiffness to be zero. This can lead to difficulty in determining the default penalty stiffness and may cause numerical problems during the analysis.
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# Use of Lagrange multiplier degrees of freedom by the various methods
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Using Lagrange multipliers to enforce contact constraints can add significantly to the solution cost, but they also protect against numerical errors related to ill-conditioning that can occur if a high contact stiffness is in effect. Abaqus/Standard automatically chooses whether the constraint method makes use of Lagrange multipliers, based on a comparison of the contact stiffness to the underlying element stiffness. Table 38.1.2–1 summarizes the use of Lagrange multipliers. Lagrange multipliers are not used for the default contact stiffnesses associated with the penalty and augmented Lagrange approximations of hard contact. Any Lagrange multipliers associated with contact are present only for active contact constraints, so the number of equations may change as the contact status changes.
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Table 38.1.2–1 Use of Lagrange multipliers in constraint enforcement methods.
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<table><tr><td rowspan="2">Constraint Method</td><td colspan="2">Use Lagrange Multipliers</td></tr><tr><td>Yes</td><td>No1</td></tr><tr><td>Direct, hard contact</td><td>Always</td><td>Never</td></tr><tr><td>Direct, exponential softened contact</td><td>If $k > 1000k_e$ </td><td>If $k \leq 1000k_e$ </td></tr><tr><td>Direct, linear softened contact</td><td>If $k > 1000k_e$ </td><td>If $k \leq 1000k_e$ </td></tr><tr><td>Direct, tabular softened contact</td><td>If $k > 1000k_e$ </td><td>If $k \leq 1000k_e$ </td></tr><tr><td>Penalty, hard contact</td><td>If $k_{penalty} > 1000k_e$ </td><td>If $k_{penalty} \leq 1000k_e$ </td></tr></table>
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<table><tr><td rowspan="2">Constraint Method</td><td colspan="2">Use Lagrange Multipliers</td></tr><tr><td>Yes</td><td>No1</td></tr><tr><td>Augmented Lagrange, hard contact</td><td>If $k_{penalty} > 1000k_e$ </td><td>If $k_{penalty} \leq 1000k_e$ </td></tr><tr><td colspan="3"> $k = \text{slope of pressure-overclosure relationship}$ $k_{penalty} = \text{penalty stiffness}$ $k_e = \text{underlying element stiffness}$ $^1$ Lagrange multipliers are always used, regardless of the constraint enforcement method or stiffness, in the following cases: design sensitivity analyses, direct steady-state dynamics analyses, analyses using the quasi-Newton method.</td></tr></table>
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# 38.1.3 SMOOTHING CONTACT SURFACES IN Abaqus/Standard
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Products: Abaqus/Standard Abaqus/CAE
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# References
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• “Defining general contact interactions in Abaqus/Standard,” Section 36.2.1
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• “Defining contact pairs in Abaqus/Standard,” Section 36.3.1
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• \*CONTACT
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• \*CONTACT PAIR
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• \*SURFACE PROPERTY ASSIGNMENT
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• \*SURFACE SMOOTHING
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# Overview
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With the finite element method, curved geometric surfaces are naturally approximated as a faceted group of connected element faces. This section discusses methods to improve faceted surface representations for purposes of contact computations based on knowledge of idealized initial surface geometry. Other types of surface smoothing are discussed in “Smoothing master surfaces for the finite-sliding, nodeto-surface formulation” in “Contact formulations in Abaqus/Standard,” Section 38.1.1, and “Using the small-sliding tracking approach” in “Contact formulations in Abaqus/Standard,” Section 38.1.1.
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The use of a faceted surface geometry rather than the true surface geometry can significantly contribute to contact stress inaccuracy in contact interactions, especially when the magnitude of the differences between the faceted and true surface is not small with respect to the deformation of the components in contact. Contact stress output is of primary importance in many applications; for example, the distribution of contact pressures can be used to identify wear patterns and peak pressure values to determine relative lives of machine parts.
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Abaqus/Standard offers techniques to improve the accuracy and robustness of contact computations based on comparisons between the initial faceted geometry and a more idealized initial geometry of the same surface. In some cases you may know that the idealized surface is (exactly or approximately) cylindrical, spherical, or toroidal. Abaqus/CAE identifies cylindrical, spherical, and toroidal surfaces automatically. When creating finite element models as CAD representations in the 3DEXPERIENCE platform, idealized surface representations are often available for selection.
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# Smoothing of common curved surface geometries
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One method of surface smoothing applies to surface regions that are roughly axisymmetric, roughly spherical, or part of a toroidal surface. The smoothing method applies to general contact and surfaceto-surface contact pairs. For example, the pin insertion model in Figure 38.1.3–1 could benefit from this smoothing: the body of the pin is cylindrical, the head of the pin is hemispherical, and the hole is
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conical. Surface-to-surface contact smoothing would also be effective if the surfaces were not perfectly axisymmetric, spherical, or toroidal; for example, if the pin body were slightly elliptical.
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<details>
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<summary>text_image</summary>
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a
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b
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</details>
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Figure 38.1.3–1 Surface-to-surface contact model with surface smoothing.
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# Effects of contact surface smoothing
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The impact of contact smoothing based on comparison of initial faceted surface geometry to initial idealized surface geometry can be demonstrated by a simple model. Figure 38.1.3–2 shows the initial mesh geometry for a two-dimensional model of concentric cylinders with an interference fit. The concentric cylinders are modeled with first-order elements of different sizes.
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Discrepancies between the true surface geometry and the faceted surface geometry result in noise in the contact pressure solution if surface smoothing is not used. If the interference distance and resulting deformation distance is small with respect to the geometry discrepancy, this noise can have a significant effect on the accuracy of the solution. Although surface-to-surface contact typically handles these discrepancies better than node-to-surface contact, it is not unusual for the maximum deviation from the analytical pressure solution to be upward of 100%. The effects of the noise become less apparent for larger deformations, but they are never completely eliminated.
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# Applying smoothing of common curved surface geometries to general contact
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Smoothing of common curved surface geometries for general contact is enabled by surface property assignments. Surface property assignments specify which surfaces are to be smoothed and the smoothing method to be used. The underlying geometry correction methods are the same for general contact and contact pairs:
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• The circumferential smoothing method is applicable to surfaces approximating a portion of a circle in two dimensions or a portion of a surface of revolution in three dimensions.
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<!-- source-page: 789 -->
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<details>
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<summary>natural_image</summary>
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Technical wireframe diagram of a mechanical component with a conical tip and grid pattern (no text or symbols)
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</details>
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Figure 38.1.3–2 Initial mesh geometry for interference fit model.
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• The spherical smoothing method is applicable to surfaces approximating a portion of a sphere in three dimensions.
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• The toroidal smoothing method is applicable to surfaces approximating a portion of a torus in three dimensions (i.e., a circular arc revolved about an axis).
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For each surface, you must specify the appropriate geometry correction method and either the approximate axis of revolution (for circumferential or toroidal smoothing) or the approximate spherical center (for spherical smoothing). For toroidal smoothing, you must also specify the distance of the center of the circular arc from the axis of revolution, and the line joining point $\mathrm { ( X _ { a } , Y _ { a } , Z _ { a } ) }$ and the center of the circular arc should be perpendicular to the axis of revolution.
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# Input File Usage:
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\*SURFACE PROPERTY ASSIGNMENT, PROPERTY=GEOMETRICCORRECTION
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data lines to define smoothing regions (see below)
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Use the following data line to apply circumferential smoothing to a surface with an axis of symmetry passing through points $( \mathrm { X } _ { \mathrm { a } } , \mathrm { Y } _ { \mathrm { a } } , \mathrm { Z } _ { \mathrm { a } } )$ and $( X _ { \mathrm { b } } , \mathrm { Y _ { \mathrm { b } } , Z _ { \mathrm { b } } ) }$ :
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$s u r f a c e , C \mathrm { I R C U M F E R E N T I A L } , X _ { a } , Y _ { a } , Z _ { a } , X _ { b } , Y _ { b } , Z _ { b }$
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Use the following data line to apply spherical smoothing to a surface with a spherical center at point $( \mathrm { X } _ { \mathrm { a } } , \mathrm { Y } _ { \mathrm { a } } , \mathrm { Z } _ { \mathrm { a } } )$ :
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$s u r f a c e , \mathrm { S P H E R I C A L } , X _ { a } , Y _ { a } , Z _ { a }$
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Use the following data line to apply toroidal smoothing to a surface with an axis of symmetry passing through points $( \mathrm { X } _ { \mathrm { a } } , \mathrm { Y } _ { \mathrm { a } } , \mathrm { Z } _ { \mathrm { a } } )$ and $( X _ { \mathrm { b } } , \mathrm { Y _ { \mathrm { b } } , Z _ { \mathrm { b } } ) }$ with the center of the revolved circular arc at a distance R from the axis of symmetry:
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surface, TOROIDAL, $X _ { a } , Y _ { a } , Z _ { a } , X _ { b } , Y _ { b } , Z _ { b } , R$
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Repeat the data lines as many times as necessary to define the appropriate geometry corrections for all surfaces in the contact domain.
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# Abaqus/CAE Usage:
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Contact surface smoothing can be applied only to native geometry models in Abaqus/CAE. By default, Abaqus/CAE automatically detects all circumferential, spherical, and toroidal surfaces in the general contact domain that can be smoothed and applies the appropriate smoothing.
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Use the following option to prevent automatic surface smoothing of a model:
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Interaction module: Create Interaction: General contact (Standard): Surface Properties: Surface smoothing assignments: Edit: toggle off Automatically assign smoothing for geometric faces
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Use the following option to manually apply smoothing to a surface:
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Interaction module: Create Interaction: General contact (Standard): Surface Properties: Surface smoothing assignments: Edit:
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Select surface, click the arrows to transfer surface to list of smoothing assignments.
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In the Smoothing Option column, select REVOLUTION to apply circumferential smoothing, select SPHERICAL to apply spherical smoothing, select TOROIDAL to apply toroidal smoothing, or select NONE to prevent smoothing of the surface.
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# Example: Pin-in-hole with general contact
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To improve contact pressure accuracy for the model in Figure 38.1.3–1, contact smoothing can be applied to both the master and slave surfaces. Two different geometric correction methods are required for the pin (the slave surface), so additional surfaces are defined corresponding to regions of the slave surface. Spherical smoothing is defined for the tip of the pin. Since the body of the pin and the hole share an axis of revolution, circumferential smoothing is applied to both of these surfaces. This surface smoothing definition applies even if the cross-sectional shapes of the pin and hole deviate from perfect circles.
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* SURFACE INTERACTION, NAME=FRICTION1
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* CONTACT
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* CONTACT INCLUSIONS
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PIN, HOLE
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* CONTACT PROPERTY ASSIGNMENT
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, , FRICTION1
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* SURFACE PROPERTY ASSIGNMENT, PROPERTY=GEOMETRIC CORRECTION
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PIN_TIP, SPHERICAL, $X_b$ , $Y_b$ , $Z_b$ PIN_BODY, CIRCUMFERENTIAL, $X_a$ , $Y_a$ , $Z_a$ , $X_b$ , $Y_b$ , $Z_b$ HOLE, CIRCUMFERENTIAL, $X_a$ , $Y_a$ , $Z_a$ , $X_b$ , $Y_b$ , $Z_b$
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