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.