Page 511 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 511
MULTISCALE MODELING AND METHODS: APPLICATION IN ENGINEERING,
BIOLOGY AND MEDICINE (MS-80)
Asymptotically based simulation of the Stokes flow in a layer through
periodic flexural plates made of beams
Maxime Krier, maxime.krier@itwm.fraunhofer.de
Fraunhofer ITWM, Kaiserslautern, Germany
Coauthors: Julia Orlik, Grigory Panassenko
Stokes fluid is flowing through a spacer fabric, a porous layer between two parallel hyperplanes
with periodically distributed parallel beam latices, which are orthogonal to the hyperplanes. The
flow direction is parallel to the hyperplanes and orthogonal to the latices. The top and the bottom
of the spacer fabric is insulated for the in- and outflow. The thickness of the spacer fabric is
assumed to be one, while the thickness of the latices or porous layers is a small parameter ε. The
fluid viscosity is assumed to be ε3E, where E is the Young’s modulus of the beams. Fluid-solid
interaction is considered in the structure. A dimension reduction as ε → 0 was considered in [1]
and the latice-layer is replaced in the paper by its mean surface with a condition: the pressure
jump through the surface is proportional to the biharmonic operator in the surface applied to
the velocity trace at this surface. The normal component of the limit macroscopic velocity field
is an H2-function of the lattice mean-plane variable and the limit problem is non-local in time.
This corresponds to the non-stationarity of the initial problem.
For the numerical computations, a further dimension reduction is performed (see [2], [3],
[4]), reducing the elasticity problem to beam equations on one-dimensional lattices and further
to a linear algebraic system with 6 unknown degrees of freedom in the nodes of the lattice.
Using the equivalence of finite dimensional interpolated norms on segments of the lattice and
in the 3D-domain spanned between periodic latices (hexaedral mesh), [5], the nodal solution on
the lattices is extended by Q1 interpolation into the fluid part and the limit problem is solved
staying with just 6 degrees of freedom in the lattice nodes. The convergence estimates from the
corresponding analysis will be used to estimate the numerical accuracy of the reduced dimen-
sion algorithm. Finally, local stresses in the beams and the fluid pressure will be reconstructed
as in [2] with the help of interpolated and extended piece-wise polynomial function sequences
which are strongly convergent to the solution.
References
[1] J. Orlik, G. Panasenko, R. Stavre, Asymptotic analysis of a viscous fluid layer separated by
a thin stiff stratified elastic plate, Applicable Analysis, 100:3, 589-629, (2021)
[2] Griso, G., Khilkova, L., Orlik, J., Sivak, O.: Asymptotic Behavior of Stable Structures
Made of Beams. J. Elast (2021). https://doi.org/10.1007/s10659-021-09816-w
[3] Orlik, J., Panasenko, G., Shiryaev, V.: Optimization of textile-like materials via homoge-
nization and dimension reduction. SIAM Multiscale Model. Simul., 14(2), 637–667 (2016)
[4] Griso, G., Migunova, A., Orlik, J.: Asymptotic Analysis for Domains Separated by a Thin
Layer Made of Periodic Vertical Beams. J. Elast., 128(2), 291–331 (2017)
[5] Falconi, R., Griso, G., Orlik, J.: Anisotropic extension of the unfolding tools, in preparation,
2021
509
BIOLOGY AND MEDICINE (MS-80)
Asymptotically based simulation of the Stokes flow in a layer through
periodic flexural plates made of beams
Maxime Krier, maxime.krier@itwm.fraunhofer.de
Fraunhofer ITWM, Kaiserslautern, Germany
Coauthors: Julia Orlik, Grigory Panassenko
Stokes fluid is flowing through a spacer fabric, a porous layer between two parallel hyperplanes
with periodically distributed parallel beam latices, which are orthogonal to the hyperplanes. The
flow direction is parallel to the hyperplanes and orthogonal to the latices. The top and the bottom
of the spacer fabric is insulated for the in- and outflow. The thickness of the spacer fabric is
assumed to be one, while the thickness of the latices or porous layers is a small parameter ε. The
fluid viscosity is assumed to be ε3E, where E is the Young’s modulus of the beams. Fluid-solid
interaction is considered in the structure. A dimension reduction as ε → 0 was considered in [1]
and the latice-layer is replaced in the paper by its mean surface with a condition: the pressure
jump through the surface is proportional to the biharmonic operator in the surface applied to
the velocity trace at this surface. The normal component of the limit macroscopic velocity field
is an H2-function of the lattice mean-plane variable and the limit problem is non-local in time.
This corresponds to the non-stationarity of the initial problem.
For the numerical computations, a further dimension reduction is performed (see [2], [3],
[4]), reducing the elasticity problem to beam equations on one-dimensional lattices and further
to a linear algebraic system with 6 unknown degrees of freedom in the nodes of the lattice.
Using the equivalence of finite dimensional interpolated norms on segments of the lattice and
in the 3D-domain spanned between periodic latices (hexaedral mesh), [5], the nodal solution on
the lattices is extended by Q1 interpolation into the fluid part and the limit problem is solved
staying with just 6 degrees of freedom in the lattice nodes. The convergence estimates from the
corresponding analysis will be used to estimate the numerical accuracy of the reduced dimen-
sion algorithm. Finally, local stresses in the beams and the fluid pressure will be reconstructed
as in [2] with the help of interpolated and extended piece-wise polynomial function sequences
which are strongly convergent to the solution.
References
[1] J. Orlik, G. Panasenko, R. Stavre, Asymptotic analysis of a viscous fluid layer separated by
a thin stiff stratified elastic plate, Applicable Analysis, 100:3, 589-629, (2021)
[2] Griso, G., Khilkova, L., Orlik, J., Sivak, O.: Asymptotic Behavior of Stable Structures
Made of Beams. J. Elast (2021). https://doi.org/10.1007/s10659-021-09816-w
[3] Orlik, J., Panasenko, G., Shiryaev, V.: Optimization of textile-like materials via homoge-
nization and dimension reduction. SIAM Multiscale Model. Simul., 14(2), 637–667 (2016)
[4] Griso, G., Migunova, A., Orlik, J.: Asymptotic Analysis for Domains Separated by a Thin
Layer Made of Periodic Vertical Beams. J. Elast., 128(2), 291–331 (2017)
[5] Falconi, R., Griso, G., Orlik, J.: Anisotropic extension of the unfolding tools, in preparation,
2021
509
