Problem addressed

The designers have questioned whether the spider subassembly of the seeker is capable of withstanding the accelerations it will be subject to under normal operating conditions without deforming beyond a given tolerance.

The critical deformation parameter is, in this case, the relative angle of deformation of the normal to the secondary mirror. This tolerance ensures that the optical path will be within specification.

Solution approach

The goal here is to ensure the spider remains within spec under worst case operating conditions of the seeker. In this problem, the distributed loading forces of intrest are due to two sources: inertial effects and motor torques. (Note that we are assuming that thermal effects are inconsequential.)

The solution has two steps:

First we calculate the maximum possible accelerations of each of the components using a rigid body dynamics model of the seeker assembly. Inputs to this problem (i.e., the rigid body problem) are thus the geometric shapes, mass property parameters, the kinematic relations between the components, and maximum motor torques and external forces. Experiments are then performed to determine the maximum possible accelerations for each rigid body component.

The second stage of the analysis uses the accelerations determined in the rigid body analysis, together with the (local) mass properties of the components to define a loading function for a finite element analysis of the spider assembly. That is, for each finite element, the intertial forces are calculated by integrating the product of mass and the acceleration over the volume of the element. The resulting FEM problem is solved and the maximum deformations thus calculated.

The technical issues

The scenario above requires the interaction of several different models of physics, which are, unsurprisingly, not mutually consistent. For instance, the rigid body model precludes deformation, while the elasticity model precludes motion, and so forth. For this reason (and for others), modelers for these (and other) domains are typically standalone programs, and translations between their representations are often difficult, and sometimes impossible (both for theoretical and practical reasons).

A primary design goal of the SimLab environment is to represent a variety of mathematical and physical models in a uniform framework -- a framework that represents the higher level semantics of the components. In the case of SimLab, the environment is capable of representing algebraic objects, operations and transformations using the Weyl/SPL Transformation Environment. Beyond algebraic computation, discrete topological computation is supported, both for creating Guaranteed Quality Meshes , as required for solving distributed parameter problems, and for the Chains Programming Language, which enables algebraic specification of physical problems in terms of discrete topological objects such as cell complexes and k-chains. The scenario illustrates how these concepts can be used in concert as a problem solving environment to address the problem of maximum deformation in the spider assembly.

The practical issues

In order to create the analysis, we must first aquire all the necessary data, and translate it into a form that we can compute with. In the case of this example, the geometry data is the most difficult to deal with, the largest and so forth.

The geometry data was aquired in the form of triangulated boundaries of solids, which were created by Alpha_1, an integrated graphics, design, modeling and manufacturing system which is being used as the design tool for the mechanical parts in this project. Alpha_1 created the data files in a format suitable as input to a stereolithography system. In order to use the data within SimLab, the data files were translated to a standard representation, which can be used by all components of the SimLab environment. This translation, however, is only the first required. Because the the original triangulation was created for stereolithography, the shapes of the triangles are not optimal for performing analysis. In particular, triangles with poor aspect ratios are known to give poor performance in finite element analyses. The following snapshot from of the original spider mesh illustrates a part of the Alpha_1 triangulation, showing triangles with very large aspect ratios.

So the first step is to re-triangulate the surfaces, using the guaranteed quality mesher. This may be done by computing (an approximation to) the C1 discontinuitites in the surface, partitioning the surface by them, and computing the boundaries of the resulting open sets. This is illustated in the following image.

This is done using topological computations in Chains. The following illustration shows a single surface computed by these methods, together with its boundary. The triangulation shown is a subset of the original triangulation received from Alpha_1.

These boundaries are then passed to the mesher , which computes a Delaunay triangulation of each of the surfaces. In this case, the resulting mesh looks as follows. (Note that there are fewer triangles now, and that they have good aspect ratios.)

Performing an analysis

Once the geometric data has been massaged to a form suitable for analysis, we may perform a variety of numerical computations based on this discretization. For instance, for some applications, (not this one) we may wish to solve a steady state heat transfer problem. An example of how this can be done in SimLab is shown here.

To be continued

The maximum deformation analysis proposed above has not yet been completed. In particular, because we wish to perform a finite element analysis on the spider, which is a three dimensional solid, we must have the ability to create a 3D guaranteed mesh of the volume. This is a current area of research in our group (see).