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One of the most challenging problems in dynamic concurrent multiscale
simulations is the reflectionless transfer of physical quantities between the
different scales. In particular, when coupling molecular dynamics and finite
element discretizations in solid body mechanics, often spurious wave reflections
are introduced by the applied coupling technique. The reflected waves are
typically of high frequency and are arguably of little importance in the domain
where the finite element discretization drives the simulation.
In this work, we provide an analysis of this phenomenon.
Based on the gained
insight, we derive a new coupling approach, which neatly separates high and low
frequency waves. Whereas low frequency waves are permitted to
bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbitrary frequencies. The separation of
wavelengths is achieved by employing a discrete $L^2$-projection, which acts as a
low pass filter. Our coupling constraints enforce matching in the range of this projection. With respect to the numerical realization this approach
has the advantage of a small number of constraints, which is computationally
efficient. Numerical results in one and two dimensions confirm our theoretical
findings and illustrate the performance of our new weak coupling approach.
We present an algorithm that constructs parametrizations of boundary
and interface surfaces automatically. Starting with high-resolution triangulated
surfaces describing the computational domains, we iteratively
simplify the surfaces yielding a coarse approximation of the boundaries
with the same topological type. While simplifying we construct a function
that is defined on the coarse surface and whose image is the original
surface. This function allows access to the correct shape and surface normals
of the original surface as well as to any kind of data defined on it.
Such information can be used by geometric multigrid solvers doing adaptive
mesh refinement. Our algorithm runs stable on all types of input
surfaces, including those that describe domains consisting of several materials.
We have used our method with success in different fields and we
discuss examples from structural mechanics and biomechanics.
The purpose of the paper is to apply monotone multigrid methods
to static and dynamic biomechanical contact problems.
In space, a finite element method involving a mortar
discretization of the contact conditions is used.
In time, a new contact--stabilized Newmark scheme is presented.
Numerical experiments for a two body Hertzian contact problem
and a biomechanical knee problem are reported.