[Algorithms] Numerical methods

[Jamie F. <j.f...@re...>]

Cheers, I'll look into them when I get time :)

Jamie


"Graham S. Rhodes" wrote:

> Here are a few of the references that I use:
>
> "Computational Dynamics" by Ahmed Shabana is a decent book on, well,
> computational rigid-body dynamics with full discussion on many many joint
> constraints but not collision detection. It has a fair discussion of
> numerical methods, but it does not analyze the error terms sufficiently.
>
> "Computational Geometry: Algorithms and Applications" by M. de Berg et al.
> This book has some decent discussions on the development of robust geometric
> algorithms that handle degenerate cases well. Although I use the book as
> background, I haven't really tested their algorithms.
>
> One my favorite references on numerical methods is:
>
> "Computational Fluid Mechanics and Heat Transfer" by Tannehill, Anderson,
> and Pletcher. The order of the authors is random for each edition (there are
> two so far). This will be one of my references for the papers I'll be
> preparing.
>
> I know this sounds like a strange reference, but it has one of the best
> discussions I know of on the fundamental nature of numerical errors in
> discrete integration schemes for differential equations. Chapters 2 and 3
> are introductions to DE's (especially PDE's due to the nature of the
> material of the topic of the book), and have nothing really to do with
> fluids. Chapter 4 analyzes truncation error in a bit more detail, and
> studies the stability issues of a laundry list of equations, using a variety
> of different difference formulas.
>
> There is some discussion in the book about how to deal with discontinuities.
> In fluids, discontinuities are shock waves, contact surfaces (two regions of
> fluid that move at different velocities at a common boundary in an inviscid
> flow----such as the interface between water and air at the ocean). But some
> of the rules apply elsewhere, including when you have cracks in a rigid or
> nonrigid body, and when there are collisions in a dynamics problem. The
> trick is detecting the discontinuities in an automated and robust manner.
> Obviously, it is hard to detect collisions in a robust manner while keeping
> time steps large enough for games. (Well, even without dealing with time
> steps). In fluids, shock capture methods are pretty good at finding
> discontinuities, but as with using penalty methods in dynamics there tends
> to be a general mushiness/springiness with oscillations at the
> discontinuity---second order accurate methods are required to come close to
> tightly modeling the discontinuity. Shock fitting methods actually model the
> geometry of the shock explicitly, and this is similar to restarting the
> integration of a dynamics problem at the point of collision. Much nicer if
> you can do it fast enough, harder to code. And you still have the problem of
> intersecting the geometries. (In fluids, the geometry problem involves
> moving the shock geometry until the flow properties on both sides satisfy
> the "Rankine-Hugoniot" equations----required to satisfy the second law of
> thermodynamics for physically consistent shocks. Enough of this tangent!)
>
> Graham Rhodes
>
> > -----Original Message-----
> > From: gda...@li...
> > [mailto:gda...@li...]On Behalf Of Jamie
> > Fowlston
> > Sent: Wednesday, August 16, 2000 4:55 AM
> > To: gda...@li...
> > Subject: Re: [Algorithms] XGDC conference
> >
> >
> > Can you recommend any books on the topic? I avoided the numerical methods
> > lectures while at university, and so far it's been the most
> > useful thing I could
> > have done there....
> >
> > Jamie
> >
> >
> > "Graham S. Rhodes" wrote:
> >
> > > Wow,
> > >
> > > Lots of attendees here! I appreciate the feedback folks. Starting to get
> > > excited. I'm planning to propose a talk on predicting and
> > managing numerical
> > > error for stable physics simulation for games. The XGDC topic list on
> > > xgames3d.com has me listed with a title of "Advanced Physics
> > Programming,"
> > > but really the idea is to introduce formal techniques for analyzing the
> > > errors introduced by numerical techniques, the way that the errors
> > > propogate(sometimes leading to instability and blow-ups), and
> > how to control
> > > the errors by designing or selecting the right solution scheme.
> > Sounds a bit
> > > boring, but just about everything I've seen related to game physics
> > > simulation has skipped over this, and it is essential to
> > achieving the most
> > > robust physics simulations.
> > >
> > > I'm also going to submit at least one proposal to GDC on a
> > related matter.
> > > By the time GDC rolls around hopefully I'll have some more interesting
> > > examples, such as tricky collision detection examples.
> > >
> > > Graham Rhodes
> > >
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> >
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