Models of solid optics

Solid optics are characterized by geometric topology that lacks lightweighting or discrete stiffening. Examples are lenses, solid mirrors, prisms, and windows.

5.1.3.1 Two-dimensional models of solid optics

Some solid optics exhibit mechanical behavior that can be well approximated under the assumptions of plate or shell behavior. In such cases, the elastic stiffness of a 2D, solid optic model is defined by membrane, bending, and transverse-shear stiffnesses. The dimensional parameters on which these stiffnesses depend are the thickness of the optic and the transverse shear factor. For solid optics, the transverse shear factor should be specified as 0.8333.

2D models can provide excellent predictions of global elastic behavior for static and vibration analyses. An important limitation of 2D-element optic models, however, is that they do not predict deformation effects in the direction through the thickness of the optic. Therefore, their rigid-body motions and global elastic deformations are represented by the midplane of the optic and not necessarily that of the optical surface. Differences between the behavior of the midplane of an optic and its optical surface can be caused by mount-induced loads and thermoelastic growth through the thickness of the optic. Furthermore, mount-induced loads will show greater local deformations in 2D-element models than may actually exist at the optical surface of the actual hardware. Therefore, the analyst should choose this method of modeling a solid optic only when it is reasonable to assume that such effects are not significant to the overall goal of the analysis.

Plate-element meshes can also be used to model components where a reasonable representation of stiffness is desired but accurate displacement

predictions are not required. This is often the case when the components being modeled are far enough from the regions of primary interest that accurate representation of their elasticity is not required. A lens to be modeled as part of a lens barrel model is shown in Fig. 5.3(a). However, suppose displacements are not required of the lens shown in the figure.

A model that correctly represents its stiffness may be required to obtain useable displacement results elsewhere in the system. The lens has a relatively constant thickness approximately equal to t0, as shown in Fig. 5.3(a), and can be

reasonably represented by the plate mesh shown in Fig. 5.3(b). The stiffness of the lens shown in Fig. 5.3(c), however, may not be well represented by a plate mesh due to the inability of such a model to predict potential deformations such as those shown. Such a model may have to be constructed of 3D solid elements, as described in the next section, in order to provide a reasonable approximation of its stiffness.

5.1.3.2 Three-dimensional element models of solid optics

Components whose elastic behavior cannot be accurately represented by plate assumptions require solid-element formulations that use the full 3D representation of Hooke’s law. Examples of such components are thick lenses, thick solid mirrors, and prisms. Fig. 5.4 shows some examples of such models.

The construction of solid-element models deserves a few guidelines to be followed in most cases. Solid-element models of lenses and mirrors should have at least four trilinear elements through their thicknesses. Such a minimum resolution is required in most cases to provide a reasonably accurate prediction of the variation in stress states through the thickness of the component. In many cases, more than four elements will be required. The number of elements required is dictated by the variation of displacements through the thickness of the component and the elements’ ability to represent them.

t

(a) (b) (c)

Figure 5.3 Modeling of lenses with 2D models: (a) lens with relatively constant thickness t0, (b) corresponding 2D-element mesh, and (c) elastic behavior in a lens that would not be represented by a 2D-element mesh.

(a) (b)

Figure 5.4 Examples of 3D solid models: (a) lens and (b) Porro prism.

20 elements

Number of elements through the thickness

Figure 5.5 Axisymmetric wedge model.

0 0% 1 0% 2 0% 3 0% 4 0% 5 0% 6 0% 1 2 3 4 5 6 7 8

Number of Elements Through the Thickness

% Er ro r in N at u ra l Fr eque nc y P re di ct ion

Figure 5.6 Frequency error verses resolution.

An axisymmetric solid mirror with a diameter-to-thickness ratio of 10 is modeled as a 5-degree wedge as shown in Fig. 5.5. In MSC.Nastran, a model of 8-noded hexahedron elements with a constant radial mesh resolution of 20 and a variable through-the-thickness mesh resolution of 1 through 8 elements was used. The plot of the model shown in Fig. 5.5 illustrates four elements through the thickness. In Fig. 5.6, the percent error in the first axisymmetric free–free natural frequency is plotted verses mesh resolution. In Fig. 5.7, the gravity-induced

Gravity

Simply Supported at Edge

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 1 2 3 4 5 6 7 8

Number of Elements Through the Thickness

% Er ro r in Po w er P re d ic ti o n

Figure 5.7: Power error verses resolution.

(a) (b)

Figure 5.8: (a) Resulting aperture with mesh lines on aperture, and (b) resulting

aperture without mesh lines on aperture for a mesh of an optic created by an automeshing technique.

amplitude of the error in the Zernike power term computed with a simply supported edge condition is plotted verses mesh resolution. From the results shown in Figs. 5.6 and 5.7 the use of four or five elements through the thickness gives around 0.1% error in the natural frequency and static displacement results.

The use of automeshing algorithms to generate meshes of highly symmetric optical components, as shown in Fig. 5.4, has shortcomings in practice. Automeshing routines will commonly generate nonsymmetrical meshes for even the most symmetric structures. Such asymmetries in element meshes can generate nonsymmetrical results for problems with symmetric behavior. Automeshing routines, on the other hand, are not without usefulness––they can be useful in situations involving very complicated geometry not meshable by six- sided and five-sided solid elements.

When automeshing any optical model, extra care should be taken to give forethought to any aperturing that may be applied in the processing of the results. If the mesh layout does not contain mesh lines along such aperture or obstruction shapes as shown in Fig. 5.8(a), then chopping as shown in Fig. 5.8(b) will occur if aperturing or obstructing of the finite element results is performed. Chopping will cause misrepresentation of optomechanical behavior and result plots that

appear of questionable validity. Enforcing such mesh lines, however, may be difficult in some software that does not easily allow manipulation of CAD geometry.

The use of the four-noded, constant-strain tetrahedron element should be strictly avoided. The formulation of this element assumes a constant state of strain throughout its volume, resulting in a mesh that is too stiff for useable displacement results. If tetrahedron elements must be used, then ten-noded tetrahedron elements should be employed.