Hessm: writing ITHACA-SEM page

{{preliminary}} 

[[Category:Software]]

== Synopsis ==

[[File:ithaca-sem.png|300px|right|ithaca-sem box]]

[https://mathlab.sissa.it/ITHACA-SEM ITHACA-SEM]: In real Time Highly Advanced Computational Applications with Spectral Element Methods - Reduced Order Models for Nektar++ , is a C++ package for the Model Order Reduction
that uses
simulations from the spectral/hp element software [https://nektar.info Nektar++].
It uses [http://eigen.tuxfamily.org Eigen] to perform the matrix decompositions required for parametric model order reduction.
The GitHub repository is available here: [https://github.com/mathLab/ITHACA-SEM ITHACA-SEM].

== Requirements ==

The requirement is an installation of Nektar++ in version 4.4.0.

== Features ==

As of Feb. 2019:
* allows using a variable Reynolds number (specified by the kinematic viscosity) as parameter
* perform offline simulation in ITHACA-SEM or use precomputed Nektar++ *.fld files as snapshot solutions
* computes a POD ROM
* computes ROM parameter sweeps

== References ==

ITHACA-SEM has been used in the following publications, i.e., either the c++ version or a previous python3 version, which is now listed under 'deprecated' in the GitHub repo.

* Hess M.W., Rozza G. (2019) A Spectral Element Reduced Basis Method in Parametric CFD. In: Radu F., Kumar K., Berre I., Nordbotten J., Pop I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham, DOI https://doi.org/10.1007/978-3-319-96415-7_64

* M. Hess, A. Quaini, and G. Rozza, “Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature”, 2019, submitted, https://arxiv.org/abs/1901.03708.

* M. Hess, A. Alla, A. Quaini, G. Rozza, and M. Gunzburger, “A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions”, 2018, submitted, https://arxiv.org/abs/1807.08851.

* M. Hess, A. Quaini, and G. Rozza, “A Spectral Element Reduced Basis Method for Navier-Stokes Equations with Geometric Variations”, 2018, submitted, https://arxiv.org/abs/1812.11051.

== Links ==

GitHub repository: https://github.com/mathLab/ITHACA-SEM

Website: https://mathlab.sissa.it/ITHACA-SEM

== Contact ==

[[User:Hessm| Martin Hess]]

ITHACA-SEM

2019-02-04T11:28:02Z

Hessm:

ITHACA-SEM

2019-02-04T11:23:09Z

Hessm:

ITHACA-SEM

2019-02-04T11:09:13Z

Hessm:

{{preliminary}} 

[[Category:Software]]

[[File:ithaca-sem.png|300px|right|ithaca-sem box]]

[https://github.com/mathLab/ITHACA-SEM ITHACA-SEM]: In real Time Highly Advanced Computational Applications with Spectral Element Methods - Reduced Order Models for Nektar++ , is a C++ package for the Model Order Reduction
simulations with the spectral/hp element software [https://nektar.info Nektar++].
It uses [http://eigen.tuxfamily.org Eigen] to perform the matrix decompositions required for parametric model order reduction.

2014-07-14T13:01:10Z

Hessm:

[[Category:benchmark]]
[[Category:PDE]]
[[Category:parametric 2-5 parameters]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:affine parameter representation]]

==Description==

A '''coplanar waveguide''' (see <xr id="fig:coplanar"/>) is a microwave semiconductor device, which is governed by maxwell's equations.
The '''coplanar waveguide''' considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with 0.5mm thickness are buried in a substrate with 10mm thickness and relative permittivity
<math> \epsilon_r = 4.4 </math> and relative permeability <math> \mu_r = 1 </math>, and low conductivity <math> \sigma = 0.02 S/m </math>. The low-loss upper layer has low permittivity <math> \epsilon_r = 1.07 </math> and
<math> \sigma = 0.01 S/m </math>. The whole structure is enlosed in a metallic box of dimension 140mm by 100mm by 50mm. The discrete port with 50ohm lumped load imposes 1 A current as the input to the one side of the strip. The voltage along the discrete port 2 at the end of the other side of coupled lines is integrated as the output.

<figure id="fig:coplanar">
[[File:CoplanarWaveguideScaled.jpg|frame|<caption>Coplanar Waveguide Model<ref>M. W. Hess, P. Benner, "[http://www.mpi-magdeburg.mpg.de/preprints/2012/MPIMD12-17.pdf Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]", IEEE Transactions on Microwave Theory and Techniques, DOI 10.1109/TMTT.2013.2258167 , 2013.</ref></caption>]]
</figure>

==Data==

Considered parameters are the frequency <math> \omega </math> and the width <math> \nu </math> of the middle stripline.

The affine form <math> a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) a^q(u, v) </math> can be established using <math> Q = 15 </math> affine terms.

The discretized bilinear form is <math> a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) A^q </math>, with matrices <math> A^q </math>.

The matrices corresponding to the bilinear forms <math> a^q( \cdot , \cdot ) </math> as well as the input and output forms and H(curl) inner product matrix have been assembled
using the Finite Element Method, resulting in 7754 degrees of freedom, after removal of boundary conditions. The files are numbered according to their
appearance in the summation.

The files are numbered according to their appearance in the summation and can be found here:

The coefficient functions are given by

:<math> \Theta^1(\omega, \nu) = 1 </math>

:<math> \Theta^2(\omega, \nu) = \omega </math>

:<math> \Theta^3(\omega, \nu) = -\omega^2 </math>

:<math> \Theta^4(\omega, \nu) = \frac{\nu}{6} </math>

:<math> \Theta^5(\omega, \nu) = \frac{6}{\nu} </math>

:<math> \Theta^6(\omega, \nu) = \frac{6 \omega}{\nu} </math>

:<math> \Theta^7(\omega, \nu) = -\frac{6 \omega^2}{\nu} </math>

:<math> \Theta^8(\omega, \nu) = \frac{\nu \omega}{6} </math>

:<math> \Theta^9(\omega, \nu) = -\frac{\nu \omega^2}{6} </math>

:<math> \Theta^{10}(\omega, \nu) = \frac{16 - \nu}{10} </math>

:<math> \Theta^{11}(\omega, \nu) = \frac{10}{16 - \nu} </math>

:<math> \Theta^{12}(\omega, \nu) = \frac{10 \omega}{16 - \nu} </math>

:<math> \Theta^{13}(\omega, \nu) = -\frac{10 \omega^2}{16 - \nu} </math>

:<math> \Theta^{14}(\omega, \nu) = \frac{16 - \nu}{10} \omega </math>

:<math> \Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2 </math>

The parameter domain of interest is <math> \omega \in [0.6, 3.0] * 10^9 </math> Hz, where the factor of <math> 10^9 </math> has already been taken into account
while assembling the matrices, while the geometric variation occurs between <math> \nu \in [2.0, 14.0] </math>. The input functional also has a factor of <math> \omega </math>.

There are two output functionals, which is due to the fact, that the complex system has been rewritten as a real symmetric one. In particular the computation of the output

<math> s(u) = | l^T * u | </math> with complex vector <math> u </math> turns into <math> s(u) = \sqrt{ (l_1^T * u)^2 + (l_2^T * u)^2 } </math> with real vector <math> u </math>.

==Origin==

The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano project].

==References==

<references/>

==Contact==

'' [[User:hessm|Martin Hess]]''

Coplanar Waveguide

2014-07-14T13:00:40Z

Hessm: link to matrices corrected

[[Category:benchmark]]
[[Category:PDE]]
[[Category:parametric 2-5 parameters]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:affine parameter representation]]

==Description==

A '''coplanar waveguide''' (see <xr id="fig:coplanar"/>) is a microwave semiconductor device, which is governed by maxwell's equations.
The '''coplanar waveguide''' considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with 0.5mm thickness are buried in a substrate with 10mm thickness and relative permittivity
<math> \epsilon_r = 4.4 </math> and relative permeability <math> \mu_r = 1 </math>, and low conductivity <math> \sigma = 0.02 S/m </math>. The low-loss upper layer has low permittivity <math> \epsilon_r = 1.07 </math> and
<math> \sigma = 0.01 S/m </math>. The whole structure is enlosed in a metallic box of dimension 140mm by 100mm by 50mm. The discrete port with 50ohm lumped load imposes 1 A current as the input to the one side of the strip. The voltage along the discrete port 2 at the end of the other side of coupled lines is integrated as the output.

<figure id="fig:coplanar">
[[File:CoplanarWaveguideScaled.jpg|frame|<caption>Coplanar Waveguide Model<ref>M. W. Hess, P. Benner, "[http://www.mpi-magdeburg.mpg.de/preprints/2012/MPIMD12-17.pdf Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]", IEEE Transactions on Microwave Theory and Techniques, DOI 10.1109/TMTT.2013.2258167 , 2013.</ref></caption>]]
</figure>

==Data==

Considered parameters are the frequency <math> \omega </math> and the width <math> \nu </math> of the middle stripline.

The affine form <math> a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) a^q(u, v) </math> can be established using <math> Q = 15 </math> affine terms.

The discretized bilinear form is <math> a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) A^q </math>, with matrices <math> A^q </math>.

The matrices corresponding to the bilinear forms <math> a^q( \cdot , \cdot ) </math> as well as the input and output forms and H(curl) inner product matrix have been assembled
using the Finite Element Method, resulting in 7754 degrees of freedom, after removal of boundary conditions. The files are numbered according to their
appearance in the summation.

The files are numbered according to their appearance in the summation and can be found here:
[[Media:Matrices.tar.gz|Matrices.tar.gz]]

The coefficient functions are given by

:<math> \Theta^1(\omega, \nu) = 1 </math>

:<math> \Theta^2(\omega, \nu) = \omega </math>

:<math> \Theta^3(\omega, \nu) = -\omega^2 </math>

:<math> \Theta^4(\omega, \nu) = \frac{\nu}{6} </math>

:<math> \Theta^5(\omega, \nu) = \frac{6}{\nu} </math>

:<math> \Theta^6(\omega, \nu) = \frac{6 \omega}{\nu} </math>

:<math> \Theta^7(\omega, \nu) = -\frac{6 \omega^2}{\nu} </math>

:<math> \Theta^8(\omega, \nu) = \frac{\nu \omega}{6} </math>

:<math> \Theta^9(\omega, \nu) = -\frac{\nu \omega^2}{6} </math>

:<math> \Theta^{10}(\omega, \nu) = \frac{16 - \nu}{10} </math>

:<math> \Theta^{11}(\omega, \nu) = \frac{10}{16 - \nu} </math>

:<math> \Theta^{12}(\omega, \nu) = \frac{10 \omega}{16 - \nu} </math>

:<math> \Theta^{13}(\omega, \nu) = -\frac{10 \omega^2}{16 - \nu} </math>

:<math> \Theta^{14}(\omega, \nu) = \frac{16 - \nu}{10} \omega </math>

:<math> \Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2 </math>

The parameter domain of interest is <math> \omega \in [0.6, 3.0] * 10^9 </math> Hz, where the factor of <math> 10^9 </math> has already been taken into account
while assembling the matrices, while the geometric variation occurs between <math> \nu \in [2.0, 14.0] </math>. The input functional also has a factor of <math> \omega </math>.

There are two output functionals, which is due to the fact, that the complex system has been rewritten as a real symmetric one. In particular the computation of the output

<math> s(u) = | l^T * u | </math> with complex vector <math> u </math> turns into <math> s(u) = \sqrt{ (l_1^T * u)^2 + (l_2^T * u)^2 } </math> with real vector <math> u </math>.

==Origin==

The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano project].

==References==

<references/>

==Contact==

'' [[User:hessm|Martin Hess]]''