Beyond Parametric - New Approach to Geometric Constraint Solving

Editor's Notes

• #2: PARAMETRIC:Approach to solid modeling initiated by PTC in the middle of 80s and implemented in Pro/ENGINEER (now renamed to Creo Parametric). After a decade of triumph of this approach all leading CAD vendors implemented this approach in their products:Autodesk – in AutoCAD and Inventor;Dassault – in CATIA and SolidWorks;Siemens PLM – in NX and SolidEdge. Nowadays it became de facto standard of solid modeling.Beyond PARAMETRIC: New approach to solid modeling initiated by Cloud Invent based on it’s proprietary Cheetah geometric constraint solver. It can not only dramatically improve productivity (being implemented in existing CAD applications), but it opens a way to real unification of parametric and direct solid modeling (being implemented as a 3D sketcher, which provides in 3D space both convenience of explicit modeling and flexibility of parametric modeling). This is to be a NEW CAD REVOLUTION.
• #3: The main hero of this story is the Cheetah solver of Cloud Invent. This presentation point out its position on the landscape of the nowadays CAD and indicates its potential for the future of the CAD industry.
• #4: Initially PTC was called SPG Consulting Corporation, later renamed to Parametric.Some aspects of the fundamental ideas behind what eventually became Pro/ENGINEER were already being implemented by Matra Datavision, Intergraph and others. What separated PTC from these other vendors was the overall completeness of Pro/ENGINEER as a solid modeling application.Dr. Samuel P. Geisberg was born in Leningrad, USSR (now St. Petersburg, Russia) in 1936. Geisberg earned a Ph.D. in mathematics and became a professor of mathematics at Leningrad State University.
• #5: One of the guiding principles of the PTC approach to CAD was that Pro/ENGINEER has to be implemented as a solid-modeling system (at that time it was cool). This solid-based approach of PTC was established on the following two main closely related technologies: • Parametric 2D sketching (also known as dimension-driven sketching) • Feature-based solid modeling (also referred as history-based solid modeling) In the approach of PTC to create a 3D solid model user typically starts with creating 2D profile of the object (in other words, 2D section or 2D sketch). This 2D sketch could be then converted into a solid model by translating it through space on a given distance (this is done using the extrude tool), or revolving it around a center line (using revolve tool), or using some other operation of this kind.Part is a solid model that contains at least one feature (but it can contain more features). Further features could be added to the part in the same way as the first feature was created – by selecting some 2D section, creating some 2D sketch there, and applying, for instance, extrusion.
• #6: The 2D CAD in the epoch preceding the Parametric Revolution was coordinate-driven (contrarily to the dimension-driven CAD of the parametric epoch). It means that while operator was creating points and lines in 2D sketch the software recorded the coordinates of all end points in the drawing. Parametric Revolution freed designers from bothering about coordinates. The adepts of parametric approach claimed that engineers are not thinking in terms of coordinates. They are thinking in terms of geometric constrains and dimensions (linear dimensions and angles). This is exactly what was provided to a user by Pro/ENGINEER. In the Sketcher of Pro/ENGINEER operator never works with coordinates of points. He/she draw points, lines, curves, set dimensions between different geometric entities, and set geometric constraints (like parallel, perpendicular, or tangent constraints). The software itself writes equations from these constraints. In these equations coordinates of points are the unknowns (but these coordinates are always hidden), the dimensions are the parameters of equations that define the solution (that’s why this approach is also called “dimension-driven”). With parametric approach, when you change some dimension, you don’t need to change manually anything in the section. The system automatically regenerates the section (i.e. solves corresponding system of equations with new dimension parameters) and recalculate coordinates of all points.
• #7: First Solver writes equations (non-linear in general). The unknowns in these equations are the coordinates of points corresponding to different entities (and also radiuses, angles, etc.). The equations are received from user defined constraints (like “perpendicular”, “tangent”, “equal”, etc.), constraints from geometry (when, for instance, the ends of some line segments are coincide), and also from a lot of “weak dimensions” (created by Autodim ).Next Prune starts its work. Prune is the part of the code that inspects equations one by one and throw away equations that contradict with the equations that were already accepted.When the Prune work is finished and Solver collected 𝑛 independent equations with 𝑛 unknowns, it solves this system of equations.Typical Solver uses Newton method to solve this system of equations. To solve the system Solver receives some initial guess (usually it is the solution of the previously resolved section), then in the point of this initial guess it linearizes the system, and this linear system is solved using some standard method of linear algebra.If the initial system was non-linear (this is typically the case), then the computed solution is used as the initial guess for the next step of the Newton iterations. On this next step our non-linear system is again linearized in this new point of approximation, and the new linear system is solved. Newton iterations are finished when residuals became small enough.First Solver writes equations (non-linear in general). The unknowns in these equations are the coordinates of points corresponding to different entities (and also radiuses, angles, etc.). The equations are received from user defined constraints (like “perpendicular”, “tangent”, “equal”, etc.), constraints from geometry (when, for instance, the ends of some line segments are coincide), and also from a lot of “weak dimensions”, created by Autodim .Next Prune starts its work. Prune is the part of the code that inspects equations one by one and throw away equations that contradict with the equations that were already accepted.When the Prune work is finished and Solver collected 𝑛 independent equations with 𝑛 unknowns, it solves this system of equations.The Solver uses Newton method to solve this system of equations. To solve the system Solver receives some initial guess (usually it is the solution of the previously resolved section), then in the point of this initial guess it linearizes the system, and this linear system is solved using some standard method of linear algebra.If the initial system was non-linear (this is typically the case), then the computed solution is used as the initial guess for the next step of the Newton iterations. On this next step our non-linear system is again linearized in this new point of approximation, and the new linear system is solved. Newton iterations are finished when residuals became small enough.
• #8: Standard solvers are not able to solve really big systems of equations and non-stable even when these systems are not so complicated (having about thousand of variables).And now imagine what might be the difference were we have a method that requires only 𝑂(𝑛) amount of memory and solves the system in 𝑂(𝑛) arithmetic operations (compare it with 𝑂(𝑛2) amount of memory and 𝑂(𝑛3) arithmetic operations required for standard matrix methods). It would open the way for resolving really huge sections (i.e. sections with hundreds of thousands of entities), and to implement truly 3D Sketcher.But all this is an unrealizable dream for the solver of PTC. From its early years Pro/ENGINEER experienced difficulties with complex parts and sections. Now, after more than 20 years, the situation is practically the same – not because developers of PTC are lazy, but because the current approach to solver leaves no chances to improve the situation with performance.The situation with competitors of PTC is not better. None of the leaders of the CAD industry could manipulate with really big sections and really complex parts and assemblies. It seems to me that none of them could imagine that algebraic systems, resulted from geometric constraints, might be solved using 𝑂(𝑛) of memory and 𝑂(𝑛) arithmetic operations. Standard solvers are not able to solve really big systems of equations and non-stable even when these systems are not so complicated (having about thousand of variables).And now imagine what might be the difference were we have a method that requires only 𝑂(𝑛) amount of memory and solves the system in 𝑂(𝑛) arithmetic operations (compare it with 𝑂(𝑛^2) amount of memory and 𝑂(𝑛^3) arithmetic operations required for standard matrix methods). It would open the way for resolving really huge sections (i.e. sections with hundreds of thousands of entities), and to the work with truly 3D Sketcher.But all this is an unrealizable dream for the Solver of PTC. From its early years Pro/ENGINEER experienced difficulties with complex parts and sections. Now, after more than 20 years, the situation is practically the same – not because developers of PTC are lazy, but because the current approach to Solver leaves no chances to improve the situation with performance.The situation with competitors of PTC is not better. None of the leaders of the CAD industry could manipulate with really big sections and really complex parts and assemblies. It seems to me that none of them could imagine that algebraic systems, resulted from geometric constraints, might be solved using 𝑂(𝑛) of memory and 𝑂(𝑛) arithmetic operations.
• #9: Characteristic feature of geometric constraint systems of equations is that after linearization they have very sparse matrix. Solvers of the leading CAD vendors don’t utilize this evidence. Cheetah solver not only fully exploits this important information, but makes use of some other specific features of geometric constraint systems as well. Mathematical methods implemented in Cheetah solver seems to be not applied to geometric constraint problems previously.
• #10: Meanwhile we are using beta version 0.12 of FreeCAD as the last (for the end of 2011) stable release of this parametric modeler. We implemented our Cheetah solver in the Sketcher of FreeCAD (instead of their solver). This opens for us a way to test our Cheetah solver in different complicated situations.Cheetah solver is not a ready to use software yet. It is on the way of research and development, but the main roadmap is already clear…
• #11: This slide contains four video demonstrations (screen-shots from Creo Parametric and FreeCAD with Cheetah inside). Demo 1 consists of Video 1 and Video 2.Demo 2 consists of Video 3 and Video 4.Video 1 and Video 3demonstrate how solver of Creo Parametric (former Pro/ENGINEER) works. Video 2 and Video 4 demonstrate the work of Cheetah solver inside FreeCAD application.Both in Demo 1 and Demo 2 we use mirror operation to obtain a sequence of more and more complicated sketches until we achieve a sketch that is not resolvable by the solver of Creo Parametric. After that we repeat the same sequence of operations with Cheetah solver implemented in the sketcher of FreeCAD application.Comparing Video 1 and Video 2 from Demo 1 (consequently, Video 3 and Video 4 from Demo 2) we can see a vivid difference between the solver of Creo Parametric of PTC and our Cheetah solver.Instead of the solver of PTC we can compare Cheetah with a solver of any other commercial CAD application – the result will be similar.
• #12: The left two charts demonstrate the growth of calculation complexity (upper chart) and growth of memory requirements (down chart) with the growth of number of entities in the sketches of Demo 1 (Video 1 and Video 2 on the previous slide). The orange bar chart corresponds to the solver of PTC, the blue bar chart corresponds to Cheetah solver (it is practically invisible because both time and memory requirements are very low comparing to those of solver of PTC).The left two charts present the similar information for Demo 2 (Video 3 and Video 4 on the previous slide).
• #13: It is clear to everybody in the CAD community that the industry after a decade of incremental improvements is waiting for some big ground-breaking technologies to come. During the last very years main CAD vendors are started to introduce slowly some solutions. Some of the companies even declared (may be, without real foundation behind these declarations) that they revolutionaries the CAD industry. In April, 2008 Siemens PLM introduced their Synchronous Technology (certainly, as a “revolutionary” technology that changes solid modeling). They have been working on ST for several years (now ST 4 is already presented) and they really moved some steps forward to the unification of the direct modeling and parametric approaches to solid modeling. But still it is not that revolutionary Great Unification the CAD community is looking for. ST approach (and similar technologies of Autodesk and PTC) preserves behind the scene the same history tree of features as well as the direct-modeling presentation of a solid object. What they managed to do is to “synchronize” the changes in these two models (whether you are doing changes using direct modeling tools, or you are using parametric tools). This is a nice trick but still this is a compromise (preserving all the problems of the feature-based approach), not a real revolution. What is the verse thing with the technologies similar to ST is that this trick is rather restricted – only some local modifications of solid models are possible with such an approach.
• #14: See more details on our site www.cloud-invent.com