Colloquium

特別講義 Applied Geometry and Discrete Mechanics I

Dr.Mathieu Desbrun

5月26日(月) 16時30分

The emergence of computers as an essential tool in scientific research has shaken the very foundations of differential modeling. Indeed, the deeply-rooted abstraction of smoothness, or differentiability, seems to inherently clash with a computer’s ability of storing only finite sets of numbers. While there has been a series of computational techniques that proposed discretizations of differential equations, the geometric structures they are simulating are often lost in the process.

In these two lectures, I will review structure-preserving discretizations of mechanics. In the first lecture on "Spatial Discretizations", I will introduce recent work on the development of discrete exterior calculus of differential forms, where mixed finite elements and simplicial complexes are used for computations in order to automatically preserves a number of important geometric structures, including Stokes' theorem, integration by parts (with a proper treatment of boundaries), the de Rham complex, Poincare duality, Poincare's lemma, and Hodge theory. Maxwell's equations will used as an example for which such a discrete calculus is particularly relevant and computationally beneficial.

In the second talk, I will introduce the notion of "Structure-preserving Time Integrators", where optimality of the space-time trajectory (i.e., Hamilton's principle) yields a set of conditions on the path at each time step, from which a time integrator falls out naturally. Various schemes and their properties (momentum preservation, symplecticity) will be discussed. With a discrete geometry-driven calculus and variational integrators, key defining properties like symmetries, algebraic topologic and geometric invariants are intrinsically respected in the numerical realm, often avoiding classical flaws such as numerical viscosity and energy blowups.