MIT Design Laboratory Current Research
Solution of Nonlinear Systems
A fundamental problem in computer aided design is the robust and efficient
computation of all real roots of systems of nonlinear polynomial or more
complex equations. Such problems often occur in computing
intersections, distance functions, and extrema, as well as in
engineering design. In order to isolate all roots within a given
domain, several methods are being studied: the first method projects
control polyhedra onto a set of coordinate planes; the second employs
linear optimization; and the final method utilizes an interval Newton
technique.
Interval arithmetic implementation is a basic component of these
methods leading to computational reliability.
Robust Geometric Modeling
Full automation of the design of complex structures requires
reliability of the geometric computations involved in the design
process.
However, this reliability is not present in current modelers due
to floating point operations that lead to inconsistent or unstable
numerical computations.
The goal of this project is the development of a solid modeling system
based on interval splines.
Interval splines differ from classical splines in that the real
numbers representing control point coordinates are replaced by
intervals which in combination with rounded interval
arithmetic can guarantee the robustness of geometric computations.
Praxiteles
In order that the design and manufacture processes be interactive,
algorithms developed in the MIT Design Laboratory for
shape creation, advanced interrogation, visualization, fairing, and
inspection have been integrated into an executive system, called
Praxiteles. Praxiteles, named after the sculptor of the statue
of Hermes in Olympia, was developed for use on the Silicon Graphics IRIS
4D series workstations which provide for graphical output of numerical
computations and allows for input and output via the IGES format which
provides a standard for the exchange of geometric data with other CAD
systems.
In addition, since many systems are limited to cubic piecewise
polynomial geometries defined in terms of B-splines (for efficient and
stable processing), Praxiteles facilitate the
exchange of curve and surface geometry between different geometric
modeling systems through advanced methods for approximate
conversions of high degree Bezier and B-spline surfaces to lower degree
representations with guaranteed bounds on the approximating error.
Advanced Methods for Inspection
Inspection is the process of verifying shape conformance of a measured
manufactured part with a toleranced geometric description of a design
object. Since manufacturing processes such as milling and casting are
inaccurate, the need for accurate inspection methods is fundamental,
particularly for objects that require a very high level of precision.
Three processes for inspection take the form of feature extraction, which
is the ability to extract performance related geometric features from
a mathematically represented design or as-built model; of
localization, which is the optimal positioning of a mathematically represented
design model with respect to a measured manufactured artifact; and of
local error analysis, which is used for planning remachining
operations.
Idealization for Mesh Generation Using MAT
This project concerns the generation of coarse and fine meshes on
multiply connected 2D and 3D domains based on the medial axis transform (MAT).
The MAT or skeleton or symmetric axis of a shape is a point set
consisting of points that are equidistant from two or more points on
the boundary contour.
Using the MAT, we can extract some important shape characteristics and
their length scales so that we can create a coarse subdivision of a
complex surface and then select a local element size to generate fine
triangular meshes within individual subregions.
The MAT allows us to carry out these processes in an automated manner.
Thus, our approach can lead to integration of automated finite element
(FE) mesh generation schemes into CAD systems.
Advanced Techniques for Milling Operations
Propeller blades are manufactured by numerically controlled (NC) milling
machines. When a ball end-mill cutter is used, the cutter radius must be
smaller than the smallest concave radius of curvature of the surface to
be machined to avoid local overcut (gouging). Gouging is the one of the
most critical problems in NC machining of free-form surfaces.
Therefore, we must determine the distribution of the principal
curvatures of the surface, which are upper and lower bounds on the
curvature at a given point, to select the cutter size. We are
developing a robust algorithm for contouring curvature of a free-form
surface using our work on the solution of nonlinear equations.
We then subdivide the surface into regions of similar order of
magnitude curvature which can be used for both tool selection and tool
path generation.
The objective of this project is to advance the state of knowledge in
efficiently representing, interrogating, visualizing, updating and
manipulating vast amounts of geophysical data.
Our work involves concepts from interval arithmetic, differential
topology, algebraic topology, all applied in computational geometry
representations and algorithms.
We are currently addressing the problems of multidimensional physical
data representation using advanced higher dimensional geometrical
concepts and of uncertainty in data representation and computation by
invoking the concept of interval n-dimensional B-splines and simplex
splines and the theory of interval arithmetic.